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Proceedings of the Indian Academy of Sciences (Mathematical Sciences) Editor S G Dani Tata Institute of Fundamental Research, Bombay Editorial Board S S Abhyankar, Purdue University, West Lafayette, USA Gopal Prasad, University of Michigan, Ann Arbor, USA K R Parthasarathy, Indian Statistical Institute, New Delhi Phoolan Prasad, Indian Institute of Science, Bangalore M S Raghunathan, Tata Institute of Fundamental Research, Bombay S Ramanan, Tata Institute of Fundamental Research, Bombay C S Seshadri, SPIC Science Foundation, Madras V S Varadarajan, University of California, Los Angeles, USA S R S Varadhan, Courant Institute of Mathematical Sciences, New York, USA K S Yajnik, National Aeronautical Laboratory, Bangalore Editor of Pubiications of the Academy V.K. Gaur C-MMACS, NAL, Bangalore Subscription Rates (Effective from 1989) AH countries except India 1 year 3 years 5 years (Price includes AIR MAIL charges) US$75 $200 $300 India 1 yea r Rs. 75 Annual subscriptions are available for Individuals for India and abroad at the concessional rate of Rs. 25/- and $20/-respectively. Ali correspondence regarding subscription should be addressed to The Circulation Department of the Academy. Editorial Office: Indian Academy of Sciences, C V Raman Avenue, Telephone: 3342546 P B No. 8005, Bangalore 560080, India Telex: 0845-2 178 AC AD IN Telefax: 91-80-3346094 1995 by the Indian Academy of Sciences. All rights reserved.' The "Notes on the preparation of papers" are printed in the last issue of every volume. Proceedings of the Indian Academy of Sciences Mathematical Sciences Volume 105 1995 Published by the Indian Academy of Sciences Bangalore 560 080 Proceedings of the Indian Academy of Sciences (Mathematical Sciences) Editor S G Dani Tata Institute of Fundamental Research, Bombay Editorial Board S S Abhyankar, Purdue University, West Lafayette, USA Gopal P*rasad, University of Michigan, Ann Arbor, USA K R Parthasarathy, Indian Statistical Institute, New Delhi Phoolan Prasad, Indian Institute of Science, Bangalore M S Raghunathan, Tata Institute of Fundamental Research, Bombay S Ramanan, Tata Institute of Fundamental Research, Bombay C S Seshadri, SPIC Science Foundation, Madras V S Varadarajan, University of California, Los Angeles, USA S R S Varadhan, Courant Institute of Mathematical Sciences, New York, USA K S Yajnik, C-Af MACS, NAL, Bangalore Editor of Publications of the Academy V K Gaur C-MMACS, NAL, Bangalore Subscription Rates - 1996 All countries except India $100 I Price includes AIR MAIL charges) India Rs 150 Annual subscriptions are available for Individuals for India and abroad at the concessional rates of Rs. 75 - and $30 respectively. All correspondence regarding subscription should be addressed to The Circulation Depart- ment of the Academy. Editorial Office: Indian Academy of Sciences, C V Raman Avenue, Telephone: 334 2546 P B No. &005, Bangalore 560080, India Telex: 0845-2178 ACAD IN Telefax: 9 1-80-334 6094 i; 1995 by the Indian Academy of Sciences. All rights reserved. The "Notes on the preparation of papers" are printed in the last issue of every volume. Proceedings of the Indian Academy of Sciences Mathematical Sciences Volume 105, 1995 VOLUME CONTENTS Number 1, February 1995 The structure of generic subintegrality ......... Les Reid, Leslie G Roberts and Balwant Singh 1 Flat connections, geometric invariants and energy of harmonic functions on compact Riemann surfaces ........................................... K Gurupmsad 23 Fibred Frobenius theorem .............. Pedro M Gadea and J Munoz Masque 3 1 On infinitesimal /z-conformal motions of Finsler metric ................. , ......... ....................................... H G Nagaraja, C S Bagewadi and H Izumi 33 A bibasic hypergeometric transformation associated with combinatorial identities of the Rogers-Ramanujan type .................. . ........... U B Singh 41 Some theorems on the general summability methods ........... W T Sulaiman 53 Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays ...................................... ............... F R K Kumar and S K Sen 59 Control of interconnected nonlinear delay differential equations in .................................................................... ....... E N Chukwu 73 A note on integrable solutions of Hammerstein integral equations ............ .............................................. ..... K Balachandran and S Ilamaran 99 On over-reflection of acoustic-gravity waves incident upon a magnetic shear layer in a compressible fluid ........................................ P Kandaswamy 105 Number 2, May 1995 Badly approximate p-adic integers A G Abercrombie 123 Uncertainty principles on certain Lie groups A Sitaram, M Sundari and S Thangavelu 135 On subsemigroups of semisimple Lie groups D Kelly-Lyth and M McCrudden 153 Induced representation and Frobenius reciprocity for compact quantum groups Arupkumar Pal 157 Differential subordination and Bazilevic functions S Ponnusamy 169 ii Volume contents Convolution integral equations involving a general class of polynomials and the multivariable //-function K C Gupta, Rashmi Jain and Paw an Agrawal 1 87 On L 1 -convergence of modified complex trigonometric sums Satvinder Singh Bhatia and Babu Ram 193 Absolute sumrnability of infinite series C Orhan and M A Sarigol 201 Solution of a system of nonstrictly hyperbolic conservation laws K T Joseph and G D Veerappa Gowda 207 Oscillation in odd-order neutral delay differential equations Pitambar Das 219 Surface waves due to blasts on and above inviscid liquids of finite depth C RMondal 227 Generation and propagation of SH-type waves due to stress discontinuity in a linear viscoelastic layered medium P C Pal and Lalan Kumar 241 A proof of Howard's conjecture in homogeneous parallel shear flows - II: Limitations of Fjortoft's necessary instability criterion Mihir B Banerjee, R G Shandil and Vinay Kanwar 25 1 Number 3, August 1995 Lifting orthogonal representations to spin groups and local root numbers Dipendra Prasad and Dinakar Ramakrishnan 259 Irrationality of linear combinations of eigenvectors Anthony Manning 269 On the zeros of $ - a (on the zeros of a class of a generalized Dirichlet series-XVII) K Ramachandra 273 A note on the growth of topological Sidon sets K Gowri Navada 28 1 Characterization of polynomials and divided difference P L Kannappan and P K Sahoo 287 A theorem concerning a product of a general class of polynomials and the H-function of several complex variables VB L Chaurasia and Rajendra Pal Sharma 291 Certain bilateral generating relations for generalized hypergeometric fun- ctions Maya Lahiri and Bavanari Satyanarayana 297 A localization theorem for Laguerre expansions P K Rathnakumar 303 Degree of approximation of functions in the Holder metric by (e, c) means G Das, Tulika Ghosh and B K Ray 315 Volume contents iii The algebra A p ((0, oo)) and its multipliers Ajit Iqbal Singh and H L Vasudeva 329 Reflection of P-waves in a prestressed dissipative layered crust Sujit Bose and Dipasree Dutta 341 Computer extended series solution to viscous flow between rotating discs .- N M Bujurke, N P Pai and P K Achar 353 Number 4, November 1995 The Hodge conjecture for certain moduli varieties V Balaji 371 Equivariant cobordism of Grassmann and flag manifolds Goutam Mukherjee 381 Local behaviour of the first derivative of a deficient cubic spline interpolator Surendra Singh Rana 393 On the partial sums, Cesaro and de la Valee Poussin means of convex and starlike functions of order 1/2 Ram Singh and Sukhjit Singh 399 Uniqueness of the uniform norm and adjoining identity in Banach algebras S J Bhatt and H V Dedania 405 Weakly prime sets for function spaces H S Mehta, R D Mehta and B B Mishra 411 Oscillation of higher order delay differential equations P Das and N Misra 417 Nontrivial solution of a quasilinear elliptic equation with critical growth in U" Ratikanta Panda 425 An axisymmetric steady-state thermoelastic problem of an external circular crack in an isotropic thick plate Rina Bhowmick and Bikash Ranjan Das 445 Some characterization theorems in rotatory magneto thermohaline convec- tion Joginder Singh Dhiman 46 1 Subject Index.... 471 Author Index 476 Proc. Indian Acad. Sci. (Math. Sri.), Vol. 105, No. 1, February 1995, pp. 1-22. Printed in India. The structure of generic subintegrality LES REID, LESLIE G ROBERTS* and BALWANT SINGH* Department of Mathematics, Southwest Missouri State University, Springfield, MO 65804, USA * Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, Canada, K7L 3N6 f School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India MS received 21 January 1994 Abstract. In order to give an elementwise characterization of a subintegral extension of Q-algebras, a family of generic Q-algebras was introduced in [3]. This family is parametrized by two integral parameters p ^ 0, N ^ 1, the member corresponding to p, N being the subalgebra R = Q [ { y n \ n ^ N } ] of the polynomial algebra Q[x l9 ...,x pt z]mp + l variables, Jx.z""'. This is graded by weight (z) = 1, weight (x ) = i, and it is shown in [2] to be finitely generated. So these algebras provide examples of geometric objects. In this paper we study the structure of these algebras. It is shown first that the ideal of relations among all the y^s is generated by quadratic relations. This is used to determine an explicit monomial basis for each homogeneous co'mponent of JR, thereby obtaining an expression for the Poincare series of R. It is then proved that R has Krull dimension p + 1 and embedding dimension N + 2p, and that in a presentation of R as a graded quotient of the polynomial algebra in N + 2p variables the ideal of relations is generated minimally by 1 elements. Such a minimal presentation is found explicitly. As corollaries, it is shown that R is always Cohen-Macaulay and that it is Gorenstein if and only if it is a complete intersection if and only if N + p ^ 2. It is also shown that R is Hilbertian in the sense that for every n ^ the value of its Hilbert function at n coincides with the value of the Hilbert polynomial corresponding to the congruence class of n. Keywords. Subintegral extensions; subrings of polynomial rings. Introduction Let A B be an extension of commutative rings containing the rational numbers Q. In [3] an element beB is defined to be subintegral over A if there exist integers p > 0, N ^ 1 and c l9 . . . , c p eB such that g n := b n -f f= l ( " )c f b w ~ eA for all integers n ^ N. With this definition the extension A B is subintegral in the sense of Swan [7] if and only if every element of B is subintegral over A [3, 4]. In [3] the tuple (0, p, N; 1, c l9 . . . , c p ) with the above properties was called a system of subintegrality for b over A. There was an extra parameter 5 which we can take to be in the present discussion, and the 1 represents c . In [3] we assumed that 1 2 L Reid et al N ^ s + p. Here (as in [4]) we adopt the conventions that for any element b in a ring, b = 1 and ( n }b n ~ l : = if i > n. Then it suffices to assume that JV > 1. By [3, proof W of (4.2) <iv)=>(i)] (note also [4, (1.1)]) if b has a system of subintegrality for some N ^ 1, then b has a system of subintegrality with N = 1. Systems with N > 1 are still of interest, however, since freedom in the choice of N may result in a simpler system of subintegrality. Let x j,..., Xp, z be independent indeterminates over Q, and let x = 1. For n ^ let and let R:=QlVn\*>N]S:=Qlx l ,...jc 9 z]. Then z is subintegral over R (N) with system of subintegrality (0, p, AT; 1, x l9 . . . , x p ). Furthermore this setup is universal for subintegral elements together with their systems of subintegrality, in the sense that given any extension of commutative Q-algebras A B with beB having a system of subintegrality (0,p,N; I,c l9 ...,c p ), the homomorphism (p:S->B given by <p(x f ) = c { . and <p(z) = b satisfies (p(y^ = g n and cp(R (N) )^A. Such universal extensions played a crucial role in [3]. The rings R (N} have an interesting algebraic structure, which we discuss in the present paper. First of all R (N} and S are graded by weight (x f ) = f, weight (z) = 1, which imply that weight (y n ) = n. In 1 we find relations (1.2) of degree two (but not necessarily homogeneous) among the y n9 where degree means deg(y n ) = 1 for all n ^ 1, and is to be distinguished from weight. We show in (2.2) that these quadratic relations generate the ideal of all relations. These quadratic relations include those used in [2] to prove that R (N] is a Q-algebra of finite type, although in [2] we did not find a complete set of relations. In (2.1) we use the quadratic relations to obtain an explicit monomial basis for R ( "\ the weight k part of R (N \ from which we obtain in (2.8) the Poincare series of R (N} for arbitrary p and N (generalizing both [4, (4.4)], which handles the case N = 1, and [4, (4.7)], which is the case p = 1, N arbitrary). In 3 we use the quadratic relations to eliminate all but a finite number of the y n , obtaining thereby our main result (3.2) which gives a minimal presentation of K (N) as a graded Q-algebra of finite type. Of course, after eliminating these variables, the relations among the remaining variables are no longer all quadratic. From (3.2) we derive several corollaries ((3.3)-(3.7)) on the nature of R (N) : (3.5) says that R (N) is always Cohen-Macaulay, which was a surprise to us; (3.6) says that R (N) is Gorenstein if and only if it is a complete intersection if and only if N + p ^ 2. In 4 we give an alternative proof of the linear independence of our basis for Rf } . This method is more complicated but also more precise than the argument of 2. We conclude the paper by studying in 5 the Hilbert function of R (N \ We find the minimal number d of Hilbert polynomials needed to express the Hilbert function of R (N \ and show that if p ^ 2 then R (N] is Hilbertian, meaning that the value of its Hilbert function at n coincides with the value of the Hilbert polynomial corresponding to the congruence class of n modulo d, for every n ^ (rather than just for n 0). The non-negative integers are denoted by Z + , and \_a\ is the integral part of the real number a (i.e. the largest integer ^ a). 1. The quadratic relations Let R (N} S be the universal extension as defined above. Let T be an indeterminate The structure of generic subintegrality over S, and let F(T)= H-f =1 ( p z~' (so that y n = z n F(n)). Then we have the following (generalizing [2, (1.2)]). Theorem 1.1. Let k be an integer > 2p, and let < d l < d 2 < < d p + r ^ fc/2 be any p + 1 distinct integers. Let d be any integer < d ^ /c/2, distinct from the d t . Then P+I jdJk-d^ E fl iW*-d O- 2 ) /=i /or some rational numbers a t . Proof. Note that we have d t < k - d t (1 ^ i ^ p), d p + j ^ k - d p+ 19 and the p + 1 pairs (dfrk di) are distinct (as unordered pairs). First consider the case d p + 1 <k d p + l so that each pair (d f , fc~d ) consists of two distinct integers. Let / = {d l9 ...,d p+1 , k d p+ !,..., /c di}. For p + 2<i<2p-f-2 define d i = k d 2p + 3 ^ i , so that / = Wi<i^2/+2- The set / contains 2p + 2 distinct integers. For 1 ^ i ^ 2p + 2 let TT,- be the interpolating polynomial of degree 2p+ 1, which is 1 at d f and at the remaining elements of /. Let G(x) = ^ J f^ 2 n i (x)F(d i )F(k-d i ) and H(x) = F(x)F(k-x). Then G(c) = H(c) for all ce/. The polynomial G(x) is of degree < 2p+ 1 in x, whereas H(x) is of degree 2p in x. These two polynomials (with coefficients in the integral domain Q [x !,..., x^z" 1 ]) agree at 2p + 2 values of x, hence are equal. Setting x==d, a { = Ki(d) + K 2 p+i-i(d) (K i < p+ 1) and multiplying by z k yields (1.2). . Now consider the case d p+l =k d p+l . Let / = {d l ,...,d p+1 ,k d p ,... 9 k d 1 }. For p + 2<f<2p+l define d i ^k-d 2p + 2 , h so that / = {d^^^zp + i- The set ^ contains 2p 4- 1 distinct integers. For 1 ^ i < 2p 4- 1 let n L be the interpolating polynomial of degree 2p, which is 1 at d t and at the remaining elements of /. Let G(x) = Hf^ 1 n i (x)F(d i )F(k - d^) and H(x) = F(x)F(k-x). Then G(c) = H(c) for all eel. The polynomials G(x) and H(x) are both of degree < 2p in x. These two polynomials (with coefficients in the integral domain Q[x 1 ,...,x p ,z~ 1 ]) agree at 2p+ 1 values of x, hence are equal. Setting x = d, a = n t (d) + n 2p+2 -i{ d ) C- < l ' < P) a p+i = n P + i( d \ and multiplying by z k yields (1.2). 1 COROLLARY 1.3. (a) Ifk^Ip then the monomials of degree <2 and weight k in the y t span a vector space V kt2 of dimension p-f-1, and any set o/p-f 1 distinct monomials of degree ^2 is a basis for this vector space. (b) If k^2p+l then any set of distinct monomials of degree ^ 2 and weight k is linearly independent. (c) In any relation (1.2) all the a t are uniquely determined and nonzero. Proof. The monomials y h 7i7 k _ 1? . . . , y d y k - d (d = min(|_fc/2J, p) are linearly independent by [4, proof of (4.1)] from which (b) follows. It also follows that if fc ^ 2p then V kt2 is of dimension ^p+1, and by "(1.2) any p+1 elements span. Thus (for k^2p) dim V k ^ 2 = p + 1, and (a) and (c) follow. (Note that (c) is vacuous unless k^2p + 2.) Examples 1.4. Here are a few examples of the quadratic relations (obtained using a computer program that we wrote): 4 L Reid et al forp=l: (1.4.1) 74 = 47^3- 37^ - (1.4.2) 7 5 = 37 1 74~27 2 7 3 (1.4.3) 7i7 5 = 47 2 7 4 -37* and for p = 2: (1.4.4) 7s = 207 2 7 6 - 647 3 7s + 457^ (1 A5) 7 9 7i = 207377 - 647 4 7 6 + 457^ (1.4.6) 7io = (63/5)7 2 7 8 - (128/5)7 3 7 7 These examples illustrate the following. Theorem 1.5. (1) The quadratic relations are translation-invariant, i.e. if p+i jdjk-d^ Z *fW*-di t=l then also for any integer j^O (with the same a t ). (Homogenize by putting in y if necessary.) (2) If the di are consecutive integers, then the coefficients a t in (1.2) are integers. Proof. (1) In (1.1) replace d t by d\ = d t + j (1 < i ^ p + 1), d by d -f ; and k by fc + 2j. Then also d t is replaced by d; = d +; (p + 2<i<2p + 2 or p-|-2<i<2p+l respectively in the two parts of the proof of (1.1)). Formula (1.2) becomes where < = 7r;(d+j) + ^' 2p+3 _.(d-f 7) for l^Kp + 1 (respectively aJ = 7r 2 P +2-i(^+^) for K^'<P and a p-fi = 7C p+i(^+;)X < being the interpolating polynomial of degree 2p 4- 1 (respectively degree 2p) which is 1 at d'., and at the remaining d'y Obviously n'^c +j) = n t (c) for all real numbers c, from which it follows that a\ = a. for all f, proving (1). (2) If the d t (1 <i^2p + 2, resp. l<i^2p-l-l in the two cases) are consecutive integers, then the Lagrange formula for the n t (when evaluated at any integer) is (up to sign) the product of two binomial coefficients. Thus the n t (d) are integers, hence also the a h proving (2). B Example (1.4.6) shows that in general the a { need not be integers. We can arrange to have the d t consecutive by taking c = [_k/2\ and {y c y k - c , y c -iy k - e+l9 . . . , 7 c - P 7fc-c+ P } as the set of quadratic monomials on the right-hand side of (1.2). 2. The Poincare series of Determining the Poincare series of R (N) is essentially the same as determining the dimension of the Q-vector space R ( f\ the weight k part of R (N \ for every k. In fact, The structure of generic subintegrality 5 we do more. Namely, using a basis interchange technique, we find in the following theorem an explicit monomial basis for Theorem 2.1. R[ N ) has Q-basis if Proof. If p = the result is trivial. For then y t = z l for all i and R (N] = Q|V | i ^ AT). lik^N then JRjf has basis y fc and J^ >Jt contains only y fc , since we must have d= 1. If /c = then R ( <f } has basis y = 1 and & NtQ contains only the empty product 1 since we must have d = 0. If < A; < AT then JRj^ = and ^ Njk is empty. Hence assume p^ 1. First consider the case N= 1. In [4, (4.1)] a basis {z*GJ(fc)| te^jj for jR 1J (there denoted simply as R k ) is obtained. The definition of this basis is quite technical, so we will not recall its definition completely. It suffices to note that 3~ k is a set of integers indexing all sequences of the form a t = (a l9 a 2 ,...,a k ) with ^a t ^ ^a k ^ p, afc _ 1 = a fc , and IX- ^- Also, in the proof of [4, (4.1)] the above basis is put in one-to-one correspondence with another basis of R( 1] that consists of monomials in the y's. Under this bijection, z k G' l (k\ for a t = (a 1? a 2 , . . . , a k ), corresponds to y ai y ak _ j^, where /? fc ^ a^ is chosen so as to make the weight o^-f ^ a*- 1 + /? fc = /c (remember that some of the a's can be 0, and that y =l). But (omitting the y 's, renumbering the remaining y's and noting that N + pl=p) this is just the basis ^ 1>k claimed for . N = 1 in the statement of the theorem. Now, for general AT, if y^j is a factor of a monomial in the y's of weight /c, with i and ; both ^ AT + p, then the quadratic relations (1.2) can be used to replace y^- by a linear combination of }';+j> JNyi+j-NiyN+lJi + j-N-l'' '-^yN + p-iyi+j-N-p+l (note that z+j N p-fl^N-fp l^N) from which it follows that ^^ spans R ( ^\ Thus it suffices to prove the linear independence of & N>k . This we prove by induction on N. The idea is to produce a basis for R^ ~ 1 } that contains @ N ^ k as a subset. Hence suppose that N^2 and $ N - ltk is a basis for R^ -1) . We have *N-i f fc={yj 1 y J2 ---7iJAf-l<^<---<^-i<^id-i<^ + P--l if </ >!,=!'; = /:}. Let % = &N,k r( &N-i,k ( = those elements of ^-i.fc that do not contain any y^^'s). Let (f be the set of those elements of 38 N - ltk which contain a certain number of y^-i's, say e^l of them, and which have the largest subscript i d satisfying i d ep^N + pl. Let $ be obtained (elementwise) from <^ by replacing each y N _ 1 by y N + p _ j and decreasing the highest subscript accordingly. The theorem follows from (2.1.1) Claim (2.1.2) ^ = ^11*? (2.1.3) (& N - ltk - )v<f is a basis for R ( k N ~ 1} . Proof of (2.1.2). Obviously # ^ N ^ and ^ c Jf N fc . Furthermore, any element that contains e > 1 y^+p-i's (or one y N + p _i and one y with subscript >N + p 1) is obtained uniquely by the above transformation from an element of <? , and any element of @ N that contains at most one y#+ P -i and has all other subscripts L Reid et al -l is in #. Thus ^ Nk ^^^jS. It is obvious that #n<^ = 0, which proves (2.1.2). Proof of (2.1.3). Let and let <?' be the set of those elements of Sf which contain a certain number of y/v-i's, say e^l of them, and which have the largest subscript i d satisfying i d -ep^N + p-1. Then JVijc^N,* c #' and g r o . Let p: <T II N tk ->'oUO! Ntk be the map which is identity on ^ N>fc and is defined on <f' as follows: if ye<f ' then write y = y N ~i8y c with c^ N -\-2p- 1 and <5 a monomial in y^-i, )>#,..., y^H-p-i, and define p(y)==5 i y N+p _ 1 y c _. p . Further, for such a y = yN-i<5y c e<f Put ^ = ^ II#, and for z^l let ^ = {p(y)\yeS! i . 1 }. Then each f is a subset of , p is a bijection from 2 i onto ^i_ l5 and ^11^ = ^- for f0. Let Q.j {yt<$ i \y N _. l appears exactly to power; in y}. Then Q i = 11^ f</> and for i, j ^ 1 we have p( _ u) ^ fj _ t with equality if; ^ 2. Let ye y with; ^ 1. We claim that S(y) ^j-i. This is clear for i = 0. If i^l then y = p(j8) with j?6^^ 1J+1 , and clearly S(y)= {p(a)|aeS(j8)}. So the claim follows by induction on i. Now, the set (y,p(y)}uS{y) has p + 2 elements, and by (1.1) and (1.3) (c) any p -f 1 of these elements form a basis for the vector space spanned by this set. So, as S(y)^^ iJ _ 1 , the sets {y}^& ltj ^ i and {p(y)}^j^ ij , 1 span the same vector space. Therefore, since 2 { can be obtained from S i _ 1 in stages by changing u))u^ starting with the highest h, it follows that each spans the same space. In particular, ^ U^ and <^II# = & Ntk span the same space. The former being a part of a basis for #*-!,*, (2.1.3) is proved. B COROLLARY 2.2. The idea/ o/a// relations among the y's is generated by the quadratic relations (1.2). Proo/. Only the relations (1.2) were used to reduce the set of all monomials of weight k in the y's to the basis 39 Ntk . H COROLLARY 2.3. Let V ktd be the subspace of R ( * } spanned by monomials of \veight k and degree ^d in the y t (deg y t - = 1 for all i ^ 1) as in [4, 2]. Then V k%d has Q-basis of those monomials in & N ^ k of degree ^d. Proof. The indicated elements are linearly independent since they are part of the basis & Ntk . Therefore it suffices to prove that they span K M . To do this we may assume that p ^ L If yflj is a factor of a monomial in the y's of weight /c, and degree ^d with i and; both >N + p, then as in the proof of (2.1) the quadratic relations (1.2) The structure of generic subintegrality 1 can be used to replace y^- by a linear combination of ? +^7^ +./-#, y N+1 y i+j . N , l9 ... 9 y N+p ^ 1 y i+ j^ ff . p+1 (note that i+j- N -p+ l^N + p-l^N and that the quadratic replacement does not increase degree), from which it follows that the claimed elements span F M . COROLLARY 2A (cf. [4, (2.1)]) We have dim V k d = { } for k 0. More precisely, dim V kd { } if ' \ P / ' V p ) and only if k^m, where m is defined as follows: (1) if p^l and d^2 then m = (N 4- p l)d; (2) ifd = then m = 1; (3) in all other cases m = N or m = accordingly as N> 1 or N = 1. Froo/. Case (2) is trivial. For, if d = then =0, and the only product of \ P / degree zero is the empty product which is 1. So assume that d^l. Then if the y of highest weight is removed from each element of the basis of V kd described in (2.3), this basis is put in one-to-one correspondence with a subset of the monomials of degree less than or equal to d 1 in the p variables y N9 y N+ 19 . . . , y N + p ~ ^ If k is large enough we obtain in this manner all monomials of degree less than or equal to d1 in yN,yN+i 9 .'. 9 yN+ p -i. Since there are ( 1 such monomials, the first \ P / part is proved. Assume now that we are in case (1), i.e. p ^ 1 and d ^ 2. Then a monomial M of degree ^d 1 iny N9 y N+1L9 ... 9 y N+p ^- L corresponds to an element of our basis if and only if fc wt(M) is bigger than or equal to any subscript occurring in M. The most critical case is y^Vp- 1 which requires k (d l)(N + p l)^ N + p 1, or k^(N + pl)d = m, proving case (1). The proof of case (3) is an easy and straightforward verification. H Example 2.5. Here is an example to illustrate the algorithm in the proof of (2.1). Let N = p = 2. Then dim Q R ( v = 31 9 dimQ^ = 10. Monomials in the /s will be represented by listing the subscripts, thus (1,1,2,7) represents y^y?- ^ e ^ ave * = {(1, 10), (1,1,9), (1,2,8), (1,1,2,7), (1,2,2,6)} and # = {(11), (2,9), (2,2,7), (2,2,2,5), (2,2,2,2,3)}. To understand the example it is not necessary to list the elements of # 1(11 - S > explicitly. We have ^ = f UV = 00 U0 01 II0 02 with 00 = {(11), (2, 9), (2, 2, 7), (2, 2, 2, 5), (2, 2, 2, 2, 3)}, ^ 01 = {(1,10), (1,2,8), (1,2,2,6)} and 02 = {(1, 1,9), (1,1,2,7)}. The following table shows how the transformation proceeds using the linear relation among y,p(y) and S(y): y = Replaced by p(y) = Using 5(y) = (1,1,9) (1,3,7) (1,10), (1,2, 8) (1,1,2,7) (1,2,3,5) (1,2, 8), (1,2, 2, 6) (1,10) (3,8) (11), (2, 9) (1,2,8) (2,3,6) (2, 9), (2, 2, 7) L Reid et al (1,2,2,6) (2,2,3,4) (2, 2, 7), (2, 2, 2, 5) (1,3,7) (3,3,5) (3, 8), (2, 3, 6) (1,2,3,5) (2,3,3,3) (2, 3, 6), (2, 2, 3, 4) The first two rows show how ^ 02 is transformed into p(@ Q2 ) an( * the next three rows show how @ 01 is transformed into p( i)- This S ives ^i=-io u ^ii with ^ n =p(S 02 ) = {(l,3,7),(l^ (11), (2,9), (2,2,7), (2,2,2,5), (2,2,2,2,3)}. Finally, the last two rows show how S n is transformed into p(Q^\ giving 2 = 20 = p(^ i)U 10 = { (3, 3, 5), (2, 3, 3, 3), (3, 8), (2, 3, 6), (2, 2, 3, 4), (1 1), (2, 9), (2, 2, 7), (2, 2, 2, 5), (2, 2, 2, 2, 3)} = tUV = # 2il ^ Note that for fixed i, j the order in which elements of 2 tj are transformed into those of is immaterial. The basis of K 3ill given by (2.3) is {(11), (2,9), (3,8), (2,2,7), (2,3,6), (3,3,5)}. The calculation of the Poincare series is now just a matter of counting # Ntk . The number of partitions of k as sums of integers each ^ N and ^ N -f p 1 is the coefficient of t k in 1 77- (2-6) Allowing one integer > N -f- p 1 is the same as finding the partitions of the integers from to k N p as sums of integers each ^ N and ^ N -f p 1 (adding one more integer, which will be greater than N + p1, to each partition to bring the sum up to k), and the number of such partitions is the coefficient of t k in ^ (2-7) Adding (2.6) and (2.7) yields Theorem 2.8. Let P(t) be the Poincare series for the ring R (N \ i.e. where H(k) = dim Q R( N \ Then By a similar argument, using x to keep track of the number of terms added, we obtain that dim F M is the coefficient of x d t k in 1 + i v+N + p ^ (2-9) 3. Relations ideal and the structure of In this section we determine the structure of R (N) by finding a minimal presentation for it as a graded Q-algebra. We show that R (N) has Krull dimension p + 1 and The structure of generic subintegrality 9 embedding dimension N + 2p 9 and that in a presentation of R (N) as a graded quotient of the polynomial algebra in N + 2p variables the ideal of relations is generated minimally by f ) elements. As corollaries, we show that R (N) is always Cohen-Macaulay; that R (N} is Gorenstein if and only if it is a complete intersection if and only if N + p^2 (which happehs exactly in the three cases p = 0, N"= 1; p = 0, N = 2; p=l = JV); and that R (N) is regular if and only if p = 0, N=l. Let B=Q[T N9 T N + i9 ... 9 T 2N + 2 p-i] ^ e the polynomial ring in N + 2p variables graded by weight (T t ) = i, and let cp: B-+R (N) be the Q-algebra homomorphism given by (p(T t ) = y t for all L Let A = Q[T^ T N + l9 ... 9 T N + P ] 9 let M be the 4-submodule of B generated by l,r Ar+p+1 ,...,r 2Ar +2p-i an( * * et M' = <p(M). Then M' is the ^-sub- module of R (N} generated by 1, y N+p + 19 ... i y 2 N+ ZP- 1 where ^' = Q[y^ 7* + 1> (We will see later that M' = Lemma 3.1. W 7 e /iat;e y^M' and yf/^eM' /or a// i, / ^ N. Proof. We prove the first part by induction on f. Clearly we have y f eM' for N^i^2N + 2p-L Let i^2N + 2p. Then i-N-p^N + psoby (1.2) y ( - belongs totheQ-spanofy N y _j V ,y N+1 y f JV l9 ,..,7 JV + p y I .. Ar ^ p . Now 7 i . JV ,y l .. jV _ 1 ,... 9 y l .. w . p eM / by induction, since i>i N^i N p'^N. Therefore y^-eM', and the first part is proved. Now, if at least one of f and j is < N 4- p then y^eM' by the first part. On the other hand, if both z and 7 are > N } p then z +7 N p 4- 1 > N + p 1 so by (1.2) ytfj belongs to the Q-span of y i +j 9 y N y i+J - N9 ... 9 y N + p _ 1 y i+j , N _ p + 1 (just y i+j if p = 0) and these p+ 1 monomials belong to M' by the first part. So y By the Lemma we can write, for i 9 j^N + p+ 1, y -yj = a' a\p' h eA'. We may assume that a',/^ are homogeneous of appropriate weight so that the expression is homogeneous of weight i +j. Lift a', p' h to homogeneous elements a, p h of A of the same weight and let 2N + 2p-l P tj =T t Tj-tt- X ftT A . Then P y is homogeneous of weight z + j. Theorem 3.2. T/ie graded Q-algebra R (N) has Krull dimension p-f 1 anJ embedding dimension N 4- 2p, flm/ /i^5 <2 minimal presentation with N -f 2p generators and I relations. M^^ precisely, the Q-algebra homomorphism (p:B-*R (N} is surjective and the ideal ker(cp) of B is generated minimally by the I 1 elements P^ Proof. By [2, (1.4)], or by (3.1) above, R (N) is generated by y This means that cp is surjective, and R (N) is a Q-algebra of finite type. Now, since the quotient field of R (N) is Q(x 1 ,...,x p ,z) by [4, (5.2)], we get dim(R (A ) = p+ 1. (That dim(J^ (N) ) = p+ 1 also follows independently from (3.3) below.) We show next that the set {P^N 4- p + 1 < i ^ j ^ 2N + 2p - 1} generates ker(<p) minimally. To do this, let / be the ideal of B generated by this set. 10 L Reid et al Minimality. Since the P tJ are homogeneous, it is enough to show that no P tj belong to the ideal generated by the remaining ones. Suppose for some ij we ha\ Pgj = Z (r S )#a jyf rs P rs with /rs ejB - We may assume that each / rs is homogeneous wit weight (/) = i+j r s (negative weight means the element is zero). L< frs(T r T s Since AT + p + 1 ^ i, j ^ 2N + 2p 1 and Q tj is of degree at most one i TJV +P+ i ?2N+2p- 1 the term T i T j is present on the left hand side. Let us look fc this term on the right hand side. First of all, T i T j cannot appear in any of the tern f rs T r T s because (r,s) ^(,7) is an unordered pair. It follows that T i T j must come froi one of the terms / rs Q rs . Since N 4- p + 1 < z, j < 2N + 2p 1 and Q rs is of degree at mo; one in T N + p+l9 ... 9 T 2N + 2p - 1 , m order for T i T j to appear in the term f rs Q rs it necessary for/ rs to contain a term which is a nonzero rational times T t or T} or 7^7 Accordingly, we would get i +j r s weight (/ rs ) = i or j or i+j whence r + s = or i or 0. This is a contradiction, since r -f 5 ^ 2N 4- 2p + 2. This proves the minimalil of the generators. Generation. By construction, we have / ker(</>). So we have the surjective ma \I/\B/I -+R (N} induced by cp. We have to show that \l/ is an isomorphism. Note th; M is a free ^-module of rank N + /?, with basis T := l 9 T N+p+l9 ... 9 T 2N+2p . 1 . 11 module M is graded by weight (T ) = i. Let (: M - B// be the restriction of the natur; map B-+B/I to M. Given any polynomial in B, we can reduce it modulo / to a element of M. This means that is surjective. Now, let <r = \l/. Then <r: M -> J R ( ^ V) an y4 -linear map which is homogeneous of degree zero and is surjective. Now, denotir by P L (t) the Poincare series of a graded ^.-module L and writing R = R (N \ it is enou^ to prove that P R (t) = P M (t). For, since a is surjective, this would show that a is a isomorphism whence also \f/ is an isomorphism. Now, by (2.8) we have On the other hand, since A is the polynomial ring Q[T N ,T N+1 ,... 9 T N+P ~] with weig] .) = i, we have p (A _ XU Therefore, since M is A-free with basis 1, T N + p+l ,... 9 T 2N+2p _ l and weight (T.) = we get l * . .N + p+ 1 Now, it is checked readily that P R (r) = P M (t). This completes the proof of the equali Finally, we show that the embedding dimension of R (N} is N + 2p. Recall that f a finitely generated graded ring C = @ k ^ Q C k with C a field its embedding dimensic emdim(C) is the minimal number of homogeneous C -algebra generators of C, equivalently the minimal number of homogeneous generators of the ide C+ = *^iCfc. ^ n our situation we have R (N) = B/I with / generated by the P The structure of generic subintegrality 11 AT + p + 1 < i <; < 2N + 2p 1. For such f, 7 we have i+j^2N + 2p + 2. Therefore in the expression Pij=T t Tj-ai-xl p p ~ + \fi H T h we have aeX + and each p h eA + . This shows that /s^ Therefore by (graded) Nakayama the minimal number of homogeneous generators of the ideal R ( + } of R (N) is the same as that of the ideal B+ of B, which is AT -f 2p, since B is the polynomial ring in N + 2p variables. This proves COROLLARY 3.3. The ring A' *=Q[y N ,y N+l9 ...,y N+p ] is the polynomial ring in p + 1 variables over and R (N) is a finite free A'-module with basis Proof. The restriction of the isomorphism <r.M-+R (N) to A is a Q-algebra isomorphism of A onto A', sending T t to y t (N ^ i ^ N + p). This implies the first part. The second part follows since (T(T t -) = y f (i = or AT + p + 1 < i < 2 AT -h 2p 1). H COROLLARY 3.4. A Q~basisfor R (N) in terms of monomials in y^ Consequently, a Q-basisfor R ( ^ is which can also be written, for comparison with (2.1), as j=l Proof. Immediate from (3.3). H COROLLARY 3.5. The sequence y N ,y N +i,...,y N+p is R (N} -regular, and the ring R (N) is Cohen-Macaulay. Proof. The regularity of the sequence is immediate from (3.3). Therefore the localization of R (N) at the irrelevant maximal ideal R ( * ) of R (N} is Cohen-Macaulay. It is well known that this implies that R (N) is Cohen-Macaulay (e.g. [1, (33.27)]). COROLLARY 3.6. The following three conditions are equivalent: (1) R (N) is Gorenstein; (2) R (N) is a complete intersection; (3) A/ + p<2. 12 L Reid et al Note that, since N ^ 1, (3) occurs in exactly the following three cases: p = 0, N = 1; Proof. (l)o(3): Since R (N) is graded, it is well known that R (N) is Gorenstein if and only if its localization at the irrelevant maximal ideal R is Gorenstein (e.g. [1, (33.27)]). Let C denote this localization and put D = C/(y N , y N + 19 . . . , y N+p ). Then, since C is Cohen-Macaulay and y N9 y N+l9 ...,yN+ P is a regular C-sequence by (3.5), C is Gorenstein if and only if D is Gorenstein. Let m be the maximal ideal of D. Then, since dim(D) = 0, D is Gorenstein if and only if ann(m), the annihilator of m, is a 1-dimensional space over D/m. Now, it follows from (3.2) that m is generated minimally by S N+p+19 ... 9 8 2N + 2p . 19 where 8 i denotes the natural image of y, in D. Consider two cases: Case 1: m=0. In this case D is Gorenstein, and this case occurs <=> Case 2: m ^ 0. Then ann(m) c m. IfN + p + Ki,j<2A/' + 2p-l then, as noted in the proof of (3.2), we have P y = T^T,- - a - Zj^iJ J^i ^T h with aeA 2 + and each P h eA + . It follows that m 2 = 0. Thus m^ann(m) whence ann(m) = m. So D is Gorenstein om is generated by one element o2N + 2p-l = iV-fjp+l<>iV-fp = 2. (2)o(3): Since dim(JR w ) = p-f-l and R (N} = B/I with / minimally generated by homogeneous elements, R {N) is a complete intersection if and only if fN+p\ JV+2p=p-j-l-H 1. The solutions of this equation with integers p ^ 0, N ^ 1 are exactly those given by N + p ^ 2. COROLLARY 3.7. The ring R (N) is regular if and only i/p = 0, JV= 1. Proof. R (N} is regular oemdim(K (JV) ) = dim(R (JV) )<*> N + 2p = p + 1 oJV + p = 1 op = 0, Example 3.8. We illustrate the structure theorem (3.2) by computing P lV explicitly in the cases p=l = JV andp = l, ]V=2. First, let p= 1 = JV. In this case B = Q[T l5 T 2 , T 3 ], A = Q[T l5 T 2 ], 4' = Q[y ls y 2 ], M' is the ^'-module generated by I,y 3 , and there is only one relation P 33 . To find it we have to express yl as an A'-linear combination of 1, y 3 . We da this by eliminating ? 4 ,y 5 among the relations (1.4.1), (1.4.2), (1A3) obtaining 3 1'2 '2 *1'3 *1'2'3 as the desired linear combination. So P 33 = T\ - 3T^T^ + 47^ 4- 4T^T 6T T T A similar computation for the case p= 1, JV=2 gives R (2) s Q[T 2 , T 3f T 4 , T 5 ]/(P 44 , P 45 , P 55 ) The structure of generic subintegrality 13 4. The independence of 3$ Nyk In this section we give a new proof of the linear independence of & Ntk , which does not depend upon the proof of (2,1). The matrix approach used here gives additional insight into the nature of .jR (N) . In particular, we obtain a sharpening of the independence part of (2.1), in that we prove that a specific minor of a certain matrix is nonzero. Our matrix theoretic techniques are perhaps of interest in their own right. Before stating our result precisely (Theorem (4.1)) we would like to describe more carefully the relationship between the two bases ^ 1>fc and <& k := {z k G' t (k)\te3~ k } of JRj^. In 2 we noted that y k is a set of integers indexing (as t ranges over ^ k ) all sequences of the form a t = (a ls a 2 , . . . , a fc ) with ^ o^ < a 2 ^ - ^ a fc ^ p, a k _ L = a k , J> < k. In [4, 3] we also introduced monomials b t = x ail x OL2 "-x Xk (with x = l). If we wish to write an element M of ^ 1>fc (or more generally, any monomial M of weight k in the y t ) as a linear combination of & k we just expand M in terms of monomials b t . Then the coefficient of z k G' t (k) in M is the rational coefficient of b t (ignoring the power of z). See [4, (3.7) (5)], and for some explicit examples [4, (4.3)]. We shall think of the basis element z k G' t (k) as also being indexed by the monomial b t . Put ^ ifc = W^-xJ'IO^Kp, a q ^2, Z? =1 (z + N-l)a f </c}. Then ^ has the same cardinality as 3B Ntk . An explicit bijection between 3S' N k and 3& Ntk is given by x? 1 x5 a -xj^yj r % a +1 - -y^_ 2 y^-i^ where e^JV + ^-'l is chosen to yield weight k. Give the set {x^x^ 2 -x a q q \Q^q^p,a q '2z2} the reverse lexicographic order and let 8' N k have the induced order. Let & Nk be given the order corresponding to . that of &' N k under the above-mentioned bijection between & Ntk and &' Ntk . This done, let C be the matrix over Q whose ij entry is the coefficient of the jth element of & k in the expression of the ith element of & Ntk written as a Q-linear combination of ^ k . The linear independence of 38 Ntk follows immediately from the following theorem. Theorem 4.1. Let p ^ and let be the matrix (with entries in Q) defined above. Let r\ be the submatrix o/( consisting of the columns corresponding to 3^' N ^ Then det(^) ^ 0. Our first attempt to prove the linear independence of the ^ Njfc was by proving (4.1), but this turned out to be somewhat elusive. So we ended up proving (2.1) using the basis interchange technique given in 2. However, we were still intrigued by the equality card( # Ntk ) = card( &' N k ), and we were finally able to prove (4.1), showing that this equality is not a coincidence. This gives an independent, but more difficult, proof of (the linear independence part of) (2.1). In the proof of [4, (4.1)] (the case N = l)'the matrix ( = r\ in this case) was triangular with nonzero entries down the diagonal so non-singularity was easy to establish. We have not been able to find such a simple argument in the case N > 1. The following example will help explain the meaning of (4. 1), as well as illustrate (2.3). Example 42. Let JV = p = 2, fc=10. Then J" 2>10 = {l^x^x^XpX^x^xJx^x!} and in the corresponding order # 2tlo = {y 10 , y 2 y s , y^y 6 , y|y 4 , y*, y 3 y 7 , y 2 y 3 y 5 , y\y\> y^yj. Then K 1(U has basis {y 10 }, F 10>2 /F 10 ,i has basis {y 2 y^y^i}, ^10,3/^10,2 has basis fe'Wa^?^}' "10.4/^10,3 has basis (vly^ylyli and "io, 5 /"io,4 has basis {yf}- The complete list of monomials corresponding to & k is {l,XpXpXpXpXpxJ,xJ,x^, 1 * 2* X^X-2? X^X*2^ X^X^i X]X>2i X-^ X^ X-^X^ X^ X^X^t X-^X^ X-^jC^ Xr-^X,^ 2* 12' 12' 2 / so the matrix is 9 by 26. Monomials of degree greater than 5 can be omitted since all entries in their columns will be 0. This leaves l.xxxxxxxxxxx 14 L Reid et al X 2 x i x i x i x l x 2 x i x 2> x 2} so ^ e non-trivial part of f is 9 by 15. We shall not write this matrix down, but the possibly nonzero entries by degree considerations (a row of degree d can have nonzero entries only in a column of degree ^d) are indicated by *'s, and only the subscript digits are indicated for the row indices (x being 10). The column indices of ^' 2tlo (i.e. the columns of rf) are underlined. 2tlo X * 28 * * * 226 * * * * 2224 * * * * * 22222 * * * * * * 37 * * * 235 * * * * 2233 * * * * * 334 * * * * * * * * * * * * * * * * * Theorem (4.1) in this case is sharper than (2.1) in that there are several other maximal minors that could be nonzero. The proof of (4.1) will now occupy the rest of this section. The various constructions involved are illustrated by Example (4. 12) below, to which the reader might refer while working through the proof. Suppose p = 0. Then ^ = {z fe } and ^i^ = {l}- Further, ^ N , fc = if 0<k<N and & Nik = {y k } otherwise. So ( is either the Oxl empty matrix or the 1x1 identity matrix, and (4.1) holds trivially in either case. Similarly, (4.1) is trivial in case /c = 0. Assume therefore that p^ 1 and k^ 1. The integers p, k and N ^ 1 are fixed in what follows. Let d = \_(k/N)]. In the notation of [4, (3.5)] let s/' = {(a l5 . . .,a d )eZ d 1 <!< a d _! = a d < p}. For i > 1 let a t be the number of times i occurs in (a l9 ...,a d ). Then the correspondence (a !,..., a d ) <-+ (a !,..., a p ) identifies s#' with the following subset U of (Z + ) p : d, 3; with a^ 2 and a i = For a = (a l9 . . . , a f define the Define K = of a to be wt(a) = = {(0,...,0)} and f- 1 (W + * - l)flj. Let for 1</<P define . Put ^ = We use the reverse lexicographic order on U. Namely, (a l9 . . . , a p ) < (b^ . . . , b p ) (or (a l9 ...,a p ) "precedes" (b l9 ... 9 b p )) if the last nonzero entry of (a l5 ... 9 a p ) (b l9 ... 9 b p ) is negative. Let Fand P^have the induced order. This order is such that the elements of V j , l (resp. Wj-t) precede those of V j (resp. Let S = Q[x l5 ...,x p ,T] and let the ith element of W define F (T) = i (where x = l). If (a l9 ... 9 a p )i& - JF(N + p - Note that JF(n) = (7 n ) r = i, and that F t (k) is the fth element of a Ntk (with z set equal to 1). The reason for decreasing the last index in defining the elements of W is to take The structure of generic subintegrality 15 into account the adjustment of the last index to obtain weight k when defining the elements of Ntk . If (b lt ... 9 b p ) is the ;th element of V then x* l x^ 2 ---xj* is the jth element of S8' Nk (where the latter has the same order as before). Let r = card(F) = card(H / ). Let M(T) be the rxr matrix (Afy(r)) 1<jJ<r with M;;(r)eQ[r] the coefficient of x^x* 2 --^ in F (T), where (b l9 ... 9 b'J is" the jth element of V. (Note that the rows of M are indexed by W and that the columns are indexed by V.) By the discussion preceding Theorem (4.1), M -(/c) is the coefficient of z k G' t (k) (t corresponding to the jth element of S8' N k ) in the expansion of the z'th element of & Ntk . Therefore ?/ = M(fe), so (4.1) is equivalent to M(/c) being invertible. If p=l then M(k) is lower triangular with nonzero entries down the diagonal, hence trivially invertible. The argument that follows is needed only for p ^ 2. Note that if j corresponds to an element of V h then deg r Afy(TXfc. (4.3) Therefore deg T det(M(T))<<5:= /rcard(F ft )= /i-card(^). (4.4) /=! A=l Our intention is to show that M(k) is invertible by finding S roots for det(M(T)), each less than fc, and then showing that the coefficient of T 8 in det(M(T)) is not identically zero. The d roots will be found by obtaining coincidences of the rows of the matrix M(s\ as s ranges between N and k 1. We begin by proving a few lemmas. Lemma 4.5, Let M(T) be an rxr matrix with entries in Q[T]. Let peQ. Let ^ be the set of rows of M(T) and let f be the set of all nonempty subsets of %,. Suppose there exists a subset S of f such that (1) The sets in S are disjoint. (2) For each Ee<?, all the rows in E coincide when T is specialized to \JL. Lei c = c(S) = Z e ,(card(E) _ i). Then (T- tf divides det(M(T)). Proof. It is clear that rank(M(/z)) < r c. By elementary row and column operations over Q[T] the matrix M(T) can be reduced to a diagonal matrix D(T) with diagonal entries {/i(T),...,/ r (T)}. (This is well known, and easily proved using that Q[T] is an Euclidean domain.) Then (since the same operations can be carried out with T set equal to ILL) we have rank(5(ju)) = rank(M(/x)) < r c. Thus (T ^) divides at least c of the' /r Since (up to a nonzero scalar) det(M(T)) = det(D(T)) = U\ = l /., the lemma follows. M Before stating the next lemma we introduce some notation. For a=(a l9 . . . , a p )e(Z + ) p put / = yj l yy +1 -7j i v p . 1 . Then @ N , k = {y a y k - wt(a} \aeW}. Since the rows of M(T) correspond to & Ntk , those of M(s) correspond to & N , k (s):= {y a y s - wt (a)\ a eW}. Here the elements y a y s - wt (a) are treated as symbolic monomials with s wt(a) allowed to be negative. Given symbolic monomials y a y t , y b y u with a,be(Z + ) p and t, weZ, we say they are formally equal if at least one of the following two conditions holds: (1) (a, t) = (b, u); (2) both t and u belong to the set {0} u [AT, N + p - 1] and /y f and /y tt coincide as formal monomials in 7 N ,...,7 N+p _ 1 on replacing y by 1. We say that a row R of M (s) is labeled by a symbolic monomial y fl y, if the symbolic monomial in 16 L Reid et al @ Ntk (s) corresponding to R formally equals y a y t . Clearly two rows of M(s) labeled by the same symbolic monomial are equal. Let 2;= {(b l9 ... 9 b j9 Q 9 . . . , 0)e(Z + H^O}. For beQj put E(6)= Wn{fr-*,|0< i < j} where e Q = (0, . . . , 0) and for 1< i < p, e = (0, . . . , 1, . . . , 0) is the standard basis vector with 1 in the ith place. Lemma 4.6. Let beQj. Then the rows of M(wt(b)) wWc/i are labeled by /( = /y ) are precisely those indexed by E(b). Moreover, ifb, ceQj with b^c and wt(b) = wt(c) thenE(b)r\E(c) = 0. Proof. It is clear that the rows of M(wt(b)) indexed by E(b) are labeled by y b . Let R be a row of M(wt(b)) which is labeled by y b . Let a be the element of W corresponding to R. Then the symbolic monomial of & Ntk (wt(b)) corresponding to is y a y wt(b )- wt(a y Comparing the subscripts and exponents of this symbolic monomial with those of y b we conclude that aeE(b). This proves the first part. Now, let 6, ceQ/ with wt(fc) = wf(c) = s, say. Suppose E(b) and E(c) have a common element, say a. Let R be the row of M(s) indexed by a. Then jR is labeled by y b as well as by y c whence we get b = c. H Lemma 4.7. For an element b of Qj the following three conditions are equivalent: (1) card(E(b)) ^ 2; (2) b-e^W for some i, Q^i< j; (3) b - ep Wandb- e t e W for some i, < i<j- Moreover, if any of these conditions holds then wt(b) < k. Proof. Assume (2). Then b e^Qh for some fc, i^h^j. Since b eieWj we have \vt(b - ej) < wt(b - ed ^ k - (N +j - 1)< k ~ (N + h - 1) whence b - e^ W h . This proves (2)=>(3). Also, the inequality wt(b-ej)<k-(N+j-l) gives \vt(b)<k. The implications (1)=>(2) and (3)=>(1) are trivial. Put Q = {beU'^Qj\caid(E(b)) ^ 2}. Lemma 4.8. The product U bQ (T- w^fc)) "^^" 1 divides det(M(T)). Proof. Writing Q(s) = {beQ\ wt(b) = s}, it is enough to prove that U bfe Q (s) (T-w^fr)) 6 "^ 5 "" 1 divides det(M(T)) for every s. But this is immediate from (46) and (4.5), since rows labeled by the same symbolic monomial are equal. \ Lemma 4.9. % beQ (card(E(&))-l) = <5. Proof. For fceQ n Q,. put E'(b) = { (ft, b - e t ) \ ^ f < ;, b - e,e W}. It follows from (4.7) that card(F(fc)) = card((fe)) - 1. Let <$ =U beQ E'(b). The second projection induces a map rj:^^W. Let ae W} and let i be an integer with ^ i <j. Then a + e eQ by (4.7). It follows that //~ V) = {(a + e* a)|0 < i <j}. Thus there are exactly j elements in the fibre of r\ over each element of Wj. Therefore we get ^ beQ (csird(E(b)) 1) = ) = card(^) = J, , 7 card(^.) = 5. Now, since deg r det(M(T)) ^6 by (4.4), and since (4.7)-(4.9) taken together exhibit 8 roots of det(M(T)) each less than /c, it remains only to show that det(M(T)) is not 'The structure of generic subintegrality 17 identically zero. We do this by showing that the coefficient of T* is not zero. Let a tj be the coefficient in M {T) of T h if j corresponds to an index in V h (by (4.3) h is highest power of T with a potentially nonzero coefficient in M i} {T)). It then suffices to show that det((0- -))^0. For l^i, j^r (where as before M is rxr) let H t (T) = F(T) ai F(T + If 2 - F(T + p - l) fl *, where (a l9 ..., a p ) is the fth element of W, and let r y be the coefficient of x^x^ 2 ---*^ in HN), where (fe^...,^) is the jth element of W. Let h be the index for which (b l9 ... 9 b p )eW h . Then, since (fc l5 ...,l +b h , 0,...,0) is the corresponding element of V h and since FT) = Hi(N)F(Tc) for some integer c, we get <r y = (l/fc!)T y . So it suffices to prove that det(i) ^ (where t = (r y )). Rearrange the rows and columns of i by reordering W by degree (where degree (a l9 . . . , a p ) = IX-). Then T is lower-block triangular with degree blocks down the diagonal. It suffices to show that each of these blocks has a nonzero determinant. Therefore for u (Q^u^d I) let S u be the submatrix of t with rows and columns indexed by elements of W and V of degree u. It suffices to show that The matrix S u is obtained as follows: Let W(u) be the elements of W of degree u, and let r(u) = card( W(w)). For 1 ^ i, j ^ r(u) let (a l9 ... 9 a p )be the fth element of W(ii) and let (b !,..., b p ) be the jth element of W(u). Define an r u x r u matrix L U (T) by setting the (i, ;') entry to be the coefficient of x*'x* 2 xj* in H t (T). Then S u = L M (AT). Thus it suffices to show that W is not a root of det(L M (T)). Since we are now dealing with the homogeneous case we can replace F(T) by F(T) = ^? = A lx and H^T) by Hi(T) = F(T) ai F(T + l) fla - F(T + p - l) ap without changing L U (T). We now note that H-(T) is divisible by T l (T + l) a2 - - - (T 4- p - l) flp , or equivalently, the ith row of L u is divisible by T fll (T-h l) fl2 (T + p - l) ap . Factoring out these entries from the rows of L U (T) we obtain a matrix K U (T) which can be defined directly as follows: let /T-l\ G(T) = f =1 (l /f) )x, (so that TG(T) = F(T)) and define L t (T) = G(T) fll G(T + - l) ap . Then the (i 9 j) entry of X U (T) is the coefficient of x^x^ xj* in Li(T). Noting that the roots of the factors (T+ i) flf are all <0, it suffices to prove that N is not a root of det^R^T)). In fact, det(K u (T)) is a nonzero constant, as we show next. For a = (a l9 ... 9 a j9 Q 9 ... 9 Q)eQj define aw(a), the augmented weight of (2, to be N +7 - 1 + Zf = t (N + i - l)fl . Also define aw(0) = 0. If ae W then aw(a) is the weight of the corresponding element of V, and aw(a)^k for all a^W. Now, order the elements of W(u) by augmented weight with small weights coming first, and order elements of the same weight by reverse lexicographic order as was done previously. This ordering is such that (4.10) if for j < i we decrease a t by one and increase a j by one then we get an earlier element in the ordering. Furthermore W(u) is a leading segment in the set W(u) of all elements of degree u in (Z + ) p (where W(u) is ordered in the same manner). The matrix K U (T) can be constructed with W(u) ordered in this way without changing the value of det(X M (T)). Now we shall work with W(u). Let r(u) = czrd(W(u)) and let K U (T) be the f u x f u matrix whose (i, 7) entry is the coefficient of x^x* 2 - x b p p in L t (T):= G(T) ai G(T -h I)" 2 G( T + p l) ap where (a l9 ... 9 a p ) and (b l9 ... 9 b p ) are respectively the fth and the jth 18 L Reid et al elements of W(u) (for convenience of notation we are changing the 'meaning of L rather than introducing a new symbol). Let u = 1. If we take out the factors l/i from the columns then K^T) is reduced to the matrix J= ( J . If we subtract each row of J from the next (performing the operations in the order replace pth row by pth (p l)st, replace (p l)st by (p l)st (p 2)nd etc.) and use the . 1 tj .. /T-H-A /T+i-2\ r binomial identities 1 ( = then J row reduces to V j-i ; V j-i / V j-2 ; ) where J' = I ) - (Performing row operations in this J'J LV j-1 /JKI^P-I manner was suggested to us by Sue Geller.) Continued row reduction of thisjype (subtracting from a row Q-linear combinations of previous rows) will reduce K X (T) to an upper triangular matrix with ones down the diagonal. We conclude that det(J) = 1 whence det(K 1 (7'))= 1/p!, a nonzero constant. Now, let E = (E tj ) by any p x p matrix with entries in Q[T]. If R t is the ith row of E let us identify R t with the element E il x i +E i2 x 2 + -- + E ip x p of Q[T,x l5 ...,x p ]. Let /() be the f u x r u matrix whose (i, j) entry is the coefficient of x^x^ x^ in R^Rf -'R a p p , where as before (a l9 . . . , a p ) and (& 19 . . . , b p ) are respectively the tth and the jth elements of W(u). This construction is such that f u (K 1 (T)) = K u (T). Furthermore if we change E into a matrix E' by row operations of the above type (i.e. subtracting from a row Q-linear combinations of previous rows) then because of (4.10) f u (E) is changed into /(') by row operations of the same type. We have that / of an upper triangular matrix is upper triangular, so K U (T) can be converted to an upper triangular matrix with nonzero constant entries down the diagonal by a succession of row operations in which from a given row we subtract a Q-linear combination of previous rows. These row operations leave invariant the subspaces spanned by the first i rows (1 ^ i ^ f J. Since W(u) is an initial segment of W(u) we conclude that detX M (T) is a nonzero constant, completing the proof of (4.1). H Exampfe4.il. If JV = 3, p = 4, then in reverse lexicographic order we have (1, 0, 2, 0) < (0, 1, 2, 0) < (0, 0, 3, 0) < (2, 0, 0, 1) with augmented weights respectively 18, 19, 20, 18. Therefore if we order reverse lexicographically instead of by augmented weights the argument above will fail for k = 18 since then W(3) will not be an initial segment of W(3). Example 4.12. Let us return to (4.2), where N = p = 2, /c = 10. Here we have K = {(0,0)}, ^ = {(2,0), (3,0), (4,0), (5,0)}, V 2 = {(0 9 2\ (1,2), (2,2), (0,3)}, W* = {(0,0)}, W l = {(1,0), (2,0), (3,0), (4,0)} and W 2 = {(0, 1), (1, 1), (2, 1), (0,2)}. The rows of M(10) are indexed by the monomials # 2flo ylyl^ly^} as noted in ( 4 - 2 )- Thus the rows of M(s) are indexed by j {y& v 2 ^-2' 72^-^ rive* y^s-s^ y^y s -3^ y2yiy s - 5 > yly*y s -v yly s -6}- The polynomial det(M(T)) is of degree card(7 x ) + 2card(F 2 ) = 4 + 2-4 = 12, and we have Q = {(1,0), (2,0), (3,0), (4,0), (0,1), (1,1), (2,1), (3,1), (0,2), (1,2)}. Taking 6 = (1,0) we get (&)= {(0,0), (1,0)}. This corresponds to the pair y s >y 2 y s . 2 , indexing the first two rows, which become equal when we set s = 2. The complete set of row coincidences is obtained similarly and is given by the following table: The structure of generic subintegrality 19 b E(b) elts of * tfft (r\ 2 1 0\ / rows roots of det(M(T)) (1,0) (0,0), (1,0) 7 S , 7 2 7 s-2 1,2 2 (2,0) (1,0), (2,0) y 2 y s _ 2 (> 7 2 y s 2,3 4 (3,0) (2,0), (3,0) 727s -A v^v I- 5 '2's 6 3,4 6 (4,0) (3,0), (4,0) y3y 7 2 y s 4,5 8 (0,1) (0,0), (0,1) 7 S > 7 3 7 5-3 1,6 3 (1,1) (1,0), (0,1), (1,1) 7 2 7 s - 2 ^W.-^Ws-S 2,6,7 5,5 (2,1) (2,0), (1,1), (2,1) 727s -4 ^2^3^-5^273^-7 3,7,8 7,7 (3,1) (3,0), (2,1) 727 s - j, 7 2 7 3 y s - 7 4,8 9 (0,2) (0,1), (0,2) 7s7 s -3 P yly s 6,9 6 (1,2) (1,1), (0,2) 7 2 7 3 7 S -s>/&-6 7,9 8 By direct computation det(M(T)) turns out to be 2 4 3 5 (r-9)(T-8) 2 (r~7) 2 (T-6) 2 (T-5) 2 (T-4)(r-3)(T--2), which is in agreement with the roots (together with multiplicities) obtained from the above table. We have that det(M(T)) does not vanish at T= 10, as claimed. Now we shall illustrate some features of the last part of the proof. Here G(T) = xj + ((T - l)/2)x 2 , and G(T + 1) = x t + (T/2)x 2 , so K^T) = K^T) = (I (T- l)/2\ v " . We have W(3) = { (3, 0), (2,1)} and W(3) = {(3,0), (2,1), (1,2), (0,3)}. \1 (T/2) J The respective augmented weights of the elements of W(3) are 8 ( = 4-2), 10 ( = 2-2 + 2-3), 11 (= 1-2 + 3-3) and 12 ( = 4-3). The last two have weights greater than 10 and so are not included in W(3). lip = 2 the reverse lexicographic ordering is also an ordering by weight, but this need not be the case for larger p, as we saw in (4.11). Set R l = G(T) and R 2 = G(T+ 1). Then the matrix K 3 has rows {jRj, .R 2 2 , jRjjR 2 , ^2)9 ( or more precisely the 4x4 matrix obtained by taking the coefficients of {xpX 2 x 2 ,x 1 x 2 ,X2} in these polynomials). The rows of K 3 will be denoted as {r 1 ,r 2 ,r 3 ,r 4 }. The row operation that reduces K^T) to upper triangular form is to replace {Ri 9 R 2 } by {R 1 ,R 2 R 1 }. Then / 3 ({^i,^ 2 ~^i}) has row corresponding tr\ /Z?3 l?2/n p \ n fn n \2 / p p \3\ fp3 D2p p3 p p2 0/?2p _i LO |r\.^ y i\^^/\ 2 ^iJ> ^-ll^2 -*^1/ 1^2 -*^1/ J 1 1> 12 *^l> ^1^2 ^1^2 ' ^i ^2 ~ 3-^1-^2 "^" ^^ 1^2 ~ ^?} so the row operation to reduce K 3 to upper triangular form (with nonzero diagonal entries) replaces {r 1? r 2 , r 3 , r 4 } by (r l9 r 2 r l9 r 3 2r 2 + r 1? r 4 3r 3 H-3r 2 + r x }. The matrix K 3 is the upper left 2x2 submatrix of K 3 , to which these row operations restrict, so det K 3 is also a nonzero constant. If we had used weight 11 rather than 10, then K 3 would have been the upper left 3x3 block of K 3 , which also has determinant a nonzero constant, for the same reason. 5. Hilberty polynomials The graded ring R (N) has Hilbert function H given by H(n) = dim Q /?Jf. We consider the problem of expressing H(n) as one or more polynomials in n. The Hilbert function 20 L Reid et al of a graded ring which is standard (i.e. finitely generated over a field by elements of weight 1) is given for n by its Hilbert polynomial. Our ring R (N) is finitely generated but is not standard except in the trivial case p = 0,.JV= 1. For such a ring there exist, by [5, Corollary 2], a positive integer d and polynomials H , H ly . . . , H d _ t such that H(n) = Jf j(n) if n and n == i (mod d ). (*) In general, it is of interest to quantify precisely the condition "w 0". In particular, in the standard case, if the Hilbert function coincides with the Hilbert polynomial for all n ^ then the ring is called a Hilbertian ring. So we may call a general finitely generated graded ring Hilbertian if (*) holds for all n ^0. In our first result (5.1) we show that R (N} is Hilbertian if p ^ 2, and determine the minimal d satisfying (*). If p = then H(n) =1 if n = or n ^ N 9 so in this case (*) holds with d= 1, H = 1, and R (N) is Hilbertian if and only if W= 1. Now, in general, to say that R (N) is Hilbertian is the same as saying that its Hilbert function H is a quasi-polynomial in the language of [6, (4.4)]. The integer d appearing in (*) is then a quasi-period of H . Theorem 5.1. Let d = lcm(JV,AT+l,...,Ar + p--l). // p>2 then H is a quasi- polynomial with minimum quasi-period d, and in particular R (N) is Hilbertian. If p = 1 then the function H given by fl(n) = H(n) for n^l and fi(0) = H(0)- 1=0 is a quasi-polynomial with minimum quasi-period d. Proof. Let P(t) = ^ H(n)t n and P(t) = n %H(n) n , whe r e we put H = H if p ^ 2. Then by (2.8) we have and P(t) = P(t) - 1 = t N /((l - r)(l - r*)) if p = 1. In either case write P(t) = with f(t\ g(t) polynomials without a common factor. Then deg f(t) < deg#(t) and the zeros of g(t) are the dih roots of 1. So by [6, (4.4.1)] H is a quasi-polynomial with quasi- period d. To prove the minimality of d, we claim first that d is the 1cm of the orders of the roots of unity which occur as zeros of g(t). This is clear if p= 1. Hence assume that p ^ 2. If 1 is a root of unity as well as a zero of 1 t + t N+p then 1, - 1 and JL N+P are three roots of unity whose sum is zero. This is the case if and only if (1, -/I, 1 N+P } are the three cube roots of unity. Thus A is a primitive cube root of unity, so >L is a primitive sixth root of unity and 1 N+ p = (- 1) 2 is the other primitive cube root of unity, whence N + p = 2 (mod 6). Obviously 1 - 1 + t N+p has no repeated factors, so if N 4- p s 2 (mod 6) then we can cancel the cyclotomic polynomial 1 - 1 + 1 2 of primitive sixth roots of unity once, otherwise there is no cancellation. The cancellation still leaves us with roots of unity of order 2 and 3 as zeros ofg(t\ proving our claim. _ Now let D be the minimum quasi-period. Then we can write P(t) in the form P(t) = ^~o^oHj(j + Di)t j+Di for some polynomials #,, Multiplying by l~t D amounts to differencing the coefficients (except in low degrees) so (l-t D ) e P(t) is a polynomial in t for some positive integer e. Therefore the roots of unity that occur as zeros of g(t) must have orders which divide D. Thus d divides D, proving the minimality of d. The structure of generic subintegrality 21 Theorem 5.2. The polynomials H t in (5.1) are all of degree p. Proof. This is seen by examining the partial fraction expansion of P(t). We have that 1 is a root of the denominator of P(t) of multiplicity p-f 1, and that all other roots are of smaller multiplicity. Setting X = AT in the well-known expansion x n (in which the coefficient of X n is a polynomial in n of degree r 1), we see that a root k of multiplicity m of the denominator contributes a polynomial of degree m 1 to each of the Hj. Thus 1 contributes degree p to each Hj and the other roots contribute a lower degree, so the highest degree terms cannot cancel leaving all the Hj of degree p. M Now, we give an example to show that the various Hj need not be distinct. Consider the case N = 2, p = 3, where our Poincare series l-t + t 5 has partial fraction expansion a b 1/8 2/9 with a of degree 3 and b of degree 1 which need not be stated explicitly. The power series expansions of l/(l + 2 ) and l/(l+t-K 2 ) are of periods 4 and 3 respectively, with coefficients in each period being 1,0, 1,0 and 1, 1,0 respectively. The "non-polynomial" contribution to the various H(i) are given by the following table (with rows corresponding to t l for i = 0,1,2,... and columns corresponding respectively to the roots of order 1,2,4,3): I J I 1 1 1 1 ] I -1 -1 2 1 L 1 -1 3 ] [ -1 1 4 ] I 1 1 -1 5 -1 6 1 -1 1 7 1 -1 8 1 1 9 -1 1 10 1 -1 -1 11 -1 12 [ 1 1 1 The polynomials coincide if and only if the rows are the same. By inspection of the table we see that the period is indeed 12, as given by (5.1), and that H l = # 7 , H 3 = H 9 , 22 L Reid et al and # 5 = H 11 , with the polynomials Hj (0<j<ll) being otherwise distinct. Tl equality of the /f/s here comes from the O's in the power series expansion of tl cyclotomic polynomial of primitive fourth roots of unity. Note that the possibility are determined only by the columns corresponding to roots of order 4 and 3. Obvious the first column plays no role in deciding on the cases, and the second does not eith< since whenever entries in columns three and four are equal, so are the entries column two. t t 2 t 3 19 By explicit computation we obtain H (t) = 1 H 1 1 , H 1 = H 1 = D T-O InrT" J.H'T 1 5t t 2 t 3 I j. e tc. with the polynomials all of degree 3 as claimed by our theorem, ar 48 48 144 F with polynomials equal and distinct as claimed above. The coefficients of t 2 and are the same in all polynomials, which can be explained by the fact that only tl root 1 has multiplicity greater than two, and the coefficient of t is periodic with peric 2 since only the root 1 has multiplicity 2. In another example that we have worked out, equality of the various Hj arose a seemingly accidental way from primitive roots of unity of order other than powe of two. The general situation seems to be quite complicated. References [1] Herrmann M, Ikeda S and Orbanz U, Equimultiplicity and Blowing up (New York: Springer- Verh 1988 [2] Reid Les, Roberts Leslie G and Singh Balwant, Finiteness of subintegrality, in Algebraic K-Thec and Algebraic Topology, P Goerss and JF Jardine (eds) (Kluwer) 1993, pp. 223-227 [3] Roberts Leslie G and Singh Balwant, Subintegrality, invertible modules and the Picard Group, Comp Math. 85 (1993) 249-279 [4] Roberts Leslie G and Singh Balwant, Invertible modules and generic subintegrality, J. Pure Af Algebra 95 (1994) 331-351 [5] Shukla PK, On Hilbert functions of graded modules, Math. Nachr. 96 (1980) 301-309 [6] Stanley Richard P, Enumerative Combinatorics, Volume I, (Wadsworth and Brooks/Cole) 1986 [7] Swan RG, On seminormality, J. Algebra 61 (1980) 210-229 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 1, February 1995, pp. 23-29. Printed in India. Flat connections, geometric invariants and energy of harmonic functions on compact Riemann surfaces K GURUPRASAD Department of Mathematics, Indian Institute of Science, Bangalore 560012, India MS received 1 September 1993; revised 23 March 1994 Abstract. A geometric invariant is associated to the space of flat connections on a G-bundle over a compact Riemann surface and is related to the energy of harmonic functions. Keywords. Principal G-bundle; flat connections; Chern-Simons forms; energy of maps; harmonic maps. Introduction This work grew out of an attempt to generalize the construction of Chern-Simons invariants. In this paper, we associate a geometric invariant to the space of flat connection on a S[/(2)-bundle on a compact Riemann surface and relate it to the energy of harmonic functions on the surface. Our set up is as follows. Let G = Sl/(2) and M be a compact Riemann surface and ->M be the trivial G-bundle. (Any Sl/(2)-bundle over M is topologically trivial). Let # be the space of all connections and J^ the subspace of all flat connections on this G-bundle. We endow on # the Frechet topology and the subspace topology on 3?. Given a loop tr.S 1 -#", we can extend a to the closed unit disc <r:Z) 2 -># since <g is contractible. On the trivial G-bundle E x D 2 -*M x D 2 we define a tautological connection form 9" as follows Clearly restriction of $* to the bundle E x {t}-+M x {t} is 5(t)VteD 2 . Let K(&*) be the curvature form of S*. Evaluation of the second Chern polynomial on this curvature form K(& ff ) gives a closed 4-form on M x D 2 , which when integrated along D 2 yields a 2-form on M. This 2-form is closed since dim M = 2 and thus defines an element in H 2 (M,R)&R. It is seen that this class is independent of the extension of a. We thus have a map where !(#") is the loop-space of #". We assume that the genus of M^2. The energy E(f) of any smooth function /:M-G is defined using the Poincare metric on M and the bi-invariant metric on G = SU(2) given by the Killing form. 23 24 K Guruprasad Any smooth function f:M-+G defines a flat connection &>/ = /* (A*) on the trivial bundle M x G- M, where ju is the Maurer-Cartan form on G. By a result of Hitchin ([H]), the loop in V is given by G f (t) = -(o f -f (cos t)o) f + (sin t)(*o)f)) for te[0, ITT], where *;A 1 (M,^)->A 1 (M, ( ^) is the Hodge star operator, is actually a loop in OF if and only if / is harmonic. (9 is the Lie- Algebra of G). The main result of this paper is Theorem ///:M~G is a harmonic map, then 1. Construction of the basic geometric invariant In this paper we suppose M is a compact Riemann surface of genus with Lie algebra # = su(2) and n:E-+M is the trivial G-bundle on M. V is the space of connections and ^ is the subspace of all flat connections on E-*M. D 2 is the closed unit disc in R 2 and dD 2 = S 1 is the unit circle. O(J ? ') = Map(5 1 ,J 5 ') is the loop-space of J 5 ". Given a loop crrS 1 -* J* we extend cr to <r:Z) 2 -# (# is contractible). On the trivial bundle x D 2 - M x D 2 , let 9" be the tautological connection defined in the introduction. Let K($ ff ) be the curvature 2-form of the connection 9 ff . Let C 2 be the second Chern polynomial on ^. For the Lie algebra 9 = su(2\ C 2 is essentially the determinant. More particularly C 2 (A)= ~(l/47i 2 )det(A) for Aesu(2) (cf. [KN], Chap. XII). Now an easy computation shows that C 2 (A) = -^trace(,4 2 ) for Evaluation of C 2 on K(9 ff ) gives a closed 4-form C 2 (K(9 ff )) on x D 2 which projects to a closed 4-form C 2 (K( ff )) on M x D 2 . Integrating C 2 (K(S ff )) along D 2 yields a closed 2-form on M(dim M = 2) and thus defines a cohomology class in H 2 (M, R) i.e. R. D 2 J We outline the proof of the following lemma (cf. [G], 1 and [GS], 2,3). Lemma LI. J D2 C 2 (K(& <r )) is independent of the extension of cr.S 1 -^ to d:D 2 :-^^. Proof. Let a l ^& 2 be two extensions of a with corresponding connection forms 9^, &2 and curvature forms (&*), ($*) on the bundle E x D 2 -M x D 2 . On x Z) 2 we have Flat connections, geometric invariants ... 25 where TC 2 ($l), TC 2 ($ a 2 ) are the Chern-Simons secondary forms with respect to SJ, 2 respectively (cf. [CS, 3]). We can easily check that C 2 (K(&*)) - C 2 (K(9*)) is an exact form on E (cf. [G, 1]). Since n*:H 2 (M,R)-+H 2 (E,R) is an isomorphism it follows that {C 2 (K ($*))} = {C 2 (K (<[))} eH 2 (M,R) and this proves the lemma. We thus have a map 1)2 where Q^) is the loop-space of 3F. It is easy to check that %(G &') where crcr' is the composite of two loops in 3F. We call this map % the geometric invariant. 2. Energy of functions and a class of special loops We recall the definition of energy of a function. Let X and Y be Riemannian manifolds. Given a smooth map /:X-> Y, the energy density of / is a function e(f):X-+R defined by where ||d/(x)|| denotes the Hilbert-Schmidt norm of the differential d/(x)e T*(x) T /(JC) ( Y). If A" is compact and oriented, the energy of/, denoted by E(f) is given by Qy/2 (/)(x)dx) M / where dx is the volume form of X with respect to its Riemannian metric. / is harmonic if it is a critical point of the energy functional. Using the Poincare metric on the compact Riemann surface of genus ^ 2 and the bi-invariant metric on G = 517(2) given by the Killing form, we can define the energy (/) of a smooth function f:M-+G by the above formula. Any smooth function f:M^G defines a flat connection co f = /*(//) on the trivial bundle E -> M where is the Maurer-Cartan form on G. In the case of the trivial bundle E-+M, clearly the space of all connections # can be identified with the space A 1 (M,^) of all ^-valued 1-forms on M. For any smooth functipn /:M-G, consider the loop in ^ given by a f (t) = (co f + (cos t)a> f + (sin t)(*co f )) for te[Q, 2n], where *: A 1 (M, #) -* A 1 (M, 9) is the Hodge star operator. By a result of Hitchin ([H]), we know that a f ([Q, 2ri]) cJ 5 " iff /is harmonic, i.e. dy is a loop in ^" iff /is harmonic. 26 Guruprasad 3. Relation between the geometric invariant and the energy of harmonic maps We prove the following result Theorem 3.1. // f:M>G is a harmonic map, then x( a r) -- (/) 4n Proof. At the outset we show that the closed 2-form which represents i(a f )eH 2 (M, R) is (*&>! AcOi + *co 2 Aco 2 -f *o> 3 Aco 3 ) where 2n We extend the loop <7y in J^ to a map d f :D 2 -^^ in an obvious way. We drop the suffix / and simply use a and S in the computations that follow. Let (s, r) be the polar coordinates on D 2 = {(s, f), ^ s ^ 1, < t ^ 27i}. Set <r(s,t) = scr(t). We now compute the curvature K( ff ) of the connection form d ff on the bundle x D 2 -> M x D 2 . A where K(v(s> f) is the curvature of <r(s, f)) and d E and ^3 are respectively the exterior differentials on E and D 2 . If we set as a form on M for each feS 1 , then after a straightforward calculation (see [G], Lemma 4.1), it follows that J D2 C 2 (K(S' 1 )) is cohomologous to the form Now so that i L a(t) A CL> 2 (t) A (co 2 i(co 3 4- cos to> 3 -hsint*co 3 ) + c os ta> 2 -f sin t*o> 2 ) -f i(co 1 + cos tco a -f- sin t*co 2 ) + costo 3 -f Flat connections, geometric invariants ... 27 .e. Now oi(t) = (co 1 + cos to! = (o> 2 -f cos tco 2 + sin t*o> 2 ) = (o> 3 + cos to> 3 + si +cost*co 1 )A(a> 1 - >! 4- cos 2 t*(D Aco 1 = *<y 1 Aa> 1 . Similarly d at It follows that f^ C 2 ((3*)) is cohomologous to the form 1 = *G> 3 Aco 3 . A c^ 4- *G> 2 A co 2 4- *co 3 A co 3 )dt S i = (*&>! A a?! 4- *o> 2 A co 2 4- *o> 3 A co 3 ). 27T Thus the closed 2-form on M representing %(o f )H 2 (M,R) is (*o> 1 A co x 4- 2;r *co 2 A co 2 4- *ct) 3 A o> 3 ). To prove that %(G f ) = -- E(f\ w ^ check using local coordinates that the forms 471 2n dt A ~y(t) A dt and -- g(/)(w)dw (dm is the volume form on M) are equal at any arbitrary point. 471 Since any left translation in G is an isometry, for any weM, ||d/(m)|| = ||d(L /(wrl o/)(m)|| where L /(wrl :G->G is left translation by /(m)" 1 . We can therefore assume that / maps some point weM to the identity element in G, i.e. /(m) = 1. Since we intend to use local coordinates to prove the equality of forms, we can go to the universal cover D 2 of M with Poincare metric and assume /:D 2 -G and /(m) = 1 for some fixed meD 2 . Since there exist an isometry of D 2 which maps the origin to m, we can assume /(O) = 1 and check equality of forms at the origin. At the origin we have dx'dx dy'dy K Guruprasad 28 and where , are the usual coordinate vector fields. Let dx and dj; be the dual 1 -forms ox dy Clearly at the origin *dx = dy and *dj; = dx. Since dm = dx A dy we have dy , 4-*co 2 Ao> 2 -f *co 3 Ao> 3 ) , 1= -- e(f)(m). dx dy) 4n We prove that 2n If o)j = djdx 4- bjdy (1 <; < 3, a j5 b } are functions on D 2 ) then *(0j = cijdy. - bjd* for 1< j ^ 3 so that *co j A co,- = - (aj 4- b*)dx A dy for 1 < j < 3 =* r-Mi A a*! H- *co 2 A o) 2 + *o) 3 A co 3 ) = -- (aj + &J + a 2 -h ^ + a^ 4- bl)dx A dj; 271 2n For/:D 2 -^SC/(2) with /(O) = 1 2 dx By definition of Maurer-Cartan form ?x ; i.J d M\ -^1 V \ dx J ^\ dx J -\ dx ) ! The pairing (A, B)h-trace(/lJ3) for A, Besu(2) gives the Killing form on su(2) so that dx 1 dx Similarly dx Flat connections, geometric invariants . . . Noting that f*fij = a>j(l ^j < 3) we have 29 Now Therefore Similarly Thus Therefore we have ( (*a>! A a> 1 + *a> 2 A o> 2 + *o>3 In other words A dy. ( (*co 1 A co 1 + *co 2 A o) 2 + *o> 3 A co 3 ) J = e(f)(m)dm Consequently x(<r f ) = E(f) and the theorem follows. 4n Acknowledgement It is a pleasure to thank A R Aithal, I Biswas and N Hitchin for helpful discussions. References [CS] Chern S S and Simons J, Characteristics forms and geometric invariants. Ann. Math. 99 48-49 (1974) [G] Guruprasad K, Flat connections, geometric invariants and the simplectic nature of the fundamental group of surfaces. Pac. J. Math. 162 no. 1, (1994) [GS] Guruprasad K and Shrawan Kumar, A new geometric invariant associated to the space of flat connections. Compos. Math. 73 199-222 (1990) [H] Hitchin N J, Harmonic maps from 2-torus to the 3-sphere. J. Differ. Geom. 31, 627-710 (1990) [KN] Kobayashi S and Nomizu K, Foundations of Differential Geometry, Vol. II Interscience Publications, (1969) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 1, February 1995, pp. 31-32. Printed in India. Fibred Frobenius theorem PEDRO M GADEA and J MUNOZ MASQUE* C.S.I.C., I.M.A.F.F, Serrano 123, 28006-Madrid, Spain *C.S.I.C, I.E.C., Serrano 144, 28006-Madrid, Spain MS received 27 April 1994 Abstract. We give a version of Frobenius Theorem for fibred manifolds whose proof is shorter than the "short proofs'* of the classical Frobenius Theorem. In fact, what shortens the proof is the fibred form of the statement, since it permits an inductive process which is not possible from the standard statement. Keyword. Frobenius theorem Theorem. Let n:M -> N be a submersion, dim N = n, dim M = m, + n, and let E c V(n) be an involutive sub-bundle of rank r of the vertical bundle V(n) of K. Given a point y eM and any coordinate system (x ly ... 9 x n ) on a neighbourhood of x = n(y \ there exist functions (yi,.-.,y m ) on M such that: (i) (x 1 K 9 ... 9 x n n t 9 y l9 ... 9 y m )isa coordinate system on an open neighbourhood U ofy , (ii) Proof. By induction on r. For r = 1 there exists an open neighbourhood U Q of y and a non-singular vector field Y such that r(l/ ,) = < Y>. Since n is a submersion, given (x l5 ...,x w ) there exist functions (y' v . * . <> y' m ) on M satisfying (i) and, since Y is vertical, we have Y=^ l =1 f i (d/dy' l ). As Y is non-singular, we can apply the theorem of reduction of vector fields to normal form (see [1, Lemma 2]) by considering (x !,..., x n ) as parameters, thus obtaining a system (x 1 TT, . . . , x n n; y 1 , . . . , y m ) such that Assume r> 1. There is an open neighbourhood 17 of y on which E admits a basis: r(/ ,E) = <Y 1 ,..., 7 r >. Applying the above case to Y l9 we obtain a system (*! TC, - . . , x n ir, j/ 1? . . . , y'J such that Y 1 = d/dy' v Let so that Y' 2 , . . . , Y' r span an involutive sub-bundle E f c E of rank r 1. In fact, as E is involutive, we have for 2 ^ j, j < r: [y;,y"j]= i f\? k = E k=2 As 7^;) = 0, one has [FJ, Y;] (j/J = 0. Hence / y = 0. Let Tt'iM-^JV x [R be the submersion 7r' = (7c,/ 1 ). Since Y / j(y\) 9 we have E c ^(Tr'). Let x n+ ! :N x IR -> [R be the projection onto the second factor, which makes (x l9 ...,x n ,x n+l ) a coordinate system on N x (R. By the induction hypothesis, there 31 32 Pedro M Gadea and J Munoz Masque exist functions 0>2>-->JC) satisfying conditions (i) and (ii) with respect to E 1 . Consequently, there exists an open neighbourhood U of y such that HI^^Y!, 3/3/2,... ,3/dj/'>, and from x n +iri^y' v we deduce Y 1 = d/dy\ + ^ 2 fi(d/dy f f). Substituting Y\ =d/dy\ + !?-,+ 1/^ 3 / 8 ??) for Y l9 we also have T(U, E) = < FI , 3/3/2, - > 3/3y, >, and since E is involutive for every 2 <;* < r, we have: Applying both sides to y\ we conclude ^!=0. Hence 3/ +r /3jJ = 0; that is, (/r + 1 / J depend only on (x x TT, . . . , x n TT; y\ , ^ + 1 , . . . , /^). Consequently, there exists a change of coordinates which reduces Y\ to Y^ d/dy^ Now, writing 3>j = yJ, 2^j^r, we have F(17, ) = <3/3y 1 , a/3); 2 , . . . , d/3)> r >, thus finishing the proof. Acknowledgement This work is supported by DGICYT (Spain) through Grant No. PB 89-0004. Reference [1] Lundell A T, A short proof of the Frobenius theorem, Proc. Am. Math. Soc. 116 (1992) 1131-1133 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 1, February 1995, pp. 33-40. (D Printed in India. On infinitesimal /t-conformal motions of Finsler metric H G NAGARAJA, C S BAGEWADI and H IZUMI* Department of Mathematics, Kuvempu University, B.R. Project 577 115, India *Sirahata 4-10-23, Fujisawa 251, Japan MS received 14 May 1993; revised 9 September 1994 Abstract. The conformal theory of Finsler spaces was initiated by Knebelman in 1929 and lately Kikuchi [7] gave the conditions for a Finsler space to be conformal to a Minkowski space. However under the fc-condition, the third author [4] obtained the conditions for a Finsler space to be /i-conformal to a Minkowski space. The purpose of the paper is to investigate the infinitesimal fc-conformal motions of Finsler metric and its application to an H-recurrent Finsler space. We obtain the following results. A. Theorem 2. 1 . If an HR-F B space is a Landsberg space, then the tensor F l hjk is recurrent. B. Proposition 3.3. An infinitesimal Ji-conformal motion satisfies L x G l jk - PJ C. Proposition 3.6. An infinitesimal /i-conformal motion satisfies L x P l jk = pC l jk . D. Theorem 3.7. In order that an infinitesimal /i-conformal motion preserves Landsberg spaces, it is necessary and sufficient that the transformation be an infinitesimal homothetic motion. E. Theorem 3.8. An infinitesimal /i-conformal motion preserves *P-Finsler spaces. F. Theorem 3.10. An infinitesimal /2-conforma3 motion preserves /i-conformally flat Finsler spaces. G. Theorem 4.1. An infinitesimal homothetic motion preserves H-recurrent Finsler spaces. H. Theorem 4.2. If an H-recurrent Finsler space admits an infinitesimal homothetic motion, then Lie derivatives of the tensor F! ,. and all its successive co variant derivatives by x l or n J K y l vanish. Keywords. Infinitesimal h-conformal motion; h-conformal tensor; infinitesimal homothetic motion. 1. Preliminaries 1.1 Berwald connection Let F n be an n-dimensional Finsler space with the Finsler metric F(x,y). The metric and angular metric tensors are given by y := didjF 2 /2 and hif^g^ lil^ where 33 34 H G Nagaraja et al We use the following: + 8jg kh - d h g jk ) 9 djG\ G<:=iy' fc //. (1.1 Two types of covariant derivatives for a vector X' are given by (a) X' k := d k X l + G l hk X\ d k := d k - G"d m , (b) X l ]k :=d k X\ d k :=- Jt' and the Cartan tensor is defined by C l hk := ^g im d k g mh . This connection is known as th Berwald connection, which is not metrial, that is, ', = 2PW. (cf. [2]) (1.2 When a Finsler space satisfies the condition P l . k = 0, the space is called a Landsben space. The curvature tensor H^. fc is defined by H^-d^ + G-G^-M, (1.3 where j | /c means the interchange of indices/ and k in the foregoing terms. We see LJI . _ Lri Tji . _ pi JJT . _ fji ~ ~- n hj . n hji , where the index means the transvection by y. The Ricci identities are denoted by / Q \ Hpi nri rrt T" m TH* 7 m JL/m npi {a.j A h' i'k h'k'i mik h m hik ifc h\m y In the theory of conformal transformation, the h-conformal tensor F^ - k is define^ by ([4], (4.15)) 1 __ hjH mk g im ~j\ k) + H(g hj 5[ - g hk fy .2 Lie derivative We consider an infinitesimal extended point transformation in a Finsler spac generated by the vector X = u*(x)3 f , i.e. J x'^x' + u'dt, ^^y + ^/Vdt. (Li The well-known commutation formulae ([1], [5], [6], [8-10], etc.) involving L: Infinitesimal h-conformal motions 35 and covariant derivatives are given by (a) L x T, k - ' HJk where (b) L x H' HJk = Ai, j:k + AfG l hkm -J\k, (L7) In the usual way we raise or lower indices by means of the metric tensors g ij or g^. 2. An HR-F n space A Finsler space F n is said to be an H-recurrent Finsler space (denoted by an HR-F W space), if the Berwald curvature tensor H l hjk satisfies the relation #<*., = * H U ffU*. (2-1) where K m is a nonzero vector. As y l k = 0, we have H l j:k = K k Hj, H ij:k = K k Hij and from (1.5) we obtain *!*. = ^ U + -(P-jAg* - 9 hJ H tk P -j\k) -Af'PaJ-jl*}. (2.2) Thus we have Theorem 2.1. If an HR-F n space is a Landsberg space, then the tensor F l h . fc is recurrent. 3. An infinitesimal A-conformal motion 3.1 An i.c.m. The condition for an infinitesimal transformation (1.6) to be an infinitesimal conformal motion (denoted by an i.c.m.) is that there exists a function <f) of x such that 1|| L x g jk -2</)(x)g jk , L x g jk = - 2ct>(x)g jk . (cf. [1], [5], etc.) (3.1) $] If the function (j> is a constant, the i.c.m. (3.1) is called an infinitesimal homothetic motion (denoted by an i.h.m.) and when = 0, the (3.1) called an infinitesimal isometric motion (denoted by an i.i.m.). It is well known that an i.c.m. (3. 1) satisfies L x C jk = and L x y l = 0. We can easily see 1 1 (c) L x f = - 4>l\ L x lj = </,, (d) L x h] = 0, L x (g ih g jk ) = 0. (3.2) 36 H G Nagaraja et at Since the Lie derivative is commutative with d k or d k , we see from (1.1) (a) and (3.1) Transvecting the above equation by /'/ we have L x G L = B ih <t> h , F A :=//~iFV". ( 3 - 3 ) Differentiating (3.3) by y j and /, we get (a) L X G] = Bf < Bf := djB ih = d}y h + <5*/ - Using (3.2) we have PROPOSITION 3.1. An infinitesimal conformal motion satisfies the following: Loth __ f\ . T "Dih - f\ j T D^ ...u-r f\ y_J sss \J 4~r JL/ y Jj " \ J 4r-r LJ y fj .. ==:: \J. 3.2 An i.a.m. If an infinitesimal transformation (1.6) satisfies L x G\ k = 0, then the transformation is called an infinitesimal affine motion (denoted by an La.m.). First, we shall show Theorem 3.2 ([1], (VII), Theorem 5.1). In order for an infinitesimal transformation be homothetic, it is necessary and sufficient that the transformation be conformal and affine motion at the same time. Proof. We see from (3.4) = y'y^L^G^ = B l Q (j) h = 2B ih (t> h = F 2 (2fl h g ih )(j> h . Transvecting the above equation with 2/ f / fc g ik , we have F 2 (j) k = 0. Q.E.D. Remark., This theorem was first proved by Takano (Japanese, 1952). 3.3 An i.h-c.m. If we impose the Ji-condition on the vector <,-, i.e. FC h u (j) h = #!&, 4> 1 := ^ . (cf. [4], 3) (3.5) n~ 1 the transformation is called an infinitesimal h-conformal motion (denoted by an i./2-c.m.). Because the function $>i(x) is proved to be a function of x only (see [4], Lemma 3.2), we get Infinitesimal h-conformal motions 37 (a)F5 t /r}=-V-*& (b) F 2 (d k C?)<t> h = F = FdMtf) - Wfr = - WJ + h(lj + Ajy. (3.6) Using the above calculations, we obtain A',, = L x G} k = Pj 6' k + Pk S' } - g jk p l - fayj, A} = L X G] = pSfj + 4> 3 y ~ yjP\ (3.7) FC h l ' I Hence we have PROPOSITION 3.3. An infinitesimal h-conformal motion satisfies L x G' jk = Pj Sf k + p.d'j - p'g jk - The vector PJ is called an associated vector with a vector 4>j and satisfies the conditions: (a) FC h jk p h = (t>ih jk , (/i-condition) (b) pj\ k -d kPj -C h jk p, = Q. (Cartan's covariant derivative by /) (3.8) A vector which satisfies (3.8) (a) (b) is called an /i-vector. PROPOSITION 3.4 ([4], Proposition 3.4). Let VI(K, y) be a vector in a Finsler space. If v t satisfies the conditions v t \ k = and FC* k v h = v 1 hj k , then the function v l and the vector *v l :=v i v 1 l i are independent of y. Here we shall show Lemma 3.5. We have Proof. We see 3 _ih is^ih a /^"i p f^ih/^ \ l/ 01 ^/^ L ^i^^ /^ ^ = - 2 ^fc . ^mS'* d -"' 5 ' C *W = ~" 2C m C hjk + 9 O m L hjk , and using d m C hjk = d k C hjm , (3.6) (a) and '(3.8) (b), we get )-C hJ J k (Fp m )} From (1.7)*(a) we see L x C} kl - (L x C l jk ), = A^C"- AC> mk - Ayl - A?d m C l jk . 38 H G Nagaraja et al In consideration of L x C Jk = and transvecting the above equation by y l , we have L^ = LA, = A'.Ci - AJ-CU - AlC} m - W*, Substituting (3.7) into the above equation, we have LxP 1 * = <Mty* + Wj + hff} + *oCj + F 2 p m S m C ]k . Using Lemma 3.5, we obtain ^=Wo+wq*=/>q*- (3 - 9) Thus we have PROPOSITION 3.6 An infinitesimal h-conformal motion satisfies Remark. If we denote the deformed tensor (cf. [1]) of P l jk with respect to an i./i-c.m, (1.6) by Pjfc, we see This means that the deformed space of a Landsberg space (P l jk = 0) is not necessarily a Landsberg space. However we can state the following. Theorem 3.7. In order that an infinitesimal h-conformal motion preserves Landsberg spaces, it is necessary and sufficient that the transformation be an infinitesimal homothetic motion. Proof. It is sufficient to show fa = 0. In fact, we have It is evident that the theorem holds. Q.E.D. 3.4 *P-Finsler space If the tensor *^:=^j fc -Aq k vanishes, the space is called a *P-Finsler space (cf. [3]). The *P-condition: P} k = kC l jk is invariant under any fc-conformal change of Finsler metric. From (3.9) we have L x Pj = pC s , P ; :=Pj,. (3.10) Using (3.10) we see Q = L x (Pj-lCj) = (p-L x X)Cj 9 M = p. (3.11) This means L x *P( k = 0. Hence we have Theorem 3.8. An infinitesimal h-conformal motion preserves *P-Finsler spaces. Infinitesimal h-conformal motions 39 3.5 An h-conformally flat Finsler space If a Finsler space is ft-conformal to a Minkowski space, the space is called an h-conformally flat Finsler space. An /i-conformally flat Finsler space is proved to be one of *P-Finsler space (cf. [4], (5.2)). Here we shall show Lemma 3.9. In a *P-Finsler space an infinitesimal h-conformal motion satisfies **-& (3J2) Proof. Differentiating (3.11) w.r.t. y h we have L x k h = p h . Next from (3.2) we see n Hence we have L x */ ; = ( d] -- ^- )p = <,, Q.E.D. \ " V On the other hand, we know the theorem ([4], Theorem 6.6): The necessary and sufficient conditions for a Finsler space to be /i-conformally flat are that dfl l j k = and nj, w = and ^ is an h- vector, where fl **hkr = ^l*Ijik ~^~ *^hk^*ml ' ' ^k ==: ^k Hok^m- (j.lj) The parameter H( k and the tensor nj^ are invariant under an /z-conformal transformation and these are independent of j;. We shall show Theorem 3.10. An infinitesimal h-conformal motion preserves h-conformally flat Finsler spaces. Proof. It is sufficient to prove L x Tl i jk = Q. We see LjBjJ = from Proposition 3.1. Moreover, we have from (3.4) (b) and (3.12) Lx^jk = L x(G l jk ~~ Bfk *^ft) = B'jk^H ~ B^h = 0. It is easy to prove n[ kl = 0. Q.E.D. 4. An infinitesimal homothetic motion in HR-F n spaces In this section we shall consider an i.h.m. only, that is, L x g Lj = 2cg L p L x g ij = 2cg ij \ c = constant. (4. 1 ) From Theorem 3.2 and (1.7) (b), we have L x H l hjk = 0. From (1.7) (a) and (2.1) we see r L/t r / IX" r/i \ / r jy- \ rji r\ L X^hjk.m - L x(^m^hjk) - ( L X&m)H hjk - U, 40 H G Nagaraja et al which means L x K m = and L X (H^ M - K m H l hjk } = 0. Thus we have Theorem 4.1. An infinitesimal homothetic motion preserves H-recurrent Finsler spaces and satisfies L x K m = 0. An i.h.m. (4.1) satisfies L x lj = clj, L x f=-cf, L x h] = Q. (4.2) From Proposition 3.6 we see L x H=-2cH. (4.3) . Moreover we see from (2.2), (4.2) and (4.3) After some calculations we obtain L x F l hjk = 0. *U + *- L * F U = 0- Hence we have Theorem 4.2. If an H-recurrent Finsler space admits an infinitesimal homothetic motion, then Lie derivatives of the tensor F l hjk and all its successive covariant derivatives w.r.t. x l or y l vanish. References [1] Yano K, The theory of lie derivatives and its applications (1957) (Amsterdam: North-Holland) [2] Rund H, The differential geometry of Finsler spaces (1959) (Berlin: Springer Verlag) [3] Izumi H, On *P-Finsler spaces, I, II, Mem. Defence Acad. Japan 16 (1976) 133-138; 17 (1977) 1-9 [4] Izumi H, Conformal transformations of Finsler spaces. II. An /i-conformally flat Finsler space, Tensor N.S. 34 (1980) 337-359 [5] Izumi H, On Lie derivatives in Finsler geometry, Symp. on Finsler Geom., at Naruto, 1980. [6] Sinha R S, On projective motions in a Finsler space with recurrent curvature, Tensor N. S. 21 (1970) 124-126 [7] Kikuchi S, On the condition that a Finsier space be conformally flat, Tensor N. S. 55 (1994) 97-100 [8] Pande H D and Kumar A, Special conformal motion in a special projective symmetric Finsler space, Lincei - Rend. ScLfis. mat. e nat. 58 - Maggio (1975) 713-717 [9] Kumar A, On projective motion in recurrent Finsier spaces, Math. Phi. Sci. 12 (1978) 497-505 [10] Sinha R S and Chowdhury V S P, Projective motion in recurrent Finsler spaces, Bull. Cal Math. Soc. 75 (1983) 289-294 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 1, February 1995, pp. 41-51. () Printed in India. A bibasic hypergeometric transformation associated with combinatorial identities of the Rogers-Ramanujan type U B SINGH Department of Mathematics and Astronomy, Lucknow University, Lucknow 226 007, India MS received 29 December 1993; revised 16 March 1994 Abstract. During the last five decades, a number of combinatorial generalizations and interpretations have occurred for the identities of the Rogers-Ramanujan type. The object of this paper is to give a most general known analytic auxiliary functional generalization which can be used to give combinatorial interpretations of generalized ^-identities of the Rogers-Ramanujan type. The derivation realise the theory of basic hypergeometric series with two unconnected bases. Keywords. Auxiliary functions; unibasic hypergeometric series; bibasic hypergeometric series; < g-hypergeometric identities. 1. Introduction The two celebrated Rogers-Ramanujan identities oo n n 2 -f an oo IT r =n(i-T l , ll<i. (i) "=0 (qiq) n =i n = (a + l)(mod 5) where a = or 1, were first given by Rogers [12] in 1894 and then rediscovered (without proof) by Ramanujan in 1911. In 1916, MacMahon ([11]; 7, Chap. Ill) gave the following combinatorial inter- pretation of these two identities: "The number of partitions of n into parts that differ by at least 2 with each part > a is equal to the number of partitions of n into parts = (a + l)(mod 5), where a may be either or 1". In 1917, while scanning some old volumes of the Proceedings of the London Mathematical Society, Ramanujan came across the remarkable papers of Rogers [12-14] which not only contained analytical proofs of these identities but also contained other similar identities for the moduli 7, 10, 14, 15, 20 and 21. In 1919, in a joint paper, Rogers and Ramanujan [15] gave several proofs of these identities which are based on the general transformation formula: n = o (l-a)(q;q) n n = o(q;q) n proved by them. 41 42 U B Singh Later, in 1929, Watson [22] gave an elegant and straightforward proof of these identities with the help of the following transformation formula connecting a terminating well-poised 8 O 7 and a terminating Saalschiitzian 4 <S> 3 series: , f^M^^ggl ) n [_aq/c,aq/d,efq n /a J (aq/e,aq/f\q) In 1936, with the help of certain difference-equations, Selberg [17] obtained, besides a number of other identities, the Rogers-Rarnanujan identities (1) by means of his auxiliary function in which where fe is real and > - 1- In 1947, Bailey [6,7] outlined a technique of obtaining a large variety of trans- formations of basic hypergeometric series from which he deduced known as well as new identities of the Rogers-Ramanujan type on different moduli by specializing the parameters suitably. Shortly afterwards, Slater [19,20] made a systematic use of Bailey's technique to give a list of 130 identities of the Rogers-Ramanujan type involving prime factors 2, 3, 5 and 7 in the moduli. A generalization of Rogers-Ramanujan type of identities in a different direction was given by Alder [3] in 1954. He used Selberg's auxiliary function (4) to prove the following generalizations of the Rogers-Ramanujan identities (1): (6) , fc(mod2/c-f 1) M0,l (mod2fc+l) where G k tii (q\ k^2, are certain polynomials which reduce to q" 2 for k = 2, the Rogers- Ramanujan case. Singh [18] extended these results of Alder by giving r-generalizations of the above two identities with the help of a transformation theorem for basic hypergeometric series given by Sears ([16]; 4). In 1974, Andrews [4] obtained another analytic generalization of the Rogers- Ramanujan identities (1) with the help of Selberg's auxiliary function (4) by using the ^-difference equations A bibasic hyper geometric transformation 43 iteratively. Later, he [5] considered the auxiliary function ''-i)/2 n >o (I- a)(q 9 aq/b, aq/c; q) n and showed that it is equal to (aq/b, aq/c; q) n Ml ..... mfc _ , > o (q; q) mi . . . (g; #) mk _ t \ be X < 2< /c ~ 1 ) mi + ... + 2mk-2 + Wfc-i~mJ-f (raj +m 2 } 2 + ... +(m l + ... + m k _ 2 ) 2 ("9") In. 1980, Bressoud [8] also obtained an analytic generalization of the Rogers- Ramanujan identities (1) by considering the following auxiliary function for < r ^ k with Mi = m + m i+ 1 + . . . + rn k _ i : -!- z M (aa/bca) ^M^.-. v *!/ ' *^m>c~i . c . (ID and also gave a combinatorial interpretation of these identities in the following form: Let <5, r, k be integers satisfying d = or 1, < r < (2fc -I- <5)/2. Let B^ r ^(n) denote the number of partitions of n such that, if f i denotes the number of times i appears as a part in the partition, then / ^ r 1, f i + f i+ i ^ k 1 for all z and / s 4- / i+ x = k 1 implies that if t + (i + l)/ i+ x = r 1 (mod 2 5). Also let A krd (ri) denote the number of partitions of n in which no part is = 0, r(mod 2k + <5). Then, for each positive integer n, ^>) = **,,>) (12) In another paper, Bressoud [9] gave a further analytic generalization of the Rogers-Ramanujan identities by using the following auxiliary function: H ^ t i *4> v ^r^); *)'<?)= 7; I c - A 4- 1 (13) He also proved that ?; q) mi ...(q\ q) mtc . 44 U B Singh >/b 2k+ , _,; ) }, (14) and gave a very general combinatorial interpretation of the identities obtained by him. A close examination of the auxiliary functions (from Rogers-Ramanujan to Bressoud) stated above raises some very natural questions of the type: (i) Is it necessary to take a and a r , simultaneously, in the auxiliary function as has been done by Bressoud? (ii) Is it necessary to take two related bases q and q r instead of two general unconnected bases q and q^l Since the general transformation theory for basic hypergeometric series with two unconnected bases has already been developed in 1967 by Agarwal and Verma [1,2], the object of the present paper is to establish a general bibasic transformation formula similar to (14) with a parameter 1 in place of </ and then discuss a few interesting particular and limiting cases of this transformation. 2. Notation For |g|<l, let A generalized multibasic hypergeometric series, whenever it converges, is defined as ..o (q,b;q) n t=i(d,;q,) n where R = r + r 1 + ... + /- mJ S = s + s 1 + ... + s m) a = (a l! ...,a r ),fc = (6 1> ...,b s ),c ( c .i."- c ,,,M = (4.i.---.O The superscript (m + 1) in the <l>-symbol denotes the number of bases in the series. 3. We shall first prove the following general bibasic transformation formula: A bibasic hyper geometric transformation 45 Theorem. For \q\ < I 9 \q^\ < 1, [a,b 9 c 9 a l9 ... 9 a 2k _ l : 4ixA 2m + 3 /i / / , /7 L aq/b 9 aq/c, aq/a t , . . . , ag/a 2fc _ x : ^/A, - ;q) M ^ 2 ..4aq/a 2 ^^ l9 ... 9 b 2m ;q l )M l (- -J + (5/2)M, + M 2 + ... + M k _ x (m- l ;<?,4i; i (15) M ( - = m f 4- m. + 1 4- . . . + m k _ 1 , m, r and fc are positive integers. Proof. By the ^-analogue of Saalshiitz's theorem ([10]; eqn. (1.7.2)), we have (b 9 c,q) H wi i \ v" 1 \ "i * " j ^f ' "^ s n^ (a^f/ft, aq/c; q) n \bcj s = o (4, 04/6, a<?/c; <?) 5 Using (16) in the left hand side of (15), one can easily write it in the form (q 9 aq/b 9 aq/c 9 q) mh _ i n n(k-m k - 1) n n(m- 1 l-'-^fc-l^l^ Putting n = m k _ 1 -h t, the last expression is equal to l , . . . , a 2 ,_ , ; q) mu _ i (a; q) 2mk _ i , aq/c 9 aq/a l , . . . , i ! 46 V B Singh f jm-t-t/r-k-mic-i _m-l ) _ * g *1 We now iterate the procedure used in transforming the left hand side of (15) to the form (17) (such that the parameters b, c are "shifted out" from the <l> series). Then, after (k - 2) iterations, we find that the left hand side of (15) can be written in the following form: ^ i^;^-^,^;^.^^^ -v^k-ytto-i''^-^...**^^^ ^ xj - l + - +m /a 2k _ 3 ,fl^ 1 ? Wk - l + - +mi /a 2Jk . 2 A, 7 fl m fc -i + ...4-mi . -(l+mk-1+...+mi) 77 ^l+wik- 1 +...' + mi /J" ^k-l^ ' ^1 V A '~"^1 V^ -i-...-m 1 -m Introducing the summatory symbols M f , it is easy to see that the last expression is equivalent to the right hand side of (15). A bibasic hypergeometric transformation 4. Particular cases 47 We shall now discuss a few interesting particular and limiting cases of the above transformation (15). Case I. Let us take /c = 2, w = 0, r=l, A = a and q 1 = q in (15) and make b,c,a l9 a 29 a 3 -+ oo. Then, with the help of a well-poised 6 <X> 5 summation formula ([1 ; 0]; eq. (2.7.1)) and some simplification, we get the transformation (2) proved earlier by Rogers and Ramanujan [15]. Case II. We now consider the following auxiliary function which is a multiple of the left hand side of (15): If we first take w = 0, A = fl r , q 1 =q r in (18) and then make b,c,a 1 ,...,a 2/c _ 1 ->oo, we get which is equivalent to Selberg's auxiliary funption (5). Case III. If we take m 0, A = a, q 1 = q, r = 1 and a 2A _ x = q~ N in (15), then the inner series on the right hand side of it can be summed up by a well-poised 6 O 5 summation formula ([10]; eq. (2.7.1)). We easily get the following identity: /-/-// / V a, - ^/a, aq/b, aq/c 9 aq/a^ , . . . , aq/a 2k ^.^ bca 1 ...a 2k _ 1 a 2k~s a 2 k -4'> 9) mi ^ti^-^a^^.^^^'--^^ aq/aM)^ (20) which is seen to be equivalent to the identity ([15]; Theorem 4) due to Andrews. 48 U B Singh Case IV. Again, let us take /I = </, q 1 = q r (0 < r ^ k) in (15) and make The inner series on the right hand side of (15) is then summable by a well-poised 6 $5 summation formula ([10]; eq. (2.7.1)) and we get the following transformation: ! * a 2fc _ 4 ; ) m . (21) k , 52fc _ 4 The transformation (21) is easily seen to be equivalent to Bressoud's transformation (14). If, in addition to these changes, we make in (18), we get the auxiliary function (10) due to Bressoud. However, if we make all these changes in (15), then, on making use of a well-poised 6 O 5 summation formula ([10]; eq. (2.7.1)), we get the transformation (11) due to Bressoud. Case V(a). Let us take k = p - 2, m = 2, r = 1, A = a, q l = q, a 2p-7 = 6 ' a 2p-6 = X ' a 2p-5 = ~" X in (15), and transform the inner series on the right hand side of the resulting trans- formation by another transformation ([21]; eq. (1.3)). Then, we easily get a general transformation which is seen to be equivalent to the result ([21]; eq. (4.1)) due to Verma and Jain. In (15), let us first replace q by q 2 and then take k = p - 2, A = a, q^ = <j 2 , m = 2, r = 1, We can now transform the resulting inner series in (15) by the transformation formula A bibasic hyper geometric transformation 49 ([21]; eq. (1.4)). We thus get a general transformation which is easily seen to be equivalent to the result ([21]; eq. (4.3)) due to Verma and Jain. Case V(b). Let us now take k = p - 3, m = 3, r = 1, J, = a, q l = q, in (15), and then transform the inner series on the right hand side of the resulting transformation by another transformation ([21]; eq. (1.5)), we thus get a general transformation equivalent to the result ([21]; eq. (4.4)) due to Verma and Jain. Next, we first replace q by q 3 in (15) and set k = p - 3, m = 3, r = 1, A = a, q l = q 3 , i = y>b 2 = yq, b 3 = yq 2 , Then, by using the transformation formula ([21]; eq. (1.6)), we get a general trans- formation which is also seen to be equivalent to the result ([21]; eq. (4.5)) due to Verma and Jain. Case VI. In (15), let us replace q and q 1 by q 2 and q 2N+1 , respectively, where AT is a positive integer and make 5,c,a 1 ,...,a 2;c _ 1 ->cx),c 1 ,...,c 2m ->oo. (22) Then, on setting a = q 2 , we easily get the following interesting identity which is believed to be new: ... ,>o f nm + k/rlAfi+n > x (3 + 2Nm + m)n 2 + 2{2M 1 +(2N + l)M l m-N-k+l}n * L * -l m+klr q*q- 2N ' = lim 3 <|) 2 a, 6-* oo 0,0 /I Km 3 <l> 32 ^0,0 (23) where ^* = 50 U B Singh Again, if we replace q and q^ by q 3 and q*, respectively, in (15), take the limits indicated in (22) and then set a = <? 3 ; we obtain the following interesting identity which is also believed to be new: = lim 3 <X> 2 .0,0 -A lim 3 <D 32 L ab aft -0,0 (24) where ^ 5. Conclusion We have not tried to list all the special cases of our general result but have only drawn attention to the fact that multidimensional transformations of bibasic hyper- geometric series perhaps provide the best way of unifying the enormous number of partition - theoretic analytical identities. We hope to exploit this viewpoint in a future communication. Acknowledgement This work is supported by a grant (No. PDF/93/2084) from the National Board for higher Mathematics. References [1] Agarwal R P and Verma A, Generalized basic hypergeometric series with unconnected bases, Proc. Cambridge Philos. Soc. 63 (1967) 727-734 [2] Agarwal R P and Verma A, Generalized basic hypergeometric series with unconnected bases (II), Q. J. Math. 18 (1967) 181-192; Corrigenda, ibid. 21 (1970) 384 [3] Alder H L, Generalizations of the Rogers-Ramanujan identities, Pacific J. Math. 4 (1954) 161-168 [4] Andrews G E, An analytic generalization of the Rogers-Ramanujan identities for odd moduli, Proc. Nat. Acad. Sci. (USA) 17 (1974) 4082-4085 [5] Andrews G E, Problems and prospects for basic hypergeometric functions. Theory and applications of special functions (R Askey, ed.), (New York: Academic Press) (1975) 191-224 [6] Bailey W N, Some identities in combinatory analysis, Proc. London Math. Soc. 49 (1947) 421-435 [7] Bailey W N, Identities of the Rogers-Ramanujan type, Proc. London Math. Soc. 50 (1949) 1-10 [8] Bressoud D M, An analytic generalization of the Rogers-Ramanujan identities with interpretation, Q. J. Math. 31 (1980) 385-399 A bibasic hyper geometric transformation 51 [9] Bressoud D M, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc., No. 227, 24 (1980) 1-54 [10] Gasper G and Rahman M, Basic hypergeometric series, (Cambridge: University Press) (1990) [11] MacMahon P A, Combinatory Analysis, Vol. 2, (New York: Cambridge University Press) 1916 (Reprinted: Chelsea, New York 1960) [12] Rogers L J, On the expansion of some infinite products, Proc. London Math. Soc. 24 (1893) 337-352 [13] Rogers L J, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894) 318-343 [14] Rogers L J, Third memoir on the expansion of certain infinite products, Proc. London Math. Soc. 26 (1895) 15-32 [15] Rogers L J and Ramanujan S, Proofs of certain identities in combinatory analysis, Proc. Cambridge. Philos. Soc. 19(1919)211-216 [16] Sears D B, Transformations of basic hypergeometric functions of special type, Proc. London Math. Soc. 52 (1951) 467-483 [17] Selberg A, Ober einige arithmetische identitaten, Avh. norske vidensk, Akad. 8 (1936) 1-23 [18] Singh V N, Certain generalised hypergeometric identities of the Rogers-Ramanujan type, Pacific J. Math. 1 (1957),1011-1014; 1691-1699 [19] Slater L J, A new proof of Rogers' transformations of infinite series, Proc. London Math. Soc. 53 (1951) 460-475 [20] Slater L J, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. 54 (1952) 147-167 [21] Verma A and Jain V K, Transformation between basic hypergeometric series of different bases and identities of Rogers-Ramanujan type, J. Math. Anal. Appl. 76 (1980), 230-269 [22] Watson G N, A new proof of the Rogers-Ramanujan identities, J. London Math. Soc. 4 (1929) 4-9 11 /? Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 1, February 1995, pp. 53-58. Printed in India. Some theorems on the general summability methods W T SULAIMAN Department of Mathematics, Mosul University, Iraq MS received 20 January 1993 Abstract. In this paper a new theorem which covers many methods of summability is proved. Several results are also deduced. Keywords. Summability methods. 1. Introduction Let Da n be an infinite series with partial sums s n . Let cr d n and q 6 n denote the nth Cesaro mean of order <5(<5 > 1) of the sequences {s n } and {na n }, respectively. The series La n is said to be summable (C,<5) with index /c, or simply summable |C,<5| fc , k^ 1, if or equivalently z-xi k <- M=l Let {p n } be a sequence of real or complex constants with The series Ea w is said to be summable \N,p n \, if 00 where t n = P~ 1 p n - v s v (t_ 1 =0). We write p = {p n } and f PnH M=<p:p n >0 and Pn Pn 53 54 W T Sulaiman It is known that for peM, (1) holds if and only if (Das [4]) IAJ I I n I Pn-v < 00. DEFINITION 1 (Sulaiman [5]) For peM, we say that I,a n is summable \N 9 p n \ k9 k 3* 1, if 1 < 00. In the special case in which p n = A r n x , r > 1, where A r n is the coefficient of x n in the power series expansion of (1 x)~ r ~ l for |x| < 1, |N,pJ k summability reduces to |C,r| k summability. _ The series Ea w is said to be summable \N 9 p tt \ k9 k^ 1, if where (Bor[l]) v-0 If we take p n = 1, then |JV,p B | k summability is equivalent to |C,l| k summability. In general, these two summabilities are not comparable. We set . = MO and assume that P n , L/ n , R n and W n all tend to oo. DEFINITION 2 (Sulaiman [6]) Let {p rt }, {^f n } be sequences of positive real constants such that qeM. We say that is summable \N 9 p n ,q n \ k ,k^l,\f t?=l < 00. Clearly \N,p n9 l| k and |7V, l,<? B | k are equivalent to |AT,p rt | k and |N,^ n | k respectively. We prove the following: I Some theorems on the general summability methods 55 [ Theorem 1. Let [p n } 9 {q n } 9 {u n } and {v n } be sequences of positive real constants such that q,veM y q n =: 0(v n ), {pJP n R k n _ 1 v k n } nonincreasing and that a n ^ if v n ^ c. Suppose {e n } is a sequence of constants and write W n _ 1 G n L"^ 1 U r _ i v n _ r a r . If ' f =7+iP r /? r _ 1 u*_ r _i w '-i V t t -^ " " -i ;-,-i . \* 5 B l*<oo, (3) oo \fe~l (4) \ \V nn - k-l (5) then the series ~La n e n is summable \N,p a ,q a \ k , k^l. 2. Lemmas Lemma 1 (Sulaiman [6]). Let qsM, then forQ<v V ^L = o(r- 11 )- ,=rne r V ; Lemma 2. {pJP n ^- k n -^ k n } nonincreasing implies Froo/ Since therefore {pJPnR*-^ is nonincreasing. We have m nlA/7lf 2r m ZPn l^r^w-ri J V . V J i r * -- - - = < L + L ( = Ji+J2> say Z i =r+l = i 56 W T Sulaiman 3. Proof of theorem 1 Write R " r=l then, by Abel's transformation , i \b r \ |e,l In order to prove the theorem, by Minkowski's inequality, it is therefore sufi to show that V _ p " -f* <co, r = 1,2,3, 4,5, 6, ^J p ok "' r = l f n K -l where k > 1. Applying Holder's inequality, " n-1 lc-1 x I |A r D n _ r | r=l Some theorems on the general summability methods 57 = 0(1) E -1 fn- I l=i 1 n ,, ")fc-l p r q n - r -i n n / W \ k ^1*10,1* = 0(1). m+1 n m+1 V _ ^! _ p k y "^ m + 1 x y ?= *-l n m+1 n-1 Pn pk < y P^ y Dfc r n,5^ ^ D n Ls P r P \ k ~ 1 ( W \ k is completes the proof of the theorem. 58 W T Sulaiman 4. Applications Theorem 2. (Bor [1] and [2]). // nu n = 0(U n \ U n = O(nu n \ then the series E0 n is summable |C, l| fc if and only if it is summable |JV,i*J fc , k ^ 1. Proof. (=>) follows from theorem 1 by putting p n = 1, q n = 1, v n = 1, and e n = 1. (<=) follows from theorem 1 by putting q n = 1, u n = 1, i7 B = 1, and e n = 1. Theorem 3. (Bor and Thorpe [3]). Let {p n }, {u n } be sequences of positive real constants. If p n U n = 0(P n u n ) and_P n w n = O(p n U n \ then the series a n is summable |NPnlfc whenever it is summable \N^u n \ k9 k^l. Proof. Follows from theorem 1 by putting q n = 1, v n = 1 and e n = 1. Theorem 4. // the sequences {p n }, {q n }, {U B }, {v n }, satisfy the conditions of theorem 1 except (3H5) and if Pn U n = 0(P n u n ), P n u n = 0(p B l/J and P^_ A = 0(v n U n ^ , ), tten the series Sa B is summable \N,p n ,q n \ k whenever it is summable \N,u n ,v n \ k ,k^ 1. Proof. Follows from theorem 1 by putting e n = 1. COROLLARY 5 Let {q n }, {u n } be sequences of positive real constants such that geM, U n = 0(nu^) and nu n ~O(U n ). Then the series *La n is summable \N,q n \ k whenever it is summable \N 9 u n \ k , Proof. Follows from theorem 4, by putting p n = 1, v n = 1, and making use of lemma 1. COROLLARY 6 If the sequences {p n }, {<?}, {u n }, {v n } satisfy the conditions of theorem 1 except (3)-(5), and if p n U n = O(P n u n ) and P n u n = O(p n U n \ then sufficient conditions that Za n s n is summable \N 9 p n9 q n \ k whenever it is summable \N 9 u n ,v n \ k9 k^ 1 are Proof. Follows from theorem 1. References [1] Bor H, On two summability methods, Math. Proc. Cambridge Phil Soc. 97 (1985) 147-149 [2] Bor H, A note on two summability methods, Proc. Am. Math. Soc. 98 (1986) 81-84 [3] Bor H and Thorpe B, On some absolute summability methods, Analysis 7 (1987) 145-152 [4] Das G, Tauberian theorems for absolute Norlund summability, Proc. London Math. Soc. 19 (1969) 357-384 [5] Sulaiman W T, Notes on two summability methods, Pure Appl. Math. Sci. 31 (1990) 59-68 [6] Sulaiman W T, Relations on some summability methods, Proc. Am. Math. Soc. (to appear) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 1, February 1995, pp. 59-71. Printed in India. Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays F R K KUMAR and S K SEN Supercomputer Education and Research Centre, Indian Institute of Science, Bangalore 560012, India MS received 3 November 1993; revised 19 April 1994 Abstract. A symrnetrizer of a nonsymmetric matrix A is the symmetric matrix X that satisfies the equation XA = A'X, where t indicates the transpose. A symrnetrizer is useful in converting a nonsymmetric eigenvalue problem into a symmetric one which is relatively easy to solve and finds applications in stability problems in control theory and in the study of general matrices. Three designs based on VLSI parallel processor arrays are presented to compute a symmetrizer of a lower Hessenberg matrix. Their scope is discussed. The first one is the Leiserson systolic design while the remaining two, viz., the double pipe design and the fitted diagonal design are the derived versions of the first design with improved performance. Keywords. Complexity; equivalent symmetric matrix; Hessenberg matrix; symmetrizer; systolic array; VLSI processor array. 1. Introduction A symmetrizer [3, 7, 14, 16, 19, 20] of an n x n nonsymmetric matrix A is the solution X satisfying the equations XA = A*X and X = X 1 . A symmetrizer is used in trans- forming a nonsymmetric matrix into an equivalent symmetric matrix [14, 20] whose eigenvalues are the same as those of the nonsymmetric matrix and is useful in many engineering problems, specifically stability problems in control theory and in the study of general matrices [14]. Let '11 '12 '22 '23 (1) be a lower Hessenberg matrix with b iti+1 ^0 for i = l(l)n - 1, where i = l(l)n - 1 denotes i = l,2,...,n- 1. Also, let x f be the i-th row of the symmetrizer X for i = n (~ 1)1. Then, from XB = B'X, we write the serial algorithm [3] as follows STEP 1: Choose x n 7^0 arbitrarily. 60 F R K Kumar and S K Sen STEP 2: Compute x M-1 ,x n ^ 2 ,...,x 1 recursively from 1 j = M _l(-l)l As an illustration, consider 3 -4 -1 2 -4 2 -1 6 -2 5 3 -2 4_ Choose x 4 = [1 2 1]. x 3 ,x 2 , and then x l are computed following the foregoing algorithm. Hence the symmetrizer is " 1-8438 3-8750 2-0000 1-0000 3-8750 3-2500 1-5000 -2-0000 2-0000 1-5000 -5-0000 0-0000 _ 1-0000 -2-0000 0-0000 -1-0000_ It can be seen that the symmetrizer is not unique because if we choose x 4 = [1 1 1 1] then we get a different X. 2. Leiserson systolic design The single assignment algorithm [10] for computing a symmetrizer of the Hessenberg matrix B in Equation (1) is as follows. for f:= 1 to n do for 7": = to n do read BpJ]; for fc: = 1 to n do read X[n, k]; for i: = n 1 down to 1 do begin for j : = 1 to n do Y p,;, 1] : = 0.0; for j:= 1 to n do begin for k:= 1 to n do YpJ,k+ 1]:= Y[U,k] + Xp + U]*J3[/cJ]; for p:=i+ 1 to n do YpJ. + 1]:= ypj,n + p] - i + 1]; write end end. The implementation [6, 15, 17, 18] of this single assignment code a 4 x 4 matrix on the Leiserson systolic array depicted in figure 1 is straightforward by using the re- ; ming technique [10]. The allocation of the diagonals of the Hessenberg matrix to the rocessing cells (Type I) of the linear string of processors is as shown in figure 2. The nspecified output of PE 5 in figure 1 is ignored while its unspecified input is zero. Symmetrizing a Hessenberg matrix 61 TO SYSTEM BUS /HOST y\ = x ;B INTERFACE PROCESSOR AND MEMORY PEL Jl'EJ C M^t Here X- => elements of veciorXj| ; so do Yj and Xj_j a. b . 1 1 BASIC CELLS OPERATIONS x. . TYPE I x := x. o i TYPE II x := x . + a . b . o i 11 TYPE III x := x . / a . o 1/1 if x j= ' 'henx o :=- if a. = . then X Q := x. if x .= then * Q '^' and y o - y. if a.= and b.=. then x := x. y = y. + a. x. o 'i 11 Figure 1. Systolic array cells system for a 4 x 4 matrix symmetrization. Figure 2. Systolic array cell (Type I) allocation for the diagonals of a 4 x 4 Hessenberg matrix. Figure 3 displays how the pumping of the row vector x i+l and the matrix B into Type I cells is done for the matrix-vector multiplication while figure 4 demonstrates the array consisting of Types II and III cells to generate a symmetrizer row by row. The pumping will be done elementwise in Types II and III cells. The notations x "> k^ y |9 in figure 4, each of which has 2n 1 elements including tag bits are given as .. = [h.. o b tj o ... o 62 F R K Kumar and S K Sen X- u 42 u b 33 u b 4l U b 32 b 23 b 22 o O u 31 b 21 o b !2 o b u o o o u o o o 1 1 I 1 i PE1 ~ * PE2 ~ " PE5 . * PE3 _ * PE4 ^ (a) Just before the first time cycle o u o o b 43 u b 3 4 ^42 b 33 b 4. b 32 o b 23 U b,, U 22 U o 31 b 2l b !2 u o ^11 J 1 1 1 i * PE1 . ^ *JL PE2 ~* PE3 . PE4 PE5 o yi I (b) Just after the third time cycle 44 PE1 u X o PE5 PE2 PE3 PE4 y y (c) Just after the ninth time cycle PE1 u PE2 ^i. PE3 u o PE5 PE4 o o o (ci) Just after thirteenth time cycle Figure 3. 1-D systolic array for vector Hessenberg matrix multiplication. and = o _ x o ... o y\~\ This notation is used to conserve space. A lower (or upper) Hessenberg matrix of order n needs n + 1 cells of Type I. Denoting these cells PE 1 , PE 2 , . . . , PE n+l following the same notation (and connection) as in figure 1, the diagonal consisting of only one element b nl is positioned appropriately Symmetrizing a Hessenberg matrix 63 '22 x} * U 32 X" u u 00 U U -b 43 X* 8 pairs -b X 4 u u of tag bits ~ b 44 X 4 7 P airs 7 P airs 4 of tag bits of tag bits "34 K If u X . => elements of the vectorx so does b: : l * J Figure 4. Systolic array for generating a 4 x 4 symmetrizer row by bow. to be pumped into PE l , the next diagonal (just above the foregoing diagonal) consisting of the elements b n _ 1 1 , b n2 is allocated to PE2. The third diagonal with elements k/i-2 i'^n-i 2'^n3 is assigned to PE 3 and so on. Figure 2 illustrates the allocation of the diagonals of the 4 x 4 Hessenberg (symbolic) matrix B = [b. .] to different cells. The generalization to an n x n matrix is immediate. However, the diagonals could have been allocated in the reverse order, i.e., the diagonal having the elements b 12 ,b 23 , i> 34 , -">b n _ 1 n could have been allocated to PE^ , the principal diagonal to PE~ 2 , and so on. Both the allocations are functionally identical. We, however, use the former allocation. In a one-dimensional Kung-Leiserson systolic array [4], the elements of the vector x flow from left to right (figure 3) for row vector-Hessenberg matrix multiplication. This array consists of Type I cells, viz., inner product step (ips) cells. The matrix elements flow into the top and the solution elements appear from the left of the cells. Here half the cells are active at any one time. It is, however, possible to orient the data flow so that the cells are all active simultaneously [8, 11]. Note that the number of cells depends on the bandwidth (number of diagonals) of B and not on the size of B. The summation with a negative sign, viz., the result of Step 2 of the algorithm ( 1) is computed using Type II cells as shown in figure 4. The values y n = x n B,y n _ 1 =x n - 1 B, ...,yi. =\ 1 B to which the results is to be added are pumped into the cells from the left while the terms of the summation are pumped into them from the top. The single division by b. i+1 is then carried out in Type III cell, one of which only is needed to be used irrespective of the size of B. The elements of the row- vector of the symmetrizer X which are output rhythmically one after the other by this Type III cell are then fed back as indicated in figure 1. This row- vector is then used in the computation of the remaining row-vectors of X recursively. 64 F R K Kumar and S K Sen 3. Double pipe and fitted diagonal designs The Leiserson systolic model [6,9, 12] needs 2n -f 1 cells and 4n + 1 time cycles to obtain a row of the symmtrizer. Here we discuss two designs - one called the double pipe construction method, based on introducing a second pipe while the other, called the fitted diagonal method, on reducing the number of diagonals of the matrix B. While mapping the single assignment algorithm in 2, the double pipe design aims at minimizing the time complexity while the fitted diagonal design the number of cells. 3.1 Double pipe construction method This method [18] uses n + 1 cells comprising two pipes - the first one consisting of odd labelled cells P 1 ,P 3 ,..., while the second one the cells PE 2 ,PE 4 ,..., where n is the order of B; in addition, it uses one adder and one delay cell (figure 5). It computes a symmetrizer B in time cycles where f ] indicates the upper integral part. The double pipe concept increases cell efficiency and removes tag bits. It minimizes the hardware delay that exists before the start of actual computation. The data flow and the architecture of the n + 1 cells are illustrated in figures 6 and 8, respectively. Split the Hessenberg matrix B i.e., write B = Bl 4- B2. Bl contains only odd diagonals of B, where the first diagonal contains only the element b ni , the second the elements b n _ ll ,b n29 and so on while B2 the even diagonals. The remaining elements of Bl and B2 are zero. Since x B = y f , we have x (Bl + B2) = y . If we allow x.Bl = y B1 and x.B2 = y B2 then y B1 + y B2 = y.. Figure 6 depicts the flow and com- putation of y B1 and y B2 . The array needs no dummy elements, viz., the tag bits. Pipe 1 cells. Pipe 2 requires (n + 1) - ips cells. The time complexity * u - ^ fw+H 1 f 5n + 3 l to obtain a row is 2n + -f 1, i.e., . contains b 41 b 42 b 43 b 44 b 34 b 31 b 32 b 33 b 23 b 21 b 22 b 12 > ' , 1 ' 11 l jfri .J m\ PEl PE3 PE5 "* \ < " ' s~\ PE2 PE4 -H>FI \vi*e Figure 5. Double pipe array for vector Hessenberg matrix multiplication. y = y + b-i\, v = v I I 1 2t2 "2 '2 * V i y i + bit X 3 v = v + K x 1 ! 31 *2 *2 b 222 *3 b 32 v = v '3 '3 ^3 -3 b 43 4 y^x y 4 +b J4 3 V = V "4 "4 Figure 6. Data flow for double pipe method. J 42 43 32 h 22 b 21 b, D 34 b 23 b, 2 4 I 1 x . y i f * ^ 1 * d v ' 3 X A ^AA X 3 - V 4 Figure 7. Fitted diagonal method and data flow. 66 F R K Kumar and S K Sen . . .. 1 1 1J 1 Figure 8. Architecture of ips cells for Leiserson and double pipe methods. 3.2 Fitted diagonal method This method [15] consists in halving the number of diagonals, and hence the number of ips cells used is half of that required in the double pipe method. The number of i -I diagonals can be reduced to - by fitting two adjacent diagonals into one. Let d k+ 1 and d k be two adjacent diagonals, of JB, of length k + 1 and k, respectively A fitted diagonal A f is defined by interleaving the elements of d k+1 and d k , as where length (d,) = length(d fc+ x ) + length(dj = 2k + 1. Therefore, if the bandwidth n -f 1 of B is even then B is transformed to fitted diagonal v\ -4- 1 matrix B F with bandwith - . For a 5 x 5 Hessenberg matrix B = [6 r ], b 2l Sl 34 '54 55_ Symmetrizing a Hessenberg matrix 67 the bandwidth is odd then B F will have n- + 1 fitted diagonals where the diagonal is fitted with one additional diagonal of - zero elements. For a 4 x 4 enberg matrix B, B F = '43 gure 7 illustrates the fitted diagonal method for a 4 x 4 Hessenberg matrix. A ction in the number of PEs in this method necessitates some minor modifications e ips cells. Each of the input vectors x. +1 and that of the output vectors x. = y : be kept in each of the PEs for two time cycles as shown in figure 9. time complexity is the same as that for Leiserson systolic model but the number Es is + w, where the last n PEs do the same job as the last n PEs in :rson systolic model of figure 1. This number is about 75% of those required for ;onventional Leiserson systolic model. If (n -f 1) is odd the diagonal left is fitted Here all the gates are multi-input gates Figure 9. Architecture of ips cell for fitted diagonal method. 68 F R K Kumar and S K Sen Table 1. Time complexity for a row and number of PEs for the designs. Method Time complexity Number of PEs (for computing a row) 1. Leiserson Systolic Method w 1 =2n+l w,+2 2. Double Pipe Construction w 2 = 2n + 3 . + f^ ! 1 3. Fitted Diagonal Tn-hll w I _L, IT w 3 + 2n 3 1 2 r n -*>22 X^ 32 X* o o O 01) PE6 -b 43 JLJL jM 34 X. => elemen ts of the vector X ; so does b: Figure 10. Modified systolic array for generating a 4 x 4 symmetrizer row by row. with an additional diagonal of null elements. However this reduction in the number of processors needs some minor modification required for the processing elements. The first (n + 1) PEs owing to the elements of vectors x and y must be kept in each processor for two clock cycles. The time complexity is the same as that of Leiserson systolic model but the number of processors is n+n r which is roughly half of that for conventional Leiserson systolic model. We present, in table 1, a comparison of time complexity to compute a row of the symmetrizer and number of PEs for the proposed three designs. Symmetrizing a Hessenberg matrix 69 8L 70 E R K Kumar and S K Sen 4. Scheduling and total time complexity In the Leiserson systolic model, computation of a row of a symmetrizer requires 2/i + Wi time cycles (where w 1 =2n-f 1). Repeating this process for all the rows independently, the total number of time cycles required is (n l)(2n + w x ). Even though this total number of time cycles is O(n 2 \ it is still expensive. After w 1 -fn time cycles Type I cells (figure 1) are totally idle. A new pumping process is scheduled every w x + n time cycles. Therefore, the total number of time cycles to obtain the symmetrizer is T l =(n l)(n + w x ) 4- n which reduces the number of time cycles by n 2 - 2n. The same number of time cycles T is required in fitted diagonal method even though, in this case, number of PEs is reduced by , compared to that in Leiserson systolic model. Similarly, in the double pipe construction method, the symmetrizer is computed with the number of time cycles T 2 = (n 1)1 4- (n -f 3) If we use programmable systolic chip, then Types II and III cells are modified as in figure 10 and the cells architecture is as depicted in figure 11; Type II cells (except the last) have two output gates. The switch value is always assigned zero. The controller sets one for particular clock counter values, e.g., for Leiserson model of 4 x 4 matrix symmetrization, the processor PE6 controller sets the switch value one from 6th time cycle to 12th time cycle so that the data is pumped to the division cell directly. Type III cell gets input from any one of the gates. This modification reduces the tag bits in Type II and III cells. It also reduces the time complexity by (n ~2)(n 3)/2. 5. Conclusions The systolization procedures, i.e., all the three designs can also be easily extended to the general serial algorithm [14] to compute a symmetrizer of an arbitrary square matrix. The bandwidth will, however, be more. We hope that such a systolization will enormously reduce the complexity of computing an error-free symmetrizer [19, 20]. This error free symmetrizer will produce a more accurate equivalent symmetric matrix [14, 19] than what an approximate one does. It can be seen that when a real non-symmetric matrix has one or more pairs of complex eigenvalues then the equivalent symmetric matrix will be a complex one, Jacobi-like methods [1, 2, 5, 13] have been developed for computing eigenvalues, some of which are complex, of a complex symmetric matrix. These methods obviously make use of the "symmetry" property which results in a significant reduction in computation. Acknowledgement The authors thank the referee for his comments which have significantly helped in revising the paper. Symmetrizing a Hessenberg matrix 71 References [1] Anderson P and Loizou G, On the quadratic convergence of an algorithm which diagonalizes a complex symmetric matrix, J. Inst. Math. Its Appl 12 (1973) 261-271 [2] Anderson P and Loizou G, A Jacobi-type method for complex symmetric matrices (Handbook), Numer. Math. 25 (1976) 347-363 [3] B N Datta, An algorithm for computing a symmetrizer of" a Hessenberg matrix, (unpublished) [4] Dew P M, VLSI architectures for problems in numerical computation, (ed.) D J Paddon, Super- computers and Parallel Computation, New Series No. 1 (ed.), The Institute of Mathematics and Its Application Series, 1984 [5] Eberlein P J, On the diagonalization of complex symmetric matrices, J. Inst. Math. 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J. 12 (1969) 156-157 [14] Sen S K and Venkaiah V Ch, On computing an equivalent symmetric matrix for a nonsymmetric matrix, Int. J. Comput. Math. 24 (1988) 169-80 [15] Suros R and Montagne E, Optimizing systolic networks by fitted diagonals, Parallel Computing 4 (1987) 167-174 [16] Taussky O, The role of symmetric matrices in the study of general matrices, Linear Algebra Appl. 5 1(1972) 147-154 [17] Ullman D J, Computational Aspects of VLSI, (1984) (Standford Univ.: Computer Science Press) [18] Uwe S and Lother T, Linear systolic arrays for matrix computations, J. Parallel and Distributed Computing 1 (1989) 28-39 [19] Venkaiah V Ch and Sen S K, Computing a matrix symmetrizer exactly using modified multiple modulus residue arithmetic, J. Comput. Appl. Math. 21 (1988) 27-40 [20] Venkaiah V Ch and Sen S K, Error-free symmetrizers and equivalent symmetric matrices, Acta Applicande Mathematicae 21 (1990) 291-313 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 1, February 1995, pp. 73-98. Printed in India. Control of interconnected nonlinear delay differential equations in E N CHUKWU Mathematics Department, North Carolina State University, Raleigh, NC 27695-8205, USA MS received 1 September 1993; revised 26 September 1994 Abstract. Our main interest in this paper is the resolution of the problem of controllability of interconnected nonlinear delay systems in function space, from which hopefully the existence of an optimal control law can be deduced later. We insist that each subsystem be controlled by its own variables while taking into account the interacting effects. This is the recent basic insight of [13] on ordinary differential systems. Controllability is deduced for the composite system from the assumption of controllability of each free subsystem and a growth condition of the interconnecting structure. Conditions for a free system's controllability are given. One application is presented. The insight it provides for the growth of global economy has important policy implications. Keywords. Large-scale systems; delay equations; decentralized control; growth of capital stock; depression. We motivate the problem with a simple economic system derived by Kalecki [24] and reported in [1]. He argued that the dynamics of capital stock x(t) of a firm is given by x(t) = a x(t) + a^(t - fc) + bu(t\ (1.1) where a i9 i = 0, 1, are constants, bu(t) is a sum of two terms a constant multiple of autonomous consumption and a trend term. The crucial assumption for (1.1) is that the net capital formation x(t) is given by I(t\ the investment function. To obtain (1.1), Kalecki assumes that the investment decision B is given by where a, c, k are constants, is a windfall which may be time varying. The income (or output) is y, x(t) denotes the stock of capital, and e is the trend term. The delay h represents the time lag between the decision to invest and the deliveries of capital equipments. One can interpret (1.1) as a system whose growth can be controlled by autonomous consumption and windfalls. For example one can ask whether it is possible to grow from a 3% growth rate x(i) = 3t/100 = <(t) te[ h, 0], to 10% growth rate, x(t+ T) = Wt/lQQ = \j/(t\ te[-/i,0] in time T, by using u as a control. To motivate a nonlinear system of the form (1.1) which is interconnected by the so-called "solidarity" function inspired by [14], [20], [21] we argue as follows. Let Z denote aggregate demand consisting of consumption (C), investment (I), net exports (X) and government outlay (G). These differentiable functions are related as follows: (1.2) 73 74 EN Chukwu where (1.3) and y T is the current after-tax income, r=r +/ 1 (y). (1.4) T >0 is the level of non-income taxes, and f^y) is the income taxes. , e L >0. (1.5) the part of income that is spent on other countries' products, X is autonomous net exports, and R is the real rate of interest. Public expenditure is G = / 3 G>(r-/0) + i;(r), (1.6) where / 3 is public consumption which is dependent on the previous high income, and v(t) public investment. Investment is autonomous, i.e. it does not depend on income, but on "animal spirits" of entrepreneurs: I(r) = Io(0- (1-7) Thus Z(t) = (Z (t) + f 3 (y(t - h)) + v(t) + cy(t) - cT Q (t) ~ cf^y) - f 2 (y(t)} - e, R, (1.8) where Z (t) = I -f X -h C . From the model equations of money demand and supply [14] we deduce that "HHA- since where fc is a fraction of income, r > is measured in dollars. Here M is the nominal value of money supply which is controlled by the Central Bank, P is the price level. The real demand for money is denoted by L/P. The symbol j is autonomous real money demand. With R in (1.9) we deduce that / e k\ Z(t) = Z (t) + y(t)l c - j + f 2 (y(t-h)) - Following Allen [1] we postulate that -dy(t)/dt= - X(y(t)-Z(t)\ where /I is a constant. Thus .v(r)=-A(l + ^-, ; - /r)) + fl I fl _y I + ^(f) + ; iZo ( f ) _ l c T (t). (1.10) Delay differential equations in W ( ^ 75 Denote "solidarity functions," by ^ -cT (t) + ( f "o d "private initiative" by en the dynamics of income is ( e k \ y(t) = - A^ 1 + -|- - c\y(t) + A(c/i(y(t)) + f 2 (y(t)) + Jl/ 3 ' +PW + 0W- (Lll) [t is an interconnected nonlinear system whose controllability is investigated for values of p and q. The type of result which we shall prove in Theorem 2.2 when plied to the special system (1.11) will now be stated. Suppose (l.lla) 1 B(t)u(t) = p(t) + <?(0; then the dynamics of gross national product is e k 1 + _i_ __ c s possible for national income to be controlled by w, a combination of government 1 private controls. Using this, for example, we can steer a growth rate of 3% i.e., [t) = 3r/100, *[- /z,0], to a growth rate of 10%, \j/ 2 (t + T) = 10r/100, re[- /i,0], ;ime T provided B(t) * on [T- *, T], B(t) * 0, re[0, T- *] (Condition (ii), Theorem 2.2), < 00, the combined effect of the coefficient of solidarity and private initiative is "strong", I nontrivial. This is (iii), Theorem 2.2, the condition of the essential uriforni mdedness of the generalized inverse of B(t). Also there is a condition of how big q should be compared with p(t): Theorem 2.4 (ii), see Remark 2.4: "private initiative" should dominate "solidarity". t is proper to consider x i (t) = (x[(t) 9 ...,x i n (t)) to be the value of n capital stocks 76 EN Chukwu with strategy u l = (u\ ...wj,), Bw l '(0 = P;W 1 ^ w}(0 < 1, which is located in an isolated region S f . They are linked to / other such regional systems in the country and the "interconnection" or "solidarity functions", or government intervention givenby Here q t describes the action of the whole system on its fth interconnected subsystem (St\ x l (t) -A^x\t - h) = 4 jc'(t) + A^t ~-h) + Bu l (t). (S^ Thus 1 x i (t-h) + B i u(t) + q i . (1.12) Thus formulated we are interested in using the firms strategy u l and government interventions q t to control the growth of capital stock on which the wealth of a nation depends. Theorems 3.1 and 3.2 can be stated loosely as follows. If a regional economy is well behaved, carefully weighted government interventions q t can maintain the country's economic growth. Even if a regional economy is not controllable the intervention of solidarity function can render the system controllable. (See Remark 3.1 and Theorem 3.2.) Implications of controllability questions for the control of global economy are pursued elsewhere in [5], [8]. The issue of optimality is apparent in [8]. 1. Introduction For linear free systems, criteria for W ( 2 } controllability have been provided in [2]. For nonlinear cases a similar investigation was recently carried out in [8]. Recently Sinha [16] treated controllability in Euclidean space of large scale systems in which the base is linear. We extend the scope of the treatment in [16] by treating large scale systems with delay when the state space is W and the base system is not necessarily linear. We state criteria for controllability of the free subsystem by defining an L 2 control that does the steering both in the linear and nonlinear case. We prove that such a control exists as a solution of an integral equation in a Banach space. For this, we use Schauder's fixed point theorem. Assuming that the free subsystem is controllable and the interaction function has a certain growth condition we prove the controllability of the interconnected system. We begin with a simple system. A linear state equation of the z'th subsystem of an interconnected control system can be described by, x l (t) = A^x^t) + A 2i x l (t - h) + A 3i y l (t) + A 4i y l (t -h) + B t u l (t\ (1.13) where x l (t)eE nt is the n r dimensional Euclidean state vector of the fth subsystem, u t eE mi is the control vector and A u A 2i B t A 3i A 4i are time invariant matrices of appropriate dimensions. Also, y l (t) is the supplementary variable of the ith subsystem and is a function of its own Euclidean -state vector x l (t) and other subsystem state vector x j (t) ; = 1, ...,/. We express this as follows: i y l (t) = M,,x'() + Z M.jX^t) (1.14) where M u , M.. (j = l,2,...,l, j ^ i) are constant matrices. Delay differential equations in W ( ^ 77 By substituting (1.14) into (1.13) we obtain the state equation of the overall interconnected system, where H i = A U + A 3i M ii> G i= A 2l' i In (1.15), m f (t) and e ( (t /i) describe the interaction, the effects of other subsystems on the z'th subsystem. This can be measured locally. The decomposed system (1.1.5) can be viewed as an interconnection of / isolated subsystems x'(t - h) + B M (t), with interconnection structure characterized by which does not depend on the state variables x l (t). We now consider the more general free linear subsystem, and the decomposed large scale system, x i W = L.( ) x;) + B i .u i () + ^.(t,-/ l ), x = f,J =!,...,/, (/,) k where L.(t,xj)= f J-A We assume B.(t) is an n. x m. continuous matrix. The linear operator (/>->L.(t, 0) is described by the integral in the Lebesgue-Stieltjes sense, where (t,0)-f/ (,i?) is an n. x n. matrix function. It is assumed that ->^(, 0), te, is continuous for each fixed 0e[ /z,0] and O-^ri^O) is of bounded variation on [ /t,0] for each fixed te. Also, ff (f, 0) = 0, 9 ^ 0, fy (t, 6) = ^(r, h\6^h and -> iy E .(t, 0) is left, continuous on ( h, 0). It is assumed that where p.(t) is locally integrable. These conditions also hold for r\ ir Throughout the sequel E r is the r-dimensional Euclidean space with norm |-|. The symbol C denotes the space of continuous functions mapping the interval [ /i,0], 78 EN Chukwu h > 0, heE into E n with the sup norm || ||, defined by || || = sup |<(s) -h^s^O The controls are square integrable functions tieL 2 ([<Mi],JE w ), t^E, t^xy and L 2 is the space of measurable functions u defined on finite intervals [Mi] for which \u\ 2 is summable. If re[tr, tj, we let x t eC be defined by x,(s) = x(t -f s), /i < s < 0. With L 2 as the space of admissible controls, the state space is either E n on W ( \ the Sobolev space of absolutely continuous functions x: [ /i,0]-^" with the property that t-+x(t)<=L 2 ([-h,0],E n ). Thus if or, teE and fieW^d-h^lE*), u'eL^Mj, mi ) there is a unique absolutely continuous function x (% a, (f>\ u l ) = x f :[a h, tj] ~~> F" which satisfies (L.) or (/.) a-e on [cr,^] and the initial condition x^ = whenever the earlier conditions in rj., ^ and B i are satisfied. Also xj(', <r, , u^e^ 1 } ([- /i,0], n O forteCMj. DEFINITION 1.1 The system (L f ) is controllable (respectively Euclidean controllable) on the interval [Mj if for each </ f , ^eW^^C- /i,0],E Wf ) (respectively ^ 6^ ( 2 1) ([- /i,0], w O), xJeJE" 1 , there is a controller M l "eL 2 ([cr ) t 1 ], mt ) such that x^(% a, ^', w l ") = </> f and xl^cr^u 1 )^^ 1 (resp ^(f^tr,^,!^) = **!) If (L.) is controllable on every interval [a,tj, ^ >cr-f-/i, we say it is controllable. If (L.) is Euclidean controllable on every interval [or, tj], t i > o we say it is Euclidean controllable. For the free subsystem (L.) the following controllability theorem is available in [2, p. 616]. PROPOSITION 1.1 In (L.) let B*(t) denote the Moore-Penrose generalized inverse of B.(t), teE. Assume that f-B. + (t) is essentially bounded on [t^h^t^. Then (L.) is controllable on an interval [cr, tj] wit/i t i >a + h if and only if rank B.(t) = n. on [t x fc, 1 1 ]. An easy adaptation of the argument in [2] yields the following result on the system (/;). Theorem 1.1. Consider the interconnected decomposed system (/.) in which Bf(t) is essentially bounded. Suppose rank B t (i) = n. on [t l h 9 t l ]. Then (/ f ) is controllable on [cr, tj, t x ><r + h. Proof. Let ^ f (t,s) be the fundamental matrix solution of Then has rank n i9 so that (7 ) is Euclidean, controllable. Here B* is the algebraic adjoint of A This is proved by letting fieW*? = H* 2 1} ([- fc.0], "'), x^eE 11 /, and by defining a control Delay differential equations in W ( } 79 . -*(* (7, 0,0) pi-* -1 L l * 1 ' s0iS ' s s j' s x l '(t, <r, </>,0) is the solution of (L ) with u l = 0. Using the variation of parameter Aerifies that u l indeed transfers 4> l to x[ in time ^ h. Thus there is a control 2 ([<r, tj - h], m ') such that x (^ - fe, <r, 0, u') = \I/ L (- h). We extend u l and x f to iterval [a,^] so that >n [ti - Mil To do this note that ^(t,a-t)^ Jri-/t ow define r + j tl _^ yt ' a '-+/- * p 1 "* * I i ^ ' * j t _ A ' ' a r ' j,,-' / '' ' a v r , M i/ t , f 1 '* / + X >/ y (t, /t)x-'(t h) + \ f/ y (, a t)x jj* \ L J t~~h "' + f ' Jti-fc ! h ^ t ^ t l . Because of the smoothness properties of x 1 , x j and \l/\ u l is indeed opriate. Thus the controllability of the composite system can be deduced from of the subsystems so long as the interconnection is as proposed. We now turn attention to the nonlinear situation. 80 E N Chukwu 2. Nonlinear systems Consider the general nonlinear large scale system, x(t) = /(f, x t , u(t}) + B(t, x t )u(t) + g(t, x t , v(t)), (2. 1) where /: x C x m ->E" is a nonlinear function #:E x C x E m ~+E n is a nonlinear interconnection and the n x m matrix function B:ExC->E nXm is possibly nonlinear. Conditions for the existence of a unique solution x(% or, 0, M), when weL 2 , 0eC([ h, 0], ") are given in Underwood and Young [18]. It is shown there that (</>, u) - x t (" o" 5 w)eC is continuously differentiable. These conditions are assumed to prevail here. Indeed we have, Lemma 1.1. For t/ze 5ysrem x(t)=f(t 9 x t9 u(t)) -f B(r,x t )w(t) assume that (i) : x C-> nXm is continuously differentiable. (ii) TTiere exist integrable functions N h N t :E-^[0 9 oo), i = 1,2, SMC/I /or tsE anrf (/>eC([- /i,0],E n ). Here an<i in t/ie s^ue/ D^( ) is the Frechet derivative of g with respect to the ith variable. (in) /(t, v) is continuously differentiable for each t. (iv) /(, 0,o)) is measurable for each <j> and CD. (v) For each compact set KaE n there exists an integrable function M f :E-[0, oo), i = 2, 3 such that l|D 2 /(t,^G))||<M 3 (t) VteE, Then for each ueL 2 there exists a unique solution x to Remark 1.1 Note that we have The system (2.1) may be decomposed as B t (t, x? t )u l (t) + ^ y (t, xf, t? (t)), i = 1, . . . , I, (2.2) where Let Z i = w Z m i . = Delay differential equations in [/(t, x,, )] T = [(/, (t, x, 1 , u 1 )) T ,. ..,(f,(t, x l , u') T ], / g i (t,x,,v i (t))= 81 t, x,, t/) = [ 01 (t, x, 1 , v 1 (0) T , . . . , te,(r, xj, i>'( ten we can view (2.1) with decomposition (2.2) as an interconnection of / isolated bsystems (S ; ) described by the equations ( (t s x l V(tX (S f ) th interconnecting structure characterized by 9i (t, x t , v'(t)) = t g. .(t, x/, D f (t)) = g t (t, x,, v'(t)). mditions for the existence and uniqueness of solutions are assumed. In particular ^^i(t,x t ,v l (t)) is assumed integrable. First we shall state the conditions for controllability of each isolated subsystem To do this we define a matrix H t : H.= (L) r each <'eC([- A,0], n ') = C ni . Here B* is the transpose of JJ.. leorem 2.1. In (S^ assume that i) there is a continuous function N* t (t) such that ||JJ*(t,^')|| ^ JV*.(t) V^'eC 11 '; i) H t - in (L) tos a bounded inverse', i) /zere exist continuous functions G.^C" 1 ' x E m ^E + and integrable functions :E^E + j = l,...,q such that r a// (t, (j),u(t))eE x C nf x mi , w/zer^ t/ie following growth condition is satisfied: limsupr- j=l hen (Si) is Euclidean controllable on [cr, ^]. emar/c 2.1 Condition (iii) is a growth condition which should be compared to a liform bound imposed on /by Mirza and Womack [15, Theorem C] when treating slay equations. Such growth conditions have a long history: see [9, 4, 3, 22]. In ] one sees the consequences of the growth condition. 82 EN Chukwu Proof. Let fie W ( }\ x\eE n '. Then the solution of (S,.) is given by xty + ffJ-MX te[-fc,0]; i (s))ds+-fB i (s,xi)u i (5)d S , t*a. (2.3) J <r Now define a function u* on [cr,^] as follows: (2.4) where x'( ) is a solution of (S ) corresponding to u l with initial function <t> 1 . Such a solution exists as earlier remarked if w exists as an L 2 function. Since t-^BfaQ) is continuous, w', as defined, is L 2 . Introduce the following space with norm h IIOMII = li4>ll + iNl2, (<M)eX, ? where ati \l/2 |u(s)| 2 ds ) . T / We show the existence of a positive constant r , and a subset A(r ) of X such that where ^i(*i> r o) = {^[ Mi]-*^ 1 " continuous = <, ||J ^^relV,^]}, | 42(*ir ) = {t*eL 2 (0,t 1 [,E m O:(i)ltt(t)|<r a.e. i* 1 ^[cr,^] and fi (ii) | tt (t + s) tt(r) \ 2 dt -* as s -> uniformly with t Jar H respect to ueA 2 (t 1 ,r )}. It is obvious that the two conditions for A 2 ensures that A 2 is a compact convex subset of the Banach space L 2 ([1 1, p. 297]). Define the operator Ton X as follows: where I /(*, X^ M (5))ds + f ' B (S, Xj)(5)d5, Jar J<r ^ CT, (2.5) (2.6) Obviously the solutions x'(-) and u (-) of (2.3) and (2.4) are fixed points of T; i.e., T(x\ u l ) = (x\ u l ). Using Schauder's fixed-point theorem we shall prove the existence of such a fixed point in A. Let G y (r) = sup {G y (0', W ''): || (<l>\ u l ) \\ < r}, Delay differential equations in W ( ^ 83 where G^. are defined in (iii). Because the growth condition of (iii) is valid there exists a constant r > such that or See a recent paper by Do [22, p. 44]. With this r define A(r ) as described above. To simplify our argument we introduce the following notation: (t 1 -a), 1}, a = If (x'.a'Jeyl^oX from (2.5) and (2.6), we have that j " f a/s)Gy(jcJ, u'(s))ds I f f " a.(s)G y (r )dsl J*! Jr J Also rtl aj (5)G..(X s ,M(s))ds < ^o + ^ = 2ro We now verify that f 11 - 2 } ff uniformly with respect to veA 2 (t l ,r Q ). Indeed f' 1 f' 1 j. vt * s vt ^L as where I- \' 1 ffax^uWds j ^ 84 E N Chukwu Because t -> B*(t, x t ) and t - x t are continuous, we assert that indeed fc \v(t 4- s) v(t)\ 2 dt ->0 as s->0. This proves that i?e/4 2 and we have completed the proof that T maps A(r Q ) into itself. We next prove that T is a continuous operator. This is obvious if are continuous since w -*(,) is continuous. To prove continuity in the general situation we argue as follows. Let (x l 9 u l \ (x ri ,u fi )eA(r ) and T(x i 9 u i ) = (z(t) 9 v(t)) 9 T(x' l ",w'') = (z'(t),z/(t)), where v(t) is as given in (2.6) corresponding to u l and v'(t) is also given by (2.6) corresponding to w f . Also z(t) is given in (2.5) corresponding to u l and z'(t) corresponds to w. Then \v(t)-v'(t)\ 2 = - Bf(t, - / i ( Sj x7,w(s))ds B*(t,xJ") [" Jff (2.7) f J<T |/ f (s,x7,w(s))| 2 ds Since m-B*(t,x^) and u->/.(t,x",u) are continuous given >0, there is a fj>0 such that if |u - w| < r, then |B*(t,j - B*(t,x; w )| < E, and |/.(t 5 xj u ; u) -/.fexj 1 ", w)| < e, tefV.tJ. Divide the interval [ff,tj] into two sets e x and e 2 ; put the points at which \u(t)-w(t)\<ri to be e! and the remainder e 2 . If we write ||u-w|| L2 = y, then T 2 = f;l(f)-wW| 2 dt5=J e2 iu(t)-fflWI 2 d^^ 2 mese 2 so that mese 2 <y 2 /?7 2 . Consider the integral 1 = Then 4v 2 Delay differential equations in for some R. This last estimate is deduced from the fact that if (x, w), (x', u f )eA(r ) this implies that sup{/ t -( )} < K, for some jR. On using this last estimate in (2.7) we deduce that if \u w| < rj; then Thus !b-'H 2 = ws)-t/(s)| 2 ds v 2 /? 2 f' 1 1 ^- N*(s)ds . ^7 Jcr J Because y 2 = ||u w|| 2 and JV*. is integrable i; and v' can be made as close as possible if u and w are sufficiently close. We next consider the term |z(t) z'(t)|. We have + ||B (s,x J )-B (s,x' f )|||i;(s)|ds+ Because of this inequality and an argument similar to the above, z, z' can be made as close as possible in A i if u, w are sufficiently close. We have proved that T is continuous in w. It is easy to see that T is continuous in x, the first argument, and thus, by a little reasoning based on the continuity hypothesis on / 4 and B h that T(x, u) is continuous on both arguments. To be able to use Schauder's fixed point theorem, we need to verify that T(A(r )) is compact. Since A 2 (t 1 ,r ) is compact we need only verify that if (x,u)eA(r ) and (z, v) = T(x, u) then z as defined in (2.5) is equicontinuous for each r . To see this we observe that for each (x,u)eA(r ) and s 1} s 2 e[<M 1 ], s 1 <s 2 , we have < ft I |s 2 - Sl I + I" /s)G v (r )ds (2.8) HGy(r )|s 2 -sj 86 EN Chukwu In the above estimate we have used the fact that ft = max ||fi.(s,0)||, and It now follows that the right hand of (2.8) does not depend on particular choices of (x,). Hence, the set of the first components of T(A(r )) is relatively compact. Thus T(A(r Q )) is compact which by an earlier remark proves that T is a compact operator. Gathering results we have proved that T:,4(r )-,4(r ) is a continuous compact operator from a closed convex subset into itself. By Schauder's fixed point theorem there exists a fixed point (x, u) = T(x, w), given by (2.3) and (2.4). r/ (s,xX(5))ds+ r J <r J <r Euclidean controllability is proved. We now consider the criteria for controllability in W\ for the system (SO- It is well known that with L 2 controls the natural state space of (2.1) is Conditions on existence and uniqueness of solutions of variants of (2.1) are treated by Melvin [27, 28] and recently by Chukwu and Simpson [29]. Since the optimal control of the linear system has been extensively studied in W ( ^ it seems appropriate to treat the nonlinear case. The growth rate we desire in economics is a function. Theorem 2.2. In (S t ) assume (i) Conditions (i) and (Hi) of Theorem 2.1. (ii) ranfc[B f (t,] = n. on [t, ~h,tj for each (iii) The Moore Penrose generalized inverse of ., Bf (t,)-is essentially uniformly bounded on [t l h 9 t l ] 9 for each eC te[t l h 9 t l ]i (iv) -.+ (,) is continuous. Then (S^ is controllable on [a, r^, with t^xr + h. Proof. First we show that (S f ) is Euclidean-controllable on [Mi -ft]- For this we let ^'eW, x\eE". The solution x (0 of (S ) with x' ff = (^ is given by (2.3). Since hypothesis (ii) is valid, the matrix B i (t l -h 9 x i ti _ k )Bf(t 1 -h 9 ^ ti _ h ) (where B*( ) is the algebraic adjoint of .( )) has rank n r Since t-.(t,xp is continuous, there exists some e > such that for each s, ^ s < e, B.(t 1 - h - s,x* ti _ h _ s )B*(t l -h-s, x l ti _ h _ s ) has rank n r As a consequence of this ' B(s,x s )B*(s,x s )ds+ ^ " B(s,x s )B*(s,x s )ds Jti-h-e has rank n h since the last integral is positive definite and H^ - h) is positive semidefinite. By Theorem 2.1, (S f ) is Euclidean controllable on [cr,^ - ft], t x > a 4- h, so that given any f , ^eW there exists a u i 6L 2 ([a 5 t 1 -/i], mi ) such that the solution of (Si) satisfies x l ff = 1 ", x'(* ; - /i, tr, <^ f , w f ) = ^ f ( - /i). We conclude the proof by extending M and x f (% a, <', u ) = x f ( ) to the interval [<r, fj t l >o -f /i so that Delay differential equations in W ( * } 87 ^(t - *! ) = /,(*, *;', u l (f)) + JB,(t, x t V(*X (2.9a) 5. on [t x ' /i, tj, where X'*(T) = ^'(T f j), f j fc < T < f t on the right hand side of .9a). Because of the rank condition (ii) we may define a control function u l as follows: u*(t) = B+ (t, x ; w ) *lf'(t -t.)- at, x? , u'(t)) , (2.9b) r L<fc J r r x h^t^ t l . That such a u exists can be proved as follows: We define the llowing set A 1 (r )=luL 2 ([t l -h 9 t l l,Er<):\\u( )\\ L2 <r and with P - -+ 1 Jti-/i J follows from [7, p. 297] that A = ^4^^) is compact. Let T be a map on ,4 defined ; follows here (2.11) /e shall prove that there is a constant r such that with A = A l (r Q ), T:A-+A, where T is continuous. ecause of [7, p. 297] and [7, p. 645], T is guaranteed a fixed point, that is hich implies that (2.9a) and (2.9b) hold. Observe that A is a compact and convex ibset of the Banach space L 2 . Because of a result of Campbell and Meyer [25, . 225], and hypotheses (i) and (ii) of the theorem, the generalized inverse t - B + (t, {) is ontinuous and therefore uniformly bounded on [t 1 h,t 1 ']. Since the growth ondition (iii) is valid there exists a r > such that t CjG y (r ) + rf<r , ;=i >r some d. With this r define A = A^TQ). Now introduce the following notations max f" |a,(s)l<U ll,ll, sup l Ix'J + I^OUI^H^, sup 88 E N Chukwu Let we A. Then where q 1 I a/t)G y (x|,u(t)) ;=i J a/*)G y (r )"| j =1 J Therefore IMI<0 + | ll^il L2 + llM G u( r o) 1^1 Wli 2 + U a jllGy(r ) Therefore We have proved that T:A-*A 9 if we can verify the second condition. Now ti-fc where { (t) = ^(t-t 1 )-/ i (t,x|,u(t)). The function k(t) = B+ (t, x l t ).(t) is measurable in t, and is in L 2 . We can therefore choose a sequence {/c n (t)} of continuous functions such that i: asn->oo. Jti-h Therefore 11/2 1/2 Qti 11 \k n (t + s)-k n (t)\ 2 dt ti-h J Qti "11/2 \k H (t)-k(t)\ 2 dt\ . We choose n large so that the last and first integral on the right hand side of this inequality are less than an arbitrary s > 0. Also s can be made small enough for the second integral to be less than e > 0. This verifies the first part of the assertion T:A-+.A. We now turn to the problem of continuity. Let (u\ (u')eA(r ), T(u) = (t?), T(w') = (t/). Delay differential equations in W ( ^' 89 Then |(t) - v'(t)\ < Bt(t,$)W l (t - !)-/,(*, *T) - B + (t, x') llB+fcxjj- + ii B; (t, x;<) - B. + (t, xj) ii i/ f (t, Since M-B j ' t "(t,x") and u(f) -/,(*, x t ,u(t)) is continuous given >0 there exists an i] > such that if |(t) - u'(t)\ < q then |/,(t,xj",u(t))-/,(t,x|"',u'(t))|<e VteC^ -fc,^] si. Divide I into two sets gj and e 2 and put the points at which \u(i) u'(t)\ <r\ to be ej and the other to be e 2 . If we set ||u ' || 2 = y, then |M(t)- U '()| 2 dt so that mes e 2 ^ y 2 /rj 2 . A simple analysis shows that |iAll2 + 2^ + n + 2 mese 2 {sup|/()|} 2 -f4/? +2 ^sup{|/ i ()|} It follows from these estimates that || v v' \\ 2 can be made arbitrarily small if || M u' \\ 2 is small. This proves that T:A-+A is a continuous mapping of a compact convex subset of L 2 with itself. By Schauder's fixed point theorem [11, p. 645] T has a fixed point: With this u in V \jf l (t - rj = 0.(t, x r , w(t)) + ;(, x r )w(t) (19a) is satisfied. The proof is complete. Remark. Condition (iv) can be removed by employing an argument similar to the earlier proof. 90 EN Chukwu In Theorem 2.2 we have stated conditions which guarantee the controllability of each isolated free subsystem (S t ). Next we assume these conditions and give an additional condition on the interconnection g i which will ensure that the composite system (2.2) is controllable. It should be carefully noted that g t (t,x t ,J(t))= g tj (t,xi 9 v?(t)) # is independent of x 1 , the state of the ith subsystem, though it is measured locally in the (S t ) system. Theorem 2.3. Consider the interconnected system in (2.2). Assume that (i) Conditions (i)-(iii) of Theorem 2.1 are valid: Thus each isolated subsystem is Euclidean controllable on [<M t ]. (ii) For each i, j = 1, . . . , /, i ^j tfifc*,, '(*))= I ^M W satisfies the following growth condition: There are continuous functions and L 1 functions /},-:- + y = 1, ..., g such that y (xX) forall(t,x tt it\ w/iere /or some constants c limsupfr- c J sup{G y (x,,'(t)):||(x f , < (t))K'-}V +00 - r-*oo \ j=l / Then (2.2) is Euclidean controllable on [a, t x ]. Theorem 2.4. In (2.1) and (2.2), assume that (i) Conditions (i) - (iv) o/ Theorem 2.2 JioW. (ii) For each i,j= I,...,/, 1=5*7, ^t satisfies the growth condition: there are continuous functions and L 1 functions a -: E -+ E" 1 ", j = 1, . . . , q such that /or a// (r, (j>, u\ v l \ where fij < a,, and /or some constants c j9 limsup(r~ J ^suptGy^u 1 ): ||(<M )Kr} ) = +00. r-oo \ j-1 ' Delay differential equations in W ( ^ 91 Then (2.2) is controllable on [cr, rj, t l >a + h. Remark 2.4 The condition (ii) of Theorem 2.3 and Theorem 2.4 is similar to the growth condition of Michel and Miller in [23, Theorem 5.8.4 (ii), Theorem 3.3.5 (iii), Theorem 3.3.2 (iii), Theorem 2.4.20 (iii)]. The condition states that the external (government intervention g t on (S f ) (in forms of taxation, money supply, investment, etc., i.e. q { = g t (t 9 M /P, T, V)) should be dominated by some "power" foPy of the firm, "power" measured as a function of (I ,C ,X ,y). This condition that g t is sufficiently "small" is a nonlinear generalization of the requirement in the linear pursuit game, that IntPoQ. The firm's control set (or initiative) should dominate the government's. This is a necessary and sufficient condition (on the control sets) for controllability. See Hajek [10, p. 61] for the genesis of this idea. It settles this century's basic problem: How much (in comparison to private effort (i.e. autonomous consumption, investment, export, money holding) should government intervention (i.e. q(M /P , T ,v)) be in the economy. The nonlinearity of (2.2) has been well motivated in our introduction. The interconnectedness is natural and essential in the economic application. As a control action of government, q(t) = A[(e 1 /r)(M /P )-cT (t) + v(t)'] 9 in (1.10) and (1.11) is (realistically) not linear in w(0 = (w 1 ,w 2 ,w 3 ,w 4 ) = (M ,Po^o 5 V). We combine the fiscal and the monetarist views. The modern debate of macroeconomics, particularly of Lucas critique [12] makes the incorporation of q(t) very reasonable. (see [17, Macroeconomics in the Global Economy, Chapter 10]. The argument demands a game theoretic formulation for the .dynamics of income. This is well spelled out in Mullinex [26, p. 91]. Wt are therefore compelled to insert a nontrivial g { . Mathematicians often object to and scoff at the full rank of B, but the economic insight of Tinbergen in [17, p. 5, 90] shows how essential this "classical non-degeneracy assumption is in executing monetary and fiscal policies to achieve a target with several dimensions. Proof of Theorem 2.3. The proof parallels that of Theorem 2.1. The integral equation of (2.2) corresponding to (2.3) is f J <r The control function corresponding to (2.4) is defined by lt(t) = B*(t, XJ)Jf- * |"xi - < (0) - [*/! (5, X< , t/(s))ds - I*'' g This control steers ft to x( in time ^ . The additional sum f ff # (s, x* , i? (s))ds is utilized in the estimates by using condition (ii), noting that ft < a . Just as in the proofs of Theorems 2.1 and 2.2 under the conditions of Theorem 2.3 the system (2.2) is Euclidean controllable. The operator T is defined as in the proof of Theorem 2.1 with the 92 EN Chukwu modification that where - j Just as before we prove that T has a fixed point: T(x f ,u') = (x',w')> so that (2.2) is Euclidean controllable. To prove this we suppose that G.. (r) = sup (G y (<M): II (<M) II <r}. Because of the growth condition in (ii), there is some r such that c f G..(r ) + d ^ r j=i for some c f , d. With this we deduce the estimate ', | + |tf'(0)| _t J" (/) + j8 J -(s))G y (r )ds | + |^(0)| + J^ 2 J" a,(s)G y ( since jSj < a^ by condition (ii). With this, G,(r ))<l(^+ f C;GjJ .(r ) / J/C \ ;=1 2r In the same way, we have ||z|| < -. Thus T is a continuous operator since (t, </>) -* J5 (t 5 </>), (t, 0, t>) ->/ (t, 0, u) and (t, 0, w) -*-/i(t, </>, w) are continuous and u -> x(-, u) is continuous. The general situation follows an argument that yielded (2.7) and the subsequent inequalities. For equicontinuity the inequality rs 2 q (2.8) has an extra term a j( s )G -(r )ds due to #.. The reasoning is as before. J Si J =1 The interconnected system is Euclidean controllable. Proof of Theorem 2.4. Our proof here parallels that of Theorem 2.2. From Theorem 2.3 we conclude that (2.2) is Euclidean controllable on [V,^ /i], so that given any $*, ( ^ there exists a u l eL 2 ( [a, 1 1 ~ h] 9 E mi \ such that the solution x of (2.2) satisfies = < and x l (f-/i,ff,0 i ,u) = ^(~fc). Delay differential equations in W ( ^ 93 The control u and the solution x l (',a,(l)\u) are extended on the interval [<r,ti],fi >cr + h so that * W = /,(', 4 u W) + ^(t, xj, !>'(*)) 4- B. (t) ii(t) for t l - ft < f ^ t l9 where x(t) = \j/(t - t^\ t 1 - h ^ t ^ t r Define a control (2.12) The various estimates that lead to the proof of the existence of a fixed point carry through with a,- replaced by 2a 7 - (since /?, < a,-) and f (t) defined by In all the calculations one remembers that once v is chosen and fixed, u is allowed to q vary with its constraints. With minor modification caused by adding /^(OGy.fc', w l ()) j= i the rest of the proof is completed as in the case when g i == 0. Remark. An economic interpretation may define /?,- as a measure of government intervention while <x is a measure of the firm's reaction. To ensure controllability 3. General nonlinear systems In (2.2) it is very important that the system is of the form in which some term is linear in u. Here we consider the more general situation x(t)=f(t,x t ,u(t)\ (3.1) where f:ExCxE m -+E n is continuously differentiate in the second and third arguments, and is continuous, and it also satisfies all the conditions of Lemma 1.1. Details of the proof of the following is contained in Chukwu [8]. Theorem 3.1. In (3.1) assume that: (i)/(t,0,0) = O (ii) The system (3.2) is controllable on [a, t x ], where t^a + h, and where D 2 f(t, 0, 0)z t = L(t, z t \ D 3 /(r, 0,0)t; = B(t)v. Then 06lnt^(t,tr), (3.3) where is a solution of (3.1) with x ff = 0} (3.4) is the attainable set associated with (3.1). 94 EN Chukwu Remark. The argument in the proof is as follows. The solution of (3.1) with x ff = is The mapping can be demonstrated to be Gateaux differentiable with Gateaux derivative where z(t,Q,v) is a solution of (3.2). Because F(Q):L 2 ->W ( v is a surjection because of Condition (ii), all the requirements of Corollary 15.2 of [19, p. 155] are met. Therefore F is locally open which implies (3.3). We shall next investigate the large scale system *'' =/;(,*;,"') + t g u (t,x>,tf(t)) (3.5) j ~i where f t and g tj are as defined following (2.2). Thus we investigate the interconnected system (2.1) * W = f(t, x t , u(t)) -f g(t, x t , v(t)) (3.6) where / and g are identified following (2.2). We state the following result. Theorem 3.2. Consider the large scale system (3.6) wit/z its decomposition (3.5) where (ii) f i9 g { satisfy all the requirements of Lemma 1.1. (iii) Assume that the linear variational system z i (t) = L.(t > z;)4-B l '(t)^(t) (3.7) of x i (t)=/ i (t,x> i (t)) ) (3.8) where D 2 f t (t 9 0, 0)zj = L.(t, zj), is controllable on [a, t ^ t l > a -f h. Tfeen t/ie interconnected system (3.5) fs locally null controllable with constraints. COROLLARY 3.2 Assume (i) Conditions (i)-(iii) o/ Theorem 3.2. (ii) TTie system ,0) (3.9) zs globally exponentially stable. Delay differential equations in W ( ^ 95 Then the composite system is (globally) null controllable with controls in Proof of Theorem. By Theorem 3.1, Oelnt^.(f,cr) for t>a + h, (3.10) where t s/ i is the attainable set associated with (3.8). Let x be the solution of (3.5), with x l = 0. Then ,0,"V")W = P/ i (s,xX J. j-i Thus, if we define the set ff (t p a) = {xV,0,tt^e^^^ we deduce that ,cr). Because /.(, 0, 0) = y (f, 0, 0) = and because x l (t, 0, 0, 0) = is a solution of (3.5), QeH.(t 19 o). As a result of this and (3.10) we deduce ^cH^,*). (3.11) There is an open ball B(0,r) center zero, radius r such that The conclusion follows at once. Using this one deduces readily that Oelnt^, the interior of the domain of null controllability of (3.5), proving local null controllability with constraints. Proof of Corollary 3.2. One uses the control u l = Oel/j t;' = Oel/ f to glide along the system (3.5) and approach an arbitrary neighborhood of the origin in "'). Note that Because of stability in hypothesis (ii) of (3.9) every solution with u l = is entrapped in (9 in time a ^ 0. Since (i) guarantees that all initial states in this neighborhood can be driven to zero in finite time, the proof is complete. Remark. Conditions for global stability of hypothesis (ii) are available in Chukwu [6, Theorem 4.2]. 96 EN Chukwu Remark 3.1. From the condition OeInt^(t,0-)c:H.(,<7), (3.11) we deduced that OeIntH c (t,a), (3.12) is of fundamental importance. If the condition ,<7), (3.13) fails, the isolated system is "not well-behaved" and cannot be controlled (3.12) may still prevail and the composite system may be locally controllable. To have this situation we require OeIntG.(,<7), (3.14) where f f r ' G (t,<r) = In words, we require a sufficient amount of control impact (i.e., (3.14)) to be brought to bear on (S f ), which is not an integral part of S . Thus knowing the limitations of the control u i eU i a sufficient signal gM> x^ v l ) is despatched to make (3.14) hold. And (3.12) will follow. Remark 3.2. The same type of reasoning yields a result similar to Theorem 3.2 if we consider the system ^^..^^ 1 ^-^ 1 ^)), (3.15) where (^...,xr^o,x; + V..^^ Also conditions (ii) and (iii) of Theorem 3.2 are satisfied. If we consider (3.16) instead of (3.5) we can obtain the following result. Theorem 3.3. In (3.16) assume (i)-(iii) of Theorem 3.2. But in (3.7) L,(t,zj) and 5'( ar^ defined as D 2 f s (t, 0, 0)zJ = L,(t, zj), D 3 (/,a, 0, 0) + 9i (t, x t , 0)) = B'(t). Then (3.16) fs locally null-controllable with constraints. The proof is essentially the same as that of Theorem 3.1. We note that the essential Delay differential equations in W^ 97 uirement for (3.16) to be locally null-controllable is the controllability of x(t) = L(t, x, ) + (B, (t) + B 2 (t))u(t), (3.17) ere he isolated system (3.8) is not "proper" (and this may happen where B^t) does : have full rank on [0",^], txr + h, the "solidarity function" g t can be brought bear to force the full rank of B = B l + B 29 from which (3.16) will be "proper" ;ause (3.17) is controllable. Even if B^ has full rank and (3.8) is proper, the large le system need not be locally null controllable. The function has to be so nice that -f B 1 has full rank. An adequate "proper" amount of "regulation" is needed in the m of a "solidarity function" g t . .n applications it is important to know something about g i and to decide its squacy. It is possible to consider g { as a control and view a differential game. Considered in this way the control set for g t can be described. the linear case see Chukwu [7]. knowledgements e author is grateful to a referee whose criticism made him to include more detailed )ofs of Theorems 2.3 and 2.4. Remark 2.1 and 2.4 now add more understanding the investigation. ferences ] Allen R G D, Mathematical economics (London: McMillan) (1960) ] Banks H T, Jacobs M C and Langenhop C E, Characterization of the controlled states in W of linear hereditary systems, SI AM J. Control 13 (1975) 611-649 ] Balachandra V K and Dauer J P, Relative controllability of perturbations of nonlinear systems, J. Optimi. Theory Appl. 63 (1989) ] Chukwu E N, Controllability of delay systems with restrained controls, J. OptimL Theory Appl 29 (1979) 301-320 i] Chukwu E N, Mathematical controllability theory of the growth of wealth of nations proceedings, first world congress of nonlinear analysts, in: Tampa, Florida (ed.) V Lakshmikanthan. 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Sci.), Vol. 105, No. 1, February 1995, pp. 99-103. tinted in India. iote on integrable solutions of Hammerstein integral equations K BALACHANDRAN and S ILAMARAN Department of Mathematics, Bharathiar University, Coimbatore 641 046, India MS received 4 November 1993; revised 12 March 1994 Abstract We derive a set of sufficient conditions for the existence of solutions of a Hammerstein integral equation. Keywords. Hammerstein integral equation; Caratheodory condition; Lusin theorem; Scorza Dragoni theorem; Schauder fixed point theorem. ntroduction 5 of the most frequently investigated integral equations in nonlinear functional lysis is the Hammerstein equation f J o 9 s)f(t,x(s))ds e[0,l]. (1) h an equation has been studied in several papers and monographs [1-6]. Existence )rems for eq. (1) can be obtained by applying various fixed point principles. In Banas proved an existence theorem for (1) using the measure of weak non- ipactness. On the other hand Emmanuele [5] established an existence theorem the same equation using Schauder's fixed point theorem. In this paper we shall ve the existence of solutions of the following nonlinear Hammerstein equation \ J t,s)f(t 9 x(a(s)))ds * 6 [0,1] (2) suitably adopting the technique of [5]. The result generalizes the result of [5]. Existence theorem >rder to prove existence theorem for (2) we shall first prove the following theorem: iorem L Assume that flieJ^fl), 1] and a^^Ofor all fe[0, 1]. /:[0, 1] x R-+R satisfies Caratheodory condition and there exist a 2 eL 1 [0, 1] and b 2 > such that for a.e. te[0, 1] and all xeR. 99 100 K Balachandran and S Ilamaran (iii) /c:[0, 1] x [0, 1] -+R + is measurable with respect to both variables and is such that the integral operator Kx(t) = k(t, Jo s)x(s)ds maps L^O, 1] into itself. (iv) a: [0,1] ~>[0, 1] is absolutely continuous and there exists a constant M>0 such that v'(i)^M for all te[0,l]. s. Then there exists a unique a.e. non-negative function (peL l [0 9 1] such that <p(t) = ^f + l f * ftfc s) [a 2 (s) + 101 l-0iJo Define a function ^:[0, l]-v,R by 1 k(t s s)a 2 (s)ds. Put B = , 1] : || x K r> where r = 5 J ii H-. , Define an operator FiL^O, ll^L^O, 1] by Fx(t) = -^ + ~^ From our assumptions for xeB r we have - f 1 ~J <:j-^Ja 1 (f)d + y-^-j' 1 f * k(t,s)[a 2 (s) + l 1 Hf f 1 1 <- Ui() + k(t,s}a 2 (s)ds dt l-oJoL Jo J (l-*i) (1 + Thus we have F(B r ) <= B r . If we define B p + = {xeB r :x(r) >0 a.e.} then F(B+) c B r + . Also B* is a complete metric space, since B r + is a closed subset of L l [Q, 1]. Hammerstein integral equations 101 Now for any two elements x,j;eB r + we have - Fy u lJO 1 f 1 k(t, Jo At /i m,. (1-bJM On applying contraction fixed point theorem we get a fixed point for F. This proves Theorem 1. Theorem 2. Assume that (i) #:[0, 1] x R -+R satisfies Caratheodory conditions and there exist a^L^O, 1] and b l >Q such that for a.e. te[0, 1] and for xejR and \g(t,x(t)-g(s,x(s))\^(o(\t-s\) where o>(|t s|)-*0 as t-+s. (ii) /:[0, 1] x R-+R satisfies Caratheodory condition and there exist a 2 eL 1 [0, 1] and b 2 >0 such that for a.e. fe[0, 1] and all xeR. (iii) fc:[0, 1] x [0, 1]->R + satisfies Caratheodory condition and is measurable with respect to the second variable. Also the integral operator Kx(t) = f k(t, s)x(s) ds maps L 1 [0, 1] into itself. Jo (iv) a: [0, 1] ->[0, 1] is absolutely continuous and there exists a constant M such that (v) fci + ~ < 1. Then (2) has a solution in L^O, 1]. Proof. Since all the assumptions of Theorem 1 are satisfied, there exists a unique a.e. non-negative function cp such that f Jo s . First let us assume cp = Ll[0 t] in L^O, 1]. In this case, if we take 102 K Balachandran and S Ilamaran then Jo and so j>(r) = 0. Therefore <p = Ll[01] is the solution of (2). Now, assume that <p 96 Ll[0tl] . Define a set Q in L^O, 1] by Then clearly Q is nonempty, bounded, closed and convex set in I/fX), 1]. Define an operator HiL^O, 1] ->/[(), 1] by f 1 H x(t) = g(t, x(t)) + kit, s)/(, x(cr(s))) ds. Jo Then according to our assumptions H is continuous and for xeQ, we have \Hx(t)\^a 1 (t) + b i \x(t)\+ /c(t,s)[a 2 (s) + b 2 |x Jo f 1 Jo t>S ^ Therefore H(Q)c:Q. Now we shall prove that H(Q) is relatively compact. Using Lusin's and Scorza-Dragoni's theorems [see 5] for each positive integer n there exists a closed set A n c [0, 1] such that m(A c n ) <(l/n) and a^^cpl^k U nX[0 ,i] are uniformly continuous. Now let (y k ) be a sequence in Q. For t',t"eA n we have - Hy k (f)\ ^ \g(f, y k (t')) - g(t",y k (t")\ + [ 1 |/c(^s)~^, Jo This proves that (Hy k ) is a sequence of equicontinuous functions on A n . Also for every teA n we have f 1 \Hy k (t)\ ^ a,(t) -f 6 lV (r) 4- fc(t,s)[a 2 (s) + h 2 <p(cr(s))] ds. Jo Because of the continuity of a l and cp on the compact set A n and fe on the compact set A n x [0, 1] the sequence (Hy k ) is equibounded on A n . By applying the Ascoli-Arzela theorem we get for each n there exists a subsequence (y m ) of Cy fc ) such that (Hy^ is a Cauchy sequence in the space C(A n ) of all equicontinuous and equibounded functions on A tt . Now, given >0, there exists ^>0 such that $ A (p(s) ds < (e/4) whenever m(A) < 5. Choose a positive integer N such that (I/AT) < 3. Then m(^) < 5. Therefore J <p(t)dt<-. Hammerstein integral equations 103 Also for sufficiently large h' and /i" since (Hy k ) is a Cauchy sequence in C(A N ). Hence ') - Hy-) || Ll[0>1] = \Hy k(h . } (t) - fl> J < for sufficiently large h f and /z". Therefore (Hy k(h) ) is a convergent subsequence of the sequence (Hy k ) in L^O, 1]. This proves the relative compactness of H(Q). Applying the Schauder fixed theorem we get a fixed point for H. This proves our theorem. References [1] Appell J, On the solvability of nonlinear, noncompact problems in function spaces with applications to integral and differential equations, Boll. Un. Mat. It., B6 (1982) 1161-1167 [2] Banas J, Integrable solutions of Hammerstein and Urysohn integral equations, J. Aust, Math. Soc., 46 (1989) 61-68 [3] Corduneanu C, Integral equations and applications (1991) (Cambridge: Cambridge University Press) [4] Deimling K, N onlinear functional analysis (1985) (Berlin: Springer Verlag) [5] Emmanuele G, Integrable solutions of Hammerstein integral equations, Applicable Analysis (to appear) [6] Gribenberg G, Londen S O and Staffens O, Volterra integral and functional equations (1990) (Cambridge: Cambridge University Press) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 1, February 1995, pp. 105-122. Printed in India. On over-reflection of acoustic-gravity waves incident upon a magnetic shear layer in a compressible fluid P KANDASWAMY Department of Mathematics, Bharathiar University, Coimbatore 641 046, India MS received 10 August 1992 Abstract. A study is made of over-reflection of acoustic-gravity waves incident upon a magnetic shear layer in an isothermal compressible electrically conducting fluid in the presence of an external magnetic field. The reflection and transmission coefficients of hydromagnetic acoustic-gravity waves incident upon magnetic shear layer are calculated. The invariance of wave-action flux is used to investigate the properties of reflection, transmission and absorption of the waves incident upon the shear layer, and then to discuss how these properties depend on the wavelength, length scale of the shear layers, and the ratio of the flow speed and phase speed of the waves. Special attention is given to the relationship between the wave-amplification and critical-level behaviour. It is shown that there exists a critical level within the shear layer and the wave incident upon the shear layer is over-reflected, that is, more energy is reflected back towards the source than was originally emitted. The mechanism of the over-reflection (or wave amplification) is due to the fact that the excess reflected energy is extracted by the wave from the external magnetic field. It is also found that the absence of critical level within the shear layer leads to non-amplification of waves. For the case of very large vertical wavelength of waves, the coefficients of incident, reflected and transmitted energy are calculated. In this limiting situation, the wave is neither amplified nor absorbed by the shear layer. Finally, it is shown that resonance occurs at a particular value of the phase velocity of the wave. Keywords. Over-reflection; gravity waves; magnetic shear layer. I. Introduction During the last decade, considerable attention has been given to the phenomenon of over-reflection (wave amplification) of a hydrodynamic or hydromagnetic gravity wave incident upon a shear layer in an incompressible homogenous or stratified fluid. It has been known that the reflection coefficient for waves of one kind or another incident upon a shear layer can be greater than unity. This implies that more energy is reflected back towards the source than was originally emitted. This phenomenon known as over-reflection (wave amplification) occurs in various hydrodynamic and hydromagnetic fluid models under different conditions. Several authors including Booker and Bretherton [3], Jones [7], Breeding [4], Jones and Houghton [8], Acheson [1,2], McKenzie [9], Eltayeb and McKenzie [6] and Kandaswamy and Palaniswamy [10] have studied various aspects of the critical layer for internal gravity waves in a shear flow, critical layer for internal gravity waves in a shear flow, critical-level behaviour and over-reflection of a hydrodynamic 105 106 P Kandaswamy or hydromagnetic gravity wave incident upon a shear layer in an incompressible homogenous or stratified fluid. The over-reflection of internal gravity waves by a finite layer of constant shear separated by two uniform streams of incompressible fluid has been investigated analytically by Eltayeb and McKenzie [8] and numerically by Jones [7] and Breeding [4]. Mckenzie [8] has studied the reflection and refraction of a plane acoustic-gravity wave at an interface separating two fluids in relative motion. He predicted the phenomenon of over-reflection for pure acoustic waves provided the shear flow speed exceeds the horizontal phase speed of the incident gravity wave. A discussion of this result implies that the gravity waves can extract energy and momentum from the mean flow along with the idea of a critical layer at which the energy and momentum of gravity waves are absorbed into the mean flow. Acheson [2] has investigated the phenomenon of over-reflection for a variety of different systems involving waves propagating towards a shear layer. He studied the reflection of hydromagnetic internal gravity waves travelling in an incompressible fluid towards a vortex-current sheet with special attention to the relationship between over-reflection and critical layer absorption. Recently, the over-reflection of hydromagnetic gravity waves in a compressible stratified fluid was considered by Kandaswamy and Palaniswamy [10]. In spite of these works, attention is hardly given to the phenomenon of over-reflection of hydromagnetic waves in a compressible fluid. The main objective of this paper is to study the phenomenon of over-reflection of acoustic-gravity waves incident upon a magnetic shear layer in an isothermal compressible electrically conducting fluid in the presence of an external magnetic fluid. The invariance of the wave-action flux is used to investigate the properties of reflection, transmission, and absorption of the acoustic-gravity waves incident upon the magnetic shear layer, and then to discuss how these properties depend on the wavelength, length scale of the shear layer, and the ratio of the flow speed and the phase speed of the waves. Special attention is given to the relationship between the wave amplification and critical-level behaviour. The over-reflection is due to the fact that the excess reflected energy is extracted by the wave from the external magnetic field. For the case of very large vertical wavelength, the coefficients of the incident, reflected and transmitted energy are calculated. In this limiting situation, the hydromagnetic acoustic-gravity wave is neither amplified nor absorbed by the magnetic shear layer. It is also shown that resonance occurs at a particular value of the phase velocity of the wave. 2. Basic equations The basic hydromagnetic equations governing the unsteady motion of an isothermal compressible electrically conducting fluid in the presence of an external magnetic field H are in standard notation (Chandrasekhar [5]): Du [(VxH)xH] (2.1) Dp ~J + p(V-ii) = (2.2) Dp 2 Dp = c 2 (23) Dt Dt ( } Over-reflection of acoustic-gravity waves 107 = (H-V)u-H(V-u) (2.4) V-H = (2.5) where where u is the Eulerian velocity vector, p the fluid density, g the acceleration due to gravity, n the magnetic permeability, p the hydrodynamic pressure, and c the constant speed of sound. The equilibrium configuration is given by u = (0,0,0), H = (/f (z),0,0), p = p , p = PO and g = (0,0, - g) where H 05 p ,p represent the basic magnetic field, density and the pressure respectively. In view of these results, the basic equations yield _5po =0 = apo (27ab) dx dy (2.8) dz whence it follows that Po = P(4 Po = PoW and - 1 = (2.9abc) Po Sz On the above equilibrium configuration, we superimpose a small disturbance of the form u = (w, u, w), H = (H Q (z) + ft x , h y , h z \ p = p + p', p = p + P'- (2.10abcd) We assume that the disturbances are small enough compared to the initial state so that higher-order terms in perturbed quantities can be neglected. We then substitute (2.10abcd) in (2.1)-(2-5) and invoke linearization so that the resulting equations reduce to a set of linear partial differential equations. This system admits plane- wave solutions in which all perturbed quantities /may be written as /(x, y, z, t) = /(z)exp [i(foc + ly - cot)], (2. 1 1) where (fc, /) and co are constants, and the former represents the wavenumber and the latter denotes the frequency of the wave. Elimination of all perturbed variables but w leads to the equation -/?+ (a> 2 -A 2 k 2 )(Q-co*)dz Q dz , (2.12) Q 108 P Kandaswamy where Q = (co 2 -A 2 k 2 )(a) 2 -<x 2 c 2 )-l 2 (o 2 A 2 , t (2.13) with the Alfven velocity A, the wavenumber a and the Brunt- Vaisala frequency N being given by Po Invoking the transformation w = </> exp I J, (2.12) assumes the form d? + [(co 2 - A 2 k 2 )(Q - co 4 ) dz Q J dz" ^ /? Q d (co 2 -A 2 k 2 )(Q-(o*) 4 2(co 2 -A 2 k 2 )(Q-a> 4 )dz Q V 2 (2.14abc) or - - (2 ' 15) dz In the next section we calculate the reflection and transmission coefficients for a hydromagnetic gravity wave incident upon a magnetic shear layer. 3. Reflection and transmission of hydromagnetic waves by a magnetic shear layer We consider the problem of a hydromagnetic gravity wave incident upon a magnetic shear layer specified by A\^ z<0 (region I) A 2 = A\z, A\-=-A\, 0<z^L (regionll) (3.1abc) it A\, z ^ L (region III) A gravity wave from region I incident upon the magnetic shear layer (region II) gives rise to a reflected wave in region I, a transmitted wave in region III, and two waves, one moving upward and the other moving downward going in region II. In region I, (2.15) reduces to the form O, (3.2) and the corresponding solution has the form 0(z) = /exp(ia ri )z + Kexp(- iza 2l ), (3.3) Over-reflection of acoustic-gravity waves 109 where / is the amplitude of the incident wave and R that of the reflected wave, and <x zi is given by + lcoAN, (3.4) with Q, = (co 2 - Alk 2 )(cD 2 - aV) - l 2 co 2 Al (3.5) If we take a zi as the positive root of (3.4), the choice of the signs in (3.3) ensures that the incident wave transports wave energy upwards (towards the magnetic shear layer) and the reflected wave carries wave energy downwards (away from the magnetic shear layer). In region II, (2.15) takes the following form * 2 + j ^ , 62 Q' 2 iw dz 2 L (o 2 -zA 2 2 k 2 Q 2 -co 4 Q 2 ]dz 2 (a> 2 -zA 2 2 k 2 ) (3.6) where Q 2 represents the expression (2.13) with A 2 replaced with zA 2 2 . Making use of the transformation $ = ^Q 1 2 ' 2 (Q 2 - co 4 )~ 1/2 , (3.6) becomes z -^ -I _ + _l_i^ i_i ^ _. Q ,3 j, dz 2 z 7 ^ I - ^ I ^ ^ (3.8abc) 4a 2 + ~&^ { Qz + " 2C2(N2 ~ 92/ 2) + ^ 2 ^' 2 /e 2 )}. (3.10) Using the transformations Z = ri (z-z ci ), (lllab) 110 P Kandaswamy (3.7) can be transformed into the confluent hypergeometric equation - a This admits two independent solutions in the form -+..., (3.13) ( 3 - 14 ) | where Si=-lB*Z*, B t = -^ 7 4A (3.15ab) i k = 1 j f 1 . \ \ i Ht *ll(l2-- +n~^Tl)' (3 ' 16) Thus, the solution of (3.7) can be written as W 2 (Z) (3.17) where -+r^M (I!) 2 i (3.18) (2I) 2 + s; 2 + 4; t - z 2 + ... (3.19) and D! and D 2 are amplitude constants. Using the transformation 4> = \l/Q l 2 /2 (co 2 - A\zk 2 )~ 112 , (3.6) reduces to (3 . 20) Over-reflection of acoustic-gravity waves 1 1 1 where z = lf> C 2= 2^ , (3-21ab) ,--ra & i 2|_e2 <a 2 -A 2 2 zk 2 j Q 2 (a> 2 -k 2 c 2 )(o> 2 -A 2 2 zk 2 ) ( } Q 2 c> 2 -A 2 2 k 2 z i 2| x>2 v*2 7 162 (o) 2 Alk 2 z Invoking the transformation j ^ (Z\ 2 = 72(2- z C2 )> (3.24ab) (3.20) assumes the form Z^ + (l-ztf-(~J 2 Wo. (3.25) This gives two independent solutions in the form '1 .\ /I . V3 . -h] (3.26) (3.27) where (3.28ab) - , wl-/, (3.29) Therefore, the solution of (3.20) can be written as V 4 (Z) (3.30) 112 "" P Kandaswamy where e,-[, '1 2~ 72 (I!) 2 3 + v ( y '*+ \ < 22 - l - A 2 2 zk 2 ) f V2 / Y2 A2 / "1 *[ i+ V z+ zi+ -]' 8Z (3.32) and D 3 and Z) 4 are amplitude constants. The transformation ^=-\l/(Q 2 ~o} 4 r 1/2 (^ 2 ~ A 2 2 zk 2 )- 1/2 reduces (3.6) to the following form: J: ^ j_ ( J3?3 _ Vs \ j, _ n n 33) V,-' *'*'/ where z ej = ^, c 2 = l ~ ac ' (3.34ab) C3 A 2 ' 3 a 2 (co 2 -k 2 c 2 ) l p -If-Jlii K3 ~2Lc, 2 -^ (3.35) + ^- y -^- (3.36) yt-p-wi -^-.--^^ 1-2! ( ^: 2 ; 2 ^ 2)2+ ^ ] 4a2 ' -,+a 2 C 2 (N 2 -^r i u 2 ' 5? IJL 2 J (i o'7\ (co 2 -A 2 zk 2 )(0 2 -co 4 ) ' Equation (3.33) can be transformed to the following form -0 (3.38) by means of the transformation \ji = e~ (Z}2) \j/(Z\Z = y 3 (z C3 -z\ which has two Over-reflection of acoustic-gravity waves 113 independent solutions where where = ~ z / 2 - 4 - 1/22 - 2 W S (Z) = e~ z / 2 (Q 2 - co 4 )- 1/2 (o> 2 - A 2 2 zk 2 )- 1 W 6 (Z) = e~ z/2 (Q 2 - co 4 )- 1/2 (ro 2 - A 2 2 zk 2 )- 112 . (I!) 2 (2!) 2 ' = (l/s log Z + S 3 , (3.40) ,= B k Z\ B k = - \ . (3.41) Therefore, the solution of (3.33) can be written as W 6 (Z) (3.43) I . +7 2 +y ' 3 2 +7 (144) feZ-l 3/2 + 4/3-- )Z 2 + ... , (3.45) and D 5 and D 6 are amplitude constants. For region III, (3.15) takes the following form d 2 ^ J fi 2 a 2 f / / 2 \\ -f / 2 o) 2 ^ 2 AT 2 U = (3.46) and has the solution of the form 0= Texp(fea, 3 ), (3.47) 114 P Kandaswamy where T is the amplitude of the transmitted wave and <x, 3 is given by g i = I 2 4 co 2 -fc 2 ~co 4 > 2 - A 2 2 k 2 ) Q 3 + a 2 c 2 N 2 - + / 2 a> 2 ^ 2 N 2 . (3.48) Again the choice of the sign of a r3 ensures that the transmitted wave transports wave energy upwards. To simplify the calculations in the following discussion, we shall consider the low- frequency approximation, so that the dispersion equations (3.4) and (3.48) appropriate to region I and region III are approximated by By using the boundary conditions at z = and z = L, namely the continuity of the vertical component of velocity and continuity of pressure, we determine the amplitudes of the reflected and transmitted waves. Since the vertical component of velocity w is continuous, and from the relation w = <jf>e (/ * /2)ir , </> is also continuous. It follows from the equations of motion that the pressure p t can be obtained in the form . J . (3.50) dz we put w = cf)e ( P /2)s to transform (3.50) into the form "I (0 -~ ft) 4 )! 1 <t> L W V 2* a 2 Therefore, the boundary conditions are equivalent to atz = 0,L (3.52) where the square bracket denotes the jump in the quantity inside the square bracket. Utilizing the boundary conditions (3.52), and when A\ < c\ yields [$] 0)Z , = which implies / + R = D l W^G) + D 2 W 2 (ty> (3.53) Texp(fLa Z3 ) = D, W,(L) 4- D 2 ^ 2 (L), (3.54) fd^l and the condition =0 leads to 2 W' 2 (Q). (3.55) a sj Texp(iLa Z3 ) = ?1 D 1 W^(L) + 7l D 2 W" 2 (D- (3.56) Over-reflection of acoustic-gravity waves 1 1 5 m the above equations we obtain R = D l JX, ^(0) - 7l ^(0)] + P 2 rx, W 2 (Q) - 7l W' 2 m I 0^x^(0) + y^m + D^W^ + ^W'^rf DI i*,W 2 (L)- yi W 2 (L) = -, 1.3. JO) D 2 i^W,(L)- 7l W\(L) rexp(ia, t L) = D l W,(L) + D 2 W 2 (L) ' I + R D^V\ the case A\ > c\, the above expressions can be written as D 2 (3.60) rexp(ia 23 L) _ DI W,(-L) + D 2 W 2 (-L) the above results can be written in the compact form K = Mi 4, 2 (L) A\<c\, and ^ = Mi^ (3.64) / M,l+ d ,2 Tex P (ia, t L) = ^ W,(- L) + W 2 (- L) I + R 5^(0)+ W 2 (0) A\> c\, where d h = i z , ^.(0) - V i W' m (Q\ d rn = ia zi W n (0) + y , W' n (Q), n = 1, 2 (3.66) ^."yx^W-fa^^D (3.68) ^^Vi^- 1 -)- ^.(-L) (3.69) i the following notations were used: (3.71) (3.72) 116 P Kandaswamy 4 Wave amplification and critical-level behaviour We investigate the properties of reflection, transmission and absorption of hydro- magnetic acoustic-gravity waves incident upon a magnetic shear layer, and discuss how these properties depend on the wavelength, the length scale of the shear layer, and the ratio of the flow speed and the phase speed of the waves. We use the invariance of the wave action flux to prove some general properties. The wave energy flux is _ __ _ | j E = Mz = -(P f *9 s + PJ*) = - Re(A<?*), (4.1) where the asterisk denotes the complex conjugate and Re stands for the real part. Using (2.15), the expression for becomes where I p c = p</ z - (4.4abc) We next define the wave action flux M as the ratio of the wave energy flux and the local relative frequency by ,4.5, -co 2 It turns out that ^ = forz/z cr ,r=l,2,3. (4.6) dz This means that M is independent of z except at the critical levels where it is discontinuous. The invariance of M is closely linked to the invariance of both the vertical component of the total energy flux and of the horizontal component of momentum. If A\<cl and A\<c*A\>c\ and A\ >c\,A\* and A\<c\ (4.7abc) and there is no critical level inside the magnetic shear layer. In view of the invariance of M over the whole domain, it turns out that M = g^ (/4 '"" c ' )( ^" C lJ [ |/| 2 -|R| 2 ] in region I (4.9) 2co 2 a 2 c 2 -A i Over-reflection of acoustic-gravity waves 111 M z * ^ 3 _1 1 ?__ | y|2 j n re gj on in (4.10) These results combined with the invariance of M give -- LI ' IJ - m - (4.11) Since a 2i and a, 3 are positive and terms within the square bracket are either positive or negative, the amplification of wave is impossible. On the other hand, if A\>c\ and Al<cl,A\<c\ and A\>c\,A\<c\ and A\>c (4.12abc) there exists a critical level inside the magnetic shear layer. We use the following approximate solution near z = z ci : + 2; 1 7,(z-z ci )}, z<z ci (4.13a) z tl -z)] + D 2 {[1 +; 1 y 1 (z tl -^jlogCy^z^ -z) + m] -2/ iyi (z Cl -z)}, z>z ci . (4.13b) We then calculate the values M b of M below the critical level and the value M a of M above the critical level. These values are We obtain a zi from (4.7) and (4.11), and a Z3 from (4.8) and (4.12abc) so that they are given by (c 2 A 2 \( A 2 r 2 \ (r 2 r 2 \ C22 - ( * (4.16) (4.17) The results combined with (4.10) give the total energy flux in the shear layer: 118 P Kandaswamy The term on the left-side of (4.18) represents the total energy flux into the shear layer whereas the first two terms on the right-side denote the total energy flux out of the layers. The term on the right side of (4.18) is negative whenever the critical level exists. Thus, if the critical layer exists, the wave is amplified. The solution near the critical level z = z C2 is given by 2/ 2 y 2 (z-z C2 )}, z<z C2 , (4.19a) -2j 2 y 2 (z~z C2 )}> z>z C2 . (4.19b) The value M b of M below the critical level and the value M a of M above the critical level are (4 - 20) We obtain a zi from (49) and (4.20) and a Z3 from (4.10) and (4.21) in the form -- (4.22) 2> (A 2 c^)( c 2 + A 2 } (c 2 c 2 ) These relations combined with (4.19) yield the total energy flux a *' The term on the left hand side of (4.24) represents the total energy flux into the shear layer whereas the first two terms in the right hand side denote the total energy flux out of the layer and the last term on the right hand side of (4.24) is negative whenever the critical layer exists. Thus the conclusion is that if the critical layer exists, the wave is amplified. Finally, at the critical level z = z C3 , the equation for the total energy flux is found to be This implies that the total energy flux into the shear layer is equal to the total energy flux out of the shear layer. So, the wave is not amplified in this case. Over-reflection of acoustic-gravity waves 5. Reflection coefficient for large vertical wavelength 119 A large vertical wavelength normalized by the thickness of the shear layer corresponds to y l L 1. The approximate solution can be obtained from (4.13). The following results can be found from (3.62)-(3.65): where R 1 ) 2 z Cl (z Cl ^ 1 y 1 ) 2 z ci (z ci - L)] = a 2l ; 1 y 1 (; 1 ); 1 z ci + 1)4- z Cl 4- IJ- z Cl -L z ci inj\y 1 - - z ci l^ z ci tl / X 2 \" 1 1 Ari-7 - +Ji7i( 1 f +(;iri) 2 ^ -^~ z ci \ c i / J [i / ^1 2 \" 1 1 JiVi. - +7i7i M -- 1 +(/i7 1 ) 2 2c, ^~ Z Cl \ C l / J 120 P Kandaswamy z 3 te/i7i(/iyi(- L - Zci )j + C/ 1 y 1 ) 2 (L Also, we find , * vl^ v * +s 2 (l - ^-lAy^-L) J for^<c*. (5.2a) cl ))] -si = ia z l tilogf^-lj + iTt-^y^L-zJ-lAy^, L \ ^i / J t 2 . (5.2c) ^ S 7i7i"^0 *e results (5.1) and (5.2) reduces to the form / ^2\ / A 2\ a, 3 a Zl (L-z Cl )log( 1 j- + ia Z3 ( 1 ---f + ia zi \ C l / \ C l J /r o\ a S3 a Zl (L - z ci )logfl - -f J - ia Z3 f 1 - -| J + ia zi and 'T 1 -r^r ( 5 - 4 ) In the limit L~ z ci ->0, we obtain R a,, -|-a ? ,(l 04?/c?)) ,. - x 7 = a -a fl-UVJn* and Over-reflection of acoustic-gravity waves 121 view of results (5.5) and (5.6), we conclude that the wave is neither amplified nor sorbed by the layer. Expressing the reflection and transmission co-efficients for the :al energy flux as a function of A 2 and c\ we obtain T 2 = ' (5.7) (5.8) (5.9) ;re i 2 is the ratio of the transmitted energy flux to the incident flux in the moving id. Also we choose the sign of <x zi and a Z3 so It follows from (5.7) that |R//|->oo provided id. Also we choose the sign of <x zi and a Z3 so that the above result is positive. 42 A2 is result reveals that resonance occurs (that is |JR/T| n of the square root of (5.10) is taken. oo) only when the negative Discussion and conclusion is clear from the above analysis that if (4.7abc) and (4.8abc) are satisfied, there is critical level within the magnetic shear layer. Consequently, the amplification of dromagnetic wave is impossible. On the other hand, if the condition (4.12abc) is satisfied, there exists a critical level thin the magnetic shear layer. The wave action flux is found to be invariant ery where in the fluid medium except at the critical level. In view of (4.12abc), the ive incident upon the shear layer is over-reflexed, that is, more energy is reflected ck towards the source than was originally emitted. In the present hydromagnetic alysis, the mechanism of the over-reflection is due to the fact that the excess reflected ergy is extracted by the wave from the external magnetic field. When the vertical wavelength is very large, y 1 Ll the incident energy /, the lected energy R and the transmitted energy T satisfy results (5.5) and (5.6). It is ident from these results that the wave is neither amplified nor absorbed by the agnetic shear layer. Finally, result (5.7) reveals that \R/T\ -> oo provided the phase velocity of the wave negative and given by l/2 ius resonance occurs at this value of c^ . And this quantity c x can be expressed as (6.2ab) 122 P Kandaswamy where (63ab) 2c 2 ^) ) (6.4) f (6-5) fc, = - (A*X'c*)/(/l> + 2cM 4 ), (6-6) j. a 2 = -[(co 4 /a 4 + c 4 - 2a 2 c 2 /a> 2 )Al - (A\ + 2Al)c* J\ -Al(ca 4 /a. 4 + A*-2Al(o 2 /a 2 ) + 2c 2 (Al + Al)-\. (6-7) b 2 = -[2c*AlA 2 3 + Alc*-2c 2 AlAll (6-8) X. X = 2co 2 (c 2 - A 2 3 )/x 2 + A* - 2c 2 (6.10) and c is the constant speed of sound. References [1] Acheson D J, The critical level for hydromagnetic waves in a rotating fluid. J. Fluid. Mech. 53 (1972) 401-415 [2] Acheson D J, On over-reflexion, J. Fluid. Mech. 77 (1976) 433-472 [3] Booker J R and Britherton F P, The critical layer for internal gravity waves in a shear layer. J. Fluid. Mech. 27 (1967) 513-539 [4] Breeding R J, A nonlinear investigation of critical levels for internal atmospheric gravity waves. J. Fluid. Mech. 50 (1971) 545-563 [5] Chandrasekhar S, Hydrodynamic and hydromagnetic stability (Oxford: University Press) (1961) [6] Eltayeb I A and McKenzie J F, Critical-level behaviour and wave amplification of a gravity wave incident upon a shear layer, J. Fluid. Mech. 72 (1975) 661-671 [7] Jonos W L, Reflexion and stability of waves in stably stratified fluids with shear flow: a numerical study, J. Fluid. Mech. 34 (1968) 604-624 [8] Jones W L and Houghton D D, The coupling of momentum between internal gravity waves and mean flows: A numerical study, J. Atmos. Sci. 28 (1971) 604-608 [9] McKenzie J F, Reflection and amplification of acoustic-gravity waves at a density and velocity discontinuity, J. Geophys. Res. 77 (1972) 2915-2926 [10] Kandaswamy P and Palaniswamy E M, Over reflexion of a hydromagnetic gravity wave incident upon a magnetic shear layer. Proc. Indian Natn. Sci. Acad. 53 (N.4) (1987) 499-505 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 2, May 1995, pp. 123-134. Printed in India. Badly approximable />-adic integers A G ABERCROMBIE Department of Pure Mathematics, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, UK MS received 7 July 1994 Abstract. It is known that the p-adic integers that are badly approximable by rationals form a null set with respect to Haar measure. We define a [0, l]-valued dimension function on the p-adic integers analogous to Hausdorff dimension in R and show that with respect to this function the dimension of the set of badly approximable p-adic integers isl. Keywords. Diophantine approximation; p-adic numbers; Hausdorff dimension. Introduction A real number x is called badly approximable if, roughly speaking, there are no rationals p/q such that x p/q is small compared with q~ 2 .lt is well known (see [5]) that the set of badly approximable real numbers has Lebesgue measure zero and Hausdorff dimension 1. As might be expected, we can in an analogous way define the set of badly approximable p-adic integers. It is known (see [6]) that this set is a null set with respect to Haar measure on the group Z p of all p-adic integers. In this paper we describe a natural analog of Hausdorff dimension applicable to the space of p-adic integers and we show that with respect to this dimension the dimension of the set of badly approximable p-adic integers is 1. The proof of this result makes use of an approximation scheme for p-adic numbers developed by Mahler in [7], the essential features of which are recalled in the course of 3 below. We also exploit a method initiated by Billingsley in [2], and further developed by the author in [1], for comparing Hausdorff-like dimension functions defined with respect to arbitrary non-atomic measures. The basic facts about this method are explained in 4. With the aid of Mahler's scheme we construct a measure with respect to which the set of badly approximable numbers has measure 1. We then apply Billingsley's method to complete the proof. 1. Notation and preliminary remarks We denote by N the set of strictly positive integers ajid write N = Nu{0}. For a natural number N we denote by [AT] the set 124 A G Abercrombie If z is a complex number we shall always write x = Re(z), y = Im(z). For any real j; we denote by U >10 the set We denote by T the modular group SL 2 (Z), and by I the identity of T. As usual, we let F act on the upper half-plane U in the following way. For in F and z in U we put az We denote by R the standard fundamental region for this action of T given by | R = RuR 2 where and It is easy to check that for any f in R the expression is a positive definite quadratic form in r and s. We may therefore define a positive- valued function $,* oji R x R by setting For a fixed prime p, we denote by Z p the ring of p-adic integers with the usual valuation | | p . Thus a typical element p of Z p is a sequence (pJ ne N , where each p n is an element of the additive group Z/p n Z, and for each n the natural homomorphism Z/p"+ 1 Z -+ Z/p"Z sends p n+ 1 to p n . Given p = (p n ) neNo , p' = (p;) neNo in Z p we define and We define |p| p = p v where v = v(p) is the least integer in N such that p v+l is different from zero. We say that p, p' are congruent modulo p*, and write p = p' (modpO, $ We equip Z p with the topology induced by the metric d(p, p' ) = IP - P V The space Z p is homeomorphic to the topological product [p]<, where [p] is equipped with the discrete topology. Therefore Z p is compact. Badly approximate p-adic integers 125 e put set 53 is a basis for Z p consisting of closed open sets. An element of 93 will be called kere. The reader will observe that the sphere B h (p) is the set of all p' in Z p with p h . In sections 4-5 below we shall persistently abuse notation by writing p h in 50fB h (p). ;t (Z p , ^, $ be a probability space on Z p , where ^ is the <r-algebra generated by 23. ^ be any probability measure on Z p that is non-atomic, i.e. /^({p}) = for all p . Suppose y > 0. For 9 > and M c Z p , write : the infimum is taken over all coverings of M by subsets of S of the form ;p (0 ):ieN} such that n(B h .(p (i} )) < d for all ieN. The (not necessarily finite) limit s for all M. For a simple proof the / as thus defined is an outer measure see [2], 36, 141. It can be shown ([2], pp. 136-137, 141) that for each M c Z p there exists a ue real number A = A M (M) such that / = oo for all y < A and ^ = for all y > A. e define ^:23(Z ;7 )->R by ?/(^(a)) = p"' 1 for all aeZ p , /ieN . Then by the itheodory-Hopf extension theorem ([4], 13, Theorem A), rj can be extended to lability measure on Z p , also denoted by rj. The measure r\ is clearly translation- riant and therefore by the Haar uniqueness theorem ([3], pp. 309-310) it coincides Haar measure on Z p . We call \(M) the Hausdorff dimension of M. This imology is appropriate because, as is proved in [2], p. 140, Hausdorff dimension can be defined by the same procedure with Lebesgue measure in place of Y\. tatement of the result each positive real number i let us say that a p-adic integer p is badly approximate nd write peJ(r) if for all a, b in Z we have us say that p is badly approximate if it is badly approximate (t) for some T > 0. denote the set of badly approximate p-adic integers by J. Thus J= U 'to- T>0 is well known (see for example [6], Th. 4.23) that r\(J) = 0. Thus it is of interest to rmine A^(J). Our purpose in this paper is to prove the following: oremll. We have A(J) = 1. 126 A G Abercrombie We first recast this result in a more convenient form. For in R and t > let denote the set of p such that for all a, b in Z, and write J,= |J J (T). T>0 Since (^) 2 is positive definite, a simple computation shows that J is identical with J^ for each Therefore Theorem 2.1 is a consequence of the following result, which, though more detailed than Theorem 2.1, appears to be no harder to prove. Theorem 2.2. There is a constant C depending only on p such that for any % in R and any K in N we have \(j f ( P - K - c ))>i-~. Theorem 2.2 is analogous to the following result on Dio'phantine approximation in R. Call r in R badly approximate if there is a constant T such that \a + br\^tb~ l for all a, b in Z. Then we have: Theorem 23. The Hausdorff dimension of the set of badly approximable real numbers is 1. This was established by V Jarnik in [5], a pioneering paper in which dimension theory was applied for the first time in the study of Diophantine approximation. The proof of Theorem 2.3 depends on a special feature of R, namely the availability of an appropriate continued fraction algorithm. It turns out that badly approximable real numbers are those whose simple continued fractions have bounded partial denominators. To prove Theorem 2.2 we shall use an approximation scheme for p-adic integers developed by K Mahler in [7]. As Mahler points out, his scheme is a working substitute for a continued fraction algorithm in the sense that it yields all "good" approximations to a p-adic integer p, that is all potential counterexamples to (2.1). Lemma 3.2 below is a more precise statement of this fact. As we shall see, the badly approximable p-adic integers can be nicely characterized in the language of Mahler's scheme. In the early days of research on Hausdorff dimension it was notoriously difficult to find sharp lower bounds for the dimension of sets like J(i). It is now, in many cases, much easier, thanks to a method developed by P Billingsley which we review briefly in section 4 before applying it to the present problem. 3. Mahler's approximation scheme Given a p-adic integer p we define, for each n in N , an integer E n = E n (p) by means of the relations Badly approximate p-adic integers 127 \En~P\p** P n 3 easy to check that exactly one integer E n satisfies these two relations. 7 i\ in R, and for each n in N define a complex number Z n = Z n (p) by setting rther for each n in N let z n = z n (p) = x n + ty n = x n (p) -f r> n (p) the unique element of R that is equivalent to Z n under the action of F on U . luppose that tie element of F satisfying I write o for each n in N write i for each n in N write Q T-l T ** - 1 -l *n' > can now state the fundamental results due to Mahler on which our proof of ^orem 2.2 will be based. nma 3.1. ([7], p. 12). For any p in Z p and any n in N we have nma 3.2. ([7], p. 51 (Theorem 18)). Let a,binZ satisfy en 128 A G Abercrombie Lemma 3.3. ([7], p. 15). The subset M(p) of GL 2 (Z p ) defined by is a finite set and the determinant of each element of M(p) is p. Moreover a matrix Q is in M(p) if and only if pQ~ 1 is in M(p). Note. It turns out that M(p) is independent of the choice of . However we do not require this fact. Lemma 3.4. ([7], p. 14). For each n in N we have Lemma 3.5. ([7], p. 14). For each n in N the integers a n and b n are relatively prime. We now derive some consequences of the preceding lemmas. Lemma 3.6. The set of badly approximate p-adic integers coincides with the set of | those p such that y n (p) remains bounded as n goes to infinity. More precisely, for each T > we have Proof. It suffices to prove the second statement. Suppose p is in J 5 (r). Then by the definition of J 5 (r) we have for each n in N that \^n + b n p\ p ^x(^(a n9 b n )r 2 . (3.1) By the definition of a n , b n we have P~ n ^\a n + b n p\ p . (3.2) By Lemma 3.1 we have (s(a n ^ n )r 2 = y n (p)p~ n . (3.3) Combining (3.1), (3.2) and (33) we have which proves that J^t) is included in the set of those p such that y n (p) never exceeds To prove the reverse inclusion, suppose that p satisfies y n (p) ^ i~ 1 for all n in N . Let a, b be any integers and define h = h(a, b) by the relation Then using Lemmas 3.1 and 3.2 we have > A)) 2 Badly approximate p-adic integers 129 t p is in J s *(r) as required. 2 3.7. There exists a constant C depending only on p such that for any p in Z p , ver y n (p) > p c we have y n+i=p iy n (3.4) n ^ 1 we have y n _ v = p -^y n . (3.5) By Lemmas 3.3 and 3.4 we have for any p in Z^ and any n in N that Q is in M(p). We write 'a a' a fixed complex number z, the subset Y(z) of [0, 2n) defined by = {|argQz|:QGAf(p)} :e, by Lemma 3.3. Now suppose z is in R. We see easily that when y is large h we have |arg^z| < \ for all Q satisfying a/? ^ 0. Moreover if a = we see that whenever y is sufficiently large. Thus there is a constant C such that for z in R, , and Q in M (p) we have flz in R only when /J = 0. then, since by Lemma 3.3 we have det 1 = p, we find that either a = p or p = p. establishes (3.4), and if n ^ 1 the same argument with z n _ x in place of z n shes (3.5). 2 3.8. Suppose that for some, n in N we have y n (p)>y n+ i(p)>p c > C is the same constant as in Lemma 3.7. Then we have In view of Lemma 3.7 we need only show that y n + 2 ^ Py n + i We know, using ia 3.7, that y n = py n+l . Therefore by Lemma 3.4 we can write iven z n+1 in R there is just one choice of a' such that z n = Q M+1 z n+1 is in R. fore the relation y n+2 = py n+ x would imply 130 A G Abercrombie so that and then each component of T n+2 would be divisible by p, which contradicts Lemma 3.5. We conclude that y n+2 *py n+1 as claimed. 4. Billingsley's lower bound for the dimension of a set A version of the following lemma, relating to subsets of [0, 1], is proved in [2], pp. 144-145, and the proof carries over to the present setting without significant alteration. Recall that we agreed to abuse notation by writing p j in place of Bj(p). Lemma 4.1. For any non-atomic Bar el measures A, fj, on Z p and for any S^OJf Note: if either of the real numbers a, b is either or 1, then log a/log b is defined equal to 0, 1 or oo according as a > b, a = b or a < b. The logarithms can be taken to any positive base except 1, and in what follows we shall take all logarithms to the base p. In order to apply Lemma 4.1 to the problem at hand we need to construct a measure v on Z p such that and such that The construction of such a measure is made possible by the following result, which is a special case of Lemma 5.2 in [1]. If is any sphere, we denote by a(u) the set of maximal proper subspheres of u. Lemma 4.2. Suppose that u':<B\{Z p } -> [0, 1] satisfies y '() _ i for all u in 93. Then there is a unique Borel probability measure ji on Z p f*(u)n'(v) = fi( v ) I for all u, v in 93 with v in a(u). Badly approximable p-adic integers ' 131 5. Proof of Theorem 2.2 Let K be a fixed integer greater than 0, and let C be the constant whose existence is guaranteed by Lemma 3.7. For p in Z p write One checks easily that y n (p') is actually determined by p' n . We show that if then #(*n(p))>l. (5.1) Suppose the contrary. Then for every maximal subsphere p' n of p n _ 1 we have logy n (p')>K + C. But there are at least two maximal subspheres p' n contained in p n (in fact there are p of them) and therefore there are at least two (I in M(p) satisfying As in the proof of Lemma 3.8 any Q, satisfying must be of the form and there is just one choice of a'eZ such that Qz n _ x (p) is in R. Thus we have arrived at a contradiction and must conclude that (5.1) holds as claimed. We may therefore define a function v' = v' K on 93\{Z p } with values in [0,1] as follows. Case(i). If K-l + C and log y n (p) > K + C, we set Case(ii). If K-l and log y n (p) ^ K -h C, we set 132 A G Abercrombie Case (iii). If log y n _ x (p) lies in the complement of the interval (K - 1 -h C, K + C], we set One checks easily (using the definition of t n (p)) that v' satisfies the hypotheses of Lemma 4.2, and so there is a probability measure v on Z p satisfying (4.1). To check that v is non-atomic, choose p in Z p , so that P=C\Pn- rieNo We must show that v(p n ) goes to as n goes to oo. By (4.1) and straightforward induction we have V(P)= n v(pj. Kj^n Now by Lemma 3.7 we cannot have both and Hence for infinitely many n case (ii) of the definition of v' does not apply and for such n we have Therefore v(p n ) -0 as required, so v is non-atomic. We now verify that v(J(p~*- c )) = 1. If p is in the complement of J(p' K " c ) then by Lemma 3.6 for some n in N we have logy n (p) > K + C. Choose N to be the least integer with this property. Then by Lemma 3.7 we have v Therefore case (i) of the definition of v' gives v'(p N ) = 0, so also v(p N ) = 0. Thus the complement of J(p" x ~ c ) is covered by elements of 93 each of which has measure zero with respect to v. Since is countable we have v(J(p" K ~ c )) = 1 as claimed. Our next objective is to show that for all p in J(p~ K ~ c ) we have l-L (5 .2) 2K' ( } For p in J(p~ K ~ c ) let H = H(p) be the subset of N consisting of those n for which Let p be in J (p " K ~ c ), and choose n in H (p). By the choice of p we have log y n ^ K + C, and also by Lemma 3.6 we have log y n + x ^ K + C. Thus by Lemma 3.7 and the fact that K 1 4- C < logy n we have Badly approximable p-adic integers 133 2 we then have log y n+ 1 > C, and since log >> > logy n+ 1 Lemma 3.8 implies that n + 2 = - 2 + logy,, > K - 3 + C ming in this way we find that h h = 0, . . . , K. A further application of Lemma 3.7 shows that s the difference between consecutive elements of H(p) is at least IK, so we JV-l-^ (5.3) ZA if p is in J(p~ K ~ c \ we have for each n in N either AMnN . is in ff (p\ or ise. Therefore for each N in N we have e clearly have log?/(p N )= -AT, using (5.3) and the fact that t n (p) ^ 1 we have logv(pj 11 N 2X* ore letting N go to oo we have (5.2). e v(J(p~ K ~ c )) = 1, we certainly have >w an appeal to Lemma 4.1 with A = v, jn = rj, and <5 = 1 - (1/2-K) completes the 134 A G Abercrombie Acknowledgement The author wishes to thank the SERC for financial support. References [1] Abercrombie A G, The Hausdorff dimension of some exceptional sets of p-adic integer matrices, J. Sumber Theory (to appear) [2] Biilingsley P, Ergodic theory and information (1965) (New York: John Wiley) [3] Cohn D L, Measure theory (1980) (Boston: Birkhauser) [4] Halmos P R, Measure theory (1950) (New York: Van Nostrand) [5] Jarnik V, Zur metrischen Theorie der Diophantischen Approximationen, Prace mat.-fiz. 36 (1928/9) 505-543 [6] Lutz E, Sur les Approximations Diophantiennes lineaires p-adiques, Actualites Sci. Ind. 1224 (1955) (Paris: Hermann) [7] Mahler K, On a geometrical representation of p-adic numbers, Ann. Math. 41 (1940) 8-56 ndian Acad. Sci. (Math. Sci.), v o*- 105, No. 2, May 1995, pp. 135-151. ited in India. jrtainty principles on certain Lie groups A SITARAM, M SUNDARI and S THANGAVELU Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore 560 059, India MS received 22 April 1994; revised 26 September 1994 Abstract. There are several ways of formulating the uncertainty principle for the Fourier transform on R*. Roughly speaking, the uncertainty principle says that if a function / is 'concentrated' then its Fourier transform / cannot be 'concentrated' unless / is identically zero. Of course, in the above, we should be precise about. what we mean by 'concentration'. There are several ways of measuring 'concentration' and depending on the definition we get a host of uncertainty principles. As several authors have shown, some of these uncertainty principles seeni to be a general feature of harmonic analysis on connected locally compact groups. In this paper, we show how various uncertainty principles take form in the case of some locally compact groups including (FT, the Heisenberg group, the reduced Heisenberg group and the Euclidean motion group of the plane. Keywords. Fourier transform; Heisenberg group; motion group; uncertainty principle. reduction i are several ways of formulating the uncertainty principle for the Fourier brm on R w . Roughly speaking, the uncertainty principle says that if a function 'concentrated' then its Fourier transform / cannot be concentrated unless lentically zero. Of course, in the above, we should be precise about what we by 'concentration'. There are several ways of measuring 'concentration' and iding on the definition we get a host of uncertainty principles. As has been i in [1], [2], [4], [9], [12], [13], [17] etc, some of these uncertainty principles to be a general feature of harmonic analysis on connected locally compact >s. We continue these investigations in this paper to see how various uncertainty iples take form in the case of some locally compact groups including R", the nberg group, the reduced Heisenberg group and the Euclidean motion group \ plane. In a forthcoming paper [14] we consider semi-simple Lie groups and [nore general eigenfunction expansions on a manifold with respect to some c operator. e way of measuring concentration is by considering the decay of the function at ty. In this context, a theorem of Hardy for the Fourier transform on R says the /ing: *em 1. (Hardy) Suppose f is a measurable function on R such that (1.1) 135 136 A Sitaram et a! where a, /? are positive constants. If a/? > r/ien / = a.e. //' a/? < r/iere are infinitely many linearly independent functions satisfying (1.1) and if %[} = then f(x) = O~ a * 2 . For a proof of the above theorem see [3]. A more general theorem due to Beurling, from which Hardy's theorem can be deduced, can be found in [10]. In this paper we establish an analogue of the above theorem for the Heisenberg group 3? n (see 2 for the precise formulation). We also prove Hardy's theorem in the case of R n , n ^ 2 and show that though the exact analogue for the reduced Heisenberg group fails, a slightly modified version continues to hold. In the final section we prove an analogue of Hardy's theorem for the Euclidean motion group of the plane. Another natural way of measuring 'concentration' is in terms of the supports of the function / and its Fourier transform /. If / is non-trivial and compactly supported then / extends to an entire function, and so / cannot have compact support. A non-trivial extension of this result due to Benedicks [1] says: If /eL^R") is such that m{x:f(x) ^ 0} < oo and m {:/() ^ 0} < oo then / = a.e. Here m stands for the Lebesgue measure on R B . This result of Benedicks has been extended in [2], [12], [4] etc. to a wide variety of locally compact groups. In particular, one has the following result for the Heisenberg group: Theorem 2. (Price-Sitaram) Let /eL 1 nL 2 (Jf ). Suppose that m{tEU:f(z, t) ^ 0} < oo for a.e. zeC" and m{AeR*: / (A) ^ 0} < oo. Then f = a.e. In the above /(A) stands for the group Fourier transform on & n and R* means M\{0}. Roughly speaking, the above theorem says that if /eL 2 pfJ is concentrated in the t direction then /(A) cannot be concentrated. It is the concentration in the t direction, not that in the z direction, which forces the spreading out of the Fourier transform. In fact, as was shown by Thangavelu in [17], we can have L 2 functions with compact support in the z variable for which / is also compactly supported. The special role played by the t variable in the above theorem (as well as in our Hardy's theorem in 2) should not come as a surprise. The Fourier transform on jf is more or less the Euclidean Fourier transform as far as the t variable is concerned. If one goes through the proof of the above theorem, one observes that it is a con- sequence of the corresponding theorem for the Euclidean Fourier transform in the t variable. In view of the preceding remarks one would like to have an analogue of the above theorem which respects the z variable. We formulate and prove such a theorem in 3. We will show that when / has compact support in the z variable then /(A) (as an operator) cannot have 'compact support'. We will give a precise meaning to this statement in 3. We now turn our attention towards quantitative versions of the uncertainty principle, namely uncertainty inequalities. The classical Heisenberg-Pauli-Weyl uncertainty inequality for the Fourier transform on R B says that H/!l2^C n (J|x| 2 |/(x)| 2 dx)(J|^| 2 |7(ai 2 da. (1-2) For a proof of (1.2) with the precise value of C n we refer to [6]. A version of the above inequality for the Heisenberg group was established by Thangavelu in [17]. Here we are concerned with local versions of the above inequality for the Heisenberg group. Uncertainty principles on certain Lie groups 137 For the Fourier transform on R n one has the following local uncertainty inequality: For any measurable E c R n , and < 6 < |, C e m(E) 2d J/(x)| 2 |x| 2 "'dx. (1.3) An analogue of the above inequality is known on the Heisenberg group. The following result is proved in [13]. Theorem 3. (Price-Sitaram) Let 0e[0,|). Then, for each feL^L 2 (3f n ) and measurable E c R*, one has r (1.4) (In the above tr stands for the canonical semifinite trace and dju is the Plancherel measure on $t n see 2.) Again we observe that the t variable plays a special role. As in the case of the Euclidean Fourier transform one would like to have an inequality which is more symmetric in all the variables. In 4 we formulate and prove a local uncertainty inequality with the right hand side being (1.5) where |w| 4 = |z| 4 + t 2 and Q = (2n + 2) is the homogeneous dimension of 3tf n . From the local uncertainty inequality we will also deduce a global inequality similar to the classical Heisenberg-Pauli-Weyl uncertainty inequality. Finally, for various facts about the Heisenberg group we refer to the monographs of Folland [6] and Thangavelu [19]. We closely follow the notations of the latter which differ from the former by a factor of 2n. 2. Analogues of Hardy's theorem for R" and Jf Before we prove Hardy's theorem for the Heisenberg group, consider the case of R", n > 2. The proof of Hardy's theorem (for n = 1) depends heavily on complex analysis. As we have not found a reference in the literature for the higher dimensional case of Hardy's theorem we take this opportunity to present a proof which follows easily from the one-dimensional case via the Radon transform. Theorem 4. Let f be a measurable function on U n and a,/? two positive constants. Further assume that \f(x)\ ^ Ce~^\ |/({)| ^ Ce~\ x, { 6 R. (2.1) // #/?>;, then / = a.e. If a/?<, there are infinitely many linearly independent solutions for (2.1) and if a/? = , / is a constant multiple of e~ a|x|2 . Proof. As mentioned above, we will use theorem 1 . So, assume that n ^ 2. We use the Radon transform to reduce the problem to the one-dimensional case. Recall that the 138 ASitarametal Radon transform Rg of an integrable function g on R" is a function of two variables (co, s) where coeS n ~ 1 and se(R and is given by I g(x)dx. (2.2) [ ) = S j where dx is the Euclidean measure on the hyperplane x.a> = s. Actually, for each fixed I co, the above makes sense for almost all selR which may depend on co. However for \ functions with sufficient rapid decay at infinity it makes sense for all 5. For various properties of the Radon transform we refer to [5] and [8]. Our definition of the Fourier transform of a function / on R n is : (2.3) Then it can be easily seen that where seSR, coeS M ~~ 1 and (Rf)~ stands for the Fourier transform of Rf in the s-variable alone. From the definition of the Radon transform Rf and the relation (2.4), the conditions on / and / translate into conditions on Rf and OR/)~. For each fixed co, we therefore get \Rf((D,r)\^Ce~* r \ reU (2.5) (2.6) By appealing to Hardy's theorem for R we conclude that for a/? > , Rf(co, .) = 0, for almost all CD. In view, of the inversion theorem for the Radon transform this implies/ = a.e. When a/J = , (K/T(a>, s) = f(sa>) = A(^)e~^ where A is a measurable function on the unit sphere S n ~ l . Because /eL 1 (U n \ f is continuous at zero and by taking s -* we \ obtain A(a) = /(O). Hence /(<*) = f(G)e~^ 2 so that /(x) = Ce~ a|x|2 for some constant C. f If aj? < |, the n-dimensional suitably scaled Hermite functions >^ satisfy (2.1). We now consider the case of the Heisenberg group tf n = C" x R. The multiplication law of the group 3f n is given by ? (z 9 t)(w, 5) = (z + w, t + 5 + ^lm(z.w)), (2.7) where z,weC n , t,s6R. Then ^f rt becomes a step-two nilpotent Lie group with Haar measure dzdt. In order to define the group Fourier transform we need to recall some f facts about the representations of the Heisenberg group. For each AeR*, there is an I irreducible unitary representation TC A of jff n realised on L 2 (IR n ) and is given by ] MZ, t)0)({) = e^e^+^W + y\ (2.8) where z = x -f 13; and </>eL 2 (lR rl ). A theorem of Stone- von Neumann says that all the infinite dimensional irreducible unitary representations of ^ are given by TI A , AeIR*, | (up to unitary equivalence). The Plancherel measure dfi = |A|"dA is supported on R*. \ (There is another family of one-dimensional representations of 3tf. n which do not play , a role in the Plancherel theorem.) Uncertainty principles on certain Lie groups 139 Given a function /, say in L l (3^ n \ its group Fourier transform / is defined to be the operator valued function -I. /(A)= /(z,t)7t i (z,t)dzdt. (2.9) J Jt'n (The above integral being interpreted suitably). For each AeR*, /(A) is a bounded operator on L 2 (R W ). A simple calculation shows that /(A) is an integral operator with kernel }(, 77) given by (2.10) where we have written /(z, t) = /(x, y, t) and ^ 13 / stands for the Fourier transform of / in the first and the third set of variables. For / in L 1 nL 2 (Jf n ) a simple calculation shows that |^ 3 /(z,A)| 2 dz, (2.11) (for a suitable constant C) where \\-\\ HS is the Hilbert-Schmidt norm. From this and the Euclidean- Plancherel theorem, the Plancherel theorem for the Heisenberg group follows: 11/111 = CJ ||/W||J s dAi(A), (2.12) JR* where d^u(A) = |A| n dA and C n is a constant depending only on the dimension. We now state and prove the following analogue of Hardy's theorem for j^ n . Theorem 5. Suppose f is a measurable function on 3tf n satisfying the estimates \f(z,t)\^g(z)e-^\ zeC", reR, ^ (2.13) \\fW\\Hs <*-**> te*> ' ( 2 - 14 ) where geL 1 n L 2 (C M ) and a, /? are positive constants. Then, if a/? > , / = a.e.; z/ a/? < J are infinitely many linearly independent functions satisfying the above estimates. Proof. For a function / on Jtf n define /* to be the function /*(z, r) = /(z, ~ t) and let /*3/* stand for the convolution of / and /* in the t- variable. Then, a simple calculation shows that f -1 f (/* 3 /*)(z,ry'dzdt = . J Jfn J C -f.i Jc n (2.15) 140 A Sitaram et al which, in view of (2.11), equals C~^n/(A)|| 2 s . Define a function ft on R by = (f**f*)(z,t)dz. c" Then one has Now the conditions (2.13) and (2.14) on / and / translate into the conditions (2.18) where /?' can be chosen so that a/?' > J or < according as a/? > or < |. If a/J > i then a/?' > J, so that Hardy's theorem for R implies that A = a.e. This means || f(X) \\ 2 HS = for all AeK* and consequently / = a.e. by the Plancherel theorem for 2tf n . If aj? <i then any function of the form g(z)h k (t) where h k is a suitably scaled Hermite function satisfies the hypothesis of the theorem. The following is the exact analogue of Hardy's theorem for 3tf n . COROLLARY 6 Suppose f is a measurable L 1 -function on J^ n and i/(z,t)|<Ce- a(! * |2+ltl2) , zeC", teR (2.19) \\fW\\Hs <Ce-*\ AeR* (2.20) for some positive constants a and /?. If a/? > J, then f = a.e. // ajJ < |, t^en r/i^r^ are infinitely many such linearly independent functions. We shall now consider the case of the reduced Heisenberg group <#^ ed = C" x S . The multiplication law is as in (2.7) except for the understanding that t is a real number modulo 1. The reduced Heisenberg group J^ T n Qd is also a step two nilpotent Lie group with Haar measure dzdt where dt denotes the normalized Lebesgue measure on S 1 . For each meZ* = Z\{0}> there is an irreducible unitary representation 7c m of Jf r n ed realized on L 2 (R n ) and is defined exactly as in (2.8). As in the case of <#* , we get (up to unitary equivalence) that all the infinite dimensional irreducible unitary representations of jf^* are given by 7r m , meZ*. Apart from this there is a class of one dimensional representations, n 0tb9 a, beR n given by n afb (z,t) = e 2 * i(ax+by) for (z,t)e d . (2.21) The dual & T * d can be thought of as the disjoint union of Z* and R 2n . The Plancherel measure is the counting measure on Z* with a weight function C\m\ n (for a suitable constant C) and the Lebesgue measure on R 2 ". (This is in sharp contrast to the case of Heisenberg group.) Given / in L 1 (tf * ed ), we can write /(z,t)= V k (z)e ik < (2.22)- Uncertainty principles on certain Lie groups 141 as a Fourier series in the central variable t. (Here / can be thought of as the LMimit of the Cesaro means of the right hand side of (2.22).) Hence, as in the case of 3 n , if we compute the group Fourier transform /(m), meZ* we see that it is an integral operator with kernel K(, Y\) given by 4 (2.23) where ^ ^ _ m stands for the Fourier transform of x F_ m in the first set of variables. Therefore, for /eL 1 nL 2 (^ ed ), a simple calculation shows that ||/(m)||| s = |m -"Il^-J^c-,, meZ*. (2.24) Remark 7. We will now show by an example that the exact analogue of Hardy's theorem on J^J ed is not valid. Since t varies over a compact set in this case, one might be tempted to consider the following analogue of Hardy's theorem: Suppose / is a measurable L 1 -function on Jf^ ed and / satisfies the following estimates: \f(z,t)\^Ce~^\ \\f(m)\\ HS ^Ce-^\ zeC",meZ*, . (2.25) for positive constants a, ft. Then if ajS > , / = a.e. However, the following demonstrates that this is 'not the case. Observe that as / satisfies (2.25), / belongs to L 1 nL 2 (^ ed ) and the series in (2.22) converges to / in L 2 -sense. Now take /(z, t) = e"~ aN V kr , for some /ceZ*. Using (2.24) one can see that / is a non-trivial function satisfying the conditions (2.25). However the following, which can be viewed as a "sort of" uncertainty principle still holds: Suppose / is a measurable L^function on Jf ed satisfying |/(z,r)|<a(z)/f(t), zeCVeS 1 (2.26) (2.27) where a is any function with reasonably rapid decay at infinity, /? is any function that vanishes to infinite order at some point tQ^S 1 and y is a positive constant. Then / = a.e. . Remark 8. Since S 1 is compact the point r can be "viewed" as the point at infinity and therefore condition (2.26) can be thought of as the analogue of the decay of the function at infinity. 3. An uncertainty principle for the Heisenberg group In this section we formulate and prove an uncertainty principle for the Fourier transform on the Heisenberg group. In the uncertainty principle stated in theorem 2 as well as in the analogue of Hardy's theorem the Fourier transform has been considered as a function of the continuous parameter L The properties of the given function / as a function of the t variable are reflected in /(A) as a function of L But if we want to 142 A Sitaram et al investigate how the properties of / as a function of z are affecting /(A) one has to view the Fourier transform as a function of two parameters, one continuous and the other discrete. ^ To justify the above claim let us write down the formula for /(A) when / is a radial function. In what follows, by a radial function we mean a function which is radial in the z variable. In order to state the formula we need to introduce some more notation. For each multi index aefoT let <D a (x) stand for the normalized Hermite functions on W. For AeR* we let <&(x) = |A| n/4 O a (W 1/2 *) and define P k (X) to be the projection of L 2 (U n ) onto the eigenspace -spanned by {:|a| = /c}. By <p(r) we denote the scaled Laguerre function ^W^Lr^iWr 2 )^* 1 ' 4 ^, (3-1) L\~ 1 (t) being the fcth Laguerre polynomial of type (n - 1). Now let /(z, t) be a radial function and write /(r, t) in place of /(z, t) when |z| = r. Then we have the following formula for the Fourier transform of /: (3-2) where the coefficients jR fe (A,/) are given by KW-Mr. . (3.3) -\-n In the above /(r, 1) stands for the Fourier transform of /(r, t) in the t- variable and C n \ is a constant. From the above formula it follows that we can identify /(A) with the sequence of functions {JR k (A,/)}. The support properties of / as a function of t are reflected on the properties of R k (hf) as a function of A. Likewise, one expects that the z support of / will influence the properties of R k (A,/) as a function of k. We will show { that this is indeed the case. \ More generally we consider the Fourier transform /(A) as a family of linear ; functional F(A,a) on L 2 (IR n ) indexed by (A,a)eR* x M". For each (A, a) the linear j functional F(A, a) is given by | F(A,a)cp = ((p,/(A)cD^ ^> 6 L 2 (R-). (3.4) 1 With the above notations the uncertainty principle stated in theorem 2 can be restated r as follows. If m {t:/(z, t) 0} < oo for a.e. z and m{A:F(A, a) ^ 0} < oo then / = 0. Now | to state our uncertainty principle let [ (3.5) and B(A)={a:F(A,a)^0}. (3.6) Then we have the following result. Theorem 9. Suppose /eL 1 nL 2 (jf J is swcfe tto m(yl(A)) < oo and B(X) is finite for a.e. Uncertainty principles on certain Lie groups 143 Before going into the proof of the theorem we make the following remarks concerning the statement of the theorem. If there exists a compact set K c C" such that /(z, f) = whenever z$K and re(R then it follows that A (A) is compact for each /I and hence m(A(%))< oo is satisfied. The condition B(A) is finite simply means that /(1)<I>^0 only for finitely many a and consequently there is a k = k(X) such that/(A)P ; -(A) = for all j>/c. Let Sj* be the span of {<D*:|a =k}. Then it has been observed by Geller in [7] that S k are the analogues of the spheres \x\ = r in R n . In other words we can think of S as a sphere in L 2 (U n ) of radius (2k + n)|A|. This view has turned out to be fruitful in other problems also as can be seen from [18]. Thus we can let B k to be the span of {3>:|a| ^ ^} which is the analogue of a ball in R n and the condition f(X)Pj(X) = for j>k simply means that /(1) = in the orthogonal complement of B k in L 2 (tR"). Let us say that /(/I) has compact support in B k when the above holds. With this definition we can restate the above theorem in the following form. Theorem 10. Let /6L 1 nL 2 (Jf n ). Suppose for each X the Fourier transform f(X) is compactly supported. Then /(., X) cannot have compact support for each A unless /=o. We now come to the proof of theorem 9. We need to use some facts about the special Hermite expansions for which we refer the reader to [19]. If /eL 2 (C") then we have the expansion f = (2n)-" Z/x^v (3.7) In the above <^(z) = LpHiM 2 )e~ (1/4)|z|2 and / x (p k stands for the twisted convolution (/ x <p h )(z) = f f(z - w)<p>)^ 2 ><^>dw. (3.8) Jc n The functions <p k are eigenfunctions of the operator (3.9) with eigenvalues (2k + n) and /-/x <p k is the projection of L 2 (C W ) onto the /c-th eigenspace of the operator L. We also have for any m L m (f x q> k ) = (2k + n) m f x (p k (3.10) and in view of the orthogonality the relation \\L m f\\ 2 2 = (2nr 2n (2k-+n) 2M \\fx<p k \\ 2 2 . (3.11) fc = We need the following proposition in order to prove theorems 9 and 10. 144 A Sitaram et al PROPOSITION 11 Suppose /GL 2 (C") is such that \\f x cp k \\ 2 ^ Ce~* (2k+n) for some a >0. Then f is real analytic. Proof. By the Sobolev's embedding theorem it is easy to see that / is in C(C n ). We want to apply an elliptic regularity theorem of Kotake-Narasimhan to prove the proposition (see [1 1], theorem 3.8.9). In view of their theorem it suffices to show that for any positive integer m ||L m /|| 2 <M m+1 (2w)! (3.12) holds with some constant M. Under the assumption on /, the relation (3.1 1) gives n I (2k + n) 2m e~ 2 * (2k+n) . (3.13) * = o The series can be estimated by which gives the estimate ||L m /|| 2 ^C 2m+1 (2m)l (3.15) which is more than what we need. Now we can give proofs of theorems 9 and 10. Define a radial function Gj(z, t) by i(z)\l\ n dL (3.16) : it follows from (3.2) that \ G/AHC,,*-' 1 ^ (3.17) ; C n is some constant which we do not bother to calculate. Setting #,. = /* G ; and !- taking the (group) Fourier transform we get \ djW =/(A)Gj(A) = C n e-f(X)P.()i\ (3.18) Now fix L Then under the hypothesis of the theorem we have g^X) = for ; > k which in view of (2. 1 1 ) means that for a.e. z in C"gj(z) = for j > k where we have set gj(z) to stand for g.(z> A) the Fourier transform of g. in the t- variable. Recalling the definition of the convolution g j = f* G j on tf n and taking the Fourier transform in the f-variable we get with the same notation as above where the A-twisted convolution is given by f**,Gj(z)=\ f^z-^GjMe'W^dv. (3.20) Uncertainty principles on certain Lie groups 145 Let /j[(z) =f\2~ 1 \l\' (i/2) z). Then it follows from the definition of G j that /Jx^O(4 (3-21) Under the hypothesis of either of the theorems we have (f\ x <pj)(z) = for j > k. This means that f\ satisfies the conditions of proposition 1 1 and consequently /(z, /) is real analytic for a.e. X as a function of (x 9 y). But then the set {z:/(z,A) ^0} cannot have finite measure unless /(z, A) = for a.e. z. This implies / = and hence theorem 9 follows. It is clear that the hypothesis of theorem 10 implies that of theorem 9. Hence both theorems are proved. 4. Some uncertainty inequalities for the Heisenberg group In this section we establish a local uncertainty inequality for the Fourier transform on Jf B and deduce a global inequality too. As we have remarked in the previoussection we consider the Fourier transform f(X) as a family of linear functional F(/i, a) indexed by (A,a)eR* x M". From the definition of f(A,a) it follows that tr(/w*/a))=z ii/w<i>iii! = z iifou)ii 2 , (4.1) a a where ||F(A,a)|| is the norm of the linear functional F(A.,ct). In this notation the uncertainty inequality of theorem 3 can be written as [ ||F(A,a)|| 2 d/i(A)<C e m(^) w f \f(z, J A J Jf n ,t)\ 2 \t\ 2 dzdt. (4.2) a A In the next theorem we will prove an inequality which is more symmetric in both variables. Let v be the counting measure on F^" and let a = \JL x v on R* x I\T. We now prove the following inequality. We let Q = (2n 4- 2) and |w| 4 = |z| 4 + t 2 for w = (z, r)e#V Theorem 12. Gzi?en 0e[0,), /or eacfe /eL 1 nL 2 (^f J and c R* x N" wft/i (j() < oo one has (4.3) C e depends only on 6 and Q. Proof. Let r > be a positive number to be chqsen later. We write f = g + h where 0(w) =/(w) when |w| < r and #(w) = otherwise. We then have [ JE \\m\\ n*iii 2 = (4.4) .E Since 146 A Sitaram et al where \\$(X)\\ is the operator norm of g(l) on L 2 (r) and as \\(X)\\ ^ H^ we obtain (4.6) where we have applied Cauchy-Schwarz to get the second inequality On the other hand by the Plancherel theorem = C. |/j(w)| 2 dw J J#n f = CJ l/i(w)| 2 |w|- I J^' / Jr n (4.7) Therefore, we have proved the inequality : jfn (4.8) Minimizing the right hand side by a judicious choice of r we get the inequality (49) This completes the proof of the theorem. ' CaSe K " e D W deduCe a global ^certainty inequality from the We need some more n tation Let S> be ct v HeiSenber 8 8 rou P and let W) be the Hermite operator whose spectral decomposition is given by (410) Uncertainty principles on certain Lie groups 147 For the definition of & we refer to [16] and we remark that when /= 1, H(A) = A 4- x 2 on U n . The relation between < and H(X) is given by (J?/HA) = /(A)H(A), . (4.11) for any reasonable function / on $F n . We can define any fractional power J^ v by the equation )^ (4.12) where (ff (A)) y is given by the decomposition k (4 (4.13) fc = We can now prove the following global uncertainty inequality for jf n . Theorem 13. For f in L 2 (Jf J, < y < Q/2 one has |/(w)| 2 |w| 2 Mw)( f |JSf y/2 /(w)| 2 dw ) (4.14) .tfn / \J.#n / where K is a constant. Before going into the proof of the above inequality the following remarks are in order. When y = 1 the above inequality reduces to l/(w)| 2 |w| 2 dwVf |^ 1/2 /(w)| 2 dw) (4.15) / \ J.^n / and this is the analogue of the classical uncertainty inequality for the Fourier transform on 1R". The analogy can be seen clearly if we write the inequality (1.2) in the form \f(x)\ 2 \x\ 2 dx ( |(-A)^ 2 /(jc)| 2 dx I. (4.16) / \ J / The inequality (4.15) is valid even if we replace |w| by |z| as was shown in [17] and then a precise value for K can also be obtained. Now we prove theorem 13. As in the case of the previous theorem the proof is modelled after the proof in the Euclidean case. Let E r denote the set E r ={(A,a):(2|a| + n)U|^r 2 }. (4.17) We claim that a(E r ) ^ Cr Q . To see this we first note that r= U U {A:(2|a| + n)|A|<r?}-x{a} (4.18) and therefore lk + n)\2.\^r 2 }. (4.19) 148 A Sitaram et al Since /i{A:(2fc + n)W sr 2 } < Cr Q (2/c + n)-"- 1 and S w=k l < C(2k + n)"~ l we get <r(E r ) < O e (2/c + n) - 2 $ Cr c (4.20) fc=o and this proves the claim. Let E' r stand for the complement of E r and write = C, f ||/(A)||f, s dAi(A) (4.21) JR = C. W,a)|| 2 da F r r ! Applying the local uncertainty inequality to the first integral with 9 = y/Q < j and ^ making use of the claim we obtain I * |/(w)| 2 |w| 2 Mw. (4.22) For the second integral one has the following chain of inequalities: |tdff (4.23) f ll/*illidcr<r- J > f J F J ) f jfn Thus we have obtained the inequality H/lli<CJr 2y ||/(w)| 2 |w| 2 Mw + r- 2y [|^ /2 /(w)| 2 dw|. (4.24) v. J J J Minimizing the right hand side we obtain ll/ll! W fl/! 2 M 2 MwV [|jS^/ 2 /(w)| 2 dw\ (4.25) V J / \ J / which proves the theorem. 5. The Euclidean motion group In this section we shall state and prove an analogue of Hardy's theorem for the Euclidean motion group, M(2). The group G = M(2) is the semidirect product of Uncertainty principles on certain Lie groups 149 S0(2)( ~ S 1 ) and R 2 ( ~ C). A typical element of G is denoted by (z, a) and this element acts on (R 2 as r(z)r(a) where r(z) is the translation by zeC(~[R 2 ) and r(a) is the rotation by an angle a, ^ a ^ 2n. The multiplication law is given by the composition of such maps. Haar measure on G is dzda where dz is Lebesgue measure on C(~ [R 2 ) and da is the normalized Haar measure on SO(2)(cS 1 ). For any unexplained terminology and notation in this section see [15]. For aelR 4 " =(0, oo), we have the unitary irreducible representation U a of G as operators in ^ (L 2 (S 1 ) ) defined by (U(z, a)0)(0) = e^^W - a), (5.1) where (j>eL 2 (S l \ 0^9^2n and <.,.> is the inner product on R 2 . Here one is identifying ae!R + with (0,a)eC. The Plancherel measure fj, on G is supported on this family of representations parametrized by IR + , and is given by a da, where da is Lebesgue measure on IR 4 ". The Fourier transform / of feL 1 (G) is a function on IR 4 " taking values in 1 )), and is defined by f(a) = U(f) = [ l/(z, a)/(z, a)dzda (5.2) J M(2) (the integral interpreted suitably) and therefore we have (5.3) C S0(2) The following is an analogue of Hardy's theorem for the Euclidean motion groupM(2): Theorem 14. Suppose f is a measurable function on G satisfying the following conditions for some positive constants a, /? and C: ^Ce~^\ . (z,0)eG, (5.4) \\f(a}\\ HS <Ce~^\ aeU + . (5.5) // aj? > , then f = a.e. Remark 15. Since functions on [R 2 can be thought of as functions on G invariant under right action by SO (2), Hardy's theorem for IR 2 shows that is the best possible constant. Proof. For neZ, define % n on SO(2) as % n (Q) = e in$ . It is enough to show that if ^ n */*7 m = for all n,m. This is because if / is a L^function (or more generally a distribution) and x n *f*X m ^ s zero f r a ^ w meZ, then / is itself zero. A simple calculation shows that if / satisfies (5.4) and (5.5) then for all n, m, % n *f*x m also satisfy (5.4) and (5.5). For n,meZ, define . r(0), 150 A Sitaram et al Observe that if h = X n *f*x m then h belongs to L* m (G). Therefore it is enough to prove the theorem for a function h in L* m (G). It is easy to check that if heL l nm (G) then h(a) maps x m ^L 2 (S 1 ) to a multiple of x n and is zero on the orthogonal complement of x m . In fact, Therefore Using the transformation property of h, it can be shown that for a.e. 9 and y in [0, 2n) where J^/z denotes the Euclidean Fourier transform of h in the C( ~ R 2 )- variable z. Thus from (5.5) and (5.6) it will follow that: l&M&yn^Ce-* (5.7) for eC(~ R 2 ) and a.e. y in [0,24 But h also satisfies (5.4). Using the analogue of Hardy's theorem for R 2 (~C) we conclude that h(. 9 y) = Q for a.e. y in [0,24 This implies that h a.e. References 1 [I] Benedicks M, On Fourier transforms of functions supported on sets of finite Lebesgue measure, ; J. Math. Anal Appl. 106 (1985) 180-183 [ [2] Cowling M, Price J and Sitaram A, A qualitative uncertainty principle for semi-simple Lie groups, J. [ Australian Math. Soc. A45 (1988) 127-132 ^ [3] Dym H and McKean H P, Fourier series and integrals (1-972) (New York: Academic Press) J [4] Echteroff S, Kaniuth E and Kumar A, A qualitative uncertainty principle for certain locally compact ' groups, Forum Math. 3 (1991) 355-369 ! [5] Folland G B, Introduction to partial differential equations, Mathematical notes (Princeton: Princeton Univ. Press) 1976 No. 17 [6] Folland G B, Harmonic analysis in phase space, Ann. Math. Stud. (Princeton: Princeton Univ. Press) No. 122(1989) [7] Geller D, Spherical harmonics, the Weyl transform and the Fourier transform on the Heisenberg group, Can. J. Math. 36 (1984) 615-684 [8] Helgason S, The Radon transform, Progress in Mathematics (1980) (Birkhauser) No. 5 [9] Hogan J A, A qualitative uncertainty principle for unimodular groups of type I, Trans. Am. Math. Soc. 340(1993)587-594 [10] Hormander L, A uniqueness theorem of Beurling for Fourier transform pairs, Arkiv fur Matematik 29 (1991)237-240 [II] Narasimhan R, Analysis on real and complex manifolds (1985) (North-Holland: North-Holland ! Math. Lib.) 35 * [12] Price J and Sitaram A, Functions and their Fourier transforms with supports of finite measure for / certain locally compact groups, J. Func. Anal 79 (1988) 166-182 I [13] Price J and Sitaram A, Local uncertainty inequalities for locally compact groups Trans Am. Math. I Soc. 308 (1988) 105-1 14 F ' f [14] Pati V, Sitaram A, Sundari M and Thangavelu S, Hardy's theorem for eigenfunction expansions, preprint (1995) Uncertainty principles on certain Lie groups 151 Sugiura M, Unitary representations and harmonic analysis, An introduction (1975) Tokyo: Kodansha scientific books Taylor M E, Noncommutative harmonic analysis, Math. Surveys and Monographs, American Mathematical Society (1986) No. 22 Thangavelu S, Some uncertainty inequalities, Proc. Indian Acad. ScL, Math. Soc. 100 (1990) 137-145 Thangavelu S, Some restriction theorems for the Heisenberg group, Stud. Math. 99 (1991) 11-21 Thangavelu S, Lectures on Hermite and Laguerre expansions, Math. Notes (1993) (Princeton: Princeton Univ. Press) No. 42 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 2, May 1995, pp. 153-156. Printed in India. On subsemigroups of semisimple Lie groups D KELLY-LYTH and M McCRUDDEN Department of Mathematics, University of Manchester, Oxford Road, Manchester, M139PL, UK MS received 30 August 1994; revised 17 October 1994 Abstract. In this paper we classify the subsemigroups of any connected semisimple Lie group G which are K-bi-invariant, where G = KAN is an Iwasawa decomposition of G. Keywords. Lie group; semisimple; subsemigroup. In a recent investigation of the support behaviour of certain Gauss measures on a connected semisimple Lie group (see [KM]), we encountered the following question. Let G be a connected semisimple Lie group with Lie algebra g having a Cartan decomposition g = t + p (in the usual notation of Helgason [He]), and let K be the analytic subgroup corresponding to t. Can one classify the subsemigroups S of G such that K^Sl Here "subsemigroup" means only a subset of G which is closed under the group multiplication. In this note we show that this problem has a very simple answer. To describe this, we let be the decomposition of g into its simple ideals g 7 -, 1 ^j ^ n, and we recall that [9i? 9;] = for all 1 < i <j < n, and g^ and g 7 - are orthogonal w.r.t. the Killing form on g. If QJ = tj + PJ is a Cartan decomposition for g^, then g has a Cartan decomposition g = t + p, where t = t!+ 1 2 + +! andp = p 1 4-p2 + " + ? Let N be a subset of {1, 2, . . . , n} and form Q N := t 4- / e tfPj. It is easy to see that g N is a reductive subalgebra containing t, and if G N is the corresponding analytic subgroup, then G N = (Yl jeN G j )(Yl j ^ N K j \ where G j9 Kj are the analytic subgroups determined by g^ and t j9 respectively. Our question raised above is now answered by the following result. Theorem. Let G be a connected semisimple Lie group and let S be any subsemigroup of G bi-invariant under K. Then S = G N for some subset N of { 1, 2, . . . , n}. In the special case when G is simple (and noncompact) this theorem tells us that K is a maximal proper subsemigroup of G. This special case therefore implies the observation of Hilgert and Hofmann that SO(2) is a maximal proper subsemigroup of SL(2, R) ([Hi H], Corollary 4.20, p. 49) and extends the theorem of Brun (see [B] or [He], Exercise A.3, p. 275) that K is a maximal proper subgroup of G, in the simple case. 153 154 D Kelly-Lyth and M McCrudden PROPOSITION 1 Any subgroup of a connected semisimple Lie group G which contains K is of the form Proof, (i) Let xeG\K and let H x denote the subgroup of G generated by K and x. We may write x = x 1 x 2 ...x n , where XjeGj for 1 <y ^ rc, and set N x *={l^j^ n: x^Kj}. Since x determines Xj up to translation by a central element, and the centre of G lies inside K, x determines N x uniquely. For each j <=N X , H x r\G j contains Kj and XjKjXT 1 . As G 7 is simple, Brun's theorem implies that the normaliser of K j in Gj is K p hence H x n G j ^ K p so by Brun's theorem again, H x r\Gj = Gj. It follows that H x contains G Nx . As G Nx clearly contains K and x, we conclude that H x = G Nx . (ii) Now let H be an arbitrary subgroup of G containing K and with H^K. Then H=\jG Nx ~G N , xeH . where N = [j xeH N x . D Given semisimple g with Cartan decomposition g = t + p, we choose a maximal abelian subspace a p of p and denote by the set of all roots of g relative to a p (see [He], p. 263, and note that we follow the notation there except that a p replaces t) po and the subscript on g and the subspaces of g is dropped). We write N K (a p ) for the normaliser of a p in X. PROPOSITION 2 There exists fceN x (a p ), and some m^ 1, such that for all Xea p , Proof. For each aeE, let r a :a p ->a p denote the reflection in the hyperplane (YEa p : a(Y) = 0} w.r.t. the restriction to o p of the Killing form on g. We choose a basis of simple roots (a l9 . . . , a z } from S, and set s = r i or a 2 or > which is a Coxeter element of the Weyl group W of (g, a p ). Since a x , . . . , a z are linearly independent in a*, we have 5(7) = Y for Yea p if and only if r a .( Y) = Y for all 1 <; < / (c.f. [Ca], Proposition 10.5.6, p. 165). Hence s(Y) = Y if and only if a/ Y) = for all 1 <j < /, which is equivalent to Y=0 since a^..,,^ span a*. Hence the linear map / - s: a p -* a p is invertible. Let the order of s be m, then from the identity (/ + s + - + s m ~ l ) (I - 5) = I - s m = and the invertibility of / - 5, it follows that on a p , Because W can also be realised as N K (a p )/C K (a^ where C K (a p ) is the centraliser of a p in J, we can find keN K (aJ such that s = Adfc| a ". Then (1) gives that for all } \ On subsemigroups of semisimple Lie groups 1 55 X + M(k)(X) + M(k 2 )(X) 4- - + Ad(/c w ~ ^(X) = 0, gives the result. D 3LLARY 1 exists keN K (a p ) and m^ 1 such that for each aeA = expa p , a- 1 =(fca) M " 1 fc" III + 1 . D DLLARY 2 ; connected semisimple Lie group G, any K-bi-invariant subsemigroup of G is a >up containing K. Let S be a K-bi-invariant subsemigroup and let xeS, then x = /qa^ for some ind k lt k 2 eK. Hence x" 1 =/cJ 1 (3~ 1 /c~ 1 eS by Corollary 1. Also leS and so D roof of the theorem stated earlier is now immediate by Propositions 1 and 2, lary 2. -fc. We note the following consequence of the theorem. If G is a connected mple Lie group and C is any K-bi-invariant subset of G, then there is some ;uch that C r is a neighbourhood of the identity in G(C), the subgroup of G ited by C. , by the theorem above, G(C)= \JC\ s=l - is Haar measure on G(C), there exists neN such that A(C") >0. But we may C = KDK for D^A, and by Proposition 2, Corollary 2, j result now follows because C n C~ n is a neighbourhood of the identity in G(C), /], bottom of page 50. )\vledgements >f the authors (DKL) thanks the Science and Engineering Research Council for financial assistance during the completion of some of this work. Both authors I like to thank the referee for a number of helpful suggestions. ences Brim J, &ur la simplification par les varietes homogenes, Math. Ann. 235 (1977), 175-183 Carter R W, Simple groups of Lie type (1972) (London, New York: J. Wiley) 156 D Kelly-Lyth and M McCrudden \ [He] Helgason S, Differential geometry, Lie groups and symmetric spaces (1978) (New York, London: Academic Press) [HiH] Hilgert J and Hofmann K H, Old and new on Sl(2). Manuscripta Math. 54 (1985) 17-52 [K M] Kelly-Lyth D and McCrudden M, Supports of Gauss measures on semisimple Lie groups, Preprint j Math. Zeit. (to appear) [W] Weil A, LMntegration dans les groupes topologiques et ses applications (Paris, 1953) [ Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 2, May 1995, pp. 157-167. Printed in India. Induced representation and Frobenius reciprocity for compact quantum groups ARUPKUMAR PAL Indian Statistical Institute, Delhi Centre, 7, SJSS Marg, New Delhi 110016, India MS received 13 September 1994 Abstract. Unitary representations of compact quantum groups have been described as isometric comodules. The notion of an induced representation for compact quantum groups has been introduced and an analogue of the Frobenius reciprocity theorem is established. Keywords. Induced representation; compact quantum group; Hilbert C*-module. Quantum groups, like their classical counterparts, have a very rich representation theory. In the representation theory of classical groups, induced representation plays a very important role. Among other things, for example, one can obtain families of irreducible unitary representations of many locally compact groups as representations induced by one-dimensional representations of appropriate subgroups. Therefore, it is natural to try and see how far this notion can be developed and exploited in the case of quantum groups. As a first step, we do it here for compact quantum groups. First we give an alternative description of a unitary representation as an isometric comodule map. This is trivial in the finite-dimensional case, but requires a little bit of work if the comodule is infinite-dimensional. Using the comodule description, the notion of an induced representation is defined. We then go on to prove that an exact analogue of the Frobenius reciprocity theorem holds for compact quantum groups. As an application of this theorem, an alternative way of decomposing the action of SU q (2) on the Podles sphere S^ is given. Notations. 3tf,3C etc, with or without subscripts, will denote complex separable Hilbert spaces. 08(3?) and ^ O pf) denote respectively the space of bounded operators and the space of compact operators on 3? . sf, J 1 , # etc denote C*-algebras. All the C*-algebras used in this article have been assumed to act nondegenerately on Hilbert spaces. More specifically, given any C*-algebra s#, it is assumed that there is a Hilbert space Jf such that j^^(JT) and for weJT, a(u) = for all asjtf implies u = 0. Tensor product of C*-algebras will always mean their spatial tensor product. The identity operator on Hilbert spaces is denoted by /, and on C*-algebras by id. For two vector spaces X and Y 9 X (x) alg Y denote their algebraic tensor product. Let sf be a C*-algebra acting on Jf. The subalgebras {ae^(Jf): and {aE^(^C)\ab,bae^^bE^} of J'(jr) are called respectively the left multiplier algebra and the multiplier algebra of j/. We denote them by LM(<$/) and M(j/) respectively. A good reference for multiplier algebras and other topics in C*-algebra theory is [4], See [9] for another equivalent description of multiplier algebras that is often very useful. 158 , Arupkumar Pal \ i 1. Preliminaries 1.1 Let si be a unital C*-algebra. A vector space X having a right j^-module j structure is called a Hilbert ^-module if it is equipped with an j^-valued inner product f that satisfies i (i) <x,y>* = <y,x>, I (ii) <x,x0, (iii) <x,x> = 0=>x = 0, (iv) <x,yb> == <x,;y>f? for and if ||x|| : = || <x, x> || 1/2 makes X a Banach space. Details on Hilbert C*-modules can be found in [1], [2] and [3]. We shall need a few specific examples that are listed below. Examples, (a) Any Hilbert space Jf with its usual inner product is a Hilbert C- module. (b) Any unital C*-algebra j/ with <a,&> = a* b is a Hilbert ^-module." / (c) Jf J3/, the 'external tensor product' of 2tf and ja/ 9 is a Hilbert ^-module. (d) #(jf, Jf ), with <5, T> = S* T is a Hilbert (jf )-module. j 1.2 We have seen above that & #(jf ) and #(Jf , Jf jf ) both are Hilbert modules. It is easy to see that the map SiZu^ a^Zi^ a f (-) from Jf a ^PO to #~) extends to an isometric module map from ^<g>#(Jf)' to , Jf JT), i.e. 9 obeys Thus & embeds J?(jf) in ^(jf,^ jf). Observe two things here: first, if ^ = C, 9 is just the identity map. And, & is onto if and only if 2ff is finite-dimensional. \ The following lemma, the proof of which is fairly straightforward, gives a very useful * property of 8. i Lemma. Let O f be the map $ constructed above with ^ replacing Jf,z= 1,2. Let jff 2 ) and xeJ^ l (jf). T/ien & 2 ((S fd)x) = (S I)^(x). 1.3 For an operator Te@(J{f jf), and a vector we^f, let T tt denote the operator I;M> T(w v) from jf to jf jf . It is not too difficult to show that T u ^(^f ^ (Jf )) if Te LAf( (^) ^ W). Define a map (T) from ^f to S" L (T u ). Then Y is the unique linear injective contraction from to J^,^JW) for which 9(^(T)(u))(i;) = T(ut?) Vwe^^eJT, Te (^) (Jf)). Here are a few interesting properties of this PROPOSITION Let W:LM(a (je)aW)^0(je,jea(jr)) be the map described above. Then we have the following: (i) *maps isometries in LM(^ (^) ^f (jf )) onto tte isometries in Induced representation for quantum groups 159 (ii) For any Te (iii) Ifjtf is any C*-subalgebra of &(&) containing its identity, then Te LM(^? pf ) .a/) if and only if range ^(T)^ Proof, (i) Suppose T6LAf(# pH#PO) is an isometry. By 1.2, <(!>, (7 = <9" 1 (TJ,d" 1 (rj> = < T u , ?;>= <M,i7>7 for M,i;eJf. Thus (T) is an isometry. Conversely, take an isometry 7i:Jf -> J"f ^pf) and define an operator T on the product vectors in ^f JJT by T(uv) = S(7i(w))(i?),& being the map constructed in 1.2. It is clear that T is an isometry. It is enough, therefore, to show that T(|w><i;|S)6 # Pn &(&) whenever SeJ'pf ) and u,v are unit vectors in Jf such that <w,u> = or 1. Choose an orthonormal basis {e t } for Jf such that ^ = M, e r = where O if <u, i;> = 0, if <ii, !;>=!. Let 7c . = e f |^)7i(^). Then T(|M><i?|S) = S|e I .)<e r |7c I . 1 S where the right- hand side converges strongly. Since TufcJeJf ^(Jf), it follows that S 7r a *?!;,.! converges in norm. Consequently the right-hand side above converges in norm, which means T(|w> (ii) Straightforward. (iii) Take T= |w><u|(g)a,w,i;e^f,aej3f. For any Since *F is a contraction, and the norm closure of all linear combinations of such T's is > we have ran S e ^CT) ^ 3e ^ for all Assume next that' TeLM(^ (^f) ja^). Then T(|w><u|/)e^ pf)^ for all Hence ^F(r(|M><tt|/))(M)eJf jtf, which means, by part (ii), that f j/ for all uetf. Thus^ range ( T) e -jf ^. To prove the converse, it is enough to show that T(\u)^v\a)G^ Q (^) $0 whenever aestf and u,veJ4f are such that <w, i?> = or 1. Rest of the proof goes along the same lines as the proof of the last part of (i). H 1.4 Let jf 19 jf 2 be two Hilbert spaces, <$/ being a C*-subalgebra of J'pf f ) contain- ing its identity. Suppose <p is a unital *-homomorphism from s4 1 to j/ 2 . Then id(j)\ Sa\-+S<l)(a) extends to a *-homomorphism from ^ (^f) 6 s/ 1 to ^ (J^) j/ 2 . Moreover {((W0)(a))fc:^eJ > (Jf')j/ 1 ,fc6^ (Jf)^ 2 } is total in ^ (Jf)j/ 2 . Therefore id extends to an algebra homomorphism by the following prescription: for all fl PROPOSITION Let (j> be as above, and } i be the map T constructed earlier with JT replacing Cfr. Then 160 Arupkumar Pal Proof. It is enough to prove that (<w|iW<)^TO)) = (^ Rest now is a careful application of 1.2. 1.5 Consider the homomorphic embeddings 12 : and 13 :^ (^f)(g)^ 2 ->^ (^ ? )(g)^ 1 ja/2 given on the product elements by respectively. Each of their ranges contains an approximate identity for j*i j/ 2 , so that their extensions respectively to LM(^ (JH ^i) and j/ 2 ) are also homomorphic embeddings. PROPOSITION Let X F 1 , X F 2 ^ as * n ^e previous proposition, and let be the map* with replacing d. Let Se LM(^ pf ) s/J, Te LM(^ (^f ) s/ 2 ). Then Proof. Observe that for M 1 ,...,u ll 6^f,(Y 1 (S)(u l ),*i(S)("j) Therefore X F 1 (S)W is a well-defined bounded operator from jf j/ 2 to 3df sf i st 2 - Ta5ce an orthonormal basis {ej for ^f. Define Sj/s and T f /s as follows: This converges to ^ 12 (S)^> 13 (T)(e i w) as n--oo. On the other hand, which implies lim^^acCPJS) W)(P,, W)V 2 (T)(,)) = ^((T Therefore Let P w :=I? =1 |e ><^|. Then CF^S) id)(P n zW)^F 2 (T)(^) = CF 1 (S) W)(Z 7<II ^ Ty Hence for i?6Jf I9 wejf 2 , Induced representation for quantum groups 161 2. Representations of compact quantum groups 2.1 We start by recalling a few facts from [6] on compact quantum groups. DEFINITION Let jtfbea separable unital C*-algebra, and u:<$/-+jtf<s/bea unital *-homomorphism. We call G = (X, /u) a compact quantum group if the following two conditions are satisfied: (i) (id IJL)IJL = (n id)^ and (ii) {(a I)n(b):a 9 bej/} and {(/ a)^(b):a,be^} both are total in \JL is called the comultiplication map associated with G. We shall very often denote the underlying C*-algebra s$ by C(G) and the map /x by U G . A representation of a compact quantum group G acting on a Hilbert space 3 is an element n of the multiplier algebra M(# p?) C(G)) that obeys 7C 12 7T 13 = (id}ji)n, where 7c 12 and 7i 13 are the images of TC in the space M(^ pf ) C(G) C(G)) under the homomorphisms </> 12 and 13 which are given on the product elements by: A representation TC is called a unitary representation if THT* = / = n*n. One also has the notions of irreducibility, direct sum and tensor product of representations. As in the case of classical groups, any unitary representation decomposes into a direct sum of finite-dimensional irreducible unitary representations. Let A(G) be the unital *-subalgebra of C(G) generated by the matrix entries of finite-dimensional unitary representations of G. Then one has the following result (see [8]). Theorem. ([8]) Suppose G is a compact quantum group. Let A(G) be as above. Then we have the following: (a) A(G) is a dense unital *-subalgebra of C(G) and n(A(G)) A(G) alg A(G). (b) There is a complex homomorphism &:A(G)-+C such that (B id)iJL = id = (id (c) There exists a linear antimultiplicative map K: A(G) -* A(G) obeying m(idK)n(a) = s(a)I = m(jc id)n(a), and /c(/c(a*)*) = a for all aeA(G\ where m is the operator that sends abto ab. The maps B and K in the above theorem are called the counit and coinverse respectively of the quantum group G. 2.2 Let G = (C(G),/i G ) and H = (C(H\u H ) be two compact quantum groups. A C*- homomorphism from C(G) to C(H) is called a quantum group homomorphism from G to H if it obeys (0 (f))u G = /^0. One can show that if G, H are compact quantum groups, then H is a subgroup of G if and only if there is a homomorphism from G to H that maps C(G) onto C(H). 162 Arupkumar Pal 2.3 Let G = (jtf , /i) be a compact quantum group. From now onward we shall assume that st acts nondegenerately on a Hilbert space Jf* , i.e. j/ is a C*-subalgebra of J^pf) containing its identity. We call a map TT from J-f to 3tfsf an isometry if <7c(M),7c(i?)> = <M,i;>/ for all u,i?e^f. If TC:^ -+Jf <s/ is an isometry, then 7izW: u ah->7t (w) a extends to a bounded map from tf j^ to Jf ^ s/. n is called an isometric comodule map if it is an isometry, and satisfies (7izW)7r = (/^)TL The pair (Jf , TC) is called an isometric comodule. We shall often just say n is a comodule, omitting the 3? . The following theorem says that for a compact quantum group isometric comodules are nothing but the unitary representations. Theorem. Let n be an isometric comodule map acting on jjf. Then *P~ ^TC) is a unitary representation acting on $?. Conversely, if TC is a unitary representation of G on Jf , then (Stf, *(&)) is an isometric comodule. We need the following lemma for proving the theorem. Lemma. Let (Jtf, n) be an isometric comodule. Then Jf decomposes into a direct sum of finite dimensional subspaces 2 = @3tf a such that each Jf a is n-invariant and TT| #I is an irreducible isometric comodule. Proof. By 1.3, there is an isometry n in LM(^ (^) jtf) such that *F(A}= n. Using 1.4 and 1.5, we get rt 12 rt 13 = (M/)A where 7c 12 = ^ 12 (7i), 7T 13 = < 13 (A), 12 an d 13 being as in 1.5 with s^ l = stf 2 = <$/. Let </ = {aej/:h(a*a) = 0}, From the properties of the haar state, J is an ideal in jaf. For any unit vector u in ^f, let Q(M) = (M/i)(rt(|tt><tt|/)A*). Then 6(")* = G(")eo(^)- If 6() = 0, then |A(|u><u|/)**| 1/2 * (^)^ Therefore A(|w><|/)A*e (J?)^. It follows then that |w><w|/eJ > ( e ^ ? )J zr . This forces u to be zero. Thus for a nonzero w, )(w) 7^ 0. Choose and fix any nonzero u. Then Thus A(j2()/) = (Q(w)/)A. If P is any finite-dimensional spectral projection of Q(u\ then (P I) = (P I)rt t which means, by an application of part (ii) of 1.3, that nP (Pid)n. Standard arguments now tell us that n can be decomposed into Induced representation for quantum groups 163 ect sum of finite-dimensional isometric comodules. Finite-dimensional comodules, rn, can easily be shown to decompose into a direct sum of irreducible isometric adules. The proof is thus complete. H 'oofof the theorem: Let rt be a unitary representation. By 1.3, *(ft) is an isometry J^toJjf C(G\ Using 1.4 and 1.5, we conclude that *(n) is an isometric comodule. >r the converse, take an isometric comodule n. If n is finite-dimensional, it is easy ee that V ~ I (TI) is a unitary representation. So, assume that n is infinite- insional. By the lemma above, there is a family {P a } of finite-dimensional actions in $(#?) satisfying P*P f = 8*pP*> I,P* = I> nP* = (P a id)n Va (2.1) that n\ Pyje = 7cP a is an irreducible isometric comodule. n\ Pt ^ is finite-dimensional, fore ~ ifa |p 2jf ) is a unitary element of LM(^ (P a ^f) ,o/f= ^(P a Jf)^. Let us 'te *~ 1 (n) by n. Then the above implies that in the bigger space S&(tf Jf ), (rt(P a /))*(7i(P a /)) = P a / = (7i(P a 7))(7r(P a /))*. second equality implies that 7i(P a /)TI* = P a / for all a, so that TUT* = /. We tdy know by 1.3 that 71*71 = / and by 1.4 and 1.5 that n i2 n l3 =(id]Li)n. Thus it lins only to show that neM(^ (J^)^). It is enough to show that for any and aes/, (Sa)7te^ (J^)^. Now from (2.1) and 1.3, A(P a /) = for all a. Therefore (S^)^/)* = (5a)7i(P a /)6J > (^f) j</. Since a) n is the norm limit of finite sums of such terms, (S a)7ce^ (J ( f ) ^. Thus TT is itary representation acting on Jf . Sfext we introduce the right regular comodule. Denote by L 2 (G) the GNS space ciated with the haar state h on G. Then stf is a dense subspace of L 2 (G). One can see that $#$tf can be regarded as a subspace of L 2 (G)stf. Consider the = (hid)iJL(a*b) = fc(a*h)7 11 a, feej/. Therefore ja extends to an isometry from L 2 (G) into L 2 (G) j^. Denote y M. The maps (/^)5R and (Wfd)5R both are isometries from L 2 (G) to ?) s ^/ and they coincide on sf. Hence (/ /x)9? = (91 id)SR. Thus 51 is an letric comodule map. We call it the right-regular comodule of G. By theorem 2.3, (9?) is a unitary representation acting on L 2 (G). This is the right-regular represen- >n introduced by Worono wicz ( [8] ). inally let us state here a small lemma which is a direct consequence of the T-Weyl theorem for compact quantum groups. Lemma. {ueL 2 (G)\yi(u)eL 2 (G) alg C(G)} = A(G). iduced representations lis section we shall introduce the concept of an induced representation and show Frobenius reciprocity theorem holds for compact quantum groups. Throughout 164 Arupkumar Pal this section G = (C(G), U G ) will denote a compact quantum group and H = (C(H), H H \ a subgroup of G. We start with a lemma concerning the boundedness of the left convolution operator. 3.1 Lemma. Let G = (c/,^) be a compact quantum group. Then the map L p :<$&-+jtf given by L p (a) = (p id)^(a) extends to a bounded operator from L 2 (G) into itself. Proof. The proof follows from the following inequality: for any two states p and p 2 \ on stf, we have Pi((p2* fl )*(P2* a )) < P2*Pi( a * a ) Vaej/, I i where p f * a: = (p id)fi(a). * | I 3.2 Let 7i be a unitary representation of H acting on the space Jti? .n: = *(ft) is then an isometric comodule map from tf to Jf C(H). Consider the following map from J? L 2 (G) to ^ L 2 (G) C(G): J where 91 G is the right-regular comodule of G. It is easy to see that this is an isometric comodule map acting on ^f L 2 (G). Let p be the homomorphism from G to H (cf. 2.2). Let ^ = {ue^f L 2 (G): (7 L , )u-(id p)nl}u for all continuous linear functional p on C(H)}. Then /9r keeps 2tf invariant; the restriction of /SR G to 2f is therefore an isometric comodule, so that X F~ l ((/ 9^)1^) is a unitary representation of G acting on #*. We call this the representation induced by n, and denote it by ind ,;! or simply by indTt when there is no ambiguity about G and H. Let 7i t and 7t 2 be two unitary representations of H. Then clearly we have i (i) ind A x and ind n 2 are equivalent whenever ft^ and A 2 are equivalent, and (ii) md(n 1 n 2 ) and ind n { ind n 2 are equivalent. ;? Before going to the Frobenius reciprocity theorem, let us briefly describe what we f mean by restriction of a representation to a subgroup. Let T! G be a unitary representa- tion of G acting on a Hilbert space tf . We call (uip)7t G the restriction of TT G to H and denote it by 7i G|fr . To see that it is indeed a unitary representation, observe that x F((idp)7i G ) = (/p) v F(7t G ) whicft is clearly an isometric comodule. Therefore by 2.3, TI G|H is a unitary representation of H acting on 3tf Q . Denote *(n G ) by TI G and ' f )by7i G ' H . 3.3 Theorem. Let n G and n H be irreducible unitary representations ofG and H respec- tively. Then the multiplicity of H G in ind G 7t H is the same as that of n H in n GlH[ . Proof. Let /(7r G|H ,7i H ) (respectively ,/(7t G , ind7t H )) denote the space of intertwiners between TT G| * and n H (respectively T! G and ind i! H ). Assume that 7t G and rt H act on Jf and Jtf o respectively. Jf C(G) can be regarded as a subspace of Jf L 2 (G) and hence TT G as a map from tf into jf L 2 (G). Since TE G = V(fi G ) is unitary, we have for Induced representation for quantum groups 165 Thus 7r G :Jf -Jf L 2 (G) is an isometry. Let S:Jf -Jf be an element of /(A C|H , TI H ). (S /)TT G is then a bounded map from Jf into J^ L 2 (G). Denote it by f(S). It is not too difficult to see that f(S) actually maps Jf into Jf , and intertwines TT G and ind n H . /:S(->/(S) is thus a linear map from </(TT G|H , n H ) to /(7i G , ind A"). We shall now show that / is invertible by exhibiting the inverse of / Take a T: Jf -> jf that intertwines TT G and ind TT H . For any we^f , T": = w| /) T is a map from jf to L 2 (G) intertwining TI G and the right regular representation SR G of G, i.e. <R G T u = (T u <S)id)n G . Now, TT G is finite-dimensional, so that 7t G (Jf )c jf fl/0 A(G). Hence 9* G T u (Jf ) c L 2 (G) fl ^(G). By 2.5, r w (jf ) c X(G). Since this is true for all ueJf , T(jf ) ^oflZff^(^)'- Therefore (7 e G ) T is a bounded operator from 3C to ^ . Denote it by 0(T). For a comodule TI and a linear functional p, denote (id p)n by 7i p . Let p be a linear functional on C(/f). Then n*g( T) = rf(I e G ) T = (/ fiG)(7r^ id) T=(l B G ) (/ L p . p ) T = (/pjp)T. On the other hand, since T intertwines n G and ind7T H , we have g(T)(7t GlH ) p = ^(T)(/ P)TT G|H = ^(T)(/ p)(I P)TT G = (/ e G ) Tn G p = (/ e G ) U9l^ | ,)T = (/pop)7: Thus n*g(T) = g(T)(n GlH ) p for all continuous linear functional p on C(ff),' which implies 0(T)e/(# G|H ,7i H ). The map T\-^g(T) is the inverse of/. Therefore /(TC G|H ' n H ) ^ J(ft G , ind TT H ), which proves the theorem. H COROLLARY 1. For any unitary representation n G of G and n H of H, the spaces J(fi G{H ,n H ) and , ind n H ) are isomorphic. COROLLARY! Let H be a subgroup of G and K be a subgroup of H. Suppose TC is a unitary representa- tion of K. Then ind^tf and ind^indf A) are equivalent. 3.4 Action of S U q (2) on the sphere S^ has been decomposed by Podles (see [5] ). Here we give an alternative way of doing it using the Frobenius reciprocity theorem. Let us start with a few observations. Let u be the function z\ >z, zeS 1 , where S l is the unit circle in the complex plane. Then u is unitary, and generates the C*-algebra C(S 1 ) of continuous functions on S 1 . Let a and /? be the two elements that generate the algebra C(SU q (2)) and obey the following relations: The map p:ai->u, /?i >0 extends to a C*-homomorphism from C(SU q (2)) onto It is in fact a quantum group homomorphism. By 2.2, S 1 is a subgroup of SU q (2). For any ne{0, 1/2, 1, 3/2, . . .}, if we restrict the right-regular comodule 91 of SU q (2) to the subspace tf of L 2 (SU q (2)) spanned by {a*'^""':,- = 0,l,...,2n}, (3.1) then we get an irreducible isometric comodule. Denote it by u (n \ It is a well-known fact ([6], [7]) that these constitute all the irreducible comodules of SU q (2). If we take 166 Arupkumar Pal the basis of 3 n to be (3.1) with proper normalization, the matrix entries of u (n} turn out to be 2n-)A /-\ / (__ nr r ( 2i-r where (fc) _,:= 1 + 4-2 + ,-*.+ . .. + Since w (;i) | sl = (J p)u ( "\ matrix entries of (n) | s ' are given by Therefore if n is an integer then the trivial representation occurs in u (n) \ sl with multiplicity 1, and does not occur otherwise. Consider now the action of SU q (2) on S 2 q0 . Recall ([5]) that C(S,? ) = {aeC(SU q (2)): (pid)n(a) = Ia} and the action is the restriction of ju to C(S| ). From the above description, C(S^ ) can easily be shown to be equal to {aeC(5^ ): L p . p (a) = p(l)a for all continuous linear functionals p on C(S 1 )}. Therefore when we take the closure of C(S^ ) with respect to the invariant inner product that it carries and extend the action there as an isometry, what we get is the restriction of the right-regular comodule SR of SU q (2) to the subspace tff = {ue L 2 (SU q (2)): L p . p (u) = p(I)u for all continuous linear functionals p on CCS 1 )}, which is nothing but the representation n of SU q (2) induced by the trivial representation of S l on C. Hence the multiplicity of u (n) in n is same as that of the trivial representation of S 1 in u (n) \ si which is, from (3.2), 1 if n is an integer and if n is not. Thus the action splits into a direct sum of all the integer-spin representations. Acknowledgements The author expresses his gratitude to Prof. K R Parthasarathy for suggesting the problem and for many useful discussions that he has had with him. This work was supported by the National Board for Higher Mathematics (India). References [1] Jensen K K and Thomsen K, Elements of KK-Theory, (1991) (Boston: Birkhauser) [2] Kasparov G G, Hilbert C*-modules - Theorems of Stinespring and Voiculescu, J. Operator Theory, 4(1980)133-150 [3] Paschke W, Inner Product Modules Over B*-algebras, Trans. Am. Math. Soc., 182 (1973) 443-468 Induced representation for quantum groups 1 67 [4] Pedersen G K, C*-algebras and Their Automorphism Groups, (1979) (London: Academic Press) [5] Podles P, Quantum Spheres, Lett. Math. Phys. 14 (1987) 193-202 [6] Woronowicz S L, Twisted SU(2) Group. An Example of a Noncommutative Differential Calculus. Publ RIMS, Kyoto Univ., 23 (1987) 117-181 [7] Woronowicz S L, Compact Matrix Pseudogroups, Commun. Math. Phys. Ill (1987) 613-665 [8] Woronowicz S L, Compact Quantum Groups, Preprint, 1992 [9] Woronowicz S L, Unbounded Elements Affiliated with C*-algebras and Noncompact Quantum Groups, Commun. Math. Phys. 136 (1991) 399-432 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 2, May, 1995, pp. 169-186 Printed in India. Differential subordination and Bazilevsc functions S PONNUSAMY School of Mathematics, SPIC Science Foundation, 92, G. N. Chetty Road, Madras 600017, India Present address: Department of Mathematics, PO Box 4, Hallituskatu 15, University of Helsinki, FIN 00014, Helsinki, Finland MS received 24 December 1993; revised 22 April 1994 Abstract Let M(z) = z n H ,N(z) = z"-\ be analytic in the unit disc A and let A(z) = N(z)/zN'(z). The classical result of Sakaguchi-Libera shows that Re(M'(z)/JV'(z)) > implies Re(M(z)/JV(z))>0 in A whenever Re(A(z))>0 in A. This can be expressed in terms of differential subordination as follows: for any p analytic in A, with p(0) = 1, l.+ z 1+z p(z) + J.(z)zp'(z) X implies p(z) -< , for Re/l(z) > 0, zeA. 1 z 1 z In this paper we determine different type of general conditions on A(z), h(z) and $(z) for which one has p(z) + l(z)zp'(zK/i(z) implies p(z)<^(z)<h(z\ zeA. Then we apply the above implication to obtain new theorems for some classes of normalized analytic functions. In particular we give a sufficient condition for an analytic function to be starlike in A. Keywords. ^Differential subordination; univalent; starlike and convex functions. 1. Introduction Let / and g be analytic in the unit disc A. The function / is subordinate to 0, written f<g,or f(z) < Q(Z\ if g is univalent, /(O) = 0(0) and /(A) c g(A). Define d = {/: /(O) = /'(O) - 1 = 0}, j/ k = {/: /(z) = z +*a k+ iz k+l + }, and s/' = {/: /(O) = 1}. Let A(z) be a function defined on A with ReA(z) >rj > 0, zeA and let pe sf'. Then a recent paper [8, Theorem 1] establishes the following: Re[p(z) + A(z)zp'(z)] > p implies Rep(z) > , for zeA. (1) Let /i and A satisfy | Im /z(z) | < Re>l(z), zeA and let pe d'. Then a result of Miller and Mocanu [5, Theorem 8] shows that Re[>(z)p(z) + ;i(z)zp'(*)] > implies Rep(z) > 0, for zeA. (2) (2) is equivalent to (1) if we take /*(z) = 1 in (2) and /? = r\ = in (1). 169 170 S Ponnusamy Let M and N be analytic in A, with M'(0)/AP(0) = 1 and let /? be real. If N maps A onto a multisheeted starlike domain with respect to the origin, then from [4, Theorem 10] we get Re < 8 (or > B resp.) implies Re 7 < p (or > /? resp.), for zeA. * N (z) N(z) (3) A well-known condition for a function pe,o/ subordinate to q is that [6] under some conditions on q(z). Suppose we let p(z) = z/'(z)//(z) and q(z) = 2(l+z)/(2z\ then we get I - 2 r ^ . <T> forzeA. Similarly it follows from a result of Mocanu et al [7] that for , Re[p(z) + zp'(z)] > implies | argp(z) | < < n/3, for zeA, where lies between 0.911621904 and 0.911621907. This improves the relation (2) whenever jj,(z) = A(z) = 1 for zeA. However the example M(z) = z/'(z), N(z) = /(z) and = 3/2 [or M(z) = zf\z\ N(z) = z and ]S = resp.] in (3) suggests that there may exist some conditions on M and N so that forzeA (4) for some h t (/ = 1, 2) to be specified. Thus it is interesting to ask whether there exist such conditions for our implication. V By writing (4) in terms of differential subordination, in this article we determine some / new sufficient conditions on A(z), /? . and h t (z\ (i= 1,2) for Re[p(z) + l(z)zp'(z)] > Pi to imply p(z) is subordinate to h&z). Some interesting applications of this are given. In particular they improve the previous works of different authors [1, 8, 9, 12]. All of the inequalities in this article involving functions of z, such as (2), hold uniformly in the unit disc A. So the condition Tor zeA' will be omitted in. the remaining part of the paper. 2. Preliminaries Let /e,c/ and S* = {/es/: /(A) is starlike}. Then for y > and ft < 1, we say /e(y, j?) if, and only if, there exists geS* such that where all powers are chosen as principal ones. Differential subordination and Bazilevic functions 171 Denote by S^y,/?), the subclass consisting of those functions in B(y,/?) for which geS* can be taken as the identity map on A. As usual we let J3 1 (l,j8) = /?(jS) and BjCO, /?) = S*(0). From (1), for < /? < 1, y > and for /efi^y, A we easily have In Lemma 1 of section 3 below, we obtain a more general result which improves the above inequality. Lemma 1 has been used in [9] to obtain new sufficient conditions for starlikeness. We use the following two lemmas in our proofs. Lemma A. [5] Let F be analytic in A and let G be analytic and univalent on A, with F(0) = G(0). IfF is not subordinate to G, then there exist points z eA and Co e ^A, and m^lfor which F(\z\ < |z |) c G( z < z |), F(z ) = G(C ), and z F(z ) = mC G'(Co). Lemma B. [5, 6] Let 0. c: C and /ef ^f fc^? analytic and univalent on A except for those CedA/0r w/zzc/i Lt z ^^(z) = oo. Suppose that \l/: C 2 x A-~C satisfies the condition 'A(4(C),mC<?'(0;z)^ (6) w/zen <gr(z) is ,/imte, m^k^l and \ C | = 1. // p an^? g are analytic in A, p(z) = p(0) -f p fc z k + -, p(0) = ^f(O), and further if then p(z) < q(z) in A. Suppose that pe^ f with p(z) = 1 +p k z*H , and ^(z) = (l+z)/(l z). Then the condition (6) reduces to \l/(ix,yiz)tn (7) when x is real and 3; ^ - k(\ + x 2 )/2. Except for Theorems 5 and 6, we, in our results, consider the situations where k = 1. 3. Main results We now state and prove our main results. Lemma 1. Let pej/' , aeC with Rea ^ (a & 0), /? < 1 be such that (8) l], (9) where d <5(Rea) is an increasing function of Rea with (1 +Rea)/(l + 2Rea) ^ <5 < 1. estimate cannot be improved in general. 172 S Ponnusamy Proof. We use the well-known result of Hallenbeck an Ruscheweyh [2], namely, 1 f 2 p(z) + azp'(z) X h(z) implies p(z)<- z~ 1/a h(t)t lf *~ 1 dt (11) a Jo " and h a convex(univalent) function with h(0) . 1. If we let 2 p l | 2(1 -]8) 1 z then h is convex and univalent on A, h(Q) = 1 and Re h(z) > p. For this choice of h, the condition that (8) implies - in fact is equivalent to - Therefore from a straightforward calculation, Inequality (8) implies p(fK 2-l +2(1 -* where (j> defined by (12) \ = 1 4 V = n = !na + l Jo l~ ;, zeA is convex in A. Let so that Then, for |z| = r and < t < 1, we have .Re a This implies that and so W- l-r 2 t 2 rt Rea l-r 2 + rt* t ,*Rea' (Note that if Rea < 0, rt Rca need not be less than one and the above will not work.) Therefore, we have 1 Jol+rt Re " for|z| = r, Differential subordination and Bazilevic functions 173 Observe that the series K(r) is absolutely convergent for 0<r<l. Suitably rearranging the pairs of terms in K(r) it can be shown that 1/2 < K(r) < 1. In particular for r -> 1 ~ the above inequality reduces to Re 0(z) ^ K(r) >K(1) = <5(Re a), where 6 is as in (10). Next we show that 5 satisfies the inequality (1 + Rea)/(l + 2Rea)^ 5 < 1. Since 2/J 1 4-2(1 f$)<t>(z) is the best dominant for (8), we obtain taking A(z) = a, with Rea > YI in (1), }> 2ft- 1+2(1 - 21 -, -f- rj Thus making rj -Rea + , we get Kn\-MK 1 + Rea ^2 + Rea' This from (12) proves (9). To complete the proof we need only to show that the bound in (9) cannot be improved in general. For this we let = 2j8 - 1 + 2(1 - / Then q is the best dominant for (8), because it satisfies the differential equation q(z) + xzq'(z) - 2/J - 1 + ^^ h(z). Therefore the function q(z) shows that the bound in (9) cannot be improved. Q D Remark. In fact the second assertion, namely, 1 + Rea ~ ' can be seen directly. If Re a > 1, then Rea 1 1+Rea 2(l + 2Rea) 1 + Rea : 2 + Rea' 174 S Ponnusamy Similarly if < Re a < 1, then Re a (Re a) 2 1 2 4(1 +Rea) 4(1 + Rea)(l + 2Rea) 1 Re a l+Rea 2(2 -f Using Lemma 1 in particular for Re a -* 0, a ^ 0, /? = 0, one has Re{p(z) + azp'(z)} > implies Rep(z) > 0. In the next result we improve this relation by showing that the same conclusion may be obtained under a weaker hypothesis on p. Theorem 1. Let a be a purely imaginary number, i.e., a = za 2 , a 2 real. Let Q be the unique- function that maps A onto the complement of the ray {it: t^2~ 1 (a 2 ~ 1 a 2 )} whenever a 2 >0({ft: t^2~ 1 (a 2 " 1 a 2 )} whenever a 2 <0). Ifpejtf' satisfies then Rep(z) > 0. Proof. If we let i/^(r, s) = r + as, then subordination becomes is analytic in A and the above The conclusion of the theorem will follow from Lemma B and (7) if we can show that \l/(ix, jO2(A) when y ^ - (1 +x 2 )/2 and x-real. Suppose that a = ia 2 , with a 2 > 0, then i/^'x, y) = i(x + <x 2 y) and x -h x - a 2 (l + x 2 )/2 for all x-real. This shows that for a 2 >0, a 2 < 0. Hence the theorem. . A similar conclusion holds for the case D However the special case of the following lemma improves the conclusion of Lemma 1 further at least for aeC such that | Im a | ^ ^(Re a - Y\) for a suitable fixed r\ > 0. Lemma 2. Let X be a function defined on A satisfying (13) Differential subordination and Bazilevic functions 175 such that and let zeA /ffo) -( + ' + 2V ) L 9 -2 > /9+lSiy 2 10 6e such that 2p f (rj) + 77 ^ 0. Ifpes/' satisfies (14) (15) (16) :|argp(z)|<7r/3. Proo/. Note that /?'()?) < if, and only if, r\ ^ ,/3/2. Now using (1), (16) implies Rep(z)>: Since 2jS / (^) + rj ^ 0, this Inequality further implies Rep(z) > in A. If we let fll = {coeCiReco > j8'(^)} and q(z) = [(1 + z)/(l - z)] 2/3 s then ^(A) equals : | arg co | < Tr/3}. Then for \l/(r, s; z) = r + >l(z)s, (16) can be rewritten as So to prove the lemma we need only to show that p -< q. If p is not subordinate to q, then by Lemma A there exist points z eA and Co and m ^ 1 such that p( | z | < | z 1 ) c 4(A), p(z ) = (C ) and z p'(z Q ) = mCo9 ; (C )- We first discuss the case p(z ) 7^ which corresponds to a point on one of the rays on the sector #(A). Since p(z ) ^ 0, ( ^ 1. Next by letting X and 7 be the real and imaginary parts of A(z ), respectively, from (13) and (14), we find that X- .(17) Further if we set ix = (1 + Co)/(l Co) an d use the above observations, we obtain For x ^0, ReiA(p(z ),z oP '(z );z ) = |2/3 ^r-upo^ it;( <o 3|x| 176 S Ponnusamy 2, if x > . ,, ifx<0 . Therefore, for x ^ 0, since A(z ) satisfies (17) and m ^ 1, we obtain where withtHx| Since is the maximum for f(t\ we have This implies that ^(P^oX ^oP'fcofc^o) l jes outside fl>, contradicting (16). Hence we must have p^q when p(z ) ^ 0. Now consider the case p(z ) = which corresponds to the corner of the sector q(A). Observe that the sector angle of ^(A) is 2n/3 and so p(\z\ = |z |) cannot pass through such a corner without itself having a corner and hence the case p(z ) = cannot occur for the present form of our lemma. This completes the proof. D Lemmas 1 and 2 yield improvements on most of the results of [8]. As an equivalent form of Lemma 2 we state Theorem 2. Let /Ffo) be as defined by (15) so that 2p'(rj) + rj^Q. Let M(z) = z" -f - - - and = z"+:.be analytic in A and such that for some aeC, N satisfies Im odV(z) zN'(z) Then arg Proo/. Consider the function p{z) = M(z)/N(z) and let A(z) = aJV( Z )/zAT'(z). Then by hypothesis, pe ^' and all the conditions of Lemma 2 are satisfied. Now it is elementary to show that and hence Theorem 2 follows from Lemma 2. Differential subordination and Bazilevic functions 177 COROLLARY 1. jjj ;. fa -L Ifpej/' and if k is a function defined on A such that ^ then Re{p(z) 4- A(z)zp'(z)} > implies |argp(z)| < n/3. Proof. If we let ^ = ^/3/2 then in this case fi'(ri) = Q in (15) and the corollary now follows from Lemma 2. D COROLLARY 2. Let feB^y, 0). Then we have (i) Re /^Y> 2$(l/y) -1, / V z I (ii) <7t/3, For the function F defined by F y (z) = - f~ l f y (t) dt, we have z Jo (iii) ReF(z)( ) > 2<5(l/(y 4- c)) - 1, for y and c real with 0<y + c. \ z J (iv) argF'(z)( j <7r/3, /or y and c 5wch that < 7 -f c ^ 2/^/3. Proo/. Proofs of the above inequalities follow from Lemma 1 and Lemma 2 using the techniques of [8]. ^ Theorem 3. Let r\>Qbe such that where /J'fa) is as defined in (15). Let pes/' andv.^r\. Suppose r , . .. Re[p( Z ) + a Z p'(z)] Then we have Re[p(z) + fpp'(z)] > /P(iX |argp(z)| < 7i/3 IW Rep(z)>2(l- Proof. Observe that '(z) = 1 -- ]p(z) + -[ a a 178 S Ponnusamy Now Lemmas 1 and (19) yield (20) Taking /(z) = ff in Lemma 2 and a = 77 in Lemma 1, respectively, the theorem follows. D We note that, using Lemma 1 and Theorem 2, we can construct several new examples. The result even for the special case a ^ y/3/2 where (3'(^/3/2) = could not be found in the literature. ' and aeC with | Ima|< ^(Rea - v/3/2), we have Re{/'(z) + oz/"(z)}>0 implies | arg /'(z) | < 7T/3 and Re f'(z) > 2<5(Re a) - 1. We can use Corollary 1 to improve the result obtained by Yoshikawa and Yoshikai in [12, Theorem 4] concerning the transformation ) = /(z)exp-z-' , for (21) of the well-known y-spiral-like functions. His result proves that for | y \ < n/2, From Corollary 1, with A(z) = 1/c, we see that we can improve the above implication to zf'to l- implies arg zF'(z) F(z) < 7T/3 whenever <n/3; or, equivalently, if 1 arg -- < Tt/3, then 3/2 c 2 , , , v z/'(z) e*-e-^z . rzF(z)l e v ----< implies e iy -- f(z) l+z * [_ F(z) J i-t-z We next prove the following lemma and then apply this to derive Theorem 4. Lemma 3. Let a* 0.407 - - be t/ze root of the equation a* = tan[(27i-37ra*)/6] (22) Differential subordination and Bazilevic functions 1 79 and 6 = a*7i/2. Suppose that /? is the smallest positive root of the cubic equation Further let F(z) be a complex function that satisfies |argf(z)|<a*7r/2. (23) J/pec/' satisfies Hh (1 -flzp'(z) - (]S + (1 - j8)p(z))]} >0 (24) Rep(z) > m A. Proof. First, we write and F(z) = X + iY = ReF(z) + ilm F(z). Let us now apply Lemma B. Then for all x, y reals and zeA, we have From this it is easily verified that Rei^(ix, 37; z) < - (Kx 2 + Sx + T) for all x real, j> < (1 +x 2 )/2 and all zeA, where Therefore Re^(ix, 3;; z)< if, as usual, Rx 2 + Sx + T ^ for all real x. The second inequality holds if and only if S 2 ^4RT. By performing further algebraic simplifications, it can be easily seen that this is indeed equivalent to 7 1 < (tan (a*7u/2))AT, i.e., | argF(z) | < a*n/2, where the required identity to claim this is . 180 S Ponnusamy Since this automatically follows from the hypothesis, the desired conclusion now follows from Lemma B with Q={coeC: Reo)>0} and (7). Therefore the proof is complete. D \. Theorem 4. Letfestf and /? be as stated in Lemma 3. Suppose that for a ^ ^73/2, -. (26) This implies feS*(0). Proof. Suppose that / satisfies (26). Then taking p(z) = f(z) and rj = *Jl/2 in Theorem 3, we obtain Re {/'(z) + (j3/2)zf'(z) } > (27) and | arg /'(z) | < n/3. Thus from [6, Theorem 5] we get arg where a* is as in (22). Now we need only to show that (27) implies /eS*(/?). For this we let Then by performing differentiation and some algebraic simplifications, (27) deduces to where The theorem now follows from Lemma 3. Taking a= 1 in the above theorem we obtain the following. COROLLARY 3. Let fi be as in Lemma 3. Ifges/ satisfies - 2-(2-73)(21n2-l) then the Alexander Operator I(g) defined by o is in S*(j8), where j8 is as in Lemma 3. Differential subordination and Bazilevic functions 181 Observe that a little computation shows that /? is slightly bigger than the value btained in [9, Corollary 3]. Further the above corollary favours the existence of family of analytic functions, containing non-univalent functions, mapping onto '*(/?) c 5* under the Alexander Operator. "heorem 5. Let a be a real number with a < 2/kd. Let M(z) = z n + a n + k z n + k + and J(z) = z n + be analytic in A (n ^ 1, k ^ 1) and let N satisfy <5, (0 < <5 < 1/n). (28) A 2 Voo/. If we let Q = {coeC: Re co < )8}, j5 x = 2$ + fc<5a/2 + k(5a, A(z) = N(z)/zN'(z) and (z) = (l-/? 1 )" 1 (M(z)/N(z))-jS 1 ), then p(z)= l+p k z k + is analytic in A and the ondition (29) implies iA(p(z),zp'(z);z)eQ /here i//:C 2 x A->C is ^(r,s;z) = jSi +(l~)8 1 )[r-faA(z)s]. Since N satisfies (28), we have Re /l(z) > & in A. If X is real and y ^ - fc(l + x 2 )/2 then Dr this ^ we have ince a < 0, i.e. i^(z'x, j;;z)^Q. Hence (7) is satisfied and Lemma B leads to Rep(z) > 0. liis shows the first part of (30). Since l-a>0, this proves (1 -a)Re(M(z)/N(z)) > I a)/?!. Moreover, from this and (29) we easily have the second inequality of (30). lence the theorem. D :OROLLARY 4. ^t|A|<l andfes/ k .(i)If Re{(l 4- Az)[(l .+ aAz)/'(z) + a(l + Azjz/^z)] } < j? (31) ' 182 S Ponnusamy then for fax + 2(1 + |1|)<0, p ->-ri\ Ree /(2) Proo/. For the proof of (i) we choose M(z) = z/'(z) and N(z) = z/(l + /z). Then and M'(z) + a = Since /GJ^ satisfies (31), we have n M(z) 2/3-h 2 4- whenever (5 < 1 - A . But ^ can be chosen as close to 1 |A| as we please and so we can allow 6 - 1 | X | from below. Thus making (5->l 11| we establish our claim. The proof for the case (ii) follows on similar lines taking M(z) = z/'(z) and N(z) = ze A ~. D Similar arguments used in Theorem 5 would help us to prove the following more general result. Theorem 6. Let a be a complex number with Rea< 2n/k6. Let M(z) = z" + a n+k z n+k H and N(z) = z n -\ be analytic in A (n ^ 1, k ^ 1) and let N satisfy Re(aN(z)/zJV'(z)) < (5, (Rea/n < 5 < - 2/fc). , /5 mplies COROLLARY 5. Let aeC be such that Rea < 2m/fe, w/iere m is a positive integer and let (i> 1. satis/}; Proof. The corollary follows from Theorem 6 taking M(z) = (/(z)) m and N(z) = z m . D In the following theorem we generalize the concept of a-close-to-convexity [1] when a is a complex number. Differential subordination and Bazilevic functions 183 rheorem 7. Let M(z) = z" 4- and N(z) = z" 4- be analytic in A and suppose that V satisfies Re(JV(z)/zJV'(z))x5, (Q<6<l/n). (33) Further let k be a complex number satisfying \lmk\^^/DS, 0<D<:(<S"f 2Rek). (34) Then mplies n M(z), (35) -j^i) and define {36) hen pe j/'. From (36) and (35), we obtain, as before, Re ^(p(z), zp'(z); z) > 0, where iA(r, 5; z) = - /J + kft + (1 - jSJ [kr + If we can show that Re \l/(ix, y, z) ^ when y^ (l+x 2 )/2 and x any real, the equired conclusion is immediate from Lemma B and (7). But for this \l/ we obtain iy (34), we deduce that Rei/^ix, 3;; z) < and so the proof is complete. D Examples. Let M(z) = z" + and N(z) = z n H be analytic in A. Then for k eC with Im k | < J&(S + 2ft), and Re(AT(z)/zAT / (z)) > <5 > 0, Theorem 7 shows As a special case of Theorem 7, let /eja/ and keC with | Imfe | < ^/D < v /l+2Rek. n this case, Theorem 7 leads to Re(k/'(z) + z/"(z)) > implies Re f'(z) > l+2Refc-D' 184 S Ponnusamy In particular, this yields Re(fc/'(z) 4- zf"(z)) > ft implies Re f'(z) > (ft < Re fc) provided | Im fc | < jl+2fi. This simple fact for /} = has been used in [9, Theorem 3] to obtain an affirmative answer to a problem of Mocanu (for details see [9]). Problems Suppose that pe^', ft<\,p = /? + (! ~^)[2^(Rea)-vl ] and H be defined by r i -p) (l-a)^ ; Now by setting | z| = 1, i.e., z = e ie and H(|z| = 1) = U + iV, we easily obtain This, upon simplification for the case a real, yields the parabola T/2 _-2(l-p)r .a(l-p)"| -4(l -__ + __,_ and so for real a, the function H maps the unit disc \z\<\ into the convex domain, say >, bounded by the above parabola. Observe that the domain D contains {coeC: Rcaj! + ((a + 2)5(a) -( + !))} for ]8<1. Also from the sharp subordination relation (11) and a little manipulation we have the following implication ' and p(z) + azp'(z) X H(z) implies p(z) -< provided Re a > 0. From this, it is interesting to note that the same bound in Lemma 1 may be obtained under weaker hypothesis, though the images of A under p, respectively under the stated conditions on h and H, are different. Here h is as in the proof of Lemma 1 and H as above. Problem L Find a (convenient) function G(z) such that G(A) c Jff(A) for which /ej/and/'(z) + a z/"(z)xG(z) implies /eS*? D For a < - 2, let : Re(/'(z) + az/"(z)) < /?}, (/? > 1). Differential subordination and Bazilevic functions 185 For fetf, a < - 2 and Re ] (1 - a) + a/'(z) I < /?, by Theorem 5, we have ( z \ z 2 + a "' v ' 2 + a ' However, for aeC, Re a < 2, Theorem 6 yields fejtf and Re (/'(z) + az/'(z)) < jB implies Re /'(z) > . 2 + Rea In particular for a < 2 and /? ^ a/2, /eP(a, /?) implies Re/'(z) > and further it is easy to show that Although a function fejtf such that Re/'(z) > in A is univalent, Krzyz [3] showed that such a function need not be starlike in A. As pointed out in [10] there are functions, say / in jtf satisfying the condition \f'(z) 1 1 < 1 in A, but they are not in general starlike in A. However the natural problem is the following: Problem 2. Find certain subsets Q of the left half plane, such that /eS*, whenever /'(z) 4- az/"(z) belongs to Q for all zeA and a< 2. In particular, under what conditions on /? and a, z(F*G)'(z) is starlike in A whenever F and G belong to P(a, /?). Here * between two functions denotes Hadamard convolution. D For (5^0, define a ^-neighborhood of /(z) = z + a 2 z 2 4- e j/ by k\a k -b k \<S /c = 2 ^-neighborhoods were introduced by Ruscheweyh [11], who used this to generalize the result that N x (z) c 5*. Now for O 0, let U(a) = {fes/: R(f'(z) + xzf'W > 0, zeA}. It is known [9] that, R (a) c 5* at least when a ^ 0.4269 . Using Lemma 1, it is seen that if /ejR(l) then Re/ / (z)>21n2- 1 and hence proceeding as in [11], it is not difficult to show that A^ina-iW 1 )) c R ()- Interestingly Ruscheweyh proved that if/ is in S*(/3) then there is no value of 5 > such that N 6 (S*(P)) c S* for any < j8 < 1. In spite of this, it seems reasonable to ask the following: Problem 3. Do there exist some conditions on a and <5 such that N d (R(a)) c 5*? If so, what is the best possible d for a suitable fixed a? D 186 S Ponnusamy Acknowledgements The author is grateful to Prof. V Singh for the present form of Lemma 1 and Dr Sankaran for his help in preparing the manuscript. This work has been done with the support of National Board for higher Mathematics and the results of this paper is a part of an internal report (1990). References [1] Chichra P N, New subclasses of the class of close-to-convex functions, Proc. Am. Math. Soc. 62 (1977) 37-43 [2] Hallenbeck D J and Ruscheweyh S, Subordination by convex functions, Proc. Am. Math. Soc. 52 (1975) 191-195 [3] Krzyz J, A counter example concerning univalent function, Mat. Fiz. Chem. 1 (1962) 57-58 [4] Miller S S and Mocanu P T, Second order differential inequalities in the complex plane, J. Math. Anal. Appl. 65 (1978) 289-305 [5] Miller S S and Mocanu P T, Differential subordinations and Inequalities in the complex plane, J. Differ. Equ. 61 (1987) 199-211 [6] Miller S S and Mocanu P T, Marx-Strohhacker differential subordinations systems, Proc. Am. Math. Soc. 99 (1987) 527-534 [7] Mocanu P T, Ripeanu D and Popovici M, Best bound for the argument of certain analytic functions with positive real part, Prepr., Babes-Bolyai Univ., Fac. Math., Res. Semin. 5 (1986) 91-98 [8] Ponnusamy S and Karunakaran V, Differential Subordination and Conformal Mappings, Complex Variables: Theory and Appl. 11 (1989) 79-86 [9] Ponnusamy S, Differential Subordination and Starlike Functions, Complex variables: Theory and Appln. 19(1992)185-194 [10] Ponnusamy S, Convolution of Convexity under Univalent and Non-univalent Mappings, Internal Report (1990) [1 1] Ruscheweyh S, Neighborhoods of univalent functions, Proc. Am. Math. Soc. 81 (1981) 521-527 [12] Yoshikawa H and Yoshikai T, Some notes on Bazilevic functions, J. London Math. Soc. 20 (1979) 79-85 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 2, May 1995, pp. 187-192. Printed in India. Convolution integral equations involving a general class of polynomials and the multivariable //-function K C GUPTA, RASHMI JAIN and PA WAN AGRAWAL Department of Mathematics, M.R. Engineering College, Jaipur 302017, India MS received 23 December 1993; revised 16 August 1994 Abstract. In this paper we first solve a convolution integral equation involving product of the general class of polynomials and the H-function of several variables. Due to general nature of the general class of polynomials and the H-function of several variables which occur as kernels in our main convolution integral equation, we can obtain from it solutions of a large number of convolution integral equations involving products of several useful polynomials and special functions as its special cases. We record here only one such special case which involves the product of general class of polynomials and Appell's function F 3 . We also give exact references of two results recently obtained by Srivastava et al [10] and Rashmi Jain [3] which follow as special cases of our main result. Keywords. The convolution integral equation; multivariable H-function; general class of polynomials; Laplace transform. 1. Introduction On account of the usefulness of convolution integral equations, a large number of authors, notably Srivastava [5], Kalla [4], Buschman et al [1], Srivastava and Buschman [8], Srivastava et al [10] and Rashmi Jain [3], have done significant work on this topic. In the present paper we develop generalizations of results of the last two papers referred to above. Also, Srivastava and Buschman [7, pp. 34-42 and 4.3] have discussed extensively such family of convolution integral equations as those considered here and in the works cited above. We start by giving the following definitions and results which will be required later on. (i) A general class of polynomials [6, p. 1, eq. (1)] S M TY! = V - N ' k Y fc N 019 n 1^ W L-^J /j . * > n u, i, z, . . . (.i'*-) where M is an arbitrary positive integer and the coefficient A Ntk (N, k ^ 0) are arbitrary constants real or complex. On suitably specializing the coefficient A Ntky S^ [x] yields a number of known polynomials as its special cases. These include, among others, Laguerre polynomials, Hermite polynomials and several others [12, pp. 158-161]. (ii) A special case of the ./f -function of r variables [11, p. 271, eq. (4.1)] 187 188 K C Gupta et al r H h 2, , i), ..,; - - - ; (o, D, ( Or equivalently [10, p. 64, eq. (1.3)] H (1.2) (1.3) lt,,...,fc r =0 where nm-cf+7'% i=l,...,r) (1.4) = = = 1= For the convergence, existence conditions and other details of the multivariable H-function refer the book [9, pp. 251-253, eqs. (C.2)-(C8)]. (iii) The following property of the Laplace transform [2, p. 131] J -' u vv> "j " j \"j \^*W holds provided that / (0 (0) = 0, i = 0, 1 , 2, . . . , n - 1, n being a positive integer, where L{/(x);s} = I V s */Mdx=/(5). (1-7) Jo (iv) The well-known convolution theorem for Laplace transform L j [V(x - tt)0() du; s j = L{/(x); s}L{g(x); s} (1.8) Uo J holds provided that the various Laplace transforms occurring in (1.8) exist. 2. Main result The convolution integral equation _z r (x~u)_ (2.1) Convolution integral equations has the solution given by 189 = r^x- Jo where Re(J - p - //) > 0, Re(p) > g (i) (0) = (i = 0, 1, . . . , / 1), / being a positive integer and Ej is given by the recurrence relation or by and ^ is least J? for which A B ^ where M + 2 + 1 oo... o o ... o =i ;=! 1=1 = r(l - - yf fe ( ) and AT 'N (2.3) (2.4) (2.5) (2.6) -i (2.7) -1 (2.8) (2.9) Proof. To solve the convolution integral equation (2.1) we first take the Laplace transform of its both sides. We easily obtain by the definition of Laplace transform and its convolution property stated in (1.8), the following result z r x dx f(s) = g(s). (2.10) 190 K C Gupta et al Now expressing the Sjjf [( z r+ t )x] and H involved in (2.10) in series using (1.1) and (1.3), changing the order of series and integration and evaluating the x-integral, we obtain /( S ) = (2.11) where A(fc l9 . . . , k r+ x ) is defined by (2.6). Now making use of the known formula [10, p. 67, eq. (2.3)], we easily obtain from (2.11) (2.12) where 1 B is defined by (2.5). Again, (2.12) is equivalent to B=O (2.13) If /i denotes the least B for which k B ^ 0, the series given by (2.13) can be reciprocated. Writing ( 2 - 14 ) B=O J j=o eq. (2.13) takes the following form: f(s) = s>- l +*ZEjS-Wg(sn. (2.15) J=o (2.15) can be written as L{f( X );s} = LJ Ej*"'^ 1 ^\L{^( X )i3} (2.16) u-o ro + /-Ai-p) J [on using (1.6)]. Now using the convolution theerem in the RHS of (2.16) we get (2.17) oj=o Finally, on taking the inverse of the Laplace transform of both sides of (2.17) we arrive at the desired result (2.2). 3. Special cases If we put r = 2 in (2.1) and reduce the H-function of two variables thus obtained to Appell's function F 3 [9, p. 89, eq. (6.4.6)] we find after a little simplification that the convolution equation given by Convolution integral equations 191 (3.1) r (l)\p/ r (2h has the solution (3.2) where Re(/ - p - /*) > 0, Re(p)>0, Z^X-M)^!, |z 2 (x-w)|<l, (0 (0) = (i = 0, 1,...,/ 1), / being a positive integer and E 7 - are given by recurrence relation (2.9) or (2.4) and \JL is least B for which A B ^ where in (3.3) (3.3) and o, N' .M. 'N" M (3.5) will reduce to A 0t0 which can In the main result if we take N = (the polynomial be taken to be unity without loss of generality), we arrive at a result given by Srivastava etal [10, p. 64, eq. (1.1)]. Again, if we put r = 1 , p = q = 0, z 2 = 1 in the main result, and further reduce the Fox's //-function thus obtained to exp( zj [9, p. 18, eq. (2.6.2)] and let z 1 -*0, the Fox's H-function reduces to unity and we arrive at a result which in essence is the same as that given by Rashmi Jain [3, pp. 102-103, eqs (3.5), (3.6)]. Acknowledgement The authors are thankful to the referee for making useful suggestions. References [1] Buschman R G, Koul C L and Gupta K C, Convolution integral equations involving the //-function of two variables, Glasnik Mat. Ser. Ill 12 (1977) 61-66 [2] Erdelyi A, Magnus W, Oberhettinger F and Tricomi F G, Tables of integral transforms (1954) (New York/Toronto/London: McGraw-Hill), Vol. I [3] Jain R, Double convolution integral equations involving the general polynomials, Ganita Sandesh, 4 (1990) 99-103 192 K C Gupta et al [4] Kalla S L, On the solution of certain integral equations of convolution type, Ada Mex. Cienc. Tecnol. 2(1968)85-87 [5] Srivastava R P, On certain integral equations of convolution type with Bessel-function Kernels, Proc. Edinburgh Math. Soc. 15 (1966) 111-116 [6] Srivastava H M, A contour integral involving Fox's ^-function, Indian J. Math. 14 (1972) 1-6 [7] Srivastava H M and Buschman R G, Theory and applications of convolution integral equations (1-992) (Dordrecht/Boston/London: Kluwer Academic Publishers) [8] Srivastava H M and Buschman R G, Some convolution integral equations, Proc. Ned. Akad. Wet. Ser A77, Indagationes Math. 36 (1974) 211-216 [9] Srivastava H M, Gupta K C and Goyal S P, The H-Functions of One and Two Variables with Applications (1982) (New Delhi: South Asian Publ.) [10] Srivastava H M, Koul C L and Raina R K, A class of convolution integral equations. J. Math. Anal. Appl 108 (1985) 63-71 [11] Srivastava H M and Panda R, Some bilateral generating functions for a class of generalized hypergeometric polynomials, J. Reine Angew. Math. 283/284 (1976) 265-274 [12] Srivastava H M and Singh N P, The integration of certain products of the multivariable H-function with a general class of polynomials, Rend. Circ. Mat. Palermo, 32 (1983) 157-187 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 2, May 1995, pp. 193-199. Printed in India. On L ^convergence of modified complex trigonometric sums SATVINDER SINGH BHATIA and BABU RAM Department of Mathematics, M.D. University, Rohtak 124001, India MS received 20 July 1993; revised 21 June 1994 Abstract. We study here L 1 -convergence of a complex trigonometric sum and obtain a new necessary and sufficient condition for the L 1 -convergence of Fourier series. Keywords. L 1 -convergence of modified complex trigonometric sums; L 1 -convergence of Fourier series; Dirichlet kernel; Fejer kernel. 1. Introduction It is well known that if a trigonometric series converges in L 1 to a function /eL 1 , then it is the Fourier series of the function /. Riesz [1, Vol. II, Ch. VIII 22] gave a counter example showing that in a metric space L we cannot expect the converse of the abovesaid result to hold true. This motivated the various authors to study L 1 -convergence of trigonometric series. During their investigations some authors introduced modified trigonometric sums as these sums approximate their limits better than the classical trigonometric series in the sense that they converge in L 1 -metric to the sum of the trigonometric series whereas the classical series itself may not. Let the partial sums of the complex trigonometric series be denoted by S H (C,t)= Z c k J \k\*n If the trigonometric series is the Fourier series of some /eL 1 , we shall write c n = f(n) for all n and S H (C 9 t) = S H (f, t) = S n (f). If a k = o(l) as fc-oo, and ^^ =1 k 2 \A 2 (a k /k)\< 60, then we say that the series ] =1 a k <!>fc(x), where k (x) is cos fcx or sin foe, belongs to the class R. Kano [2] proved that if ZfcLi fl fc*k( x )' belongs to the class IR, then it is a Fourier series or equivalently, it represents an integrable function. Ram and Kumari [3] introduced modified cosine and sine sums as and *.(*)= 7/ 194 Satvinder Singh Bhatia and Babu Ram and studied their L 1 -convergence. The aim of this paper is to study the //-convergence of the complex form of the above sums. Let - sin ~ t 2 ^ m=l ~ 2sm- 2 cos cos n + - \t 2 V 2; 2sin- and 4sin 2 - 2 sin(n+ denote the Dirichlet's kernel, the conjugate Dirichlefs kernel, and the conjugate Fejer's kernel respectively. Let E n (0 :=: Z!Uo eik '- Then the first differentials D' n (t) and D' n (t) of D H (t) and D n (t) can be written as where E' n (t) denotes the first differential of E n (t). The complex form of th^e above modified sums is g n (C, t) - S n (C, t) -f n n _ M . n . We introduce here a new class R* of sequence as follows: Definition. A. null sequence <c a > of complex numbers belongs to the class #* if /clogfc< oo, k=l < 00. (1.1) (1.2) 2. Lemmas. The proof of our result is based upon thefollowing lemmas, of which the first three are due to Sheng [4]: Lemma L \\D f n (t)\\, =4/7r(nlogn) + o(n) Lemma 2. \\ D' n (t) \\ t = o(n log n). Lemma 3. For each non-negative integer n, there holds Modified complex trigonometric sums 195 ||c n ;W + c_ M / _ n (0|| 1 =o(l), n-oo if and only ifnc n log\n =o(l), |n|->oo, where <c n > is a complex sequence. Lemma 4. (i) There exist positive constants a and /? such that 1 aOognXII^WH^ (ii) || K' n (t)\\i = o(4 Proof. The existence of ft follows from the fact that || /)() II i = o(logrc). Further, we have 2n\\K n (t)\\ 1 ^ p n (r)dr Jo i n r * "i - rZ Zd-cosM7 n+ lfc = oU=o J for some constant M, the last step being the consequence of the relation " = 1 logi; = logrc!. Using Sterling's asymptotic formula n\ ~^/2nnn n e~ n , we then have This completes the proof of (i). To prove (ii) we have, \D' n (t)\ = kcoskt and so This implies that \K' n (t)\dt = o(n). /n DiiBferentiating ^ n (t) we get W^Zi-W-Z where ln (t) = {cos t - cos(n + l)t} ( 4sin 2 ~\ 196 Satvinder Singh Bhatia and Babu Ram ( t 3n (0 = {2sinrsin(w+ l)f}/(w 4-1)1 2 sin- 1 Obviously, \L jn (t)\ = o(\t\' 2 ) for 7 = 1,2, and (n + l)|S 3ll (t)| = o(|f|~ 3 ). Using these estimates, we get Combining the above estimates, we infer that ||XJ,00||i = o(n). Lemma 5. Let n ^ 1 and < e < n. Then there exists A & > such that for alls^\t\^n (i) \E' n (t)\^A e n/\t\, (ii) |E'_ B (t)|<4 8 /i/|t|, (iii) \D' n (t)\^2A t n/\t (iv) |^(t)|^^ n/|t|. Proo/ We have Since | M (OI < AJ\t\ for some constant ^4 e , we have Since L n (t) = (-i)E:(-t), we obtain \E'_ n (t)\^A e n/\t\. The other two inequalities follow from D'(i) = E'(t} + E'_ n (t) and 2i5;(t) = ;(t) - '.(). 3. Main theorem We prove the following result. Theorem. Let c n eR*. Then there exists f(t) such that lim g n (C 9 1) =f(t)for allQ<\t\^ n, (3.1) -*oo v ' f(t)eU(T) and \\ g n (C, t) -f(t) \\ , ='o(l) as n -* oo, (3.2) II S n (f, t) -/(t) || j = o(l) as n -> oo if and only iff(n) log | n \ = o(l) as | n \-> oo. (3.3) Modified complex trigonometric sums 197 Proof. We have, by using Abel's transformation, n (C,r) = S n (C,)4 -[c n+1 J n + 1 - c _ (n+ !,_ = 2 By Lemma 5, we get I k=l and fc=3 where y4i_ is a suitable constant. These imply that *=i exists and thus (3.1) follows. Further, for t ^ 0, we have f(t)-g n (C,t) = 2 ; = 2 ' Thus =l C -*~ C *W) < 00, sW \K'(t)\dt n + 198 Satvinder Singh Bhatia and Babu Ram But, by Lemma 4, Also + E \E'_ k (t)\dt. by the hypothesis of the theorem. Lemma 1 and Lemma 2 imply that Therefore, A 2 T +0(1) ^V-^ fclog/c = o(l), by the hypothesis of the theorem. Since g n (C, t) is a polynomial, it follows that /eL^T), which proves the assertion (3.2). We notice further that n+1 and -(/(n+i);w-/(- = II fc,(C, t) -$.(/) || 1 /-, Since ||/-?.(C,t)|| = o(l), n->oo, by (3.2), and by Lemma 3, B /(-n)'_ n (t)|| 1 = o(n), n->oo if and only if J(n)log|n| = o(l), |n|-*oo, the assertion (3.3) follows. Modified complex trigonometric sums 199 References [1] Bary N K, A Treatise on Trigonometric Series (1964) (London: Pergamon Press), Vol. II [2] Kano T, Coefficients of some trigonometric sums, J. Fac. ScL Shinshu Univ. 3 (1968) 153-162 [3] Ram B and Kumari S, On L 1 -convergence of certain trigonometric sums, Indian J. Pure Appl Math. 20-(1989) 908-914 [4] Sheng Shu Yun, The extension of the theorem of C V Stanojevic and V B Stanojevic, Proc. Am. Math. Soc. 110 (1990) 895-905 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 2, May 1995, pp. 201-205. Printed in India. Absolute summability of infinite series C ORHAN and M A SARIGOL* Department of Mathematics, Faculty of Science, Ankara University, Ankara 06100, Turkey * Department of Mathematics, Erciyes University Kayseri 38039, Turkey MS received 4 November 1993; revised 22 April 1994 Abstract. It is shown in [4] that if a normal matrix A satisfies some conditions then |C, l| k summability implies \A\ k summability where k ^ 1. In the present paper, we consider the converse implication. Keywords. Normal matrix; |C, l| fc summability; \A\ k summability. 1. Introduction By uJJ and tJJ we denote, respectively, the Cesaro means of order a (a > 1) of the sequences (s n ) and (rj, where (s n ) is the partial sums of the series Ex n and r n = nx n . The series Ex n is then called absolutely summable (C, a) with index k, or simply summable |C,a| fc ,k>l,if (1) n= 1 Since t^ = n(u^ u < ^_ l ), [3], condition (1) can be written in the form Z n- l \? H \ k <ao. (2) n=l Let A =(a nv ) be a normal matrix, i.e., lower-semi matrix with non-zero diagonal entries. By ( TJ we denote the ^-transform of the sequence (s n ), i.e., We say that the series x n is summable \A\ k , k ^ 1, if i^-^r.-T.^l^oo. (3) M=l Given a normal matrix A = (a nv ), we associate two lower-semi matrices A (d nv ) and A = (d) as follows: 201 202 C Orhan and M A Sarigdl &nv = "nv-an-l.v for H= 1,2,... If A is a normal matrix, then A' (a' nv ) will denote the inverse of A. Clearly, if A is normal then A = (d nv ) is normal and it has two-sided inverse A' (a' nv \ which is also normal (see [2] ). Note that, if A is normal then n n 'n n T n = Z fl mA = Z Z ^m*. = Z ** y = w = i = r t' = and &T n ^ = T n ~ T a ^= Z (<^~^-i, t X = Z V. K-i. = 0), y=0 y=0 which implies *= I CAT,., (r_ 1= 0). (4) y = In connection with the absolute summability we have the following theorem. Theorem A. Suppose that, for k ^ 1, X (1/^)1^1 =0(l/n) and \&J v-1 n = v then ifl<x n is summable C, 1 | fc , it is also summable \ A | fc , where M nv = d nv d n v+ 1 [4]. Furthermore it is shown in [4] that the conditions of Theorem A are satisfied whenever A is (C,a), a>l. This deduces that |C, l| fc summability implies |C,a| fc , k ^ 1, a ^ 1, summability which is a well-known result. We may now ask what conditions should be imposed on A = (a nv ) so that the converse implication holds in Theorem A. It is the object of this paper to answer this question, 2. The main result Theorem B. Let A = (a nv ) be a normal matrix such that (i) l=0(vaJ 9 (iii) Z ( + 2)| tf w+2i | = 0(i + l). (5) v = i If I,x n is summable \A\ k , then it is also summable |C, 1 | k ,fc ^ 1. Absolute summability of infinite series 203 Proof. By T n and t n we denote the A-transform and (C, l)-mean of the series I,x n and the sequence (nxj, respectively. Then it follows from (3) that (n+l)- 1 i)=l n v-2 Zv 1 I y=0 r=0 By considering the equality n Z ^nk^ = ^t; 5 /c = u where <5 Ml) is the Kronecker delta, we have vd' vv + (v + l)4; +lfi; = i;/fl ui; + (t; + 1)(- & v+ i tV /a vv a v+ +ltU+l -a v + ltV /a vv a v+ltV+l ']-' l/a vv and so n(n+l)- 1 (l/OAT n _ 1 +a' 10 AT_ 1 (n+l) * which implies, by virtue of (5i), (5ii) and (5iii), that -1 To prove the theorem, it is enough to show that oo Z n " 1 l w nil k< for z=l, 2, 3. 204 C Orhan and M A Sarigdl Now it follows from Holder's inequality that m+l f+l ( n ~ l B ? 2 n ""*" 1 { l ,? 1 C m m+l = i^ (. t = 1 and Finally, Z "-'W-O =l Hence the proof of the theorem is completed. 3. Applications Let (p n ) be a sequence of positive real numbers such that P n = p + P! H ----- hp w , P_ 1 =p_ 1 =0. The Riesz (weighted mean) matrix is defined by a nv = p v /P n for < v ^ n and <2 ny = for v > n. From now on, we suppose that A = (a nu ) is a weighted mean matrix with P n -+co and n - oo. Hence if no confusion is likely to arise, we say that Ex rt is summable |/?,pj k ,fc > 1, if (3) holds. With this notation we have COROLLARY 1 Let (p n ) be a sequence of positive real numbers such that P n = 0(np n ). Then if Sx rt is summable \R,p n \ k9 it is also summable |C, l| k ,fc > 1. Proof. Applying Theorem B with A = (a nv ), a weighted mean matrix, we see that (5ii) clearly holds and (5i) is reduced to the condition P n = 0(np n \ On the other hand, a small calculation reveals that and if V=n otherwise. Absolute summability of infinite series 205 Thus we get v-i and so the proof is completed. COROLLARY 2 Let (p n ) be a sequence of positive real numbers with np n = 0(P n ). Then if Sx n is summable |C, l| k , it is also summable \R,p n \ k , (k ^ 1). Proof. Apply Theorem A. Now the next result which appears in [5] is a consequence of Corollaries 1 and 2. COROLLARY 3 Suppose that (p n ) is a sequence of positive real numbers such that np n = 0(P n ) and P n = (nP n ). Then the summability |C, l| fc is equivalent to the summability \R 9 p n \ k ,k^ 1. Acknowledgement This paper was supported by the Scientific and Technical Research Council of Turkey (TBAG-CG2). References [1] Flett T M, On an extension of absolute summability and theorems of Littlewood and Paley, Proc. London Math. Soc. 7 (1957) 113-141 [2] Cooke R G, Infinite Matrices and Sequence Spaces (Macmillan) (1950) [3] Kogbetliantz E, Sur les series absolument sommables par la methode des moyennes arithmetiques, Bull. Sci. Math. 49 (1925) 234-256 [4] Orhan C, On absolute summability, Bull. Inst. Math. Acad. Sinica 15 (1987) 433-437 [5] Orhan C, On equivalence of summability methods, Math. Slovaca 40 (1990) 171-175 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 2, May 1995, pp. 207-218. Printed in India. Solution of a system of nonstrictly hyperbolic conservation laws K T JOSEPH and G D VEERAPPA GOWDA TIFR Centre, P.B. No. 1234, Indian Institute of Science Campus, Bangalore 560012, India MS received 30 November 1993; revised 17 June 1994 Abstract. In this paper we study a special case of the initial value problem for a 2 x 2 system of nonstrictly hyperbolic conservation laws studied by Lefloch, whose solution does not belong to the class of L functions always but may contain ^-measures as well. Lefloch's theory leaves open the possibility of nonuniqueness for some initial data. We give here a uniqueness criteria to select the entropy solution for the Riemann problem. We write the system in a matrix form and use a finite difference scheme of Lax to the initial value problem and obtain an explicit formula for the approximate solution. Then the solution of initial value problem is obtained as the limit of this approximate solution. Keywords. System of conservation laws; delta waves; explicit formula 1. Introduction The standard theory of hyperbolic systems of conservation laws assumes usually the systems to be strictly hyperbolic with genuinely nonlinear or linearly degenerate characteristic fields, see Lax [6] and Glimm [1]. But many of the hyperbolic systems which come in applications do not satisfy these assumptions and such cases were studied by many authors [3, 5, 8]. In all these papers solutions are found in the sense of distributions, say in the class of L functions. In a very interesting paper, Lefloch [7] considered a system of conservation laws, namely dv d - + (a(u)v) = ^ dt dx with initial conditions M(X, 0) = M O (X), i?(x, 0) = v (x\ (1.2) where a(u) = f'(u) and /: R -> R is a strictly convex function. For systems of this type generally there is neither existence nor uniqueness in the class of entropy weak solutions in the sense of distributions. He has shown that when u Q L 1 (R)r\BV(R) and v eL co (R)^L l (R) (1.1) and (1.2) has at least one solution (u, v)eL*(R+,BV(R)) x given by 207 208 K T Joseph and G D Veerappa Gowda u(x,t) = (/*)' | where 3; = y (x, minimizes min [ P ,, -oo<y.<ooLJ-co and /* is the convex dual of f(u) and M(R) is the space of bounced borel measures on R. Further he proved that if w satisfies (1-3) dx in the sense of distributions for some K , then the problem (1.1) and (1.2) has one and only one entropy solution. If we take (U L ifx<0 -, A [U R ifx>0 then (1.3) is equivalent to saying R\ <P ^ and this will be true for some K Q and for all <peCJ(#), q> ^ 0, iff U L ^ w^. In fact for the Riemann problem, i.e., when the initial data for (1.1) is of the form (1.4) U R ,V R ix>, Lefloch [7] has given an infinite number of solution for the case U L < U R . In this paper we study a criteria to choose the correct entropy solution. Classically, vanishing viscosity method or proper numerical approximations are used to choose the correct entropy solution. Following Hopf [2], vanishing viscosity method was used by Joseph [4] to pick up the unique solution for the Riemann problem when f(u) = w 2 /2 in (1.1). It was shown that in the case, U L < U R , which is the case of non- uniqueness, the v component of the vanishing viscosity solution is !i? L , if x < u L t 0, if u L t <x<u R t V R , if x > u R t. In other words in the rarefaction fan region of u component, the v component is zero. In the present paper, we consider the special case f(u) = log [ae u + be'"'], a + b = 1, a > 0, b > are constants in (1.1). Then we have (1.5) ae u -be~ u - v =0, u ~ u Nonstrictly hyperbolic conservation laws 209 and study the unique choice of solution. Here we use a numerical approximation of Lax [6], which he used to pick the correct entropy solution for a scalar conservation law. For the Riemann problem, we show that in the rarefaction fan region of M, the v component is zero, see Theorem 1. These examples suggest a uniqueness criteria at least for the Riemann problem. Before stating our main results let us introduce the difference approximation. To do this first we note that (1.5) can be written in the matrix form + fre-^)] x = 0, (1.6) where A u Am Let Ax and A be spatial and time mesh sizes and let A n k ~A(k&x,n&t) 9 fc = 0, 1, 2,..., n = 0,l,2,... (1.8) and following Lax [6], define the difference approximation where the numerical flux g(A 9 B) is given by g(A 9 B) = log[ae A + be-*]. (1.10) Here we can take At = Ax = A, since the characteristic speed of the eigenvalues ae u .\)Q~ U A! = A 2 = - - of (1.5) which are less than one in modulus. Then we note that (1.9) ae u + be " and (1.10) become ^ = ^-\4-lo g [a^ : i + ^-<" 1 ]~log[^^ 1 +^-^^] (1.11) with initial conditipn When o (1.11) is nothing but the Lax scheme for the scalar equation u t 4- (log[ae u + be~ u ~}) x = 0. With the notations 5 = (log|> u * 4- be~ UR ] - loglX^ + &e~ Ul -])/K - U L ) (1.13) and x , x of* -be~ u * ae^-be^ R(UL> UR, VL, V R) = S ( V R " V L) -- UR ^ , _ UR ^ + ^ . . -^ ^ (1-14) a^ R -f fce * a^ L -f oe i - we shall prove the following results. 210 K T Joseph and G D Veerappa Gowda Theorem 1. Let (W A (X, t),v\x, t)) be the approximate solution o/(l.l) defined by (1.11) and (1.12) with Riemann initial data (1.4), then lim (U A (X, t), t; A (x, t)) = (u(x, t), i;(x, t)) m /z sense of distributions and (u(x,t\v(x,t)) is given by the following explicit formula: (i) When U L > U R , then (u(x, t), v(x, t)) = {U L + (U R - u L )H(x - st), V L + (V R - v L )H(x - st) where H(x) is the Heaviside function. (ii) When U L < U R , then v2 - \a t-x K, *>*) (iii) P^/ien U L = MR = u, then ., if X > ae UR -be~ UR Theorem 2. Let t/ie im'fia/ data u(x) and i;(x)eL 00 (R)nL 1 ( J R). T/ien (w A (x,r)X(x,r)) defined by (1.11) and (1.12) tends to (w(x,r),z;(x,t)) m the sense of distributions and is given by t + x- y (x, t) a t - x + y Q (x 9 1) d f t?x,t=- 3; = j; (x, t) maximizes max Here /*(A) is the convex dual of f(u) = \og[_ae u + be~ u ~] and is given by 2. Proof of Theorem 1 As a first step in the proof of Theorem 1, we obtain (W A (X, t), ^ A to do this we recall from (1.11), (1.12) and (1.4), , 0) explicitly. In order Nonstrictly hyperbolic conservation laws for n= 1,2,3,. ..,/c = 0,l, 2,..., with '11. if*< .. Let us set then, (2.1) becomes Let Taking summation in (2.4) from k to oo, we have 211 (2.1) (2.2) (2.3) (2.4) (2.5) Following Lax [6], we use the nonlinear transformation, D = log in (2.6), and obtain log E = log - * + log [0e x *(E^ (2.6) (2.7) Simplifying this we get . k k where a = ae AR (ae A * -\-be~ AR )~~ l and /? = We note that a + /? = /. It can be easily seen that the solution El of (2.8) is given by ? -iC From (2.2), (2.3), (2.5) and (2.7) we get '1 s 1 (2.8) (2.9) (2.10) (2.11) 212 K T Joseph and G D Veerappa Gowda Using (2. 11) in (2. 10) we get where (2.13) and <= >^^^ , (2-14) Here we used the notation S(n,k,q) = ^n + k-2q-\n+k-2q\). Now o o (2-15) By the transformations (2.3), (2.5) and (2.7) we get, Componentwise this becomes By using Stirling's fonnula, n!f-Y(27cii) 1/2 , we get n n -, asn,<j,n-<?-o. Let t = nA, x = kA, y = (n + k - 2$)A be fixed, then Nonstrictly hyperbolic conservation laws t+x-y ^ qA = , (2q - n) A = x - y. . 213 (2.19) We have, lim A log 0J = max Alog ( H ] + Aq loga + A(n - <?)logft A-+0 0<(r + ;c-y)/2<f L \#/ Also as A ->0 in the above fashion, we have , -.U R ) - 1 \og(ae UR + be' UR ) (2.20) (2.21) Alogf") W x\og f ft + x- y y + *-/2 ( t -. x + y\-*+'V* LV 2 ) ( 2 ) J and hence from (2.19)-(2.21), we get , lim A log 9 n k = max L-l/2(y-\y\)(u L -u R ) A-*0 x-t^y^x + t + (x-y)u R -tlog(ae u * -loga + -log 6 log . (2.22) Let y (x, t) be the value of j; for which maximum is attained on the RHS of (2.22). An easy calculation shows that the following is true. Lemma. Let y (x,t) be a point where maximum is attained on the RHS of (2.22), then y (x,t) is given by the following: (i) Let U L > U R , then x a(u L )t, if x < st x a(u R )t, if x > st. Let U L < U R , then )x - a(u L )t, if x < a(u L )t 0, ifa(u L )t<x<a(u R )t x - a(u R )t, if x > a(u R )t, where a(u)~f(u) = (ae u -bt and s is given by (1.13). 214 K T Joseph and G D Veerappa Gowda From the above lemma and (2.22) we have if U L >U R , then lim A ^ Alog0J| ^x, t), where - (x-a(u L )t)(u L -u R ) + u R a(u L )t - Hog(ae" '1 A 1 (x,t) = 2 if * < st a(u R )tu R - t log(ae UR + be" UR ) + 1~*K) flog a 2 J ~(V 2 If U L < U R , then lim A ^ Alog&jJ == A 2 (x, t)> where (x a(u L ) t)(u L M A 2 (x,t} = if x < a(u L )t, t -\- X t _ X XU R - 1 log [ae" R 4- be~ MR ) -I- - log a 4- - log fc + t log t (t-hx) (t-x). /t-x u R a(u R )t - 1 log ( t log a Again from (2.13), (2.14) and (2.19), we get limA^ = - A-0 ffi Nonstrictly hyperbolic conservation laws 215 no- where ,y (x, t) maximizes the RHS of (2.22). Using the lemma we have the following: If U L > U K , then A-0 where If U L < U R , then where (x-a(u L )t(v L -v R )) v R \_ae UR (a(u L ) - l)r + be' u *(\ -(x-a(u L )t)(v L -v R ) + M^ v R be ~ UR (t -h x) + i^ Now it follows that, if U L > U R roo Km (u*(y,t)-u R )dy=Ai(x,t\ A-*0 J x f* lim (i; A (y,r) A-^Ojjc and if W L < U R lim I (M A (y,t)-M A->Oj;c lim I (*\y,t)- A-^O J* Hence be~"*(l +a(u , ux>st. , if x < ^( , if x > a(u R )t. 216 K T Joseph and G D Veerappa Gowda ~ 3X v > if U L < U R dx dB 2 ox in the sense of distribution as A-+0. An easy calculation shows that dA,_\u L , U R - dx dx U R , if x > st, dA 2 dB 1, fbt + xl -log -- , * L^ t~xj ,-~v R )ll-H(x-st)l if x < a(u L )t tfa(u L )t<x<a(u R )t ifx>a(u R )t, t? L , ifx<a(u L )t 0, ifa(u L )t<x<a(u R )t V R , if x > a(u R )t. Proof of (iii) is similar. The proof of Theorem 1 is complete. 3. Proof of Theorem 2 To prove Theorem 2, we first note that the approximate solutions are defined by Al = Al~ l +log[ae j4 - 1 + be~ A * ] log[ae^ +be~^+ 1 ]> (3.1) with / t ,o n \ (3.2) where u = w(fcA) 5 v% == v(kA). Following Lax [6], let us introduce and use the nonlinear transformation We get as before n + 1 ___ k whose solution is 00 ,JU" (00 Z - + \ j 1 Nonstrictly hyperbolic conservation laws ^= z In terms of the original variable A n k , we have Carrying out the explicit calculations as before, we get n / \ r oo Q ya<b"-'expj_ 2 217 "2 = log Zo W J Z n = j ~~ Now let x = fcA, t = nA, 3; = (n + k 2<?) A be fixed and let A -* 0. Lax has shown that 2 where y = )> (x, t) maximizes max (3.3) where Again the same analysis of Lax [6] gives Too oo / lim U A (X, r)dy = lim A 0J = i? (z)dz. Here again 3; = y (x,t) maximizes (3.3). Since $v*(y,t)dy is a sequence of bounded function converging to J* (jct) t; dx for a.e. (x,-t), it follows that ^x,?) converges to 3 f 00 ^x I in distribution. The proof of Theorem 2 is complete. Acknowledgements One of the authors (KTJ) thanks the Indo-French centre for the Promotion of Advanced Research, New Delhi, for the financial support to visit France, where he had several fruitful discussions on this problem, and Prof. C. Bardos and Prof. O. 218 K T Joseph and G D Veerappa Gowda Pironneau for their hospitality. The authors are also thankful to the unknown referee for his constructive criticism which improved the presentation of the paper. References [1] Glimm J, Solutions in the large for nonlinear systems of equations, Commun. Pure Appl. Math. 18 (1965) 697-715 [2] Hopf E, The partial differential equation u t + uu x - uu xx1 Commun. Pure Appl Math. 3 (1950) 201-230 [3] Isaacson E L and Temple B, Analysis of a singular Hyperbolic system of conservation laws, J. Differ. Equ. 65 (1986) 250-268 [4] Joseph K T, A Riemann problem whose viscosity solutions contain ^-measures, Asymptotic Analysis 7 (1993) 105-120 [5] Keyfitz B and Kranzer H, A system of non-strictly hyperbolic conservation laws arising in elasticity theory, Arch. Ration. Mech. Anal 72 (1980) 219-241 [6] Lax P D, Hyperbolic systems of conservation laws II, Commun. Pure Appl Math. 10 (1957) 537-566 [7] Lefloch P, An existence and uniqueness result for two nonstrictly hyperbolic systems, in Nonlinear Evolution Equations that Change Type (eds) Barbara Lee Keyfitz and Michael Shearer, IMA (1990) (Springer- Verlag) Vol. 27 126-138 [8] Liu T P, Admissible solutions of hyperbolic conservation laws, Am. Math. Soc. Mem. (1981) (AMS, Providence RI) Vol. 240. Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 2, May 1995, pp. 219-225. Printed in India. Oscillation in odd-order neutral delay differential equations PITAMBAR DAS Department of Mathematics, Indira Gandhi Institute of Technology, Sarang, Talcher, Orissa, India MS received 15 January 1994; revised 16 June 1994 Abstract. Consider the odd-order functional differential equation (x(t) - ax(t - T)) (W) + p(t)f(x(t - a)) = 0, (*) where 0<a<l, T, ae(0,oo), peC([0,oo), (0,oo)), f<=C l (R,R) such that / is increasing, xf(x) > for x ^ and / satisfies a generalized linear condition liminf ,-.0 IVdx Here we prove that every solution of (*) oscillates if liminf a"-" 1 p(s)d5>-(l -a)(n- !)!( -? ] \ - J t - ff m e \n-ij This result generalizes a recent result of Gopalsamy et al [6]. Keywords. Functional differential equations; oscillation of all solutions. 1. Introduction In a remarkable result Ladas [4] proved that every solution of the first-order delay differential equation (j) = 0, (1) where p,cre(0, oo) oscillates (i.e., every solution has an unbounded set of zeros in (0, oo)) if and only if P<r>. (2) e The result was extended by authors in [5] for general odd-order differential equation x (n \t) + px(t-a) = 9 (3) replacing (2) by The first result was further improved (see [7]) for equations with variable coefficients with the statement that 219 220 Pitambar Das P 1 liminf p(s)ds>-, I /> t--oo Jt-ff ** and limsup I p(s)ds>- t-OO Jt tr & L are respectively sufficient and necessary conditions for every solution of where peC([0, oo),(0, oo)), to be oscillatory. But a similar extension for has not been proved yet. Recently, Gopalsamy et al [6] proved that iminff (t-s)-- t-oo >- (6) implies that every solution of the odd-order differential equation (x(t) - ax(t - r)) (n) + p(t)x(t - a) = (7) oscillates, where ^ a < 1. Indeed, for a = and p(t) = pe(0, oo), (6) reduces to pa" > n!, that is, n n which is the sufficient condition for oscillation of (3). In view of the condition given in (4), the lower bound of p l/n (<r/n) in (8) is comparatively larger than that of (1/4 In this paper we prove a result, a particular case of which shows that all solutions of (7) are oscillatory if f r 1 f n I"" 1 lim inf (f 1 ' l p(s) ds > -(1 - a)(n - 1)! < > . (9) When p(t) = p(0, oo) and a = 0, the above condition reduces to '(n!) 1 /". (10) In view of the known inequality i/ Functional differential equation 1/n __ i 221 (11) where C(nl,r) is the (r+l)th binomial coefficient in the expansion of (1 + l/(n I))"" 1 , our condition is weaker than that of (8). We give examples to support our claim. 2. Main results Consider the odd-order nonlinear functional differential equation WO x(t ~ T)) (n) + P(t)f(x(t - (7)) = 0, (E) with the assumptions that p 6 C(R + ,R + \fe C(R, R) such that / is increasing, x/(x) > for x + 0, |/(x)| - oo as (x| - oo, ^ a < 1, (H) n > 1 is an odd integer and T, <re(0, oo). Let <5 = max{t, a] and <^eC([jT <5, T],R). By a solution of (E) in [T, oo), we mean a function xeC([T, oo), R) such that x(0 = ^(0 T-d^t^T, (x(t) - ox(t - t))e C (n) ([T, oo ), R) and x(0 satisfies (E) for t^T. As usual, a solution x(t) of (E) is called oscillatory if it has zeros for arbitrarily large t and nonoscillatory, otherwise. We say (E) is generalized sublinear if / satisfies lim inf *-o superlinear if lim inf ,-o and linear if lim inf dx dx dx <1 (12) (13) which includes the cases f(x) x*, < a < 1, l<a<oo and a = 1, respectively. In what follows, we list the following two results for our use in sequel. Theorem 1 ([3], Lemma 1). Suppose that 0eC (n) ([T, oo),(0, oo)) such that g (i) (t) has no zeros in [T, oo) (i = 1,2, . . .(n - 1)) and g (n \t) ^Ofart^T. If fie(0 9 oo) then (n-1)! " Theorem 2 ([7], Theorem 2.1.1). //]8e(0, oo), QeC([T, oo),(0, oo)), T>0 and f r lim inf f-oo Jf- lim inf Q(s)ds>-, 222 Pitambar Das then the first-order differential inequality has no eventually positive solutions. Our main theorem is as follows. Theorems. Suppose that (H) holds and f satisfies (13). Then (9) implies that every solution of(E) oscillates. Proof. Since (9) holds, there exists < e < 1 such that (l-e) 2 liminf| <r n - 1 '- Ji-a/ii (14) To the contrary, assume that x(t) is a nonoscillatory solution of (E). Let x(t) > for t ^ t . (The case for x(t) <Q 9 t^t Q may be treated similarly.) Setting z(t) = x(r)-ax(t-t), (15) from (E) it may be observed that z (n \t) < for f > t -f <r. Consequently, there exists 7> r + o- such that z (i) (r) (i = 0, 1, 2, 3. . . (n - 1)), has no zeros in [T, oo). Suppose that z(t)<0, t^T. Since n is odd, z (n) (r)<0, f^T implies that z'(t) < 0, > T. On the other hand, let If n = oo, there exists a sequence of real numbers <O?L i such that t n - oo, x(r w ) -+ oo as n -* oo and x(s) < x(t n ) for s < *. From (15) we see that - ax(t n - T) > (1 - a)x(O, which further gives lim z(t n ) = oo, n-*oo a contradiction to the fact that z(t) < 0, r> T. In case /i is finite, there exists a sequence O*>*i such that t B -^oo, x(tJ->/i as n-^oo. Since <x(t ll -T)>* s:1 is a bounded sequence of real numbers, it admits a convergent subsequence. Let <s n >* =1 be the subsequence for which <x(s w -t)>J . 1 converges to a real number A. Clearly A< ji. Again x(s w ) -+ ^ as n ~> oo. Now lim zfa.) = lim (x(s n ) - ox(s n - T)) > (1 - a)A, n-^oo n-*oo that is, which is a contradiction to the fact that z(t) is negative and decreasing function. Hence f z(t) < 0, t ^ T is impossible. Let z(t) > 0, t > T. Clearly, it follows that z'(t) < 0, t > T. Indeed, otherwise, z'(0 > 0, t ^ T implies that lim inf z(t) > and consequently Functional differential equation 223 lim inf x(t) = lim inf (z(t) + ax(t - T)) > 0. t-*QO t-+00 Integrating (E) from T to t and using the above observation along with the fact that (9) implies r p(s)ds=oo, we see that z (w 1) (r)-> oo as t-+co. Consequently, z(f)-+ oo as ~ao, a contradiction. Further z (n \t) < implies that Consequently, and lim z(t) r-*oo If k > 0, then repeating the argument applied earlier we lead to a contradiction. Hence k = 0. From (13) it follows that lim inf iminff ) = 1. y-o \dyj Taking yCO = z<n ~ 1) (0> and from the definition of limit infimum it follows that for every e > there exists a large positive number M such that >(l-e) fortttM. (16) AyJ From (15) we see that t), t^T. (17) The repeated application of (17) on it, as per the idea in the paper of Gopalsamy et al [6] results in ^ = From the above inequality it follows that there exists M^T+Ni such that (18) , (1-a) In Theorem 1, replacing g(t) by z(t - a/n) and ft by ( - \a we get \ n / or. (19) 224 Pitambar Das Using (19) in the inequality obtained by replacing t by t a in (18) we get x(t - (r) > Kz (n -"(t-a/n\ t ^ maxfMj + <r, T+ 3d} = T , where (20) Since / is increasing, f(x(t-v))^f(Kz (n - l \t-(r/n)), t^ T . (21) From (E) and (21) it follows that z (n \t) + p(t)f(Kz (n ~ l \t - a In)) < 0, t ^ T . (22) Multiplying both sides of (22) by (f(y)\ where y = z (n ~ l \t), we obtain (f(y(t))) -H ( P(t) }f(Ky(t a In)) ^0, t^ T . (23) Set Now z {n " 1] (t) > 0, t ^ T implies that H(t) > 0, t^ T . From (23) and (16) it follows that dt Hence H(t) is an eventually positive solution of the differential inequality given in Theorem 2, where and =<7/n. But, by (14), lim inf Q(s) ds = lim inf ) K(l - e)p(s) ds > -, J-*oo Jt <r/n t-*cb Jt-fffn & a contradiction to Theorem 2. Hence (E) cannot have a nonoscillatory solution. This completes the proof of this theorem. Example. Consider the equation Since (6) fails to hold, Theorem 4.1 of Gopalsamy et al [6] is not applicable, but (9) holds and hpnce Theorem 3 shows that every solution of it oscillates. Functional differential equation 225 Remark. In view of the inequality n n\e\n-lj J 2 it follows from (10) that (24) ,nj 2V n or in particular, (25) 3 implies that every solution of eq. (3) oscillates. Indeed, n > 1 and odd gives that Since the arithmetic mean exceeds the geometric mean r(n r) < I - I for every r and hence 2 Consequently, using Binomial theorem we get d -d/)) (26) n \ 2) 2\ nj 2\ n Now (24) follows from (11) and (26). Since n ^ 3, (25) follows from (26). Acknowledgement This research was supported by the National Board for Higher Mathematics, Department of Atomic Energy, Government of India. References [1] Das P, Oscillation criteria for odd order neutral equations, J. Math. Anal Appl. (to appear) [2] Das P, A note on a paper of Shreve, J. Math. Phys. Sci. 27 (1993) (to appear) [3] Das P, Oscillation of odd order delay differential equations, Proc. Indian Acad. Sci. (Math. Sci.) 103 (1993) 341-347 [4] Ladas G, Sharp conditions for oscillations caused by delays, Appl. Anal. 9 (1979), 95-98 [5] Ladas G, Sficas Y G and Stavroulakis I P, Necessary and sufficient conditions for oscillations of higher order delay differential equations, Trans. Am. Math. Soc. 285 (1984) 81-90 [6] Gopalsamy K, Lalli B S and Zhang B G, Oscillation of odd order neutral differential equations, Czech. Math. J. 42 (1992) 313-323 [7] Ladde G S, Lakshmikantham V and Zhang B G, Oscillation Theory of Differential Equations with Deviating Arguments (1987) (New York: Marcel Dekker) [8] Swanson C A, Comparison and oscillation theory of linear differential equations (1968) (New York and London: Academic Press) 'roc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 2, May 1995, pp. 227-239. D Printed in India. lurface waves due to blasts on and above inviscid liquids of finite depth C R MONDAL Department of Mathematics, North-Eastern Hill University, Shillong 793 022, India MS received 21 July 1992; revised 22 April 1994 Abstract. For the problem of waves due to an explosion above the surface of a homogeneous ocean of finite depth, asymptotic expressions of the velocity potential and the surface displacement are determined for large times and distances from the pressure area produced by the incident shock. It is shown that the first item in Sakurai's approximation scheme for the pressure field inside the blast wave as well as the results of Taylor's point blast theory can be used to yield realistic expressions of surface displacement. Some interesting features of the wave motion in general are described. Finally some numerical calculations for the surface elevation were performed and included as a particular case. Keywords. Surface waves; inviscid liquid; asymptotic expansion; blast theory; surface elevation. . Introduction 'he problem of surface waves caused by the interaction of a blast-generated shock /ave with an ideal incompressible fluid has been analysed by Rumiantsev [9], Kisler 4] and Sen [11], mainly when the fluid is infinitely deep. The problem of waves iroduced by explosions above the surface of a shallow liquid has also been touched ipon by Kranzer and Keller [5] as an application of the asymptotic Caucny-Poisson yave theory for fluids of finite depth. This treatment, however, did not include the fleets of the time variation of the pressure distribution on the surface. Choudhuri 1] and Wen [14] considered the case where the disturbance is over any arbitrary egion of the free surface and the water is of uniform finite depth by the method of aultiple Fourier transforms. In both the cases the method of stationary phase was ipplied to obtain the approximate expression for the potential function and surface levation for large values of time and distance. Mondal and Mukherjee [8] considered he corresponding problem by Hankel transform method and finally the approximate xpressions for the potential function and inertial surface elevation were obtained or large distances and times by the method of stationary phase. The basic simplifying assumption in this problem is that the large difference between he densities of the gas and the fluid make the fluid displacements too small to affect he motion of the gas, which is supposed to be known. Here we present the three- limensional problem of the generation of waves due to explosions above the surface >f a fluid of constant finite depth due to the incident shock and of the area on which t acts. After deriving the formal solution of the problem in terms of infinite integrals n the usual manner, we use the known asymptotic expansions of the Bessel function ind Kummer's confluent hypergeometric function alongwith the method of stationary 227 228 C R Mondal phase to find approximate expressions of the velocity potential and the surface displacement (= ) integrals of large times and distances from the pressure area. For the pressure field inside the blast wave, we first make use of an expression closely &+ resembling the first term of Sakurai's [10] approximation scheme. It is also easy to see that the expressions of ( in the form of infinite series may be obtained by the same methods as used by Sen [11], but these will not be deduced here. Instead, we describe the more tractable features of the asymptotic wave motion in its general form as well as special forms which use the results of the Taylor point blast theory, and then place our results on a more realistic footing. 2. Formulation of the problem We assume that surface waves are excited when the spherical shock wave due to a point blast in the gas interacts with the fluid surface. An expanding circular region of pressure is formed on the free surface as a consequence. Using cylindrical X coordinates (r, 6, z), we write the governing equations as follows: f " For t > 0, (1) (2) .dt dt 2 dz dt' <p(r,0,0) = 0, <p,(r,0,0) = , (5) = on*--*. (6) dz The conditions (5) are equivalent to the conditions (p = 0, C = at t = 0, since p is finite and r-Oast-0-K 3. Formal solution We assume a solution of (2) of the form f 00 ' q> = A(k, t)J (kr)coshk(z -h Wscch Jkfcdfc Jo so that (6) is satisfied. Surface waves due to blasts , 229 Substituting for (p in (4), we obtain the following differential equation for A(k,t): d f ro(0 kp- 1 a/ St J Q here 'he solution of this equation is g fro(5) A A Q (k)cos(crt Q o f r si Jo An integration by parts then gives A = (k/p) Pcos|>(r - s)]ds | r <S) o/(a,s)J (fox)da. (7) Jo Jo le velocity potential is therefore f coshfc(z-Hh) f r f ro(s) <p = p~ 1 kJ (kr) dk cos<r(t-s)ds oc/(a,s)J (/ca)da. Jo coshkh J J (8) "he surface displacement is then determined by (3): C= -(gpr 1 r*kJ (kr)dk f sma(t~s)ds f^ a/(a, 5 )J (/ca)da. (9) Jo Jo Jo . Asymptotic representation of cp and ^ for a uniformly expanding pressure area Ve adopt the following model for f(r,t) because it closely resembles the first term >f Sakurai's [8] approximation scheme for the determination of the pressure field aside the blast wave /(r, = (t + t i r n F(r/r (t)) 9 r < r (t) (10) ^here n(> 1) is non-integral, and r x is the time taken (from the moment of the xplosion) by the shock front to just reach the surface. Also, at high pressures, l(t 4- tj = the radius of the shock front at time (t + t t ) cc(t + t^ 215 Ref. [6] From this result, one can obtain the expression for r (t): 230 C R Mondal Here, however, we assume, alongwith the model (10), that r (t) = ut, u = constant, (11) for convenience of analysis. Then, from (8) f 1 f 00 (p = p~ 1 af(a)da kJ Q (k Jo Jo x rl(s)(t l + s)~ n co$0(t-s)J Q (kotr Q (s))ds. (12) Jo To evaluate (12) asymptotically for large r and t, we first replace J (/car (s)) by its integral representation, f* irfi J (/car (s)) = (2/7r) cos (/caws sin 0)d0, (13) Jo and J (fcr) by the first term of its asymptotic expansion for large fcr, J Q (kr) - (2/nkr) 1 ' 2 cos(fcr - n/4). (14) The resulting s-integral is expressed in terms of a function T n (ia,t,ti) defined as follows: ) = e iflfl s 2 (t,+s)- n e ias ds Jo s $ I ( f t l +tl) . (15) Here sin0}, 7 =1,2. (16) and ifi denotes Kummer's confluent hypergeometric function. In place of (12), we have now ' fl f 1/2 Re F(a)da Jo J o x [exp{irP.(fe)} + exp{i>Q.(fc)}]dk, (17) where Q/k) = a(t -f t t )r" ' + (- iykiiat 1 r" 1 sin[0 + k - (ir/4r)] and K > is such that Kr 1, and the stationary point(s), if any, lies in (K, oc). Surface waves due to blasts 231 The function Qj(k) has no stationary point for ; = 2 in < k < oo, and none either in the same interval for; = 1, since ut l r. Therefore, the part of the fe-integral arising from Qxp{irQj(k)} in (17) is O^" 1 ), as r- oo. The function P;(fc), on the other hand, has one and only one stationary point, k = kj (say), when 5 + 5'" (' where (5 yi is Kronecker's delta function. To show this, we note that (i) Pj(k) is continuous in < fe < oo, (ii) Pj(k) is strictly monotone decreasing in < k < oo, since PJ(/c) < therein, (iii) P'.(k)-*2i 1 + ( I) j uctt 1 r~' l sm0 = a 1 (say) as /c->0 + Pj(k)-+( I) j uat 1 r~ 1 sind 1 =a 2 (say), as fc-*oo, < for both 7, since ut^ r~ l 1. These conditions make P'.(k) vanish once and only once in < k < oo, when ( fl i)min > 0> that is, when T > - H - S Jl9 as stated above. A similar argument shows that the equation [Pj(/c)] M=0 = has one and only one non-negative real root fe = k Q (say), independent of j, when t > 1/2 and hence, under the condition (19) as well. Since ut 1 r~ l 1, an approximate value of kj may be obtained by putting kj = k Q + ej (20) in the equation P^.(/c) = 0, whence e.^( l)- /+1 uat 1 r~ 1 sin0/PJ(fe ). (21) Applying the method of stationary phase to evaluate the fe-integral of (17), we obtain i: x coshfc^ 2 ' x Re X TJjafal t, I, )expi{rP ; (fe ; ) - 7t/4} j (22) (npu- 2 /2)q> - aF(a)da o o The asymptotic expansions of the functions ^ for large arguments [Erdelyi, [2] I, 6.13.1 (2)] show that (t + tj-^^x.+.o, (23) where we suppose n < 2, a restriction required for the Taylor point blast theory. 232 c R Mondal Using the approximations (20) and (23), we get for (22), the expression -7 P f*' 2 2 (npu 2 /2)(p~\ jp(a)da d<9 Jo Jo o x cosh [fe (z + ft)] sech fc ft . (24) j By [Erdelyi, [2], I, 7.12. (45)], we have |**/2 (2/7t) sin{k r + (- I) j k utasme}d9 Jo = sin(fe r)J (fe ua) + (- iy(2/)cos(/c r) s 0(0 (fc itfa) (25) when T > 1/2 + ~ so that both ^ and fc 2 exist, the Lommel function s M (k utx) cancels out in the j-sum of (24). The asymptotic expression for <p thus finally becomes F (fe uOsin(/c r), (26) where (27) )= | Jo o The result (26) holds under the conditions rr (r)r (t 1 ), k rl, *>- + (28) A similar process applied to (9) gives for the surface displacement the asymptotic expression U 2 t 2 (29) under the same conditions (28). If F(x) = D, < x < 1, the limiting value of C, as ft-, oo, equals the corresponding value of C for the case of infinite depth [Sen [1 1], eqn. (69)]. 4.1 An illustrative case When a concentrated explosion of constant total energy E takes place in a still atmosphere of density Po , Taylor's formula for the maximum pressure (which happens Surface waves due to blasts 233 to be on the shock front) is ^ax = 0- 14 HVo(' + 'l)~ 6 } 1/5 (30) when the ratio of specific heats of air is about 1.4. If we adopt this law of pressure for an approximation in the present case while retaining the hypothesis r (t) = ut for a relatively small spread of the pressure area, we have n = 6/5, and F(R) = 0-141 (E 2 po) 1/5 for all R so that Equation (29) then gives (E 2 o 3 } 1/5 ut -' 141 - 'I W-^-M*.*)"*^) (31) subject to the conditions (28). 5. Wave elevation due to a Taylor point blast above the fluid surface At the outset, we transform the general expression (9) for C as follows: Writing f 00 _(0p)-itlm vkJ tyr)e iff(st+ti) dk Jo x f r 2 Q (st)'(t 1 +str n e- ia(st+tl) F(kr (st))ds. (32) Jo j' Jo (33) We follow the same procedure as shown in 4, it being assumed that the function F(kr Q (st)) is sufficiently well behaved, and it does not make T n strongly oscillatory or singular for large t. The latter is a pre-requisite for the applicability of the method of stationary phase [Stoker, [12], 6.8]. Then asr' = (r/7z)->oo . (34) where P(k) = (t + tjr- ^(gktanhkh) 112 - k + 7i/4r, P"(fc) = PJ(jfc) as obtained from (18), and k = fc is the non-negative real root of P'(k) = 0. 234 c R Mondal This approximation holds under the conditions (t + tj^/ghr'^l, rr (t), fc rl. (35) For large t, T^fco^i) approximates to ). (36) Therefore C^^prr^VI^Mr 1 ' 2 ^ (37) This result is used below to determine the wave height caused by a Taylor point blast above the fluid surface. 5.1 Pressure inside a blast wave: Taylor's formula For an intense explosion of constant total energy E occurring at a point 0' at a height H above the ground, the pressure p(r,z,t) inside the expanding spherical blast wave and the radius R(t) of the shock wave at time t from the moment of the explosion are given by the following formulae due to Taylor [13]: (38) (39) ,, * 2y Py+1 ^-i"| /ifoH-fr - -- ~ y + li_ 7 7 J (40) (41) Here n = (7y - l)/(y 2 - 1), y = ratio of specific heats of air ~ 1.4. z = depth of a point vertically downwards from 0'. r = distance of a point P from the perpendicular O'O on the surface. The surface pressure distribution in the present problem may therefore be taken as (42) , r>r f where r^t) = J R 2 (t + t 1 )~ J R 2 (t 1 ) (43) and t 1 = time taken by the shock to reach the surface. 5.2 Adjustment of Taylors formula to the wave problem The pressure model (10) without the one for r (t) results from the above when H = 0. The same model may be retained when H is small compared with r (t) or R. To this purpose, a Lagrange expansion (MacRobert [7], 54) of p Q (r,t) is useful. Surface waves due to blasts 235 Writing (44) /i(\/w=/2(AO we get and f ( \ f i v * fH\ 2m d" 1 " 1 i m! \r y dju"" 1 020- v ) Also ^ 2 -# 2X - m Consequently, m=i/=o m!/! x (tj(t + t 1 )} 4(m+1)/5 F m (^L ) L when ^ < 1; Po (r, = 0, when jU > 1 , (47) since R(t + tj = (W26r 2l5 (E/p ) 1/5 (t + r,) 2 / 5 Here ' (49) (50) 5.3 Asymptotic wave height Subject to the validity of the linearised wave theory, the asymptotic expression for C under the conditions (35) is C * - 0-129(E*pV s /gp r ){k /\P"(k )\} l > 2 (t + 1,)- 6 ' 5 m=i /=o ml II 236 C R Mondal where /(aV(fc)da. ( 52 ) The last result shows that a good approximation to ( for small H is obtained if only the first term / ! 2 (fco r oW) * n the square bracket is retained. Further evaluation of C can be accomplished, it seems, only by numerical methods. 6. Some characteristics of the motion In both the expressions (29) and (51) for the wave elevation , fc rl. The factor cosk r in both therefore changes its sign rapidly so that we may regard fc r as the phase and the co-factor of cosfc r as the amplitude of ( in either expression. The phase is not directly affected by the velocity parameter u in (29) or by r (t) in (29) and (51). Since dfc /dt is positive as per (18), the degree of oscillation of level at any point becomes more rapid with time. Since (t 2 k Q h) at first diminishes with t (up to dt the value given by the equation 2T 2 PJ(fc ) = /i) and then increases with it, the oscillation at any point in shallow water is somewhat more rapid at first and less rapid thereafter than what would happen if the sea were deep. Denoting t^/gh/(2r) by T O , we have for * - K(T O ), P;(k )[or P"(/c )] ~ (T O /T)[P;(IC) or (say) and equation (PJ(fc)) l|SBO = 0) may be written as t/^(K) = 1, where [X/c)]- 1 =(7^i/2r)[{tanhfe/i/^} 1/2 + {fcVtanhfcfc} 1/2 sech 2 fcfc]. (53) The amplitude of C in (29) then varies as From (53), it appears that kh = 0(t 2 ) and P"(K) = O(K~ l ) when Kh(or T) 1. As n is usually > 1, one finds that the amplitude ->0 as t- oo when F(x) is 0(x~ i ) or of a higher order of smallness as x -> oo. The times of maximum amplitude at any point are given by 2r (a n tanha n ) 1/2 ^ = -7=...^. . . ....,, > n=l,2,3,..., where satisfy the equation Surface waves due to blasts 237 Therefore, the points of maximum amplitude at a distance r travel outwards with the corresponding constant velocities tanh a n + a n sech 2 a n The amplitude at any point becomes nearly zero at times 2r fc n sech 2 b n where * = >, n=l,2,3,... satisfy the equation F(KUIJ/(K)) = Q. These points of minimum amplitude travel outwards with the corresponding constant velocities tanh b n -h b n sech 2 b n The values of a n and b n increase with n. Hence, the outer rings spread out faster than the inner ones. A similar discussion may be given for (51). 7. A particular case Let Therefore F(k ut) -fW Jo - r k ut Jo D . k^ut Then (29) gives = ut, k rl. (54) By (18), we have a 112 It + t 1 " [4feMtanh^)(sech 2 kh)(l + k tanh kh) A,, .rz 4r(fetanhfc/i) 3 . (55) 238 C R Mondal 0-006- 0-00<<- 0-002- 0- - 0-002 - 0-00* -0-006 - 0-008 -0-01 - 0-012 - 0-OU - Q.Q16 - 0-018 Figure 1. Variation of C 1 with r. u = 0-05, n = 1, g = 32, r, = 0-5, t = 2, h = 1. Now let us take 2 V/ 2 / 3 , cos [k ut--n . nk utj \ 4 Using (55) and (56) in (54), we get r~/i \ 1 /2 "1 r~ *-! r I V ul -^i 1 V^o ut ) C = | -777T II , , . , - t + tj JL'* (56) D * L rk% 2 J L (t + 1 ! ) n J Lr(fc tanh k, x [(4fc /itanhfc /zsech 2 /c /i)(l (tanh fc ft -f fc ^ sech 2 k ft) 2 ] ~ 1/2 x cos(k r)cos(fc u 37i/4), r > wt. -1/2 (57) The variation of _ D t with r as shown in figure 1. Acknowledgements The author is grateful to Professor A R Sen for his help in the preparation of this paper. This work was supported by a Research Fellowship of the CSIR, New Delhi. The author is thankful to the referee for suggesting several improvements which have since been incorporated in the paper. Surface waves due to blasts 239 References [1] Chaudhuri K S, Appl Sci. Res. 19 (1968) 274-284 [2] Erdelyi A, Higher Transcendental Functions, Vols I & II (New York: McGraw-Hill) (1953) [3] Geisler J E, Linear theory of the response of a two-layer ocean to a moving hurricane, Geophys. Fluid Dynam. I (1970) 249-272 [4] Kisler V M, Prikl. Math, Mek (Translated as Appl Math. Mech. 24 (1960) 496-503 [5] Kranzer H C and Kellar J B, Water waves produced by explosions, J. Appl Phys. 30 (1959) 398-407 [6] Kynch G J, Blast waves in: Modern Developments in Fluid Dynamics (ed.) L Howarth, Vol. 1 (1953) pp. 146-157 [7] MacRobert T M, Spherical Harmonics (Pergamon) (1947) [8] Mondal B N and Mukherjee S, Water waves generated at an inertial surface by an axisymmetric initial surface disturbance, Int. J. Math. Educ. Sci. TechnoL 20 (1989) 743-747 [9] Rumiantsev B N, Prik. Math. Mek. (Translated as Appl Math. Mech.) 24 (1960) 240-8 [10] Sakurai A, Blast wave theory in: Basic developments in Fluid Dynamics (ed.) M Holt, Vol. 1, p 320, (Academic Press) (1965) [11] Sen A R, Surface waves due to blasts on and above liquids, J. Fluid Mech. 16, pt i, 65-81 (1963) [12] Stoker J J, Water Waves (New York: Interscience) (1957) [13] Taylor G I, Proc. R. Soc. London A201 (1950) 159 [14] Wen S L, Int. J. Math. Educ. Sci. TechnoL 13 (1982) 55-58 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 2, May 1995, pp. 241-249. Printed in India. Generation and propagation of 577-type waves due to stress discontinuity in a linear viscoelastic layered medium P C PAL and LALAN KUMAR* Department of Applied Mathematics, Indian School of Mines, Dhanbad 826 004, India *NIC Unit, Computer Centre, District Collectorate Building, Dhanbad 826001, India MS received 19 August 1993 Abstract. In this paper the generation and propagation of S/f-type waves due to stress discontinuity in a linear viscoelastic layered medium is studied. Using Fourier transforms and complex contour integration technique, the displacement is evaluated at the free surface in closed form for two special types of stress discontinuity created at the interface. The numerical result for displacement component is evaluated for different values of non- dimensional station (distance) and is shown graphically. Graphs are compared with the corresponding graph of classical elastic case. Keywords. SH-type waves; stress discontinuity; 1. Introduction The usefulness of surface waves and its investigations in isotropic elastic medium have been well recognised in the study of earthquake waves, seismology and geo- physics. Wave propagation in a layered medium has been studied extensively by many people, especially in the last two decades. Various approximate theories have been proposed to predict the dynamic response of layered medium and one of them is due to Sun et al [10]. Nag and Pal [7] have considered the disturbance of SH-type waves due to shearing stress discontinuity in an isotropic elastic medium. In another paper, Pal and Debnath [8] have considered the propagation of SH-type waves due to uniformly moving stress discontinuity at the interface of anisotropic elastic layered media. Due to the effect of viscosity, gravity plays an important role in the propagation of surface waves (Love, Rayleigh, etc.). The viscoelastic behaviour of the material is described by the mechanical behaviour of solid materials with small voids. The linear viscoelasticity generally displayed by linear elastic materials is termed as 'standard linear solid', if elastic materials are having voids. Kanai [5] has discussed the Love-type waves propagating in a singly stratified viscoelastic layer residing on the semi-infinite viscoelastic body under the conditions of the surface of discontinuity. Sarkar [9] considered the effect of body forces and stress discontinuity on the motion of SH -type waves in a semi-infinite viscoelastic medium. The propagation of Sff-waves in nonhomogeneous viscoelastic layer over a semi-infinite voigt medium due to irregularity in the crustal layer has been discussed by Chattopadhyay [1]. He has followed the perturbation technique as indicated by 241 242 P C Pal and Lalan Kumar Eringen and Samuels [4]. The viscoelastic behaviour of linear elastic materials with voids has been considered by Co win [2]. The present paper considers the generation and propagation of SH-type wave due ^ to shearing stress discontinuity at the interface of two homogeneous viscoelastic media. Fourier transform method combined with complex contour integration technique is used to evaluate the displacement function at the free surface for two different types of stress discontinuity. Numerical results are obtained for a case only with the aid of viscoelastic model as considered by Martineck [6]. Results are shown graphically and are found to be in good agreement with classical elastic case. Since the material of the earth is viscoelastic of a standard linear type, certain seismic observations and calculations may be explained on this basis. Thus the problem considered here is of interest in the theory of seismology. 2. Formulation of the problem and basic equations Let us consider a viscoelastic layer of standard linear type (I) of thickness h lying over a viscoelastic half-space (II). The origin of the rectangular co-ordinate system is taken at the interface. The wave-generating mechanism is a shearing stress discontinuity which is assured to be created suddenly at the interface. The geometry of the problem is depicted in figure 1. As the SH-typt of motion is being considered /here, we have u = w = and v = v(x,z,t). The displacement v is also assumed to be continuous, bounded and independent of y. The only equation of motion in two-layered viscoelastic media in terms of stress components is given by where fi t are related to shear moduli and n' t to viscoelastic parameters. Substituting (2.2) in (2.1), the resulting equations of motion become Assuming that the stress functions are harmonic and decrease with time, we have r- and correspondingly where co is the frequency parameter. i SH-type waves due to stress discontinuity y FREE SURFACE ' / / / / / / //J//T'}) f / / J> j I J * /// Figure 1. Standard linear viscoelastic layered model. With the help of (2.5), (2.3) becomes o^K-W | d2V \ ** . aj (dx 2 + dz 2 )' sre G ] = ^ ^ j( . J=12. 243 (2.6) Method of solution us define the Fourier transform V(^ z) of V(x, z) by refore (3.1) (3.2) pplying the above transformation into (2.6), it is found that V(,z) satisfies the ition d 2 v 244 P C Pal and Lalan Kumar where ' ^ 2 + ^> /-1.2. (3 ' 3) Thus for the layers (I) and (II) we have )e' 4x d (3.4) =^-\ 211 J (3.5) The boundary conditions of the problem under consideration are (i) stress component must vanish on the free surface i.e. (T yz ) 1 =0atz=-h forallt>0 (3-6) (ii) displacements must be continuous at the interface i.e. V 1 = V 2 at z = for t > (3.7) (iii) stress components (shearing) must be discontinuous at the interface z = i.e. (V)i = ( T y*)2 = SM*' * at z = 0, for all x and t, (3.8) where S(x) is some continuous function of x to be chosen later. The above boundary conditions determine the unknown constants A 19 A 2 and B 2 . After simplifying we have at the free surface (z = h) ni (3.9) where U() is an unknown function related to S(x) by r r / \ I CV \ ( "& \A C\ 1 C\\ and which is associated with the reflection coefficient in the two media. 4. Determination of unknown function We now consider two different forms of the function S(x) to determine U(Q. ^ Case I. Let S(x) = P, |x|^a ; = 0, elsewhere. . (4.1) SH-type waves due to stress discontinuity 245 This case implies that the stress discontinuity is created in the region a ^ x ^ a. Hence exp(-ic)cbc (4.2) Pi From (3.9) and (4.2) we have n foo / p i$xi ^pi&2\ (x9 - h ) = -r-i m (f - f- \ lpi Jo V nit J (x x = x - a, x 2 = x 4- 0). (4.3) Here we wish to evaluate the integral for a few values of m, say m = 0, 1, 2 only. So we have where = / 01 + / 02 ;(say) (4-5) p x2 /sinjx i _sin^ 1 \ . Jo V ^i ^i / = / n +/ 12 ;(say) ( 4 -6) /a . rvf^i-^V' 9 "^ 2 Jo V ^i ^1 / ~' p K3 /sin|x 1 _sin^ 1 \ _i lljl)ld ^ Jo V ^i &li J = / 21 +/ 22 ;(say). ( 4 - 7 ) To evaluate / 02 . / ii.- ir i2' J 2i' / 22. we use the method of contour integration and / 01 is directly evaluated from the Table of Integral Transforms by Eradelyi [3]. Thus, we have v= 246 P C Pal and Lalan Kumar where K (0) is a modified Bessel's function of argument 9 and of order zero. / =2 I w*"*-*-! ~ SJn X 2 ) f -u/2)[(a>ih 2 /0 2 )-Z 2 where X A = (xjh\ x 2 = (x 2 //i) and wft/crj > co/i/cr 2 r _ 12 _ 02 -- where where -Ri Figure Z Complex contour integration in 4-plane. (49) (4.10) (4.11) (4.12) SH-type waves due to stress discontinuity 247 The integrals in J l5 / 02 ^i 1^12^-- ^ ave branch points at f = coh/<r l9 coh/ff 2 and a simple pole at = 0. The path of the contour integration is shown in figure 2. Hence Case II. Let S(x) = P/i(5(x), -oo^x<oo. (4.14) Factor ft is multiplied on the right side because both sides should maintain the dimension of stress. Now (4.15, Therefore, in this case, we have Ph f 1 (x,-fc) = 5 Re noiPi J f J Ph _ (4.16) In this case also, we evaluate the integral on the right-hand side of (4.16) for a few values of m only, say m = 0, 1, 2. Hence V l (x, - h) = -4 [/o + *i + 7 2 + -1 ( 4 - ! 7 ) J = [1-f Ke~'"' I ]d Jo >7i = / 01 +/ 02 (say) (4.18) ' cos^xe~ nih _ 2rjih 2 -3,,^-.,^ o ^7i = / n +/ 12 (say) (4.19) J o *7i = / 21 +/ 22 (say). (4.20) Just like case I, we can evaluate / 01 ,'o2^n>^.i2--- as foU ws: (4.21) '-f J c 248 P C Pal and Lalan Kumar etc. '02= -4 / u =-4 0>/t/<72 e-t*e~ 7 + - or O) '? J (4.22) (4.23) (4.24) by The integrals in (4.22), (4.23), (4.24) are valid only when (oh/a, > coh/a 2 . Hence, m this case the displacement component on the free surface z = - h is given e~'Ph (4.25) h = 37-5 Km I - 0-2 II - 0-5 III - 0-7 IY - 2-0 0-1 0-1 0-1 1-0 VISCO -ELASTIC ANALOGY *5 2 2-5 3 3-5 4 4*5 5 5-5 6 6-5 7 7-5 Figure 3. Variation of displacement with distance from the source. SH-type waves due to stress discontinuity 249 Numerical results and discussion umerical calculations are performed here for case II only using Gauss quadrature rmula and the table of integral transforms (Eradelyi [3]). The values of Kv 1 x 10~ 2 , here K = nalp^e^/P are tabulated for different values of x and Q.^ =a)/2/<7 1 and jeping 1 2 <z>h/(j 2 constant. The values of non-dimensional parameters f! 1 and f! 2 e taken from a viscoelastic model considered by Martineck [6]. For comparison graph corresponding to isotropic case is drawn (figure 3) and is found to be in good ;reement with viscoelastic analogy up to a certain value of x. From the curves so awn, it is inferred that the displacement v l decreases as x increases and the rate of :crease slows down after a certain distance. :knowledgement ne of the authors (LK) is thankful to Sri S B Singh, NIC, Dhanbad, for his technical :lp during the preparation of this paper. rferences I] Chattopadhyay A, The propagation of SH-waves in nonhomogeneous viscoelastic layer over a semi- infinite Voigt medium due to irregularity in the crustal layer, Bull Calcutta Math. Soc. 70 (1978) 303-312 >] Cowin S C, The visco-elastic behaviour of linear elastic materials, J. Elasticity 15 (1985) 185-201 J] Eradelyi A, Table of Integral Transforms Vol. 1 (New York: McGraw-Hill) (1954) I] Eringen A C and Samuels C J, On perturbation technique in wave propagation in a semi-infinite elastic medium, J. Appl. Mech. 26 (1959) 491-503 5] Kanai K, A new problem concerning surface waves, Bull. Earthquake Res. Inst. 39 (1961) 359-366 5] Martineck G, Torsional vibration of a layer in a visco-elastic half-space, Acta Tech. CSA V Bratislava ' 24(1970)420-429 7] Nag K R and Pal P C, Disturbance of SH-type due to shearing stress discontinuity in a layered half space, Geophys. Res. Bull. 15 (1977) 13-22 ?] Pal P C and Debnath L, Generation of SH-type waves in layered anisotropic elastic media, Int. J. Math. Sci. 2 (1979) 703-716 )] Sarkar A K, On SH-type of motion due to body forces and due to stress discontinuity in a semi- infinite viscoelastic medium, Pure Appl. Geophys. 55 (1963) 42-52 )] Sun C T, Achenbach J D and Herrman G, Stress waves in elastic and inelastic solids, J. Appl. Mech. 35 (1968) 467-485 roc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 2, May 1995, pp. 251-257. ) Printed in India. proof of Howard's conjecture in homogeneous parallel shear flows - : Limitations of Fjortoft's necessary instability criterion MIHIR B BANERJEE, R G SHANDIL and VINAY KANWAR Department of Mathematics, Himachal Pradesh University, Shimla 171 005, India MS received 4 May 1994 Abstract. The present paper on the linear instability of nonviscous homogeneous parallel shear flows mathematically demonstrates the correctness of Howard's [4] prediction, for a class of velocity distributions specified by a monotone function U of the altitude y and a single point of inflexion in the domain of flow, by showing not only the existence of a critical wave number k c > but also deriving an explicit expression for it, beyond which for all wave numbers the manifesting perturbations attain stability. An exciting conclusion to which the above result leads to is that the necessary instability criterion of Fjortoft has the seeds of its own destruction in the entire range of wave numbers k > k c a result which is not at all evident either from the criterion itself or from its derivation and has thus remained undiscovered ever since Fjortoft enunciated [3]. Keywords. Shear flows; Introduction tie point of inflexion theorem of Rayleigh [5] and the semicircle theorem of Howard >] impose necessary restrictions on the basic velocity field U(y) and the complex ave velocity field c = c r -h ic t which are accessible to an arbitrary unstable (c f > 0) ave in the linear instability of nonviscous homogeneous parallel shear flows and it is ' interest to have a similar restriction on the growth rate kc t possible for such an istable wave, fe being the wave number and y being the altitude. In his pioneering >ntribution (1961; henceforth referred to as Ho), Howard established one' such timate in the form Max ~, (1) Flow domain \ ^ / id considering its inability to provide the correct qualitative result for the case of ane Couette flow with dU/dy constant, which is known to be neutrally stable with : f - as k -> oo remarked "This estimate is not usually sharp for example, the ouette flow with dU/dy constant, is known to be neutrally stable but in most cases will probably give the correct order of magnitude of the maximum growth rate. It is ifficient to show that c t must approach zero as wavelength decreases to zero given e boundedness of dU/dy, but there is likelihood that infact kc t .~0 as fc- oo, and ith sufficient assumptions the still stronger statement that all waves shorter than 251 252 Mihir B Banerjee et al some critical wavelength are stable is probably true, as illustrated by the examples of Drazin and Holmboe cited in I". A rigorous mathematical proof of the first part of this conjectural assertion of Howard, namely that kc t -> as k -> oo, was given in an earlier paper by Banerjee et al [1] under the restriction of the boundedness of d 2 U/dy 2 in the concerned domain of flow and the present paper which is in continuation to the earlier one mathematically demonstrates the correctness of the latter part of this assertion, namely that all waves shorter than some critical wavelength are stable, that is c { = when k>k c where fc c is some critical value of k for the class of velocity distributions specified by a monotone function 17 of the altitude y and having a single point of inflexion in the domain of flow [2]. An exciting conclusion to which this latter part of Howard's assertion leads to is that the basic assumption c t ^ in Fjortoffs derivation of his necessary instability criterion breaks down, for the class of velocity distributions as specified in the preceding paragraph, in the wave number range k>k c where k c has the same meaning as given in the abstract, thus rendering the derivation of the criterion invalid. This invalidity assumes striking proportions for the wave with wave length zero, that is fc~oo, in which case Fjortoffs necessary criterion of instability is actually a sufficient criterion of stability as will be shown later. What is really surprising is that it has taken such a long time to discover this wave number dependence of Fjortoffs necessary instability criterion but it may, possibly, be expected on the ground that neither the Fjortoffs discriminant (d 2 U/dy 2 ) (U - U s ) which is to be negative somewhere in the domain of flow for any general velocity distribution U(y) and negative everywhere in the domain of flow except being zero at the point of inflexion of U(y) in the present context, involves any wave number implicitly or explicitly nor the derivation of the criterion itself shows any restrictivity with respect to some wave number in the set of all admissible wave numbers k > where U s =U(y s ), yi<y s <y 2 and d 2 l//dy 2 = at y = y s with 17 being twice continuously differentiable in y 1 < j; ^ y 2 . Proof of Howard's Conjecture. To facilitate reference to Ho, we shall make use of the same notation here and denote the basic velocity field by U(y) while the Rayleigh stability equation that governs the linear instability of nonviscous homogeneous parallel shear flows is (Ho; equation (5.1) with /? = and n = 1) ^L- k ^H V^ 2 / Q n) dy2 k H iTTT-- - (2) The boundary conditions are that H must vanish on the rigid walls which may recede to oo in the limiting cases and thus H(y l ) = H(y 2 ) = Q. (3) Multiplying equation (2) by H* (the complex conjugate of H) throughout and integrating the resulting equation over the vertical range of y with the help of the Howard's conjecture in homogeneous parallel shear flows - II 253 boundary conditions (3), we derive k 2 \H\ 2 )dy + - - dy = 0, (4) C yi where D stands for d/dz. Equating the real and the imaginary parts of both sides of equation (4), we obtain I V2 2 i 1,2 I L/|2\ J.VI and (|DH| 2 + F|H| 2 )dy+J X (U _ C) 2 + C 2 d y = Q > (5) -c r ) 2 (6) Rayleigh's theorem, which states that a necessary criterion of instability (c t > 0) is that the velocity distribution U(y) must have at least one point of inflexion at some y = y s where y l <y s <y 2 and U s = U(y s ) follows from equation (6) while Fjortoft's more stronger theorem, which states that a necessary criterion of instability is that (U U s ) < at some point y = y q ^ y s (obviously) k "^ where y l < y q < y 2 and U a = U(y s ) 9 (7) follows from equation r Jyi 2 - dj; = 0, (8) yi \ ~ C r) + C i which is obtained by multiplying equation (6) throughout by the constant factor (c r U s ) after cancelling c i > from both sides of it and then adding the resulting equation to equation (5). Further multiplying equation (2) by d 2 H*/dy 2 throughout, we get U-c ,9, and substituting for d 2 H*/dy 2 from equation (2) in the last term of equation (9), we derive upon integrating this latter resulting equation over the range of y with the help of the boundary conditions (3) f 2 r ya (\D 2 H\ 2 + k 2 \DH\ 2 )dy-k 2 \ Jyi Jyi C 254 Mihir B Banerjee et al Equating the real part of both sides of equation (10), it follows that J,, )W y (, . ,dy-0. (ID 2 L ) ~T" c and adding to equation (11), the equation which follows from equation (6) since c t > 0, we obtain -7T-A dy- U s being the value of 17 at y = y s where y 1 < y s < y z . Writing equation (13) in the form (\D 2 H\ 2 + k 2 \DH\ 2 )dy- we deriv^ that a necessary criterion of instability is that /d 2 l/\ 2 /d 2 l/\ \d7j ~" ^s) + ,2 > at some point y = y p ^ y s (obviously) ~ ~" s ,2 = p s if* where y 1 <y p <y 2 . (15) f The necessary instability criterion expressed by inequality (15) imposes another independent restriction, one being imposed by Fjortoft on Fjortoft's discriminant (d 2 U/dy 2 )(U - U s \ and is valid for any general velocity distribution U(y). Howard's conjecture in homogeneous parallel shear flows - // 255 We shall presently show the importance of this necessary instability criterion inestablishing the conjecture of Howard for a specific class of velocity distributions. Consider the class of velocity distributions specified by a monotone function U of the altitude y and a single point of inflexion in the domain of flow j^ ^y ^y 2 . If instability is to manifest in such flows then Rayleigh's criterion implies that yi<y s < y 2 and Fjortoft's more stronger criterion implies that d 2 U dy 2 (U - U s ) ^ everywhere in y l ^ y < y 29 (16) with equality only where y = y s [2]. It may be noted that for a U(y) belonging to this class (d 2 Ufdy 2 )(U U s ) can either be < or > everywhere in the domain of flow with equality only where y = y s , and it is Fjortoft's criterion which shows that only those flows can possibly be unstable for which (d 2 U/dy 2 )(U ~U S )^0 everywhere in the domain of flow with equality only where y = y s . Thus, a necessary criterion of instability can be derived from inequalities (15) and (16) in the form d 2 U dy 2 It/ -17, d 2 U dy 2 k 2 >0 at some point y - y ^ y s (obviously) where y 1 <y p <y 2 - Hence, if fc 2 >/c 2 = Max y( & 3>s)eFlow Domain d 2 u d 2 U \TT TJ I dy 2 \ U U s\ (17) (18) then the basic assumption c t > is not tenable and we must have c t = which implies stability since Rayleigh's equation (2) and boundary conditions (3) are invariant under complex conjugation. It is clear from the above mathematical analysis that the conjecture of Howard remains valid even for a larger class of velocity distributions U(y) which have a single point of inflexion at some y =* y s where y 1 < y s < y 2 and for which (d 2 U/dy 2 )(U L/ s )< everywhere iny l ^ 3; < y 2 with equality only where y = y s . The following two theorems are, thus true: Theorem 1. All nonviscous homogeneous parallel shear flows, with velocity distributions specified by a monotone function U of the altitude y and a single point of inflexion in the domain of flow, are stable against all infinitesimally smalt perturbations in the wave number range k>k= Max Domain \ ( d 2 U\ 2 ' (< [y 2 ) d 2 U TJ TJ 1 [ dy 2 U u s\ 256 Mihir B Banerjee et al Theorem 2. All nonviscous homogeneous parallel shear flows with velocity distributions U(y) specified by a single point of inflexion in the domain of flow and the constraint (d 2 U/dy 2 )(U U s ) ^ everywhere in y l ^ y ^ y 2 with equality only where y y s are stable against all infinitesimally small perturbations in the wave number range fe > fc c = Max domain d 2 U |77 77 1 dy 2 \ U U S\ An Example. Consider a sinusoidal flow with U(y) = siny(y 1 ^y ^y 2 ) such that J>i <0<} ? 2- Rayleigh's necessary instability criterion is thus satisfied and hence we cannot draw any conclusion regarding stability or otherwise of the flow. Now, let y 2 - y l < n. Then, since ( -p- }(U-U S )=- smy(smy - sinO) = - sin 2 }; ^ everywhere in y l ^ y ^ y 2 , with equality only where y = y s = (origin being the only point of inflexion in the flow domain) Fjortoft's necessary instability criterion, in addition to Rayleigh's, is also satisfied and hence we cannot draw any conclusion, regarding stability or otherwise of the flow, as before. Further, since according to the present criterion d 2 U k 2 must be greater than zero at some point, other than the point of inflexion obviously, as a necessary criterion of instability, we see that it is satisfied only for fc 2 < 1. Hence, for k 2 > 1 the flow must be stable. This simple counter-example to Rayleigh's necessary instability criterion was given by Tollmien [6] and incidentally it also serves the purpose of a counter-example to Fjortoft's necessary instability criterion in the light of our present work. For velocity distributions U(y) belonging to the class for which Theorem 1 is valid, we obtain a necessary criterion of instability for the wave with wave length zero (that is k -> oo ) from inequality (15) as (U U s ) > at some point y = y p ^ y s (obviously) where y l <y p <y 2 > ( 19 ) dy 2 and hence if d 2 U dy 2 (V U s ) ^ everywhere in y a < y ^ y 2 with equality only where y = y s , (20) Howard's conjecture in homogeneous parallel shear flows - // 257 then we must have c- = which implies stability. It is to be noted that Fjortoft's necessary criterion of instability, which is given by (U U s ) < everywhere in j^ < y ^ y 2 with equality only where y = y s , (21) in the present context, has actually become a sufficient criterion of stability for the wave with k ~> oo and this is in accordance with Banerjee et a/'s [1] theorem on the rate of growth of an arbitrary unstable perturbation. We state this result in the form of a mathematical theorem as follows: Theorem 3. Fjortoft's necessary criterion of instability, for all nonviscous homogeneous parallel shear flows with velocity distributions specified by a monotone U function U of the altitude y and a single point of inflexion in the domain of flow, is actually a sufficient . condition of stability for the wave with k-+ao and this result is in accordance with the * prediction of Howard [4] and its subsequent confirmation by Banerjee et al [1]. References [1] Banerjee M B, Shandil R G and Kanwar V, A proof of Howard's conjecture in homogeneous parallel shear flows, Proc. Indian Acad. Sci. Math. Sci. 104 (1993) 593-596 [2] Drazin P G and Howard L N, Hydrodynamic stability of parallel flow of inviscid fluid, Advances in Applied Mechanics (1966) (New York: Academic Press) vol. 9 [3] Fjortoft R, Application of Integral Theorems in deriving criteria of stability of laminar flow and for the baroclinic circular vortex, Geofys. Publ. 17 (1950) 1-52 [4] Howard L N, Note pn a paper of John W Miles, J. Fluid Mech. 10 (1961) 509-512 [5] Rayleigh J W S, On the stability or instability of certain fluid motions, Proc. Lond. Math. Soc. 9 (1880) 57-70 [6] Tollmien W, Ein allegemeines kriterium der instabilitat kminarer gesch verteilungen, Nachr. Akad. Wiss. Goettingen, Math. Phys. Kl. 50 (1935) 79-1 14 Lifting orthogonal representations to spin groups and local root numbers DIPENDRA PRASAD and DINAKAR RAMAKRISHNAN* Mehta Research Institute, Allahabad, 2 1 1 002, India, and Tata Institute of Fundamental Research, Bombay, 400 005, India * California Institute of Technology, Pasadena, C A 9 11 25, USA MS received 10 October 1994; revised 16 December 1994 Abstract. Representations of D* /fc* for a quaternion division algebra D k over a local field k are orthogonal representations. In this note we investigate when these orthogonal representations can be lifted to the corresponding spin group. The results are expressed in terms of local root number of the representation. Keywords. Orthogonal representations; spin groups; local root numbers. Let D be a quaternion division algebra over a local field k. Then Df/k* is a compact topological group, and all its irreducible representations are finite dimensional. It can be seen that, in fact, all the irreducible representations are orthogonal, i.e. for any irreducible representation V of D/fc*, there exists a quadratic form q on V such that the representation takes values in 0(V). Using the natural embedding of O(V) in SO(V C) given by Q\- >(0,det0), we get a homomorphism of D/k* into 5O(KC). In this note we investigate when this can be lifted to the spin group of the quadratic space V C. The results are expressed in terms of the local root number of the representation V, or of the corresponding two dimensional symplectic representation of the Weil-Deligne group. We recall that by a theorem of Deligne [Dl] the local root number of an orthogonal representation of the Weil-Deligne group W' k of a local field fe is expressed in terms of the second Stiefel- Whitney number of the representation, or equivalently in terms of the obstruction to lifting the orthogonal representation to the spin group. In our case we have a symplectic two dimensional representation of the Weil-Deligne group and its root number is being related to the lifting problem for the orthogonal representation of the quaternion division algebra. The formulation of Deligne's theorem is very elegant and has important global consequences. We, however, have not succeeded in making such an elegant formulation of our results and have neither succeeded in any global application. As the problem is trivial in the case of an archimedean field, we will confine ourselves to the non-archimedean case only. We have been able to treat the case of only those non-archimedean fields with odd residue characteristic; we will tacitly assume this to be the case all through, and let q denote the cardinality of the residue field of fc, and a) the unique non-trivial quadratic character of F*. Lemma 1. Any finite dimensional irreducible representation of D%/k* is orthogonal. Proof. If xi-x denote the canonical anti-automorphism of DJ such that x-x = Nrd(x) where Nrd(x) is the reduced norm of x, then as an element of Df/k* 9 x = x~ l . By the 259 260 Dipendra Prasad and Dinakar Ramakrishnan Skolem-Noether theorem, ,x and x are conjugate, and therefore x is conjugate to x ~ 1 in DjJV/c*. By character theory, this implies that every representation of D%/k* is self-dual. Now it can be proved that for any irreducible representation V of DJ/k*, there exists a quadratic extension L of k such that the trivial character of L* appears in V\ see Lemma 2 below for precise statement. Since every character of L* appears with multiplicity ^ 1 in any irreducible representation of Djf, cf. Remark 3.5 in [P], the eigenspace corresponding to the trivial character of L* is one-dimensional. The unique non-degenerate bilinear form on V must be non-zero on this one-dimensional sub- space, and therefore the bilinear form must be symmetric. The following Lemma follows easily from the construction of representations of D; it can also be proved using the theorem of Tunnell [Tu]. Lemma 2. Let n be an irreducible representation of D%/k* associated to a character of a quadratic extension K of k. Let L be the quadratic unramified extension of k if K is ramified, and one of quadratic ramified extensions if K is unramified. Then the trivial representation of L* appears in n. The trivial representation of K* appears in n if and only if K/k is a ramified extension of k, and q = 3 mod (4). The proof of Lemma 1 shows more generally that a self-dual irreducible representa- tion V of a group G must be orthogonal if we can find a subgroup H such that the restriction of K to H is completely reducible and contains the trivial representation of H with multiplicity one. From this remark, one gets the following Proposition. PROPOSITION 1 Every irreducible, admissible, self -dual, generic representation V of GL(n,k), k non- archimedean, is orthogonal for any n^l. Indeed, the theory of new vectors for generic representations of GL(n,fc) (cf. [J-PS-S]) gives the existence of an open compact subgroup C such that the space of C-invariant vectors in V is one-dimensional. According to a program begun by Carayol in [C] for the GL(2) case, representations of D* where D is a division algebra over a non-archimedean field, together with corresponding representations of GL(n) (assumed to be supercuspidal) and W k are expected to appear in the middle dimension cohomology (H"" 1 ) of a certain rigid analytic space. Considerations with Poincare duality suggest the following conjecture generalising lemma L Conjecture. Let D* be the multiplicative group of a division algebra central over a non-archimedean local field k. Let a n be the representation of W' k associated by the local Langlands correspondence to n. Then whenever a n is self-dual, symplectic, and trivial on the SL(2, C) factor of W h ,n}& orthogonal. The following Proposition calculates the determinant of a representation of D*/k*, and implies in particular that the determinant is never trivial; this was the reason why we have to consider the representation K C of D*/k* instead of just V. PROPOSITION 2 Let n be an irreducible representation of D*fk* associated to a character of a quadratic Orthogonal representations 261 extension K of k. Then where L K if K is the quadratic unramified extension of k or if K is ramified with q~\ mod (4); if K is ramified with q = 3 mod (4), then L is the other ramified quadratic extension. Proof. Since the kernel of the reduced norm map is the commutator subgroup of Djf, we can write det(Tc) as ju Nrd for a character ju of k*. As n is self-dual, its determinant is of order < 2, and by class field theory, \JL is either trivial or is co /fc , for a quadratic extension E of fe. For any quadratic extension M of fc, write the decomposition of n as M*-module as *= Z M I ju'^a-ieb-v (i) Me* MS* where a and h are integers ^ a, b ^ 1, v is the unique character of M*//c* of order 2, and X is a finite set of characters of M*/fc* of order ^ 3. Since the dimension of n is known to be even, a = b. It follows that the determinant of TI restricted to M*/k* is trivial if and only if the trivial representation of M* does not appear in n in which case ^ is trivial on the norm subgroup Nrd(M*). Lemma 2 now easily completes the proof. Remark 1. It should be noted that self-dual representations n of/)* not factoring through D*/fc* need not be orthogonal. For instance, for fe = R, n = p(g)det(p)~ 1/2 , where p is the standard two-dimensional representation of D*, is a symplectic rep- resentation of D*. It will be interesting to characterize self-dual representations of Djf which are orthogonal. Lemma 3. Let SO(2nH-l, C) correspond to the quadratic form q = x 1 x 2 + ...+ X 2n- 1 X 2n + X 2n + 1 > an ^ ^ ^ associated maximal torus. For characters Oh, . . . , &,) o/fln abelian group G, /et TT be the representation of G with values in SO (In -f 1, C) given by x*~+(Xi (x), Xi l (x), 1 2 (x\ X2 l (x), "-,Xn ( x )> Xn * (x), V- Then the representation nofG lifts to Spin (2n+l 9 C) if and only if U" = l Xi = V 2 f or some character p of G, i.e. if and only if Il" =1 Xi is trivial on the subgroup G[2] = {geG\2g=l}. Proof. The proof is a trivial consequence of the fact that the spin covering ofSO(2n + 1, C) when restricted to the maximal torus T = {(z 1 , z 1 , z 2 , z^ l , . . . , z n , z n " 1 , l)|z.eC*} is the two-fold cover of T obtained by attaching v/TL^. Lemma 4. A homomorphism n:D%/k* -*SO(n) can be lifted to the corresponding spin group if and only if n restricted to K*/7e* can be lifted for any quadratic extension K of k. Proof. As the two sheeted coverings of a group G are classified by H 2 (G, Z/2), one needs to prove that an element of H 2 (D*/k*,Z/2) is trivial if and only if its restriction to # 2 (K*//c*, Z/2) is trivial for all quadratic extensions K of k. Let D* be the image in D/k* of the first congruence subgroup of D under the standard filtration. Then since 262 Dipendra Prasad and Dinakar Ramakrishnan the residue characteristic of k is odd, //'(*, Z/2) = if i>0. It follows that H 2 (D*//c*, Z/2) = H 2 (D*//c*D*, Z/2). Now D*//c*D* is the dihedral group: - F* /F* - D*//c* /)? -t Z/2 - 0, where F fl is the residue field of k. Dividing Df/k* D* by the maximal subgroup H'of odd order of F* 2 /F*, we again get the dihedral group D r = D*/k*D*H f with H 2 (D*/fc*, Z/2) s ff 2 (D* //c* D*H', Z/2): -+Z/2 r ->>,- Z/2 -+0. Clearly Z/2 Z/2 c D r , and it can be seen from the explicit description of cohomology of dihedral groups, cf. [Sn, page 24], that H 2 (D r ,Z/2) injects into H 2 (Z/2@Z/2, Z/2) H 2 (Z/2 r , Z/2) under restriction. An element of H 2 (Z/2 Z/2, Z/2) is zero if and only if its restriction to all the three Z/2's in Z/2 Z/2 is zero. These three Z/2's come from the three quadratic extensions; also, Z/2 r comes from the quadratic unramified extension, proving the proposition. The following Lemma summarizes the information we need about the characters of irreducible representations n of D*/k*, for k non-archimedean, cf. [Si, pages 50-51] where he calculates the characters of representations of ?GL(2, k). Lemma 5. For K a quadratic extension of k, let n = n x be the representation of D*/k* attached to a character % of K*. Then we have the following table K/k condM dim (ft) cond(Tc) unramified / ramified 2f V 1 te+D^- 1 V 2/ + 1 Let L be any quadratic extension of k, and x the unique element of L*/k* of order 2. Denote by & n the character of n. Then we have: 2. // L = X and K/k unramified, 3. // L = K and K/k ramified, where We now begin analysing the lifting of orthogonal representations of D*/fc* to spin groups. PROPOSITION 3 s. Jf? Let K be an irreducible representation of DJ/fe* with values in 0(V) associated to f a quadratic extension K of k. Then the associated representation with values in SO(VQ lifts to the spin group, Spin(V C), when restricted to L*//c* for L a quadratic extension of k different from K if and only if co( - 2) = - 1 if K is a ramified extension, Orthogonal representations 263 and co( I/" 1 = 1 if K is the unramified extension where 2f is the conductor of the representation n. (We recall that a> is the unique non-trivial quadratic character of F*.) Proof. Let L = fc(.x ) with x^efc*. Clearly x is the unique element of L*/&* of order 2. As n is self-dual, whenever a character \JL of L* appears in n, so does /x~ 1 . Let us now write the decomposition of n as L*-module as n= Z /* Z A^+fl-l+fc-v (i) Me* /*e* where a and ft are integers ^ a, fe ^ 1, v is the unique character of L*/k* of order 2, and Jf is a finite set of characters of L*/&* of order ^ 3. Since the dimension of n is even, a = b. Note that v(x ) = 1 except in the case when L is a quadratic unramified extension of k with q = 3 mod (4) in which case v(x ) = 1. By Lemma 3, the representation n of L*/k* with values in S0( V@C) lifts to the spin group, Spin (7C), if and only if As X Q has order 2 in L*/k*, all the characters of L*/fc* take the value 1 on x . Let r be the number of characters IJL from X such that /J(X Q ) = 1, and let 5 be the number of characters u, from Z such that /i(x ) = 1. From Lemma 5, the character of TT at x is zero. Assuming that L is not the quadratic unramified extension with q = 3 mod(4), so that v(x ) = 1, we have from the decomposition of n as in (i) dim (TT) = 2(r + s) + 2a (ii) 7t (x ) = 2(r- 5 ) = 0, (iii) From (ii) and (iii), dim(Tr) = 4s -f 2a. (iv) Also, From (iv) and (v), and using Lemma 5 for the dimension of TT, it follows that if K is a ramified extension of fc, and L is not the quadratic unramified extension of k with q = 3 mod (4), the representation n restricted to L*//c* lifts to the spin group if and only if q == 5 mod (8) or q = 1 mod (8). Similarly, when K is the quadratic unramified extension of /c, the representation n restricted to L*/fc* lifts to the spin group if and only if q = 3 mod (4) and/ even. Finally, if L is the quadratic unramified extension of k with q = 3 mod (4), then the representation n restricted to L*/k* lifts to the spin group if and only if q = 1 mod (8) as follows from a similar analysis. All these conclusions combine to prove the proposition. We next consider the lifting of a representation n of DJ//c* associated to a quadratic field K when restricted to K*/k*. In this case the obstruction to lifting is related to the epsilon factor of n. We will assume that the reader is familiar with the basic properties of the epsilon factor for which we refer to [T]. We, however, do want to state two theorems about epsilon factors which will be crucial to our calculations; the first due to Deligne 264 Dipendra Prasad and Dinakar Ramakrishnan [D2, Lemma 4.L6] describes how epsilon factor changes under twisting by a character of small conductor, and the second is a theorem of Frohlich and Queyrut [F-Q, Theorem 3]. Lemma 6. Let a and /? be two multiplicative characters of a local field K such that cond(a) ^ 2 cond(j8). For an additive character if/ of K, let y be an element of K such that a(l +x) = *l/(xy) for all xeK with val(x)^ cond(a) if conductor of a is positive; if conductor of a is 0, let y = 7r~ cond( ^ where n k is a uniformising parameter of k. Then Lemma 7. Let Kbea separable quadratic extension of a local field k, and \j/ an additive character of k. Let \I/ K be the additive character of K defined by \l/ K (x) - i//(tr x). Then for any character i of K* which is trivial on k*, and any x eK* with tr(x ) = In the next proposition we analyse the lifting of a representation n of D$/k* associated to a quadratic field K when restricted to K*/k*. PROPOSITION 4 Let n be an irreducible representation of D%/k* with values in 0(V) associated to a character x of K* for a quadratic extension K of k. Then the associated representation with values in SO(VC) lifts to the spin group, Spin(VC), when restricted to K*//c* if and only if B(K) = - co(2) if K is ramified, and o>( - l) f (n) = 1 if K is unramified and the conductor of n is 2f. Proof. The proof of this proposition is very similar to that of Proposition 3. Since the proof is essentially the same in the case when K is unramified or ramified, and in fact since the unramified case is much simpler, we will assume in the rest of the proof that K is ramified. Since k has odd residue characteristic, K*/k* has exactly one character of order 2 which is an unramified character of K* taking the value 1 on a uniformising parameter n K of K; denote this character by v. We fix n K such that n k = n 2 K belongs to k so that K = /c(y/7cj. Clearly % is the unique element of K*/fc* of order 2. Let us now write the decomposition of n as *-module as in Proposition 1: *= Z ^ Z H'^a'lQb'v (i) where a and b are integers ^ a, b ^ 1, and X is a finite set of characters of K*/k* of order ^ 3. Since the dimension of n is (q + l)q f ~ \ it is in particular even. Therefore a = b. By Lemma 3, the representation n of K*//c* with values in S0( F C) lifts to the spin group Spin(F C) if and only if As n K has order 2 in K*/k*, all the characters of K*//c* take the value 1 on n K . Let r be the number of characters u from X such that u(n K ) = 1, and let s be the number of Orthogonal representations 265 characters /z from X such that n(it K ) = - 1. Therefore from the decomposition of n as in (i) we get, dim(Tr) = 2(r -f s) + 2a (ii) 2(r-s), (iii) - 1 ^*- (iv) / From (ii) and (iii), dim(Ti) - Q n (n K ) = 4s + 2a. (v) Using Lemma 5 for the character of n at n K we get and as dim(Tr) = (q + l)q f ~ *, we get from (v) that (q + 1)*/' * + 2G / o)(2)x(7EK) = 4s + 2a. (vi) We next calculate the epsilon factor &(n). As the associated representation of the Weil group is induced from the character % of K*. Here ^ fc is any additive character of fc, and \// K is the additive character of K obtained from \l/ k using the trace map from K to k. We now use the theorem of Frohlich and Queyrut to calculate e(%, \I/ K ). As the restriction of % to k* is a) K/k and not the trivial character, we cannot directly apply this theorem. However, a slight modification works. For this observe that as k has odd residue characteristic, the quadratic character co K/fc of /c* is trivial on 1 -I- n k (9 k where & k (respectively (9 K ) is the maximal compact subring of k (respectively K). Also, since K is a ramified extension, Use this isomorphism to extend <o K/k from to <P and then extend this characterof 0-k* to K* in one of the two possible ways. Denote this extension of co x/k to K* by a>. As the conductor of o> is 1, by Lemma 6, e(7t) = e(r &-co~ x , il/ where j; is the element of K* with the property that %&(! + x) = i/^xy) for all x with val(x) ^ f cond ^, therefore y = n^ (2f+1) a (x) + higher order terms. It follows that 266 Dipendra Prasad and Dinakar Ramakrishnan From the definition of epsilon factors, x = and therefore, Comparing with the definition of G x , we get G x ^o) Klk (a (x)'n k )'e(o) Klk9 \l/ k ). Using (vii), Finally, we can use (vi) to give the value of s as follows: 4s + 4a = (g + !)</" 1 4- 2a + 26(n). We note that by Tunnell's theorem, the trivial character of X* appears in n if and only if e(7c)-e(7rcu jc/fc )= ~o> K/k (- 1). But since TI s TC <8> &>*/* and (TI) = 1, the trivial character of X* appears in TT, i.e. a = 1, if and only if co x/k ( 1) = 1. Now the proposition can be deduced by a case-by-case analysis depending on the values of co(2) and a>(-~ 1). Propositions 3 and 4 can now be combined using Lemma 4 to give the following theorem. Theorem 1. Let n be an irreducible representation of DJ/fc* wif/i values in 0(V) associated to a character i of K* for a quadratic extension K of k. Then the associated representation wi'rfi values in SO(V C) lifts to the spin group, Spin(VC), if and only if co(-2)= - I and &(n) = a)(- 1} if K is ramified, and a>(- Vf~ l = lande(K)= I if K is unramified and the conductor of n is 2f. Remark 2. We do not know when an orthogonal representation of a connected compact Lie group can be lifted to the spin group, say in terms of the highest weight of the representation. The question is interesting for finite groups too, for instance the symmetric group all whose representations are known to be orthogonal, or for finite groups of Lie type. References [C] Carayol R Non-abelian Lubin-Tate theory, in Automorphic Forms, Shimura varieties, and L-functions, Perspect. Maih. 10 (1990) 15-40 [Dl] Deligne P, Les constantes locales de liquation fonctionnelle de la fonction L d'Artin d'une representation orthogonale. Invent. Math. 35 (1976) 299-316 [D2] Deligne P, Les constantes des equations fonctionelle des fonctions L, Modular functions of one variable II, Led. Notes Math. 349 (1973) 501-597 [F-Q] Frohlich A and Quey rut J, On the functional equation of the Artin L-function for characters of real representations. Invent. Math. 20 (1973) 125-138 Orthogonal representations 267 [J-PS-S] Jacquet H, Piatetski-Shapiro I and Shalika J A, Conducteur des representations generiques du groupe lineaire, C. R. Acad. Sci. Paris, Sir. Math. 292 (1981) 611-616 [P] Prasad D, Trilinear forms for representations of GL(2) and local ^-factors, Compos. Math. 75 (1 990) 1-46 [Si] Silberger A, PGL(2) over p-adics, f Springer- Verlag) Lect. Notes Math. 166 (1970) [Sn] Snaith V, Topological methods in Galois representation theory, Canadian Mathematical Society series of monographs and advanced texts (A Wiley-Interscience Publication) (1989) [T] Tate J, Number theoretic background, in Automorphic Forms, Representations, and L-functions, Proc. Symp. Pure Math. 33 (1979) 3-26 AMS, Providence, R.I. [Tu] Tunnell J, Local epsilon factors and characters of GL(2), Am. J. Math. 105 (1983) 1277-1307 Froc. Indian Acad. Sen. (Math. Sci.), Vol. 105, No. 1; August 1995, pp. 269-271. Printed in India. Irrationality of linear combinations of eigenvectors ANTHONY MANNING Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK MS received 25 November 1994 Abstract A given n x n matrix of rational numbers acts on C" and on Q". We assume that its characteristic polynomial is irreducible and compare a basis of eigenvectors for C" with the standard basis for Q". Subject to a hypothesis on the Galois group we prove that vectors from these two bases are as independent of each other as possible. Keywords. Irrationality; Galois group; eigenvectors. A square matrix AeGL(n, Q) can be considered as acting on Q" and on Q" C = C". The action on C n is best understood in terms of eigenvectors and that on Q" in terms of the standard basis e l , . . . , e n where e { = (<5 fj )] = l . We shall study the possibility of linear dependence (over C) between vectors from these two bases. An eigenvector corresponding to an irrational eigenvalue clearly cannot lie in Q". But can it lie in K C where V is some codimension one subspace of Q"? How many of the coordinates of an eigenvector can be rational? And could a non-zero C-linear combination of r eigenvectors lie in V C where V is some codimension r subspace of Q rt ? Because we can work with A conjugated by a change of basis matrix in GL(n, Q) it is sufficient to consider these questions for subspaces spanned by vectors of the standard basis of Q". To avoid rational eigenvalues let us assume that the characteristic polynomial %(A) is irreducible over Q. Since x(A) is separable there are n distinct eigenvalues and A is diagonalizable. Moreover, there is no ^-invariant subspace of Q". Now if there was an ^-invariant subspace U then eigenvectors in U C would be linearly dependent over C on vectors that form a Q-basis for U, and we have avoided this type of possibility of linear dependence by the hypothesis that %(A) is irreducible. Let F denote the splitting field extension of #(,4) over Q and F denote the Galois group of this extension. Then F acts on the set of roots ot%(A). This action is transitive [1, p. 66]. When 1 ^ r < n we call the action r-homogeneous if, for any two subsets consisting of r roots, there is an element of F that takes the elements of the first set to those of the second. Certainly the action of F is 1 -homogeneous. It is r-homogeneous if and only if it is (n - r)-homogeneous. If F is the symmetric or alternating group on n symbols then the action is r-homogeneous for each r. Theorem. Suppose that the characteristic polynomial %(A) of AeGL(n, Q) is irreducible, so that the eigenvalues a l , . . . , a n of A are distinct. Consider a matrix BeGL(n, C) whose i-th row is a left eigenvector of A corresponding to the eigenvalue a,-, l^i^n. Let T denote the Galois group of the splitting field extension F:Qof %(A), and fix 1"^ r < n. If, for each q with 1 ^ q ^ r, the action of F on {a t , . . . , a B } is q-homogeneous then every r x r minor of B has non-zero determinant. 269 270 Anthony Manning Remark L The determinant of an r x r minor of B is equal (at least up to sign) to the determinant of a matrix obtained from B by replacing the n r rows not in that minor by the n - r vectors of the standard basis that do not correspond to any of the r columns in the minor. Thus the theorem asserts that any set of r eigenvectors and n r vectors ^ from the standard basis is independent. (Independence over F is equivalent to independence over C since both are equivalent to the vanishing of the determinant.) Two corollaries follow immediately. COROLLARY 1 // F is the symmetric or alternating group on n symbols then any set of n vectors taken from among the standard basis vectors and eigenvectors corresponding to different eigenvalues is independent over C. COROLLARY 2 An eigenvector of A cannot lie in F C when V is a codimension one subspace o/Q n . "Si Remark 2. If x = (x l , . . . , xJeC" has x l , . . . , x r+ t eQ then x = f l + I] =r + 2 x j e j v c for the codimension r subspace V of Q" spanned by fi,e r + 2 ,...,e n where \ / 1 : = Z^iJx j e J eQ n . Thus Corollary 2 implies that no eigenvector can have two \. coordinates rational; the conclusion of the Theorem implies that a C-linear combina- tion of r eigenvectors can never have r + 1 coordinates rational. However, the precise number of coordinates that are rational can change if we change the basis of Q". Remark 3. If '0 -1\ A = 100 010 1 \0 1 0] then i(A) = x 4 x 2 -f 1 is irreducible and A has eigenvalues a, a" 1 for a a square root of (1 4- z\/3)/2. Then Galois group of %(A) is the Klein four group, which is not 2-homogeneous. So this case does not satisfy the hypotheses of the theorem if r = 2. A has left eigenvectors v l = (1, a, a 2 , a 3 ) and v 2 = (1, - a, a 2 , a 3 ) corresponding to the eigenvalues a. But then v l + v 2 = 2e l + 2a 2 e 3 so that these four vectors are linearly dependent and the conclusion of the theorem is not satisfied. Thus some hypothesis on the Galois group is needed. Remark 4. The theorem arose from work on hyperbolic total automorphisms. Here A is assumed to have only integer entries and det(,4) = 1. Then A induces an automorphism A of the quotient group R W /Z", which is the n-dimensional torus T". A vector subspace of R" that has a basis in Q" or Z" corresponds to a lower-dimensional torus in T n , If A has no eigenvalue of modulus 1 the toral automorphism A is called hyperbolic. A hyperbolic^ has elaborate dynamical properties: on the one hand, for some xe T" the orbit {A k x:keZ} is dense in T", on the other hand the periodic (i.e. finite) orbits are the orbits of rational points (i.e. points of Q n /Z n ) and these form a dense subset of T n . (See Theorems 3.3 and 6.2 of [2] or 1.1 1 of [3].) Study of these dynamical Irrationality of combinations of eigenvectors 271 properties uses R" = E S E U where E s = {veR":A k v-+Q as fe->oo} and E u = {veR": A k v-+Q as fc- oo }. Our theorem gives algebraic conditions under which the projections of E s and E u to T n are in general position with respect to the lower- dimensional tori. Let % r denote the characteristic polynomial of the automorphism induced by A on the homology group H r (T"). Then the roots of % r are products of r distinct roots of %(A). If x r is irreducible then the action of F is r-homogeneous, which helps in checking the hypothesis of the theorem. Proof of Theorem. We work in F" where (A OL 1 I)...(A~ z n l) = and each A a,-/ has nullity one. For each), choose a left eigenvector VjF" corresponding to a,-. Any element a of the Galois group F is a field isomorphism a:F-+F that leaves Q fixed pointwise. <r induces a permutation of {a 1? ...,a n } and we shall write *fy) = **(,,<;> Now <j induces a Q-linear map <j:F"->F n . Up to multiplication by constants, <j permutes the eigenvectors of A because d(v^A = d(VjA) = fffajVj) = (7(ctj)a(Vj) so that 9(vj) = c(<?J)v n(ff)(j) for some non-zero c(<rJ)eF. Now suppose, if possible, that r vectors from {v i9 ...,v n } and n rfrom {e i9 ... 9 e n } in F n are linearly dependent over F and that r is the least number for which this is true. By Remark 1 it suffices to find a contradiction to the existence of such vectors. By renumbering if necessary we can assume that the vectors are v 19 ..., v r9 e r + l9 ... 9 e n . By the dependence there are /? t , . . . , /? w eF, not all zero, with Since r is the least possible, /?,. ^ for 1 ^ j ^ r. Since the Galois group F is r-homogeneous we can, for k = 0, 1, . . . , n r, find c^eF for which the permutation n(a k ) maps { 1, . . . , r} to {k + 1, . . . , k -j- r}. Apply each a k to (1). This gives j=l j=r+l The n r + 1 vectors on the left hand sides of (2) all lie in the (n - r)-dimensional subspace of F" spanned by e r + 1 , . . . , e n so they are linearly dependent over F. Thus, for some fc^n r, 2 r j= 1 G k (f} j )c((r k J)v 1t(ffk)( j ) is a linear combination of I* r j=1 ff m (fij) C K J K(<r w) u)> ^ m < fe. Now ff k (vj) = c(a k J)v n(ffk)U) = c(a k j)v r+k when; = (n(a k ))' 1 (r + fe). For this value of 7 it is a k (^ j )c(a k ,j ') that is the (non-zero) coefficient of v r+k and so v r +k i s a linear combination of v 1 , . . . , v r+k _ x , which contradicts the independence of the eigenvectors, and so completes the proof. References [1] Garling D, A course in Galois theory (Cambridge: University Press) (1986) [2] Smale S, Differentiable dynamical systems, Bull Am. Math. Soc. 73 (1967) 747-817 [3] Walters P, An introduction to ergodic theory (New York: Springer) (1982) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 3, August 1995, pp. 273-279. Printed in India. On the zeros of > (l \s) a (on the zeros of a class of a generalized Dirichlet series - XVII)* K RAMACHANDRA School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India MS received 10 October 1994; revised 14 December 1994 Abstract. Some very precise results (see Theorems 4 and 5) are proved about the a-values of the /th derivative of a class of generalized Dirichlet series, for / ^ / = I (a) (1 Q being a large constant). In particular for the precise results on the zeros of C (/) (s) - a (a any complex constant and / ^ / ) see Theorems 1 and 2 of the introduction. Keywords. Riemann zeta function; generalized Dirichlet series; derivatives; distribution of 1. Introduction The object of this paper is to prove the following two theorems. Theorem 1. Let 5 = ( log f } } ( log- 1 . There exists an effective constant e >0 V V lo 8 2 /A 2 / such that if & is any constant satisfying < e ^ e , then the rectangle 1 / 3 ^t^(2k + 2)n I log- contains precisely one zero of C (0 ( s )> provided I exceeds a constant 1 = I (z) depending only on e. This zero is a simple zero. Moreover this zero does not lie on the boundary of this rectangle and further lies in + s). Here as usual s = a 4- it and k is any integer., positive negative or zero. Theorem 2. Let S = (loglog 15) (log 15)" 1 and a any non-zero complex constant. There exists an effective constant & >Q such that if e is any constant satisfying < ^ e , then the rectangle ^t^ T + where T =(Im log + 7E/ + 2fc7r)(log 15)" S contains precisely one zero of provided I exceeds an effective constant 1 = I (a, s) depending only on a and e. This zero is a simple zero. Moreover this zero does not lie on the boundary of this rectangle and further lies in <r^l(S + e). * Dedicated to Prof, Paul Erdos on his eighty-first birthday 274 K Ramachandra Here k is any integer, positive negative or zero. Remark. In [1] we dealt with slightly different questions on the zeros in <r>^ of (0 (s) a where a is any complex constant and / is any fixed positive integer. Interested reader may consult this paper. However the results of the present paper deal with large / and are more precise. The main ingredient of the proof of Theorems 1 and 2 (and the more general results to be stated and proved in 3 and 4) is the following theorem (see Theorem 3.42 on page 116 on [2]). Theorem 3. (Rouche's Theorem). // /(z) and g(z) are analytic inside and on a closed contour C, and \g(z)\ < \f(z)\ on C then /(z) and /(z) -f g(z) have the same number of zeros inside C. Remark 1. In what follows we use s in place of z. Remark 2. It is somewhat surprising that we can prove (with the help of Theorem 3) Theorems 4 and 5, which are much more general than Theorems 1 and 2. These will be stated in 3 and 4 respectively. Remark 3. Theorems 4 and 5 can be generalized to include derivatives of C L functions and also of C function of ray classes of any algebraic number field and so on But we have not done so. 2. Notation {l n }(n = 1, 2, 3, . . .) will denote any sequence of real numbers with A x = 1 and ^ A n+ 1 - A n ^A where A( ^ 1) is any fixed constant. {a n } (n = 1, 2, 3, . . .) will denote any sequence of complex numbers with ^ = 1 and \a n \ ^ n A . k will be any integer, positive negative or zero. S n (n ^ 2) will denote (loglog A B )(log l n )~ l 3. A generalization of Theorem 1 Theorem4 Let w -l ^ V V toSAio // log- 1 ) . Also let A n+ ! < Xl for alln>l. There exists an effective constant A no / such that if e is any constant satisfying < e ^ 6 , then the rectangle where T = ( Im log I -^ ] J 1 log-^ ) , contains precisely one zero of the analytic \ V o / / \ ^no / function Generalized Dirichlet series 275 provided I exceeds an effective positive constant / = l Q (A,s,n ) depending only on the parameters indicated. This zero is a simple zero. Moreover this zero does not lie on the boundary of this rectangle and further lies in a^l(d + e). Remark. Theorem 1 follows by taking n = 2, A n = n and a n = 1 for all n. The following lemma will be used in this section and also while applying Theorem 5 of 4 to deduce Theorem 2. Lemma 1. For any (5>0 the function (logx)x'* (of x in x^ 1) is increasing for 1 <x^exp(5 -1 ) and decreasing for x^exp^" 1 ). It has precisely one maximum at x = exp(<5~ 1 ). Remark. The maximum value is (ed)~ 1 - The proof of this lemma is trivial and will be left as an exercise. To prove Theorem 4 we apply Theorem 3 to and where J n = a n (a^)~ l . It suffices to prove that f(s) + g(s) has its zeros as claimed in Theorem 4. Lemma 2. The zeros of f(s) are all simple and are given by S = S Q where for all possible values of log( a' no+1 ). If s = <T O 4- it Q then ( **n + ' i HO ' i lo g-r log/ Bo and Also / A n+1 \- -<+i))(log-f I V 2 no / Proo/ The proof is trivial. Lemma 3. For a ^ 200 >4, w /I 276 K Ramachandra where Proof. The proof follows from Y i 'if log M JrJ " and the fact that ' A. Remark. Hereafter we write a = 5 Q l and Also we remark that the condition cr ^ /(<5 - e) is the same as a ^ 1(6 e) with a change of s. Lemma 4. Let S = S(a). Then for <j^l(d s) we have, provided / > J = 1 (A, , n ), which is effective. To prove this lemma it suffices to prove that This will be done in two stages. We have (by Lemma 3) -)- z w^ In Lemma 5 we prove that exp(6~ 1 )< A no+1 and so by Lemma 1 it follows that (log A H )A n ~* is decreasing for n ^ n + 2. Hence it suffices to prove that This will be done in Lemma 6. This would complete the proof of Lemma 4 since for all large n ( logA ; \-5+e n \l A n Generalized Dirichlet series 211 is less than a negative constant power of /l n . Lemma 5. We have Proof. Since for < x < 1 we have - log(l - x) > x, it follows that 5 = ( log ( 1 I 1 : j-^ 2 - 1 11 ( log-y^ A+.V This proves the lemma. Lemma 6. We have -}(T^}' <I - t- 1 / \ A n + 2/ Proof. We have l no + 2 < l* Q+l and also for < x < 1 we have log(l + x) < x. Using these we obtain and so and since (log A no + 1 ) ~ l < 5, we obtain Lemma 6. Lemmas 2 and 4 complete the proof of Theorem 4. 4. A generalization of Theorem 2 Theorem 5. Let d ni be the maximum of 6 n taken over all n for which a n ^Q and n>l. Suppose that for all n^ 1, n 1 we have 6 ni - d n ^ A" 1 and also k ni -e^A~ l . We further suppose that \a ni \^A~ l and put d ni = d. There exists an effective constant such that for all e satisfying Q<s^e ,the rectangle {a ^ 1(5 ~ el T -n(\oglJ-^t^ T + n(\og AJ- l } where T = (Im log( fl Bl ) + 2/c7i)(log>l ni )~ ^ contains precisely one zero of the analytic function Z ^ n=2 provided I exceeds an effective constant / = 1 (A, e, n x ) depending only on the parameters indicated. This zero is a simple zero. Moreover this zero does not lie on the boundary of 278 K Ramachandra this rectangle and further lies in Remark. Theorem 2 follows by taking X n = n and a n = ( - 1) /+ 1 a' 1 for all n > 2. Note that the maximum of d n occurs when n = 15. It is necessary to check that <5 15 > <5 16 . In fact we have * e = 15-21. ...logjo^V =0434357... and log^jfe 1 =0434455..., by using tables. To prove Theorem 5 we apply Theorem 3 to and where the asterisk denotes the restrictions n ^ 1, n l . Lemma 1. The zeros of f(s) are all simple and are given by s = s where s = (log(- a ni ) + HoglogAjaog A Wi )- 1 /or a// possible values of log( - a ni ). // s = <T O 4- zt 05 <T O and t Also Remark. We write <r = <5 / and ^^'"Mlogla^lXlogA^)" 1 + 5. The condition <r ^ /(^o e) is the same as o- ^ /(5 - fi) with a change of e. Proof. The proof is trivial. Lemma 2. For a^l(d s\ we have Proo/. LHS is trivially not more than for all a ^ 200 A. This proves the lemma. Lemma 3. We have for ^^1(8 e), Generalized Dirichlet series 279 Proof. Using log h n = U n ) 6n we obtain, by Lemma 2, By the hypothesis of Theorem 5 we see that 6 d n ^ A ~ * (note also that A Wi so that <5 ^ - - - r-j if k n i ^ e e ) and so Lemma 3 is proved. Lemmas 1 and 3 complete the proof of Theorem 5. Open questions 1) How much can one generalize Theorems 1 and 2? 2) Whatever the integer constant / ^ 1 and whatever the complex constant a, prove that C (0 (s) a has infinity of simple zeros in a > |, (more precisely T simple zeros in (cr ^ | + (5, T ^ r ^ 27) for some absolute constant 6 > 0). References [1] Balasubramanian R and Ramachandra K, On the zeros of C'(s) - a, Acta Arith. 63 (1993) 183-191 [2] Titchmarsh E C, The theory of functions (second edition) (1939) (Oxford University Press) Proc. Indian Acad Sci. (Math. Sci.), Vol. 105, No. 3,- August 1995, pp. 281-285. Printed in India. A note on the growth of topological Sidon sets K GOWRI NAVADA School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India MS received 31 October 1994; revised 14 December 1994 Abstract. We give an estimate for the number of elements in the intersection of topological Sidon sets in R" with compact convex subsets and deduce a necessary and sufficient conditions for an orbit of a linear transformation of R" to be a topological Sidon set. Keywords. Topological Sidon sets; growth of sets. Given a locally compact abelian group G, a subset A of the dual group X is called a topological Sidon set if any be/ 00 (A, namely any bounded complex- valued functions on A, is the restriction to A of the Fourier transform of a complex bounded Radon measure on G. These sets play an important role in harmonic analysis ([LR], [M]). When G is compact, X is discrete and the notion of topological Sidon sets coincides with that of Sidon sets. ([LR], [M].) For any topological Sidon set A as above there exist c ^ 1 and a compact subset K of G such that any be/ 00 (A) is the Fourier transform of a measure which is supported on K and has norm at most c \\ b \\ ^ . When this condition holds for a c ^ 1 and a compact subset K, A is called a (c, K) topological Sidon set. Sidon sets are known to be 'thin' set ( [LR], [M], [P] ). Further, estimates are known for the number of elements in intersections of Sidon sets with finite subsets (see Theorem 3). The purpose of this note is to give the similar estimate for the number of elements in intersections of topological Sidon sets in R m with compact convex subsets. Let / denote the Lebesgue measure on R m . For a set E we denote by | E \ the cardinality of E. Then our result shows in particular the following. Theorem 1. Let weN. Then for any compact set K c= R m and c ^ 1, there exist ad>0 and a neighbourhood UofQeR such that for any (c, K) topological Sidon set A of R m and any convex subset A of R m we have We deduce from the theorem the following criterion for orbits of linear transform- ations to be topological Sidon sets. COROLLARY Let A:R m h->R m be a linear transformation and veR m . Then {A"(v)\neN} is an infinite topological Sidon set if and only if v is not contained in any A-invariant subspace of R m on which all the eigenvalues are of absolute value at most 1. 281 282 K Gowri Navada While the estimate as in the theorem is adequate for the above corollary, it seems worthwhile to note that our argument below gives not just existence of a neighbour- hood (/, but a concrete way of choosing such a neighbourhood. This is of some interest since the right hand side would typically be big when U is small and so for getting a better estimate one would be interested in choosing U as big as may be allowable. We shall prove the following stronger version of theorem 1. Theorem 2. Let raeN. Then for any c ^ 1 there exists a d > such that the following holds: for any compact set K of R m , any(c, K) topological Sidon set A of R m and any convex subset A of R m we have \ A n4| <dlog(/(X + 3 U)/l(U)), where U = {AeR m |sup X6KuB |I?L ! AfXjl < l/47ic}, B being any basis of R". We shall now recall a result from [LR], on which our proof of Theorem 2 is based, prove some preparatory results and then proceed to prove the theorem. A finite subset A of a discrete topological group X is said to be a test set of order M, where M ^ 1, if \A 2 A' 1 1 ^ M\A\. Theorem 3 [LR]. If E^X is a Sidon set with Sidon constant K ^ 1, then \A r\E\ ^ 2>c 2 eM log | A | for test sets of order M such that \A\^2. The following proposition signifies that any countable set close to a topological Sidon set is again a topological Sidon set. It is just a higher dimensional version of Lemma 3 of Ch. VI of [M] and is deduced analogously, as indicated below. PROPOSITION 1 Let A = {l n }%L ! be a (c, K) topological Sidon set in R m and p>lbe given. Let e be such that (1 - ec)~ 1 = p and let < 6 < & and W = {A6R m |sup X6X |f = l x^l < 6/4n}. For each n let X n EA n + W. Then A' = {^}^ =1 is a topological Sidon set and further any function b in /(A') is the restriction to A' of the Fourier transform of a measure jueM(R m ) wit/i Proof. SinCe A is a topological Sidon set, A is a coherent set of frequencies, (cf : [M], Theorem I of Ch. VI for a proof in the case m = 1. The proof actually holds in general) We now argue as in the proof of the assertion (a)=>(c) in Theorem X of Ch. IV of [M]: The argument there shows that for the set W as above {^ n -h W}^L l are mutually disjoint and if. H:A -f VT-*A x W is the (well-defined) map such that H(A n + w) = (i n , u\ for all neN, we W, then for each geB(\ x W), gHeB(A + W) and _ Let bel*(A') be given. Let/(A n + u) = b(X n \ Vn, Vwe Wand let/ be the restriction of /to A. Since A is a (c, K) topological Sidon set, there exists a measure /ieM(R m ) such that u = 'f on A and \\u\\ ^cll/H^. Then /z x d yields an element of B(A x ^; we denote it by g and put v = gHeB(h + W). Then Hencethereexistsameasurev6M(R m )suchthat ||v|| <pc||/|i 00 andv = particular \\v\\ < pcll&H^ and v = b on A 7 . Let {x i , . . . , x m } be any linearly independent set in R m with m elements. Then any translate of the set {=1^10 <*<!} is called a parallelopiped in R m ; further if {xj , . . . , x m } is an orthogonal set, then such a parallelopiped is called a box. Growth of topological Sidon sets 283 PROPOSITION 2 Let A be a compact, convex subset of R m with nonempty interior. Then A contains a parallelopiped P such that l(A) ^ (2m) m /(P). Proof. By a suitable translation, we can assume that Oe A We define an orthogonal set {*!,..., x m } in R m and a linearly independent subset {j>i,...,>' m } of A by induction as follows. Let x l = y^eA be an element of maximum norm. Assume that for some fc < m 1 , an orthogonal set {x 1 , . . . , x k } and a linearly independent subset {y 1 , . . . , y k } are chosen. Let P fc :R w -~><x 1 ,...,x k > 1 be the orthogonal projection map onto the subspace of R m orthogonal to {x t , . . . , x k }. Choose x k + 1 to be an element of maximum norm in P k (A)\ since A has nonempty interior x k+1 ^0. Let y k+l eA be such that PfcOWi) ***!- Clearly {x 19 ... 9 x k+i } is an orthogonal set and {^,...,^+1} are linearly independent. By induction this yields the sets {x i9 ...,x m } and {y l , . . . , y m } as desired. Let / be the box generated by {i 19 ...,x m }, i.e. / = {ZJ" asl f i x |0<f I .< 1}. Let J = {I?L ! f f x t . I - 1 < r < 1}. Then l(J) = 2 m /(/). If -ae.4 and (a l , . . . , aj are the coordi- nates of a with respect to the vectors x ls ...,x m , then |a.| < ||x ( .|| Vi and hence aeJ. Therefore ^4^J and consequently l(A) ^ /(J) = 2 m /(/). Let P be the parallelopiped generated by {y 1 /m, . . . , y m /m}, i.e. P = {IJ" = j t f ^/m|0 < t ^ 1}. Since A is convex and OeA it follows that P c 4. The matrix of the transformation x <-)>; is lower triangular with diagonal entries equal to 1. Therefore /(P) = m m /(/). Hence we get l(A) ^ (2m) m /(P). Proof of Theorem 2. Write A = { A rt } rt = 1 . Let B be any basis of R m . Let Oe(Q, l/c) be arbitrary and let ee(0, l/c). Let I7 a = {A6R m |sup Jce ^ uB |Zr.i^l< e / 4 ^}; We a PPty Proposition 1 to p = (1 -ec)' 1 and e, and U e as above. Clearly, U e is a convex, compact and symmetric neighbourhood of 0. Applying Proposition 2 to U e , we get a parallelopiped P^I7 e such that l(U e )^(2m) m l(P). Let {z 1 ,...,z m } be such that P is a translate of {Z^-zJO^ t^ 1}. Let L be the lattice generated by {z 1? ...,z m }. If we choose ^e(A n -h U e )r\L, then A' = {^}^ =1 is a coherent set of frequencies with respect to (1,F), where F is a fundamental domain of the annihilator L of L. By Proposition 1, A' is a topological Sidon set and any fceL(A') is the restriction to A' of the Fourier transform of a measure ueM (R m ) with || \JL || ^ pc || b || w . This implies that A' is a (pc, F) topological Sidon set ( [M] ). Since L = R m /L and F is a fundamental domain for L in R m , this is equivalent to saying that A' is a Sidon set in L with Sidon constant pc. Now let A be a compact, convex subset of R m . Put A -h U e = B and B + U e = C. We shall prove that CnL is a test set with associated constant (18m)" 1 . We have I7 a ) + (C + l/ e ) - (C + C 4- U is a convex, compact subset of R m with nonempty interior. Applying Proposi- tion 2 we get a parallelopiped P l c C + I7 e such that /((C 4- I7 e ) + (C + U e ) - (C + 17,)) ^ (6mr /(Pi). Then (6mr/(P 1 ) ^ (6m) m /(C + 17.) = (6m) m /(B + L7 + U e ) ^ (6m) m 3 m /(B), because B contains a translate of U e . These inequalities and the fact that U e contains 284 K Gowri Navada P yields that This proves that C n L is a test set as claimed. By applying Theorem 3 to C n L we now get that |An A| ^ I An(4 4- U 9 )\ ^ |A' n(A + 2l7 a )| ^ d, log|Ln(4 + 2U 9 )\ 9 where d l = 2e(pc) 2 (18m) m . Then |An A\ ^ d, log(l(A + 2U e + U e )fl(P)) ^ d\og(l(A + 3U 9 )/l(U 9 )) where d is a constant depending on c, e and m. By letting -* 1/c we get the required result. The following theorem is analogous to Theorem II in Ch. VI of [M]. Theorem 4. // {A n }^ = l is a sequence in R m such that for some a > 1 we have for all large n > IUii + 1 II ^ a ii ^ n II then {A n }J = t zs a topological Sidon set. This can be deduced from the following lemma in the same way as Theorem II in Ch. VI of [M] from the analogous lemma there. Lemma. If {^ n }^ l is a sequence in R m such that U H + 1 \\ > 6||AJ|, Vn, and if {b n }* =i is any sequence in T, then there exists a point seR m such that \\s\\ ^ 1/HAj || and Proof. Let 1 = (a 1 , . . . 9 a m ) be a nonzero element in R m . Let jB be a ball in R m with radius I/ 1| A || and centre at x . Let jS = (a 1 /||A|| 2 ,...,a m /||A|| 2 ). Then the points x are contained in the boundary of B and each of the two line segments joining x to x jJ is mapped onto T by the map x-><x,A>. Therefore given any feeT we can find a point yeB such that <jU> = b and B(y, 1/2||A||) c B. By induction we choose balls and points y n eB n such that B n+1 c B(j; w , l/6||A n || ) c B n , Vn as follows: Let B, be the ball with centre at and radius = 1/UAJ. Let y l eB l be such that <y 1 ^ 1 > = b 1 and ^OJi.l/fiPilD^Bi. Suppose B n and y n have been chosen satisfying the above conditions. Let B n+1 be the ball with centre at y n and radius = 1/IU W+1 ||. Then B n+l ^B(y n ,l/6U n \\)^B n . Choose y n+1 eB n+1 such that <y ll+1 ,A ll+1 > = i ll + 1 and (yn+i>V6|K +1 ||)c: M+1 . Let 5 be the point of intersection of {#}. Then 56 n J B(y n ,1/6 ||/y ) also. Then for all n, || s - y u \\ ^ 1/6 |UJ| and hence KS,^ >- b n \ = > - O n U n > I ^ 1; which proves the lemma. Proof of the Corollary. There exists a unique largest ^-invariant subspace V of R m such that all eigenvalues of A on V are of absolute value at most 1 . Suppose v e V. Using Jordan decomposition it is easy to see that there exists a c>0 such that II A n (v) || ^ cn m ~ L for all n. Let r n = cn m ~ 1 and B n the ball with centre at and radius r n . If {A n (v)}^ l = A is an infinite topological Sidon set then A n (v\ neN, are all distinct and hence by Theorem 2 above, we have, n^|B n nA| ^ dlog(/(B n + 3U)/l(U)) for some compact neighbourhood U of 0. Therefore there exists a constant D such that n^Dlogr n for all n. Since r n = cn m ~ 1 this implies that n/logrc is bounded which is a contradiction. Growth of topological Sidon sets 285 Now suppose that v$V. Using Jordan decomposition one can see that there exists a c> 1 and an integer k ^ 1 such that \\A n+k (v) \\^c\\ A n (v) \\, for all large n. It follows from Theorem 4 that A is a finite union of topological Sidon sets. Since A is uniformly discrete it is a topological Sidon set. Acknowledgements The author thanks Prof. S G Dani for suggesting the problem and for many helpful discussions and also wishes to thank the National Board for Higher Mathematics for the financial support. Thanks are also due to the referee for useful comments enabling improvement of the text of the paper. References [LR] Lopez J M and Ross K A, Sidon Sets, Lecture Notes in Pure and Applied Mathematics, 13; (New York: Marcel Dekker) [M] Meyer Y, Algebraic numbers and harmonic analysis (Amsterdam, London: North-Holland Publishing Company) (1972) [P] Pisier Gilles, Arithmetic characterization of Sidon sets; Bull. Am. Math. Soc. 8 (1983), 87-89 i Printed in India. Characterization of polynomials and divided difference P L KANNAPPAN* and P K SAHOO f * Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Department of Mathematics, University of Louisville, Louisville, Kentucky, 40292, USA MS received 24 June 1994; revised 22 February 1995 Abstract For distinct points x ! , x 2 , . . . , x n in & (the reals), let / [x l , x 2 , . . . , x n ] denote the divided difference of/ In this paper, we determine the general solution f,g: &-+ of the functional equation for distinct x^ ,x 2 , . . . , x n in $ without any regularity assumptions on the unknown functions. Keywords. Characterization of polynomials; divided difference; distinct points; unknown functions. Let ^ be the set of all real numbers. It is well-known that for quadratic polynomials the Mean Value Theorem takes the form Conversely, if /satisfies the above functional-differential equation, then/(x) = ax 2 + bx + c (see [1] and [4] ). A cubic polynomial satisfies the following functional equation + y + (1) where/ [x, y, z] denotes the divided difference of/. Recently, Bailey [2] has shown that if the above functional-differential equation holds, then / is a cubic polynomial. For distinct points x x , x 2 , . . . , x n in ^, the divided difference of / is defined as ffx x ~l * = - where c 1 ,x 2 ,...,x n ;/] ]x 1 ,x 2 ,...,x n ] Lx 1 ,x 2 ,...,x n ;/ J 1 x 1 xl 1 Y n-2 v.n 2 /(*!> /(x.) . and 1 X, X? 287 288 PL Kannappan and P K Sahoo This definition of the divided difference is the same as the one given in [2]. In explicit ! > *] can be written as 1=1 j*i \j~ i 1 Bailey, generalizing a result of Aczel [1], has shown in [2] that if /: 9* -+5R is j a differentiable function satisfying the functional equation L /[x,y,z] = M* + y + 4 (2) j (which is a generalization of functional-differential eq. (1)), then / is a polynomial of \ degree at most three. In Bailey's proof the differentiability of/ plays a crucial role. In I [2], Bailey wrote "One is also led to wonder if fl_x l , x 2 , . . . , x n ] = h(Xi + x 2 H ----- h *) ; and f continuous (or perhaps differentiable) mil imply thatf is a polynomial of degree no \ more than n. At this point we have no answer" In this paper, we provide an answer to this j problem. Our method is simple and direct. Further, we do not impose any regularity conditions like continuity, differentiability or boundedness on/ etc. For characteriz- \ ation of polynomials with mean value property, the interested reader should refer to M> [2], [3], [4] and [5] and references therein. i Lemma. Let S be a finite subset of SR symmetric about zero (that is, S = S) and let j /, g : 9t -> 5R be functions satisfying the functional equation I f(x)-f(y) = (x-y)g(x + y) for all x,>;e5R\5. (3) \ Then f(x) = ax 2 + bx + c and ()>) = ay + b (4) for xeSR\S and ye 91, where a, b, c are some constants. Proof. Putting y = x in (3), we obtain f /(*) -/(-*) = 2x0(0), forxe<R\S. (5) I Changing y into y in (3), we get which after subtracting (3) from it and using (5) gives, (x + y)fo(x - y) - ^(O)) = (x - y)(g(x + y)~ g(0)\ for x, ye<R\S. (6) Fix a nonzero ue$R. Let t?e9l such that (uv)/2$S and put x = (w-ht;)/2 and y = (w - o)/2. Then x + y = u and x - y = v and by (6) to get (^W~^(0)) = %( W )~0(0)), forp6^\(2Sw), (7) where 2S u denotes the set {2s + u\seS} u {2s - u\seS}. For each fixed w, (7) shows that g is linear in u, that is of the form av 4- b, except on the J\ finite set 2S w. To conclude that g is linear on % (reals), one has to note that, if one f takes two suitable different values of w, which is now treated as a parameter, the 1. exceptional sets involved are disjoint and so g(v) = av + b for all real v with the same constants everywhere. \ Characterization of polynomials and divided difference 289 Substituting this for g in (3) yields f(x) - ax 2 - bx = f(y) - ay 2 - by, for x, yeW\S. (8) Choosing any ye9?\S in (8) yields that f(x) = ax 2 -f bx + c for X6SR\F, for some constant c, which is the required form of/ in (4). This completes the proof of the lemma. Theorem. Let /, g:$l - 9? satisfy the functional equation /[x 1 ,x 2 ,...,x ll ]=0(x 1 +x 2 + -"+x ll ), (FE) /or distinct x 1 , x 2 , . . . , x n , tfzar is, /or x t ^ x 7 -(i ^ j, ij = 1, 2, . . . , n). Then f is a polynomial of degree at most n and g is linear, that is, a polynomial of first degree. Proof. It is easy to see that if/ is a solution of (FE), so also/(x) + ZJ[ I Q a k x k . So, we can assume without loss of generality that /(O) = = f(y^ =-'=f(y n _ 2 ) for 3^ , y 2 ,..., y n _ 2 distinct and difTerent from zero. Obviously there are plenty of choices for 0, j; l , . . . , y n _ 2 . Putting in (FE),(x,0,y 1 ,...,^ II _ 2 ) and (x,0,^y 1 ,...,y n _ 3 ) for (x 1 ,x 2 ,...,xj, we get (9) V k=i / and (10) respectively for x = 0, y i - - y n _ 2 and y ^ x. Now (10) can be rewritten as f(x) where /(x)': = - ^-- - for x, y + 0, y t , . . . , y n _ 3 . Then by Lemma and the x(y 1 -x)--(y n _ 3 ~-x) arbitrary choice of 0, y^ , . . . , y n _ 3 we get that g is linear and /(x) is quadratic. Hence by (9) / is a polynomial of degree at most n. This proves the theorem. Remark. The same conclusion can be obtained without using the Lemma as follows. Subtracting (10) from (9), we have 3^ .Vn-2 f(x) where L(x) = - ^ - -, for x,y = 0, y l9 y 2 , . .. 9 y n . 2 . Interchanging x and x(y l x)--(y n _2 x) y in (11) and adding the resulting equation to (11), we get / n-3 \ / if-2 (x-y)g[x + y+ X y k )^(x-y n -2)9(x+ Z \ k=l / V k=l / n-2 \ -(y-y*-2)9\y+ I ^ V k=i / 290 P L Kannappan and P K Sahoo y rt _ 2 intheabove,we obtain (x-y)G(x + y) = xG(x)-yG(y) (W ~. where f G(x) = g(x + yi + -~+y n ^ + 2y n _ 2 ) l forx,)^0,-;y._ 2 ,(j^-;y._ 2 ),/.^ \ for x*Q 9 y H . 29 (yi-y n . 2 ) 9 ... 9 (y H . 3 -y n _ 2 ). Replace); by -y in (12) and subtract the resultant equation from (12) and use (13) to get (x -I- y)(G(x - y) - G(0)) = (x - y)(G(x + y) - G(0)) (14) (y n - 3 - y n - 2 ). As in the Lemma, it can be shown that G is linear so that g is also linear, g(x) = C& + b. This g in (9) shows that / is a polynomial of degree at most n. Acknowledgements We express our thanks to Prof S G Dani and the referee for comments which improved the presentation of the paper. This research is partially supported by grants from the University of Louisville and the University of Waterloo. References [1] Aczel J, A mean-value property of the derivative of quadratic polynomials without mean values and derivatives. Math. Mag. 58 (1985) 42-45 [2] Bailey D F, A mean- value property of cubic polynomials without mean value. Math. Mag. 65 (1992) 123-124 [3] Cross G E and Kannappan P L, A functional identity characterizing polynomials. Aequationes Mathematics 34 (1987) 147-152 [4] Haruki Sh, A property of quadratic polynomials. Am. Math. Mon. 86 (1979) 577-579 [5] Kannappan P L and Crstici B, Two functional identities characterizing polynomials. Jtinbrant Seminar on Functional Equations, Approximation and Convexity, Cluj-Napooa (1989) 1 75-180 f I Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 3, August 1995, pp. 291-296. Printed in India. A theorem concerning a product of a general class of polynomials and the //-function of several complex variables V B L CH AURASIA and RAJENDRA PAL SHARMA Department of Mathematics, University of Rajasthan, Jaipur 302 004, India MS received 16 August 1994 Abstract. A theorem concerning a product of a general class of polynomials and the H-function of several complex variables is given. Using this theorem certain integrals and expansion formula have been obtained. This general theorem is capable of giving a number of new, interesting and useful integrals, expansion formulae as its special cases. Keywords. H-function of several complex variables; general class of polynomials; expansion formulae; integrals. 1. Introduction and the main result Srivastava [3, p, 1, eq. (1)] introduced the general class of polynomials [/] / _ yj\ S:M= I ( *.*, n=o,u,... (i) a' = a ' where m is an arbitrary positive integer and the coefficients A na ,(n, a' ^ 0) are arbitrary constants, real or complex. By suitably specializing the coefficients A nt0l ,, the poly- nomials S[x] can be reduced to the well-known classical orthogonal polynomials such as Jacobi, Hermite, Legendre, Laguerre polynomials, etc. For the H -function of several complex variables defined by Srivastava and Panda [4; see also 6, p. 25 1], we derive the following theorem: The main theorem v (l-yr + "-" 2 F 1 (2a,2^;27;y)= a k y k (2) k = then fi / i \ / 1 \ 2 F, (a,/3;y:f-;}> 2 F! y-a,y-/?;y + 5 ;y Jo \ L J \ / (3) 291 292 V B L Chaurasia and Rajendra Pal Sharma where h ( > 0, Re ( 1+ ]T hid^/fi ^ arbitrary positive integer and the coefficients A naL ,(n, a' ^ 0) are arbitrary constants, real or complex. 2. Proof of the main theorem To prove the main theorem, we have (2, p. 75) (4) where a fc is given by (2). Now, multiply both sides of (4) by S[y h ]H(z 1 y h \...,z r y hr ) and integrate with respect to y between the limits and 1, we have f 1 Jo 1 1 = Z O fc = O (5) Express the H -function of several complex variables using [6, p. 251] and a general class of polynomials by [3, p. 1, eq. (1)] on the right of (5), then interchange the order of integration and summation which is permissible under the conditions mentioned in (3) and evaluating with the following result f j Jo [lJ/Wl] ( _-\ Z\ *^ /.' , ~ F-A.,1 a' = a (6) where /z ; >0, R e (r + 1 + ZJ = : ^w/^)) > , |arg(z,.)| < T t n/2, T,>0,i = l,...,r, 7 = 1,..., w <0 and m is an arbitrary positive integer and the coefficients ,4 B a ,(n, a' > 0) are arbitrary constants, real or complex. We arrive at the required result. Theorem concerning a product of a general class 293 3. Applications If we put a = y in the main theorem, the value of a k in (2) comes out to be equal to (f!) k and the result (3) yields the following interesting integral i: ' V" V (~ n) - A MM / / ^^ ,/T. / ^ ^ \_-k-~ ha' :h l ,...,h r ~\, \_-k-hu! - I:*!,...,/!,], (7) where h, > 0, Re(jS) < 1/2, Re(l + ZJ x hffl/Sf) > 0, |arg(zj)| < T f 7c/2, T > 0, i= l,...,r; j= l,...,u (I) and m is an arbitrary positive integer and the coefficients A n a ,(n, a' > 0) are arbitrary constants, real or complex. Take /? = a + 1/2 and a = e (e is a non-negative integer) in (7), we have [\F (-e; Jo i (8) where h f >0, Re(l + EJ =1 Mf /f)>0, |arg(z )| < 7^/2, T >0, i = l,...,r; 7 = 1,...., w (l) and m is an arbitrary positive integer and the coefficients A n(t ,(n^ y! ^ 0) are arbitrary constants, real or complex. Now evaluating the integral on the left of (8) with the help of (6), we establish the following interesting expansion formula 294 V B L Chaurasia and Rajendrd Pal Sharma [(a): 9', . . . , 0] : = !> + te' + l) ( " )mg \ *' = o a'! /(U+l:<ii',t'');...;<ii<'V'-') provided that both sides exist. 4. Special cases (i) On taking m = 2 and A^, = (- If in (3); we have Theorem 1 (a). // n/2 oo valid under the same conditions as obtainable from (3). (ii) When m= land Theorem l(b). 1 + !) . in (3), we have /? \ ^"^ ' ' > ^/ ^ "fc fc = (9) (10) 1 Theorem concerning a product of a general class 295 then p 1-, Jo" ' ^ ' 7 2 '' " * f n + u\ /'n + u + v + a'\ (Aa* =.L?O<- "'(-*)( valid under the same conditions as obtainable from (3). " + " (iii) Letting m = 1 and A^. = Theorem l(c). in (3), we get (11) 2 f !(,/; y + ijyJaF^y- 0,7 -fty + ^ a (12) valid under the same conditions as obtainable from (3). (iv) Letting n->0, the theorem given by (3) reduces to a known theorem recently obtained by Chaurasia [1, eq. (1.2), p. 193]. (v) For n = 0, the results in (6), (7), (8) and (9) reduce to the known results obtained by Chaurasia [1, eqs (2.3), p. 194, (3.1) and (3.2), p. 195 and (3.3), p. 195]. 296 V B L Chaurasia and Rajendra Pal Sharma The importance of our results lies in its manifold generality. In view of the generality of i the polynomials S [x], on suitably specializing the coefficients A n ^. , and making a free use of the special cases of S*[x] listed by .Srivastava and Singh [5], our results can be / reduced to a large number of theorems, integrals and expansion formulas etc. involving [ generalized Hermite polynomials, Hermite polynomials, Jacobi polynomials and its \ various special cases, Laguerre polynomials, Bessel polynomials, Gould-Hopper \ polynomials, Brafman polynomials and their various combinations. i Secondly, by specializing the various parameters and variables in the H-function of | several complex variables, we can obtain, from our theorems, integrals and expansion j formulae etc. involving a remarkably wide variety of useful functions (or products of j several such functions) which are expressible in terms of , F, G and H functions of one j and several variables. Thus, the results presented in this paper would at once yield j a very large number of results, involving a large variety of polynomials and various [ special functions occurring in the literature. I; Acknowledgement The authors are grateful to Prof. H M Srivastava (University of Victoria, Canada) for his help and suggestions in the preparation of this paper. 1 References j [1] Chaurasia V B L, A theorem concerning the multi variable //-function. Bull. Inst. Math. Acad. Sinica, Vol. 13, No. 2, (1985) 193-196 [2] Slater L J, Generalized hypergeometric functions, (1966) Cambridge University Press [3] Srivastava H M, A contour integral involving Fox's H-function, Indian J. Math. 14 (1972) 1-6 [4] Srivastava H M and Panda R, Some bilateral generating functions for a class of generalized hyper- geometric polynomials, J. Reine Angew. Math. 283/284 (1976) 265-274 [5] Srivastava H M and Singh N P, The integration of certain products of the multi variable H-function with a general class of polynomials, Rendicontidel Circolo Mathematico di Palermo, Ser. //, 32 (1983) 157-187 [6] Srivastava H M, Gupta K C and Goyal S P, The H-functions of one and two variables with applications, (1982) (New Delhi, Madras: South Asian Publ.) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 3, August 1995, pp. 297-301. Printed in India. Certain bilateral generating relations for generalized hypergeometric functions MAYA LAHIRI and BAVANARI SATYANARAYANA Section of Mathematics, Manila Mahavidyalaya, Banaras Hindu University, Varanasi 221005, India MS received 30 August 1994; revised 27 January 1995 Abstract. Recently, we introduced a class of generalized hypergeometric functions 7*[J*} (x, w) by using a difference operator A x w , where A xvv /(x) = : - . In this paper an w attempt has been made to obtain some bilateral generating relations associated with 7*(x, w). Each result is followed by its applications to the classical orthogonal polynomials. Keywords. Generalized hypergeometric functions; difference operator; bilateral generating relations; classical orthogonal polynomials. 1. Introduction In the previous paper [2] we introduced a class of generalized hypergeometric functions I^](x 9 w) defined by using a difference operator as follows: _ A C Y wV (a+n ~ n , ( x _ ^[^] A *, w U* w ' (a } __ * (b V w W* ' l* w (1.1) where p+l F q denotes the generalized hypergeometric functions (see, for example, Srivastava and Manocha [8]). We also derived the following relation: I***P)( \_ ( 1+oc )n p-2-i ^ ^ w ' n,( q ) n j q.1,0 /i \ where F^(x,j;) is a double hypergeometric function (see Srivastava and Karlsson [7, p. 27(28)]). The following definitions and results given by Konhauser [1, p. 303(3)], Srivastava and Manocha [8, p. 243(1 1)] and Manocha [4, p. 687(1.3)] have been used here in regard to the bilateral generating relations for the generalized hypergeometric function I*(x, w): (1.3) where a + 1 a + 2 a-h/c A(k;l + a ) = r , 7 ,..., r~ fc = 1,2,3...); k k k (a -+ n + m + 1; 1 + a; - x); (1.4) 297 298 Maya Lahiri and Bavarian Satyanarayana y - m! x F , -m; -a -ft ^ (1.5) where Fj is an Appell function [6]. We also derived the extended linear generating relation [3] as follows: .X w' X w' wt 1-r , w, w (1.6) where F (3) is Srivastava's general triple hypergeometric series (see, e.g., Srivastava and Manocha[8, p. 69(39)]). 2. Bilateral generating relations We have derived the following bilateral generating relations for the generalized hypergeometric function /*(*, w): = M _ t\- f J7P + 2:0,0,0,1 V 1 L ! r o + m- i : o,0, 0,0 : U, 1,0], [(a,): 1,0, 1,1], - , /Z, W/Z, 1 ' (2.1) [A(k:l+/0:0,l,l,0]:-;-;--;-; 1~?' F is a generalized Lauricella hypergeometric function of 4 variables and " From (2.1), we have X In- Generalized hyper geometric functions 299 __ Y vv Y i 2:0,0,0,1 1:0,0,0,0 ):0, 1,1,0]:-; -;-;-; wt 1-t , Jl, W/J, W [using (1.5)]. This completes the proof of (2.1). Applications (i) By setting p = q and a } = bj(j = 1, 2, ... p) in (2.1), we get 5 J 5 - -,, i r (2.2) KvY t 1 V and J*(x, w) is a modified Jacobi polynomial studied by kj t 1 Parihar and Patel [5]. (ii) On taking k=l,p = q, (a,.) = (fy) and letting w -* in (2. 1), we get the known result given by Srivastava and Manocha [8, p. 133(9)]. The following results can also be deduced by using the same technique as followed in the previous result. XjF (3) W W w, - wt -;(&.):-;! [using (1.4)]. (2.3) 300 Maya Lahiri and Bavanari Satyanarayana Applications (i) By writing p = q and (a ; ) = (bj) in (2.3), we have r .-,- ,-,; (Z4) where X F 1 is Humbert's function defined in [7, p. 26(21)] and J"(x,w) is a modified Jacobi polynomial studied by Parihar and Patel [5]. (ii) Taking limit as w -> in (2.4), we obtain the result given by Srivastava and Manocha [8, p. 160(70)]. 4 + 2:0.0,0,0 Tx "1 [(a p ):l,l,0,l],[l +y:l,0, 1,1], I- 1:1,0, 0,1J, [(&,): l,l,0,l],[l+y:l, 0,0,1], w, rh; [using (1.5)]. (2.5) where h= The following applications are obvious: n! w where h = {1 +(y + 1 n! wtfr (2.6) (y + , t/z, xt/z (2-7) Generalized hypergeometric functions 301 o ooii X X 1+a;-; (2.8) Applications [As usual, we get] ; (2.9) where F^ } is a Lauricella hypergeometric function of n variables (see [8, p. 60(1)]). (2.10) Acknowledgements The authors take this opportunity to express their sincere thanks to Prof. H M Srivas- tava (University of Victoria, Canada) for his valuable suggestions and helpful criticism in the preparation of this paper. References [1] Konhauser, Joseph D E, Biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math. 21 (1967) 303-314 [2] Lahiri M and Satyanarayana B, A class of generalized hypergeometric functions defined by using a difference operator, Soochow J. Math. 19 (1993) 163-171 [3] Lahiri M and Satyanarayana B, Extended linear and bilinear generating relations for a class of generalized hypergeometric functions, Indian J. Pure Appi Math. 24 (1993) 705-710 [4] Manocha H L, Bilinear and trilinear generating functions for Jacobi polynomials, Proc. Cambridge Philos. Soc. 64 (1968) 687-690 [5] Parihar C L and Patel V C, On modified Jacobi polynomials, J. Indian Acad. Math. 1 (1979) 41-46 [6] Rainville E D, Special functions, (New York: Macmillan) (1960) [7] Srivastava H M and Karlsson P W, Multiple Gaussian hypergeometric series, (Ellis Horwood Limited, Chichester; Halsted Press) (New York: John Wiley) (1985) [8] Srivastava H M and Manocha H L, A treatise on generating functions, (New York: Halsted Press, John Wiley) (1984) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 3, August 1995, pp. 303-314. Printed in India. A localization theorem for Laguerre expansions P K RATHNAKUMAR Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore 560059, India MS received 30 October 1994; revised 7 February 1995 Abstract. Regularity properties of Laguerre means are studied in terms of certain Sobolev spaces defined using Laguerre functions. As an application we prove a localization theorem for Laguerre expansions. Keywords. Laguerre means, Laguerre series, Sobolev spaces. 1. Introduction The Laguerre polynomials L(x), of type a > 1 are defined by the generating function identity r) , |t|<l. (1.1) o The associated Laguerre functions are defined by &i(x)= L*(x)e~ x/2 x* /2 (1.2) and they are the eigenfunctions of the Laguerre differential operator / n \ \ ' ~ Moreover the normalized functions ^*(x) = I - ) <&*(x) form an or- \ V ' / thonormal basis for L 2 [(0, oo),dx]. Therefore for any fe L 2 (0, oo) we have the eigen- function expansion /=IX^M ( L4 ) o with Three types of Laguerre expansions have been studied in the literature. The first one is concerned with the Laguerre polynomials LJJ(x), a > - 1, which form an orthonor- mal basis for L 2 [(0, oo),e~*x a dx]. The second type is concerned with the Laguerre functions (1.2) which form an orthogonal family in L 2 [(0,oo),dx]. Considering / r(+ 1) Y /2 the functions /JJ(x)= - - - 1 L*(x)e~ x/2 as an orthonormal family in L [(0, co), x a dx], we get a third type of expansion. 303 304 P K Rathnakumar Several authors have studied norm convergence and almost everywhere convergence of Riesz means of such expansions. Some references are Askey-Wainger [2], Mucken- houpt [6], Gorlich-Markett [3], Markett [5], Stempak [7], Thangavelu [10]. Various results can also be seen in [12], [1]. Recently by invoking an equiconvergence theorem of Muckenhoupt for Laguerre expansion, Stempak [8] has proved the following almost everywhere convergence result for expansions with respect to f* n (x) as well as /"(x). (1) Z%(g,<e c k ) L2(dx) e ( k (x)->g(x) for almost every xe!R+ as N-oo for f <p<4 if oc> - , and for pe ( ( 1 +- ) ,4 ] otherwise. (2) !Q(#, /fc) L 2 (xMx) /fc(x)->0(x) for almost every xeR+ as JV-> oo for < p < 2a -f 3 if a > 4, and for 1 < p < oo otherwise. (Zcf. + 1) In this paper we study the twisted spherical means associated with the Laguerre expansions which we will call Laguerre means. We consider expansions with respect to the system <p(x) = Lj;(x 2 )e~* 2/2 . Then the normalized functions form an orthonormal basis for L 2 [(0, oo),x 2ot+1 dx]. We have the mapping T: L 2 [x 2a + 1 dx] -> L 2 [x a dx] defined by Tf(x) = -A=/(>/x), which is a unitary map- pmg which takes t/^(x) to l* k (x). Therefore the expansion in \f/l is equivalent to the expansion in /. We prove a localization theorem for Laguerre expansion with respect to ^J without appealing to the equiconvergence theorem. Clearly a localization theorem follows from the almost everywhere convergence result of Stempak given above, but this result only says that if / = in a neighbourhood of a point ze(0, oo), then S N f(\v) - for almost every w in this neighbourhood. But using the method of Laguerre means we could identify the set on which S N /(w) -* 0. The twisted spherical mean of a locally integrable function / on " is defined to be = I J (1-6) where d/i r (w) is the normalized surface measure on the sphere {|w| = r} in C n . Such spherical means have been considered by Thangavelu in [11], where its regularity properties are used to prove a localization theorem for the special Hermite expansion of L 2 functions on <". The special Hermite expansion of a function/ is given by (L?) A f where <p fc (z) = L n k 1 (l|z | 2 )e 1/4|z|2 . Here LJ" 1 (r) stands for the Laguerre polynomial of type n - I . Measuring the regularity of /> P (z) using a certain Sobolev space denoted by H^(R + ), he proved the following localization theorem: A localization theorem for Laguerre expansions 305 Theorem 1. (S. Thangavelu) Let f be a compactly supported function vanishing in a neighbourhood of a point ze( n . Further assume thatfu r (z)e W"^ 2 (R+ ) as a function of r. Then S N f(z)^Q as N-+ oo. By assuming certain regularity of f^ r (z) as a function of r he could also establish an almost everywhere convergence result for special Hermite expansion. In the study of y> r (z) a crucial role is played by the following series expansion: 00 ) (1.8) jt = IA T H AJI for the twisted spherical means. Here fq> k denotes the twisted convolution of/ and cp k , where twisted convolution of two functions / and g on f n is defined by "1 (e i/2/m(z.H-) dw ( 19 ) C n For a radial function / we have M(z) = (27c)-" J R k (/)( Pk (z), (1.10) where Therefore from (1.8) it follows that for a radial function/ the special Hermite expansion becomes the Laguerre expansion with respect to the family L n k ~ l {j\z\ 2 )e~ 1/4lz] *. The above observation suggests that we can also study the localization problem for Laguerre expansion with respect to the orthogonal family L(r 2 )e~ l/2r \ a > 1. What we need is something similar to twisted spherical means. Using the local co-ordinates on the sphere |z| = r in $ n it is easy to see that (1.11) for a suitable constant c n . We define the Laguerre means of order a to be fVc Jo 2n (1.12) (rzsin0) a ~ 1/2 Then T? is a bounded self adjoint operator on L 2 (R + ,x 2flt+ MX). We have the interesting formula, see [12] *. for oc> ~^,r ^0,z^0. From the series expansion for 7?/(z) in terms of (JP(Z) and 306 . P K Rathnakumar using the above formula it is easy to see that T*f(z) has the series expansion r^0,z^0,a> - 3, where <p(r) = L(r 2 )e~ 1/2r \ Here (,) a denotes the inner product in f the Hilbert space L 2 [# + ,x 2a+1 ]. Using this notion of Laguerre means we establish j a localization theorem for Laguerre series expansion for fe L 2 [(R + ,x 2a+1 dx] with j respect to the orthogonal family <p(r). Our main result is the following: \ Theorem!. Ler/eL 2 [R + ,x 2a+1 dx],a> - 4 be a funct ion vanishing in a neighbour- hood B z of a point zeU + .IfweB z is such that TJ/(w)e W ( f+ 1)/2 (iR + ), as a function of >, ; then S N /(w)->0 as N-oo. [ We use the following notation: L*(R + ) stands for the space L 2 [R + , x 2a+ 1 dx], and | the norm and the inner product in this space are denoted by || . || a and (. , .) a respectively. j 2. The Sobolev space W' 9 (R + ) * / The usual Sobolev space H*(R"), for s ^ is defined to be j H S (R") = {fe L 2 (R M ): (- A + l) s /e L 2 (R n )} d 2 d 2 using the operator A = j + - - - + T-? - Since we are interested in studying the regular- I dx 2 dx; \ ity of the function r-> TJ/(z), motivated by the expansion (1.14) we define the Sobolev space W*(R + ) using the operator L a = - ? + ^ 2 I which is a positive |_dx 2 x dx J definite symmetric operator and the <pj's form the family of eigenfunctions with (a-h 1\ k + r- 1. Also we have the normalized functions \l/l(z) forming an orthonormal basis for L 2 (1R + ). We define for 5 ^ L W s a (U + ) = {feL 2 l (R + ):L s JeL 2 (R + )}. (2.1) f where L^ is defined using the spectral theorem. In other words 30 ! f X"" 1 / I d\ / a fc = belongs to W^ if and only if, 2 z < 00. We now prove the following useful proposition which is needed for the proof of the main theorem. PROPOSITION 3 Let a > - 1 and let (pbea smooth function onR + which satisfies the following conditions (i) cp = Q near the origin in R + (if) dV A localization theorem for Laguerre expansions 307 1 -. ) as r - oo /or 7 = 0, 1, 2, 3, . . . , 2m. TTierc t/te operator M v :W S X -+ W s a+l defined by M 9 f = <p.f is a bounded operator Vs such that The proof of this proposition needs the following lemmas. Before stating the first lemma we introduce, for each non-negative integer fc, the class C k , consisting of all smooth functions on R + , vanishing near and which also satisfies the decay condition, dV / 1 \ a = O , . . , . as r-oo. The class C fc satisfies the following properties: (i) dr/ \^ r *+*+jj C k + l a C k9 (ii) If (peC k ,-<peC k+1 , r<peC k __ 13 for k> 1, (iii) If cpeC k , (p (J] eC k+j . Lemma 4. Under the above assumptions on m, cp and a we have L^^^M^ Y d - 1 L\ m with <p k>t eC k . Proof. We claim that L^ + j_ M^ can be written as a linear combination of the form with (p kt eC k . (2.2) r +fc ^ \ary ^ First we note the following relations d n 7^ Qr L~t- **-*- <> rdr Using this relation in the above we get We also use the relation, where b^ c 1 , c 2 , are constants. This can be easily proved by induction on k. We prove (2.2) by induction on m. (2.2) is clear for m = 1. Assume (2.2) for m =;'. Now, 308 P K Rathnakumar In the above computation we have used (2.3). In view of (2.6), the first term of the above is k-l /i\i /A k-i ar - I Afm-z*, z ; t + k<j+l A ar / i = r-fk<j\ r / ar + I c^riY'V I.c 2 (^Y" 2 Ii t+fc^j ar ar Now by induction hypothesis we have cp kt eC k . Note that in the second term of the above the coefficient of I j L t a is(l/r) i cp kr WQhavQ(l/r) i cp kt eC k+i aC k c: C k _ i for i > and also rcp kt eC k _ 1 . Hence the first term in (2.7) is of the required form. The / d Y second term of (2.7) can be written as -2 t+k<m M- i I 1 L' a , and <? M eC fc by induction hypothesis. Therefore ^J t _ 1 t eC k in view of (iii). Hence the second term of / d \ k (2.7) is also of the required form. In the third term the coefficient of L; is \drj M " _4_2v and <p' k ^ t H 9k,t ^k+2 c Q by induction hypothesis and in view of k. t t r V*,r ' Y (i), (ii) and (iii). Similarly - <p' ktt occurring in the fourth term belongs to C k + 2 c= C k . Also -<p k ft occurring in the fifth term C k + 1 <= C k . Therefore (2.2) holds for m =7 -h 1 also. / d\ k Thus we have T m f = L + 1 M V L~ m f = E^^M^ L*~ m f- Which proves the \ / first lemma. I A localization theorem for Laguerre expansions 309 Lemma 5. f J Z4:L 2 (R + )-> L^(R + ) is a bounded operator whenever i is a non- negative integer and i 4- 1 ^ Proof. We prove that L, is a bounded operator on L 2 (R + ) for 1 + t < 0. We first note that - This can be seen as follows. We have + _ \ Here we have used the relations (i) ^(r)=-L^} and, /::\ ra+1 _ ra+l_ r a W H ^-i H Now (2.9) follows from the definition of i/^. Let /e L 2 (IR + ). By definition L'J(r) = 4' ar and using (2.9) we get ^rS/^) (2.10) where and 310 P K Rathnakumar Therefore, d dr a \\rSj \r (2.13) Now using the expansion (2.11) we calculate, l|rr/(r)|| 2 = o Jo = |T/(r)| 2 r 2 +3 dr !o 2 ' +1 Using (2.14) and (2.15) in (2.13) we see that one can show that , dr Z4 2 ' = II/L 2 (2-14) since 1 + 1 ^ 0. Similarly one can see that (2.15) : 2 1| / 1| a for 1-hf^O. Similarly ^ c II / II a f r some constant c, whenever j + 1 < 0, which * proves the second lemma. Proof of proposition 3. We have by definition W s a = L~ S (L*(R + )). Therefore it is enough to prove that J4+iAf v L~ S :L^((R + )-* Lj +1 (R + ) (2.16) is a bounded operator. Put where !+ 1 and L~* are defined using spectral theorem. Then clearly, (2.18) for some constant c independent of/. We will also prove that, for any positive integer m Hr m /L +1 <c 1 ll/iU (2.19) for some constant c t independent of/. V, A localization theorem for Laguerre expansions 3 1 1 Assuming (2.19) for a moment choose f l eLl(R + ) and ge L* +l (U+) to be finite linear combinations of i/^'s and \l/% + 1 's, respectively. Consider the function h which is holomorphic in the region < Re(z) < m and continuous in ^ Re(z) ^ m, defined by: h(z) = (T'f l9 g l ) n+1 =(L: +l oM^L-'f l9 g l ) a+l ' (2.20) Then by (2. 18) we have, where 7i = L~ i); /i > and g l = L~.ft # x . Therefore, <^o HAIL llsi IUi and since both L~ iy and L^ are unitary operators, we get Similarly by using (2.19) we get Thus we have IM^Kcoll/JIJI^H^! . (2.21) IMm + ^KcJ/JJI^IUi ' (2-22) Since /i is a bounded function we have by three lines theorem for < t < m. In particular, IfcWKcS-^v^llAil.ll^lL^, that is, KT7i^i)l<4~ r/m 4 /m ll/ilUl^ill a+ i- (2.23) Now taking supremum over all such g^L^ +1 with ||0ill a +i^* we get H T 'fi II + 1 < cj~ r/m c r / m || /! || a . Therefore T r is a bounded operator on a dense subset of L^. Therefore it has a norm preserving extension to L*. Thus we have H^/ll a+ i^^ll/LV/eL a 2 (R + ), forO<t<m (2.24) which proves (2. 16). To prove (2.19) we proceed as follows. By Lemma (4) we have T m / = I> t+k *m M (p ktt (dr) L ^~ m ' And by Lemma ( 5 )' (j") L a' m is a bounded operator on Lf(R+), whenever fe -h (t m)< 0. Also note that since q> ktt satisfies the conditions (1) and (2) of 312 P K Rathnakumar the Proposition 3 for; = 0, M^ ( maps L 2 (IR + )- Ll + 1 ((R + ) boundedly. Thus we get ||T m /|| a+1 ^Cill/lla. This completes the proof of the proposition. 3. Regularity of r r a /(z) In this section we prove that the Laguerre means T"/(z) are slightly more regular than /, for z ^ 0. To prove this fact we use the series expansion (1.14) for T* r f(z). Let/e W s a . Then '> converges in L(R + ). We also use the following asymptotic estimates, (see [4] or [9]) (3.2) . = -$ .,. 0.3, i/^(0) /c a/2 as /c -> oo (3.4) From (1.1 4) we have r , .. * = o * V^ T -h ij for z = 0, in view of (3.2) and (3.3). Also 2 forz = (3.6) f Jo in view of (3.2) and (3.4). Comparing (3.1) and (3.5) we see that /e W^r-* T*f(z)e ^s +(a /2)-f ( i/4) comparing (3.1) and (3.6) we see that/e W 5 a if and only if T*f(z)e W^. Thus we have proved the following: Lemma 6. (i) /e W*=>r- T*/(z)e j^+</2)+(i/4) jZ ^ 0. (w) /e ^ i/ and on/y i/ r -* TJ/(0)e W^. Now we prove some properties of Laguerre means Lemma 1. (i) Iff is supported in z^b, then T"/(z) as a function of r is supported in r^b + z. (ii) If f vanishes in a neighbourhood of z then T*f(z) as a function of r vanishes in a neighbourhood of origin inU + . Proof, (i) If / is supported in z^b then the integral (1.12) vanishes unless (r 2 + z 2 -h2rzcos0) 1/2 s:b. This implies (r~z) 2 O 2 . Therefore the integral (1.12) vanishes unless |r~z|^feorr^b + z. (ii) Again if/ vanishes in a neighbourhood {| y - z\ < a], a > of z, the above integral (1.12) is zero if |(r 2 + z 2 + 2rzcos 0) 1/2 - z\^ a. Since z is fixed this says that the above A localization theorem for Laguerre expansions 313 inequality holds for r in a neighbourhood of 0. Now consider the continuous function g(r) = |(r 2 + z 2 + 2rz cos 0) 1/2 - z| - a, defined on R + . We have 0(0)= a<0. Therefore #<0 in a neighbourhood of as well. This means that for r in some neighbourhood of we have |( r 2 + z 2 4- 2rz cos 0) l/2 z| < a. Thus T?/(z) = in that neighbourhood. 4. A localization theorem for Laguerre expansions Now we are in a position to prove Theorem (2) stated in the Introduction. From (1.14) using the orthogonality of i/r J we get T*f(z)(p* k (r)r 2 * + x dr = T(a + 1" o Again from (1.14) we get, fc = T r f(z) ^(r)r 2 " + Mr o 1))- 2 r?/(z)< Hr)r 2a+ Mr. (4.2) o Here we have used the relation g LJ(x) = L" N +1 (x). We use the above representation for S' N f(z) to prove Theorem (2). The proof uses the following fact: If ge L, (R + ), then the Fourier-Laguerre coefficients to,^) a -0 as k-oo. Recalling the definition of \l/l this means that Jo f Jo 1 Also if ge W^(R+) then, _ s+a/2 /44) I o From (4.2) we get T a f (Z) xl /N 9*+^J ^451 J^~ - m* +1 (r)r 2 dr. ^- J ' !o r Let h be a smooth function on (R + ) such that h(r) * 1 on the support of T*,f(z) and h(r) = in a neighbourhood of the origin in R + . Put h(r) = -^- . Thus we get SJ/W = (T( + I))' 2 f " Mr) T?/(z)< Hr)r dr ( 4 -6) Jo Now if T}f(z)e W ( * + 1)/2 , we have by Proposition 3 fc(r) rj/(z)e ^ + 1 314 P K Rathnakumar by (4.3), as N - oo. Therefore S/(z) -*0 as N -* oo, which proves the theorem. In view of Lemma 6, if /e H^ 1/2 , then T* r f(z)e W ( f + 1)/2 , for z ^ 0. Thus we have the following corollary to the above theorem. COROLLARY 8 If fe W 1 J 2 then the conclusion of Theorem 2 holds at points z^O. Acknowledgement The author thanks Prof. S Thangavelu for suggesting this problem and also for many useful discussions he had with him. He also thanks The National Board for Higher Mathematics (India) for the financial support. References [I] Akhiezer N I, Lectures on integral transforms, Am. Math. Soc. Providence, Rhode Island., (1988) [2] Askey R and Wainger S, Mean convergence of expansions in Laguerre and Hermite series, Am. J. Math., 87 (1965) 695-708 [3] Gorlich E and Markett C, Mean cesaro summability and operator norms for Laguerre expansions, Comment. Math. Prace Mat., tomus specialis II, (1979) 139-148 [4] Lebedev N N, Special functions and their applications, (New York: Dover Publ.) (1992) [5] Markett C, Mean cesaro summability of Laguerre expansions, and norm estimates with shifted parameter, Analysis Math., 8 (1982) 19-37 [6] Muckenhoupt B, Mean convergence of Laguerre and Hermite series II. Trans. Am. Math. Soc, 147 (1970) 433-460 [7] Stempak K, Almost everywhere summability of Laguerre series, Stud. Math., 100(2) (1991) [8] Stempak K, Transplanting maximal inequality between Laguerre and Hankel multipliers, preprint. [9] Szego G, Orthogonal polynomials, Am. Math. Soc., (Providence: Colloq. Publ.) (1967) [10] Thangavelu S, Summability of Laguerre expansion, Anal. Math. 16 (1990) 303-315 [II] Thangavelu S, On regularity of twisted spherical means and special Hermite expansions, Proc. Indian Acad. ofSci., 103, 3 (1993) 303-320 [12] Thangavelu S, Lectures on Hermite and Laguerre expansions, Math, notes, 42, (Princeton: Princeton Univ. Press) (1993) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 3,- August 1995, pp. 315-327. Printed in India. Degree of approximation of functions in the Holder metric by (#, c) means G DAS, TULIKA GHOSH and B K RAY + Department of Mathematics, Utkal University, Bhubaneswar 751 004, Orissa, India + Department of Mathematics, BJB Morning College, Bhubaneswar 751 014, Orissa, India MS received 15 January 1994; revised 5 December 1994 Abstract. Degree of approximation of functions by the (e, c) means of its Fourier series in the Holder metric is studied. Keywords. Fourier series; Holder metric; Banach space. 1. Definitions and notations Let/ be a periodic function with period 2n and integrable in the sense of Lebesgue over [ TC, TI]. Let the Fourier series of/ at t = x be 1 -a + (a k coskx + b k sinkx). (1) ^ k=i Let **W = *{/(* + ') + f(x - ~ 2/(x)}. (2) Let S t (/; x) be the fcth partial sum of the Fourier series (1). Then it is easily seen that (see C9], p. 50) ' ,. (3) Let C 2K denote the Banach space of all 27r-periodic and continuous functions defined on [ 7i, TT] under the sup-norm. For < a < 1 and some positive constant K, the function space H a is given by the following: K\x-yr.. (4) The space H a is a Banach space [7] with the norm ||- || a defined by where ||/|| c = sup |/(x)| (6) -7C*$X^Jl and ,7, We shall use the convention that A/(x, y) = 0. The metric induced by the norm (5) on H a is called a Holder metric. It can be seen that ||/||, ^ (2n)'~ f ||/|| for < ft < a < 1. 315 316 GDasetal Thus (H a , ||-|| a ) is a family of Banach space which decreases as a increases, i.e. C 2;t 2 Hp 3 H a for (K J8 < a < 1. DEFINITION An infinite series 2*^ c w with partial sums {} is said to be summable (e,c)(c > 0) to sum 5, if where it is understood that C n + k = when n + k < 0. The (e,c) summability method which is a regular method of summation was introduced by Hardy and Littlewood [4] (cf. also [5] ) as an auxiliary method to prove Tauberian theorem for Borel summability. It is known [6] that, if c n = 0(1) and c-i k -i + q ( 9) C "~2 a ~2(l-/c)~ 2q U then summability of Ec n by any one of the methods (e, c), Borel exponential method (JB, a), Borel integral method (', a), a > 0, Euler method (E, q)(q > 0) and circle method (y, k)(0 < fe < 1) implies its summability to the same sum by any of the others. 2. Introduction Alexits [1] studied the degree of approximation of function of H a by the Cesaro mean of their Fourier series in the sup-norm. Since C 27C 3 H a 2 H^ for ^ j? < a < 1, Prosdorff [7] obtained an estimate for \\(r n (f) - /||^ for/e/f a , where <r n (/) is the Fejer means of the Fourier series of/. Precisely he proved the following: Theorem A ([7], Theorem 2). Let /eH a (0 < a ^ 1) and ^ ft < a. Then nOogny- 1 (a l) ' The case j? = of Theorem A is that of Alexits referred to earlier. Recently Chandra has studied the degree of approximation of functions in Holder metric by Borel's means [3] and by Euler's means [2]. Precisely, he proved Theorem B [3]. Let ^ < a ^ 1 and let /eH a . Then where B n (f) is the Borel exponential mean of 5 n (/; x). Theorem C [2]. Let ^ ft < a ^ 1 and let /eff a . Then where E q n (f) is the Euler (E, q) 9 q > mean of S n (f; x). Degree of approximation of functions 317 The object of this paper is to find the degree of approximation of functions by the (e, c)-mean of its Fourier series in the Holder metric. Denoting the (e, c)-mean of (/; x) (10) *(/,*) = <(/;x) = exp-S n+fc (/;x), where S n+x (f; x) = 0, n + k < 0, we prove the following theorems: Theorem 1. Let Q< oi ^ I and ^ ^ x. Let feH x . Then ) -/!!, = 0(1) logn Theorem 2. Let Q x ^ I and Q ^ ft <a and let feH x . Further, if *' tlogn/ "" 2 < then 2jc/2n II .(/)-//, = 0(1) (lognp 3. Additional notations and estimates We use the following additional notations: c " ck 2 \ K n (t)= h+2 Y exp (.- Icosfct - P -r t e=6(n)= - ck 2 exp( - '/='/() = 2n+l Ttlogn N=AT(n) = n ' i l ~^c F(t)=<D x (r)-*(t) (11) (12) (13) (14) (15) (16) (17) (18) (19) 318 GDasetal Estimates. We need the following estimates: If/sH a ,0<a<l,then (20) y| a ) (21) and exp(- nA(r + >7) 2 ) - exp(- nit 2 ) = 0(t + *?)exp(- nit 2 ) (23) K n (t) = exp(-nAf 2 ) + iMtt), where ^(n) = 0(e~^), c><5>0 (24) L rt (r)=0(re-^), (c>5>0) ( 25 ) If there is no confusion, we shall write throughout S as a suitably chosen positive constant not necessarily the same at each occurrence. ^ Proof of the estimates. Estimates (20) and (21) follow immediately from the definition of<3> x (r)and# a . Now and Hence (22) follows at once. Proof of (23). We put g(x) = exp( - nix 2 ). By mean value theorem for some < < 1 exp(- nA(t + ??) 2 ) -exp(- nAt 2 ) = g(t + 1?) - ^W = w'(t 4- ^), from which (23) follows at once. Proof of (24) is contained in (Siddiqui [8], p. 122), and proof (26) can be found in (Hardy [6], p. 205). Proof of (25). We have ck 2 ex p -i-sin n 00 X /cexp - xexpf- n-fl V Degree of approximation of functions 319 Proof of Theorem 1. From (3), (10) and (15), we get taking S n+t (/;x) = 0, when k< -n 2 c " / c/c 2 \ . / 1\ \ exp -- sin \n+k+- }t dt -. V / \ 2 / / We have C/C 2 \ .(')-./- 1 exp|- v sm.n + fc+ -) t ' [7r " ( ck 2 \ . / , 1\ = / E exp -- sin [n + k+-]t ^nnl k ^_ n P V / V 2; + exp(-^-)sin( + k + ^)t *=+i V n / V 2 / J / c fc2Ny \ / A 2yexp -- coskt sin \n + - }t k- 1 \ / ./ V 2 ; / c^ 2 \ . / , 1\ 1 + I exp -Isin + * + -) *=+! \ " / V 2 / J using (13) and (14). From (27), (28) and (24), we get Let Then using (12) and (19), we obtain '(*) = /.(y) = ^ p 2^n% e (t)dr + (0(n) ~ 1)(/(x) ~ /0;)) - (27) (28) 2 + (0(n) - l)(/(x) - /(y)) (29) n J 320 G Das et al L - (r)dr Using (20), (21) and (24), we get L, say. Using (31) and (32) J = Ji -/ and using (26), we get Similarly (argue as in J) We write Using (20), we get = 0(1) f Jo (30) (31) (32) (33) (34) (35) (36) Degree of approximation of functions 321 Using (21), we get f* **0(l)\x-y\*n\ dt = 0(\x-y\ a ). (37) Jo Using (20), we get = 0(1) a " 1 e"" ;ua dt JN = 0(l)e~ nAJV2 f a ~ Mr (as e~" At2 is decreasing) J N = 0(l)(e" A1 8n ) (A > however large) Using (21), we get pc. e -A*> -yn dt JN l = O(|x - y D( -T ) (A > however large) (39) \ n J as in (38). Now _H r *.,,.(. + ' J,2sinft V 2 2 fF(t) . . / 1\ . = - ^e"** sin n + -tdt + F(t) ..e-^sin + TrJ, w L2sm^ tj V = / 2ll + / 2>2 ,say. (40) 322 G Das et al Using (20) and the fact that we get "~KL * w l2sinif f v 1J L_ * = 0(1) rVv 1 *' J 1J - 1 ) f t a ~(e~ ;int2 )dt J rj dt = (n ~ x " a ) (integrating by parts). Next, we write = fVW )n\ t F "' - f . \ \ 2 ( 1 " (n + l -)tdt \ (since sin(n + %)(t -f jy) = sin(n + ^) 2 = M! 4- M 2 -f M 3 + M 4 + M 5 , say. ( 42 ) Degree of approximation of functions 323 Using (22), we have fw e -*> f"dt rt /logn\ .... M, = O(^) ^- dt=o(^) 7 = (--- (43) J 1\ J tj Using (20) and (23) pN (r + i/) a ~ l e~ Al " a (t J ? r-* a dt = O(n~ a ) (integrating by parts). ( 44 ) Using (20), we get and f2ri 4 = 0(l) t a - 1 e- A " |1 dt = J >; pN + r, 5 = 0(l) f^'^dt JN fN + r, = O(l)e" A " Jva f'Mt JN (A however large) ( 46 ) Now, we write 2M 2 = aN [N+n pn [N + n , + L + J, -L x sm ^ 324 G Das et al ,. Using (20) and the fact that it can be proved employing the argument used in proving (45) and (46) that (48) I j (49) [, By formal computation, we get ' j i As we obtain using (20) p 2 = od) r^( ,. Using (22), we get l ; \ 'M / (51) f Lastly using (20) and (23), we get - 1 df = Q/f!s^ A (53) (52) v Degree of approximation of functions Collecting the results of (42)-(53), we get 7 - ^.i^ From (40), (41) and (54), we have 325 (54) (55) Using (21), we also get = 0(|x-3>| a ) | N y = (56) Writing / = /l-/3/ar0/a (fc _ 1 7 ^ i fc J fc J fc yc 1,Z,, JJ and using the estimates (36), (37) for I 19 (55), (56) for I 2 and (38), (39) for / 3 we get -yl^-j, A>0, however large. From (35), (57), (58) and (59), we get (57) (58) (59) (60) Using (25), we get ' ^;W>* From (61), we get (writing K=K l -^K? 1 *) Collecting the results of (30), (33), (34), (52) and (62) we get Hence -L, V" (61) (62) (63) 326 G Das el al Again /eH I =><J> x (r) = 0(|rn and so proceeding as above, we get log" A , U= sup | Tt^.X^Tt (64) Theorem 1 is completely proved by combining (63) and (64). Proof of Theorem 2. We proceed as in the proof of Theorem 1 and retain all the estimates of J, K and JL As regards 7, we retain all the estimates of the components of / except the one given in (43) for M^ which contributes the estimation 0(log n/n*). By (11) of the hypothesis of Theorem 2 sin i A - rdr *- = 0(1) dr Using (65) instead of (43), it can be proved that Now using (56) and (66) (65) (66) P*it-\ (67) Proceeding as in Theorem 1 and using (67) and the estimates of I and / 3 from (57), we obtain (68) Arguing as in Theorem 1 and using (11) as employed above in the estimation of / 2 , it can be shown that (69) Now Theorem 2 follows at once from (68) and (69). Degree of approximation of functions 327 Acknowledgement We thank the referee for his helpful suggestions. References [1] Alexitis G, Ober die Annaherung einer stetigen Function durch die Cesaroschen Mitteel ihrer Fourier- reihe, Math. Annalen 100 (1928) 264-277 [2] Prem Chandra, Degree of approximation of functions in the Holder metric, J. Indian Math. Soc. 53 (1988)99-114 [3] Prem Chandra, Degree of approximation of functions in the Holder metric by BoreFs means, J. Math. Anal. Appl. 149(1990) 236-246 [4] Hardy G H H and Littlewood J E, Theorems concerning the summability of series by Borels exponential method, Rend. Circ. Mat. Palermo 41 (1916) 36-53 [5] Hardy G H H and Littiewood J E, On the Tauberian theorem for Borel summability, J. London Math. Soc. 18(1943)194-200 [6] Hardy G H, Divergent series, (1949) Oxford [7] Prossdorf S, Zur Konvergenz der Fourierreihen Holderstellger Funktionen, Math. Nachr. 69 (1975) 7-14 [8] Siddiqui J A, A criterion for (e, c)-summability of Fourier series, Math. Proc. Camb. Philos. Soc. 92 (1982) 121-127 [9] Zygmund A, Trigonometric series, (1959) (Cambridge University Press, New York) Vol. 1 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 3, August 1995, pp. 329-339. (Q Printed in India. The algebra A p ((, oo)) and its multipliers AJIT IQBAL SINGH* and H L VASUDEVA f * Department of Mathematics, University of Delhi South Campus, Benito Juarez Road, New Delhi 110021, India Department of Mathematics, Panjab University, Sector 14, Chandigarh 160014, India MS received 22 April 1994; revised 18 February 1995 Abstract. Let I = {xeR: 0<x< oc} be the locally compact semigroup with addition as binary operation and the usual interval topology. The purpose of this note is to study the algebra A p (I) of elements in Lj (/) whose Gelfand transforms belong to L p (I), where / denotes maximal ideal space of Lj(7). The multipliers of A p (I) have also been identified. Keywords. Binary operation; interval topology; Gelfand transforms, maximal ideal space. 1. Introduction Let G be a locally compact Hausdorff Abelian group and G denote the dual group of G. The algebra A p (G\ l^p^ao, of elements in L 1 (G) whose Fourier transforms belong to L p (G), and the multipliers for these algebras have been studied by various authors including Larsen, Liu and Wong [8], Reiter [10], Figa-Talamanca and Gaudry [3], and Martin and Yap [9]. The algebra ^4 p ((0, oo)) with order convolution, in short, A p (I) of elements in L t (7) whose Gelfand transforms belong to L p (I) and the multipliers for these algebras, where / is the locally compact idempotent commutative topological semigroup consisting of the open interval (0, oo) of real numbers from to oo equipped with the usual topology and max. multiplication and 7 is the maximal ideal space of L! (/), have been studied by Kalra, Singh and Vasudeva [6]. The purpose of this note is to study the algebras A p (I) of elements in LJ7) whose Gelfand transforms belong to L p (I) and the multipliers for these algebras, where / = {xeR: ^ x < 00} is the locally compact semigroup with addition as binary operation and the usual interval topology and /is the maximal ideal space of 1^(7). Whereas the algebras A p ((Q, oo)) with order convolution studied in [6] are dissimilar to the order convolution algebra L 1 ((0, oo)), the algebra A p (I) proposed to be studied in this note show similarities to the algebra 7^(7). In particular we shall see that the maximal ideal space j\(A p (I)) of A p (I) is the same as that of 7^(7). The situation is thus akin to the group algebras Lj(G) and its subalgebras A p (G) studied by Larsen, Liu and Wong [8]. It turns out that the algebras A p (I) are not regular, whereas the algebras A p (G) [8] and A p ((Q, oo)) with order Convolution [6] are regular. Moreover, the algebras of multipliers of A p (I) contain the "algebras of multipliers of L^(I). We establish below our notations and then proceed to describe the results. Let 7 = {xeR:0<x<oo}bethe locally compact semigroup with addition as binary operation and the usual interval topology. Let E = {zeC: Rez > 0} and E denote the closure of E. The measure associated with E or E shall be the usual planar measure. The Fourier transform of a measureable function /, whenever it is meaningful, shall be denoted by/ C c (7) (resp. C*(I)) shall denote the space of continuous complex-valued functions (resp. infinitely differentiable functions) with compact support in 7. The 329 330 Ajit Iqbal Singh and H L Vasudeva index conjugate to p, 1 ^p*$ oo, shall be denoted by p', i.e., p and p' are positive numbers greater than or equal to 1 such that - + r = 1. Let M(I) denote the Banach P P algebra of all finite regular Borel measures on / under convolution product * and total variation norm. Then the Banach space L (I) of all continuous measures in M(7) which are absolutely continuous with respect to Lebesgue measure on 7 becomes a com- mutative semisimple Banach algebra in the inherited product *. More specifically, for = [*f(x-y)g(y)dy, 11/11!= [ l/W|dx Jo Jo satisfy \\f*g\\ < \\f\\t \\g\\ lf The maximal ideal space 7 of 7^(7) can be identified [4] with Z and the Gelfand transform of an/e 7^(7) is its Laplace transform, i.e., The function/ is analytic in S. It, therefore, follows that LJ7) is not regular, a fortiori, no subalgebra of L t (7) under any norm with the same maximal ideal space can be regular. Clearly, for x ^ 0, the function y. - f(x -f iy) is ^/2nf x , where f x L l (R) is given by f x (t) = f(t)Q~ tx for te7 and in JR /. For these and other results that may be used in the sequel, the reader is referred to [4], [12]. Let l^p^oo. The algebras A p (I) consist of all those feL^I) whose Gelfand transforms /belong to L p (I). A p (I) form an ascending chain of ideals in 7^(7). A p (I) equipped with suitable norms become Banach algebras. These algebras do not have bounded approximate identity nor are these algebras regular. However, these algebras are semisimple. The maximal ideal space A(v4 p (7)) can be identified with Z. The above and other related results are contained in 2. A mapping T on a commutative Banach algebra A to itself is called a multiplier if T(xy) = ( Tx)y for x, y e A For results on multipliers, we refer to Larsen [7] rather than original sources. As A p (I) is semisimple, every multiplier of A p (I) is bounded and we may define a multiplier of ,4^(7) to be a bounced continuous function on 2 such that (j>fA p (I), whenever feA p (l\ where A P (I) = (f:feA p (I)} is the Banach algebra under pointwise operations and norm ||/| = ||/Hi + ||/|| p . In 3, we prove an analogue of Paley- Wiener theorem. This, in turn, helps us provide a set of sufficient conditions and a set of necessary conditions on <f) such that (t>feA p (I) whenever feA p (I). 2. The Banach algebras A f (I) As the Gelfand transform of a function in L 1 (7) belongs to C (7), it is evident that ,4^(7)= L t (7) and each A p (I) is an ideal in 7^(7). Moreover, A p (I)^A r (I) if p<r. Indeed, if fA p (I) and p < r < op, then The case r = oo is trivially true. For each p, 1 < p < oo, we define I/lllp = 11/11 1 + 11/11,, feA p (I). The algebra A p ((0, oo)) and its multipliers 331 It can be verified as in Larsen, Liu and Wong [8] that ||| ||| p defines a norm on A p (I) and that A p (I) is a commutative Banach algebra under convolution. As observed earlier the algebras LJ/) and A^(l) are identical. Since n/ii,^ u/iii + H/II oo = ni/iiicc <2ii/n 19 for feA^(I\ it follows that || || l and ||| HJ^ are equivalent norms on A X (I). The mapping :A p (I)-> L t (/) x L p (I) defined by *(/) = (/,/), feA p (I) is clearly an isometry of A p (I) into the Banach space L 1 (/)xL p (f) with the norm II (/0) II = 11/11 1 + H0lljr Thus A p (I) may be regarded as a closed subspace of L x (/) x L p (I). For each p, 1 <p< oo, the dual A*(I) of A p (I) is isometricallyisomorphic to L 00 (/) x L p ,(/)/Ker<]>*, where <3>* is the adjoint of the map <X> and p' is such that - + = 1 [see Theorem 2 [8]]. P P PROPOSITION 1 Proof. Let, for neN,u n be the function n% [Q jl/B) . Then \\u n \\ = 1. Also for zeE and r ^ ^i ,u n (z) = (e- zln -\)/(-z/n) = u 1 (z/n)znd therefore. \u n (z)\ ^min]-^, 1 [. Further for Rez^n, \u n (z)\^n( \--\z\. Observe that u n eA p (I) for p>2 and M B Indeed for p > 2, I |y|*l r T*, L.L where ' f f sSLj J0<x^l J|y|>l V^ "^i J and |y|<l Thus i/ n e L (/), p > 2, and consequently u n eA p (I). 332 Ajit Iqbal Singh and H L Vasudeva We next show that u n $A 2 (I). Indeed, ,_ r A 2 r ! n i ,112= | i<wr<fc> | '(i-^ IK = n 2/ / 1 _lVf l| tan -l (j ;/ X " V e ) J X al1 = 7t/J 2 fl--Y f -dx=00. V e J Jx>.* Again u n *u,iA 1 (l). Indeed, MU.(Z) = (A(z)) 2 and J Re2 ^ol( n ( z )) 2 |dz = J Rez ^ |(u M (z))| 2 dz ^ oo, as shown above. Also n *u n 6^ p (7) for p>l. Indeed, u^w n (z) = (u n (z)) 2 and j Rez>0 l(" n ( z )) 2 l P dz = JRe^ol(^( z ))l 2pdz = | R eopl( n (z))l'dz, where ^ = 2p>2 and the right hand side is finite as shown above. It is also a consequence of above that u n *u n *u n eA l (I). So for Now, let 1 < p < r < oo. Let, if possible, A p (I) = ^4 r (^)- Then there exists K > such that |||/||| p < Kill/ III for/6X p (J). For neN and l<s<oo, IK*uJ|| s = ]]* ||, + H 2 |l s =ll n *Jl 1 +n 2/s || 2 |l s =l+ 2/i p 2 || s . Consequently, n 2 /"-^||u 2 || p ^ K(n 2/r + ||i2 2 || r ). On letting n-> oo, the left hand side of the preceeding inequality tends to infinity whereas the right hand side tends to a finite limit. This contradiction completes the proof in this case. Now, let 1 < p < oo. Since A p (I) c A 2p (I) c ^^(7), we cannot have A p (I) = A X (I). PROPOSITION 2 Let/ be a function defined on [0, oo) such thatf, f" exists and satisfies /(O) = /'(O) = /"(O) and = lira /'(x) = lim /"(x). Further suppose t/iat f,f',f"e L t (/). Then x-*oo x-c (&)feA p (I)forp>l, j (/) i//" = on (0, c) for some c> 0, ^T) if/"' ^xisto an^ is in L^/),/"'^) = and/"'(x)^0 as x-> oo. r />) = Jo Froo/. Observe that for zel, z ^ 0, -rz _ ! ^ ^ using/(t) ->0 as t-> cc and |e~ tz | = e~ u -^0 as t -> oo. On applying the above argument to/' and in case (c) also to/", one obtains respectively, /(z) = -7/"(z) and in case (c) (a)l/(^l = ^inz)l^ll/l, The algebra A p ((0, oo)) and its multipliers 333 Now, where i|"dxdy+ |/(x + iy)|dxdy and Thus /6X P (/) for p> o -JL ^ e HP The proof from now onwards is the same as in case (a). If we write II/IU = T x H- T 2 + T 3 as above, then and ^3 < 2 1| /|| j , as above. This completes the proof. Theorem 3. A p (I) is \\ \\ r dense ideal in L^I). Proof. A p (I) is an ideal in L x (/) was observed in the beginning paragraph of 2. That it is dense in 1^(7) follows from Proposition 1 on noting that C c (/) is dense in DEFINITION 4 [BURNHAM] Let (A, || D be a Banach algebra. The subalgebra B of A is an 4-Segal algebra in case 334 Ajit Iqbal Singh and H L Vasudeva (i) B is a dense left ideal of A, (ii) B is a Banach space with respect to norm || || B , (iii) There exists C > such that ||/|| A < C ||/|| B for all feB, and (iv) There exists K>0 such that \\fg\\ B <K\\f\\Jg\\ B foT&geB. Remark 5. (i) It is clear from the foregoing that A p (I) is an L^-Segal algebra. The above proofs have been included in view of their intrinsic value even though Burnham ([2], ex. 19) cites an example of a Segal algebra which includes the one studied in this note. (ii) In view of the fact that for 1 < p < oo, A p (l) is an L^/J-Segal algebra, the following results follow from the general theory of ,4-Segal algebras [2]. Let 1 ^ p < oo. (a) The maximal ideal space A(/4 p (/)) of A p (I) is homeomorphic to , (b) A p (I) has no bounded approximate identity. (iii) It follows from (ii) (a) that for each p, 1< p < oo, A p (I) is a semisimple commutative Banach algebra. Our next result provides a characterization of A 2 (I). Theorem 6. Let /e L^I). Then feA 2 (I) iff t-+f(t)/yft is in L 2 (l\ Proof. ^~ T r |/(x + OOI 2 d*dy = f [" l7(y) 27C JO J-oo JO J-oo /*QO /*00 JO J -o 2 dt, using Plancherel Theorem. Our next result shows that C C (I) is contained in A p (I) 9 p > 2. PROPOSITION 7 Suppose 2<p<cc and p' is such that - -f = 1. Choose u and u' such that 1 < u' < p/p f P P and - + - = !. Then L^/Jn L p (/)n L up ,(I) c A p (I). So LJ/Jn 1^(1) is contained in A p (I) for p > 2. In particular, C C (I) is contained in A p (I),p> 2. Proof. Suppose /eL^/Jn L p .(I)n L up ,(I). Let xel. Since feL p .(I), f x eL p .(R). So \\7x\\p < ll/xllp' 5 using Hausdorff- Young inequality. Moreover, \\f x \\ p , < ||/|| p ,. Also \\f x \\ p p '= \f(t)\ p 'e- txp 'dt 1/ M ' p o a 00 Nl/U/ / \ |/(t)r'dt e-<*"'>'dr o / \Jo / Consequently, The algebra A p ((Q, oo)) and its multipliers 335 Hence (2n)- pl2 \\f\\ p p = 11?.+ This completes the proof. Remark 8. Though A p (I) does not possess a bounded approximate identity, yet it does always have an approximate identity as the following theorem shows. Theorem 9. The sequences {u n },{u n * u n } and { u n * u n * u n }, where u n denotes the function n #[o,i/)> act as approximate identities for A p (I) with 2<p^oo, l<p^oo and l^p^oo, respectively. In particular each A p (I) possesses an approximate identity present in A ^(I). Proof. It is well-known that {u n } is an approximate identity for L^I). We shall show that v n = u n *u n *u n ,n=l,2 9 ... is an approximate identity for A^I). It has been observed that {v n } is contained in A^(I) [Proposition 1]. Suppose feA^I). Then So \\f*v n /Hj-^Oas n-^oo. There exists a compact set K c Z such that H/l^cll ^ < e/4. Now, where /^(X) denotes the planar measure of K. Also, Choose n so large that ||/*i? n /Id </6/^(X). Consequently, for this n, H/*!?,, /Id < e. This completes the proof in the case p = 1. The proof for the case p> 1 is similar and is, therefore, not included. Remark 10. (a) It follows from Theorem 9 above that for 1 ^ p ^ oo, A^^A^I) is dense in A p (I). Since A^I) is an ideal in L^I) this gives that A^I) is dense in A p (I) which, in turn, gives that for 1 ^ r <p ^ oo, A r (I) is dense in A p (I). (b) It follows from Theorem 9 that Li(I)*A p (I) is dense in A p (I). This observation together with ([5], 32.22) implies that L t (/)* A p (I) = A p (I). Now ([5], 32.33 (a)) implies that {e a } is an approximate identity in L t (/) which is present in A P (I\ then it is an approximate identity for A p (I) as well. 336 Ajit Iqbal Singh and H L Vasudeva Finally, we state without proof the following result regarding ideals in A p (I). The proof follows from the fact that A P (I) is an L^TJ-Segal algebra and Burnham ([12], Th. 13) result on ideal theory of Segal algebras. Theorem 11. For each p, 1 ^ p < oo, the following statements hold: (i) // J l is a closed ideal in L t (7), then J = J 1 nA p (I) is a closed ideal in A p (I). (ii) // J is a closed ideal in A p (I) and J 1 is the closure of J in L 1 (/), then Jj is a closed ideal in L x (7) and J J L n A(I). 3. Multipliers of A p (I) In this section we attempt to identify the multipliers of A p (I). In view of ([7], 1.2.2) and results proved in 2 above, every multiplier T ofA p (I) is bounded and corresponds to a bounded continuous function (j> on Z. It follows from the analyticity of f,feA p (I), that is analytic on Z. Since for feA p (I) and-0 bounded and continuous on Z, \j/ = 0/is in L p (Z) with \\il/\\ p ^ ll^llooll/llp, the_problem reduces to that of identifying those <'s which are bounded continuous on Z and analytic on Z such that for each/e,4 p (7), </>/is h for some he 7^(7). Remark 10(b) further reduces it to requiring 0/to be ft for some jueM([0, oo)). Thus a multiplier on A p (I) into M([0, oo)) is in fact a multiplier of A p (I) and keeping in view 10(a), we have that for 1 ^ p < r ^ oo, a non-zero multiplier of A r (I) induces a non-zero multiplier of ^(7) via restriction. The following analogue of Paley- Wiener Theorem ([11], Th. 19.2) helps us in expressing, foifeA p (I) and certain 0's, 0/and h for some he L t (7). Theorem 12 (Paley- Wiener). Let l^r^2 and r' be the number given by - 4- -= 1. Let \l/e L r (Z) be analytic on Z. Then there exists a ge 7,^(7) such that \j/ =^ on Z. T/iis gfeL 1 (7) ij^* a=ess lim infL^L < oo and then */2n\\g\\ 1 =a and if b = X-+04- j ess lim inf|| :c ^|| r <oo then geL r ,(I) with \\g\\ r ,^=b 9 where, for xe7, x -" x \l/(y) = \ Proof. We shall modify the detailed proof given in ( [1 1], Th. 19.2). We fix x, jc' in 7 with x< x' and write J = [x,x r -]. For ae7, let T a be the rectangular path with vertices at x + ia, x ia, x' - ia and x' + ia. It follows from Cauchy theorem that J r i/^(z)e rz dz = for teR. For a,jSeR, let x F(t,j?) = fX M + i)3)e t(u+w du, AOJ) = f J |^(uV ift| r dti and Ix'-x^^'maxle^e^'} with the convention that r = l for r>0. Then )| ^ A(j?) 1/r M(r). Now, by Tonelli's theorem I A(j?)djS= f ( { JR JJ\JR ./Mdi*=IM|;<oo. So A(/J) -f A( /?) is not bounded below away from zero as P -* oo and, therefore, there is a sequence {a,-} in 7 such that a,. -* oo and A(a ; ) + A( a,-) -+ and j -> oo . We note that {a,.} is independent of t and for reJR, ^(^a^-^O and *(t, a^-^O as 7-^00. Now since ^e L r (Z), u \l/e L r (R) for almost all ue7, say, in a set S with m(7 - S) = 0. Then for ueS, The algebra A p ((0, oo)) and its multipliers 337 tt ^X[-aj^}^u^ in L rW and therefore, by HausdorflF- Young inequality, ( u \l/r in J L r .(R). So for x,x'eS, there exists a strictly increasing sequence {,-} in N with (>*[- , ]r-(*^r and (>Jt[- vfl gr^(*^r almost everywhere in R. But for each n;GJVand reK, ^(^-^iT (- Oe" and ^(^[-cc^r (- *)e'*' are the integrals of \l/(z)t tz along the vertical lines of r^ and therefore for each X[- v ^ r)e lx = ( x .^r(--t)e' x ' for almost all tejR. We take as the function )-(_ t) e tx which is independent of xeS. If r = 1, then for each xeS, (^eC W and J2n ||(>r L < ||> II i- Since for each as x~0+ in S. we have 2^11011^ ^b = ess lim infllj^llj. and x-*0 + '* /T-.. .... /T-r -,v, i, = ess lim j n . Now, for ae/, t<0 27cjo So letting a - oo, we obtain that g(t) = for t < 0. By continuity of 0, 0(0) = 0. Thus ge Ll (1\ Further \g(t)\ ^ f IIAII i for te/. Thus for xe5,(^ e L X (R) and as > is also in L^Rl iA(x + r = ^(v) = (^(~j) = J"^We" xf e-^dt. Since iA as well as the function z = x + ty -^ J^0(r)e- rr dr = 0(z), ([12], Th. 6.3) are both continuous on L, we conclude that ^ = g on Z. If r > 1, then, on using Hausdorff- Young inequality, we get f oo / r ' foo / r y/'' (2;rr 2 e-^|0(t)r'dO r/r 'dx= IU)^(-Ol r dt dx Jo \JR JO VJl? / f ^ Jo X ' *~~ ' * So the function x -> J R e " r/xr 1 0(t) | r ' dt is not bounded below away from zero as x -> oo . So there is a sequence {x^} in / with x,->oo and $ R c- r ' tx >\g(t)\ r 'dt-+0 as;~>oo. Since f-oJ^Wr 'dt ^ j- oc e~ r ' fxj |0(t)r dr for each;, we conclude that = a.e. on ( oc,0). Now, for ae/,xeS, 72^ f"| ff (t)|dt= f Jo Jo <||^|| r a 1/r S"<oo.' So ^e L, 1 ,,^/). Since I-2S = 2(I- S) has measure zero, we have that m(J S n 2S) = 338 Ajit Iqbal Singh and H L Vasu'deva as well. For xeSn2S, ^ - 2n J2n Since x/2eS, ( x/2 ^r = L r ,(R) and therefore f -> g x (t) = -j=( x \J/r( - 1) is in L 1 (I). Since V/27C m(J-Sn2S) = 0, Sn2S is dense in /. So g^eLJI) for all xe/. Consequently is defined on E and is given by = f Jo which for xeS is 271 JO since x ^e L r (.R) and ( x il/)~~e L^R) ([11], Th. 9.11). Because ^ and # are both continu- ous, we conclude that \l/ = g on I. Further ge L^]^) iff a = ess lim inf || ( x i/0" || i < oo and then ^/2n \\g\\ i = a simply JC-fO + because for any sequence {xj} in 5 with Xj-^0, \ Jo = lim I j-* oo J-* oo We note that in this case g is defined on Z and extends ^ to a bounded continuous function on . Finally, for any sequence {x^-} in S, x^O, we have, by Monotone Convergence Theorem, / " oo J ./- oo j-* oo and, therefore, vll^llr' ^ b = ess lim inf || x COROLLARY 13 Let 1 ^p < oo,$ a bounded continuous function on E, which is analytic on and ). Then / = K for some /ie ^(J) if 1 ^ p ^ 2 or p > 2 and $e L 2 , 2 (S). This ess lim inf ^ 0+ Jo JK iy)e iyt dy\dt < oo. f. We take r = pifl^p^2 and r = 2 if p > 2 so that </e L r (I) under the stated ^conditions and then apply Theorem 12 above. Theorem 14. Let 1 ^ p < oo and </> be a bounded continuous function on which is analytic on Z. The algebra A p ((Q, oo)) and its multipliers 339 (i) (f) induces a multiplier of A p (I) if (a) I ^p^2 or p>2 together with 06 L 2plp _ 2 (L) and (b) ess lim inf J* | J^(x + iy)f(x -f iy)e iyt dy |dr < oo for all feAJI). p (ii) (j) induces a multiplier of A p (I) only if ess lim inf I Jo x+iy e iyt dy dt< oo /or each'neN, where s = 2if p 2,1 if p>2 and 3 if p<2. Proof. We apply the above corollary and use the fact that u* s eA p (I) for each n, where u n = X(o,i/n] an d *s denote the sth convolution power. Remark 15. Even though A p (I) is not regular, it contains functions/ with/ vanishing nowhere on S, for instance, w* s in the proof above. Such /are bounded below away from zero on compact subsets of and thus the strict topology T on M(/4 p (/)) (i.e., strong topology on a subalgebra of the algebra B(A p (I)) of bounded linear operators on A p (I) to itself) is stronger than the topology of uniform convergence on compact subsets of I. By ([7], 1.1.6) and Theorem 9 above A p (I) is dense in (M(A p (I), T S )), where geA p (I) is identical with the multiplication operator M g given by M g (f) = #*/and, /N /\ a fortiori, (A p (I)) is dense in (M(A p (I)'), topology of compact convergence on ). Acknowledgements Ajit Iqbal Singh would like to thank the centre for Advanced Study in Mathematics, Panjab University for its generous hospitality where part of this work was carried out. References [1] Burnham J T, Closed ideals in subalgebras of Banach algebras I, Proc. Am. Math. Soc. 32 (1972) 551-555 [2] Burnham J T, Segal algebras and dense ideals in Banach algebras, Lecture Notes in Mathematics 399, (1973) 33-58 (Berlin: Springer- Verlag) [3] Figa-Talamanca A, Gaudry G I, Multipliers and sets of uniqueness of L p . Mich. Math. J. 17 (1970) 179-191 [4] Gelfand I M, Raikov D A and Silov G E, Commutative normed rings, (Chelsea, New York) [5] Hewitt E, Ross K A, Abstract Harmonic Analysis, Vol. II: Structure and analysis for compact groups and locally compact abelian groups, (New York: Springer- Verlag) 1970 [6] Savita Kalra, Ajit Iqbal Singh and Vasudeva H L, The algebra ^((0, oo)) with order convolution and its multipliers, J. Indian Math. Soc. 54 (1989) 47-63 [7] Larsen R, An introduction to the theory of multipliers, (New York: Springer- Verlag) 1971 [8] Larsen R, Liu T S and Wong J K, On functions with Fourier transforms in L p , Mich. Math. J. y 1 1 (1964) 369-378 [9] Martin J C and Yap L Y H, The algebra of functions with Fourier transform in L p , Proc. Am. Math. Soc. 24(1970)217-219 [10] Reiter H, Subalgebras of L^G), Indagationes Math. 27(1965) 691-696 [11] Rudin W, Real and complex analysis, (New York: McGraw Hill Book Company) [12] Widder D V, The Laplace transform, (Princeton: University Press) *" Proc. Indian Acad. Sci. (Math. Sd), Vol. 105, No. 3, August 1995, pp. 341-351. Printed in India. Reflection of P-waves in a prestressed dissipative layered crust SUJIT BOSE* and DIPASREE DUTTA f *S. N. Bose National Centre for Basic Sciences, Calcutta 700064, India f Women's College, Durgapur 713 209, India MS received 30 August 1994; revised 23 November 1994 Abstract. The paper deals with overall reflection and transmission response of seismic P-waves in a multilayered medium where the whole medium is assumed to be dissipative and under uniform compressive initial stress. The layers are assumed to be homogeneous, each having different material properties. Using Biot's theory of incremental deformation, analyti- cal solutions are obtained by matrix method. Numerical results for a stack of four layers - modelling earth's upper layers, show a decreasing trend in both the Reflection Coefficients R and R of the reflected P and S-waves. Keywords. Reflection; P-wave; S-wave; dissipative; homogeneous layers; Biot's theory; matrix method; reflection coefficients. 1. Introduction The study of reflection and transmission of seismic body waves through multilayered media is an important part of seismic sounding techniques. It is recognized that these studies provide a very convenient method of investigating the earth's interior. Although other approximations are possible, the simplest representation of the system of rocks beneath the earth's surface might be supposed to consist of a series of plane, parallel layers, each having its own characteristic - but constant within the layer - parameters of velocity and density [12]. Observation of propagation of stress waves in solids (or fluids) show that dissipation of strain energy occurs even when the waves have small amplitude. This dissipation results from imperfection in elasticity, loss by radiation, by geometrical spreading and scattering [5, 7-1 1, 13, 14]. A convenient measure of attenu- ation in waves is the dimensionless loss factor (or specific dissipation constant) Q~ *. It is related to the rate at which the mechanical energy of vibration is converted irreversibly into heat energy and does not depend on the detailed mechanism by which energy is dissipated. For P-waves Q" 1 is given by [12]. ^ i 2u ar =7 where V and v are the real and imaginary parts of the complex P-wave velocity. It is also known that, surprisingly, Q~ 1 is independent of frequency, pressure and tempera- ture [5]. In the focal region, prior to an earthquake, considerable tectonic thrust builds up as a uniaxial stress system. It is of some interest to investigate reflection characteristics, through a theoretical model of a stack of layers under uniaxial compressive prestress. Biot [2] has provided a detailed theory of incremental deformation of a medium in a state of prestress brought about by even arbitrary finite deformation. Later, Dahlen [3], in a limited context of initial elastic deformation arrives at identical set of equations, excepting the constitutive equations for the incremental stresses. If restricted to 341 342 Sujit Bose and Dipasree Dutta two-dimensions, Dahlerf s equations fail to reduce to the equations for incompressible medium derived elaborately by Biot. Secondly, the elastic moduli in the transverse direction also change due to the uniaxial prestress. Consequently, we adhere com- pletely to Biot's theory. For treatment of the equations for a stack of layers, we adopt a simple matrix method based on Kennett [6]. In this paper we restrict to two-dimensional propagation. 2. Formulation of the problem Consider an initially stressed, dissipative medium consisting of 'n' parallel homogene- ous layers overlying a half-space. The interfaces are ordered as Z 1 , Z 2 , . . . , Z n where the origin Z = < Z t is on a hypothetical free surface from which P-wave originate and travel downwards, ultimately as plane waves. The reflected waves are received at the same surface. To keep the analysis simple in the first instance, as is often done, we disregard stress-free condition on Z = 0, that is to say, regard the top layer Z < Z^ as semi-infinite. The topmost layer is layer number 1 and the bottom layer n + 1 and thicknesses of the intermediate layers are designated as H 2 , // 3 , ...,// (figure 1). The physical quantities associated with layer number 'm' will be denoted by symbols with suffix m. In general, if we have an isotropic elastic solid under uniform initial horizontal compression S n (tensile S n <0) parallel to x-axis, which undergoes additional infinitesimal deformation, then according to Biot [2], the incremental stresses consist of two parts: one part due to additional deformation and the other due to infinitesimal rotation o> 2 acting to rotate the initial stress system: 33' (1) ir Z "m-i m Figure 1. Geometry and schematic of the problem. Reflection of P-waves in a prestressed dissipative layered crust 343 where s tj are incremental stresses referred to axes which rotate with the medium (Biot [2], eq. (4.13)) and For infinitesimal incremental strain e tj , the incremental stress s tj will be linear functions of e tj . Assuming these to be orthotropic in nature we can write du dw e **= (3 > v du < -dz Also, after careful consideration of existence of strain-energy, (Biot [2], eq. (6.2)). The elastic constants B u , . . . , Q in general may depend on the initial stress S l ^ . Biot ([2], eq. (8.3 le)) after analysis of an incompressible medium, selects for an original isotropic compressible medium (Lame constants A,^), relations equivalent to *3i=^ B 3 3 = * + 2* Q = n. (5) A salient feature of these relations is that the moduli in the x-direction (the direction of initial stress) increases due to the initial compressive stress while those in the transverse z-direction remain unchanged. To account for dissipation in the medium A 1 and p. are to be regarded complex: A = A r 4- U f , \JL = ju r -f i^i . The two-dimensional dynamical equations of motion as obtained by Biot [2] are dx dz u dz ~ dt 2 For time-harmonic plane wave propagation of frequency / = co/2n, we may assume a factor exp [i(cot /ex)]. Insertion of (3) with (5) in (6) results in two O.D.E's for u and w, the displacement components. However, for developing a matrix method we introduce stresses T 13 = s 13 -S 11 o 2 (7) and the quantities [6] TI/ rr T o /Q\ W=iw, U = u, T = n 33 , S = T 13 . (8) Constructing the stress-displacement vector 344 Sujit Base and Dipasree Dutta eq. (6), with the aid of (7) and (3) can be written as a first order system. For subseq computational purpose we nondimensionalize all quantities: the displacements b (thickness traversed by the waves in the top layer) and stresses by ^ lr , the real pa shear modulus of the top layer. Denoting the respective nondimensional quantitie superscript *, the first order system can be written as dz* " - where JJ > 1 J. - -kz Q'-o-ss* , p Mi? Pi \j j j __ _ ._ __ I fc g fag \j is the coefficient matrix, p = k/co is the wave slowness (reciprocal of phase velocit propagation in the x-direction) and p l = (^Jp^) 112 is the shear wave velocity in topmost layer. For reflection and transmission of body waves, p remains constant ii the layers. Finally, S* x = S ll /fjL lr . The incremental boundary forces have also been carefully examined by Biot ( [2], 17.56)). In our case, where the boundaries are z = const., the components turn out t( T 13 and T 33 , so that at an interface z = z m b* is a continuous vector when per bonding is assumed. 3. Propagation in the stack In an intermediate mih layer, the solution of (10) is where b*_ 1 is the stress-displacement vector at the interface z* = z* _ 1 . Hence at z* = b* = e^b_, , where H* = z*~z*_ l is the nondimensional thickness of the mth layer. Hei recursively b* = e^We^-^-s . . . , e^b* Eb*. , All the exponentials involved above are 4 x 4 matrix exponentials. For b* we note that it consists of the down going incident P type wave and reflec up going P and S type waves (figure 1). We construct the contributions from eacl these separately and superpose. Suppressing the time harmonic term, we can write the down going incident wave Reflection of P-waves in a prestressed dissipative layered crust 345 where the predominant z-component of the amplitude has been taken to be unity Inserting m the equations of motion (6) with (3) and (5) and assuming so that 6 is the angle of incidence, we get (16) (17) and the velocity of propagation co/k is given by a quadratic equation whose roots are a 2 1 where If S n is neglected, the positive sign in (18) yields P-waves and the negative sign, S-waves. In the presence of S l 1 , the velocities are p, that is, direction dependent and the waves are not pure, in the sense that P-waves are accompanied by some transverse component and S-waves by some longitudinal component [4]. Stresses corresponding to (15) can be readily calculated from (3) and (5). We thus obtain (20) ID' Kzi [A Afsin -f (AJ + 2/iJ )cos 0} - iJKz! {^(ji* - O-SSJJcos 6 -h (/it 4- 05Sf Jsin p - the constant for all the layers - in (17), can be computed from the equation l/2 (21) which is arrived at from (16) and (18). Here 6 is given so (pjSJ is to be obtained by solving the above nonlinear equation. For up going reflected P type wave, we have to use the representations u* = ^ 2 e^ (z - Zl) e"^, w* = B 2 e i * ( *-** ) e" to . . (22) Analysis similar to the above leads to (23) .(^ _ Q-5S*! )cos B 2 * ) cos e ? + 0-5S*, )sin 346 Sujit Base and Dipasree Dutta where A 2 /B 2 = - A 1 is obtained from ( 1 7). For up going reflected S type wave we again use representation of the type (22) with amplitudes A 39 B 3 instead ofA 2 ,B 2 . We thus obtain b* similar to (23) with A 2 /B 2 replaced by A 3 /B 3 = -A l and 9 replaced by 9 s given by (24) appropriate for S type waves. Here Re means real part of. The total stress-displacement vector in the top layer is thus b*. (25) Finally, for the bottom most (n + l)th layer, only down going P and S type waves are sustained. For the former we take As in the case of b^ we obtain n+ 1 -'*M IT +Wf +1 n+1 (27) where (28) Rf+ 1 and S*+ 1 are quantities identical to RJ and SJ (cf. eq. (19)), save that ^ and /^ are to be replaced A n + 1 and n n +i- Similarly AJB 4 is given by an expression like that of A l (eq. 1 7)) save for A x , /z x , 6 we have to write ^ + 1 , /* + 1 > ^+ 1 F r ^e down going S type waves we get in a similar manner b* ^ D with a form similar to (27) except that A 49 B 4 are to be replaced by similar amplitudes A 5 , B 5 and (% + x replaced by 0+ 1 given by and A$/B 5 given by right hand side of (17) with ^,^,0 replaced by A w+1 , Thus, (29) + r D- (30) The expressions for b* and b*+ 1 from (25) and (30) can now be inserted in (14). If we denote the successive vectors [ ] in the expressions for b*, b*, b*, b*^ D , b** ID by v i * V 2 > V 3 > V 4 and V s , we get the system of equations (31) Reflection of P-waves in a prestressed dissipative layered crust 347 Solving these equations we get "reflection coefficients", R = B 2 , R = B 3 and "trans- mission coefficients", T P = B 4 , T]^ = B 5 . 4. Numerical calculations for model crust In general the earth's continental crust consists of three layers: granitic, basaltic and a thin sedimentary layer at the top. For computations of reflection (and transmission) coefficients we consider the earth's crust beneath the Indo-Gangetic plain, which lies between the Himalayas and the Peninsula. Surface wave dispersion across this region has been investigated by several investigators [1]. Inversion of these data gives the crastal and upper mantle structure of the region. Such a model of crust is given by Bhattacharya [1] and is given below: Region 1. Sedimentary 2. Granitic 3. Basaltic 4. Upper Mantle Thickness P-wave S-wave Density of layer velocity velocity (gm/cm 3 ) (km) (km/sec) (km/sec) 3-5 340 2-00 2-00 16-5 6-15 3-55 2-60 23-0 6-58 3-80 3-00 00 8-19 4-603 3-30 ia40 20JDO 4JX) 5.60 720 Figure 2. Amplitudes of reflection coefficients |l| and |R| for near vertical propagation: 6 = 1. Sujit Bose and Dipasree Dutta 550 600 650 7DO 750 8UOO 350 4.00 450 5J 4JOO 450 500 550 6JGO Reflection of P-waves in a prestressed dissipative layered crust 349 & 0. O 100 053 0.86 0.79 QJ2 0.65 038 051 0.44 037 030 100 (c) 330 4.00 430 5.00 530 f 6LOO 630 7.00 730 aoo Figure 3 (Continued). Amplitudes of reflection coefficients \R\ and \R\ for wide angle propagation: (a) = 2 (b) 9 = 5 (c) 6 = 10. The above yield the real part of Lame constants of each layer. For the imaginary parts, the loss factors Q ~ 1 of P- waves as given in Waters [12] <2 a (granite) = 311, e a (basalt) = 561 Q a for sedimentary rocks is highly disperse, so, as an example we take old red sandstone for which Q a = 93 - a figure nearing the mean of dispersal of the values. Since the role of dissipation is small, the computed values are not expected to change very much on account of actual deviation. For the upper mantle we take Q a = 849 from data discussed in Ewing et al ([5], p. 278). Further data on imaginary part of shear modulus are provided by loss factor Qp 1 of S-waves: which is obtained from the often used assumption of zero dilatational viscosity [12, 5]. For initial stress-free basalt rock, strength ^ 11,000 atmospheres and if we consider hydrostatic pressure at a depth of 40 km to be present, the approximate range of the compressive initial stress f == SJ X could be (0, 0-3). We therefore consider the parametric values = 0,0-1,0-3 and 0-5, over a slightly enhanced range. For selecting suitable frequency range, we consider the cases of seismic prospecting method of weight-dropping devices in which near vertical propagation takes place and 350 Sujit Base and Dipasree Dutta explosion seismology technique where it is wide angle propagation. In the former case, / is taken within the range of 4-20 Hz [12] with 9 = 1. In the second case the range chosen is 3-8 Hz ( [5], p. 202) with ranging from 2 to 10. In the numerical treatment of (31) we use Gauss's method for matrix inversion. The computation of the matrix exponentials in E (eq. (14)) is performed using the Cayley-Hamilton theorem. The latter requires the eigenvalues of matrices like A* (eq. (11)), which is a simple task, because of the fact that the characteristic equation for the eigenvalues A of A* reduces to a quadratic in A 2 . The solution of (21) is performed by Mullet's method. We restrict presentation of the results to R and R only. In figures 2 and 3, we present the variation of the amplitudes of these quantities with frequency/, for different values of initial stress parameter In figure 2, the results for near vertical propagation are presented. There is a general trend of diminution in the reflection coefficients for increasing , which becomes significant towards the higher frequencies in the band. The results for wide angle propagation for = 2, 5 and 10 are presented in figures 3 (a), (b) and (c) respectively. Here too, is a general trend of diminution in the reflection coefficients for increasing . The trend of diminution increases with increasing 0. It may be mentioned here that when P-waves propagate vertically in an unbounded initially stressed homogeneous medium, there is no effect of initial stress on the velocity of propagation [4]. This fact can be verified from (18), (19), with k = p = for the case. For reflections from the stack, there are no up going S-waves, A l = (verifiable by the limit 0->0 in (17)), A z = A 4 = and the reflection and transmission coefficients B 2 , J5 4 are given by a pair of equations similar to (31). 5. Conclusion The focal regions at plate boundaries of the earth prior to earthquakes are at considerable thrust due to tectonic movement. For understanding the reflection and transmission characteristics of body waves in such regions appropriate mathematical model studies are required. Herein, is considered; a stack of dissipative layers under uniaxial thrust to which the theory of incremental deformation given by Biot [2] is applicable. The governing equations can be compactly treated by matrix method, as in the case of initial stress free case, for the reflection and transmission of body waves. A numerical model study of a stack of four layers - sedimentary, granitic, basaltic and upper mantle, for near vertical as well as wide angle reflections, shows significant diminution in the magnitudes of both P and S waves. References [1] Bhattacharya S N, Crustal and upper mantle velocity structure of India from surface wave dispersion, Curr. ScL, 62 (1992) 94-100 [2] Biot M A, Mechanics of incremental deformation (1965)'(New York: John Wiley and Sons Inc.) [3} Dahlen F A, Elastic dislocation theory for a self-gravitating elastic configuration with an initial static stress field, Geophys. J. JR. Astron. Soc., 28 (1972) 357-383 . [4] Dey S, Roy N and Dutta A, Propagation of P and S waves and reflection of SV wave in a highly pre-stressed medium, Acta Geophys. Pol. 33 (1985) 25-43 [5] Ewing M, Jardetzky W S and Press F, Elastic waves in layered media (1957) (New York: McGraw-Hill Book Co) , [6] Kennett B L N, Seismic wave propagation in stratified media, (1983) (London: Cambridge University - Press) Reflection of P-waves in a prestressed dissipative layered crust 351 [7] Knopoff L and MacDonald J F, Attenuation of small amplitude stress waves in solids, Rev. Mod. Phys. 30(1958)1178-1192 [8] Kuster G T and Toksoz M N, Velocity and attenuation of seismic waves in two phase media, Part I: Theoretical formulation, Geophysics, 39 (1974) 587-606 [9] Mai A K and Bose S K, Dynamic elastic moduli of a suspension of imperfectly bonded spheres, Proc. Cambridge Philos. Soc. 76 (1974) 587-600 [10] O'Brien P N S, A discussion on the nature and magnitude of elastic absorption in seismic prospecting, Geophys. Prospect., 9 (1961) 261-275 [11] Schoenberger M and Levin F, Apparent attenuation due to intrabed multiples, Geophysics 39 (1974) 278-291 [12] Waters K H, Reflection seismology: A tool for energy resource exploration (1978) (New York: John Wiley and Sons Inc.) [13] White J E, Seismic waves: Radiation, transmission and attenuation, (1965) (New York: McGraw-Hill Book Co.) [14] White J E, Computed seismic speeds and attenuation in rocks with partial gas saturation, Geophysics, 40(1975)224-232 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 3, August 1995, pp. 353-369. Printed in India. Computer extended series solution to viscous flow between rotating discs N M BUJURKE, N P PAI and P K ACHAR Department of Mathematics, Karnatak University, Dharwad 580003, India MS received 19 July 1994; revised 28 September 1994 Abstract. The problem of injection (suction) of a viscous incompressible fluid through a rotating porous disc onto a rotating co-axial disc is studied using computer extended series. The universal coefficients in the low Reynolds number perturbation expansion are generated by delegating the routine complex algebra to computer. Various cases leading to specific types of flows are studied. Analytic continuation of the series solution yields results which agree favourably with pure numerical findings up to moderately large Reynolds number. The precise variation of lift as a function of R is established in each case. Keywords. Series solution; Pade' approximants; reversion of series; Euler transformation; analytic continuation; Brown's method. 1. Introduction Flows driven by rotating discs have constituted a major field of study in fluid mechanics for the later part of this century. These flows have technical applications in many areas, such as rotating machinery, lubrication, viscometry, computer storage devices and crystal growth processes. However, they are of special theoretical interest, because they represent one of the few examples for which there is an exact solution to the Navier-Stokes equations. This problem was first discussed by Batchelor [1] who generalized the solution of von Karman [2] and Bodewadt [3] for the flow over a single infinite rotating disc. Further this problem was discussed by Stewartson [4] who obtained approximate perturbation solution for the small Reynolds number. Later, Hoffman [5] has studied this problem using computer extended series. The numerical solutions for this problem have been obtained by Lance and Rogers [6], Mellor el a\ [7] and Brady and Durlofsky [8]. Flow between rotating and a stationary disk has been studied by Phan-Thein and Bush [9]. The problem of injection of a viscous incompressible fluid through a rotating porous disc onto a rotating co-axial disc was studied by Wang and Watson [10]. Through this span of a period of half a century, since the Batchelor-Stewartson contributions, the interaction* between physically based conjectures, numerical calculations, formal asymptotic expansions and rigorous mathematical treatment has been quite intensive. In the present paper we have used semi-analytical numerical technique to understand the effect of both injection and suction separately. For simple geometries the semi-analytical numerical method proposed here provides accurate results and have advantages over pure numerical methods like finite differences, finite elements, etc. In numerical methods a separate scheme is to be developed for calculating derived quantities. If the computation of derivatives are required the numerical scheme to be used will be very sensitive to the grid/step size. This itself will be an elaborate numerical scheme. However, this difficulty is not there in the case of series solution method. A single computer run yields the solution for a large range of the expansion quantity rather than a solution for a single 353 354 N M Bujurke et al value. In addition the method reveals an analytical structure of the solution which is absent in numerical solution. Van Dyke [11] and his associates have successfully used these series methods in unveiling important features of various types of fluid flows. Recently, in the analysis of thrust bearings, Bujurke and Naduvinamani [12] have used series analysis satisfactorily. The physical problem considered in this paper is of great importance in lubrication theory. So calculation of lift is of interest in all these cases. The present analysis is primarily concerned with possible extension of Wang's [10] low Reynolds number perturbation series by computer and its analysis. The forms of the few manually calculated functions in low Reynolds number perturbation solution of two point boundary value problem allows to propose the generation of universal functions in compact form which are solutions of infinite sequence of linear problems^ Using these universal coefficient functions we obtain series solution and calculate various physical parameters of interest. The present series, which is expected to be limited in conver- gence by the presence of a singularity, may be extended to moderately high Reynolds number by analytic continuation. The aims of the present work are two folds. First, to calculate enough terms of the low-Reynolds number perturbation series by computer so that the nature and location of the nearest singularity (which limits the convergence) can be determined accurately, second, to show that the analytic continuation can be used effectively to extend the validity of perturbation series to moderately high Reynolds number. 2. Formulation As shown in figure 1 we denote the spacing between the discs by 'd', the angular velocity of bottom disc by Q! , and that of the upper disc by Q 2 . Let the injection (or suction) at the lower disc be W( - W for suction) and let M,I?, w be the velocity components in the direction r, 0, z respectively (figure 1). The governing equations of the problem are r P uv (, v \vv z -\ = vl V 2 v - (1) (2) w I Figure 1. Schematic diagram of the problem. Computer extended series solution 355 uw r + ww 2 = - + v V 2 w (3) P (ru) r + nv z = (4) where V 2 = -r-^H ---- I-T^T, subscripts denote p.d.e. w.r. to the variable, p is the pressure, dr 2 r dr dz 2 p the density and v is the kinematic viscosity. The boundary conditions are u = 0, i; = r^ , w = + W at z = 0, (5) u = 0, i> = rQ 2 , w = 0. atz = d. (6) For similarity solution, the boundary conditions and the continuity equation suggest the transformations [10] W W -r, v = rg(ri), W=-2f(ri)W (7) a a W 2 where Y\ = z/d and A is a constant to be determined. With these transformations the equations of motion reduce to r (9) or after differentiation, we have 2R(f'g-fg') = g" (11) / f\ P(tl) = - pi - 2/ 2 H^ 2 - 2v + P Here R = ( Wd/v) is the cross flow Reynolds number. The constant P is determined by the pressure at the edge of the discs. The boundary conditions take the forms = a, (13) (14) In order to investigate the mutual interaction of rotation and injection (suction), we shall assume a and j3 to be of order of unity. This includes many interesting cases where both rotation and injection (suction) are not minor perturbations. Differential eqs (10) and (11) are solved usually by direct integration which frequently involves more than One integration process because of the two point nature of the boundary conditions. The use of series solution provides an attractive alternative approach. Not only the difficulties associated with two point boundary value problems are relieved, but also 356 N M Bujurke et al the terms of series method are capable of providing results to any desired degree of accuracy with minimum time and less storage requirement of computer. 3. Method of solution We seek the solution of (10) and (11) in power series of jR in the forms Substituting (15), (16) into (10), (11) and comparing like powers of R on both sides, we get 1 _, + ^- 1 - r ) (17) (18) The relevant boundary conditions are ) = 0, / (0)=-j /' (1) = /:(0)=o, /;,(!) =o, / B (0)=o, /(!)= o (19) ffo(0) = o, g (l) = P .(0) = 0, ^ n (l) = (20) n = l,2,3,... The solutions of above equations are 11 2 13 3 1 4 1 6 1 . "- I + I -4 I ^ + 20 I '-70 1 ' -2^ + , 4 )-^!(2 > , 2 -3^+^) (21) ~ 5f74 ~ 10>?2 " The slow convergence of the series ((15), (16)) requires large number of terms for obtaining the approximate sum. As we proceed for higher approximations, the algebra becomes cumbersome and it is difficult to calculate the terms manually. We propose Computer extended series solution 357 a systematic series expansion scheme with polynomial coefficients so that whole process can be made automatic using computer. For this purpose, we consider / and g n to be of the forms k = 2 = (!-*) Z *W (23) in (15) and (16) respectively. This expression yields exactly the above calculated terms f 1 and 0! besides this it enables us to find / ( and & for i ^ 2 using computer. We substitute (22), (23) into (17), (18) and equate various powers of r\ on both sides and obtain two recurrence relations for unknowns A^ and B n(k) in the forms A 7A A -4 __ A n) ~ ^n(k+ 1) 1n<*+ 2) T X Z 4.- 1 M lk + 2-0 P ( k + 2-0 + Z B(,-l n-2 F 4 nk+i Z Z Zi ^r(f+l-nfc-)^(-D(-2) r=l L i = j=l 2 k + i Z Z ^.--o B (-iK.-3)e*+^- 3 ) r (24) /r-2 3 nk-l-i "I + Z Z Z ^^i- t -o B (-ix.-3)n + -('+l-*-^-3) r=l i = j"l -I K = l,2,...,4n (25) where m = n r, l = 1 (fc) = /c(fe-l)(fe-2), P 3 (fc) = - (3fc(k - l)(k - 2) - (k + 2)(fc + 1) k), P 4 (k) = (2fc(k - l)(k - 2) + 6(k + l)k(k - 1) + 12), P 5 (k) = - (4(k + l)k(k - 1) + 3(k + 2)(k + l)(fc - 1) + 24), P 6 (k)=(2(k + 2)(fc+l)k + 12), P 7 (k 1 )=-2fc 1 (k 1 -l)(k 1 -2), P 8 (k 1 ) = 4(k 1 + IJJk^ki - !) + 4k 1 (k 1 - .l)(fc! -2), N M Bujurke et al t + l)M*i - l)-2M*i -0(*i -2X l)M*i - 1), <2 2 (k) = - 2k(fi - a) + 2(k + l)a - 2(0 - a), T 1 (/c)=fc, T 2 (fc)=-(k+l), T 3 (k)=-(3fc-6), T 4 (/c) = (5/c-9), -3k 1 +2), T 9 (fc,fc 1 ) = 2(-k + k 1 -l), T;(fc) = 2fca, T' 2 (k) = 2()3 - a)fc - 4(fc + l)a - 2(0 - a), T' 3 (fc) = - 4(j8 - a)(k + 1) + 2(k + 2)a + 4(j9 - a), 70 12^ ' 30 ' 4 "70 60~' " 14 = 13ff " 15= ~TQ' A VF ' A _ 4 ^^""TA /en A i4~TlA' ^15~"~^: 2 20 12 20 (/S-a) 2 20 / 14 5 or the radial velocity profile/'(j), we have he constant ,4 in (9) which is proportional to the lift is given by 4w+l 02 oo X R" w=l fc=2 (26) -0 2 (27) Computer extended series solution 359 Case (1): a = 0, /? = which corresponds to the case when both discs are stationary and the flow is due to injection only. In this case the coefficients of the series for/'"(l), which is used to calculate A, has terms which are all positive after third term (table 1). Using the computed coefficients we draw Domb-Sykes plot (figure 2) for/'"(l) (series (27)) to find the nature and location of the nearest singularity which restricts the convergence of the series. In this case singularity is found to be a square root singularity at R 1 7-9826. This singularity on the positive real axis is not a real singularity, but an indication of double valuedness of the function. This artificial restriction on convergence can be eliminated by reverting the series. This type of reversion was successfully employed earlier by Richardson [13] and Schwartz [14]. Towards this goal the reversion of the series (27) for/"'(l) is performed as follows. Consider Let /'"(!)=- 6 + Y =/'"(!) + 6= "n (28) Reverting the above series, we have where = I B n Y n 1 (29) 1 m-2 i=o fc-t+i Table 1. The coefficients a n of the series (27) for /"' ( 1) in the case of a = 0, ft = 0. No No 1 - 6-OOOOOOOOOOOOOOE-OO 14 3-6058013122139E-011 2 -7-7142857142857E-001 15 2-8303130882864E-012 3 -3-2003710575139E-002 16 2-1564675357485E-013 4 5-6407538040317E-004 17 1-6028392195738E-014 5 9-4914049735636E-003 18 1-1664258107783E-015 6 2-2591450404529E-003 19 8-3319098951704E-017 7 3-7542463046543E-004 20 5-8565624104072E-018 8 5-0483529518823E-005 21 4-0599933881679E-019 9 5-8577314766192E-006 22 2-7794211813578E-020 10 6-0972460009191E-007 23 1-881 1986978930E-021 11 5-83546416825 13E-008 24 1-2608050796818E-022 12 5-2237099823 177E-009 25 8-37540970962 18E-024 13 4-4337512718581E-010 26 5-5163592628791E-025 360 JV M Bujurke et al 0-3 0-2 0-1 0-05 0-1 0-15 1/n Figure 2. Domb-Sykes plot for series (27) in the case of a = 0, /? = 0. 0-2 Besides reversion we use Fade 7 approximants for summing the reverted series (29) which yields analytic continuation. The details about Fade' approximants are given in Appendix. These results are shown in figure 3. Case (2): a = 4, j? = 0, lower disc is rotating and the upper disc is stationary. The coefficients (a n ) of the series (27) for /"'(I) are listed in table 2. They are decreasing in magnitude and have no regular pattern of sign. We invoke Fade' approximants to achieve analytic continuation of the series (27) [11] and the corresponding results are shown in figure 4. 12 numerical [16] [2/2]Pade' approximants [2/3] " *- - Brown's method 10 20 30 40 50 60 R * Figure 3. Values of A as a function of R(cx. = 0, ft = 0). Computer extended series solution 361 Table 2. The coefficients a n of the series (27) for /"' ( 1) in the case of a = 4, ft = 0. No No 1 - 6-OOOOOOOOOOOOOOE-OO 14 4-3651028454742E-007 2 4-02857142857140E-00 15 - 3-8077592733234E-007 3 - 2-6924345495784E-002 16 H081633809731E-008 4 - 1-2306126753066E-001 17 2-754430222 1770E-008 5 1-3750340358019E-001 18 -5-0110664501736E-009 6 9-5410260700099E-002 19 - H312815967417E-009 7 8-2631707941841E-003 20 4-6747879187209E-010 8 - 1-7355299293887E-002 21 2-653 1328428935E-011 9 2-971 7102085349E-005 22 -3-8735009780784E-011 10 2-9683069 164996E-004 23 3-8598232577803E-012 11 -3-0384182421661E--005 24 2-0690368287353E-012 12 - 1-82301 18955959E-005 25 -5-1032148957020E-013 13 5-0851049208551E-006 26 -8-7096012910697E-014 Case (3): a = 0, j? = 0-5 in this case the upper disc is rotating and the lower one is stationary. The coefficients (a n ) of the series (27) for/'"(l) are listed in table 3. They are decreasing in magnitude but have no regular pattern of sign. So, as in the previous case we use Fade' approximants to sum the series. The results obtained are shown in figure 5. Case (4): a = 1, j? = 1 in this case discs are corotating (with same speed). The coefficients (a n ) of the series (27) for /"'(I) are listed in table 4. They are decreasing in magnitude and alternate in sign after llth term. Using the computed coefficients we draw Domb- Sykes plot (figure 6) for/'"(l) (series (27)) to find the nature and location of the nearest singularity which restricts the convergence of the series. In this case singularity is found to be at R = 2-579849 on the negative real axis. The bilinear Euler transformation will help in recasting the series into new series whose region of validity is increased A -2 numerical [10] [2/2] Fade' approximants [2/3] " Brown's method 10 20 30 R """* Figure 4. Values of A as a function of R(ot = 4, /( = 0). 50 60 362 N M Bujurke el al Table 3. The coefficients a n of the series (27) for /'"(I) in the case of a = 0, = 0.5. No No 1 - 6-OOOOOOOOOOOOOOE-OOO 14 3-1356929508470E-009 2 -9-4642857142857E-001 15 3-3582404952799E-011 3 -6-1091012162441E-002 16 - 1-329911951 1885E-010 4 3-78589355301 74E-002 17 1-8233492599838E-011 5 6-065764368 1146E-003 18 2-0308395587595E-012 6 6-43802385 141 8 1E-.004 19 - 6-6509 147430890E-0 13 7 5-9793 18921 1569E-004 20 -6-1403723963494E-014 8 7-6918357469503E-005 21 4-7938734279652E-014 9 - 5-9467326297197E-006 22 -7-0845088542235E-015 10 8-86224061 12891E-007 23 6-2683407300982E-018 11 5-0127735327728E-007 24 1 -37998041 66256E-0 17 12 -5-5929925021510E-008 25 6-2326271557991E-017 13 - 7-7492242493884E-009 26 -2-3804925454199E-017 compared to the original series (27). Consider the Euler transformation then R = coR /(l - co) and /'"(i)= -6+ ; n where D 1 = 6, D 2 = a 2 R , n=l (30) 8 6 4 2 -2 numericoinO] [3/4] Pade' approximates 13/3 " --- Brown's method 10 20 R - Figure 5. Values of A as a function of R(a == 0, jft = 0-5). 30 40 Computer extended series solution 363 with Table 4. The coefficients^ of the series (27) for /'"(I) in the case of a = !,/?= 1. No a n No a. 1 -6-000000000000000000 14 1-3686820085834E-006 2 -7-7142857142857E-001 15 - 1-9036823867936E-007 3 - 2-7486085343228E-001 16 8-975672269942 1E-008 4 1-48 14962475 167E-001 17 - 6-0920523719445E-008 5 - H3224033836UE-002 18 2-2416431500647E-008 6 -4-6167826401268E-003 19 -5-3363130655523E-009 7 -2-9912751714986E-005 20 1-4329467638064E-009 8 1-4864726973501E-003 21 -6-7360950377740E-010 9 - 4-73733480 18575E-004 22 2-9287892083304E-010 10 2-8876377 167786E-005 23 -9-3478798663218E-011 11 4-8008012782765E-007 24 2-6176956621819E-011 12 1-0028862586723E-005 25 -9-1174262387237E-012 13 -5-9189631736138E-006 26 3-7433180023877E-012 3a 4 (31) e j = (-R ) j ' 1 a J . 12 \ I \ I \ ' \ I I I I \ Case 4 - Case 5 nran 0-2 0-6 0-8 1/n Figure 6. Domb-Sykes plot for series (27) in the case of a = 1,0 = 1 and a = 1, ft = - 1. 364 N M Bujurke et al Brown's method Eulerised series Figure 7. Values of A as a function of K(cc = 1 , = 1) and (a - 1, f - - D- This transformation maps the dominant singularities to . remains fixed Points close to the dominant singularities are mapped far from the origin aZentoSLunitcircleinth^ shown in figure 7. Case (5V a = 1 B = - 1 in this case discs are counterotating (with same speed). The coS L s (a ) of the series (27) for/"' (1) are listed in table 5. They are decrying magnitude and alternate in sign after 10th term. Using the computed coeffic ents we draw Domb-Sykes plot (figure 6) for /'"(D (series (27)) to find the nature and oca ion of the nearest singularity which restricts the convergence of the series. In this case Table 5. The coefficients a, of the series (27) for /'"(I) in the case of a - 1, J3=-l. No *. No a. 1 2 3 4 5 6 7 8 9 10 11 12 13 _ 6-000000000000000000 - 1-571428571428600000 - 2-1210059781490E-002 -2-7858353362435E-001 - 1-4495845921663E-002 - 3-2775029096463E-003 2-0976832622669E-003 -4-9288165348855E-004 - 7-5443633168820E-006 -8-6986972673174E-006 7-667501484U31E-006 -3-5515893049259E-006 5-6651432024791E-007 14 -9-2298357740712E-008 15 3-3221879279623E-008 16 _ 1-8249751078357E-008 17 5-3594587988598E-009 18 -H691594163933E-009 19 2-6218335178891E-010 20 - 9-821 1386423030E-01 1 21 3-5209099021094E-011 22 - 1-0026682644541E-01 1 23 2-4337119787355E-012 24 -6-7971322153644E-013 25 2-241 1087467669E-013 26 - 7-1355223824464E-014 Computer extended series solution 365 singularity is found to be at R = 2-623517 on the negative real axis. As in the previous case we have used Euler Transformation to increase the region of validity. So /'"(D = -6-f f a n+l (<DR /(l-co)r= DX" 1 (32) n 1 n= 1 The variation of A with R is shown in figure 7. Equations (10)-(14) are also solved by power series method. 4. Power series method We assume power series solution to (10)-(14) in the forms 00 /= I d n (\-riT +l (33) = /?+Z b m (\-rjf (34) where > B + 0=a (35) =i and P"-W-* (36) 2Rpb n+l _ 2R _ 11+3 " (n + 4)(n + 3)(n -f 2) + (n + l)(n + 2)(n + 3)(n + 4) Expression (35) comes from the boundary conditions at r\ = and (36) and (37) are obtained from (10) and (1 1) respectively. If b l , d 1 and d 2 are known then rest of (b n } and {d n } can be found from the recursive relations (36) and (37). Effectively we have transformed a two point boundary value problem into solving a system of nonlinear equations. We wish to find b l ,d 1 and d 2 such that conditions (35) are satisfied. To solve this system of nonlinear equations Brown's method is useful. The details of this procedure are given in Byrne [18]. It is found that the series (33), (34) converge much faster and also more accurate solution with very little computer time can be obtained. It is implemented in analysing all the five cases considered. The first two coefficients of the series and lift at different Reynolds numbers are calculated. All 366 N M Bujurke et al these values are accurate to six significant figures. The number of significant figures for accuracy was determined by increasing the number of terms in the series from 30 to 350. The time taken by the computer is also comparatively less whereas other methods [9, 10, 16] require more computer time and large storage. 5. Discussion of results Here the problem of injection (suction) of a viscous incompressible fluid through a rotating porous disc onto a rotating co-axial disc is studied using computer extended series analysis. The motion of the fluid is governed by a pair of coupled nonlinear ordinary differential (10) and (11) together with the boundary conditions (13) and (14). The series expansion scheme with polynomial coefficients ((22), (23)) proposed enables in obtaining recurrence relations (24) and (25). Using these interactive relations we generate large number (n = 25) of universal coefficients ((A n(k} , k = 2, 3, . . . , 4n + 1), n = 1, 2, . . . , 25) and ((J5 B(fc) , k = 1, 2, . . . , 4n), n = 1, 2, . . . , 25). To this order there are 1300 coefficients A n(k) and 1300 coefficients B n(kr A careful FORTRAN program consisting of number of DO loops makes it possible in performing complex algebra involved. Using the universal coefficients of the series ((22), (23)) we obtain series expansion for A which is directly proportional to the lift. The coefficients a n of the series (27) for A in the case of a = 0, j? = are listed in table 1. They decrease in magnitude and have same sign after third term. Figure 2 shows the Domb-Sykes plot for series (27) in the case of a == 0, jS = 0. The slope of the curve indicates square root singularity corresponding to double valuedness of the solution (by using rational extrapolation exact position of the singularity is found to be at R = 17-9826 with an error of order 10 ~ 5 ). So the region of validity of the series (27) for A in the case of a = 0, fl = will be increased by reverting the series (by changing the role of dependent and independent variables). We use Fade' approximants for summing the reverted series (29) which accelerates the convergence and yields its analytic continuation. The results agree most favourably with results of Wang [16] (numerical), Bujurke and Naduvinamani [12] (semi-numerical) and Phan-Thien ad Bush [9] (power series). It is of interest to note that [2/2] and [2/3] Fade' approximants bracket [The Fade' approximants P^(l) and P^l) form upper and lower bounds for the numerical value of lift force [15]] the Numerical results of Wang [16] (figure 3). Double precision arithmetic used guarantees the accuracy of Fade' approximants. Also, the round off errors will be of negligible order as the Fade' approximants bracketing the numerical results are of the form where denominators are polynomials of degree ^ 4 [17]. Table 2 contains the list of coefficients a n of the series (27) for the case of a = 4, $ = 0. These coefficients decrease in magnitude but have no regular sign pattern. We invoke Fade' approximants to achieve analytic continuation of the series (27). The results agree favourably with earlier numerical findings [10]. Also, we observe that [2/2] and [2/3] Fade' approximants bracket the numerical results which are given in figure 4. The coefficients a n for the case of a = 0, jS = 0-5 are listed in table 3. In this case also coefficients are decreasing in magnitude and have no regular sign pattern. As in the previous case analytic continuation of the series (27) is achieved by using Fade 7 approximants. The [3/4] Fade' approximant is found to be very near to the numerical results [10] which are shown in figure 5. The coefficients a n of the series (27) for A in he case of a = 1, /? = 1 are listed in table 4. They decrease in magnitude and have ilternate sign after llth term. Figure 6, the Domb-Sykes plot for series (27) in the Computer extended series solution 367 Table 6. Comparison of Brown's method with optimization method. Terms (N) (required for the Terms (N) (required for the conver- R Lift (Brown's Method) convergence of Brown's Method) Lift (Optimization) gence of the optimization method) 1 6-80278 30 6-80278 50 5 2-10850 50 2-10850 100 10 1-58847 75 1-58847 200 15 1-42121 150 1-42846 500 18 1-38127 200 1-39342 750 22 1-35841 350 case of a = 1, = 1 shows the singularity on the negative real axis after extrapolation at R = 2-579849 with an error of 10~ 4 . The region of validity of the series is increased by Euler Transformation. The results obtained are shown in figure 7. In case 5 the analytic continuation is achieved exactly in the way like case 4. The results obtained are shown in figure 7. This problem is also solved by power series in conjunction with Brown's method for different cases [a = 00 = 0, a = 4)3 = 0, a = 00 = 0-5, a = 10 = l, a = 1 ft = 1] and the results obtained are shown in figures (3-5 and 7). Details of case 1 (a = 0, = 0) (table 6) corresponding to stationary disks with injection shows the efficiency of Brown's method. The series (26) representing radial velocity profiles in various cases (a = 0, = 0; a = 4, = 0; a = 0, = 0-5) are analysed using Fade' approximants and these results are shown in figures 8 and 9). It is observed that velocity attains peak values for a = 0, = 0-5 and it is much higher than first two cases. 1-5 1 ,0.5 -0-5 0-2 0-4 0-6 0-8 1 7? ' Figure 8. Radial velocity distribution /'(*/) at R = 16. 368 N M Bujurke et al 1-5 -0-5 0-2 0-4 0-6 0-8 1 7? Figure 9. Radial velocity distribution f'(r\) at R = 20. The method proposed here is quite flexible and efficient in implementing on computer compared with the pure numerical methods. Once the universal coefficients are generated rest of the analysis can be done at a stretch requiring hardly any computer time and storage. Whereas other methods [9, 10, 16] require more computer time and large storage. Acknowledgement The authors thank the referee for useful suggestions on the earlier version of the paper. Appendix Fade' Approximants The basic idea of Fade' summation is to replace a power series by a sequence of rational functions of the form N where we choose B = 1 without loss of generality. We determine the remaining (M + N + 1) coefficients A^A^A 2 ^..A N ;B^B 2 ,...B M so that the first (M + N + 1) terms in the Taylors series expansion of P^-R) match with first (M + N + 1) terms of the power EC w R n . The resulting rational function PM(&) is called a Fade' approximant. If EC w jR n is a power series representation of the function f(R) than in favourable cases Computer extended series solution 369 , pointwise as N, M - oo. There are many methods for the construction of Fade' approximants. One of the efficient methods for constructing Fade' approximants is recasting of the series into continued fraction form. A continued fraction is an infinite sequence of fractions whose (N + l)th member has the form + DjjR l+D 2 R l+D N R The coefficients D n are determined by expanding the terminated continued fraction F N (R) in a Taylor series and comparing with those of the power series to be summed. An efficient procedure for calculating the coefficients D n 's of the continued fraction (E) may be derived from the algebraic identities (8.4.2a)~(8.4.2c) [15]. Contrary to repre- sentations by power series, continued fraction representations may converge in regions that contain isolated singularities of the function to be represented, and in many cases convergence is accelerated. Based on these D M 's we get terminated continued fractions of various order from other algorithms ((8.4.7), (8.4.8a) and (8.4.8b) [15]). Fade' approximants perform an analytic continuation of the series outside its radius of convergence. It is clear that it can approximate a pole by zeros of the denominator. With branch points it extracts a single- valued function by inserting branch cuts, which it simulates by lines of alternating poles and zeros [19]. References [1] Batchelor G K, Q. J. Mech. Appl. Math. 4 (1951) 29 [2] Von Karman T, Z. Angew. Math. Mech. I (1921) 233 [3] Bodewadt U T, Z. Angew. Math. Mech. 20 (1940) 241 [4] Stewartson K, Proc. Cambridge Philos. Soc. 49 (1953) 333 [5] Hoffman G H, J. Comput. Phys. 16 (1974) 240 [6] Lance G N and Rogers M H^Proc. R. Soe. (London) A226 (1962) 109 [7] Mellor G L, Chappie P J and Stokes V K, J. Fluid. Mech. 31 (1968) 95 [8] Brady J F and Durlofsky L J, J. Fluid. Mech. 175 (1987) 363 [9] Phan-Thien N and Bush M B, Z.A.M.P., 35 (1984) 912 [10] Wang C Y and Watson L T, Z.A.M.P., 20 (1979) 773 [1 1] Van Dyke M, Q. J. Mech. Appl. Math. 27 (1974) 423 [12] Bujurke N M and Naduvinamani N B, Z.A.M.P. 43 (1992) 697 [13] Richardson S, Proc. Cambridge Philos. Soc. 74(1973) 179 [14] Schwartz L W, J. Fluid. Mech. 62 (1974) 553 [15] Bender C M and Orszag S A, Advanced Mathematical Methods for Scientists and Engineers, Third Internation Edition (1987) (London: McGraw Hill Book Co.) [16] Wang C Y, A.S.M.E. J. Appl. Mech. 41 (1974) 343 [17] Graves-Morries P R, Lecture Notes in Mathematics, (1980) (Berlin: Springer-Verlag) Vol. 765 pp. 231-245 [18] Byrne G and Hall C (eds) Numerical Solution of Systems of Nonlinear Algebraic Equations, (1973) (New York: Academic Press) [19] Baker G A, Essentials of Pade' Approximants (1975) (New York: Academic Press) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 4, November 1995, pp. 371-380. Printed in India The Hodge conjecture for certain moduli varieties V BALAJI School of Mathematics, SPIC Science Foundation, 92 G.N. Chetty Road, T. Nagar, Madras 600 01 7, India MS received 20 March 1995 Abstract For smooth projective varieties X over C, the Hodge Conjecture states that every rational Cohomology class of type (p, p) comes from an algebraic cycle. In this paper, we prove the Hodge conjecture for some moduli spaces of vector bundles on compact Riemann surfaces of genus 2 and 3. Keywords. Chow groups; Abel-Jacobi maps; moduli spaces; normal functions; Hecke correspondences. Introduction For smooth projective varieties X over C, the field of complex numbers, the Hodge conjecture states that every rational cohomology class of type (p, p) comes from an algebraic cycle. More precisely, consider the Hodge decomposition Let C P (X) denote the Chow group of algebraic cycles of codimensionp on X, modulo rational equivalence. Then one has the 'class map' tf x : C p (X) Q -> H 2p (X, Q) n Then the Hodge (p,p) conjecture states that X p x is surjective. Let C be an irreducible smooth projective curve if genus g ^ 2, and let M(n, ) be the moduli space of stable vector bundles V on C, of rank n, det V ^ & a line bundle of degree d such that (n,d) = 1. The aim of this paper is to prove the Hodge (p,p) conjecture in the case when g=--2, 77 = 3 (dimM(3,) = 8). In the case when = 2, g = 2, 3, 4, the Hodge conjecture can be proved by elementary means which we indicate at the end of the paper. The case we consider is of interest, as it gives a non-trivial family of examples where the general method of normal functions is used to prove the conjecture. Geometric descriptions given in [T] in the rank 2 case lead to elementary proofs, of the Hodge conjecture. In the rank 3 case, any such description does not give elementary proofs of the Hodge conjecture, (cf. Remark 4.3, 4.4) The Poincare-Lefschetz theory of normal functions was generalized and developed by Griffiths and Zucker and had the proof of the Hodge (p,p) conjecture as a primary goal In this paper we give a natural construction of a smooth projective variety and a proper generically finite morphism onto the moduli of rank n, degree (ng - n) bundles which plays the role of the Lefschetz pencil in the context of normal functions. From the remarks of Zucker (cf. [Z-2], pp. 266) all the known examples where normal functions have been used to prove the Hodge conjecture, more elementary methods have been 371 372 VBalaji successful (cf. [M], [Z-2], and [Sh] for a full survey of the Hodge conjecture); however, in the present case this seems unlikely. In 1, we recall some general facts. Section 2, contains a theorem giving a criterion for a variational Hodge (p, p) conjecture to hold under some stringent conditions. In 3, we give a pencil type construction in the context of moduli. Section 4, gives the proof of the conjecture for M(3, ). Some notations. Let X be a smooth projective variety defined over C the field of complex numbers. We state at the outset that our base field is C. Let C P (X) denote the Chow group of cycles of codimension p modulo rational equivalence and A P (X) c: C P (X) the subgroup of cycle classes algebraically equivalent to zero. 1. Preliminaries Lemma 1.1. (cf. [2-1] A.2) Let X and Y be smooth projective varieties, f:X-*Y be a proper generically finite surjection. If the Hodge (p,p) conjecture is true for X, then it is true for Y. Proof. We note that, /*/* = multiplication by d, both on cycles and cohomology, where d= [k(X):k(Y)']. Therefore, if yeH p * p (Y 9 \ f*yeH p > p (X,Q); so if f*y is a rational cycle Z, then implying y is a rational cycle l/d(f^z) on 7. Lemma 1.2. Let Ebea vector bundle of rank r = e+l,andletP = P(E). Letf:P -+Xbe the associated projective bundle. Then the Hodge (p, p) conjecture is true for X if and only if it is true for P. Proof. Let h be the relative ample class 0p(l), and h = c 1 (^ P (l)). Then we have the well-known decompositions of the Chow groups and cohomology groups of P, and we have the diagram: c*(p) = f*c p (x) hf*c p ~ l (x) .-0 h e f*c p ~ e (X) m i% ur 1 ur* H 2p (P) = f*H 2p (X) hf*H 2p ~ 2 (X) ... h e f*H 2p ~ 2e (X) From this diagram, the proof follows easily, noting the fact that /* is an injection both on cycles and cohomology. Lemma 1.3. Let X be a smooth projective variety, Yc+X a smooth closed subvariety of codimension r; let U c+ X be X - Y, f (resp.j) the inclusion ofY(resp. U) in X. Then we have the folio-wing commutative diagram: C~ r (Y) Jl> C(X) > C q (U) ->0 >Uy l*x I H 2 ~ 2r (Y) Gysin > H 2q (X) -* H 2 (U) Proof. This follows from the existence of the Gysin map i^ which is functorial with respect to the class map L (cf. J Milne, Etale Cohomology, Proposition 9.3, Ch. VI). 373 The Hodge conjecture DEFINITION 1.4 Let J'(X) be the pth Griffiths-mtemediate Jacobian of X based on ([G],[Z-2])andlet cohomology of a proj we have to the one induced by (4 Therefore, one has * (X) - Further, one has a similar decomposition for ,he Chow groups A>(P) = A'(X>&A'-'W- A ' ' (X) ' Combining h,s ith d. functorialit, of the Abel-J.cobi maps, we get is similar. 2. Normal functions Let f:X -*S be a proper smooth -singula: ^ for x [Z-3], [Z-4]). conditions hold: issurjectiveVseS. 374 VBalaji Then Hodge (p,p) holds for X. Proof. Consider the Leray filtration {I/} on H*(X) associated to the morphism /. Since the spectral sequence degenerates (cf. [G]), we have: L^L^L 2 . We need the following description of the Leray filtration from ([Z-3], pp. 194): L 1 = ker {H 2p (X) -> H 2p (X s )} L 2 = ker [H 2p (X) -+H 2p (X - X s )} = Im {H 2p ~ 2 (X s ) Gys ' m > H 2p (X)} (cf. Lemma 1.3) for any seS, and We need to handle the (p,p) classes in the rational cohomology of X, which come from the various parts of the Leray filtration. The primitive class i.e. the (p, p) classes lying in L 1 can be dealt with as follows: (i) Observe firstly that L l /L 2 ~ H l (S, # 2p ~V*Q). Integral (p,p) classes in L^L 2 , thus arise as cohomology classes of normal functions i.e. holomorphic sections of the intermediate Jacobian bundle, J P (X S ) -S. This is a consequence of Theorem 2.13 of [Z-4]. Our assumption (b) then ensures by [Z-l], that this normal function comes from a relative algebraic cycle on X. (ii) (p,p) classes which lie in L 2 : Note that L 2 = Im {H 2p ~ 2 (X s ) Gysin > H 2p (X)} and by assumption (a) and Lemma 1.3 of 1, since Hodge (p 1, p 1) holds for X s , (p, p) classes in L 2 come from algebraic cycles. Now for the remaining classes, in L /^ 1 , let 7 be a (p,p) class in H 2p (X\ which restricts to non-zero classes y s on X s for all seS. Let ^ d x/s denote the Chow variety (or reduced Hilbert scheme) of relative codimension p cycles of degree d on X. By the theory of Hilbert schemes, for some d 0, the natural morphism x/s is a surjection. Hence for all A ^ 1, ^ s -* S is surjective. Let V' Ad be the non-empty open subset of S for all A ^ 1, such that is flat. (Such a non-empty V exists since < a is a proper surjective morphism.) By a Baire argument, it is easy to see that H^ { V u * <; choose an se n ; . > i ^ Ad and fix this s. Consider 7 1 Js = 7 S ; then by (a) of Theorem 2.1, since Hodge (p,p) is true for X s , express y s = oc s - p where a s and p s are effective codim p-cycles on X s of degree / and m respectively. Since we are interested only in rational cohomology, we may assume, without loss of generality that / and m are multiples of d. Therefore, by choice se V l n F m , and a s e< ~ x (4 Since ^ is flat over V 1 , all irreducible >mponents of $^(V*) dominate V 1 (S being a smooth curve). Choose an irreducible The Hodge conjecture component of <f>~W which contains a, Then it is easy to see, (bv choosing a cune C through a s and taking its closure in U that we get a curve S' and a finite morphia i -* S, such that (we could assume S' is also smooth without loss of generality bv aoins to the normalization if need be). ~ . . - - I I S' -* S and there is a section for ' over S', which passes through x s . That is if X' JL+X 1 I S' > S then, there exists an effective codimension p-cycle a of degree / on X', such that 2 v = x s : where s' -* s. We can similarly get a /? of deg m over another finite extension, and \ve can therefore get T, a smooth curve, with a finite morphism r-s, such that Y JL> X I I T > S and a and /? give codimension p-cycles on Y of degree / and m respectively, s.t. oc| rt =a s , ^| yt = /? r Thus, e= [>*}>- (a -j3)] is a cohomology class which (by (*)) lies in Hence e is a primft ie cofcomo/offy class on Y; observe that fibres of V-* J are the same as those of X -S, and hence thehypothesesofTheoremll.holdforthefibresotlpT-to well. So by the first part of our proof, e comes from a codimension p-algebraic cycle , on Y. i.e. is algebraic. Since : 7- X is a proper finite surjection, by Lemma 1.1. it follows that y itself is algebraic. 3. A pencil-type construction for moduli In the discussion that follows, we describe moduli spaces of vector bundles. We remark that 376 VBalaji hyperplane section in the moduli space is not very transparent and so the usual theory of normal functions and Lefschetz pencil cannot be applied in this setting. We begin by proving a lemma which is essential in the construction. Lemma 3.1. Let Wbe a stable vector bundle of rank 2 and degree 3. Let V be a non-split extension Then V is semi-stable. Proof. This is an elementary consequence of Propositions 4.3, 4.4. and 4.6 of [N-S]. To see this, suppose that V is not semistable, then by Proposition 4.6 there exists an F, stable of rank ^ 2 such that and a non-zero element /eHom(F, V). Thus /z(JF) ^ jj.(W). Thus v/eHom(F, W\ If v o/ is zero, / must factor through & which gives an immediate contradiction. If v of is non-zero, by Proposition 4.4 of [N-S], if W 1 is the subbundle of W generated by . Since W is stable, it implies W l ^W and vo/ is an isomorphism, which gives a splitting for v, q.e.d. Let M L = M(3, L), be the moduli space of semi-stable bundles of rank 3, deg 3g - 3, A n V ^ L, g being the genus of C, i.e. deg (L) = 3# - 3, # = 2. Consider the -divisor in M L which is defined as follows: = {FeM L |/i(F)>0}. More generally, we can define for all e J(C) ? the divisor Let Hr^ be the universal family on C x M(2,L) and consider the bundle of extensions given by where p:C x M(2,L)->M(2,L). Observe that, if WeM(2^L\ then the points of P 5 lying above W are given by non-split extensions Q^Q^V-*W-+Q. (1) By Lemma 3.1, we see that bundles V obtained above are semistable. Thus we can define a morphism ^ote that since det V ~ L, det(K ( 1/3 )*) = L. Also this map is well-defined since P$ parameterizes a universal family and M(3,L) has the coarse moduli property. It is easy to see that Im <^ c (when rj = 1/3 ). Further, by ([S] Theorem IV, 2.1), the component of Im 0* in B n is of codimension at least 2 (in general for rank n it is n - 1) The Hodge conjecture 377 and therefore contains a non-empty open subset of 0^, hence by the properness of <p^ (in fact by [S], ^ is birational). The above construction of P^ can be globalized as follows: Let M(2,3) be the moduli space of vector bundles of rank 2 and degree 3. Let if -> C x Af(2, 3) be the universal family. Define P = P(R l p^"*). Then the morphism </>,, globalizes to give: (the ambiguity of 'cube roots' can be resolved to by pulling back P by the following diagram: (so in fact, (j) is well-defined as a morphism <p : P' -* M(3, L)). Define P c by the following base-change diagram: P-*?' C where Cc+Jby mapping a base point x to the fixed degree 3 line bundle L. (Note that C is in fact connected). Then <p induces a morphism We claim that is surjective. This is not hard to see since Im <t> contains the 0-divisor; further, one can easily get a point in M L - in Im 0. Now surjectivity follows from the fact that P c and M L are irreducible and is a proper morphism, such that Im $ properly contains a divisor. Since dim P c = dim + dim C = dim M L , gives a generically finite proper surjection. Remark 3.2. We remark that the above construction can be done for all ranks by using the construction of desingularization of the 0-divisor in [RV] Our variety P can be related to their but we would not go into it here. 4. Proof of the Hodge conjecture for M(3, ij) In this section we complete the proof of the Hodge ( p, p) conjecture for M(3, r\\ where deg Y\ = 1 or 2, g = 2. The strategy is to relate the geometry of M(3, 17) and M(3, L) by the Hecke correspondence (cf. [B]). 378 VBalaji PROPOSITION 4.1 Let M L = M(3, L), deg L = 30 ~ 3. Let = 2 and consider the moduli space P c construc- ted in 3. Then Hodge (p, p) is true for P c for all p. Proof. By Theorem 2.1, it is enough to prove the properties (a) and (b) in its statement for n~ l (y) for all yeC, where By 3, n~ 1 ( j;)'s are the moduli spaces P%. Since P^ is a projective bundle on M (2, ^ L) associated to a vector bundle, to prove (a) and (b) of Theorem 2.1 for P^ it is enough to check them for M(2, L) because of Lemma 1.2 and Lemma 1.5. Since M(2, L) is a 3-fold, the Hodge conjecture follows from the Lefschetz (1,1) theorem. That A 2 (M(2,cL)) has the Abel-Jacobi property follows from ([B-M] pp.78) since M(2, f L) is a rational 3-fold. We could also prove the above Proposition for P c more directly by using the follow- ing fact: By Thaddeus [T], (cf. also [N]), we could, consider the variety obtained by blowing up the curve C embedded in a suitable projective space of extensions. It corresponds to the variety M l in [T]. Denote this by Af (2, L). Then, when g = 2, it is easy to see that is a birational morphism. Since M'(2,^L) also parameterizes family of vector bundles (in fact a family of pairs!), we have a variety P^, a projective bundle associated to a vector bundle on M'(2, L) and a birational morphism Properties (a) and (b) of Theorem 2.1 are fairly simple for P . Now construct globally the variety P^ such that PC -Pc I I C - C Observe that by Theorem 2.1, Hodge (p 9 p) is true for P' c . Since P^ -> P c is a generically finite surjection, Hodge (p,p) for P c follows from Hodge (p,p) for P' c , by Lemma 1.1. Theorem 42. The Hodge (p,p) conjecture is true for M(3,??), where degrj = 1 and 2. to = 2). . We prove it for deg r\ = d = 1. Proof for d = 2 follows along identical lines. Let P^ be the moduli space of parabolic stable bundles, (7, A), V of rank 3, deg 3g - 3 = 3, det V ^ L, with parabolic structure A at xeC given by F 2 V X a subspace of dim 1, and weights taken sufficiently small (cf. [B], . . . ). Then, we The Hodge conjecture 379 have the Hecke correspondence M L where r\ is a line bundle of deg Y\ = 30 5 = 1. The morphisms \l/ and h are given by where W is obtained from the following exact sequence. T being a torsion sheaf of height 2 given by T JVJF 2 V X at x [0 elsewhere. Then it is known that ^ is a projective bundle associated to a vector bundle on M(3, YJ) (cf. [B]) and the map h (in (*) above) is generically a projective bundle over the stable points of M L . Therefore by Lemma 1.2, it is enough to prove the theorem for P x . Now P c by construction parameterizes a universal family i' -* C x P c . By the definition of and h, it is easy to see that P x ^ P>(f *), where V~ x is the bundle on P c obtained by restriction of V to x x P c , and 1^* its dual. Thus by the coarse moduli property of parabolic bundles for P^, we have a morphism $: P x - P x and the following commutative diagram: By Proposition 4.1, Hodge (p,p) is true for P c and hence by Lemma 1.2, it is true for P x . Thus by Lemma 1.1, since $ is a generically finite surjection, Hodge (p 9 p) is true for P x for all p, which proves the theorem. To prove it when deg rj = 2, we modify the parabolic structure by giving F 2 V x c K x , as a subspace of dim 2 and the rest of the argument is similar. Remark 4.3. (The Hodge (p, p) conjecture for rank 2 moduli when = 3, 4). In these cases when rank is 2, there is a geometrical picture due to Thaddeus (cf. [T]); in his notation, if d > 2g - 2, d being the degree, then the moduli space of stable pairs P , z = (d- 1)/2, dominates M (2, ) d() = i Further, when d = 20 - 1, P , z = (d - l)/2, has the property that is a birational surjection. Thus, in the case when = 3, (resp. 4) d = 5 (resp. 7), the index, f = 2 (resp. 3). Now, the variety P 2 (resp. P 3 ) is obtained by a sequence of blow-ups and blow-downs where the centres are smooth and Hodge conjecture is easily verified by using the 380 VBalaji Ibrmule-clef which expresses the Chow ring (resp. cohomology) of the blow-up i: terms of the Chow ring (resp. cohomology) of the base and the centre of the blow-uj. Then by Lemma 1.1, using 0, Hodge (p,p) follows for M(2, ). When = 5, the centre blown-up are projective bundles over S 4 C, the 4th symmetric power of C and henc Hodge (p,p) would follow, once it is known for S"C, n ^ 4. Remark 4.4. In the rank 3 case, even when = 2, the centres of blow-ups in any attemp at such descriptions seem much more complicated, vis-a-vis the Hodge conjectur< Also, it is not clear if the centres are smooth in the first place. Our proof, which i inductive, uses the simple nature of the geometry of rank 2 moduli spaces. Acknowledgements It gives the author great pleasure to thank P A Vishwanath for his invaluable help. H also thanks Profs M S Narasimhan and C S Seshadri for useful discussions an D S Nagaraj for pointing out some corrections. References [B] Balaji V, Intermediate Jacobian of some moduli spaces of vector bundles on curves, Am. J. Math. 11 (1990)611-630 [B-M] Bloch S and Murre J P, On the Chow group of certain types of Fano threefolds, Compos. Math. 2 (1979)47-105 [G] Griffiths P, Periods of integrals on algebraic manifolds III, Publ Math. I.H.E.S. 38 (1970) 125-18 [M] Murre J P, On the Hodge conjecture for unirational fourfolds, Indagationes Math. 80 (197 230-232 [N] Newstead P E, Stable bundles of rank 2 and odd degree on a curve of genus 2, Topology 7 (196 205-215 [N-S] Narasimhan M S and Seshadri C S, Stable and unitary vector bundles on a compact Riemar] surface, Ann. Math. Vol. 82, 3 (1965) 540-567 [R-V] Raghavendra N and Vishwanath P A, Moduli of pairs and generalized theta divisors, Tohoku Mat J. 46 (1994) 321-340 [SH] Shioda T, What is known about the Hodge conjecture?, Advanced Studies in Pure Mathematics - (1983) pp. 55-68 [S] Sundaram N, Special divisors and vector bundles, Tohoku Math. J. (1987) pp. 175-213 [T] Thaddeus M, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117 (1994) 3 1 7-3 f [Z-l] Zucker S, The Hodge conjecture for cubic fourfolds, Compos. Math. 34 (1977) 199-209 [Z-2] Zucker S, Intermediate Jacobians and normal functions, Ann. Math. Stud. (1984) (Princeton Uni Press, New Jersey) No. 106 [Z-3] Zucker S, Generalized intermediate Jacobians and the theorem on normal functions, Invent. Mat 33(1976)185-222 [Z-4] Zucker S, Hodge theory with degenerating coefficients: L 2 cohomology in the Poincare metric, An Math. 109(1979)415-476 Pro , Indian Aca, So, (Mat, So,), Vol. 105. No. 4, Nove.be, 1995, P, 381-391. Printed in India. Bavarian, cobordism of Grassmann and Bag manifolds Madras 600 017, India MS received 3 January 1995; revised 27 March 1995 Abstract. We considerce on real Grassrnann and flag manifolds with finite stationary point sets and Id; fl a g d; tan g enti a , representation. action closed G- u of equiv ai en ce 1. Introduction ., , LetCbeacompactLiegro^^^ denoted by (M, *), where J:G x M - M denotes^ ^ ^^^ ^ ^.^ called a stationary point rf J to, ^) = ^ or al ^ ^ A smoot h closed n-dimensional erou S (ZandS 1 w 1 thfin 1 testatonary ^^ ^^ } Two & l^ i^teLtio^point sets, anjsaid to , - * J^^ union (M u M",,^ u ^ bounds eunoidyif equivariantly. This is an cLses is denoted by Z n (G). disjoint union this becomes an abehan with diagonal action makes the ^ algebra. For a smooth closed oriente every geG, <I>,M-+ M, x^x set, we say (M", *) is an onent ( W^ <D)ona compact onen ( dTffeomorphism for which the induced diffeomorphic to (M", flby o^ (- M'.fl, by just reversing the !S e, product of G-mamfolds & commu tative .^ (M", ^ (so that for } having fini te stationary point \, and only if there is an action a group of orientation presemng 8 $/ W " +1 ) is equivanantly m S P hism. We take - (M^) . Two smooth closed onented point sets, 382 Goutam Mukherjee a representative M on which there exists an action of S l with finitely many stationary points. Thus in the case G = S 1 , the map is surjective. The aim of this paper is to consider certain natural (Z 2 ) n actions on real Grassmann and flag manifolds and S l actions on complex Grassmann manifolds with finite stationary point sets and generate elements in the kernel of s and &. Group actions with finite stationary point sets are particularly interesting, as in this case, the tangential representations of the group G = (Z 2 )", at stationary points, completely determine the equivariant cobordism class of manifolds [3]. In case G = S 1 , although the tangential representations do not determine the equivariant cobordism class of a manifold completely, they carry lot of information about the bordism structure of the manifold. As for example, Atiyah-Singer [1] and Bott [2] have shown that if S 1 acts on an oriented compact manifold M with a finite stationary point set S, then the oriented RS ^modules {T x M:xeS} determine the Pontrjagin numbers of M, (also cf. 2). For G = Z 2 x Z 2 , Conner and Floyd have described the structure of Z^(G) completely (cf. [3]). Stong and Kosniowski [4], have also derived this result from a more general consideration. They showed that Z JG) is the polynomial algebra over Z 2 generated by the class [^P 2 ,<] 2 , where </> is given by the generators T 1 and T 2 as follows. Ti([x,y,z]) = [ -x,;y, z] and T 2 ([x,j>,z]) = [x, y,z]. In particular, the kernel of 8 is trivial in this case. No neat description of Z J|C (G) for G = (Z 2 ) n , n > 2, is known. Our results show that in general the kernel of & is nontrivial. a cobordism class [M d , <p'] 2 eZ d ((Z 2 ) n ) is equivariantly decomposable if (M d , </>) is equivariantly cobordant to a disjoint union of products of lower dimensional manifolds with (Z 2 ) w action with finite stationary point sets, otherwise it is equivariantly indecomposable. The first step towards understanding the structure of Z^G) in general, would be to know the indecomposable elements in Z Hs (G), which may be considered as the generators. Unfortunately, there is no indecomposability criterion known in the equivariant case. Clearly if [M] 2 eMO Hc is indecomposable (in the non-equivariant sense) and M admits an action of (Z 2 ) n , with finite stationary point set, then [M, </>] 2 is indecomposable. But there exist some elements in the kernel of & which are indecomposable in Z :Je ((Z 2 )"). For example, it is easy to argue that [RP 3 , 0] 2 is indecomposable in Z^((Z 2 ) 3 ), where $ is given by the generators as follows. r 1 ([x,y,z,w]) = [~x,3;,z,w], T 2 ([x,y,z,w]) = [x, -y,z,w] and T 3 (|>,y,z, w]) = [x,y 9 - z,w]. By knowing enough elements in the kernel, perhaps it would be possible to get an idea about the indecomposable elements in general. We believe that all the elements in the kernel given by Theorem 1.1 and Theorem 1.2 are indecomposable. This motivates our study of these actions. To determine which real flag manifolds bound, in [9] the authors gave a partial answer to this question. Real Grassmann and flag manifolds come equipped with certain natural (Z 2 ) n actions having finite stationary point sets, to be made precise later. Although, it seems difficult to determine the unoriented cobordism class of flag manifolds, the determination of (Z 2 )"-cobordism class of flag manifolds is easy. In the present paper, which real flag manifolds and Grassmann manifolds bound equivariant- ly, is completely determined. More precisely, we prove Theorem 1.1. (a) (G n/c ,$) bounds equivariantly if n = 2L CO ( G n^ <t>) does not bound equivariantly ifn ^ 2L Theorem 1.2. (G(n l3 n 2 ,...,n s ),0) bounds equivariantly if and only if n i = n j for some Equivariant cobordism ofGrassmann and flag manifolds 383 Precise definitions of the actions <p on Grassmann and flag manifolds are given in the subsequent sections. Perhaps, by knowing sufficiently many elements in the kernel of 8 it would be possible to determine whether the unoriented (Z 2 )"-cobordism class of flag manifolds lie in the kernel of & or not, and that might lead to a complete answer to the question, which real flag manifolds bound? We also consider certain natural S ^actions on complex Grassmann manifolds to produce nontrivial elements in the kernel of (cf. Theorem 3.4). In this case, our result produce an infinitely many nontrivial elements in the kernel of e. As a consequence, we deduce that for each d > 1, J r 2d (5 1 ) is not finitely generated as abelian group. 2. Representation and cobordism In this section we briefly recall [3] the relation between tangential representations at stationary points and cobordism and a result of Stong. Let G be a finite group. Let R n (G) denote the vector space over the field Z 2 , with basis the set of representation classes of degree n. The elements in R n (G) are formal sums of n-dimensional representation classes with coefficients in Z 2 . If R*(G) = ^R n (G\ then RX (G) admits a graded commutative algebra structure with unit over Z 2 . The product is given as follows. Suppose (V l9 G), (F 2 , G) are representations. We take (^ V 2 , G) to be g(v i9 v 2 ) = (gv l9 gv 2 ). Then the product is (F 1 ,G)-(F 2 ,G) = (K 1 V 2 ,G). The identity element is the representation class of degree 0. In fact, R*(G) is the graded polynomial ring over Z 2 generated by the set of isomorphism classes of irreducible finite dimen- sional real representations of G. Consider now an action (M n , </>) with finite stationary point set S. For each xeS, we have a real linear representation of G on the tangent space to M" at x. We denote the resulting representation class by X(x)eR n (G). Since x is an isolated stationary point, it is clear that X(x) contains no trivial summand. To (M",</>) we assign the element ^ xeS X(x)eR n (G). This element is zero in R n (G) if and only if each tangential representa- tion class which occurs is present at an even number of stationary points. The correspondence (M w , $) i-^es X(x) induces an algebra homomorphism rj:Z^(G) - R+(G) with image S+(G). Stong [11] showed that for G = (Z 2 )\Z^(G)^S^(G). In other words, (M 19 0J and (M 2 > <p 2 ) are G-cobordant if and only if ^ xeSi X(x) = S ve s 2 X(y)> where ^ 6Sl X(x) and ^5, X(y) correspond to (M l5 ^J and (M 2 , <J> 2 ) respectively. In particular, if xeS JT(x) = for (M rf ,<), then [M d ,<] 2 = in Z d ((Z 2 ) n ). Thus the unoriented cobordism class [M] 2 of a manifold M on which there exists an action of (Z 2 ) M with finite stationary point set S is determined by the tangential (Z 2 )"-modules {T x M:xeS}. To deal with the oriented case of S 1 action on complex Grassmann manifolds, we need an 'oriented' version of representation ring, which is briefly introduced. Let G be a compact connected Lie group. For our purpose G will be the circle group S 1 . Let V be a (finite dimensional) oriented real representation space. If dir% V> 0, then denote by - V the same R G-module but with opposite orientation on it. If V and W are oriented [RG-modules, then V W is the oriented R G-module where G acts diagonally and the orientation is the 'direct sum' orientation. We regard the 0- dimensional vector space as having a unique orientation. Then for any two oriented [RG-modules V and 'W 9 V@W^(-\f mV '* imW (W V) as oriented PG-modules, and if dimV and dimW are positive, (~V)W^V@(-W)^-(V@W) as oriented RG- modules. Note that if dimF is odd, then V^ - V as oriented IRG-modules because 384 Goutam Mukherjee id: V-+ V is an orientation reversing isomorphism. It is now easy to check that for any two oriented RG-modules V and W,V@W^W@V. We now define the graded ring *(G) which is the analogue in the oriented case of R*(G) defined above. For n^ I denote by R n (G) the free abelian group on the isomorphism classes of oriented [RG-modules of (real) dimension n modulo the subgroup generated by elements of the form [K] + [ K]; [V~\ stands for the isomor- phism class of the oriented [RG-rnodule V. R (G) is defined to be the free abelian group on [0], the class of the 0-dimensional RG-module. Let JR*(G) = Z^o^( G ) and define as before [F]-[W] tobe[K W], where K]/F is given the direct sum orientation and diagonal G action. It is straightforward to check that this gives rise to a well-defined multiplication which makes -R^(G) a commutative graded ring with unit [0]. Note that 2x = for all xejR n (G) if n is odd. Let B be the set of all isomorphism classes of irreducible oriented RG-modules, and let B { = {xeB:dimx = i mod2}, i = 0, 1. Then it can be shown that the quotient of the polynomial ring over integers Z in the variable B by the ideal generated by {Ib'.beB^. Now suppose that (M ", 0), n > 1 is a smooth closed oriented G-manifold with a finite stationary point set S. Let xeS, then the tangent space T X M at x to M, which is an oriented vector space, is an RG-module. Since x is an isolated stationary point T X M does not contain any trivial [RG-submodule other than 0. To (M,0) we associate the element fj(M, </>) = ^^[^Mje^fG). For a 0-dimensional manifold X, the only G-action is the trivial one. We define fj(X, trivial) = \X\'[0]eR Q (G). We now state a result, which may be well-known to the experts but an explicit reference is not known and which says that the function ijf behaves well with respect to G-cobordism relation. PROPOSITION 2.1 Suppose (M, (j>) and (M ',</>') are equivariantly cobordant as oriented G-manifolds with finite stationary points. Then fj(M, 0) = rj(M\ 0') in R#(G). The proof of the above result goes along the line of the proof of the corresponding result in unoriented case, (cf. 32 of [3]). Thus by Proposition 2.1, we obtain a well-defined map fj: ^(G) -> jR + (G). It is straightforward to check that the map ?? is a homomorphism of graded rings. Moreover, it can be shown that kernel of fj consists of elements having representatives (M, <j>) where G acts without fixed point on M. In fact, for G = S 1 , kernel of fj is precisely the inverse image of Torsion (MSO^) under the map 2(cf.[10]). 3. Action on Grassmann manifolds Let O(n) denote the orthogonal group of n x n matrices. The subgroup of 0(n) consisting of diagonal matrices can be identified with (Z 2 )". Let e^e 2 ,...,e n be the standard basis of {R M , and 7} be the involution Equivariant cobordism ofGrassmann and flag manifolds 385 Then there exists an action of (Z 2 ) rt on U n given by the pairwise commuting actions of T t s. This action induces an action of (Z 2 )" on G n fe , the real Grassmann manifold of /c-dimensional subspaces in R B , and this action has finite stationary point set. A k-plane X in R n is fixed by this action if and only if Jf = <e lV e ia ,...,e. fc >=: fle where a =: 1 ^ i l < i 2 < < i k ^ n. Thus there are I I stationary points for this action. A Grassmann manifold G nk along with this action of (Z 2 ) n will be denoted by (G njk , 0). In [8], [12], it was proved that G nk bounds if and only if v(n)>v(k) where for a positive integer n, v(n) denotes the integer such that 2 V(W) divides n and 2 V( " )+ * does not divide n. In this section, the Grassmann manifolds (G n fc , 0), which bounds equivariant- ly, that is, [G nfc ,0] 2 =0 in Z k(n ^ k) ((1 2 ) n ) is determined completely. We need the following lemma. Lemma 3.1. Let G be a compact Lie group, X a closed smooth G-manifold. Let t:X-*X be a smooth fixed point free involution on X such that gt(x) = t(gx)for all geG. Then X bounds equivariantly. Moreover, if X is a smooth closed oriented G-manifold and t:X-*X is a smooth fixed point free orientation reversing involution on X such that gt(x) = t(gx)for all geG, then X is an oriented equivariant boundary. Proof. Let W=X x [- 1, 1]/-, where - is given by (x,s)~(t(x), -5). Then W is a compact manifold. An element of W is an equivalence class [x,s], xeX, se[ 1, 1]. Define an action of G on W as follows. For #eG, [x, s]e W, g [x, s] = [#x, s]. Note that (gt(x), -s) = (t(gx), -s) ~ (gx,s). Thus the above definition makes sense. Hence W is a smooth compact G-manifold with boundary and dW=X x { !,!}/ ~ is G-dif- feomorphic to X by the map [x,s]i->x when s = 1. Moreover, if X is oriented, then (x,s)h->(t(x), -5) is an orientation preserving fixed point free involution on X x [ 1, 1] (as t is orientation reversing), hence W becomes an oriented G-manifold. M Proof of Theorem 1.1. (a) Suppose n = 2k. Then if X is a k-plane in R 2 \ X\ the orthogonal complement is also a k-plane in R 2k . Thus Xt-+X L gives a smooth fixed point free involution on G 2kk which is easily seen to commute with each 7}, j = 1, 2, . . . , n. The result follows by Lemma 3.1. (b) Suppose k ^ n/2. Let A be any subset of { 1, 2, . . . , n} consisting of k elements, that is, |A| = k. We shall write elements of A in increasing order. Let e } = {e^ieA}. Thus for each such A there correspond a stationary point of G nk which is the k-plane E- spanned by the vectors in e-. Let y^ k be the canonical k-plane bundle over G n , k . Then the tangent bundle rG nk has the following description [5], i;G n ^y n , k y^ k . Thus the tangent space at any point XeG n fc is X X\ where X L is the orthogonal complement of X in R". Let Ar ; ;.= T E .G nk denote the tangent space at the fixed point corresponding to L Then the standard basis of the tangent space X ; is given by fc(n-fc) vectors [e lrj = e ir ej} r= It2 fc , where i l <i 2 <-~< i k are elements of A and je{l,2, . . . , n} - L Note that the action of (Z 2 ) n on X^ is given by the pairwise commuting actions of the involutions T a , a = 1, 2, . . . , n, thus, Ue irj ) = These give the representation class X(X) of (Z 2 ) w on X>. Let co c { 1, 2, . . . , n} be given by co = { 1, 2, . . . , k}. We claim that the representation class X(co) never occurs at any other 386 Goutam Mukherjee stationary point; in other words, if A ^ co then the representation of (Z 2 ) n at X x is not equivalent to the representation at X^. Suppose, A^O. We can choose aeco such that aA. Now a basis at X^ is given by {e y }, f e{ 1, 2, . . . , fc} and jejfc + 1, . . . , w}, where the span of e t j is a (Z 2 )"-module for any i ^ fc and; > fc. Thus the action of T x on Jf^, has ( l)-eigen space of dimension n fc, whereas the action of T a on X A has ( l)-eigen space of dimension k. If there exists a (Z 2 ) n -isomorphism between X^ and Z A , then we must have fc = n fc, which is impossible as k ^ n/2. Thus X(A) is distinct from X((o), as claimed. Hence the element ^X(X)eS k(n _. k) ((l 2 ) n ) is not zero. It follows from 2 that Remark 3.2 1. The proof of part (b) actually shows that the representation classes X(X) and Z(jti) are distinct if A ^ ju, A, // c (1,2, . . . , n}, as we have not made any use of the special choice co. 2. Note that v(n) > v(fc) is a necessary condition for (G n k , 0) to bound equivariantly. For if [G M)k , 2 ] = in Z k(n _ k) ((Z 2 )) then [G Bt J 2 = in MO k(/J _ k) , hence v(n) > v(fc) by Theorem 1.1 of [8]. Moreover, note that the above theorem produces elements in the kernel of the homomorphism s, for [G n >k , <] 2 belongs to kernel of s whenever v(k) < v(n) and k ^ n/2. 3. In the case (a), that is when n = 2fc, if /,c{l,2,...,n} such that |A| = fc, then A' = {1, 2, ...,} A has cardinality fc. In this case, one can check alternatively, that X(A) = X(A'), so that each representation class X(A) occurs twice. As a result, A X(A) = 0. It follows from Stong's theorem that [G M<k , </>] 2 = 0. Next, we consider certain natural 5 1 action on complex Grassmann manifolds CG n fc . For weS 1 , let <^ w : C" -> C" denote the unitary map defined by < w (z l9 z 2 , . . . , z n ) = (wz l5 w 2 z 2 , . . . , w"z n ). This induces an action of S 1 on the complex Grassmann manifold CG^ of fc- dimensional complex subspaces of C n . Let e 1 ,e 2 ,...,e n denote the standard basis of C". This action of S 1 on CG n tk has finite stationary point set and the stationary points are given by {<X v e ia , . . . ,e ifc >: 1 < i\ < i 2 < < i fc *S n}, where <e^e iz , . . ., e ik ) is the space spanned by {e i , . . . , e ik } (cf. [9], 4). We denote this action by 0. We now prove Theorem 3.3. a) If k or n k is even then (CG n k , 0) does not bound equivariantly. b) // n is even and k is odd then (CG Btfc , </>) bounds equivariantly. Proof, (a) In [7] it was proved that if k or n - k is even then the signature of CG n k is non-zero and [CG n k ] generates an infinite cyclic group of MS0 2k(n _ ky It follows immediately that [CG n k ,0] ^0 in ^(n-*)^ 1 )- Alternatively, one can check that X A X(A)^0 in ^(S 1 ), just as in 1.1, "and get the result, as ^^(S 1 )-^^ 1 ) is a homomorphism. Here, X(A) denote the oriented representation class at Xi-TtoCG^. (b) Let k be odd and first assume that fc = n/2. In this case X\-^X- L gives a smooth involution of CG n fc , without fixed point. This commutes with the given action of S \ as this action preserves innerproduct. We claim that this involution is orientation reversing. To see this, note that # 2 (CG 2kik ;Z) is generated by the first Chern class 6 'i(?2/c,/t) f ^e canonical k-plane bundle over CG 2kk . Let 0:f Equivariant cobordism ofGrassmann and flag manifolds 387 jFf*(CG 2/ck ;Z) denote the isomorphism induced by l:CG 2k<k -CG 2JU . Note that the involution J_ is covered by the bundle map which sends y 2fcfe to y 2kk and hence ^( c i(} J 2fc,/c)) ~ ~~ c i(>'2fc,fc)- Now, c k ^(y 2k ^ k )eH 2kZ (CG 2k ifc ; Z) is a non-zero element, there- fore there exists a unique aeZ {0} such that cf (y 2k >fc ) = a-t/, where u is a generator of # 2fc ~(CG 2kfc ;Z). But as fc is odd and d(c l (y 2k ^ k ))= c l (y 2k ^ k ), we have ( ck i(y2k,k}) ( ~ I) c i 2 (}'2*j;)- This implies 0(w) = u. Hence the involution 1 is orienta- tion reversing. The result follows from Lemma 3.1. Next consider the case k is odd, n is even and k / n/2. Let n = 2m. We regard C" as an m-dimensional right H-space, where H is the division ring of quaternions. If (z ls z 2 ) is a pair of complex numbers then it can be considered as a quaternion z t + z 2 j. Since 7'z = f/ for any complex number z, we have; (2 1 H-z 2 ./) = i 2 -hzj. Write elements of C K as (z l5 z 2 , . . . , z w ) with respect to the basis e x , <? 2m , e 2 , e 2w _ 1? . . . , e m , m + 1 and consider C" as the m-tuple of quaternions H m with basis e l +e 2m j, e 2 + e 2m _ l j,...,e m j re m+1 j. Then we can define a map/ : C 77 -> C" by j (z ls z 2 , . . , z n ) = ( z 2 , z l5 . . . , z n , z w _ x ). Note that 7 is conjugate linear and hence if X is a C-linear subspace of C" then y'PO is again a C-linear subspace of C". Moreover, we have; 2 = id. Thus j induces an involution J on CG Mjt , clearly J is a smooth involution on CG rt k . We claim that J is a fixed point free involution. For suppose, J(X) = X, XeCG nik . Then X is a left H-space and J 2 = id, so dim c X = k must be even, as dim H X = ( 1/2) dim c X\ which is a contradiction as k is odd by our assumption. Next, we claim that the action on CG nk commutes with J. To see this, note that for each e t and weS 1 , w (e f .) = w*e f . Thus if (z 1 ,z 2 ,...,z 2m ) is the coordinate of a point in C n with respect to the basis e^ e 2m , e 2 , . - . , e m , e m+l , then Hence the induced maps J and cf) w on CG W k commutes with each other for each Next, we show that the involution J is orientation reversing. Since CG Htk is path- connected, it is enough to check it at one point. Note that the orientation of CG n fc as a real manifold is given by the orientation of each /c-plane in C" considered as an oriented real vector subspace of (R 2w , with the standard orientation on IR 2 ". The oriented real basis of O i , e iy? . . . , e ik > is {e tl , . . . , e ik , >/-! *,v > V^ ^J> where l^h<h<~' <i k ^n. Moreover note that j(e r ) = e n+1 _ r ; hence Je ij ,...,ej fc = <^ M+ i _ IV . . . , e n+ 1 _ fte >. From this one can show that J is orientation reversing, as k is odd. The result now follows again by Lemma 3.1. It is proved in [9] that CG M k is an oriented boundary if n is even and k is odd. Thus the above action does not give any non-trivial element in the kernel of e. However, we can perturb the above action of S 1 on CG n k in a suitable way to generate infinitely many elements in the kernel of I Before we do that, let us have a close look at the representation class of (CG^,^) in the case n is even and fc is odd. Since rjf is a homomorphism, it is clear from the above theorem that ^([CG n>k ,<^]) = 0. Let us establish this, alternatively, by analysing the tangential representations at stationary points. This description will be useful in proving the next theorem. The tangent bundle of CG n<k has the following description [5]: TCG^y^^, where y w<k is the canonical fc-plane bundle, y^ k its orthogonal complement and y Hffc = Hom c (y lltfc ,C) is its conjugate. Let -I = {r l9 r 29 .,.,rjc{l,2,... 9 w} and A be the stationary point corresponding to L Let X> be the tangent space at A . Then X ; = < ri , . . . , e rk ) <e/j ^ r 1? r 2 ,--.,r k >, where {e ri ,.'..,e rk } is a basis of Hom c ( ; jC). Note that for each 1 , (/> w (8j) = w j j and the induced action on e j is (j>^(e^ = w"- 7 '^. Note that a natural 388 Goutam Mukherjee complex basis of X x is given by e { e p zeA, ;<A, written in dictionary ordering with respect to the subscripts. In fact, {e i e j9 ieA,jfA}, forms a basis of eigen vectors for <j) w :X x -*X^\veS 1 . Clearly, the complex representation of S 1 at X^ is the sum of 1 -dimensional irreducible complex representations of S 1 with corresponding eigen values w j '~ r . Note that since n is even and k is odd, the number of stationary points is even, moreover if A = {r 15 r 2 , . . . , r k }, then A' = {n + 1 r l9 . . . , n 4- 1 r k } is distinct from L It is now easy to check that, the assignment extends to a conjugate linear isomorphism between X^ and X^ which preserves the group action. Since dim c X A is odd, it follows that there is an orientation reversing S^equi variant isomorphism X^^X X .. Consequently, according to our definition of RJG), [X J + [X^ = 0. Since 1 c { 1, . . . , n} is arbitrary, it follows that fj(CG n ^ 0) = 0. Next, we consider a different action of S 1 as follows. We choose distinct integers v t , v 2 , . . . , v n such that |v Vj| ^ |v k v z | for any z ^ j, fe ^ / and {z,j} ^ {/c, /}. For each 1 define \l/ w :C"->C a by ^ W (z l9 z 2 , . . . , zj = (w vi z l5 w V2 z 2 , . . . , w v "z n ). As before, this induces an action ^ of S 1 on CG n fc . We claim that this action of S 1 has finite number of stationary points of CG n k . Since ^ w (e ) = w v ''e ? it is clear that for any A = {r l5 r 2 , . . . , r k } c: { 1, 2, . . . , n}, E x = <e ri . . . , e rk > is a stationary point. We shall show that these are the only stationary points of this action. Let X be a fe-dimensional subspace of C" such that i/^PO = X for all weS 1 . Let {i? ls 2 , . . . , t? k } be a basis for X. Write each ^ as a linear combination a iei . of the canonical basis vectors. Let A = {z l9 z' 29 . . . , zj c {1, 2, . . . , n} be such that e fr , i r eA, appears in the representation of Vj as above for at least one;. Clearly, I = |A| ^ fe. If we show that e ir belongs to X for each z r eA, then it will follow that / = k and X = <g. t , . . . , e ifc >. So let t? = v t and t; = r= x a r e ir , we may assume without any loss of generality that a r ^ for each r = 1, 2, ...,/. Since ^ W PO = X for all weS 1 and eX, ilr Vf (v) = 2! r , l a r w vt 'e ir eX. We may choose w l9 w 2? . . . , v^eS 1 such that det P^^ 0, where W is the / x / matrix, W= (w v s ir ). In fact, detVT= Vandermonde determinant x a certain Schur function and we can choose w ls w 2 , . . . , w h algebraically independent over Q (the field of rationals) so that the Schur function is never zero, (cf. [6]). Set u j = 1/^(1;) = U l a r w] if e ir X, 1 < j ^ /. Then we have an Ixl matrix (a rs ) = (a r wj fr ). Clearly, det(a rs ) = a 1 -fl 2 r ---a / detP^^O. Since det (a rs ) 7^0, it is now straightforward to check that for each i,eA, there exist jS l5 /? 2 - - > J?i 5 not all zero, such that e ir = ^! jssl fift. Thus e ir eX. Therefore, the action \// on CG n fc has finite stationary point set. We now prove with \j/ as above, Theorem 3A [CG^, ft = in & m - k} (S 1 ) ifn=2k and k is odd and [CG n k , | ^ m Proof. If fc orn - fe is even or if n = 2/c and fc is odd, then the proof is same as the corresponding cases of 3.3. So we assume that n is even, k is odd and k ^ n/2. Since fj: ^(S 1 ) -> R^S 1 ) is a homomorphism, it is enough to prove that rj(CG ntk , \l/) ^ 0. Let Ac{l,2,...,n} and A be the corresponding stationary point. Then from the discussion following Theorem 3.3, it is clear that the complex representation at X A is decomposed into irreducible 1-dimensional complex representations characterized by the corresponding eigen values w vj ~ v \ /eAj^A. If A'c{l,2,...,n} is distinct from A then we can always Equivariant cobordism ofGrassmann and flag manifolds 389 choose per, #* such that for every i6 A,#A, {p,}*{ij}. By our choice Iv r vj^-vj. Therefore, unlike the previous action/ there "does not e4t an S -equivanant orientation reversing isomorphism between the RS l -modules X and A A , As a result, there will be no cancellation. Hence fj(CG n k , i '" Remark 3.5. By Theorem 3.1(ii) of [9], [CG. J = if n is even and k is odd. Therefore Theorem 3.4 implies that [CG Hjk> <A] belongs to kernel of a whenever n is even k is odd and k ^ n/2. It is also interesting to note that in the case when n is even, k is odd and k * n/2, we can choose integers v l5 v 2 , . . . , v n , v ;, v' 2 , . . . , v' H , in an arbitrary way, satisfying the mentioned condition so that [CG^,^[CG rtJt ,f ], where f is same as ^ replacing v- by vj. Thus we have infinitely many nontrivial elements in the kernel of For any n ^ 3 we can choose a sequence {v r } of finite sequences v r = (v r 15 v;, . . . , v r j of length n so as to satisfy |vj - vj| * |v* - v|, for i *j 9 p ^4 and {ij} * {p, 9} and for any r and s (including the case r = s) and |vj - vj| ^ |v? - vj| for any r, s, r ^ s, and ij, f *j. For instance, choose natural numbers p l9 p 2 , . . . , p r , . . . , Pl > 1, Pr > p_ 1 f or r ^ 2, and set v r = (p r , p r 2 , . . . , p;}. Now for fc < n, let \jj r denote the action of S 1 on CG nJt defined by v r as above. We exclude the case when n = 2k and fc is odd. Then for any such choice of {v r }, [CG W fc , iA r ] ^ [CG ntfc , A S ] for r ^ s, as mentioned in the above remark and more- over, any finite number of these classes [CG n k , ^J, r ^ 1, are linearly independent over Z. This can be seen easily by applying the homoinorphism rj and comparing the monomials in Z [B ] (cf. 2 and note that all irreducible real representations of S 1 are 2-dimensional). In particular, we can take k = 1 and d > 1 and consider CG rf+ K1 so that dim R C G d + ! a = 2d. This yields, Theorem 3.6. For any d > 1, /Ymfc J 2r 2d (S 1 ) fs not finite. 4. Action on flag manifolds Let G(n 1? n 2 , . . . , n s ), n = n l +n 2 -\ ----- h n s , 5 ^ 3 denote the real flag manifold of all flags (y4 l5 A 2 , . . . , yl s ) where yl is a left vector subspace of R n , A t JLA p for z ^7, dim R A { = w , 1 < f, 7* < s, G(n l5 2 , . . ., s ) is a smooth manifold of dimension , <I<J<S f ^ Alternat- ively, it can be described as the homogeneous space OW/Ofo) x - x 0(n $ ). The group (Z 2 ) n acts on G(n l9 n 29 ... 9 n s ) by pairwise commuting involutions T a , a = l,2,...,n, having finite stationary point set. This action is induced from the actions of T a s on ST as described in the last section. The number of stationary points of the action of (Z 2 f on l9 n 2 , . . . , n s ) is nl/nj - * n s l We denote this action by (G(n l9 2 , . . . , nj, ^)- Proof of Theorem 1 .2. Suppose n f = w ; for some i +j. In this case there exists an obvious smooth fixed point free involution which interchanges the i th and thejth component of each flag in G(n l5 n 2 , . . . , n s ) which is easily seen to commute with each T a . Hence by Lemma 3.1 [G(n 1 ,n 2 ,...,n 5 ),0] 2 =0. Next suppose that n t * n j for i ^j. We may without loss of generality always wnte n,. in increasing order. Let A = (l 1 ,A 2 ,...,x s ) be a partition of {1,2... .,n}, where the subset A' has cardinality n { . We shall write elements of A' in increasing order. (i) 390 Goutam Mukherjee Let e^ = {e k :ke)J}. Then the fixed points of G(n l9 n 2 , . . . , n s ) are {((^i), < g AX > <;.* : for a11 partition A = (/.*, A 2 , . . . , /1 s ) as stated in (1)}, where < A ,> = Ai (say) is the space spanned by ;i . Thus for each X as stated in (1) there exists a fixed point of (G(n 1 ,n 2 ,...,n s ),<^>) and as before, we shall denote by X^ the tangent space to G(n t , rc 2 , . . . n s ) at the stationary point corresponding to A. Then by [5] A basis of this is given by {e^, 1 < i <] ^ 5}, where e^ = {e k e t : keA\ lel j }. The representation of (Z 2 )" on X^ is given by its action on the basis element: *-*<>" (3) otherwise Let us now consider the partition co = (co 1 , co 2 , . . . ,-co s ), where co s = {H! + n 2 -\ ----- \-n s _ l + l,...n 1 + h n s }. Then rJEa> = e Uf< ^ s i^ (4) We claim that if A 7^ co then X(A) is distinct from X(co), where X(A) is the representation class of (Z 2 ) n at X(X). To see this, suppose A ^ co. Then co ( ^ A 1 ' for some i. Choose aeco 1 such that a^ A . Let ae A J , i ^7. Then from (3) and (4) it follows that the action of T a on X^ has ( l)-eigen space of dimension n 1 H ----- \-n i _ l + n i+l -{ ----- h n s = n n { , whereas from (2) and (3) it follows that the action of 7^ on X x has ( l)-eigen space of dimension n l ~\ ----- \-n j _ l + n j + 1 -\ ----- h n s = n rij. If there exist an equivariant linear isomor- phism X x X m9 then we must have n n^n- n p that is n t = rij for i *j 9 which is impossible. Thus the representation class X(co) does not occur at any other stationary point In other words j^A)eS d ((Z 2 ) n ) is non-zero, where d = Y J i^ i<j ^ s n i n j * s ^ dimension of G(n l5 . . . , n s ). Hence [G(n l5 . . . , n s \ 0] 2 ^ 0. This completes the proof. Remark 4.1. (a) In [9] it was proved (Theorem 2.2(a)) that [G(n x , . . . , n s )] 2 = if n = nj for some i ^ j, 1 ^ i, j ^ s, or for some v(n t ) < v(n\ where v(n) is as in 3. Thus Theorem 1.2 implies that [G(n l5 n 2 , . . . , n s ), c/>] 2 is a nontrivial element of kernel of s if n t ^ n ; for i 7^7 and v(n f ) < v(n) for some L (b) To get a complete answer to the question 'Which flag manifolds bound?' it would be enough to determine whether [G(n l5 . . . , n s \ c/>] 2 belongs to kernel of g or not, in the case when n is odd and n f s are distinct. Acknowledgement The author would like to thank Dr P Sankaran for his help and many useful iiscussions. References [1] Atiyah M F and Singer I M, The index of elliptic operator-Ill. Ann. Math. 87 (1968) 546-604 [2] Bott R E, Vector fields and characteristic number. Mich. Math. J. 14 (1967) 231-244 Equivariant cobordism ofGrassmann and flag manifolds 391 [3] Conner, P E, Differentiable periodic maps. Lee. Notes in Math., 738 (Springer- Verlag) (1979) [4] Kosniowski C and Stong R E, (Z 2 )*-Actions and characteristic numbers. Indiana Univ. Math. J. 28 (1979) 725-743 [5] Lam K Y, A formula for the tangent bundle of flag manifolds and related manifolds. Trans. Am. Math. Soc. 213 (1975) 305-314 [6] Macdonald I G, Symmetric functions and Hall polynomials. (Oxford Mathematical Monographs) (1979) [7] Mong S, The index of complex and quaternionic Grassmannians via Lefschetz formula. Adv. math. 15 (1975) 169-174 [8] Sankaran P, Determination ofGrassmann manifolds which are boundaries. Bull Can. Math. 34 (1991) 119-122 [9] Sankaran P and Varadarajan K, Group actions on flag manifolds and cobordism. Can. J. Math, 45 (1993) 650-661 [10] Stong R E, Stationary point free group actions. Proc. Am. Math. Soc. 18 (1967) 1089-1092 [1 1] Stong R E, Equivariant bordism and (Z 2 ) k actions. Duke. Math. J. 37 (1972) 779-785 [12] Stong, R E, Math. Reviews, 89d, 57050 " Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 4, November 1995, pp. 393-397. Printed in India Local behaviour of the first derivative of a deficient cubic spline interpolator SURENDRA SINGH RANA Department of Mathematics and Computer Science, R.D. University, Jabalpur 482 001, India MS received 30 August 1994 Abstract. Considering a given function /eC 4 and its unique deficient cubic spline interpo- lant, which match the given function and its derivative at mid point between the successive mesh point, we have obtained in the present paper asymptotically precise estimate for s' /'. Keywords. Local behaviour; deficient cubic spline; mid point interpolation; precise estimate. 1. Introduction Let P:0 = x < jjq , < - < x n = 1 denote a partition of [0, 1] with equidistant mesh points so that h = x t x _ 1 = l/n. Let fl m be the set of all real algebraic polynomials of degree not greater than m. For a function s defined over [0, 1] we denote the restriction of s over [x^^xj by s t . The class of periodic deficient cubic splines over [0, 1] with mesh P is defined by S(3, P) = {s: 5 ,en 3 , seC 1 [0, 1], s^O) = s (J \l\ j = 0, 1}. Considering a nondecreasing function g on [0,1] such that g(x + h) - g(x) = tf (const) = JJ dg, xe[0, 1 - K], Rana and Purohit [4] have proved the following for deficient cubic splines: Theorem 1. LetfeC 1 [0, 1]. Then there exists a unique 1-periodic spline seS(3, P) "which satisfies the following interpolatory conditions, 0, i=l,2,...,n, (1.1) i + *i-i)A i=l,2,..., n. ' (1.2) It is interesting to observe that condition (1.1) reduces to different interpolatory conditions by suitable choice of g(x). Thus, if g is a step function with a single jump of one at h/2 then condition (1.1) reduces to the interpolatory condition, Considering a function /eC 4 and its unique spline interpolant s matching at the mesh points, Rosenblatt [5] has obtained asymptotically precise estimate for s' /'. For further results concerning asymptotically precise estimate for cubic spline interpo- lant reference may be made to Dikshit and Rana [3]. Similar to the result of Rosenblatt [5], we obtain in the present paper a precise estimate for s r f concerning the deficient cubic spline interpolating the given function and its derivative at mid points between the successive mesh points. It may be worthwhile to mention that Boneva, Kendall and 393 394 Surendra Singh Rana Stefanov [2] have shown the use of derivative of a cubic spline interpolator for smoothing of histograms. Without any loss of generality, we consider for the rest of this paper that the deficient cubic spline s under consideration satisfies the condition s'(0) = 0. Thus, we have from the proof of Theorem 1 that the system of equations for determining the first derivative m . = s '(x.) of the deficient cubic spline interpolant s is written as, m I .. 1 )/2 = F i , i= l,2,...,n- 1 (1.4) where F i =12{/ 2. Estimation of the inverse of the coefficient matrix Ahlberg, Nilson and Walsh [1] have estimated precisely the inverse of the coefficient matrix appearing in the studies concerning cubic spline interpolant matching at the mesh points. Following Ahlberg et al we propose to obtain here a precise estimate for the inverse of the coefficient matrix in (1.4). It may be mentioned that this method permits the immediate application to the spline to standard problem of numerical analysis (see [1], p. 34). For this we introduce the following square matrix of order n. 2b a 0-00 a 2b a a 2b - 000 000 a 2b a a 2b where a and b are given real numbers such that b 2 > a 2 . By using the induction hypothesis it may be seen easily that |DJ satisfies the following difference equation, with | \D.(a,b)\=2b\D lt _ l (a,b)\-a 2 \D ll _ 2 (a,b)\ (a,b)| = 0, |D (a, b)\ = 1 and II^M)! = 2b and for a = (b 2 - a 2 ) 1 ' 2 , 2ot\D n (a,b)\=(b + x)" +i -(b-a)" +l , b 2 >a 2 \D a (a,b)\=(n+l)b", otherwise. Further, it may be observed that the system of eq. (1.4) may be written as (2.1) (2-2) -F (2.3) where the coefficient matrix A is a square matrix of order n 1, M and F are the transposes of the matrices [m 1 ,m 2 ,...,m n _ 1 ] and \_F l ,F 29 ...,F n . 1 '] respectively. In order to determine the inverse of the coefficient matrix A we first observe that for a = - 1/2, where - r = + r)\D n (a,b)\ = 2b(l ~r 2 ") l = 2[b - (b 2 - 1/4) 1 / 2 ]. -r 2 "~ 2 )/2 (2.4) Local behaviour of the first derivative. 395 Taking 2b = 5 and a = 1/2 in \D n (a,b)\, we observe that the coefficient matrix A satisfies the following difference equation, n _ 2 (- 1/2,5/2)1 -\D n _ 3 (- 1/2, 5/2) |, (2.5) Thus, using (2.4) in (2.5) we have (5 -f r)p 2 -"\A\ = (5 + r/2) 2 - r 2 "~ 6 (5r + 1/2) 2 . (2.6) We get the elements a itj of ^4~ 1 from the cofactors of the transpose matrix. Thus, for < i^j ^ n - 2 or z =} = (cf. [1, pp. 35-38]) | ^K J = (/^r </);(- l/2,5/2)ZV 7 ._ 2 (- 1/2,5/2) (2.7) and \A\a QJ =(p-ryD n -j- 2 (- 1/2,5/2) forO<^n-2. (2.8) Thus, in view of (2.4) and (2.5), we have for < i ^y < n 2 (5 + r/2)(l - r 2 ")a oj = r j (l - r 2n ' 2 ~ 2j ), for <j < n - 2, and From the above expression, we observe that A ~ 1 is symmetric. Now considering a fixed value x such that < x < 1, we see that for fixed > and e < i/n, ;/n < 1 s the elements a fj of A~ l may be approximated asymptotically by r {j ~ il /(5 + r). We thus complete the proof of the following: Theorem 2. The coefficient matrix A of (2.3) is invertible and if A ~ 1 = (a t j), then a tj can just be approximated asymptotically by r lj ~~ il /(5 + r) and the row max norm of its inverse; that is, where r = 2 x /6 5. Remark 1. It is worthwhile to mention that the estimate (2.9) is sharper than that obtained in terms of the infimum of the excess of the positive value of the leading diagonal element over the sum of the positive values of other elements in each row. For adopting the latter approach, we observe from (2.3) that \\A~ 1 1| < 0.25 whereas (2.9) shows that the \\A~ l \\ does not exceed 1/6. Since A is invertible, it follows from the proof of Theorem 1 or more precisely (2.3), that there exists a unique spline seS(3, P) satisfying the interpolatory conditions (1.2) and (1.3). 3. Error bounds Considering a 1-periodic function /eC 4 in this section of the paper we shall estimate the precise bounds of the function e' = s' - /' where s is the deficient cubic spline 396 Surendra Singh Rana interpolant of a 1-periodic function/ which satisfies the interpolatory conditions (1.2), (1.3). Considering the interval [x _ 15 x ], we see that, since sf is quadratic, hence in the interval [x ls x ], we may write h 2 s'(x) = h(x x t _ Jw; + h(Xi - x)m._ 1 4- (x x^_ 1 )(x x)c (3.1) where the constant c t is to be determined. Using the interpolatory condition (1.2), we notice that, ) + c i . (3.2) Now applying (3.2) in (3.1), we get (3-3) Thus, replacing now m by e'(x t ) in (3.3), we have fcV(x) = (x - x _ J[h - 2(x. - x)M*<) + (x i -x)lh-2(x-x i . l W(x i _ l ) + R l (f) (3.4) where ^-(/) = (x-x^ 1 )lh-2(x i -x)lf'(x i ) + (x i -x)[h-2(x-x t _ J Now using the fact that /eC 4 , we see by Taylor's theorem that ,.(/) may be expressed as a linear combination of the values of the fourth derivative/ (4) of/. Thus, Kf(/) = fe 2 /' W + / (4) (x)(x - x _ x )(x f - x)(2x - x - x _ x ) (3.5) where x is an appropriate point in (x t _ 19 x ) which is not necessarily the same at each occurrence. Rewriting (2.3) as, A(e>( Xi )) = (F f ) - A(/'(x )) = (H X (3.6) say, we first estimate (H ,-). Thus, applying Taylor's theorem again to the right hand side of (3.6), we get (3.7) Recalling eq. (3.6) and noticing that A~ * = (a j; ), we have say, where m is a sufficiently large but fixed positive integer. We shall estimate R^ and jR 2 separately. Suppose that x is a fixed point in (0, 1) and let x = [nx]/n where [nx] denotes the largest integer less than or equal to nx. Then it is clear that as n -* oo, i ^ nx and n - i s n(l - x). Now assuming that/ (4) is monotonic, we get from Theorem 2 (3.8) where d l is some positive constant. Next, we see that the points x k for the values of fc occurring in R 2 satisfy (3.9) Local behaviour of the first derivative - 397 Thus, using the continuity of / (4) and applying the result of Theorem 2 alongwith (3.7), we have (#2)- Z 71 :(- fc3 / <4) (*)/6) =0(/i 3 ). (3.10) |lc-i|<m( 5 + r ) Combining the estimates of (R l ) and (R 2 ) and noticing that m is arbitrary, we prove the following: Theorem 3. Let s eS(3, P) be the deficient cubic spline interpolant of a I -periodic function f satisfying the interpolatory conditions (1. 2) and (1.3). Let / <4) exist and be a nonnegative monotonic continuous function. Then for any fixed point x such that < x < 1, (3.11) as n->oo. Remark 2. It may be interesting to investigate the similar precise estimate for deficient cubic spline in the case of nonuniform mesh. References [1] Ahlberg J H, Nilson E N and Walsh J L, The theory of splines and their application, (New York: Academic Press) (1967) [2] Boneva L I, Kendall D G and Stefanov I, Spline transformation, three new diagnostic aids for the statistical data analyst. J. R. Stat. Soc., B33 (1971) 1-70 [3] Dikshit H P and Rana S S, Local behaviour of the derivative of a mid point cubic spline interpolator, Int. J. Math. Math. Sd., 10(1987) 63-67 [4] Rana S S and Purohit M, Deficient cubic spline interpolation, Proc. Jpn. Acad. 64 (1988) 111-1 14 [5] Rosenblatt M, The local behaviour of the derivative of cubic spline interpolator, J. Approx. Theory, 15 (1975) 382-387 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 4, November 1995, pp. 399-404. Printed in India. On the partial sums, Cesaro and de la Vallee Poussin means of convex and starlike functions of order 1/2 RAM SINGH and SUKHJIT SINGH Department of Mathematics, Punjabi University, Patiala 147002, India MS received 3 October 1994; revised 15 May 1995 Abstract. In this paper we study certain properties of partial sums, cesaro and de la vallee Poussin means of convex and starlike functions. Keywords. Partial sums; Cesaro; de la Vallee Poussin means. 1. Introduction Let S denote the class of functions f(z) = z + a 2 z 2 H which are regular and univalent in the unit disc E = {z/ 1 z | < 1 } . Denote by S t and K the usual subclasses of S consisting of functions which map E onto starlike (with respect to origin) and convex domains, respectively. Let S t (l/2) c S t be the class of functions which are starlike of order 1/2. It is known that KcS t ( 1/2). For a given function /(z) = z + Z*= i a n z n and neN, let s rt (z,/) = z-ha 2 z 2 4- n , n(n-l) n(n-l)(n-2)...3-2-l ,. n \ 7 T \ ~~ 7 i ___________________ ft "7*" _|_ . . . _l_ ___________________-_-__-- n 7 v nV">J f , 1 ^ ' / , iw_, , ">\ W 2^ ~ ' /, i 1\/,- i ''U /O,^ w ' ff .--f) = Z | (B ~ 1) a z 2 + ( ^-^i n n and Y2W_ ^ _ , W(W-1). _ 2 , (H-i; denote, respectively, the nth partial sum, the nth de la Vallee Poussin mean, the nth Cesaro mean of first order and the nth Cesaro mean of second order of /. A function / is said to be subordinate to a function F (in symbols /(z)-<JF(z)) in \z\ < r if F is univalent in |z| < r, /(O) = F(0) and f(\z\ <r)c F(\z\ < r). For every feK the following results are well-known: (i) z/2 = s 1 (z,/)/2 = (7 ( 1 1) (z,/)/2 = 4 2 >(z,/)/2x;/(z)in [2]; (ii) (4/9)s 2 (z,/K/(z)inE[10]; (iv) v m (z,f)<f(z)mE. The fascinating result (iv) is due to Polya and Schoenberg [6] (see also Robertson [7]). In the present paper, we establish the analogue of the Polya-Schoenberg theorem for a certain transformation of the nth partial sum, s n (z, /), and the nth Cesaro mean of first order, <r(z 9 f) 9 of feK. We also prove that for every feS t (l/2) and for every positive 399 400 jRam Singh and Sukhjit Singh integer n, Re(i; n (z,/)/crJ J 2) (z,/))>0, zeE. An alternative and simple proof of a well- known result of Basgoze, Frank and Keogh [1] pertaining to subordination of the partial sums of convex functions is also given. 2. Preliminaries We shall need the following definitions and results. DEFINITION 2.1 A sequence {& M }f of complex numbers is said to be a subordinating factor sequence if, whenever /(z) = z -f ^= 2 a n z n is regular, univalent and convex in , we have OC a B b n z"</(z), (a x = l) n=l in E. DEFINITION 2.2 A sequence {c n }$ of non-negative numbers is said to be a convex null sequence if c n -* as n - oo and Lemma 2.1. (Wilf [1 1]). A sequence {b n }* of complex numbers is a subordinating factor sequence if and only if Re[l + 2* = l \> n f\ > 0, zeE. Lemma 22. Lemma 2.2 is due to Rogosinski and Szego [8]. Lemma 2.3. (Fejer [4]). Let {c n }% be a convex null sequence. Then the function ^ n-1 is analytic in E and Re^(z) > 0, zeE Lemma 2.4. Let TheiiReg ll (z)>Om. In view of the minimum principle for harmonic functions, we have z)= min Partial sums, Cesdro and . . . = min Re = mn - sin(n + , 2sin 2 (0/2) +I 4sin 2 (0/2) - (p3 ' Lemma 2.5. Let f and g be starlike of order 1/2. Then for each function F analytic in E and satisfying ReF(z)>0 (zeE), we have /(z)*F(z)g(z) ^ e 771 TT~" >0 ( zejE )- f(z)*g(z) Lemma 2.5 is due to Ruscheweyh and Sheil-Small [9]. 3. Theorems and their proofs Theorem 3.1. Let feK and let s n (z,f\ neN, denote its nth partial sum. Then S n (z,/)=4(* Z s.(t,f)dt<f(z) ZJ in Efor every n = 1, 2, 3 ____ Proof. Let f(z) = z + ^2 fl ^ b ^ in K. Then In view of the Definition 2.1, the desired conclusion will follow if and only if the sequence < 1/2, 1/3, . . . , l/(n -h 1), 0, . . . > is a subordinating factor sequence. By Lemma 2.1, this will be the case if and only if , zeE. (3.1) k=1 Putting z = re iB , 0<r<l, n^0^n and making use of the minimum principle for harmonic functions along with Lemma 2.2, we have showing that the inequality (3.1) holds and, therefore, the proof of our theorem is complete. Taking n = 1, we obtain the following well-known result (also cited in the Introduction). COROLLARY 3.1 (l/2)z</(z)m, for all feK. 402 Ram Singh and Sukhjit Singh Theorem 3.2. For all elements f of K and for all positive integers n, we have in E. This result is sharp for every n. Proof. Let f(z) = z + ^ = 2 a n z n be any element of K. Since n , 1W ^ n rc 1 , n 2 , in the light of Definition 2.1, the assertion (w/(n +1)) a(z 9 /) < /(z) in E will hold if and only if the sequence < n/(n -f 1), (n - 1 )/(n + 1), . . . , l/(n + 1), 0, 0, . . . > is a subordinating factor sequence. By Lemma 2.1, we see that this is equivalent to Re or Re 1+ -(nz + (n-l)z 2 +(tt-2)z 3 + +") >0, L n + 1 J which is true in view of Lemma 2.4. To establish the claim regarding sharpness we consider the function h(z) = z/(l z) which is a member of K. For any positive real number p., we have (n+l) sin 2 [(n+l)g/2] .(n+ I)sin0-sin(n+ 1)0 2~ 2 sin 2 (9/2) 4sin 2 (0/2) Now let 9 = = 27i/(n -f 1). Then v > *v ^ 2 n Now, if p > n/(n -h 1), then it follows that Re po^ } (z 9 h)<- 1/2 and hence (since h maps E onto the right half plane Rew> - 1/2) we conclude that pa ( n 1} (z,h) will not be subordinate to h in E. Taking n = 2, we obtain the following result of Singh and Singh [10]. COROLLARY 3.2 (2/3)cr ( 2 1) (z > /) X /(z) in E, /or every feK. In the next theorem we present a simple and interesting proof of a well-known result which was established by Basgoze, Frank and Keogh [1] in 1970. Theorem 33. Let f(z) = z + ^ =2 a n z n eK and let s n (z,f) denote its nth partial sum. Then s n (z/2,/)</(z) in Efor every n = 1, 2, 3, .... The constant 1/2 cannot be replaced by a larger one. Partial sums, Cesdro and . . . 403 Proof. Since s n (z/2 ? /) = (l/2)z + (l/2 2 )a 2 z 2 + (!/2 3 )a 3 z 3 + + (l/2 n )a n z\ the con- clusion s fl (z/2, /) -< /(z) in E will follow if and only if the sequence < 1/2, 1/2 2 , . . . , 1/2", 0, 0, . . .> is a subordinating factor sequence. In view of Lemma 2.1, this will be the case if and only if " (3.2) It is readily seen that the sequence {c k }$ defined by c = 1, c k = 1/2*, k = 1, 2, 3, . . . , 77 and c fc = if/c = n + l,w-h2,..., is a convex null sequence. Thus using Lemma 2.3 we get which in turn shows that the inequality (3.2) holds. The function h(z) = zj(l - z)eK, which maps E onto the half plane Re w > - 1/2, shows that the constant 1/2 cannot be replaced by any larger number. This completes the proof of our theorem. Egervary [3] has shown that [(n + l)nz + n(n - l)z 2 + (n - l)(n ~ 2)z 3 4- - - 4- 2- 1 is a member of S t (l/2). Using this fact and the well-known result of Ruscheweyh and Sheil-Small (Theorem 3.1, [9]) we conclude that for every /eS t (l/2) is a member of S t ( 1/2). Theorem 3.4. Letf(z) = z -f ^ =2 a n z n be any member ofS t (l/2). Then for every positive integer w, we have . Consider the function F n defined by .2, - ,3 2)(n + 3) (n + 2)(n + 3)(n + 4) "' v* '__ ' 4 i n 2 (n-l),...,3 1 '" + (n+l)(n + 2),...,(2n) J' (3.3) Obviously F n is regular in E (in fact it is an entire function), and we can write it in the form r, / x n n / n-f 404 Ram Singh and Sukhjit Singh .""(^ n* /. n-l\ 3 i 1 i + 2)(w + 3) I ~w + 4/ n 2 (n-l)(n~2),...,4 i i 2n -z". In view of (3.3) and (3.4) it is now easy to see that in E we have . ReF n (z)^F n (|z|)>F(l) = 0. In Lemma 2.5 taking /(z) = 0(z,f), g(z) = z/(l - z) and F(z) = F n (z) we get This completes the proof. References [1] Basgoze T, Frank J L and Keogh F R, On convex univalent functions, Can. J. Math. 22 (1970) 123-127 [2] Peter L Duren, Univalent functions, (New York: Sp ringer- Verlag) (1983) [3] Egervary E, Abbildungseigenschaften der arithmetischen Mittel der geometrischen Reihe, Math. Z. 42 (1987)221-230 [4] Fejer L, Uber die positivitat Von summen, die nach trigonometrischen order Legendreschen funktionen fortschreiten, Acta Litt. Ac. Sci. Szeged (1925) 75-86 [5] Goodman A W, Univalent functions, (Mariner Publishing Company) Vol. I [6] Polya G and Schoenberg I S, Remarks on de la Vallee Poussin means and convex conformal maps of the circle, Pac. J. Math. 8 (1958) 295-334 [7] Robertson M S, Applications of the subordination principle to univalent functions, Pac. J. Math. 11 (1961)315-324 [8] Rogosinski W and Szego G, Uber die Abschimlte Von potenzreihen die inernein Kreise be Schrankt bleiben, Math. Z. 28 (1928) 73-94 [9] Ruscheweyh St and Sheil-Small T, Hadamard products of Schlicht functions and the Polya-Schoen- berg conjecture, Comment. Math. Helv. 48 (1973) 1 19-135 [10] Singh S and Singh R, Subordination by univalent functions, Proc. Am. Math. Soc. 82 (1981) 39-47 [1 1] Wilf H S, Subordinating factor sequence for convex maps of the unit circle, Proc. Am. Math. Soc. 12 (1961)689-693 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 4, November 1995, pp. 405-409. Printed in India. Uniqueness of the uniform norm and adjoining identity in Banach algebras S J BHATT and H V DEDANIA* Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar 388 120, India * Department of Mathematics, University of Leeds, Leeds LS2 9JT, UK MS received 14 October 1994 Abstract. Let A e be the algebra obtained by adjoining identity to a non-unital Banach algebra (A, \\ ||). Unlike the case for a C*-norm on a Banach *-algebra, A e admits exactly one uniform norm (not necessarily complete) if so does A. This is used to show that the spectral extension property carries over from A to A e . Norms on A e that extend the given complete norm || || on A are investigated. The operator seminorm |j j| op on A e defined by ]| || is a norm (resp. a complete norm) iff A has trivial left annihilator (resp. || || op restricted to A is equivalent to I! -||). Keywords. Adjoining identity to a Banach algebra; unique uniform norm property; spectral extension property; regular norm; weakly regular Banach algebra. 1. Introduction Let A e = A 4- Cl be the algebra obtained by adjoining identity to a non-unital Banach algebra (A, \\ ||) [8]. There are two natural problems associated with this elementary unitification construction: (1) which are (all) algebra norms |-| on A e that are closely related with (e.g. extending) ||-|| on Al (2) Which properties of the Banach algebra (A 9 1| ||) are shared by the normed algebra (A & \ |)? In the present paper, it is shown that A has unique uniform norm (not necessarily complete) (resp. spectral extension property [9]) iff A e has the same. This is interesting in view of the fact that for a Banach * -algebra (A, || ||) with a unique C*-norm, A e can admit more than one C*-norm [1, Example 4.4, p. 850]. This holds in spite of apparent similarity between the defining properties || x 2 1| = ||x|| 2 and ||x*x|| = ||x|| 2 of uniform norms and C*-norms respective- ly. This main result, together with a couple of corollaries, is formulated and proved in 3. Their proofs require some properties of norms on A that are regular [5]. There are twostandardconstructsofnormson/4 e ,viz.the/ 1 -norwi||x + Al|| 1 = ||x|| +m and the operator norm \\x + M || op = sup{||xj;-f- Ay||:||y|| < 1, yeA}. In general, Hop need neither be a norm nor be complete [6, Example 4.2] . Also, in general, || - 1| OP \A 9* || || . It is easy to see that if p is any algebra seminorm on A e such that pn=IHI, then II + xl ||op < p(a + xl) < p(l) || a + Al || x . The norm || -\\onA is regular (resp. weakly regular) if the restriction of || || op onA\\- \\ OP \A = H (resp. || || op ix is equivalent to || ||). These are essentially non-unital phenomena, for if A is unital (resp. having a bai (e$> then any norm | -| on A with 1 1|< 1 (or \e t \ ^ 1) is regular [5]. It is shown in 2 that || -Hop is a norm on A e iff the left annihilator Ian (A) = {0}; and in this case, || || op is complete iff II || is weakly regular iff || || r is equivalent to || || op on A e . Throughout, A is a non-unital algebra. By a norm on A, we mean an algebra norm; i.e. a norm satisfying || xy || ^ || x || || y \\ for all x, y. A uniform norm on A (resp. a C*-norm on a * -algebra) is a norm satisfying the square property ||x 2 H = ||x|| 2 (resp. the C*~ property || x*x || = || x || 2 ) for all x. 405 406 S J Bhatt and H V Dedania 2. Weakly regular norms Let (A, \\ - 1| ) be a normed algebra. The following shows that if || || op is a norm on A e , then Hop is also a norm on A e for all norms |-| on A. The left annihilator of A is Ian (A) PROPOSITION 2.1 The seminorm \\ || op is a norm on A e iff lan(A) = {0}. Proof. Let ||-|| op be a norm on A e . Let aelanU). Then ax = (xeA), hence || a ||op = sup { || ax i| : ||x|| ^ 1, xeA} = 0, so that a = 0. Hence Ian (4) = {0}. Conversely, assume that Ian (A) = {0}. Let || a + Al || op = 0. Then ax + AX = for all xeA. Suppose A 7* 0. Then - )~ l ax = x (xeA). Define L e (x) = ex (xeA), where e = - A" r a. Then L e is an identity operator on A. Then, for xeA, L x L e = L e L x , i.e. jcey = L x L c (;y) = L c L x (j;) = exy (ye A), i.e. (xe-ex)j; = (ye A). Hence, xe = ex = x. Thus A has an identity which is a contradiction. Thus A = 0. This implies ax = for all xeA, hence a = 0. This completes the proof. PROPOSITION 2.2 (a) Let \-\be a uniform norm on A. Then \-\is regular and |-| op is a uniform norm on A e . (b) Let A be a * -algebra. Let |-| be a C*-norm on A. Then \- is regular and |-| op is a C*-norm on A e . Note that if a Banach algebra admits a uniform norm, then it is commutative and semisimple. In the above, the proof of (a) is similar to that of (b) in [4, Lemma 19, p. 67]. In the following, the proof of (1) implies (2) is along the lines of [7, Theorem 1]; whereas that of the remaining part is simple. PROPOSITION 2.3 Let (A, || - 1| ) be a Banach algebra. Then the following are equivalent. (1) || || is weakly regular (so that \\ a || op ^ || a \\ ^ m \\ a || op (aeA\for some m > 0). (2) ||a + Al|| op ^||a4-Al|! 1 ^2(2-f (3) || ||op is a complete norm on A e . If || || is regular, then m^lso that \\ a + Al || op ^ || a + A 1 1| t ^ 6(expl) || a + xl || op far all a 4- /AeA e [7, Theorem 1]. 3. Uniqueness of uniform norm and unitification A Banach algebra (A.H) has unique uniform norm property (UUNP) if A admits exactly one (not necessarily complete) uniform norm. The uniform algebra C(X) has JUUNP, whereas the disc algebra does not have. In [2] and [3], Banach algebras with UUNP have been investigated. Such an A is necessarily commutative, semisimple and the spectral radius r( = r A (-)) is the unique uniform norm. We denote the HausdorfT completion of (A, r) by U(A). The spectral radius on U(A) is the complete uniform norm on U(A). A norm |-| on A is functionally continuous (FC) if every multiplicative linear functional on A is | - (-continuous. A subset F of the Gelfand space of A is a set of uniqueness for A if ]x| F = sup {|/(x)|:/eF} defines a norm on A. Uniqueness of uniform norm 407 Theorem 3.1. A Banach algebra (A, \\ - 1|) has UUNP iff A e has UUNP. We shall need the following. The proofs are straightforward. For details we refer to [3]. Lemma A. Let \ - \ be an FC norm on any commutative algebra A. Let B be the completion of(A,\-\). Then the Gelfand space A(A) (resp. Silove boundary dA) is homeomorphic to A(B) (resp. dB). Lemma B. Let A be a semisimple commutative Banach algebra. Then the following are equivalent. (1) A has UUNP. (2) U(A) has UUNP; and any closed set F in A( U(A)) which is a set ofuniquenessfor A, is also a set ofuniquenessfor U(A). (3) U(A) has UUNP; and for a non-zero closed ideal I ofU(A) with I = k(h(I)) (kernel of hull of I), Ir\A is non-zero. Lemma C. Let A be a Banach algebra with UUNP, and I be a closed ideal such that I = k(h(I)). Then I has UUNP. Proof of Theorem 3.1. Assume that A has UUNP. Case 1. Let || || have the square property. By Proposition 2.2 (a) and Proposition 2.3, (A* II ' Hop) is a Banach algebra, || || op has square property and || || op is equivalent to || || r . Let |-| be any uniform norm on A & then 1 1 1 \A is a uniform norm on A. Since A has UUNP, l-lU=|H|.Hence|h|| p^|-|<M| 1 ^6(expl)||-i|opOn^.Thus||-||o P andH are equivalent uniform norms on A e . Since equivalent uniform norms are equal, Ml op = H on A e . Thus A e has UUNP. Case 2. In the general case, note that U(A) is an ideal of U(A e ) and, by Lemma A, the Gelfand space A(U(A e )) is homeomorphic to the one point compactifications of each of AU) and &(U(A)). Define K = {xeU(A e ):xU(A) = {0}}. We prove that K = {0}. Let xeK. Then its Gelfand transform :&(U(A e ))-+C is continuous. Since xeK, xy = (ye U(A}). We prove that x is zero on &(U(A e )). Since A (17(4)) is dense in A(l/C4 e )), it is enough to prove that x is zero on A(U(A)). Suppose there exists (j>e&(U(A)) such that <p(x) ^ 0. Since is non-zero, there exists j; in U(A) such that (f)(y) is non-zero. This implies j>(xy) ^ 0, hence xy ^ which is a contradiction. Thus K = {0}. By Lemma B, it is enough to prove that U(A e ) has UUNP; and for every non-zero closed ideal / of U(A e ) with / = k(h(I)\ A e r\Iis non-zero. Let I be a non-zero closed ideal of U(A e ) such that I = k(h(I)). We prove that / n A e {0}. Let J = I n U(A). Then, first, we prove that J = k(h(J)) in U(A). Clearly J c k(h(J)). Let xe U(A) such that xJ. Then x$I, hence there exists <t>eh(I) c A(I7(X e )) such that <j>(x) + 0. Then \j/ = (j)\u(A) is zero on J and \l/(x) ^ 0. Thus x$k(h(J)) f and so J = k(h(J)\ From K = {0}, / * {0} and IU(A) J, it follows that J ^ {0}. Since A has UUNP and J is a non-zero closed ideal of U (A) such that J = k(h(J)\AnI = A n J ^ {0} by Lemma B. Hence/n,4 e ^ {0}. Finally, we show that U(A e ) has UUNP. Note that, by Proposition 2.2 (a) and Proposition 2.3, the operator norm on U(A) e is a complete uniform norm; and is the spectral radius rv(A). itself. Further, U(A) e , is clearly isometrically isomorphic to U(A e ) via the map T: U(A) e -*U(AJ, T(a + Al) = a + ^ where e is the identity of U(A e ). By Lemma C, 408 S J Bhatt and H V Dedania \ U(A) has UUNP, hence by the isomorphism T and by Case 1, U(A e ) has UUNP. \ Conversely, if A e have UUNP, then, A being a closed ideal of A e satisfying A = k(h(A)) in A & A has UUNP by Lemma C. This completes the proof. > Following [1], a Banach *-algebra B has unique C*-norm (i.e. B has UC*NP) if ( B admits exactly one C*-norm (not necessarily complete). In spite of the apparent similarity between the square property and the C*-property of norms the above result differs from the corresponding situation in B, viz. UC*NP for B need not imply UC*NP for B e [1, Example 44, p. 850]. In fact, by [1, Theorem 4.1, p. 849], for a non-unital B with UC*NP, B e has UC*NP iff the enveloping C* -algebra C*(B) is non-unital. Like C*(J5) for J5, the uniform Banach algebra U(A) is universal for A in an appropriate sense. Unlike the case of B, it happens that A is unital iff U(A) is unital. This explains why the above result for A differs from the corresponding result for B. A Banach algebra (A, ||-||) has the spectral extension property (SEP) [9] (i.e. A is a permanent Q-algebra [10]), if for every Banach algebra B such that A is algebraically embedded in J5, r A (x) = r B (x) for all xeA; equivalently, every norm |-| on A satisfies ^(*) < 1*1 for all xeA [9, Proposition 1]. COROLLARY 3.2 Let (A, \\-\\) be a semisimple commutative Banach algebra. Then A has SEP iffA e has SEP. Proof. Let A have SEP. Then, by [2, Proposition 2. 1] and Theorem 3. 1, A. has UUNP. By [2, Proposition 2.6], it is enough to prove that A e has (P)-property; i.e. every non-zero closed ideal / of A e has an element a + Al such that r x (a + U) > 0, where rja + Al) = inf (| a + Al |: |-| is a norm on A e } 9 called the permanent radius of a + Al in ^ e [9]. Let I be a non-zero closed ideal of A e . Then J = I n >4 is a non-zero closed ideal of A by [8, Theorem 1.1.6, p. 11]. Since A has SEP, by [2, Proposition 2.6], it has (P)-property, hence there exists ae J such that the permanent radius, say r 2 (a), of a in A is positive. Then clearly r l (a)^r 2 (a)>Q. Thus A e has (P)-property. Conversely, assume that A e has SEP. Let |-| be any norm on A. Then, since A is semisimple, Proposition 2.1 implies the operator norm |-| op is a norm on A e . Since A has SEP, TAW - r A .(a) ^ IflUp ^ |<z| (aeA). Thus r A (a) ^ \a\ for all a in 4 and for any norm -| on A. Hence, 4 has SEP. This completes the proof. By [9, Corollary 2], a regular Banach algebra has SEP. In understanding the relation between UUNP and SEP, a weaker notion of regularity has been found useful in [2], viz. a semisimple commutative Banach algebra U,||-||) is weakly regular if for any proper closed subset F of the Gelfand space A(4) of A, there exists a non-zero element a in 4 such that d | F = 0. COROLLARY 3.3 wafcl l] r ^ r emisimple commu ^ive Banach algebra. Then A is weakly regular iff A e re ^ ular - Then > b y R Corollary 2.4(11)], A has UUNP and ~ A ^ f V boundar ^ of A * % Theorem 3.1, X. has UUNP. Note that MA\ ., . . lnlr ~ K ^ i^ ^^ iG A W > and dA * is dosed - ^se imply dA e = A(^ e ). Hence,agamby [2, Corollary 2.4 (II)]^ e i s weakly regular. Conversely, assume that X Uniqueness of uniform norm 409 is weakly regular. The proof of Lemma C will work for the following statement; If A is weakly regular and / is a closed ideal of A such that / = k(h(I)), then / is also weakly regular. Since A is a closed ideal of A e with k(h(A)) = A, A is weakly regular. Acknowledgement One of the authors (HVD) is thankful to M H Vasavada for encouragement and to the National Board for Higher Mathematics, Government of India, for a research fellow- ship. The authors are also thankful to A K Gaur for making available reprints of his papers. References [1] Barnes B A, The properties * -regularity and uniqueness of C*-norm in a general * -algebra, Trans. Am. Math. Soc. 279 (1983) 841-859 [2] Bhatt S J and Dedania H V, Banach algebras with unique uniform norm, Proc. Am. Math. Soc. (to appear) [3] Bhatt S J and Dedania H V, Banach algebras with unique uniform norm II: permanence properties and tensor products, (communicated) [4] Bonsall F F and Duncan J, Complete Normed Algebras, (Berlin, Heidelberg, New York: Springer Yerlag)(1973) [5] Gaur A K and Husain T, Relative numerical ranges, Math. Jpn. 36 (1991) 127-135 [6] Gaur A K and Kovarik Z V, Norms, states and numerical ranges on direct sums, Analysis 11 (1991) 155-164 [7] Gaur A K and Kovarik Z V, Norms on unitizations of Banach algebras, Proc. Am. Math. Soc. 117 (1993) 111-113 [8] Larsen R, Banach Algebras, (New York: Marcel Dekker) (1973) [9] Meyer M J, The spectral extension property and extension of multiplicative linear functional, Proc. Am. Math. Soc. 112 (1991) 855-861 [10] Tomiuk B J and Yood B, Incomplete normed algebra norms on Banach algebras, Stud. Math. 95(19 89) 119-132 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 4, November 1995, pp. 411-415. Printed in India. Weakly prime sets for function spaces H S MEHTA and R D MEHTA Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar 388 120, India MS received 3 November 1994; revised 24 April 1995 Abstract. We define and study weakly prime sets for a function space and show that it coincides with the known concept of weakly prime sets for function algebras and spaces of affine functions. Keywords. Weakly prime set; function space; function algebra; space of affine functions. 1. Introduction A function space A on a compact Hausdorff space X is a closed subspace of the space C(X) of all continuous, complex-valued functions on X separating points and contain- ing constants. If A is an algebra, it is called a function algebra. The Bishop and Silov decompositions play an important role in characterizing function algebras. Later on these decompositions were studied for function spaces [6]. Ellis [3] defined and studied these decompositions for the spaces of affine functions on a compact convex set. For a function algebra, certain decompositions finer than the Bishop and Silov decompositions have been defined and studied [6]. One such decomposition of weakly prime sets, was defined and discussed by Ellis [4] for function algebras as well as for spaces of affine functions. Here we generalize this concept for a function space, study its properties and show that it coincides with the corresponding definitions of Ellis. We also give examples of function spaces whose family of maximal weakly prime sets differ from the corresponding families of its induced algebras. 2. Function space Let X be a compact Hausdorff space. Throughout this paper we assume that A is a function space on X. For a closed subset E of X, we define fgeA\ E for allgeA\ E }. For the concepts like peak set, p-set, etc. related to a function space and for the various properties of a decomposition for a function space, we refer to [2], [5] and [7]. DEFINITION 2.1 A closed subset E of X is called a weakly prime set for A if E = Gu#, with G and H generalized peak sets for N(A\ E ), then either G = E or H = E. The function space A is called weakly prime if X is a weakly prime set for A. Remarks 2.2. (i) If A is an algebra, then N(A\ E ) = A\ E and hence Definition 2.1 coincides with the definition for a function algebra given by Ellis [4]. 411 412 H 5 Mehta and R D Mehta (ii) It can be shown that each weakly prime set is contained in a maximal weakly prime set for A. The collection of all maximal weakly prime sets for A is denoted by &(A). (iii) It is easy to check that <P(A) is finer than the Bishop decompositions for A and hence &(C(X)) = {{x} : xeX}. (iv) It can be easily verified that A is weakly prime if and only if N(A) is weakly prime. Further, for a closed subset E of X, N(A)\ E <=N(A\ E ) and so, &(A) is weaker than &(N(A)). But, in general, &(A)^&(N(A)) (see Example 2.6(i)). As in case of a function algebra, we shall show that here also every member of^(A) is a p-set and &(A) has the (GA)-property [5] for A. We shall need the following lemma. Lemma 2.3. // E is a p-set for A and F a E is a generalized peak set for N(A\E\ then F is a p-set for A. Proof. Let /.ieA 1 and e>0 be given. Then there is an open set U in X such that |ju|([7 - F) < 8. Clearly, E n U is open in E and F cz E n U. Since F is a generalized peak set for N(A {E ), there is a peak set T for N(A\ E ) such that FaTciEnU. LetfeN(A\ E ) be a peaking function for T. Define /Ton E by h = 1 on T and h = on \ T. Then/" converges pointwise and boundedly to h on E. Now let g e A. Then But j #/ n dju = 0, as/^ 1 6 A, and is a p-set for A. Thus JV Now Since e is arbitrary, | Jf^d^l = or F is a p-set for A. PROPOSITION 2.4 A maximal weakly prime set for A is a p-set for A. Proof. Let E be a maximal weakly prime set for A and let F denote the smallest p-set for A which contains E. We shall show that F is a weakly prime set for A. Let F! and F 2 be generalized peak sets for N(A\ F ) with F 1 uF 2 =F. Then E = (F 1 n)u(F 2 n) and since N(A|f),c N(X|), F x n and F 2 n are generalized peak sets for N(A [E ). Since is a weakly prime set for A, either F l nE = E or F 2 n = , i.e., either E G F t or E c F 2 . If E c F x , then c F x c F where F is a p-set for A and F 1 is a generalized peak set for N(^ (F ). So, by Lemma 2.3, F x is a p-set for /I and hence F 1 = F. Similarly, if cz F 2 , then F 2 = F. Thus F is a weakly prime set for A and by the maximally of , we have = F. Next, we show that the family &(A) characterizes a function space A in the sense that it has the (D)-property for A [5.7], i.e. if/eC(X) and/ i 6(^, )- for every Ee&(A), then Weakly prime sets for function spaces 413 /e A. Actually the Bishop's theorem can be restated as "The Bishop decomposition has the (D)-property for A". In fact the Bishop decomposition has a stronger property than the (D)-property, namely the (GA)-property. By (GA)-property for a family ^ of closed subsets of X for A [7] we mean that for ch [teb(A L ) e , suppp c F for some FeJ^, where b(A^-) e denotes the set of extreme points of the unit ball of A L . . the (D)-property, namely the (GA)-property. B each oin . We shall show that &(A) has the (GA)-property for A. Theorem 2.5. &(A) has the (GA)-propertyfor A. Proof. Let neb(A L } e , the set of extreme points of the unit ball in A L and let S = supp/z. It is enough to show that S is a weakly prime set for A. Let G and H be generalized peak sets for N(A ]S ) with S = Gu H. Let / u 1 = j/ |G , 5 // 2 = ju Hi and geA. Since p. = /^e^ 1 and G is a generalized peak set for N(A ]S ), by Lemma 2.3, we get J G 0d/* = 0. Thus ^eX 1 and hence ^e^l 1 . Also, H//H = 1 = II^JI + ll/^il- Hence /^ 1 = \JL or // 2 = /z, as j ueb(^4 1 ) e , i.e., G = Sor H = S. Thus 5 is a weakly prime set for A Examples 2.6. (i) Let X be the union of a line segment F and a sequence of disjoint solid rectangles {F n ; n = 1,2, . . . } converging to F. Let y4 be the set of all/ in C(X) such that / |FM is a polynomial of degree atmost n. Then A is a function space on X and as in [7] it can be checked that ^(^) = {F n |weM}u{{x}; xeX}. Note that, here N(A) = {fe(X):f lFn is constant, for each neM} and hence &(N(A)) = {F n :neN}u{F}. Therefore, /(A] * &(N(A)). (ii) Let T denote the unit circle in C and A(T) denote the disc algebra on T. Let Oe 4(T) be such that <D ^0 on T. Define ,4 = {*" l f:feA(T)}. Then /I is a function space on T and AT(y4) = A(T) [8]. It is clear that N(/l) is weakly prime and hence by Remark 2.2 (iv), A is also weakly prime, i.e., &>(A)= {T}. Since A(T) is a maximal function algebra on T and 4(T) S X, the algebra generated by A will be C(T). But &(C(T)) = {{xj: .xeT} by Remark 2.2 (iii) while &>(A) = {T}. 3. Space of affine functions Let K be a compact convex subset of a locally convex Hausdorff space and let A(K) denote the Banach space of all real-valued continuous affine functions on K with the supremum norm. The set of extreme points of K will be denoted by dK. Ellis [4] has defined weakly prime sets* for A(K) with the help of concepts of convexity. Now A(K) can also be looked upon as a function space on K. So we can discuss &(A(K)) for A(K). But, since the functions in A(K) are determined by their values on dK, we shall consider the space A(K) ldK . In fact, weakly prime sets defined by Ellis, are also subsets ofdK. In this section, we shall prove that &(A(K\ dK ) coincides with the family of maximal weakly prime sets as defined by Ellis. For the definitions and results regarding compact convex sets and space of affine functions, we refer to [1] and [2]. Let us recall the definition due to Ellis [4]. DEFINITION 3.1 A subset E of dK is called a weakly prime set for A(K) if E = dG for some closed face G of 414 H S Mehta and R D Mehta K and if every proper facially closed subset of G has empty interior in the facial topology of G. Equivalently, for a closed face G of K, dG is weakly prime if whenever G = Co(H l u H 2 ) for some closed split faces H 1 and H 2 of G, then either H l = G or H 2 = G. If cK is a weakly prime set, then A(K) is called weakly prime. We shall denote the family of maximal weakly prime sets for A(K) according to Definition 3.1 by & E (A(K)\ The following proposition can be easily proved. PROPOSITION 3.2 where Ce(A(K)) = {feA(K):fg ]fK eA(K\ pK for every geA(K)}, the centre of A(K\ Since Ce(y4(K)),^ is the set of facially continuous functions on cK [2, Theorem 1.4, p. 105], we immediately get the following result. COROLLARY 3.3 A subset E of oK is a facially closed subset of oK if and only if E is a generalized peak set for N(A(K) ldK ). PROPOSITION 3.4 // Ee&(A(K\ dK ) 9 then Co, the closed convex hull o/, is a closed split face ofK. Proof. Let F be the smallest closed split face of K containing CoE. Then E c Co n dK <=. F n cK = 3F, as F is a face. It is enough to show that dF is a weakly prime set for A(K) { . Let H! and H 2 be generalized peak sets for N((A(K\ K \ fF ) with dF = H l uH 2 . Then E = EncF = (H l r^E)u(H 2 r\E) and fi^nE, H 2 nE'are generalized peak sets for N((A(K\ rK ) l ). Since is a weakly prime set" for A(K\ , either H 1 nE = E or H 2 n = . Thus, either c H! or c H 2 . Now, since F is a closed split face of K 9 (A\K\ ?K ), CF = A(K\. F = A(F)^ f . So, H^ and H 2 are generalized peak sets for N(A(K\. K ) and hence by Corollary 3.3, H x and H 2 are facially closed subsets of dF, i.e., H l = 8G l and H 2 = 5G 2 for some closed split faces G l and G 2 of F. Since F is a closed split face of K, G 1 and G 2 are closed split faces of K. Now Ec:H 1 =>CoE c= CoH ! = G! . Thus we get Co c G L c F and hence G x = F, as F is the smallest closed split face containing Co, i.e., H l = dF. Similarly, if c H 2 , then we get H 2 = 5F. So cF is a weakly prime set for A(K)^ ,. If CoH is a closed face of K for H c 3X, then 3(CoH) = H and hence we get the following result. COROLLARY 3.5 IfEe&(A(K\ K ) 9 then E is facially closed. Now we prove the main result. Theorem 3.6. Proof. Let Fe& E (A(K)). We want to show that F is a weakly prime set for A(K) ldK . Weakly prime sets for function spaces 415 Let H 1 and H 2 be generalized peak sets for N((A(K) [riK ) lF ) with H l (jH 2 = F. Since Fe& E (A(K)) 9 F is facially closed [4], i.e., F = cG for some closed split face G of K. Hence A(K\ G = A(G) and so (^(&) irK ) if = A ( G \- G > Thus H ! and H 2 are generalized peak sets for N(A(G\ rc ). So by Corollary 3 .3, H i and H 2 are facially closed subsets of G. Also, by definition, F = <3G, where G is a closed face of K and F = H 1 u H 2 . Since F is a weakly prime set for A(K) 9 either H i = ForH 2 = F. Hence F is a weakly prime set for A(K) } . ,. Conversely, let Fe#(A(K) } ). Then by Proposition 3.4, CoF is a closed split face of K. Let CoF = G. Then F = cG and A(G\ dG = (A(K) lfK \ F . Suppose F = H l uH 2 , where FT ! and H 2 are facially closed in G. Then by Corollary 3.3, H l and H 2 are generalized peak sets for N((A(K)\ dK )\ F ). Since F is a weakly prime set for A(K)^ K < either H l =F OY H 2 = F. Hence F is a weakly prime set for A(K). Consequently, &(A(K) {dK ) = ^(/KK)). COROLLARY 3.7 A(K) is weakly prime if and only ifA(K)^ , is weakly prime. References [1] Alfsen E M, Compact convex sets and boundary integrals (Berlin: Springer- Verlag) (1971) [2] Asimov L and Ellis A J, Convexity theory and its applications in Functional Analysis (Academic Press) (1980) [3] Ellis A J, Central decompositions and essential set for the space A(Kl Proc. Lond. Math. Soc. 26(3) (1973) 564-576 [4] Ellis A J, Weakly prime compact convex sets and uniform algebras, Math. Proc. Cambridge Philos. Soc. 81(1977)225-232 [5] Hayashi M, On the decompositions of function algebras, Hokkaido Math. J. I (1974) 1-22 [6] Mehta H S, Decompositions associated with function algebras and function spaces (Ph. D. Thesis, Sardar Patel University) (1991) [7] Mehta H S, Mehta R D and Vasavada M H, Bishop type decompositions for a subspace of C(X\ Math. Jpn.31(l992) Ill-Ill [8] Yamaguchi S and Wada J, On peak sets for certain function spaces, Tokyo J. Math. 11(2) (1988)415-425 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 4, November 1995, pp. 417-423. Printed in India. Oscillation of higher order delay differential equations P DAS, N MISRA* and B B MISHRA Department of Mathematics, Indira Gandhi Institute of Technology, Sarang, Talcher, Orissa, India * Department of Mathematics, Berhampur University, Berhampur 760007, India MS received 25 October 1994 Abstract. A sufficient condition was obtained for oscillation of all solutions of the odd-order delay differential equation where p t (t) are non-negative real valued continuous function in [T^oo] for some T^O and <7 e(0, oo ) (i = 1, 2, . . . , m). In particular, for p t (t) = p/(0, oo) and n > 1 the result reduces to i / V (nV HZ (Pi*?) 1 ' 2 ) >("-2)!^-, \i=i / e implies that every solution of (*) oscillates. This result supplements for n > 1 to a similar result proved by Ladas et al [J. Diff. Equn., 42 (1982) 134-152] which was proved for the case n - 1. Keywords. Odd order; delay equation; oscillation of all solutions. 1. Introduction This paper was motivated by certain results of the paper [7] and [8] due to Ladas et al. In [7] authors proved that all solutions of the odd-order delay differential equation * (n) + Z P*(t-<r ) = 0, (1) i=l oscillates (i.e., every solution x(t) has zeros for arbitrarily large t) if and only if the associated characteristic equation *"+ Z p e-*-< = (2) ;=i has no real roots, where p t and 0^6(0, oo) for i = 1, 2, . . . , m. Further, it was proved that (2) has no real roots if and only if In the literature, it was observed that the odd-order differential equations of the form (T i ) = 0, (3) where p f e C( [ r, oo), (0, oo)), T ^ and <7 e(0, oo), is least studied. In this connection, we may refer, in particular, to [4], [5], [9] and the references therein. For n = 1, (3) is almost well-studied. In this case there are several results associated with its characteristic 417 418 PDasetal equation (see [3] and [7]) as well as conditions on coefficients and deviating arguments which ensures that every solution of (3) oscillates. In [8], authors proved that if p f eC([T, oo),(0,oo)), d f .6(0, oo)(i= l,2,...,m)andw = 1 then lim inf p t (s)ds > (i = 1, 2, . . . , m) (4) r-oc J f ffij 2 and 1 m f p \ - X lim inf p t .(s)ds m i=l\r-*oc J t- ffl - / 2 m [~Y P W P Ml + - X lim inf Pl .(s)ds lim inf Pj (s)ds >- (5) m *<j L\r-*oo J t-ffj / \t-*oo J f-ffi /J ^ U=i then every solution of (3) oscillates. If p t (t) = p f e(0, oo)(f = 1, 2, . . . , m) then the above result becomes i implies that every solution of (3) oscillates. In this paper an attempt has been made to obtain a similar result which shows that every solution of (3) oscillates. Our result fails to hold when n = 1. Indeed, when p^t) = p f e(0, oo), the main result of this paper shows that if \i=l then every solution of (3) oscillates. Although our result does not generalize the result of Ladas et al [8], but certainly supplements for higher order equations. 2. Main results In the beginning of this section we prove a lemma for its use in the sequel. Lemma 1. Let feC (n} (\_T, oo), (0, oo)), 7>0 such thatf (n \t) ^ 0, 1 2* T. If n is odd and (76(0, oo) then there exists T ^ T such that Proof. Since f(t) ^ and f (n \t) ^Qfor t^T, there exists T such that and / (J) W/ a+1) (r)<0 for fc <;<-!. Expanding /(t) by Taylor's theorem, there exists xe(t - a, f) such that (7) Oscillation of higher order delay differential equations 419 Similarly, expanding /<*> by Taylor's theorem we get Replacing t by t - a in the inequality (7) we get Further, using (8) in (9) along with the fact that *!(-*- 1)! < ( - ! and setting TO = 7^ + 2(j we have our proposed inequality. This completes the proof of the lemma. Theorem 1. Suppose that Pi eC([T, oo). (0, oo)X T>0 and er | (Q,oo) (i= 1,2,3,. .,m). Further if liminff p f (s)ds r-*oo J t-o)ffi A (1) and / y,-i v 1 m 9 "L 1 vi /i . / 1\|V'V = J t t UKfj and f n-\ then euery so/uion o/ (3) oscillates. Proof. On the contrary, suppose that x(t) > for t > t . Dividing (3) throughout by x 01 " 1 ^) we get . PV-D^-oxr,) x { "- "(t) By Lemma 1, there exists t t > f such that and the use of this inequality in (12) results *-> _r-_L where =; 420 P Das etal Integrating both sides of (13) from t - a)a k to t we get /x^0-o^)\ p x'-^s-^) 10g (- x"- W r^H-.,/^ x'- 1 ^) Setting and P p. k = lim inf Pi(s)ds i, j = 1, 2, . . . , m t-oo J l-o>fffc we see that m log(a fe )> -X i=l Suppose that a fc < oo for k = 1, 2, 3, . . . ,m. In this case, dividing both sides of the above inequality by a fc and using the fact that ! foroc^l, and a fc ^ 1 (since x (w ~ 1} (t) is positive decreasing) it follows that 1 <*i Summing the above inequality for k = 1, 2, . . . , m we obtain ^ - V V f a i 7^ L Z- K iPik- e fc=l i=l a /c that is, Rearranging the right hand side elements of the above inequality first along the diagonal then above and below the diagonal respectively, we get that is, i,J=l Oscillation of higher order delay differential equations 421 Since the arithmetic mean is greater than the geometric mean a ; \l/2 In view of ( 1 6), ( 1 5) reduces to m e Putting the value of K f and Kj in the above inequality we obtain i m 2 m " - (16) which is a contradiction to our assumption. Next, assume that a = oo for some i = 1, 2, . . . , m. That is, for some i = 1, 2, . . . , m. From (3) it follows that for the value of z for which (17) holds. From the inequality above (13) it follows that From (17) and (18) it follows that _ C^T.) ^ 0. (19) Integrating both sides of (19) from t - a>oJ2 to t and using the fact that x (n ~ 1} (t) > and decreasing we get < 0. (20) (n 1)1 J ,-0,^ Dividing both sides of (20) first by x (n ~ 1} (t) and then by x (n ~ 1] (t - co^/2) we have the following inequalities respectively: and 1 . (n - 1)! x<" In view of (10), (17) and (21) we obtain p(s)d5<0 (22) (23) 422 ? Das et al Using (23) in (22) along with (10) we see that l-*co A V wui/^j Replacing t by 1 4- coa t /2 in the above inequality we get lim (-iw \'' 1 < ' t->oo -^ w which is a contradiction to (23). This completes the proof of the theorem. COROLLARY 1. If p t (t) = p f e(0 5 oo) and ^6(0, oo) then implies that every solution of (3) oscillates. Proof. In this particular case n-l and hence (10) reduces to that is, (24) holds. Hence the proof follows from Theorem 1. Example 1. The equation x(t _ 1} + 6 + x(t _ 2 ) = 0, satisfies the hypotheses of Theorem 1 and hence every solution of it oscillates. But Theorem 5.2 of [8] is not applicable to it. Example 2. Consider the equation \ By Theorem 5.2 of [8], every solution of it oscillates. But Theorem 1 of this paper is not applicable to this equation. This is due to the fact that Theorem 1 holds only for n > 1 and is an odd integer. Acknowledgement Research of the first author was supported by the National Board for Higher Mathematics (Deptt. of Atomic Energy), Government of India. Oscillation of higher order delay differential 423 , 44 (1982) 134-152 I. 162 (1992) 452-475 theory of differential equations Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 4, November 1995, pp. 425-444. Printed in India. Nontrivial solution of a quasilinear elliptic equation with critical growth in R n RATIKANTA PANDA T.I.F.R. Centre, P.O. Box No. 1234, 1.I.Sc. Campus, Bangalore 560012, India MS received 10 February 1995; revised 16 June 1995 Abstract. Suppose A n w = div ( j Vw | " " 2 Vw) denotes the rc-Laplacian. We prove the existence of a nontrivial solution for the problem where /(x, t) = o(t) as t -> and | /(.x, t) \ ^ C exp(a n 1 1 \ nl(n ~ l ') for some constant C > and for all '.xeR'VfeIR with a n = naj l n l( "~ u , a> n = surface measure of S n ~ l . Keywords. Elliptic equation; critical growth; Palais-Smale condition; concentration compactness; mountain pass lemma. 1. Introduction Suppose A n u = div(| Vu\ n ~ 2 Vu) denotes the n-Laplacian. We look for a solution of the problem (-~k n u + \ur 2 u~f(x,u)u n ~ 2 in R" \ueW l < tt (R tt ) ' where f(x 9 t) =o(t) as f-0 and \f(x 9 t)\ ^ Cexp(aJt| M/(M - 1) ) for some constant C>6 and for all xelR", teR with a n = na) l n l(n ~ l \ co n =surface measure of S n ~ l . In the case where 1 c R" is a bounded smooth domain, and f(x, t) == h(x,t)exp(ai n \t\ nl(n ~ 1} ) with h(x 9 t) a lower order term in t, the problem (1.1) with Dirichlet boundary condition has been considered by Adimurthi [1] and with Neumann boundary condition by the author [9]. In case of n = 2, D M Cao [5] has shown the existence of a nontrivial solution for the problem (1.1). In this paper, applying the concentration-compactness principle of P L Lions [6, 7], we show that the functional associated with (1.1) satisfies (Palais-Smale), (in short (PS) C ) condition for all ce(0,J) for some J>0 (for definition of J see 3). Then we show the existence of a nontrivial solution for (1.1) by using Mountain Pass lemma as given in [4] and constructing a critical point of the functional with critical value in (0, J). The main difficulty here is to show that whenever a Palais-Smale sequence u m > u weakly in \Vu m \ n - 2 Vu m -^\Vu\"- 2 Vu weakly in (L^"" 1 ^")) 11 We need the following assumptions on the nonlinearity f(x, t)eC(R n x R): (/i) l/(x, 01 ^ Cexp(aJ t\n/(n - 1)) for xeR", teR, where C > is some constant. (/ 2 ) /feO^" 1 =/(x, -r)(- tr-^orxe^teRj^^isnondecreasingwith respect 425 426 Ratikanta Panda to t, for t > 0; lim = uniformly with respect to xeU": f-*0 t lim ~ = uniformly with respect to xe R". t-oo ^ (/ 3 ) There exists 0e( 0, - ) such that V n J F(x,t)^9t n ~ l f(x,t) for xeRVeR, where F(x,t) = ^f(x,s)s n ' 2 ds _ (/ 4 ) 3/(t) such that lim |xhoo /(x, t) = f(t) uniformly for t bounded, more precisely, 1 for |x|>R, where s(R) -+ as -+ oo. (/ 5 )3p>n such that f(x,t)^f(t)^C p t p -+ 1 >(p/n)S p p (l - n 0) 1 ~ (p/n) t p ~ n+1 for xeR", te!R + , where S p = w jnf u^O For we ^-"(R 11 ) let r _ c- F(u)dx, (1.2) JR where F(t) = f /(s)s w ~ 2 ds. The main results in this paper are as follows. Theorem 1.1. Suppose /(x, t) = f(t) does not depend on x and satisfies (/aH/a) and (fs)- Then (1.1) has a nontrivial solution u . Moreover I(u Q ) < (l/n) - 9. Theorem 1.2. Suppose /(x, t) satisfies (fj-(f 5 ) andf(x 9 1) ^ f(t) for fixed t with respect to xelR". Then (1.1) has a nontrivial solution. We remark that the (PS) condition is not needed for the proof of Theorem (1. 1). 2. Preliminaries and notations We shall denote j to mean J Rn dx. Define for ueW l *(te) (2.1) t|M for ueL q (U"). (2.2) The variational functional associated with (1.1) is where F(x, t) = f Quasilinear elliptic equation Let /() be as in (1.2), |u||" =["-'/() /^CXD __ inf { oo, if M 00 = <t>, 427 (2.4) (2.5) Remark. If f(t) satisfies (/ x ) and (/ 2 ) then /* > 0. Prcx?/. Suppose, on the contrary, 1% = 0. Then there exists a sequence {u m } in such that Then by (fj, (/ 2 ) and Lemma 2.3 (to be proved) where C 1? C 2 > are some constants. Thus and therefore | Vw m | n ^ l/2nC 2 , a contradiction which proves the remark. Remark. If /(() satisfies (/jH/a) then C > 0. Proo/. Suppose M ^ </). For ueM , using (/ 3 ) we get Q Since f(t) satisfies (/ t ) and (/ 2 ) we have 428 Ratikanta Panda and therefore as in the above Remark we obtain for some positive constant C 2 . Therefore by above estimate C ^ ((1/n) - 0)C 2 . This proves the Remark. D Similar to the imbedding of Moser [8] we have Lemma 2.1. Suppose ueW^"(U n ), \Vu\ n n ^ r < 1, |M| B < M < oo. Then rr n ~ 2 a m M nm/(n ~ ir i expfcjur- 1 ')- X 51II1.5 - ^ C(M,r), (2.6) J L - m = m ' J C(M, r) > is a constant independent ofu. Proof. As in Moser [8] we use the method of symmetrization. Let w* be the symmetriz- ation of u. Then u* is a radial, nonnegative and nonincreasing function. Further, 111*1'== M p , l<p<oo (2.7) J G(u)= [G(U*), (2.8) f f |Vu*T< |Vu|", (2.9) J J where G(u) is the integrand on the l.h.s. of (2.6). We have f f r jG(u)= G(*)= G(u*)+\ G(u*) (2.10) where s > is a number to be determined. First we estimate the second integral in (2.10). By the radial Lemma A. IV in [3] we have / n \ 1/1. !*(*)! <() I^LI^r 1 for x^O. . (2.11) \ n/ Thus f ^, x aT 1 !"*!!! f / a" I |*r i A G(u)= " J '+ ( " ,""M if s>n|u*| B . (2.12) Quasilinear elliptic equation 429 To estimate the first integral in (2. 1 0), let us put ] x \ n = s"e ~\v (t) = n (n ' 1 }/ " u* (x). Then |Vu*T, (2.13) exp(li;(r)r /(w ~ 1) -f)dr = - exp(ajM*r /(ll " 1) )dx, (2.14) o V J|*K.s where v = dv/dt. By Holder inequality we have 0(0) + ( I |(s)|" r (2.15) where x is some unit vector in R n . Now G(u*)< |.x|<.v (2.16) Combining (2.11), (2.12) and (2.16) we have (2.6). D Lemma 2.2. There exists = J?(n) > swch t/zat/or all ueW l -"(R n ) with \Vu\ tt n < l/a B j8e, we where C > is a constant independent ofu. Proof. By the result of Talenti [10] (or of Aubin [2]) we know that if t, s > 1, t < . w and 1/s = 1/t - 1/n, then all (pGW^(R tt ) satisfy Icpl^K^OIVcpl, (2-17) with Let us set <p = || v , where v = ((n- 1) 2 +nm)/(nm-n + 1), m^2. Then |V<p| = v|w| v ~ 1 1 Vu|. Taking s = (n/(n - l))m - 1, t = n - (n 2 (n - !))/(( - I) 2 + nm) and using 430 Ratikanta Panda Holder inequality we get f ^-D-H,-! ,,<- 1>- iff v-l) ,\ j|u| ^( v y u j /( B -l)-l (2.18) Now where C(n) is a constant dependent on n. Since r < n we get (f l)/t < (n 1 )/n. Also for all m > 2, n ^ 2 we have ((n - I) 2 + nm)/n 2 (n - 1) < m and v ^ C t (n), a constant depend- ent on n. Hence we get (Kv) Rm/(n " l) ~ 1 ^ C(j?m) w for some = fi(n) > and C = C(n) > 0. Therefore by (118) f B -l /(-.-lh_ _ ,./(. J|u|" (exp(a n | M | )- -,|u| 2 ' For m > 2 we have m/(n - 1) - 2> m/(n - 1). Thus for | Vu|^ /( "~ u < !/ ]8e we have ( I |u|"- 1 (exp( 11 |ur-- 1) )- 1 -ajul"*' 1 ') where we have used the same C to denote various constants. D Lemma 2.3. Letf(t) satisfy (/J and (/ 2 ). Suppose there exists u Q eW l ' n (R")\{Q} such that JF(M ) ^ (l/n)f |u |" and | Vu C < 1. Tten / to actta*d. Moreover /? ^ | Vii C. Proo/. Using (/ 2 ) and the hypothesis that f F(u ) ^ ( 1/n) \ u \ n n it is easy to see that there exists t (0,l] such that jF(t u ) = (l/n)J|t u r Thus /?<J|Vu | n . Let {u m } be a minimizing sequence for JJ* . Without loss of generality we can assume that J | Vu m \ n < r< 1. Denote by w* the symmetrization of u m . Then w* is a radial, nonincreasing function. Furthermore, Quasilinear elliptic equation 431 Thus {w*} is still a minimizing sequence of /^. We denote it simply by {u m } in what follows. Without loss of generality we can assume that |w m | B = 1. Thus {u m } is bounded in W l - n (R n ) and so there exists we W rlt "(R 11 ) such that for a subsequence w w -*w weakly in HP ^" u m -> w a.e. in [R w . We want to use a compactness lemma of Strauss (see Theorem A.I. of [3]). Set j t\ njl(n ~ 1} + \t\*. Then using (/J we get lim, rHoo P(t)/Q W = and using (/ 2 ), lim |fKO P(*)/Q(f) = 0. Again by radial lemma A.IV of [3] we get (2.11) and so as |x| -* oo, M m (x)-^0 uniformly in m. Further by Lemma 2.1, (fi(ii J supfi(iiJ<C. M J Thus all the conditions of Strauss' lemma are satisfied and we get lim m-*oo Since f F(w m ) = | we get u ^ 0. Now - [N"<- lim inf L m | n = lim inf Wj nm-oo J m-*oo f| J and so J* is achieved by u. If on the other hand (l/n)J| Vu\ n < fF(w), then there exists (6(0, 1) such that (1/n) J| tu\* = jF(rw). Hence r r r I < M I Vw|" < I Vw| n < lim inf | Vu m \ n = /J, J J m-oo J a contradiction which proves the lemma D Lemma 2 A Suppose {u m } <=. W^ n (U n ) satisfies \Vu m \ n <l,\u m \*<M and lim sup (I Vw m | n -h |w m | n )dx = for some R > 0, m ^y^ n J y+ B R 432 Ratikanta Panda where B R = {xeR n ; \x\ < R}. Then lim m-*x lim m-+:c Proof. Let eC%(U n ) be such that c = l for |x|<R/2; | Vc| < 4rc/R; Let c v =<;(- y). Then (2.19) for 1x1 >R and (2.20) In view of (/ t ) and (/ 2 ), given 8 > there exists C f > such that For m large enough | V( y u m )\" n < l/a n j?e and hence Lemma 2.3 gives u"dx We cover R" by balls B R , 2 (x ; ) in such a way that any point of R" is contained in at most k balls B R (.x ; ) of radius R. For large m we have |F(.x,u JR" R Making w- oc and then -+0 + in (2.21) we obtain lim m-*x (2.21) (2.22) Quasilinear elliptic equation Similarly we have lim m t-oo 3. Proof of the main results First we prove the following Lemma 3.1. Let C* be as in (2.5) and J = min(C 00 , --6M. \ ' n ) Suppose f(x, t) satisfies (AH/J. Then I(u) satisfies (PS) C condition for ce(0, J). Proo/. Let {u m } be a (PS) C sequence in W l '"(R"). That is, Then 433 (2.23) D where o(l) denotes the quantities that tend to as m -> oo and Taking <p = w m in (3.2) we obtain in From (3.1) and (3.3) we get C (Y 11 \u n ~ l ~F(Y u Y\ (X, U m ) U m r (X, U m ) J Thus using (/ 3 ), and hence in view of (3.3) {u m } is bounded. Further (3.3) gives nc (3.1) (3.2) (3-3) as desired. 434 Ratikanta Panda Thus for m large enough IVuJ^r (3.4) where re(0,l) is some fixed number, and there exists ueW l ' n (R") such that for a subsequence W M -+M weakly in ^'"(IR") ,->M a.e. in R" Vu m \ n + |H m | w )dx-+d^ in measure jVuJ"- 2 Vu m ->T weakly in (L n/( "~ 1) ((R''))' 1 - Without loss of generality we may assume Claim 2: />0. If not, suppose / 0. Then by (/J, (/ 2 ) and Lemma 2.3 Similarly I jF(x,uJI^O. So /( m ) = (l/w)l|ti m ir-jF(x,M m )-^0, which contradicts the fact that We want to apply concentration-compactness principle of P L Lions [6, 7] to the sequence {p m } where p m = | VwJ" -f | wj". Applying Lemma 1.1 of [6] we conclude that for a subsequence one of the three possibilities holds: (a) vanishing, (b) dischotomy (c) compactness. We use contradiction argument to show that only (c) compactness occurs. Step 1 : Vanishing does not occur. Suppose instead that Urn sup I (| VuJ" 4- \u m \ n ) =0 for all R > 0. Then Lemma 2.4 yields This implies, in view of (33), that i|u m || M -^0, which is not possible since />0. So vanishing does not occur. Quasilinear elliptic equation Step 2- Dichotomy does not occur. Suppose dichotomy occurs. Let Q w (t) - sup^ J function of Pm . Then {ft.} is a sequence of n bounded functions on R + . As m [6], by ex there exists Q(t) such that Q m (t) -^ W (0, I). For any > 0, e < l/(2n)"a B 0e, we c > t . Then for m large enough a - (em exists |y m }cR" such that 435 (x )d.x denote the concentration nonnegative uniformly we can assuine that , fcn_Q(t) - ^ Q(t) ^ a _ (/4 ) rf tf ^ Furth ermore, there (3.6) for t ^ t and m large enough. Also we can find t. oo such that (3.7) By computation we deduce C ' Choosing t, large enough we have (3.8) <. With m large enough so that t m > 3t 1; using (f,), (h) (3.9) _ if , _ |<tl0r x- r f |V(u u.)l"<2" [|VjJ" J J 436 Ratikanta Panda Hence applying Lemma 2.3 we get |f? m l'- 1 (exp(.|? m m r /( '- 1) )- 1 - Similarly we have for m large, Combining (3.6), (3.8) with (3.11) we get J " m Also using (/i) and (/ 2 ) we get <Ce. L <, and as in the proof of (3.10) we obtain r I [ffa w m) ~~ F( x > V m) ~ F( X > W m)l From (3.13), (3.14) and (3.15) we have (3.11) (3.12) (3.13) (3.14) (3-15) -Ce (3.16) and this proves the claim. We will now consider two cases, Case 1: {y m } is bounded, and Case 2: {y m } is unbounded. Case 1: {y m } is bounded. Quasilinear elliptic equation 437 lim 4: I(wJ Js /(wj -O(e) - 0(1) as e->0+, m-> oo. iVe have - [F(x,wJ-F(wJ] dfor<5>0, *nl^W'*)- F < w ' (3.17) lere e(rj->0 as t m ->oo. Here we have used the assumption that {y m } is bounded, so when |x| is large enough, for | wj< V<5, ms (/j) and (f s ), for t > we have - - uniformly in x and = 0. ence by (3.4) and Lemma 2.1 F(x,wJ F(.x,wJ . Similarly F(wJ (3.18) (3.19) (3.20) 438 Ratikanta Panda Thus (3. 17)-(3.20) imply [F(x,wJ-F(wJ] ^0. (3.21) Therefore we get, as desired (3.22) where 0()-0as ~*0and o(l)->0as m-oc. Cto'm5: /(w w )> C x -0(e)- 0(1). We have < f(w m ), w m > = <I*'(w J, w m > + [- 7(w J ~ <- 7(*> vv J). Arguing as in the proof of (3.21) we can prove that wr l /(" j - c 7(*, <a = o(D- (3 - 23) Also by using (3.3), (3.6), (3.11) and (3.12) we can get OJ, w m > = <J'(u J, <> +0( ) = o(l) + 0(e). Hence </ x '(wJ,w m > = o(l) + 0( ). (3.24) With w m (.x) = w m (trx) we have = (1 - ff -") I* | VwJ" + a-</-'(w m ), w m >. (3.25) J We want to choose <r m close to 1 in such a way that w w 6M oc . First we show that | Vw m |^ has a lower bound A > independent of e small enough and independent of m. If not, then there is a sequence <5 fc -y0 such that where w m (<5^) is a subsequence selected by the above process for each S k . Now, by dichotomy we have 1 *&) > I - a - & k . (3.26) On the other hand using (3.24), (/ t ), (/ 2 ) and Lemma 2.3 439 Qtiasilinear elliptic equation Thus |V Therefore which contradicts (3.26). lim |Vw m-*-oc I" > A > for & small enough. i i *"'' r" we see from (3.25) , which , (C , 1 + OW + <-(. Again, in view of (3,11) we can assume that therefore }F(w.) is bounded. Hence r<(1 + r)/2tors small enough, an, |V.I, < U + Thus, in view of (3.22), we obtain and this proves the claim. Now as in (3.24) we obtain So, in view of (/ 3 ) we have " / Therefore (3.16) and (3.27) imply (3.27) (3.28) 440 Ratikanta Panda Letting m-oc and then s-+0 we get c> C* + ((l/n) 0)oc, a contradiction. This completes the case of bounded { v w j. Case 2: { y m } is unbounded. In this case we change the role of { v m } and { w m ] and then we can still get a contradiction as above. Thus we have ruled out dichotomy and therefore by Lemma 1.1 of [5] there exists {y m } in R" such that for any s > 0, there is t = t(s) > such that (|VuJ n +|w !")<. (3.29) \x-y m \>t Claim 6: {j; m } is bounded. If not, then without loss of generality suppose y m -^ oo. Now Let ?j m be cut-off functions such that < i\ m < 1, i\ m = for \x - y m |< t; - y m \ ^ f + 1, 1 V/? m | ^ 2n. Then for < l/a n (4n)"j8e and m large, (3.30) = 1 for Then by Lemma 2.3 and (3.29) Similarly 0(e). Again as in the proof of (3.21), using the assumption y m -^ oo we obtain asm-^oo. (3.31) (3.32) (3.33) Thus I(u m )^I y '(u m )-O(8)-o(l). Again as earlier we can choose a m such that a m = 1 - 0(e) -f o(l), u m (.x) = u w (a m .x) is in M x and I(u m ) C 00 ~ Quasilinear elliptic equation 441 Taking W-+GO and then -0 we obtain c^ C, a contradiction which proves the claim. Therefore, for any > 0, there exists t = t(s) > such that :e. (3.34) |*|>r To use Strauss' lemma as in [3] we set P(s) = s ll ~ 1 /(x,s), 2(s)= exp((2a n /(l +r)) | 5 |/(- D) _ Li-o(l/m!)(2a ll /(l + r)) w |s|" /(B " 1} + |s| B , so that lim |jHao P(5)/fi(5) = 0. Also by Lemma 2.1, jQ(w m ) ^ C for some constant C > 0. Therefore by Strauss' lemma, for any bounded Borel set Q lim In particular f f lim u^" 1 /(.x,Mj= u n ~ l f(x,u). (3.35) Again, as in the proof of (3.21) we obtain (3.36) |x|>t Thus lim u^ 1 /(.x,uj= "- l f(x,u). (3.37) m--oo I I J J Claim?: w m ->uin P^ 1 ' n ([R w ). Since u m -^u weakly in ^^"(R") we have by Rellich's lemma u m -^u strongly in L n (Q) for any bounded smooth Q. In particular, f J\x Thus using (3.29) we get (3.38) As in (3.35), we have for any (peC$(R n \ ^. (3.39) Now, for any (peC?(R n ) we have, by (3.5), (3.38) and (3.39) 0=lim<I'(uJ,p> (3.40) 442 Ratikanta Panda = lim (r(u m ),u m (py m-oc r r = (pd^-h uT-V(p = \uT-V(p+ \<pT-Vu + \\u\"(p- \f(x,u)u"- l <p. (3.42) Thus |uT-V<p= |/(x,u)u"-V- J t/ J and substituting in (3.41) we get In view of (3.5) we get J^|ViU"-^f<pr-V M and hence f, x|<t |VM implies, using (3.29), || Vu m \ n -^JT-Vw. That is, lim |VuJ"=lim wi-oc j m 'X Then lim m-oc which implies r 7u m -Vw| M =0, by using an inequality \a~~b\ p ^2 p - l (\a\ p ~ 2 a-\b\ p - 2 b)'(a~b) for any a, beR", p > 2. Therefore w m -^w strongly in ^ Uw ([R w ) as desired. D Proof of Theorem 1.1: By the definition of S pJ for any e > there exists u e W l ' n (R") such that (KH/|tt,| p )<S p + e. Let i? = ((l -n9) lln /\\u e \\)\u e \ p . Then \\v t \\*=l-n0 9 Claim: $F(v )>(l/n)\v E \ n n . Choose e small enough so that C p >(p/n)(S p + 2s) p (l - ndY~ (pln \ Now S p Ikvll/kvLandso and this proves the claim. Quasilinear elliptic equation 443 Therefore by Lemma 2.2 1% is achieved by some w and / J ^ 1 n6. Then for some Lagrange multiplier AeR. By (/ 3 ) we have _ 1 w , Also we know that / >0. Thus A>0. Let W(X)=W O (A~ I/W X). Then u satisfies (1.1), fF(tt) = (l/H)J|Mr and /(w) = (l//7)J|Vwr = (l/n)J|Vw o r <(!/)- 0. This proves the theorem. D Proof of Theorem 1.2: By the assumptions we see that f(t) satisfies the conditions of Theorem 1.1. Thus there exists ueW l ' n (R n ) satisfying ~-A n u + \u\ n - 2 u = f(u)u n ~ 2 inR". - (3.43) Moreover, / x (u) <(!/) ~ 9. Let t" lwl"]- (rfi). By (/ 2 ) and (3.43) we have ft'(r)^0 forr>l. Hence / x (w) = max^ / x (rw). Further, since J(rw)--oo as r-^co, there exists t e(0,oc)_such that /(r u) = max^ /(rw). Now, by (/ 5 ) and the hypothesis that /(.x, t) = /(f) we have /(t fi) < /"'(.to*) ^ max /(m) = /*(ii). (3.44) t^O We claim that C x = /(M). Clearly C ^I%= /(M). Further, given > 0, we can find weM x such that / x (w) < C x -f e. Using (/ 2 ) we can find teR + such that Jf(x,rw) = (l/n)\tu\ n n . Again as above we can show that /(w) =max t>0 / oc (t4 Thus I^(u) = !%** I(tu) ^ I*(tu) ^ I(u) <* + , which gives the other inequality, since > was arbitrary. Therefore from (3.44) we get . I(t u)<r(u)^C<--6. (3-45) It is easy to see, using (f^ (f 2 ) and Lemma 2.3, that there exist p, a > with I(u) > a for all u satisfying \\u\\ = p. Choose t 1 > t sufficiently large so that l(tu) < for t > t^ Let F be the set of all continuous paths connecting and l^u. Define c==infsup/(u). (3-46) " 444 Ratikanta Panda Then c> a. Also c< max /(ru)< n By Mountain Pass lemma (see [4]), there exists a sequence {u m } in W lttt (R n ) such that /("J-^ /'(Hj-^0 in (^-"(R"))*. By Lemma 3.1, for a subsequence u m ~^u strongly in W 1 ' n (IR n ). Thus /(w) = c, /'(M) = 0, which implies that u ^ and u is a nontrivial solution of (1.1). This completes the proof of the theorem. D Acknowledgement The author would like to thank Prof. Adimurthi for his valuable advice and the National Board for Higher Mathematics, India for the financial support. He would also like to thank Prof. Tilak Bhattacharya for his valuable suggestions. References [1] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the w-Laplacian, Ann. Scu. Norm. Sup. Pisa 17 393-413 (1990) [2] Aubin Th, Nonlinear analysis on manifolds: Monge- Ampere equations. (New York: Springer- Verlag) (1982) [3] Berestycki H and Lions P L, Nonlinear scalar field equations, I. Existence of ground state, Arch. Ration. Mech. Anal. 82 (1983) 313-346 [4] Brezis H and Nirenberg L, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Commun. Pure Appi Math. 36 (1983) 437-477 [5] Cao D M, Nontrivial solution of semilinear elliptic equations with critical exponent in !R 2 , Commun. Partial Differ. Equ. 17 (1992) 407-435 [6] Lions P L, The concentrated-compactness principle in the calculus of variations. The locally compact case, part I, Ann. I.H.P. Anal. Nonlin. 1 (1984) 109-145 [7] Lions P L, The concentrated-compactness principle in the calculus of variations. The locally compact case, part II, Ann. I.H.P. Anal. Nonlin. 1 (1984) 223-283 [8] Moser J, A sharp form of an inequality by N Trudinger, Indiana Univ. Math. J. 20 (1971) 1077-1092 [9] Panda" R, On semilinear Neumann problems with critical growth for the rc-Laplacian, Nonlinear analysis, theory, methods and applications [10] Talenti G, Best constants in Sobolev inequality, Ann. Mat. Pura. Appl. 110 (1976) 353-372 Proa Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 4, November 1995, pp. 445-459. Printed in India. An axisymmetric steady-state thermoelastic problem of an external circular crack in an isotropic thick plate RINA BHOWMICK and BIKASH RANJAN DAS Department of Mathematics, Jadavpur University, Calcutta 700032, India MS received 22 September 1994 Abstract. A steady state thermoelastic mixed boundary value problem for an isotropic thick plate is considered in this paper. The faces of an external circular crack situated in the mid-plane of the plate are opened up by the application of temperature while the bounding surface of the plate are maintained at a constant zero temperature. Solution valid for large values of the ratio of the plate thickness to the diameter of the crack has been obtained. Expressions for various quantities of physical interest are derived by finding iterative solutions of the equations and the results are shown graphically. Keywords. Axisymmetric; steady-state; external circular crack; stress-intensity factor. 1. Introduction The strength of a material with cracks is an interesting problem in fracture as well as structural mechanics and the knowledge of the elastic stress field is potentially useful for strength estimation based upon brittle fracture theory. Several papers have appeared which treat distributions of stress in an infinite solid due to the application of temperature or normal pressure on the faces of a flat internal circular crack (Das and Ghosh [2], Lowengrub [5], Bandy opadhyay and Das [1]). The problem of an infinite body containing an external circular crack covering the outside of a circle, due to the application of normal pressure has been considered by Uflyand [12] using toroidal coordinates and by Lowengrub and Sneddon [6] from the dual integral equation point of view. Lowengrub [7] has also solved the two-dimensional plane strain problem for an external crack y = 0, | x| > 1 opened up by normal pressure, using dual trigonometric equations. Distribution of stress in a thick plate containing an external circular crack opened up by the application of pressure has been considered by Dhawan [4]. This paper determines the thermoelastic stress distribution in the vicinity of an external circular crack situated in the mid-plane of an isotropic elastic plate of finite thickness and infinite radius. The temperature, the shear component of stress tensor and the normal component of displacement vector vanish over the plane boundaries while the crack is opened up by the application of a prescribed axially symmetric temperature to its faces. The method of solution is to seek suitable representations of the potential of thermoelastic displacements and the Love function and then to reduce the problem to the solution of two pairs of dual integral equations. Finally, these dual integral equations have been further reduced to Fredholm integral equations of the second kind which are solved in terms of power series. The results are illustrated by a number of diagrams (figures 2-7). 446 Rina Bhowmick and Bikash Ranjan Das 2. Basic equations of thermoelasticity We consider the temperature and displacement fields in an isotropic elastic solid which is conducting heat. If we assume that there is symmetry about an axis, which we take to be the z-axis, then the position of a typical point of the solid may conveniently be expressed by the cylindrical polar, coordinates (r, 0, z) and the displacement vector will have the components (w rs 0, wj. The non-vanishing components of the stress tensor will In the absence of body forces or heat sources within the solid, the steady-state equations of thermoelasticity with symmetry about z-axis are (Sneddon and Berry [10], p. 125) 2(l-v)f^ + i^-^ + (1 - 2v) ^ + ^ = 2(1 + v)a ^ \ dr" r or r" J dz* or 02 or (1 _ 2v) fe + i^ +2(1 _ v) ^ + ^f^ + ^ =2(1+v)a f: \ or" r dr J dz dz \ or r J oz (1) and V 2 T = 0, (2) where T = T(r, z) is the deviation of the absolute temperature of the solid from that in a state of zero stress and strain, a is the co-efficient of linear thermal expansion of the solid, v is its Poisson ratio and y2 | | /3\ dr 2 r dr dz 2 ' 3. Boundary conditions With a suitable choice of our unit of length we can assume that the faces of the crack are described by the relations z = ,r > 1. The thickness of the plate is assumed to be <5-times the diameter of the crack. We suppose that there is no external force acting on the crack-faces and that the face z = 0+,r^lis heated (or cooled) exactly in the same way as the face z = 0-, r^l. Then following Sneddon [9] we reduce the crack problem for the thick plate r > 0, |z| ^ 8 to the mixed boundary value problem for the layer r ^ 0, ^ z ^ <5 for which the thermal and elastic conditions are: on z = 0: T(r,0) = /(r), l<r<oo (5) (^(r, 0)=0, 0<r<oo (6) u a (r,0) = 0, 0^r<l (7) <r sz (r,0)=0, l<r<oo (8) on z = <5: An axisymmetric steady-state thermoelastic problem 447 T=0, OJ1 Z =0, U z = = f(r) cr rz =o,u z :=o 26 T = , C z = , U z = T = f (r) Figure 1. <oo (10) u,M) = 0, 0<r<oo, (11) where/(r) is prescribed. We further assume that the disturbance is localized i.e. the temperature and the components of stress and displacement all vanish as ^/(r 2 4- z 2 )-* oo. Position of the crack and the boundary conditions for the plate are indicated in figure L 4. The heat conduction problem A suitable Hankel integral representation of the temperature field satisfying the Laplace's equations (2) and (9) and vanishing at infinity is taken in the form f )= J o (12) where B() is an unknown function to be determined from the boundary conditions. Conditions (4) and (5) are fulfilled if the function B({) is a solution of the set of dual integral equations where f J c f J c = 0, 0<r<l -=/(r), l<r<oo, (13) (14) To reduce the above equations to a single integral equation, we apply Sneddon's method [11] and put /j ^\ where, for the convergence of the integral, we assume that lim\l/ l (t) = Q. (^ 448 Rina Bhowmick and Bikash Ranjan Das Integrating by parts and making use of (17) we rewrite (16) in the form where the prime (') denotes differentiation. Substituting from (18) and making use of the result ( [13] p. 405) J (6-)cos(c;f)dc = 0, r<t 7, r>t we can show that 0, = 1 f r \l/\(t) , H- . , , df. l<r<oo. - 2 - 2 It is clear from (20) that the form (16) satisfies (13). Now, from (14) we have J 1 Making use of the result [13], p. 405 1 J (cr)sin(ct)dc= 0, r<t t<r we find from (21) (18) (19) (20) (21) (22) Kr<oo. (23) If we replace J (cr) by its integral representation, 2 we find that the second term on LHS is equal to Simplifying and interchanging the order of integrations the second term on LHS becomes 1 dt An axisymmetric steady-state thermoelastic problem 449 where flf(o>)= I* H^cosf^du. (24) Jo \ / Then from (23) we have = /(r), 1 < r < oo or, ^[^ = /(r), l<r<oo, which on inversion gives I ^(t\ - = _2d f 00 rf(r) or, 2 d r, KKCO. (25) where KI(U, t) = H*(t-u)- H*(t + M). (26) 5. The thermoelastic problem * The potential <D of thermoelastic displacement satisfying the Poisson equation (Nowacki [8], p. 12) V 2 <& = mT, where m = (1 + v)a/(l - v), is 2 o fr)df. (27) The Love function satisfying the Inharmonic equation (Nowacki [8], p. 17) V 4x F = 0, is sought in the form of the Hankel integral - z) - <5 sinh(^z)cosech(^<5)] J ({r)df (28) which vanishes at infinity. Using basic equations, we have - z 450 Rina Bhowmick and Bikash Ranjan Das + C(f)cosech(5)[2(l - v)sinh (5 - z) + z cosh (d - z) o (29) -z) + z sinh (<5 - z) o cosh((?z)cosech(c<5)] J (^)dC (30) o B(?)sech(^)sinh {(5 - z) J^r)^ -2/x J o r)d. (31) Equations (6), (10) and (11) are automatically satisfied. Using boundary conditions (7) and (8) we get, 1 (32) = ~ ?/(>*), l<r<oo (33) o where Following Lowengrub and Sneddon [6] we put C()= I iA 2 (t)cos(^)dt, (35) where we assume that lim^ 2 (t) = 0. (36) r-*oo Integrating by parts and making use of (36) we rewrite (35) in the form , (37) where the prime (') denotes differentiation. Substituting (35) and making use of the result (19) we have 0, 0^r<l (38) , 1 <r< oo. An axisymmetric steady-state thermoelastic problem 451 It is clear from (38) that the form (35) satisfies (32). From (33) we have m =--/(r), l<r<oo. (39) z The first term on the LHS of the above integral equation (39) becomes Replacing J (r) by its integral representation 2 the second term on the LHS of the above integral equation (39) becomes 2 Interchanging the order of integration and simplifying the above term becomes 1 f 00 dt where r / '// -A ?. (40) Thus (39) becomes m or, ] = -/W> l<r<oo, which on inversion gives 1 f 00 ' * * m d p r/(r)dr 2t+7 ^Ji 2t 2 ^ TrdrJ, V(r 2 -r 2 )' l<t<oo. (41) Assuming that /(r) is continuous differentiate in (1, oo), we integrate (41) between the limits t to oo and on making use of (36), we obtain the following Fredholm integral 452 Rina Bhowmick and Bikash Ranjan Das equation of the second kind where K 2 ( M ,f) = (42) (43) 6. Method of solution Assuming that 6 1, we can write (15) and (34) as and Using (24) and (44) we have from (26) (44) (45) 3605 6 where (46) (47) To solve the Fredholm integral equation (25) we assume a series solution in the form (48) Then from the Fredholm integral equation (25), we have 7i df ; 2j n (49) etc. An axisymmetric steady-state thermoelastic problem 453 Similarly for the Fredholm integral equation (42), we assume a series solution in the form and we obtain a set of equations of the form (49). 7. Solution for a particular type of temperature distribution: Quantities of physical interest In this section we solve the integral equations (25) and (42) for large values of (5, by giving a particular value of/(r) which is important from the physical point of view. Let /(r) be defined as /(r)=-/ H(fl-r), a>\ (51) where H(t) is the Heaviside unit function. Then t<a (52) t>a. Substituting this value in (25) we get i) For t > a: It can be shown that its trivial solution is ii) Forr<a: In this case integral equation (25) becomes which on considering terms up to 6 6 gives ., M _ 2/o t (55) 454 Rina Bhowmick and Bikash Ranjan Das + t 3a 4 Substituting the above values for ^(r) we have from (12) , , where A = ^ cos -ii ^ 11 2 a 12 ~ cos - + A 12 =~-cos" 1 - + 16 a 48 and (56) (57) (58) Similarly we get a trivial solution \j/ 2 (t) = of the Fredholm integral equation (42), for t > a. ii)' For t < a: In this case X, + HM fa 6 -* 6 4 - 4 2 120<5 6 6 where (59) (60) An axisymmetric steady-state thermoelastic problem 455 Using the method used earlier an iterative solution for i// 2 (r) is obtained in the form vhere ' , (61) '. (62a) Now we derive expressions for quantities of physical interest. Using (35) in (29) we have on the crack plane z = 0, M s (r,0) = 2(l- J (cr)cos(a)dc;. Nfow substituting the value of ^ 2 (0 from (61), we can easily find that o r r.j I ' F(/) l -N(r) vhere H -> i 1 o -i 1 - = - a" - -r- | cos ' r r 2 a(3r 2 + 2a 2 )J(r 2 -a 2 ) (3r 2 (63) (64) (65) ,(66) 456 Rina Bhowmick and Bikash Ranjan Das and E(a/r\ E(j!,fl/r), K(a/r\ F(fra/r) are elliptic integrals. The normal component of stress on z = is given by 3(a 2 -l) l)(2a 2 - a - 1) +3B 12 Q(r) '11 3 STCO where 48 The stress intensity factor is given by N= Urn r-*l- (67) (62b) (68) 0-2 0-4 0-6 0-8 1-0 -0-5 Figure 2. Variation of T(r,0)/(2/ /7r) with r for a = 1-2, 1-6, 2-0 and 6-5. An axisymmetric steady-state thermoelastic problem 457 0-2 0-4 0-6 0-8 1-0 Figure 3. Variation of T(r, 0)/(2/ /*) with r for a = 1.2, 1.6, 2.0 and 8 = 7. 0-2 0-4 0-6 0-8 1-0 Figure 4 Variation of <7(r, 0)/(2 mju/o/rc) with r for a = 1.2, 1.6, 2.0 and 8 = 5. Using (67) we have from (68) (69) Quantities of physical interest namely, the temperature and the normal components of stress and displacement on the crack plane z = have been calculated for a ==1.2 IA 2.0 and 5 = 5, 7. Variations of T(r, 0), <r Z2 (r, 0) and ,(r, 0) with r are shown graphically m figures 2-7 respectively. Rina Bhowmick and Bikash Ranjan Das 0-2 0-4 0-6 0-8 LO Figure 5. Variation of a : J(r, 0)/(2 m/*/ /rc) with r for a = 1.2, 1.6, 2.0 and 6-1. Figure 6. Variation of w,(r,0)/(2(l - v)m/ /7r) with r for a = 1.2, 1.6, 2.0 and 6 = 5. 8. Conclusions When (5 -^ oo, the problem reduces to that of an infinite medium containing an external circular crack which has been solved by Das [3]. It is found that the limiting values as 8 -> oc of the temperature, stress intensity factor and the normal components of stress An axisymmetric steady-state thermoelastic problem 459 Figure 7. Variation of w s (r, D)/(2(l - v)m/ /7r) with r for a = 1.2, 1.6, 2.0 and <5 = 7. and displacement given by (56), (69), (67) and (64) are the same as those obtained by Das. References [1] Bandyopadhyay S and Das B R, Stress in the vicinity of a penny-shaped crack in a transversely isotropic thick plate, Proc. Indian Nat. Sci. Acad. A60 (1994) 503-512 [2] Das B R and Ghosh S, Thermoelastic stresses in a thick plate containing a penny shaped crack in the mid-plane, Geophys. Res. Bull 15 (1977) 65-70 [3] Das B R, Some axially symmetric thermal stress distributions in elastic solids containing cracks-I: An external crack in an infinite solid, Int. J. Engg. Scl 9 (1971) 469-478 [4] Dhawan G K, The distribution of stress in the vicinity of an external crack in an infinite elastic thick plate, Ada Mech. 16 (1973) 255-270 [5] Lowengrub M, Stress in the vicinity of a crack in a thick elastic plate, Q. J. Appl. Math. 19 (1961) 1 19 [6] Lowengrub M and Sneddon I N, The distribution of stress in the vicinity of an external crack in an infinite elastic solid, Int. J. Engg. Sci. 3 (1965) 451 [7] Lowengrub M, A two-dimensional crack problem, Int. J. Engg. Sci. 4 (1966) 289-299 [8] Nowacki W, Thermoelasticity (London: Pergamon Press) (1962) [9] Sneddon I N, The distribution of stress in the neighbourhood of a crack in an elastic solid, Proc. R. Soc. London AIM (1946) 229 [10] Sneddon I N and Berry D S, The classical theory of elasticity Handbuch der Physik, Bd. VI. (Springer), (1958) [1 1] Sneddon I N, The elementary solution of dual integral equations, Proc. Glasg. Math. Asso. 4 (1960) 108 [12] Uflyand Ya S, Elastic equilibrium in an infinite body weakened by an external circular crack, J. Appl. Math. Mech. 23 (1959) 134 [13] Watso.n G N, A Treatise on Bessel functions (1st paperback edition) (Cambridge: University Press) (1966) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 4, November 1995, pp. 461-469. Printed in India. Some characterization theorems in rotatory magneto thermohaline convection JOGINDER SINGH DHIMAN Department of Mathematics, Government Senior Secondary School, Jalag (Kangra) H.P. 176094, India MS received 22 August 1994; revised 10 January 1995 Abstract. The present paper extends the results of Banerjee et al [2] for the hydromagnetic thermohaline convection problems of Veronis' [9] and Stern's [8] types to include the effect of a uniform vertical rotation. Keywords. Hydromagnetic thermohaline convection; uniform vertical rotation. 1. Introduction The establishment of non-occurrence of any slow oscillatory motions which may be neutral or unstable imply the validity of the principle of exchange of stabilities (PES). The validity of PES in a certain class of stability problems eliminates the unsteady terms from the linearized perturbation equations which results in notable mathemat- ical simplicity since the transition from stability to instability occurs via a marginal state which is characterized by the vanishing of both real and imaginary parts of the complex time eigenvalue associated with the perturbation. Pellew and Southwell [5] proved the validity of PES for the classical Rayleigh-Benard convection problem (RBCP). Chandrasekhar [3] in his investigations of hydromagnetic RBCP conjectured that if the total kinetic energy associated with a perturbation exceeds the total magnetic energy associated with it, then PES is valid. Sherman and Ostrach [7] established the above conjecture of Chandrasekhar for a more general problem when the fluid is confined in an arbitrary region and the uniform magnetic field is applied in an arbitrary direction. However, the result of Sherman and Ostrach is of limited value since one cannot a priori be certain when their criterion will be satisfied. Banerjee et al [1] established that for the hydromagnetic RBCP if Q^ l /n 2 ^ 1, where Q is the Chan- drasekhar number and <T I is the magnetic Prandtl number, then the total kinetic energy associated with an arbitrary perturbation which may be neutral or unstable is greater than the total magnetic energy associated with it and consequently PES is valid in this parameter regime. Banerjee et al [2] further extended these energy considerations to the hydromagnetic thermohaline convection problems of Veronis' [9] and Stern's [8] types. The aim of the present paper is to extend the results of Banerjee et al for the hydromagnetic thermohaline convection problems of Veronis' and Stern's types to include the effect of a uniform vertical rotation. 2. Basic equations and boundary conditions The non-dimensional linearized perturbation equations governing thermohaline con- vection problem in the presence of a uniform vertical rotation and magnetic field are 461 462 Jocjinder Singh Dhiman given by (cf. Gupta et al [4]). (D 2 -a 2 )(D 2 -a 2 -p!a)\v = Ra 2 9-R s a 2 (j)~QD(D 2 -a 2 )h^ TDZ (1) (D 2 -fl 2 -p)0=-w (2) (D 2 ~a 2 -p/i)<p=-w/T (3) (D 2 -a 2 -pa l /ff)h s =-Dw (4) (D 2 - a 2 - p/<r)Z = - Dw - QDX (5) (D 2 -a 2 ~paJa}X = -DZ (6) together with the boundary conditions w = = = < = Dw = Z = DX = h : at i = ,1 . (7) The various symbols occurring in the above equations are defined as follows: z is real independent variable such that ^ z ^ 1 and stands for vertical coordinate, D = d/dz denotes the derivatives with respect to z. a 2 is the square of the wave number, a is the thermal Prandtl number, i is the Lewis number, a l is the magnetic Prandtl number, jR is the thermal Rayleigh number, R s is the thermohaline concentration Rayleigh number, is the Chandrasekhar number, T is the Taylor number and P = Pr + 'Pi is a complex constant in general representing the complex growth rate. Further \v, 0, fa Z, X and h z are complex valued functions of z and stand respectively for the vertical velocity, temperature, concentration, vertical vorticity, vertical current density and vertical magnetic field. We note that R>0 and R s > for Veronis 1 configuration whereas for Stern's configuration, we have R < and R s < 0. System of eqs (l)-(7), constitute an eigenvalue problem for p for given values of a 2 ,R<R s ,Q, T, o and a l and a given state of the system is stable, neutral or unstable according to p r < or p r = or p r > 0. Further, if p r = implies p f = for all wave numbers a 2 , then the principle of exchange of stabilities (PES) is valid, otherwise we will have overstability at least when instability sets in certain modes. 3. Mathematical analysis We prove the following theorems: Theorem 1. A necessary condition for the existence of a nontrivial solutions (p, w, 0, fa /!_, X, Z) of eqs (l)-(7) with R > 0, R s > and. p = p r -f ip., p. ^ is that ^i<(-/2 + J r 3 + ^) 5 (8) where (9) (10) Rotatory magneto thermohaline convection 463 and J 4 =rf 1 |z| 2 dz. (ii) J Proof, Multiplying eq. (1) by w* (the complex conjugate of w) integrating the resulting equation over the range of z, we have , , , , , f 1 , f 1 ^ ~a~)(D~ a~ p/<7)wdz = Ra" w*6dz R s a~ w*<pdz o Jo Jo pi pi + T\ w*DZdz-g w*Z)(Z) 2 -0 2 )/z 3 dz. (12) Jo Jo Using eqs (2)-(6) and boundary conditions (7), we can write f 1 f 1 Ra 2 u'*0dz= -/ta 2 0(Z) 2 -fl 2 p*)#*dz, (13) Jo Jo pi pi -R s a 2 w*(f)dz = R s a 2 <f)(D 2 - a 2 -p*/r)^*dz, (14) Jo Jo i ri Jo i o i pi pi w*DZdz=- Dw*Zdz= Z(D 2 -a 2 -p*/a)Z*dz o Jo Jo pi pi pi + Q ZDX*dz= Z(D 2 -a 2 -p*/v)Z*dz-Q DZX*dz Jo Jo Jo 1 /! o o (16) It follows from eqs (12)-(16) that 1 pi vv*( J D 2 -a 2 )(D 2 -fl 2 -p/cr)wdz= - Ra 2 6(D 2 -a 2 -p*)0*dz o Jo i r 2_ 2__ Jo P 1 2 ^,2 1*\V*A^ nj\ Integrating various terms of eq. (17) by parts for an appropriate number of times and making use of boundary conditions (7), we have ^1+^2+^3 + ^4 + ^5 +^6 = 0> (18) 464 Joginder Singh Dhiman where 1 I 2 = - Ra 2 I= 4 o / 5 = and Equating the imaginary parts of both sides of eq. (18) and cancelling p,.( ^ 0) through- out, we have i pi pi (|Dw| 2 +<r|w 2 )dz-fRa 2 <7 Iflpdz + QTo-! |*| 2 dz o Jo Jo -<R s a 2 a I |<| 2 dz+ J o (19) |X| 2 dz = 0. (20) o o Equation (20) clearly implies that This completes the proof of the theorem. We note that expressions for J l , J 2 , J 3 and J 4 as given by eqs (8)-(l 1) respectively, represent the total kinetic energy, magnetic energy, concentration energy and rota- tional energy. In view of this, Theorem 1 can be restated as follows: A necessary condition for the existence of oscillatory motion which may be stable, neutral or unstable for Veronis" thermohaline convection problem in the presence of a uniform vertical rotation and magnetic field is that the sum total of magnetic, concentration and rotational energies must exceed the total kinetic energy or, equival- ently, if the total kinetic energy exceeds the sum total of magnetic, concentration and rotational energies, then the oscillatory motions are not allowed. The above result, no doubt yields us a condition in terms of energies of the system for the non-occurrence of oscillatory motions, however, it is of limited value, since one can Rotatory magneto thermohaline convection 465 not a priori be certain when this condition will be satisfied as it involves the unknown eigen functions of the problem. It will therefore be more useful to express this condition in terms of the parameters of the problem prescribed by the fluid properties. We PCtaKlieh tVno in tVi<a fr\11r\\x/inrr tt*A/"\rp>m establish this in the following theorem. Theorem 2. // (p,w,0,<k/z z ,Jr,Z), p = p r + ipi,Pi>Q, P r >0, R>0 and R s >0 is a solution of eqs (l)-(7) and ^ + ^ + -7 < 1, then, J l > (J 2 + J 3 + J 4 ). [_ 71 2T"7T 7C J Proof. Multiplying eq. (3) by its complex conjugate, integrating over the range of z by parts a suitable number of times and making use of boundary conditions (7), we have Since, p r ^ 0, therefore eq. (21) gives f 1 1 f 1 la 2 |>0| 2 dz< |w| 2 dz Jo T ~ J o which upon using Poincare inequality [6] f 1 ' f 1 7i 2 \ |</>| 2 dz^ |D0| 2 dz (since0(0) = 0: Jo Jo yields that f 1 1 f 1 a 2 |</>| 2 dz< 7T 2 |w| 2 dz. (22) Jo 2t~ J Further, since w(0) = = w(l) also, therefore |Dw| 2 dz. (23) IL J Combining inequalities (22)-(23), we have |w| 2 )dz. (24) Multiplying eq. (4) by /z? (the complex conjugate of h s \ integrating the resulting equation by parts a suitable number of times in the range of z, making use of boundary conditions (7) and then equating the real parts from both sides of the resulting equation, we have ri n \h,fdz o 466 Joginder Singh Dhiman r P = real part of wDh*dz LJo f 1 ^ J o w||D/L|dz i "li/2 r pi 11/2 |w 2 dz |D/i_| 2 dz o J LJ o (by Schwartz inequality). Since p r ^ 0, therefore inequality (25) implies that |D/i = | 2 dz< |w| 2 dz . (25) (26) Combining inequalities (23), (25) and (26), we get 1 f 1 5 ") i i i ) , . J- iDw| 2 dz I o 4,f. n ~ Jo (27) Now, multiplying eq. (5) by Z* (the complex conjugate of Z\ integrating by parts a suitable number of times, using boundary conditions (7) and equating the real parts of the resulting equation, we have fi (|DZ| 2 + a 2 |Z| 2 + p>|Z| 2 )dz + Q (|Djq 2 + a 2 |*| 2 -f p r < o Jo = real part of ( Z*Dwdz I o = real part off - wDZ*dz wDZ*dz 1/2 r pi |DZ| 2 dz 'o J LJo (by Schwartz inequality) which by virtue of inequality (23) and the fact that p r ^ gives |DZ| 2 <-if 1 |Dw| 2 dz - wDZ*dz J o |w||DZ|dz Rotatory magneto thermohaline convection 467 |Dw| 2 + a 2 |w| 2 )dz. (28) Inequality (28) together with the Poincare inequality \DZ\ 2 dz leads to the inequality l \Z\ 2 dz<\ r(|>wJ 2 + 2 |wi 2 )dz. (29) il J Combining inequalities (24), (27) and (29), we have " < 30 > Inequality (30) clearly implies that if /? T "1 Theorem 2 implies that if ~ + "7" 4 + 4 | ^ U then the total kinetic energy then J 1 >(J r 2 H-J 3 +J 4 ). This completes the proof of the theorem. L 2l 2 7I 4 ' ^? associated with an arbitrary oscillatory (p f ^ 0) perturbation which may be neutral (p r = 0) or unstable (p r > 0) exceeds the sum total of its magnetic, concentration and rotational energies. In particular, it follows that, in the parameter regime rH I-^H- ^ 1, the principle of exchange of stabilities is valid for the problem under consideration. Theorem 3. A necessary condition for the existence of a nontrivial solutions (p, vv, 0, 0, /i., X, Z) of eqs (l)-(7) with R < 0, jR s < and p = p r -f ip h p t ^ is that where J t , J 2 anrf J 4 are as a/y^n by egs (8), (9), anJ (11) and \9\ 2 dz. (31) Proof. Putting R = \R\ and R s = \R S \ in eq. (18) and proceeding exactly as in Theorem 1, we get the desired result. Keeping in view the fact that J 5 represents the thermal energy, Theorem 3 can be restated as follows: A necessary condition for the existence of oscillatory motions which may be stable, neutral or unstable for Stern's thermohaline convection problem in the presence of a uniform vertical rotation and magnetic field is that the sum total of magnetic, thermal and rotational energies must exceed that total kinetic energy, or, equivalently, if the 468 Joginder Singh Dhiman total kinetic energy exceeds the sum total of magnetic, thermal and rotational energies then the oscillatory motions are not allowed. Further, Theorem 3 is qualitatively of the ^ same form as Theorem 1 and possesses the same drawback. We remedy this in the ^ following theorem analogous to Theorem 2. Theorem 4. // (p,w,fl,<,/i_,jr,Z),p== j p r + /p,.,p ( .^0,p r ^O,R <0 and R s <0 is the solution of the eqs(l)-(l) and ^ + V~ + , U: 1, then 7T 27T 7T Proof. Multiplying eq. (2) by its complex conjugate, integrating by parts a suitable number of times over the range of z, using boundary condition (7) and equating the real parts of the resulting equation, we have (\D9\ 2 o f| Jo |fl| 2 dz = lwfdz. (32) o Since, p r ^ 0, it follows from eq. (32) that n ri Jo Jo which upon using the Poincare inequality M 6H2 d ~<lf l 2 n ~ J and inequality (23) gives f 1 1 f 1 a 2 |0| 2 dz< |Dvv| 2 dz Jo 27C J o It follows from inequalities (24), (29) and (33) that (33) l (34) Inequality (34) clearly implies that if then J 1 >(J 2 +J 4 -hJ 5 ). This completes the proof of the theorem. Rotatory magneto thermohaline convection 469 Theorem 4 implies that if =~ + - ~- + -^ < 1, then the total kinetic energy L TT 2;r it J associated with an arbitrary oscillatory perturbation which may be neutral or unstable exceeds the sum total of its magnetic, thermal and rotational energies. In particular it follows that in the parameter regime ~ H -r- + z ^1, the PES is valid for the L *- 2 * n J problem under consideration. Theorems 1-4 clearly provide a natural extension of the results of Banerjee et al [12] as could be easily seen by putting T = 0. Acknowledgement Thanks are extended to the learned referee for his valuable comments on an earlier version of the paper. References [1] Banerjee M B and Gupta J R, Studies in hydrodynamic and hydromagnetic stability, (Shimla, India: Silver-line Publ.)(1991) [2] Banerjee M B, Gupta J R and Katyal S P, A characterization theorem in magneto thermohaline convection, J. Math. Anal. Appi 144 (1989) 141-146 [3] Chandrasekhar S, Hydrodynamic and hydromagnetic stability, (Oxford: Clarendon Press) (1961) [4] Gupta J R, Sood S K and Bhardwaj U D, The characterization of non-oscillatory motions in rotatory hydromagnetic thermohaline convection. Indian J. Pure AppL Math. 17 (1986) 100-107 [5] Pellew A and Southwell R V, On the maintained convective mo tion in a fluid heated from below, Proc. R. Soc. London A176 (1940) 312-343 [6] Schultz M H, Spline Analysis, (New Jersey: Prentice Hall) (1973) [7] Sherman M and Ostrach S, On the principle of exchange of stabilities for the magnetohydrodynamic thermal stability problem in completely confined fluids, J. Fluid Mech. 23 (1966) 661-671 [8] Stern M E, The salt fountain and thermohaline convection, Tellus 12 (1960) 172-175 [9] Veronis G, On finite amplitude instability in thermohaline convection, J. Mar. Res. 23 (1965) 1-17 Proceedings (Mathematical Sciences) Volume 105, 1995 SUBJECT INDEX \A\ k summability Absolute summability of infinite series 201 Abel-Jacobi maps The Hodge conjecture for certain moduli varieties 371 Adjoining identity to a Banach algebra Uniqueness of the uniform and adjoining identity in Banach algebras 405 Analytic continuation Computer extended series solution to viscous flow between rotating discs 353 Asymptotic expansion Surface waves due to blasts on and above inviscid liquids of finite depth 227 Auxiliary functions A bibasic hypergeometric transformation as- sociated with combinatorial identities of the Rogers-Ramanujan type 41 Axisymmetric An axisymmetric steady-state thermoelastic pro- blem of an external circular crack in an isotropic thick plate ' 445 Banach space Degree of approximation of functions in the Holder metric by (e, c) means 315 Bibasic hypergeometric series A bibasic hypergeometric transformation as- sociated with combinatorial identities of the Rogers-Ramanujan type 41 Bilateral generating relations Certain bilateral generating relations for gene- ralized hypergeometric functions 297 Binary operation The algebra /4 p ((0, oo)) and its multipliers 329 Biot's theory Reflection of P-waves in a prestressed dissipative layered crust 341 Blast theory Surface waves due to blasts on and above inviscid liquids of finite depth 227 Brown's method Computer extended series solution to viscous flow between rotating discs 353 | C, 1 | fc summability Absolute summability of infinite series 201 Caratheodory condition A note on integrable solutions of Hammerstein integral equations 99 Cesaro On the partial sums, Cesaro and de la Vallee Poussin means of convex and starlike functions of order 1/2 399 Characterization of polynomials Characterization of polynomials and divided difference 287 Chern-Simons forms Flat connections, geometric invariants and energy of harmonic functions on compact Riemann surfaces 23 Chow groups The Hodge conjecture for certain moduli varieties 371 Classical orthogonal polynomials Certain bilateral generating relations for gene- ralized hypergeometric functions 297 Compact quantum group Induced representation and Frobenius reciprocity for compact quantum groups 157 Complexity Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays 59 Concentration compactness Nontrivial solution of a quasilinear elliptic equation with critical growth in U" 425 Critical growth Nontrivial solution of a quasilinear elliptic equation with critical growth in (R n 425 Decentralized control Control of interconnected nonlinear delay differential equations in W 73 Deficient cubic spline Local behaviour of the first derivative of a deficient cubic spline interpolator 393 de la Vallee Poussin means On the partial sums, Cesaro and de la Vallee Poussin means of convex and starlike functions of order 1/2 399 Delay equations Oscillation of higher order delay differential equations 417 Control of interconnected nonlinear delay differential equations in W ( ^ } 73 Delta waves Solution of a system of nonstrictly hyperbolic conservation laws 207 Depression Control of interconnected nonlinear delay differential equations in W ( ? 73 Derivatives On the zeros of (^ - a (on the zeros of a class of a generalized Dirichlet series - XVII) 273 471 472 Subject index Difference operator Certain bilateral generating relations for gene- ralized hypergeometric functions 297 Differential subordination Differential subordination and Bazilevic functions 169 Diophantine approximation Badly approximate p-adic integers 123 Dirichlet kernel On L 1 -convergence of modified complex trigono- metric sums 193 Dissipative Reflection of P-waves in a prestressed dissipative layered crust 341 Distinct points Characterization of polynomials and divided difference 287 Distribution of zeros On the zeros of C|^ - a (on the zeros of a class of a generalized Dirichlet series - XVII) 273 Divided difference Characterization of polynomials and divided difference 287 Eigenvectors Irrationality of linear combinations of eigenvectors 269 Elliptic equation Nontrivial solution of a quasilinear elliptic equation with critical growth in R" 425 Energy of maps Flat connections, geometric invariants and energy of harmonic functions on compact Riemann surfaces 23 Equivalent symmetric matrix Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays 59 Equivariant cobordism Equivariant cobordism of Grassmann and flag manifolds 381 Euler transformation Computer extended series solution to viscous flow between rotating discs 353 Expansion formulae A theorem concerning a product of a general class of polynomials and the H-function of several complex variables 291 Explicit formula Solution of a system of nonstrictly hyperbolic conservation laws 207 External circular crack An axisymmetric steady-state thermoelastic pro- blem of an external circular crack in an isotropic thick plate 445 Fejer kernel On L 1 -convergence of modified complex trigono- metric sums 193 Flag manifold Equivariant cobordism of Grassmann and flag manifolds 381 Flat connections Flat connections, geometric invariants and energy of harmonic functions on compact Riemann surfaces 23 Fourier series Degree of approximation of functions in the Holder metric by (e, c) means 3 1 5 Fourier transform Uncertainty principles on certain Lie groups 135 Frobenius theorem Fibred Frobenius theorem 31 Function algebra Weakly prime sets for function spaces 41 1 Function space Weakly prime sets for function spaces 41 1 Functional differential equations Oscillation in odd-order neutral delay differential equations 219 Galois group Irrationality of linear combinations of eigen- vectors 269 Gelfand transforms The algebra X p ((0, oo)) and its multipliers 329 General class of polynomials Convolution integral equations involving a general class of polynomials and the multivariable H-function 187 A theorem concerning a product of a general class of polynomials and the H-function of several complex variables 291 Generalized Dirichlet series On the zeros of ( ^ - a (on the zeros of a class of a generalized Dirichlet series - XVII) 273 Generalized hypergeometric functions Certain bilateral generating relations for gene- ralized hypergeometric functions 297 Grassmann manifold Equivariant cobordism of Grassmann and flag manifolds 38 1 Gravity waves On over-reflection of acoustic-gravity waves incident upon a magnetic shear layer in a compressible fluid 105 Growth of capital stock Control of interconnected nonlinear delay differential equations in W\ i} 73 Growth of sets A note on the growth of topological Sidon sets 281 h-conformal tensor On infinitesimal /i-conformal motions of Finsler metric 33 Subject index 473 //-function of several complex variables A theorem concerning a product of a general class of polynomials and the H-function of several complex variables . 291 Hammerstein integral equation A note on integrable solutions of Hammerstein integral equations 99 Harmonic maps Flat connections, geometric invariants and energy of harmonic functions on compact Riemann surfaces 23 Hausdorff dimension Badly approximate p-adic integers 123 Hecke correspondences The Hodge conjecture for certain moduli varieties 371 Heisenberg group Uncertainty principles on certain Lie groups 135 Hessenberg matrix Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays 59 Hilbert C*-module Induced representation and Frobenius reciprocity for compact quantum groups 157 Holder metric Degree of approximation of functions in the Holder metric by (e, c) means 315 Homogeneous layers Reflection of P-waves in a prestressed dissipative layered crust 341 Hydromagnetic thermohaline convection Some characterization theorems in rotatory magneto thermohaline convection 461 Induced representation Induced representation and Frobenius reciprocity for compact quantum groups 157 Infinitesimal /i-conformal motion On infinitesimal /z-conformal motions of Finsler metric 33 Infinitesimal homothetic motion On infinitesimal /i-conformal motions of Finsler metric 33 Integrals A theorem concerning a product of a general class of polynomials and the H-function of several complex variables 291 Interval topology The algebra X p ((0, oc)) and its multipliers 329 Inviscid liquid Surface waves due to blasts on and above inviscid liquids of finite depth 227 Irrationality Irrationality of linear combinations of eigen- vectors 269 L 1 -convergence of Fourier series On L 1 -convergence of modified complex trigono- metric sums 193 L 1 - convergence of modified complex trigonometric sums On L 1 -convergence of modified complex trigono- metric sums 193 Laguerre means A localization theorem for Laguerre expansions 303 Laguerre series A localization theorem for Laguerre expansions 303 Laplace transform Convolution integral equations involving a general class of polynomials and the multivariable H-function 187 Large-scale systems Control of interconnected nonlinear delay differential equations in W\ l) 73 Lie group On subsemigroups of semisimple Lie groups 153 Local behaviour Local behaviour of the first derivative of a deficient cubic spline interpolator 393 Local root numbers Lifting orthogonal representations to spin groups and local root numbers . 259 Lusin theorem A note on integrable solutions of Hammerstein integral equations 99 Magnetic shear layer On over-reflection of acoustic-gravity waves incident upon a magnetic shear layer in a compressible fluid 105 Matrix method Reflection of P-waves in a prestressed dissipative layered crust 341 Maximal ideal space The algebra A p ((Q, oc)) and its multipliers 329 Mid-point interpolation Local behaviour of the first derivative of a. deficient cubic spline interpolator 393 Moduli spaces The Hodge conjecture for certain moduli varieties 371 Motion group Uncertainty principles on certain Lie groups 135 Mountain pass lemma Nontrivial solution of a quasilinear elliptic equation with critical growth in R" 425 Multivariable H-function Convolution integral equations involving a general class of polynomials and the multivariable H-function 187 474 Subject index Normal functions The Hodge conjecture for certain moduli varieties 371 Normal matrix Absolute summability of infinite series 201 Odd order Oscillation of higher order delay differential equations 417 Orthogonal representations Lifting orthogonal representations to spin groups and local root numbers 259 Oscillation of all solutions Oscillation in odd-order neutral delay differential equations 219 Oscillation of higher order delay differential equations 417 Over-reflection On over-reflection of acoustic-gravity waves incident upon a magnetic shear layer in a compressible fluid 105 P-wave Reflection of P-waves in a prestressed dissipative layered crust 341 Fade approximants Computer extended series solution to viscous flow between rotating discs 353 p-adic numbers Badly approximable p-adic integers 123 Palais-smale condition Nontrivial solution of a quasilinear elliptic equation with critical growth in R" 425 Partial sums On the partial sums, Cesaro and de la Vallee Poussin means of convex and starlike functions of order 1/2 399 Precise estimate Local behaviour of the first derivative of a deficient cubic spline interpolator 393 Principal G-bundle Flat connections, geometric invariants and energy of harmonic functions on compact Riemann surfaces 23 <3f-hypergeometric identities A bibasic hypergeometric transformation as- sociated with combinatorial identities of the Rogers- Ramanujan type 41 Reflection Reflection of P-waves in a prestressed dissipative layered crust 341 Reflection coefficients Reflection of P-waves in a prestressed dissipative layered crust 341 Regular norm Uniqueness of the uniform and adjoining identity in Banach algebras 405 Reversion of series Computer extended series solution to viscous flow between rotating discs 353 Riemann zeta function On the zeros of (l * - a (on the zeros of a class of a generalized Dirichlet series - XVII) 273 S-wave Reflection of P-waves in a prestressed dissipative layered crust 341 S/f-type waves Generation and propagation of SH-type waves due to stress discontinuity in a linear viscoelastic layered medium 241 Schauder fixed point theorem A note on integrable solutions of Hammerstein integral equations 99 Scorza Dragoni theorem A note on integrable solutions of Hammerstein integral equations 99 Semisimple On subsemigroups of semisimple Lie groups 153 Series solution Computer extended series solution to viscous flow between rotating discs 353 Shear flows A proof of Howard's conjecture in homogeneous parallel shear flows - II: Limitations of Fjortoft's necessary instability criterion 251 Sobolev spaces A localization theorem for Laguerre expansions 303 Space of affine functions Weakly prime sets for function spaces 41 1 Spectral extension property Uniqueness of the uniform and adjoining identity in Banach algebras 405 Spin groups Lifting orthogonal representations to spin groups and local root numbers 259 Starlike and convex functions Differential subordination and Bazilevic functions 169 Steady-state An axisymmetric steady-state thermoelastic pro- blem of an external circular crack in an isotropic thick plate 445 Stress discontinuity Generation and propagation of SH-type waves due to stress discontinuity in a linear viscoelastic layered medium 241 Stress intensity factor An axisymmetric steady-state thermoelastic pro- blem of an external circular crack in an isotropic thick plate 445 Subintegral extensions The structure of generic subintegrability 1 Subject index 475 Subrings of polynomial rings The structure of generic subintegrability 1 Subsemigroup On subsemigroups of semisimple Lie groups 153 Summability methods Some theorems on the general summability methods 53 Surface elevation Surface waves due to blasts on and above inviscid liquids of finite depth 227 Surface waves Surface waves due to blasts on and above inviscid liquids of finite depth 227 Symmetrizer Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays 59 System of conservatiop laws Solution of a system of nonstrictly hyperbolic conservation laws 207 Systolic array Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays 59 Tangential representation Equivariant cobordism of Grassmann and flag manifolds 381 The convolution integral equation Convolution integral equations involving a general class of polynomials and the multivariable H-function 187 Topological Sidon sets A note on the growth of topological Sidon sets 281 Uncertainty principle Uncertainty principles on certain Lie groups 135 Unibasic hypergeometric series A bibasic hypergeometric transformation as- sociated with combinatorial identities of the Rogers-Ramanujan type 41 Uniform vertical rotation Some characterization theorems in rotatory magneto thermohaline convection 461 Unique uniform norm property Uniqueness of the uniform and adjoining identity in Banach algebras 405 Univalent Differential subordination and Bazilevic functions 169 Unknown functions Characterization of polynomials and divided difference 287 VLSI processor array Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays 59 Weakly prime set Weakly prime sets for function spaces 41 1 Weakly regular Banach algebra Uniqueness of the uniform and adjoining identity in Banach algebras 405 AUTHOR INDEX ! Abercrombie A G Badly approximable p-adic integers 123 Achar P K see Bujurke N M 353 Agrawal Pawan see Gupta KC 187 Bagewadi C S see Nagaraja H G 33 Balachandran K A note on integrable solutions of Hammerstein integral equations 99 Balaji V The Hodge conjecture for certain moduli varieties 371 Banerjee Mihir B A proof of Howard's conjecture in homogeneous parallel shear flows-II: Limitations of Fjortoft's necessary criterion 251 Bhatia Satvinder Singh On L 1 -convergence of modified complex trigono- metric sums 193 Bhatt S J Uniqueness of the uniform norm and adjoining identity in Banach algebras 405 Bhowmick Rina An axisymmetric steady -state thermoelastic problem of an external circular crack in an isotropic thick plate 445 Bose Sujit Reflection of P-waves in a prestressed dissipative layered crust 341 Bujurke N M Computer extended series solution to viscous flow between rotating discs 353 Chaurasia V B L A theorem concerning a product of a general class of polynomials and the //-function of several complex variables 291 Chukwu E N Control of interconnected nonlinear delay differential equations in W ( ^ 73 Das Bikash Ranjan see Bhowmick Rina 445 DasG Degree of approximation of functions in the Holder metric by (e, c) means 3 1 5 DasP Oscillation of higher order delay differential equations 417 Das Pitambar Oscillation in odd-order neutral delay differential equations 219 Dedania H V see Bhatt S J 405 Dhiman Joginder Singh Some characterization theorems in rotatory magneto thermohaline convection 461 Dutta Dipasree see Bose Sujit 341 Gadea Pedro M Fibred Frobenius theorem 31 Ghosh Tulika see Das G 315 Gupta K C Convolution integral equations involving a general class of polynomials and the multi- variable //-function 1 8 7 Guruprasad K Flat connections, geometric invariants and energy of harmonic functions on compact Riemann surfaces 23 Ilamaran S see Balachandran K Izumi H see Nagaraja H G 99 33 Jain Rashmi see Gupta KC 187 Joseph K T Solution of a system of nonstrictly hyperbolic conservation laws 207 Kandaswamy P On over-reflection of acoustic-gravity waves incident upon a magnetic shear layer in a com- pressible fluid 105 Kannappan P L Characterization of polynomials and divided difference 287 Kanwar Vinay see Banerjee Mihir B 25 1 Kelly-Lyth D On subsemigroups of semisimple Lie groups 153 476 Author index 477 Kumar F R K Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays 59 Kumar Lalan see Pal P C 241 Lahiri Maya Certain bilateral generating relations for general- ized hypergeometric functions 297 Manning Anthony Irrationality of linear combinations of eigen- vectors 269 Masque J Munoz see Gadea Pedro M 3 1 McCrudden M see Kelly- Lyth D 153 Mehta H S Weakly prime sets for function spaces 41 1 Mehta R D see Mehta HS 411 Mishra B B see Das P 417 Misra N see Das P 417 Mondal C R Surface waves due to blasts on and above inviscid liquids of finite depth 227 Mukherjee Goutam Equivariant cobordism of Grassmann and flag manifolds 381 .Nagaraja H G On infinitesimal /z-conformal motions of Finsler metric 33 Navada Gowri K A note on the growth of topological Sidon sets 281 Orhan C Absolute summability of infinite series 201 PaiNP see Bujurke N M 353 Pal Arupkumar Induced representation and Frobenius reci- procity for compact quantum groups 157 Pal PC Generation and propagation of SH-type waves due to stress discontinuity in a linear viscoelastic layered medium 241 Panda Ratikanta Nontrivial solution of a quasilinear elliptic equation with critical growth in R" 425 Ponnusamy S Differential subordination and Bazilevic c functions 169 Prasad Dipendra Lifting orthogonal representations to spin groups and local root numbers 259 Ram Babu see Bhatia Satvinder Singh 1 93 Ramachandra K On the zeros of f ^ - a (on the zeros of a class of a generalized Dirichlet series-XVII) 273 Ramakrishnan Dinakar see Prasad Dipendra 259 Rana Surendra Singh Local behaviour of the first derivative of a deficient cubic spline interpolator 393 Rathnakumar P K A localization theorem for Laguerre expansions 303 Ray BK see Das G 315 Reid Les The structure of generic subintegrability 1 Roberts Leslie G see Reid Les 1 Sahoo P K see Kannappan P L 287 Sarigol M A see Orhan C 201 Satyanarayana Bavanari see Lahiri Maya 297 SenSK see Kumar F R K 59 Shandil R G see Banerjee Mihir B 25 1 Sharma Rajendra Pal see Chaurasia V B L 291 Singh Ajit Iqbal The algebra A ((0, oo)) and its multipliers 329 Singh Balwant see Reid Les 1 Singh Ram On the partial sums, Cesaro and de la Vallee Poussin means of convex and starlike functions of . order 1/2 399 Singh Sukhjit see Singh Ram 399 Singh U B A bibasic hypergeometric transformation as- sociated with combinatorial identities of the Rogers-Ramanujan type 41 478 Author index Sitaram A Thangavelu S Uncertainty principles on certain Lie groups 135 see Sitaram A Sulaiman W T Some theorems on the general summability Vasudeva H L methods 53 Sundari M see Sitaram A 135 see Singh Ajit Iqbal Veerappa Gowda G D see Joseph K T 135 329 207 \