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Proceedings of the Indian Academy of Sciences (Mathematical Sciences) Editors S G Dani Tata Institute of Fundamental Research, Bombay A Ramanathan Tata Institute of Fundamental Research, Bombay Editorial Board Gopal Prasad, Tata Institute of Fundamental Research, Bombay 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, Institute of Mathematical Sciences, 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 Publications of the Academy G Srinivasan Raman Research Institute, Bangalore Subscription Rates (Effective from 1989) All countries except India (Price includes AIR MAIL charges) India...... 1 year US$75 1 year Rs. 75 3 years $200 10 years Rs.400 5 years $300 All correspondence regarding subscription should be addressed to The Circulation Department of the Academy. Editorial Office Indian Academy of Sciences, C V Raman Avenue, P B No. 8005, Bangalore 560080, India Telephone: 342546 Telex: 0845-2178 ACAD IN Telefax: 91-812-346094 1992 by the Indian Academy of Sciences. All rights reserved. The "Notes on the preparation of papers" are printed in the last issue of every volume. Proceedings of the Indian Academy of Sciences Mathematical Sciences Volume 102 1992 iff Published by the Indian Academy of Sciences Bangalore 560080 Proceedings of the Indian Academy of Sciences (Mathematical Sciences) Editors S G Dani Tata Institute of Fundamental Research, Bombay A Rarnanathan Tata Institute of Fundamental Research, Bombay Editorial Board Gopal Prasad, Tata Institute of Fundamental Research, Bombay K R Parthasarathy, Indian Statistical Institute, New Delhi Phoolan Prasad, Indian Institute of Science, Bangalore M S Raghunathan, Tata Institute of Fundamental Research, Bombay S Ramanan, Tata Institute of Fundamental Research, Bombay C S Seshadri, Institute of Mathematical Sciences, Madras V S Varadarajan, University of California, Los Angeles, USA S R S Varadhan, Courant Institute of Mathematical Sciences, New York, USA K S Yajnik, National Aeronautical Laboratory, Bangalore Editor of Publications of the Academy G Srinivasan Raman Research Institute, Bangalore Subscription Rates (Effective from 1989) All countries except India 1 year 3 years 5 years (Price includes AIR MAIL charges) US$ 75 $ 200 $ 300 India 1 year 10 years Rs. 75 Rs. 400 All correspondence regarding subscription should be addressed to The Circulation Department of the Academy. Editorial Office Indian Academy of Sciences, C V Raman Avenue, Telephone: 342546 P.B. No. 8005, Bangalore 560 080, India Telex: 0845-2178 ACAD IN Telefax: 91-812-346094 Proceedings of the Indian Academy of Sciences Mathematical Sciences Volume 102, 1992 CONTENTS On the frequency of Titchmarsh's phenomenon for C(s)-VIII R Balasubramanian, K Ramachandra and A Sankaranarayanan 1 Generalized parabolic sheaves on an integral projecti ve curve Usha N Bhosle 13 Non-existence of nodal solution for m-Laplace equation involving critical Sobolev exponents Adimurthi and S L Yadava 23 Solution of convex conservation laws in a strip K T Joseph and G D Veerappa Gowda 29 An example of a regular space that is not completely regular A B Raha 49 Factors for |N, p n ; 6\ k summability of Fourier series Hiiseyin Bor 53 Maximal monotone differential inclusions with memory Nikolaos S Papageorgiou 59 Remark on Gronwall's inequality J Popenda 73 Proof of some conjectures on the mean-value of Titchmarsh series III jR Balasubramanian and K Ramachandra 83 Combinatorial meaning of the coefficients of a Hilbert polynomial MR Modak 93 A note on two absolute summability methods Hiiseyin Bor 125 On deficiences of differential polynomials Anand Prakash Singh and Raj Shree Dhar 129 A note on the identity operators of fractional calculus R K Raina 141 An RSA based public-key cryptosystem for secure communication V Ch Venkaiah 147 A note to the paper "An efficient algorithm for linear programming" of V Ch Venkaiah Joachim Kdschel 155 ii Volume contents Stochastic dilation of minimal quantum dynamical semigroup A Mohari and K B Sinha 159 Almost periodicity of some Jacobi matrices Anand J Antony and M Krishna 175 Three dimensional diffraction of compressional waves by a rigid cylinder in an inhomogeneous medium B K Rajhans and S K Samal 189 Topological algebras with C*-enveloping algebras Subhash J Bhatt and Dinesh J Karia 201 C(K,X)asanM-idealin WC(K,X) T S S R K Rao 217 On the zeros of a class of generalized Dirichlet series XI R Balasubramanian and K Ramachandra 225 On L 1 convergence of a modified cosine sum Huseyin Bor 235 Addendum to the paper "generalised parabolic sheaves on an integral projective curve" jjsha N Bhosle 239 Subject Index 241 Author Index 243 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 1, April 1992, pp. 1-12. Printed in India. On the frequency of Titchmarsh's phenomenon for (s) VIII R BALASUBRAMANIAN, K RAMACHANDRA* and A SANKARANARAYANAN* The Institute of Mathematical Sciences, Madras 600 113, India *School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India MS received 4 March 1991; accepted "11 December 1991 Abstract For suitable functions H^H(T) the maximum of |(C(<r + it)) 2 ! taken over T^tt^T + His studied. For fixed crQ^o^ 1) and fixed complex constants z "expected lower bounds" for the maximum are established. Keywords. Riemann zeta-function; frequency; Titchmarsh's phenomenon. I. Introduction It is our object to prove the following three Q-theorems by applying two fundamental theorems of Ramachandra which he proved in [5] (these theorems will be stated in 2). Let G(0-, t) = |((<7 -f it)Y'\ where a- is a constant in [, 1] and z is a non-zero complex constant. Since z can be written as z = re w where r > and 6 lies in [0, 2n) 9 in order to state Q, theorems for G(<r,t) it suffices to assume r 1. We shall in what follows write z = e w . Theorem 1. We have, with z = e ie , max G(l,t)^ where y is the Euler's constant and COS0 xexplsin#tan 1 The conditions on H and T are T^H^C log log log T, T ^ T where C and T are large positive constants. Remark 1. Levinson [3] was the first to prove that when = max 2 R Balasubramanian, K Ramachandra and A Sankaranarayanan and when 9 = n max G(l, t) ^ -2 e*(log log T- log log log T) + 0(1). l<fT ft Note that A(TT) = 6/n 2 . However, around the same time Ramachandra [6] proved that max G(l, t) ^ e y A(0) log log H(l + 0(1)) when = 0, and = it. Later Ramachandra [7] extended the conditions on H to H ^ Clog log log log T (without assuming any hypothesis) and to H ^ Clog log log log log T (assuming Riemann hypothesis). These results go through for any in [0, 2n). Remark 2. This theorem as well as Theorems 2 and 3 has obvious extensions to ordinary L-functions and more generally to L-functions of algebraic number fields and so on. We do not carry out the details here. Theorem 2. Let a be a constant in (^, 1). Then max G(M)>exp( C x - i 15 a positive constant depending on a and T 113 ^H ^T. The next theorem depends on Riemann hypothesis (R.H.). Theorem 3. (on jR. H .). in Theorem 2 the condition on H can be relaxed to T^H^C log log T. ,4/so under this condition on H, there holds max c/(jr, t) : C 2 >0 is a numerical constant. Remark 1. When = 0, Theorem 3 can be upheld without assuming R.H. Remark 2. The results of this paper are inspired by the paper [4] of Montgomery who proves max G(o, t) > expj C 3 - (1 g T) " * where C 3 ^ depends on a and j < a < 1. In a recent paper [8], Ramachandra and Sankaranarayanan have obtained this result with C 3 = C 4 (oc - )*/(! - a) where C 4 > is a numerical constant. (The quantity (a )* can be replaced by 1 if we assume R.H.). However Montgomery's method does not work for short intervals [ T, T + H] and also for L-functions. On the frequency of Titchmarsh's phenomenon for ((s) VIII 3 2. Ramachandra's theorems We shall now state a special case of the main theorem of [5]. Theorem 4. Let a l = l,a 2 ,a 3 ,...bea sequence of complex numbers satisfying \a n \^ (nH) A where /f ^ 10 10 and A is a positive constant. Let F(s) = S^ =1 a n /n s (where s = a + it) admit an analytic continuation in o > 0, T ^ t ^ T + H. Then fl f T+H 2 \ y J logn 1 \ where C(A) is a positive constant depending only on A. As a corollary he deduced Theorem 5. In addition to the condition of Theorem 4 let us suppose that in the maximum of \F(s)\ be ^exp exp(JF//lOO/4). Then (M>H^>,?J-K'-S + TO) where C(A) > depends only on A, provided the LHS is interpreted in a limiting sense. Remark 1. In the reference to Ramachandra's paper the theorem proved is slightly different. But it is not hard (from his argument) to prove Theorem 4 and and deduce Theorem 5. In that reference Ramachandra uses the kernel related to exp(s 4fl+2 ) where a is a non negative integer. However in deducing Theorem 5 from Theorem 4 we have to use the kernel related to exp((sins) 2 ). Remark 2. From Theorem 4 we can also deduce that the maximum of G(<r, t) in (cr^l,T^t^T + H) exceeds the right hand side of Theorem 1. Similar remarks holds good for Theorems 2 and 3. Remark 3. For improvements of Theorems 4 and 5 see the paper [2] by Balasubramanian and Ramachandra. 3. Proof of theorem 1. We will prove Theorems 1, 2 and 3 in 3, 4 and 5 respectively. We adopt different notations in each of these sections and we will explain the notations in each of these sections in the respective sections. Lemma 3.1. Let k be a positive integer and let ^ (3.1) then (3.2) 4 R Balasubramanian, K Ramachandra and A Sankaranarayanan Proof. Follows by Euler product for C(l + s). Lemma 3.2. We have in (<r > 0, T < t < T + H) |jF(5)|exp(kC 1 loglogr) (3.3) where c l is a positive constant and 10 < H < T. Proof. Follows since it is well-known that Lemma 3.3. The conditions for the application of Theorem 5 are satisfied if H log log log T where the implied constant is a large positive constant, and k = O(logH). Proof. Follows from Lemmas 3.1 and 3.2. Lemma 3.4. Let k = kz and letn^2 and let n = ["} p m be the prime power decomposition p of n. Then T-rM/C +l)...(/C + W-l) a, = 1 and a n = f]"^ - ^rV - - (3.4) V w!p m Proo/ Follows by Euler's product for f (1 + 5). Lemma 3.5. Ler, /or each p < fc, . 7 /cos0 + (p 2 -sin 2 0)\ k I = fe ^ - __ - LJ = _ Tfew, putting n = f| p w } we /iai;e p^/c log|a n | 2 = ^^Z(- 2mlogm + 2m +0(logm)- 2mlogp 4- (fc, m)} (3 <6) where m-l (fe, m) = % log(/c 2 + t; 2 + 2kv cos 0). n 7\ y=0 V*^* 7 / Proo/. Follows from the formula log ml = m log m - m + 0(log m). Lemma 3.6. We have, + /:| 'log(l + u 2 + 2ucos6)du + o(-} (3 8) Jo v ' ]B (3 5) On the frequency of Titchmarsh's phenomenon for ($) VIII 5 Proof. We have ^ (fc,m)= V <log(/c 2 + v 2 + 2kvco$9) \\ * =o1 I f u+1 1 - log(fc 2 + u 2 4- 2few cos 0)d > Jy J f m 4- log(/c 2 + w 2 4- 2/cw cos 9)du. Jo Here the sum on the right is easily seen to be O(l/p). The integral on the right is f m / u 2 u \ 2m log k 4- log 1 4- -ry + 2-cos dw. Jo V fc k ) Here we can replace the upper limit m of the integral by / with an error 0(m/k) = 0(l/p). The lemma now follows by a change of variable. Lemma 3.7. We have, and Proof. Follows by prime number theorem. Lemma 3.8. We have, { 2wlogm-f-2m log * = <--- (3.9) logk (3.10) \og Proof. On the LHS we can replace m by / with a total error The rest is which gives the lemma. 6 R Balasubramanian, K Ramachandra and A Sankaranarayanan Lemma 3.9. We have, 1 /(!/) Y k\ log(l + u 2 + 2u cos 0)dw 2/Cp^ Jo (3.11) Proo/. Trivial. Lemma 3.10. We log|aj 2 = log logfc + y + log 1(9) + 0, (3.12) where /.(&) is as in Theorem 1. Proof. By Lemmas 3.5, 3.6, 3.7 and 3.8 we see that LHS of (3.12) is, (with an error 0(l/\ogk)), Re q q q \ q \ q Now the contribution from the first two terms (in the curly bracket) to the sum is Re l = 0, since + cos + sin 2 2 e + cos g (P 2 ~ s 'n 2 ^ V 2 - 2 * +Sm (p 2 - sin 2 0)* + cosfl The third term contributes (cos 6 log^ + sin fltan- ' (-^ -ZJIogfl-i 2 ' 1 This together with the well-known formula U f<k (l - 1/p)' 1 = eMogfc + 0(1) proves the lemma. On the frequency of Titchmarsh's phenomenon for (s) VIII 1 Lemma 3.11. For the n defined in Lemma 3.5, we have, logn= mlogp = klogk + O(k). (3.13) Proof. Replacement of m by / involves an error 0(k) by the prime number theorem. Now / = k/q and / 1\ _ . A q p I p (p[l (sin 0/p )]* -fcosS) V P/ This proves the lemma. Lemma 3.12. Set fc== [log/f/(21og logff)]. TTien /or a// H exceeding a large positive constant, we have, H Proof. Follows from Lemma 3.11. Lemma 3.13. The maximum of |((1 + s)) z \ in (a = 0, T < f < T + H) exceeds log log// 1 "' Proo/. Follows from Theorem 5. Lemmas 3.12 and 3.13 complete the proof of Theorem 1, in view of Lemma 3.10. 4. Proof of theorem 2. Lemma 4.1. Let < /? ^ 1 and // = T 1/3 . TTie/i tfc^ number of zeros of (s) m (a 2? /?, z^ constant implied by the Vinogradov symbol is absolute. Proof. This is a consequence of a deep result of Balasubramanian [1] on the mean-square of |C(i + fr)l- (See Theorem 6 on page 576 of his paper). Lemma 4.2. Letj<px<l. Then there exists a t-interval I contained of length T 5 (where 5 > is a constant depending on /?) such that the region (a ^ /?, tel) is free from zeros of Proof. Follows from Lemma 4.1. 8 R Balasubramanian, K Ramachandra and A Sankaranarayanan Lemma 4.3. Let I denote the ^interval obtained from I by removing on both sides 'intervals of length (1/100) T*. Then in (<r >, te/ ) we have Proof. Follows by Borel-Caratheodory theorem. Lemma 4.4. We apply Theorem 5 to the interval / in place of T ^ t ^ T 4- H. Then 1 /2 * max |(C(a + s))1^--k| (4.2) where n ^ H/200, k ^ 1 is any integer which is 0(log H), and a n are defined by g^. (43) Proof. It is easily seen (as before) that the conditions for the application of Theorem 5 are satisfied and hence the lemma. Lemma 4.5. Let n TTzen logfe is a positive constant. Proof. Follows by Euler's product for {(5). Lemma 4.6. Let k = [C 2 (log#) a ] where C 2 >Q is a small constant. Then, we have, and so R.H.S. of (4.2) exceeds loglogH C 3 > 15 a constant. Proof. Follows from lemma 4.5. Theorem 2 now follows from (4.2) and lemma 4.6. On the frequency of Titchmarsh's phenomenon for ((s) VIII 9 5. Proof of theorem 3 The first part of Theorem 3 follows exactly as in the proof of Theorem 2. It remains to prove only the second part of Theorem 3. We begin with Lemma 5.1. Given any t ^ 10 there exists a real number T with |t T|< 1 such that (5-1) uniformly in 1 < <r < 2. Hence logC(<J + *T) = 0((logt) 2 ) (5.2) uniformly in 1 ^ a ^ 2. Proof. See Theorem 9.6 (A), p. 184 of [9]. Lemma 5.2. Let F(s) = ((C(i + *))**) = I %* > 2), (5.3) n = i n where k^l is any integer. Let x ^ 1000, L-2\ oo /, +_ = V rn. (5.4) 5 P w/zere 8 = Q/log/c and C x is any positive constant. provided where C 2 is a positive constant. (5-6) (5.7) > exp(fe 2 log 5/4) - xZ j (5-8) x W . M>xW (5.9) 10 R Balasubramanian, K Ramachandra and A Sankaranarayanan Proof. Equation (5.5) is trivial. Equations (5.6) and (5.7) follow from log (l+y)<y and >y-y 2 /2 for Q<y<l. Equation (5.8) is trivial where (5.9) follows if x d ^ exp(/c 2 e~ Cl ) and C x is large. This leads to the condition (5.10) for the validity of (5.9). Lemma 53. We have, with k = [C 3 (logH/log logH)*], where C 3 is a certain positive constant. Proof. Follows from (5.5), (5.9) and (5.10). Lemma 5.4 The condition |aj< (nH) A is satisfied for some A>0. Proof. Follows from the Euler product for f (s). Lemma 5.5. Without loss of generality we can assume that max |f(ir)|<exp(logf/) 3 ). (5.12) Proof. Otherwise the required result follows. Lemma 5.6. The inequality (5.12) implies (subject toH^C 4 log log T where C 4 > is a large constant) that max i F (* + *') I < exp((IogH) 4 ). (5.13) a > 0, T + (H/9) t T + (8H/9) Proof. Let s = (j + zt where < cr ^ 1 and T + -- <t ^ T + . Consider the 9 9 analytic function For any real t> 10 let t* denote the real number T given by Lemma 5.1. Let R denote the rectangle with the following corners, On the horizontal sides of R we have On the frequency of Titchmarsh's phenomenon for (s) VIII 11 On the vertical sides we have, by lemma 5.5, Lemma 5,6 now follows since (by maximum modulus principle) |F(s )| = maximum of \</>(s)\ on the boundary of jR provided C 4 > is a large constant Lemma 5.1. The conditions for applying Theorem 5 are satisfied for the interval H 8H <t^T + . Proof. Follows from Lemmas 5.4 and 5.6. The second part of Theorem 3 now follows from Lemma 5.3 (by a slight change of notation). References [1] Balasubramanian R, An improvement on a theorem of Titchmarsh on the mean-square of |C(i + it)\, Proc. London Math. Soc. 36 (1978) 540-576 [2] Balasubramanian R and Ramachandra K, Progress towards a conjecture on the mean- value of Titchmarsh series-Ill, Ada Arith., XLV (1986) 309-318 [3] Levinson N, Q-theorems for the Riemann zeta-function, Acta Arith. XX (1972) 319-332 [4] Montgomery H L, Extreme values of the Reimann zeta-function, Comm. Math. Helv. 52 (1977) 511-518 [5] Ramachandra K, Progress towards a conjecture on the mean-value of Titchmarsh series-I, in: Recent progress in analytic number theory (eds) H Halberstam and C Hooley (1981) Vol. I, (London: Academic Press), 303-318 [6] Ramachandra K, On the frequency of Titchmarsh's phenomenon for f (s)-I, J. London Math, Soc. 8 (1974) 683-690 [7] Ramachandra K, On the frequency of Titchmarsh's phenomenon for ((s)-VII, Ann. Acad. Sci. Fenn. Ser. AL Mathematica 14 (1989) 27-40 [8] Ramachandra K and Sankaranarayanan A, Note on a paper by H L Montgomery PubL L'Inst. Math. (Beograd) 50 (64) (1991) 51-59 [9] Titchmarsh E C, The theory of the Riemann zeta-function, (Oxford: Clarendon Press) (1951) References added in proof 1)* R Balasubramanian, On the frequency of Titchmarsh's phenomenon for ((s) -IV, Hardy-Ramanujan J, 9 (1986), 1-10 2)* K Ramachandra, On the frequency of Titchmarsh's phenomenon for Hardy-Ramanujan J, 13 (1990), 28-33 in [1]* the author has shown that we can take C 2 = 3/4. in [2]* the author has shown that if minmaxG(U) = /(H) |/|-// tel the minimum being over all intervals / of length H, then, for all H ^ H Q (Q\ l -loglogtf | <logloglogH + 0(1) 12 R Balasubramanian, K Ramachandra and A Sankaranarayanan where 1/2 ~ c c and 5 being defined by c + is = exp(ifl). It is not hard to see that A(0) is the same as before. Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 1, April 1992, pp. 13-22. Printed in India. Generalized parabolic sheaves on an integral projective curve USHA N BHOSLE School of Mathematics, Tata Institute of Fundamental Research, Homi Bhaba Road, Bombay 400005, India MS received 2 January 1992 Abstract. We extend the notion of a parabolic vector bundle on a smooth curve to define generalized parabolic sheaves (GPS) on any integral projective curve X. We construct the moduli spaces M(X) of GPS of certain type on X. If X is obtained by blowing up finitely many nodes in Y then we show that there is a surjective birational morphism from M(X) to M( Y). In particular, we get partial desingularisations of the moduli of torsion-free sheaves on a nodal curve 7. Keywords. Generalized parabolic sheaf; projective curve. 1. Introduction In [1] we defined and studied GPBs (generalized parabolic bundles) on an irreducible nonsingular projective curve. The notion easily generalizes to a GPS ( = generalized parabolic sheaf) on an integral projective curve X. A GPS is a torsion-free sheaf E together with an additional structure called parabolic structure over disjoint effective Cartier divisors {Dj} jj , J a finite set (see Definitions 1.3, 1.4). In [1] we constructed moduli spaces for GPBs with parabolic structure of certain type over a single divisor (i.e. J = singleton). Here we consider many divisors. Moreover, X being singular, the method used in [1] fails. Therefore we generalize the method of Simpson [4] for the construction of moduli spaces. Theorem 1. There exists a (coarse) moduli space M XtJ (k,d) of semistable GPS F of rank fe, degree d with parabolic structure over Dj given by a flag ^ i :H (F^Dj) :=> F{ (F) 13 0, VyG J and weights (0, a), where a,. = dim F{ (F) and rational number a are fixed with 0<a<l. M XJ = M XfJ (k,d) is a projective variety of dimension /c 2 (01)4- l + ^j,aj(k degree Dj a^\ g = arithmetic genus of X. If X is nonsingular, then M(k,d) is normal. If further (k,d) = 1, a j = multiple of k and a is close to 1 then M(k,d) is nonsingular and is a fine moduli space. Theorem 2. Let X be the curve (proper transform) obtained by blowing up nodes {yj} jej of an integral projective curve Y,n XY :X -> Y surjection. For JE J, let Dj = n XY (X/X a,- = fc. Then there exists a surjective birational morphism f XY '.M x ^ VJ >-+M Y j<. In particular, if J' = 0, X = desingularization of 7, (k,d) = l,a close to 1, then M XfJ is the desingularization of the moduli space M Y ^ of semistable torsion-free sheaves on Y. Further, if X' is a partial desingularization of Y, obtained by blowing up y j9 jJ f 9 n XtX l: 13 14 Usha N Bhosle X-+X' 9 n x . Y :X'-+ y, then (with suitable D j and parabolic structure as above) f XY = fx' Y ofxx'- Thus M x -j> is a partial desingularization of M Y ^. There is a close relationship between torsion-free sheaves on a singular curve Y and GPS on its desingularization. An analogue of Theorem 2 holds if {yj are ordinary cusps, and hopefully also in case each y t is an ordinary rc-tuple point with linearly independent tangents. 1. Preliminaries Let X be an integral projective curve defined over an algebraically closed field fc. Let Q} X denote the dualising sheaf on X, it is a torsion-free sheaf. For a torsion-free sheaf E on X we denote by r() and d(E) respectively the rank and degree of . Let {Dj}jeJ be finitely many effective divisors on X such that supports of D } are mutually disjoint. DEFINITION 1.1 A quasi-parabolic structure on E over Dj is a flag ^ j (E) of vector subspaces of H(E(9 D .)viz. ^ J (E):F J (E) s H(E Dj ) => F{ () z> - => FftJE) = 0, DEFINITION 1.2 Let 3?(E) = {^ j (E)} jeJ . A QPS is a pair (E, ^(E)) where is a torsion-free sheaf and is a quasiparabolic structure on {JDjjeJ as above. DEFINITION 1.3 A parabolic structure on E over Dj is a quasiparabolic structure &*() (See, 1.1) together with an r-tuple of real numbers a j = (a{(E) } ---,a; j .(E)),0^a J 1 {)< <a^* () < 1, called weights associated to ^ j (E\ Let m{ = dim F{_ x (E) - dim Fj(), i = 1, , r ; . Define wt,() - SJi A m/o/(J5), wtE = SjWf/JS). Let par d(E) = d() + wi(E), par /x(E) = par d(E)/r(E). DEFINITION 1.4 A GPS (generalized parabolic sheaf) is a triple (E, ^(E), a) with $F> a as in 1.1 and 1.3. 1.5 Let K be a subsheaf of E such that the quotient E/K is torsion-free in a neighbourhood of D. Let h:K-*E be the inclusion map. Since D is a divisor and E/K is torsion-free, one has Torf(E/X, 6 D ) = and therefore h\ D :K\ D -+E\ D is an injection. Hence H(K@ p ) can be identified with, a subspace F{(K) of F J (E). Define Fi(K) = F J (K)nF{(E). This gives (after omitting repetitions) a flag ^ j (K\jeJ. The set {(x,{(K)} of weights for K is a subset of {ai(K)} defined as follows. One has F{(K) = GPS on an integral projective curve 15 Fi(E)r\F j (K) for some i, let i be largest such i. Then ot{(K): = <(). Thus a subsheaf of a GPS with torsion-free quotient gets a natural structure of a GPS. DEFINITION 1.6 A GPS(, ^(E\ a) is semistable (respectively stable) if for every (resp. proper) subsheaf K of E with torsion-free quotient, one has par n(K) < (resp. < ) par /*(). Remarks 1.7. (1) If JE/K is not torsion-free, then we may still define F^K) = image of #(K 0J,) under H Q (h\ D ) and define J^(K) by intersecting F J (K) with the flag ^ J (E). Thus we can talk of wtK. If M is the largest subsheaf of E containing K, with E/M torsion-free and r(K) = r(M) then par/*(jK)<par//(M). Then the condition of 1.6 is satisfied for every subsheaf K of E if (E,3? (),) is a semistable (resp. stable) GPS. (2) There exists a natural parabolic structure on a quotient sheaf also. Semistability and stability can also be defined equivalently using quotients instead of subsheaves, (See 3.4, 3.5 [1]). Assumptions 1.8. In this paper we want to study moduli spaces of GPS(F, J^(F),a) of the form & S (F):F J (F) => F{ (F) => 0, a j " = (0, a), < a < 1 . We also assume that for all j support of Dj is contained in the set of nonsingular points of X. Henceforth we restrict ourselves to bundles of the above type. We also assume that the base field is that of complex numbers. DEFINITION 1.9 A morphism of GPS is a morphisrn of torsion-free sheaves f:F-+F r such that Lemma 1.10. Let (F,^(jF),a) be a semistable GPS. Then there exists an integer HI dependent only on g( = arithmetic genus of X) and degree DjJeJ such that if l9 then (2) F is generated by global sections, (3) H(F)-+H(F<!) Dj ) is onto. Proof. This follows from H 1 (F) w H(X, Hom(F, <%))* and the latter is zero if pfi(F') is sufficiently large (depending on i(F\g\ For (3) we need to take F = F( Dj). (For details, see Lemma 3.7 [1]). Lemma 1.11. A morphism f of semistable GPS of same par n is of constant rank. If the GPS have the same rank and one of them is stable, than either f = or f is an isomorphism. Proof. This can be proved similarly as in Lemma 3.8 [1], COROLLARY 1.12. A stable GPS is simple i.e. its only endomorphisms are homothesies. I i 1 I 16 Usha N Bhosle PROPOSITION 1.13. The category S of all semistable GPS on X (of type described in 1.18) with a fixec par jjL~m is an abelian category. Its simple objects are the stable GPS. Proof. This follows from 1.11 and 1.12. DEFINITION 1.14 In view of the above proposition, a semistable GPS(,^,a) in S has a filtration wit! successive quotients stable GPS with par/i = m. We denote by gr(jE,^,a) th< associated graded object for the filtration. Up to isomorphism this object is independeni of the choice of stable filtration. Define an equivalence relation on S by (JE, #", a) ii equivalent to (', J^', a) iff gr(, #", a) w gr(F, ^',a). Remark 1.5. We may (for convenience) use the terminology 'a GPS E n when there ii no confusion about parabolic structure possible. 2. Construction of the moduli space 2.1 Consider semistable GPS F of type described in 1.8 with rank fc, Eule characteristic n > n^ fixed. Let P(m) be the Hilbert polynomial of F. Let Q = Q((9 n , P(m) be the quot scheme of coherent sheaves over X which are quotients of (9" and hav< Hilbert polynomial equal to P. Let 3F denote the universal quotient sheaf on Q x X Let R be the open subscheme of Q consisting of points qeQ such that & q = 3F\q x % is torsion-free and the map jff (^ n )~>H (J 2r q ) is an isomorphism. It follows tha H l (^ q ) = for qeR. For every j, let pf.R x Dj~+R be the canonical map and defim Vj = (p j )(# r lDj ). Let G(Vj) be the flag bundle over R of the type determined by th< parabolic structure over Dj. It is a relative Grassmannian bundle of quotients of rani qj. Let R denote the fibre product of {G(Vj)}j over R. Let R s (resp. jR ss ) denote th subset of R corresponding to stable (resp. semistable) GPS. Similarly we can defin< & Q s and Q ss . The quot scheme Q has a natural embedding in a Grassmannian. For m ^ M^n] the natural map H(@ n x (m)) - H(^(m)) is surjective for all qeQ. Let W = H(& x (m)] then H Q (^ q (m)) is a quotient of C" W of dimension P(m) form^M^. This gives i closed embedding Q-*Grass p(m) (C n (x) W). A point q of Q gives for each 7, '. ^-dimensional quotient of C". Hence we get an embedding g-Z = Grass P(m] (C n W) x ( x jGrass^C 1 )). This embedding is equivariant under the action o PGL(n). The PGL(n) action on Q and C" is the natural one, while on W it act trivially. On Z we take the polarization a(n aV q:)/kmx aa x - x 0a, where 1 denotes (9(1) and a is a sufficiently big integer to make all the numbers abov integers, n > ,#,. We denote a point of Z by (P,(P/)/) where P:C n W-+ U,Pj:C n ->Uj are surjectiv maps, dim U = P(m\ dim U } = q } for all j. Similarly a point of Q is denoted by (p, (PJ) GPS on an integral projective curve 17 where p: n -*F,p j ;H Q (F\D j )-*Q F j are surjections, dim gj == qjlj. For a subsheaf E of F, we define Qj = pj(H(E\Dj)). For a quotient ^f:F - G, define Qf = H(G\Dj)/q(Ker p } ). For simplicity of notation, we denote dim(Qj) (dim(g^)) by q^(E) (by ^-(G)). In particular, q j = qj(F). PROPOSITION 2.2 For a nontrivial proper subspace H c= C n of dimension h define <? H by km\hP(m) - rcdim P(H Then a point (P,(P^)) of Z is semistable (resp. stable) for PGL(n) action (with the above polarization) if and only if G H ^ (respectively < 0). Proof. See [3, Proposition 5.1.1] and [2, Proposition 4.3]. DEFINITION 2.3 Let F be a torsion-free sheaf of rank k on X. For every subsheaf E of F and m ^ integer define <r E (m) = ((n - a^ I km\h(E)P(m) - \\ J // / Lemma 2.4. Let F be a torsion-free sheaf corresponding to a point (p, (PJ)J) #/ 6- Then F is semistable (respectively stable) if and only if for every subsheaf E of F we have IE = foM < (sp. < 0) for any integer m. Proof. Let be a subsheaf of F with F/E torsion-free. Substituting P(m) = km + n, %(E(m)) = #() 4- mr(E)(r(E) = rank of E) in the expression for % E and simplifying one gets JS) - n/k + By definition F is semistable (respectively stable) if and only if the expression in the square bracket is ^ (resp. < 0). 18 Usha N Bhosle Suppose now that F/E is not torsion-free. Then there exists , c: such that F/E is torsion-free, rank E = rank . Let / = t = ? 4- S/TJ, where T^ = i /jjD . By the above argument, ^(mJ^O. We claim that % E (m) < li(m}. Using %((w)) x((w)) = /I(T) for m ^ 0, <?/) - qj (E) ^ fe(T,) we get x E (m) - * f (m) = n(aZ;/i(t;) - /I(T)) < since Lemma 2.5. There exists an integer M 2 (n) ^ M x (n) swc/z fto if (p,(pj))62 is a point satisfying the following two conditions, then the image of the point in Z is semistable (resp. stable). (1) The canonical map C"-H(F) is an isomorphism. (2) For every subsheaf E of F generated by global sections, G E (m) ^ (resp. < 0) for m> M 2 (n). Proof. Let H c C" be a subspace. Let be the subsheaf of F generated by H and let K be the kernel of the surjection H (9 X -* E. As H varies over subspaces of C n and F varies over Q, the sheaves E and hence K form a bounded family. Hence there exists M 2 (n) such that for m^M 2 (n\h 1 (E(-D j )(m)) = Q, /i 1 ((m)) = and /i 1 (K(m)) = for all such and K. It follows that dim P(H W} = i(E(m)\ Clearly dim H ^ fe(), (J [ f) = ^(:). Therefore (7 H ^0 E (m)^Q (resp. <0). Thus the image (P,(Pj)) of is semistable (resp. stable). Lemma 2.6. One can /ind M 3 (n) ^ M 2 (n) swcft t/iat /or m ^ M 3 (n) t/ie following holds. If (p,p/)) e Q is a point whose image in Z is semistable then (i) C"-+H(F) is injective and (ii) for all torsion-free quotients F-^G-*Q, one has I G ^ 0. Here I G is defined by - kh(G) 4- nr(G) Proof. Note that if H is the kernel of the map C n -+/f(F), then a Ho > contradicting the semistability of the image point in Z (Proposition 2.2). Hence (i) follows. For (ii), suppose that there exists a torsion-free quotient G with T G >0. Then h(G)<n, for h(G) ^ n implies T G ^ 0. Let H be the kernel of the composite C n ~+H(F)^H(G). Let E denote the subsheaf of F generated by H. Clearly we have r(E) 4- r(G)< /c, /i(G) ^ n - /z, ^-(G) ^ ^ - gj(jE), dim Pj(/3T) ^ q^(E\ Substituting these in the expression for T G one gets Since f/ and hence runs over a bounded family we can find M 3 (n) ^ M 2 (n) such that for m^M 3 (n), the term kh~nr(E) can be replaced by (hP(m)-~ nx((m))/m = (/iP(m)-ndimP(H(x) W))/m. Thus we get cr H >0 contradicting the semistability of the image point in Z. Lemma 2.7. There exists n^n^ such that for all semistable GPS F with Euler characteristic n^n 2 the following holds GPS on an integral projective curve 19 (1) // E c F then I E ^ where (2) // T = /or some E c F, t/ien x = 0. (3) // r < 0, then cr E (m) <0form^ M 4 (n). // T E = 0, fhen a (m) = for m ^ M 4 (rc). Proo/. (1) Let = E c Ej c= c E r = E be the Harder-Narasimhan filtration of E considered as a torsion-free sheaf only, ignoring the parabolic structure. Let Q t == EJEi _ 1 , f = 1, , r, fr = degree Q / rank Q i9 t; = inf ju f . One has JH > \JL { + ; Vz < r (by definition), fe(E) ^ ly/i^Qi) (by induction). Using Corollary 2.5 [4], this implies h(E) ^ S'>(6,) (^ + B^Bi constant. Since Q l is a subsheaf of a semistable GPS F we have + wV , w = (wF)/7c. Since n ^ l5 E r(e ) = r(JE), v = /^ o we get h(E) ^ w 4- B x ) 4- (i> + B x ) + (r(aj - l)0*(f) + w + JJJ ^ i; + (r(JS) - 1)- n/k + B 2 ,B 2 constant. Hence T ^ n(v + B 2 n//c). Therefore if v ^ n/fc B 2 (resp. < ) then T ^ (resp. < 0). We can choose n 2 large enough so that for n^n 2 , we have h l (Q(m)) = for all m ^ and for all stable torsion-free sheaves Q of rank ^k and H^n/k-B 2 . Hence if v ^ n/k - B 2 for E, then h l (Q (m) = OVf, therefore h 1 (E(m)) = and i(E(m)) = fc(jB(m)) for m ^ 0. Then T = XJE and % E ^ by Lemma 2.4. Thus T < for all E c F. (2) If T E = 0, then by the above argument one must have v ^ n/k B 2 ,x(E) = h(E) and T = X E - Thus ^ = Mm) = 0. (3) Note that T = lim m ^ ^ cr (m). Hence given e > 0, 3 M 4 (n) such that for m > M 4 (n), ;( m ) < T JE + - If TE < then choosing such that T -1- e < 0, we get o E (m) < for m ^ M 4 (n). Theorem 1. (I) Let X be an integral projective curve of arithmetic genus g over C. Let {Dj} jeJ be finitely many effective C artier divisors in X such that the support of D j does not intersect the set of singular points of X for allj, supports of Dj are mutually disjoint and degree Dj = d^jeJ. Let S denote the set of equivalence classes of semistable GPS F of rank k degree d with parabolic structure over Dj given by F J (F) = H (F (9 Dj ) ID F{(F)=>0, co-dimension of F{(F) in F J (F) equal to q^ (fixed) for j^J and weights (0,a),0<a < 1. Then S has the structure of a projective variety M(k,d) of dimension (II) // X is nonsingular, then M(fe, d) is normal If further (fc, d) = 1, q^ is a multiple of k and a is sufficiently near 1 then M(k, d) is nonsingular and it is a fine moduli space. Proof. Let w x denote the dualising sheaf of X, it is a torsion-free sheaf. Fix n > max (n 2 ,fc/i(w x )-f aSj^j) and m^M 4 (n). We keep the notations of 2.1. We shall show that a geometric invariant theoretic quotient of R modulo PGL(n) exists. Our required moduli space M(k,d) will be this quotient. jR is an open subset of Q,Q is embedded in Z (with m, n as above) by a PGL(n) equivariant embedding. We first claim that if (p, pj))eR ss (resp. R s ) then its image belongs to Z ss (resp. Z s ). This follows immediately from Lemma 2.7 and Lemma 2.5. Let F correspond to a point in R ss R s . Then F has a subsheaf which is a torsion-free stable GPS with par ii(E) = par u(F) i.e. XE ^ 0- For such an ,<T H O (E) = XE (Lemma 1.10), hence the image in Z belongs to Z ss Z s . 20 Usha N Bhosle Conversely we shall now check that if a point in Q is such that its image belongs to Z 5S , then the point is in R ss i.e. if F is the corresponding quotient, then F is torsion-free, the map C"->H(F) is an isomorphism and F is a semistable GPS. Lemma 2.6 implies that C n ^>H(F) is injective and for every rank 1 torsion-free quotient G of F, n ^ fc/i(G) + aS,^ (as T G ^ 0). We claim that H l (F) = 0. Otherwise there exists a nontrivial homomorphism F->w x . If G is the sheaf image of this morphism, h(w x ) ^ h(G) and hence n ^ kh(w x ) + aS 7 -^ contradicting the assumptions on n. Thus h(F) = n and C n - H(F) is an isomorphism. Let i be the torsion subsheaf of F, T = T O + Z/r,., support T,.supp Dj, (supp T )n(u supp ,-) = <. Taking H = /fVo),#(*A<7ir<0 gives H(T ) = 9 H(T J ) = 0. Here a<l is crucial since a a = W (A -a dim P/H)). Thus H(T) = H(r ) + ZjH(Tj) = i.e. T = 0. Suppose that F is not semistable. Then there exists a subslieaf F of F such that F is a semistable GPS with par //(F) > par /x(F) i.e. % E > 0. By Lemma 1.10, O" H O () = % > contradicting the semistability of the image point in Z. It follows that the (geometric invariant theoretic) quotient M(k, d) of R mod PGL(rc) is the same as that of Q and it exists if and .only if the quotient of image of Q in Z exists. It is well known that the latter exists. The quotient M(k,d) is a projective variety as Q is so. It is easy to check that the points of M(k, d) correspond to equivalence classes of semistable GPS (3. 15, [1]; [4]). (2) If X is nonsingular R is known to be nonsingular and hence M(k 9 d) is normal. If (fc,d) = l,a is sufficiently near to 1 and q j is an integral multiple of k, then GPS is semistable if and only if it is stable by Lemma 3.3 (or Lemma 3.17, [1]). The nonsingularity of R together with corollary 1.12 then imply that M(k, d!) is nonsingular. One can show that M(k, d) is then a fine moduli space, by proving the universal bundle on R descends to a universal bundle on M(k, d) after twisting by a line bundle (see [1], Proposition 3.18). 3. Application 3.1. Let Y be an integral projective curve with only singularities ordinary double points {yj}jeJ. Let J' c J be a subset. Let X be the curve (proper transform) obtained by blowing up {yj} jej >. Let n XY :X-^ Y be the natural morphism. Let D,- denote the divisor rc^y CfyX./eJ'. All the QPS and GPS that we consider are assumed to be of the type described in 1.8. We also assume that dimF{(F)) = r(F) forjeJ'. DEFINITION 3.2 Let a be a real number in [0, 1]. A QPS (F, J 27 ) on X is a-stable (resp. a-semistable) if for any proper subsheaf K of F with torsion-free quotient, one has (d(K) + a X dim F{ (K))/r(K) < ( ^ )(d(F) + aJ'r(F))/r(F). JJ' Remark. For < a < 1, the above condition is same as that for stability (resp. semistability) of the GPS(F, J^a) with a 7 " = (0,a)JeJ'. GPS on an integral projective curve 21 Lemma 3.3. (1) Suppose that 1 - l/J'r(F)(r(F) - 1) < a < 1. TTien (F, J 5 ") is a-semistable implies that it is l-semistable. If the QPS is l-stable then it is also oc-stable. (2) Assume that (r(F),d(F))= 1. Then the QPS is l-stable if and only if it is l-semistable. Thus under the assumptions of (1) and (2) {-stability, a-stability and a-semistability are all equivalent. Proof. This is a straightforward generalization of Lemma 3.17 [1]. PROPOSITION 3.4. Let Q denote the set of isomorphism classes of QPS (F,J*") on X of given type (3.1). Let r(F) k,d(F) = d be fixed. Let S be the set of isomorphism classes of torsion-free sheaves of rank k and degree d on Y. Let S k denote the subset of S corresponding to sheaves which are locally free at yj forjeJ'. Then (a) there is a surjective map f XY '-Q ~"* S such that its restriction to f XY (S k ) I5 a bijection onto S k . (b) (F,^) is l-stable (l-semistable) iff its image under f XY is stable (semistable). Proof. Let DJ = XJ + ZJ. Then ((n XY ).F) k(y j ) = (k(x j )k(z j )r^ = H(F (9 Dj ) = F{(F). Thus we have a surjective y -linear map (n XY )+F -+ F{(F). Let F' be the kernel of the composite of this map with the surjection F J (F)-+F J (F)/F{(F). Since d(F} = follows that if (F, J^eg then FeS. We define f XY (F, &)) = F. If F'eS k then F = n$ Y F and F{(F) = F'*:^) c: F J (F) gives the bijection. Surjectivity of/ can be proved as in 4.5 [1] while the last assertion follows exactly as in 4.2 [1]. Theorem 2. (I) Let M XtJ > be the moduli space of semistable GPS on X of type described in 3.1. Assume that a satisfies the conditions of Lemma 3.3(1). Then there is a surjective birational morphism fxY : M x ,j'-+M Y({) (= moduli space of torsion-free sheaves on Y). (II) Let Z be the desingularization of Y. Then the morphism fzy'-^zj -*M Y<i , factors as fxY fzx - V tne conditions of Lemma 3.3 are satisfied then M ZJ is a desingularization of M Y j and M XJ >(J r c J) are 'partial desingularizations. Proof. (I) This follows easily from Lemma 3.3 and Proposition 3.4 since it is easy to globalise the construction (of f XY ) to families of GPS. (See Theorem 2 [1] for details). (II) Let / Z x(F, &) = F'. Notice that n zx is an isomorphism outside J J'. Hence F' has a parabolic structure 3F' over DjJeJ' viz. Ff'(F') F{(F) f or i = Q,ljeJ'. Thus f zx (F^) = (F\^ f )eM XJf . Let f XY (F',^') = F". Then we have the exact sequences (defining F',^") Using these and n ZY = n XY n zx , one gets proving f ZY ^ f XYo f zx - The last assertion follows from Theorem 1 (II). W r.,1 22 Usha N Bhosle Acknowledgements We thank A Ramanathan and T R Ramadas for useful discussions. References [1] Bhosle Usha N, Generalised parabolic bundles and applications to torsion-free sheaves on nodal curves. (Arkiv. Math.) (to appear) [2] Mumford D, Geometric invariant theory. (New York: Academic Press) (1985) [3] Narasimhan M S and Trautmann G, Compactification of M p3 (0,2) and Poncelet pairs of conies. Pacific J. Math. 145 (1990) 255-265 [4] Simpson C T, Moduli of representation of the fundamental group of a smooth projective variety (preprint, 1990) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 1, April 1992, pp. 23-27. ;n Printed in India. 'y I . ''iff- Non-existence of nodal solution for m-Laplace equation involving || critical Sobolev exponents L ADIMURTHI and S L YADAVA j| TIFR Centre, Indian Institute of Science Campus, Post Box No. 1234, Bangalore 560012, '.j , India i|j Yi MS received 2 August 1991 ?|, iV } Abstract. In this paper we study the non-existence of nodal solutions for critical Sobolev fr exponent problem W - 2 Vu- u>-*u+u<->ui*BR ^' u = ondB(R) 'fa til V ' where B(R] is a ball of radius R in (FT t l y Keywords. Critical exponent; eigenvalue; m-Laplacian. l! 1. Introduction Consider the problem -* m u = \u\'- l u + \u\'- l umB(R) 1 u = Q ond(jR),J ' ) where B(R) is a ball in R n of radius R, A m w = div(| Vw m ~ 2 VM) and 1 < m < n, p 4- 1 = mn/(n m) is the critical Sobolev exponent for the non-compact imbedding Ho' m -* -L p+ x and l^q^p l.In this paper we are interested in the radial solutions of (1.1) which change sign. For w = 2, this problem has been discussed by many authors. It has been shown by Cerami et al [7] and Solomini [13] that (1.1) admits infinitely many radial solutions which change sign for q = 1 and n ^ 7. Atkinson et al [4, 5] and Adimurthi and Yadava [1] have proved that this result of infinitely many nodal solutions is optimal in the sense, when 3 < n ^ 6, q = 1, then (1.1) does not admit any radial solution which changes sign for all R sufficiently small. For pl <q<p, Jones [11] and Atkinsion- Peletier [3] have proved that (1.1) admits infinitely many radial solutions which change sign. It has been shown by Jones [11] for 1 < q <p 1 and by Knaap [12] for q = p 1 that (1.1) does not admit any radial solution which changes sign provided jR is sufficiently small. Atkinson et al [4, 5] have used asymptotic analysis to prove their non-existence result and Jones [11] has adapted dynamical system approach. In [1], the non-existence result has been obtained by Pohozaev's identity. In this paper, following the method used in [1], we extend the non-existence result for the general m, 1 < m < n. We prove 23 24 Adimurthi and S L Yadava Theorem. Let m~l^q^p-l. Then there exists a R >Q such that for all inB(R) does not admit any radial solution which changes sign. Remark 1. If m = 2, the above theorem gives all the above mentioned known results of the non-existence of solution which changes sign. Remark 2. For Q<q<m 1, it has been shown in [9] that for sufficiently small R, (1.2) admits infinitely many solutions, Remark 3. If q = m 1, then the above theorem is true for the range m < n ^ m 2 + m. For m<n<m 2 , Atkinson et al [6] have proved a more stronger result, namely (1.2) does not admit any positive radial solution for jR sufficiently small. 2. Proof of the theorem Since we are looking for radial solution, we can set u u(r\ r = \x\ and write (1.2) as (2.1) To study the problem (2.1), we can consider the associated initial value problem (r n ~ 1 \v'\ m ~ 2 v') = r n ~ 1 (\v\ p ~ l 4- \v\ q ~ l )v in (0,oo) dr \ (2.2) t/(0) = 0, i>(0) = y. Let v(r, y) be the unique solution of (2.2). Let < RI(J) < R 2 (y) < ... be the zeros of v(r,y). In order to prove the Theorem, it is enough to show that there exists a C > such that (2.3) for all ye(0, oo). To prove (2.3) we need the following. Lemma. We have sup {|i?(r, 7)1^(7) < V6(0,co) where } (2.5) (p-q -!)"-- 1 '""-" ' ^ ' if^<p-l. Critical Sobolev exponent problem 25 Proof. Suppose (2.4) is not true. Then there exists as y > and a fe> fe such that |u(r,y)| = k has a solution in [JRi(y), #2(7)]- Let R > R 1 (y) be the first point at which y(jR, y) = - k. Let w(r) = u(r, y) 4- /c. Then w satisfies !< w>0 (2.6) where/(w) = (|w-/c| p ~ 1 + ,|w- k\ q ~*)(w k). Let F denote the primitive of/. Then by Pohozaev's identity [8] and [10], we have mn n m (2.7) where Now observe that for - k ^ s ^ 0, g(s) is decreasing and non-negative. Therefore P "*" \ hq + 1 /7 Q\ )k (2.8) forallse[-/c,0]. For 5 > we have Claim, g (s) < for alls >0. For s > 0, let h(s) = g(s)/s q . Then Case L Let ^ = p - 1. Then Since k > 1/p, we get h(s) < 0. Case 2. Let q<pl. Then h has a maximum at ,, = (-L 26 Adimurthi and S L Yadava Since k> /c , we get h(s ) = jvp- q /(p- q -i)i/< p - q -i) ^ < and this proves the claim. Now from (2.7), (2.8) and Claim, we have gf(w-fc)r"- 1 dr+ 5 w 5? /c w > k = which is a contradiction. This proves the lemma. Before going into the proof of (2.3), we recollect some known results about the first eigenvalue for A m (see [2]). Let Q be a bounded domain with C 2 ^ boundary and let aeL(Q) be such that meas {xeQ; a(x) > 0} ^ 0. Then there exists a unique A(o,Q) > such that 2 4> in Q </ (2.9) = on 3Q admits a unique (up to multiplication by a constant) solution. Obviously, if ^ o^ < a 2 and a t -e L*(fl), then ). (2.10) Moreover, A! (!,!)-> oo as meas (Q)-*0. (2.11) Proof o/ (2.3). We claim that there exists a <5 > such that R2(y)-RM>s (2.12) for all ye(0,oo). Since m 1 < q, by the lemma there exists a C > 1 such that . sup |t?r w + |i;r" +1 ;R 1 (y)<r</l 2 (7)}<C. (2.13) Ve(0,c) Now suppose (2.12) is not true. Choose a y > such that yoX^foo)))^ (2.14) Critical Sobolev exponent problem 27 where B(R 1 (y ),R 2 (y ))= {xeU";R l (y )^\x\^R 2 (y )}. On the other hand, from (2.2), A 1 (|t?r m+1 + |i;|- m+1 ,B(/? 1 (yo)^2(7o))=l. (2.15) This, together with (2.13) and (2.10), contradicts (2.14). This completes the proof of the Theorem. References [1] Adimurthi and Yadava S L, Elementary proof of the non-existence of nodal solutions for the semilinear elliptic equations with critical Sobolev exponent, Nonlinear Anal. TMA 14 (1990) 785-787 [2] Anane A, Simplicite'et isolation de la premiere valeur propre du p-laplacien avec poids, C.R. Acad. Sci. Paris 305 (1987) 725-728 [3] Atkinson F V and Peletier L A, Oscillations of solutions of perturbed autonomous equations with an application to nonlinear elliptic eigenvalue problem involving critical Sobolev exponents, Differ. Integral Equ. 3 (1990) 401-433 [4] Atkinson F V, Brezis H and Peletier L A, Solution d'equations ellipliques avec exposant de Sobolev critique qui changent de signe, C.R. Acad. Sci. Paris 306 (1988) 711-714 [5] Atkinson F V, Brezis H and Peletier L A, Nodal solutions of elliptic equations with critical Sobolev exponents, J. Differ. Equ. 85 (1990) 151-170 * ( [6] Atkinson F V, Peletier L A and Serrin J (To appear) [7] Cerami G, Solomini S and Struwe M, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Fund. Anal. 69 (1986) 289-306 [8] Egnell H, Existence and non-existence results for m-Laplace equations involving critical Sobolev exponents, Arch. Rat. Mech. Anal. 104 (1988) 57-77 [9] Garcia Azorero J and Peral Alonso I, Multiplicity of solutions for elliptic problem with critical exponent or with a non-symmetric term, Trans. Am. Math. Soc. 323 (1991) 877-895 [10] Guedda M and Veron L, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. TMA 13 (1989) 879-902 [11] Jones C K R T, Radial solutions of a semilinear elliptic equation at a critical exponent, Arch. Rat. Mech. Anal. 104 (1988) 251-270 [12] Knaap M C (Preprint 1988) ' [13] Solimini S, On the existence of infinitely many radial solutions for some elliptic problems, Rev. Mat. Appl. (to appear) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 1, April 1992, pp. 29-47. Printed in India. Solution of convex conservation laws in a strip K T JOSEPH and G D VEERAPPA GOWDA TIFR Centre, Indian Institute of Science Campus, P.B. No. 1234, Bangalore 560012, India MS received 4 April 1991; accepted 25 October 1991 Abstract. In this paper we consider scalar convex conservation laws in one space variable in a strip D = {(x, t): < x ^ 1, t > 0} and obtain an explicit formula for the solution of the mixed initial boundary value problem, the boundary data being prescribed in the sense of Bardos-Leroux and Nedelec. We also get an explicit formula for the solution of weighted Burgers equation in a strip. Keywords. Conservation laws; boundary value problem; explicit formula. 1. Introduction We consider mixed initial boundary value problem for scalar convex conservation laws of the form x = Q (i.i) in a strip D = {(x, t): < x ^ 1, t ^ 0} with initial condition u (x). (1.2) The boundary conditions are prescribed in the sense of Bardos et al [1], Let u^(t) and u 2 (t) are any bounded measurable functions, then this condition requires u(0 + , t) and u(l , t) to satisfy the following: sup [sgn( W (0 + , t) - k)(f(u(0 + , t)) -/(fc))] ='0 (1.3) fcs/MO + .O.MO) inf [sgn(ii(l -, f) - k)(f(u(l - , t)) -/(*))] = (1.3)! fce/(u(l-,0,u a (t)) where for any real numbers a and b, I (a, b) denotes the closed interval [min(a, 6), max(a,2>)]. We assume the flux function f(u) satisfies the following two conditions />0 (AJ and lim-^Loo. (A 2 ) M-.OO M 29 30 K T Joseph and G D Veerappa Gowda Under the assumption (AJ, it can be easily checked that (1.3) and (1.3)! are equivalent to (1.3)' and (1.3); respectively, see Lefloch [3]. or a.et>0, (1.3)5, /'(u(0 + , t)) ^ and /(u(0 + , t)) /W (t)) u(l-,t) = u;(t) or a.ef>0, /'(ii(l - , t)) ^ and /(t<(0 -, t)) >/(uJ (t)) where Here /I is the unique solution of the equation /'() = (). Because of the assumption (AJ on /(w), this 1 satisfies /(A) = min/(u). U&R 1 In order to have uniqueness of solution for (1.1), it is known that, an additional condition called entropy condition should be imposed. Under the condition (AJ on f(u) this condition requires u(x + 9 t) and u(x 9 t) to satisfy u(x-,t)^u(x+ 9 t) (1-5) for every < x < 1, t > 0. Existence and uniqueness of solution of (1.1), (1.2), (1.3)' , (1.3)' 1? (1.4) and (1.5) follows from the work of Bardos et al [1], where they consider a more general problem in several space variable. In this paper we are interested in obtaining an explicit formula for the solution in the case of one space variable and f(u) satisfying conditions (A A ) and (A 2 ). The formula we derive here is an extension to the mixed initial boundary case of an explicit formula derived by Lax [5], for the pure initial value problem. In the case of one boundary, i.e., when D = {x, t):x ^ 0, t ^ 0}, this problem was studied by Lefloch [3] and Joseph and Gowda [4]. Hamilton- Jacobi equation with Neumann type boundary condition was studied by Lions [6], In one space variable they are closely related to conservation laws with Dirichlet boundary condition. This paper is organized as follows. In 2, we state the main result: In 3, we give a detailed proof of the main theorem and in 4, we study the weighted Burgers equation. 2. Statement of the main theorem Before the statement of our main theorem we introduce some notations. For each fixed (x,y,t), 0<x< 1, 0< y< 1, t>0, |i-j| ^ 1, ij = 0,l,2,..., J (x,y,f) denotes the following class of paths /? in the strip Solution of convex conservation laws in a strip 31 Each path connects the point (y, 0) to (x, t) and is of the form z = /?(s) where p(s) is piecewise linear function which are straight lines in the interior of D, and having i straight line pieces lie on jc = and j of them lie on x = 1. For the cases (ij) = (0, 0), (ij) = (2, 1), (ij) = (1,2) see figures la, Ib and Ic respectively. Denote Let f*(u) is the convex dual of f(u). /*() = max [0M- Let M (x)eL(0, 1) and u^t) and w 2 (t) are continuous bounded functions and let uf (t) and U2 (t) be defined by (1.4). Let (x, y, t) be kept fixed. For each j3e^(x, y, t\ we define We let We will see later that Q(x,y,t) is Lipschitz continuous w.r.t (x 9 y 9 t). Denote (2.3) (2 - 4) (a) s (x,t ) Cy.O) (x,t) (y,o) (c) Figure l(a-c). (x,t) Cy.O) 32 K T Joseph and G D Veerappa Gowda We will see that for a.e. (x,) there exists only one y (x 9 t) which minimises min Uo(z)dz + Q(x 9 y 9 t) (2.5) y<U LJo We shall prove the following theorem. Main theorem. Let u(x, t) be defined by K(*,0 = Qi(x,yofo*X*), ( 2 - 6 ) where Qi(x,y,t) is defined by (2.4) and y (x 9 t) minimizes (2.5). Then (i) u(x,t) satisfies w t +/(w) x = 0, in the sense of distributions and satisfies the initial condition (1.2) (ii) n(0-l-,t) and w(l ,t) exists a.e. and satisfies the boundary conditions (1.3)' and (1.3)1. (iii) for 0cfe y**^ t>Q,Q<x<l, w(x 0, exists and satisfies the entropy condition (1.5). 3. Proof of the main theorem The proof of the main theorem is broken up into several steps formulated as Lemmas. First we need some preliminaries. By definition any curve ft in #ij(x, y, t) starts at (y, 0) and ends at (x, t) and is made up of straight lines joined together at point of the boundary: x = or x =4. Let a curve ft is given and let (/f(* i) *i)> (ft(h\ ^2)- be the corners, i.e. the point of intersection of two straight lines of jS. We assume that *i 9 t 2>*3 - - - are ordered such that Note that j?(tj) can take either or 1 only, see figures la, Ib and Ic. If (x, t) is a point on the boundary i.e. if x = or x = 1, and let /Je^^x, y, t\ by convention we take t 1 = t iff for some s > 0, (t e, t) c {s: /?(s) = x}, see figure 2. For I = 0, 1, we define > * = Clearly, for fe = 0,1, 2,... Vk + i.*(x, y, t) = <f k+ 1Jk (x, y, 4 i + i >k fe y, t) = ( , y, t) = < fc (x, y, t) IJ ^, k (x, y, t), ,y,t)= Q (3-2) Solution of convex conservation laws in a strip 33 (Opt,) (0,t) (1,t 2 ) -*-2 Figure 2. jjkfoy,*), we have ^^X we have k f r^+t Z j=o tJ* J+a + Z 'or j3e < ifjj jk (x,y,t), we have k f /"'*/+ 1 =- Z jt-i r r =s - Z i J=0 U 2J-2.?+l * (3.3) (3.4) (3.5) 34 K T Joseph and G D Veerappa Gowda For f}e<#$ tk (x,y 9 t), we have where k.k(x, y, t, t, , t 2 W (s))ds t-t. 1 2/c-l For / = 0, 1, Li -7 1 < 1, i,j = 0, 1, 2, . . . , define 9 t)= min (3.6) (3.7) From (3.3H3.7) it follows that 4k+i.ft(x,j>,*)= min mn *(x,>'^)= min As.(x.j.,f) min < t 4t < . . . t 2 < ,y,t)= min mn J(jS) mn (3,8) It follows from (3.2), (3.7), (3.8) and the definition (2.3) of Q(x,y,t) that Q(x,y,t)= inf {*-<U,2,...} Since uf(s) and u 2 (s) are bounded it follows Q(x,y,t) defined by (2.3) is uniformly bounded in (x,t). Hence it follows from the assumption (A 2 ) on f(u\ that in the minimisation of (3.8) it is enough to consider t j such that t 2j - t 2j+ l ^ C > 0, where C is a constant depending only on the L norm of u+ (t) and u 2 (t) and of course on /. The reason for this is that the term Z(t 2j -t 2j+l )f*(l/t 2j -t 2j+1 )-<X) if at least one of t 2j t 2j+l j+Q, because of assumption A 2 on /. From this fact the following Lemma immediately follows. Solution of convex conservation laws in a strip 35 Lemma 3.1. Let T>Q be given, then there exists an integer N(T) depending only on T (and of course on \\u+ (OH^, || wKOIloo and f( u )) sucn that Q(x 9 y, t) = min [min {A% tk (x 9 y, t), 4J, fc (x, y, t\ fce{0,l,2,..JV(7)} 4? + i,*(*,:M), 4U + i(*>3M)}] (3.10) for all < r ^ T, ^ x < 1, ^ 3; ^ 1. Now standard arguments of Conway and Hopf [2] and Lax [5] can be used to show that Al+ 1>fc (x, y, t), A% tk (x 9 y, t\ A ktk + l (5c, j;, t) and A ktk (x 9 y, t\ defined by (3.8) are Lipschitz continuous with respect to (x 9 y 9 t). Lemma (3.1) says that Q(x,y, t) is minimum of these Lipschitz continuous functions and hence, we have the following corollary to Lemma (3.1). COROLLARY 3.1 Q(x,y,t) is Lipschitz continuous function of (x 9 y,t). To proceed further, we need to study more about A$+ itk (x,y,t\ A ktk (x,y,t), AM + 1 ( x > y, a nd A ktk (x 9 y, t). Let us take the case A k + x ik (x, y, t) and the"corresponding minimization problem ^*+ 1,*(^ y> *) = min J k + i,jt(^ y,t,t l9 t 29 .. t^t i >t 2 > ...r 4t + 2 >0 Let (*!(x,jM)> h(x>y 9 t\...t^ k+2 (x 9 y 9 t)) denote a value (t l9 t 29 ...t 4k + 2 ) for which minimum is attained. There may be several (ti 9 t 29 ...t 4k + 2 ) for which this happens. ; = l,2,...4/c + 2, define t]'(x 9 y 9 t) = mm{t j (x 9 y 9 t)}.$* Similar definition can be made for the minimization problem for J ktk + i > Jk,k an( * /jU- Let y (x, t) denote a value < y Q ^ 1, for which minimum is attained in (2.5), let j P..2) First we shall prove the following Lemma. Lemma 3.3. Let S k (x,y,t) be any set in {^, fc (x,y,t), % ktk (x,y,t\ ^ +lffc (x,y,t), #fc, k +i(x,}>,)}. Let j8 achieve minimum for minp Sk(xyt] J(/3). Let (x*, t*) be a point on P and j?J be the restriction of /? on [0, t*] and let p$eS ko (x*,y,t*) for some fc , where S ko (x*,y,t*) is one of the sets in (^ (x*,y,t*), ^ jfc (x*,y,t*), V%+ ltkt (x*y 9 t*) 9 ^ kk+1 (x*,y,t*)}. Then jS* achieves minimum for min^ eSko(x ^ yt ^J(p). The same result is true if one replaces S k (x 9 y 9 t) by %(x,y 9 t) and S ko (x*,y,t*)by <if(x* 9 y 9 t*). Proof. Suppose, on the contrary, there exists ^eS ko (x*,y,*) such that (3-13) 36 K T Joseph and G D Veerappa Gowda (x,t) represents represents ( y, 0) Figure 3. and mm ^ ?Ao( V ^J($ = j(/f). There are several cases to consider, among them ^ " S ^ bC trCated ' we te Stht Hne i tf X tf) = (0, r9) and > J f o J ' n the + bo U nda^l 2 or *> > *,>,t*)-U + i, for some fco . Let > h > , - - > * 4fco + 2 be the parameters corresponding to fl* and F , > f , > f th^ pa,ao, e , ere co^ponding , f. Th e only i ' 1 and [ straight line joining (0, t) and (1, f, ) on [f x , r]. By Jensen's inequality, we obtain Now from (3.13) and the definition of ^,(5), it follows that A0o) >/(&). Using (3.14) and the definition of /? 2 .we obtain from (3.15) > and hence (3.16) contradicts the fact that The proof of lemma is complete. By construction (3.14) (3.15) (3.16) Solution of convex conservation laws in a strip 37 Lemma 3.4. Let j8 ls i = 1,2 are minimizers for min^ e ^o +it(x y r) J(/J), t 1 ^t 2 . Then fi^ and f} 2 cannot cross with different slopes in the interior of D. The same result is true when ^'k + i.kfe.M) is replaced by tfjU + ifo,^) or ^tk(^y^t) or k (*.:Mi) or Proof. The proof follows from the previous lemma and a construction similar to the one done in the proof of that Lemma and Jensen's inequality, the details are omitted. In the next Lemma we obtain some useful properties of tf(x,y,t). Lemma 3.5. (i) Let us consider the minimization problem A%+ l tk (x, y, t) = min 0<t4 ^ 2 < <ti Jk+iA x >y> t > t i>h>->-t*k + 2)orA$ tk (x 9 y,t)=T^ each fixed t>0 ? O^y^l, t*(.,y, t) and t 1 (. ,y, ) are non-increasing function of x. ti(.,y,t) is right continuous and t^(.,y,t) is left continuous. The two functions have the same set of points of discontinuity which is countable subset of [0, oo) and except on this countable set t*(.,y,t) and ti(. ,y, t) are equal. Moreover t+ (x, y, t) = t+ (x + 0, y, t) = tr (x + 0. y, *)VO < x < I/I t 1 -(x,y,f) = tr(x-0,j;,t) = t 1 + (x-0,y,t)VO<x<l.J ( ' ) (ii) Let us consider the minimization problem Ak, k+ i(x,y,t)= min 0<t 4M . 2 <... or Al k (x,y,t)= min Jl, t (x,y,t,t l ,t z ,...t 4k ). { 0< t4. k < ... < t x For each fixed t>Q,Q^y^l,tf(.,y,t)andti(., y, t) are non-decreasing function of x. tf (. , y, t) is left continuous and t^(.,y,t) is right continuous. The two functions have the same set of points of discontinuity which is countable subset of [0, oo) and except j on this countable set tf(. ,y,t) and t^ (. ,y, t) are equal. Moreover jv ll ( ' ! \ (iii) Let us consider any of the following four minimization problems, I |f min ./+ lfjk (x, y, t, f x , r 2 , . . . 0<r 4t+2 <...<r 1 min J^^y,^^,^,...^), < r 4k < . . . < t i min Jfc, fc+ i(x,y, r, t t , r 2 , . . . 0<f 4kH . 2 <...<t 1 or min o < t 4t < . . . < tj For eac/i /Jxed 0<x^l,0<y^l,^(x,y, .) and t i (x, y, .) are non-decreasing function of t and tl (x, y, .) is left continuous and t (x, y, .) is right continuous. The two functions 38 K T Joseph and G D Veerappa Gowda have the same set of points of discontinuity which is countable subset of [0, oo) and except on this countable set t*(x,y,.\and t(x,y,.) are equal. Moreover (iv) For each fixed ^ j; < 1, t + (x, y, t) = f J" (x, y, t) a.e. (x, t). Proof. We shall prove (i). From the definition of tf(x,y,t) and using the fact that two minimizers cannot cross, in the interior of D, see Lemma 3.4, we get if x 1 < x 2 ti(x 29 y 9 t)^t^(x 29 y 9 t)^(x l9 y 9 t)^(x l9 y 9 t). (3.20) This inequality shows that tj" (. 9 y 9 1) is non-increasing and hence t J" (. , y, t) have atmost a countable number of discontinuity points. By the continuity of A% + 1>k , (3.20) implies (3.17) also. Proofs of (ii) and (iii) are similar and (iv) follows from (i), (ii) and (iii). Proof of the Lemma is complete. Now let us compute the left and right derivatives of A% tk (x,y,t) 9 A k+lik (x 9 y 9 t) 9 , y, t) and A ktk+ (x, y, t) with respect to x and t, for each fixed ^ y < 1. Denote dA = lim I ?"9 L A(xh 9 y 9 t)-A(x 9 y 9 t) im -+Q I h /t>0 d^] We shall prove the following Lemma. Lemma 3.6. For k = 1, 2, . . . 8(A k ) + (i) (ii) (iii) (iv) (v) (vi) (vii) V0<x^ 1 V \J ^ A ^ A, JVO<X<1, V0<x^l, V0<x<l, -V)<x<l, t-t;(x,y,t) 1, Solution of convex conservation laws in a strip 39 (x) (xi) Proof. We shall prove (i). By the definition of A% ik (x,y,t\ we have fc-l Z and Akjt( x >y> t ) = ~ Z /( u i" ( s ))ds Z /(2( 5 ))ds + (t-tr(x,y,t))/*l so that fe ^ft-t, + fxi 40 K T Joseph and G D Veerappa Gowda Letting /i->0, we get ^<-.'^'<F^>> In a similar way, we get Al k (x + h, y, t) - Al k (x, y,t) ( x + r,(h) \ ( h)<h - h ^ (J \t-t+(x + h,y,t))' "' Letting ft-+0 and using right continuity of tf(.,y,t) we get (3 - 22) From (3.21) and (3.22) we get (i). The proof of (i) is complete. Similar argument can be used to prove (ii)-(viii). The details are omitted. Next we compute the right derivative of A% %k (x,y,t) with respect to t. As before Letting h -> 0, we get U ' J-t;(x,y 9 t)J \t-t;(x,y,t)f j '\t-t;(x,y 9 t) As in (3.22), using the right continuity of ti(. 9 y,t) we can show x and hence at x )(/*)'L ,!L..J- ( 3 - 23 ) For convex functions, the following identity is true, namely, Using this in (3.23) we get Aik dt Solution of convex conservation laws in a strip 41 The same method can be used to prove (x) (xii). The proof of Lemma is complete. Next we shall prove the following Lemma. Lemma 3.7. For each fixed y > 0, Qt+/(Qx) = a.e(x,r). Proof. By definition and hence dx and Y - V ~r From these, it follows that AQ t0 (x 9 y,t) satisfies for each fixed 0^j;<L It follows from Lemma (3.6) that A k%k (x 9 y 9 t) 9 A ktk (x 9 y 9 t) 9 Ak tk +i(x,y 9 t) and A% +lik (x 9 y 9 t) satisfies (3.24) a.e. (x,0, for each fixed 0< j; ^ 1. Also for each fixed T, Q(x,y,t) is the minimum of a finite number of functions which satisfies (3.24), in < x < 1, < t < T, see Lemma (3.2). Now recall the following result of Conway and Hopf [2]: If {v l (x y t): i = 1,2, . . N} solves (3.24), so does v(x, t) defined by u(x, t) min v l (x, t). =1,2, ..AT Using this fact we get, for fixed < y ^ 1 = a.e. (x,r), 0<x< 1, 0<t< T. But since T is arbitrary, Lemma follows. Let (x,y,t) be fixed and let /?e#(x,;y,t). Define (3.25) then (2.5) can be rewritten in the following way. Let peV(x 9 y 9 t) and .Vofe^) be such that min [H(x,j;,t, J 8)]=H(x,j; (x,t),t, j 8)). (3.25) 42 K T Joseph and G D Veerappa Gowda Note that RHS of (3.25) is nothing but Jo we call it U(x, t) i.e., *y (x* U (x, t) = | u o 2 (*, Let j>o"(x,f) and y (x,t) be defined by (3.12). We have the following lemma. Lemma 3.8. Let t>Q,be fixed, (i) y (x, t) and yQ (x, t) are non-decreasing function of x, y (x, t) is right contiuous and }>o"(x, is '#/ continuous. The two functions have the same set of point of discontinuity and except at these countably many points, the two functions are equal Moreover, 0,0,1 ' J (ii) Suppose the minimum in (3.25) /or H(x,y,t,/l) is attained for some (x,0, t) (Vk+i tk (x 9 y (x 9 t) 9 t)). Let x* <x, and let j8* attain minimum for x*,3; (x* J Of). Moreover, forQ^x*<x if (x, ^ (x, t\ t) = t* (x*, 3> (x*, t), t), ; ^ 2 5 " ' j (iii) Suppose the minimum in (3.25) for H(x,j;, t,jS) is attained for some (x,t),t) (^fc,fc+i(x,y (x,r),r)) /et x*>x, and /et j8* attain minimum in (3.25) /or H(x*, 3;, t, jS) then /J*e^(x*, 3^o(^ 0, (i* + 1 (x*, y (x*, t\ t)). Moreover, Vx < x* * t (x, j + (x, t), = t (x*, y (x*, t, 0), 7> 2 + + * l Proof. Proof of (i) is exactly the same as the proof of Lemma (3.5). We shall prove (ii). Let us take the case where jSetfJ ffc (Jc,y (x,f),4 k^l. Since p and /?* cannot interest in the interior of D, see Lemma (3.4), it follows that j3*e# fc (x*,y (x*, t\ t) and t* (x, y* (x, t\ t) = if (x*, yoi (x*, t), t) ;> 2 and Now using part (i), we get y + (x*,t)^y" 1 (x*,t). Proof of (iii) is similar. This completeness the proof of lemma. Now we shall prove the main theorem. Solution of convex conservation laws in a strip Proof of main theorem. Following Lax [5], we introduce u N (x 9 1) = P exp j- jvf P (z) + Q(x,y, t)J>dy dy 'expj-Jvl r ' V N (x,t)= exp^-. Jo Ujv^A, l)-~ lUg Kjy. As in Lax [5], it follows that, lim u N (x y t) = Q 1 (x,y (x,t),t) N-oo lim / N (x, t) =/(Qi (*, yofe t), 0) and where y (x,t) minimizes (2.5). Also It follows from (3.29) and (3.30) that dU dx 1 ' 43 (3.29) lim U N (x, t) = U(x,t) = \ u (z)dz + Q(x, y (x, t), t) (3.30) JV-oo Jo (3.31) Next we shall show that (3.32) 44 K T Joseph and G D Veerappa Gowda We consider (U ^ = _i(M ( N)t N V N ^~W Oexpj- N[ P u (z)dz + Q(x,}>, r) 1 jdy exp{-N M (z)dz4-e(x,j;, Jo t LJo 1 f r r i) exp<-AT u (z)dz+Q(x,y 9 t)\>dy ) I LJo JJ To obtain the last equality we used Lemma (3.7) and Lemma (3.8). Now (3.32) follows from the definition of / N . From (3.31) and (3.32) we get Hence for all test functions <p(x,)eQ?[(0, oo) x (0, oo)] we get f f 1 ( w Jv<Pt+/N < Pjc)dxdt = 0. (3.33) Jo Jo Let JV-+ oo, in (3.33) and use (3.30) to get r r Jo Jo U(f>t Now we shall show that w(x, t) satisfies the initial condition. By the argument similar to be Lemma (3.1) we get given e > 0, 3<5 > such that for all e < x ^ 1 e, ^ <5, t where j/ (x,t) minimizes min But then Lax's argument [5] can be used to show lim w(x, t) = M O (X) a.e. e ^ x^ I e. Since e > 0, is arbitrary, it follows that, lim M(X, t) = M O (X) a.e. ^ x < 1. Solution of convex conservation laws in a strip 45 Next we show that w(x, t) satisfy the entropy condition (1.5). Because of Lemmas (3.5), (3.6) and (3.8) and the definition of Q l9 Lemma (3.6) it is clear that u(x0, t) exists for all < x < 1. In fact Now the entropy condition (1.5) follows from the definition of Q l9 Lemma (3.6) and the increasing nature of (/*)'. Lastly we show that u(x 9 1) satisfies the boundary condition (1.3)' and (1.3)i. Here again the existence of u(0 +,t) and u(i -,) follows as before. We shall verify that n(0 +, t) satisfies (1.3)' . The proof of (1.3)i.is similar and hence omitted. To verify (1.3)' , at t = t Q , first notice that u(x 9 t) defined by (2.6) satisfies the semigroup property i.e. where u(x, t) is defined by (2.6) with initial data is prescribed at time t l9 Q<t l <t : Because of this semigroup property and the condition (A 2 ), an argument similar to the proof of Lemma (3.1) can be used to show that u(x 9 t) = Qi(x 9 y Q (x 9 t), for some e > sufficiently small. Here y ( x minimizes min u(z, t - e)dz + g (x, y, t) o<3>aLJo J and e(x,^0 where A<l, (x,y,t)= min _ (u 1 + (s))ds + (t - [C'i - Jt 2 r Again because of the condition (A 2 ) it follows that if x ^ (5, (t t x )/*(x i/t as t 1 1 ~>0. Hence for e > sufficiently small one has u(x, t) = 2 1 (x,y (x, t), 46 K T Joseph and G D Veerappa Gowda where y (x,t) minimizes Ff y 1 min u(z,r e)dz + g(x,j;, t) i j and SO^v i t\ mir,4 (~Y \) t\ A (~Y v f^~l ^A } y y i) niiii L-**! ,o \ ' j^ /' 0,0 v * j^ / J' In this case u(x, t) satisfies (1.3y was proved in [4]. The proof of theorem is complete. 4. Weighted Burger's Equation Equations of the type ,,2\ = 0, a>-2 (4.1) u 2 are interesting, because such kind of equations appear in fluid dynamics with spherical and cylindrical symmetry and is studied by Lefloch [3] in the quarter plane x > 0, t > 0. As is observed in [3], a change of variable v (y t) = \-+l x a/2 w(x, t), y = x a/2 "** 1 (4.2) V 2 / transforms (4.1) into the Burgers equation. 'v 2 \ ~ 1=0. (4.3) Thus u(x,t) is a solution of (4.1) iff t; is a solution of (4.3). From the Bardos et al [1] formulation of the initial boundary value problem one easily gets the following formulation of the initial and boundary condition for (4.1) in D = {(x, t):Q ^ x < 1, *>0}. Initial data for (4.1): n(jc,0) = H (x)0<x<l. (4.4) Boundary condition at x = 0. or \ a.e. t > 0. (4.5) lim [x a/2 w(x, t)] < and lim x a w 2 (0 + tj > w x + x-0 x-+0 Boundary condition at x = 1: u (l _ ? t) r= uJW ) or I a.e. t > 0. (4.5)i Solution of convex conservation laws in a strip 47 Here we used the notation u^(t) = max(u 1 (t\0\ u^ (t) = min(w 2 (t),0). Entropy condition: u(x+,t)^u(x-t) (4.6) From the main theorem we get the following explicit formula for u(x, t\ the solution of the problem (4.1), (4.4) (4.5) , (4.5) t and (4.6). u(x, t) = ( -2 Jx- a/2 t;(x a+2/2 , where v(y 9 1) is given by v(y,t) = Q 1 (y,z (y,t),t\ (4.7) and z Q (y, t) minimizes min | | (r +1 ) z ^2 M ^ 2)dZi _v^_^, (u+(s)) 2 ds I , O (MI+( In (4.7), Q 1 (y, z ()>> 4 is defined by (2.2), (2.3) and (2.4) with f(u) = w 2 /2. References , \ [1] Bardos C, Leroux A Y and Nedelec J C, First order quasilinear equations with boundary conditions, , I Commun. Partial Differ. Equ. 4 (1979) 1018-1034 JL f [2] Conway E D and Hopf E, Hamilton theory and generalised solutions of Hamilton-Jacobi equations, ',; J. Math. Mech. 13 (1964) 939-986 I i|V [3] Le Floch P, Explicit formula for scalar nonlinear conservation Jaws with boundary condition, Math. f f t mech. Appl. Sci. 10 (1988) 265-287 |j [4] Joseph K T and Veerappa Gowda G D, Explicit formula for the solution of convex conservation laws f : with boundary condition. Duke Math. J. 62 (1991) 401-416 *\ [5] Lax P D, Hyperbolic systems of conservation laws II, Commun. Pure Appl. Math. 10 (1957) 537-566 |l [6] Lions P L, Neumann type boundary, conditions for Hamilton-Jacobi equations, Duke Math. J. 52 || (1985) No. 3 793-820 %' I: Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 1, April 1992, pp. 49-51. Printed in India. An example of a regular space that is not completely regular A B RAHA Statistics/Mathematics Division, Indian Statistical Institute, 203 Barrackpore Trank Road, Calcutta 700035, India MS received 3 March 1990 Abstract. A simpler example of regular space that is not completely regular is attempted. Keyword. Regular space. 1. Introduction In 1925 Urysohn [10] posed, but left unanswered, the question of whether or not regular topological spaces exist in which every continuous real-valued function is constant. Tychonoff [9], in an attempt to settle this question, produced an example of a regular space which is not completely regular. Later, making essential use of the example of Tychonoff, Hewitt [5], Novak [7], van Est-Freudenthal [3] and Herrlich [4] constructed regular spaces supporting no non-constant continuous real-valued function. Among the earliest treatises on set topology, Cech [1] gives an account of Novak's example and Vaidyanathaswamy [11] presents TychonoiFs example mentioned above. In recent times, more accessible references are Dugundji [2] and Steen and Seebach [8] which give the same examples under the names "Spiral staircase" or "Tychonoff corkscrew". This example involves the use of the uncountable well-ordered space co l . I venture, in this shortnote, on an apparently simpler construction of a regular space that is not completely regular. 2. Construction For any even integer n let T n = {n} x ( 1, 1) and X l = u neven T n = {(n,y):n even integer, 1 <y< 1}. Let {a fc ,fc^ 1} be a strictly increasing sequence of positive real numbers such that lim fc ^ 00 a jk =l.- For any odd integer n, set C Btk = {(x,y):(x n) 2 + y 2 = 0^}. fc=l,2, ... and set %2 == u nodd u ^iC n ,jc- Let a and b be two distinct points not belonging to the union X 1 vX 2 . Form the' set X = X 1 vX 2 v{a,b}. Topology of X We shall define a topology on X by describing the neighbourhoods of each of its points. For each odd integer n and each k ^ 1, all points of C nk except the point a nk = (n, a k ) 50 A B Raha are isolated. A neighbourhood of a ntk consists of all but a finite number of points of C B , k . Write ' ^ 00 f, C n = (J C tttk9 nodd. If p = (n 9 y)eX l9 consider the subset {(z,y):/i-l<z<+l}n(C n _ 1 uC n+1 ) of X. A neighbourhood of p consists of all but a finite number of points of this subset. A neighbourhood of a consists of all points of X^X 2 with first coordinate greater than some real number c. A neighbourhood of b consists of points of X^ uX 2 with first coordinate less than some real number d. The neighbourhoods describe a T l topology on X. It is not difficult to see that under this topology each neighbourhood of a point of X contains a closed neighbourhood of the same point. X is thus regular and Hausdorff. Failure of complete regularity We claim that given a real-valued, continuous function / on X 9 /(a) = /(&). Consequently X fails to be completely regular. Let us first observe that if h is a continuous real-valued function on C rt>k , the set is at most a countable subset of C M)k . Let/:Z - R be an arbitrary, continuous function. Set JB n k = {(x, j^)eC B ^:/(x, y) 7^ /(# fc )} and D n = ordinates of points in u^L x J3 B?fc }. In view of the observation above, each # k is countable and consequently, each D n is so. If D = u nodd D M , D is then a countable subset of ( 1, 1). Suppose peX t is such that pe T n and the ordinate y of p does not belong to D. Consider If ( and if ( From the structure of neighbourhoods of p it is clear that )= lim/(fl n . lfjk )= Ii Regular space that is not completely regular 51 Let qeX^ be such that qT n + 2 and the ordinate of q does not belong to D. Considerations as above will lead us to conclude that f(q)= lim/(fl n + 3ifc )= limf(a n + ltk ). k-*oo /c- oo Hence, f(p) = f(q). If G = {(n,y):n even and ye( 1, 1) D}, the above argument shows that for any peG,/(p) = lim k -, ,/(<*+ lffc ) = lim^^/fa,-!.*) where p = (n, j;). Thus / is a constant on G, say, a. Since / assumes the value a in every neighbourhood of each of a and fc, /(a) = a = f(b). The claim is thus established. A few remarks about the space X (I) The space X has the merit of playing the role of the space Q which enters into the construction, due to Herrlich [4, page 153], of a regular space on which every continuous real-valued function is constant. (II) The space X admits a proper subspace which is also a regular Hausdorff space that fails to be completely regular. To be precise take the subspace Z of X where T and {a} are disjoint closed subsets of Z. If #:Z-> (R is a continuous function which is 1 on T , it can be easily seen that g(a) = 1. As a result, Z cannot be completely regular. (Ill) At the time of the construction of the space X, the author was not aware of the existence of the paper by Mysior [6] which contains an elementary example of regular space which is not completely regular. However the space X is a different example. References [1] Cech E, Topological Spaces (1966) (Interscience Pub.) [2] Dugundji J, Topology (1966) (Allyn-and Bacon) [3] Est, Van W T and Freudenthal U H, Trennung durch stetige Funktionen in topologischen Raumen, Indagationes Math. 13 (1951) 359-368 [4] Herrlich H, Wann sind alle-stetigen Abbildungen in Y Konstant? Math. Z. 90 (1965) 152-154 [5] Hewitt E, On two problems of Urysohn, Ann. Math. 47 (1946) 503-509 [6] Mysior A, A regular space which is not completely regular, Proc. Am. Math. Soc. 81 (1981) 652-653 [7] Novak J, Regular space, on which every continuous function is constant, Cas. Pest. Mat. Fys. 73 (1948) 58-68 [8] Steen L A and Seebach Jr J A, Counterexamples in Topology (1970) (Holt, Rinehart and Winston, Inc.) [9] TychonofT A, Uber die topologische Erweiterung von Reumen, Math. Ann. 102 (1930) 544-561 [10] Urysohn P, Uber die Machtigkeit der zusammenhangenden Mengen, Math. Ann. 94 (1925) 262-295 [11] Vaidyanathaswamy R, Treatise on Set Topology (1947) Madras Proc. Indian Acad Sci. (Math. Sci.), Vol. 102, No. 1, April 1992, pp. 53-58. Printed in India. Factors for \N,p n ;&\k sommabslity of Fourier series HUSEYIN BOR* Department of Mathematics, Erciyes University, Kayseri 38039, Turkey *Mailing Address: PK 213, Kayseri 38002, Turkey MS received 17 January 1991; accepted 25 October 1991 Abstract. In this paper two theorems on \N,p n ;8\ k summability factors, which generalize the results of Bor [4] OR \N,p n \ k summability factors, have been proved. Keywords. Fourier series; summability factors; absolute summability. 1. Introduction Let # be a given infinite series with partial sums (s w ) and let (p n ) be a sequence of positive numbers such that ^1,= i>*->oo as w - (P-* = P-i = 0,z>l). (1) The sequence-to-sequence transformation 1 " (2) defines the sequence (t n ) of the (N,p n ) means of the sequence (s n ), generated by the sequence of coefficients (p n ). The series Sa w is said to be summable |N,p n | k ,/c^ 1, if (see [1]) ^(Pn/Pnr^n-tn-^^ (3) w=l and it is said to be summable \N,p n ;8\ k ,k^ 1 and <5^0, if (see [2]) _ L |t rt -f M _ r | fc <oo. . (4) n=l In the special case when p n \ for all values of n (resp. S = 0), \N,p n ; d\ k summability is the same as |C, l;<5| k (resp. |JV,p n | k ) summability. Let f(t) be a periodic function with period 2n and integrable (L) over (71,71). Without any loss of generality we may assume that the constant term in the Fourier series off(t) is zero, so that (5) 53 54 Huseyin Bor and co oo n 1 n 1 We write 1 1 f ' (p(t) = -{/(x + 1) + f(x )}, <PiW = - <p(w)du and A/l n = ! A n + 1 . 2. Quite recently Bor [4] proved the following theorems. Theorem A. Let the sequence (p n ) be such that P n = 0(np n ) (7) P n Ap n = O(p n p n+1 ). (8) // (Pi(t) is of bounded variation in (0,7r) and (A rt ) is a sequence such that Y-|A n | fc <oo (9) < oo. (10) then the series EA n (t)P n A n (np n )'~ l is summable \N,p n \ k for k^l. Theorem B. Let the sequence (pj be such that conditions (7) and (8) of Theorem A are satisfied. If Ea w is a series of complex terms such that 17=1 then the series 7La n P n h n (np n )~ l is summable \N,p n \ k for k^l. 3. The aim of this paper is to prove above theorems for \N,p n ;d\ k , with k ^ 1 and <5^0, summability. Now, we shall prove the following theorem. Theorem 1. Let the sequence (p n ) be such that conditions (7) and (8) of Theorem A are satisfied and (12) n = v n-l <?i(0 is f bounded variation in (0,7r) and (%) is a sequence such that l_t \ "*n \ ^ ^ \ 1-3/ n= 1 and then the series *LA n (t)P n X n (np n )~~ l is summable \N,p n ',5\ k for k^ 1 and <5 >0. Factors for \N,p n ;d\ k summability of Fourier series 55 Theorem 2. Let the sequence (p n ) be such that conditions (7) and (8) of Theorem A and condition (12) of Theorem 1 are satisfied. If condition (11) of Theorem B is satisfied by the series Sa n , then the series *La n P n X n (np n )~ l is summable \N,p n ;8\ k for k ^ 1 and <3 ^ 0, where (l n ) is as in Theorem 1. 4. We need the following lemmas for the proof of our theorems. Lemma 1. If (p^t) is of bounded variation in (0,7c), then vA v (x) = O(n) as n- oo. (15) This lemma is a particular case of Lemma due to Prasad and Bhatt ([5], Lemma 9). Lemma 2. ([3]). // the sequence (p n ) is such that conditions (7) and (8) of Theorem A are satisfied, then A{P n /(p n n 2 )} = 0(l/ 2 )as-. (16) 5. Proof of Theorem 2. Let (7^) denote the (N, p n ) mean of the series tf n P n A B (np n )~ 1 . Then, by definition, we have T = 1T E P. Z flMfa.)' 1 =-5- Z (P.-.P.-Ja.jPAto.r 1 . * n = i = * nt; = Then, for n ^ 1, we have that By Abel's transformation, we have 17=1 n-1 n-1 ^r^+T^ +^.3+^.4, say. To complete the proof of the theorem, by Minkowski's inequality for k > 1, it is sufficient to show that E (P./P.)* + *- 1 |T..,|*< oo, for i= 1,2,3,4. 56 Huseyin Bor Firstly, we have that n,l\ k = Z = 0(1) n*- MA, =1 = 0(1) as w-oo, by virtue of (7), (11), (13). Now, when k > 1 applying Holder's inequality, with indices k and fc', where 1/fc + 1/k 7 = 1, we have that I (P./ 1 n-1 1*-1 Pu 1 n-1 f P IR - i _ - y n pl p L = 0(1) V-2fc = 0(1) as m^ oo, by (7), (11), (12) and (13). On the other hand, since t?=l by (14), we have that m+1 Z (P./P.)' n = 2 ^-i|r n , 3 |< m+1 1 fn-1 Z (P./p-)*" 1 ^ { IP.IAA. n = 2 ^i-l (.=! Factors for \N,p n ;5\ k summability of Fourier series 57 n+l I X I* n/Pn) ~B = v+ 1 * v 1 = 0(1) (P./ = 0(1), as m-* oo, by virtue of (7), (11), (12) and (14). Finally, using the fact that A{PJ(v 2 p v )} = 0(l/t> 2 ), by Lemma 2, we have 1 n ~ 1 y=l m+1 x I ( n-v + 1 =O(D 17-1 = 0(1) as m^ oo, by (7), (11), (12) and (13). Therefore, we get that Z (P ( ,/Pn)" t+ *~ 1 l T n .t\ k -0(\) as m-f oo, for i= 1,2,3,4. 58 Huseyin Bor This completes the proof of Theorem 2. Proof of Theorem 1. Theorem 1 is a direct consequence of Theorem 2 and Lemma 1. Remark. If we take 6 = in our theorems 1 and 2, then we get Theorem A and Theorem B, respectively. Because in this case the conditions (13) and (14) reduce to conditions (9) and (10), respectively. It should be noted that in this case condition (12) is obvious. If we take p n = 1 for all values of n in Theorem 1, then we get the following corollary. COROLLARY // (p 1 (t) is of bounded variation in (0, n) and (/l n ) is a sequence such that conditions (13) and (14) of Theorem 1 are satisfied, then the series *LA n (t)A. n , at t = x is summable \C, 1; 8\ k , k ^ 1, provided that 1 dk > 0. References [1] Bor H, A note on two summability methods, Proc. Am. Math. Soc. 98 (1986) 81-84 [2] Bor H, A relation between two summability methods, Riv. Mat. Univ. Parma (4), 14 (1988) 107-112 [3] Bor H, Absolute summability factors for infinite series, Indian J. Pure Appl Math. 19 (1988) 664-671 [4] Bor H, Multipliers for |N,pJ k summability of Fourier series, Bull. Inst. Math. Acad. Sinica 17 (1989) 285-290 [5] Prasad B N and Bhatt S N, The summability factors Fourier series, Duke Math. J. 24 (1957) 103-117 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 1, April 1992, pp. 59-71. Printed in India. Maximal monotone differential inclusions with memory NIKOLAOS S PAPAGEORGIOU Department of Applied Mathematics, Florida Institute of Technology, 150 W. University Blvd, Melbourne, Florida 32901-6988, USA and Department of Mathematics, National Technical University, Athens 15773, Greece MS received 6 June 1991 Abstract. In this paper we study maximal monotone differential inclusions with memory. First we establish two existence theorems; one involving convex-valued orientor fields and the other nonconvex valued ones. Then we examine the dependence of the solution set on the data that determine it. Finally we prove a relaxation theorem. Keywords. Maximal monotone operator; resolvent; resolvent convergence topology; selection theorem; relaxation. 1. Introduction In this paper we examine maximal monotone differential inclusions with memory, defined in R N . First we consider the existence problem, and we prove two such theorems. One with a convex-valued orientor field and the other with a nonconvex valued one. Then we examine the dependence of the solution set on the data that determine it; i.e., the maximal monotone operator, the orientor field and the past history information. More precisely, we consider a parametrized family of problems, where all the above data depend on the parameter, and we examine how the solution set responds to variations of the parameter. Finally we prove a "relaxation" result, which says that under reasonable hypotheses on the orientor field, the solution set of the "nonconvex problem" is dense in that of the "convex problem". Our formulation of the problem is general enough to incorporate subdifferential systems. Among them of particular interest, because of their diverse applications, are those for which the maximal monotone operator A = dd K , with S K being the indicator function of a nonempty, closed and convex subset K of R^i.e., 8 K (x) = if xeK and S K (x) = 4- oo if x$K) and dS K (-) denotes its subdifferential in the sense of convex analysis. It is well-known (see for example Aubin-Cellina [2]), that dd K (x) = N K (x) for every xeK, with N K (x) being the normal cone to the set K at x. In this case, the corresponding "differential inclusion" is also called "differential variational inequality" and appears in mathematical economics, in the study of dynamic allocation processes (see Aubin-Cellina [2], Henry [10] and Stacchetti [18]) and in theoretical mechanics in the study of unilateral processes (see Moreau [14]). Our system has a memory feature, since the derivative of the state depends on the past history of it. We should mention, that this memory feature of our system, arises in the so-called "absorption lag" dynamic economic models. It signifies that the growth rate x(t) of the capital depends on the past history x,(-) of the capital. Finally given that every control system, after 59 60 Nikolaos S Papageorgiou "deparametrization" (union over all admissible controls of all vector fields), can be described by a differential inclusion, the systems studied in this paper incorporate hereditary control systems, monitored by maximal monotone, multivalued in general operators. Our results also extend the works on differential inclusions done by Aubin-Cellina [2], Bressan [5] and Cellina-Marchi [7]. 2. Preliminaries Let (Q, Z) be a measurable space and let X be a separable Banach space. Throughout this paper, we will be using the following notation: P f(c} (X) = {A X: nonempty, closed (convex)}. A multifunction (set- valued function) F:Q -> Pf(X) 9 is said to be measurable, if for all xeA", the Unvalued function c0^d(x,jF({0))==inf{||x z||:zeF(a>)} is measurable. Other equivalent definitions of measurability of a Pf(X)- valued ^ multifunction can be found in Wagner [20] (see theorem 4,2). We will say that G:Q->2*{0} is graph-measurable, provided that GrG = {(<o,x)el x X:xeF(e0)}e I,xB(X) 9 with B(X) being the Borel cr-field of A". For P^)- valued multifunction, measurability implies graph measurability, while the converse is true if there exists a cr-fmite measure /*() on 2, with respect to which is complete (see Wagner [20], theorem 4.2). Now let (fl,E,ju) be a finite measure space and F:Q-2*,{0}. By S we will denote the set of integrable selectors of F(-); i.e., Sp = {/e L 1 (X):f((D)eF(co>)^ a.e.}. ; This set may be empty. For a graph measurable multifunction F(-), it is nonempty if \ and only if (y-*inf{||z||:zeF(o>)}6L+. In particular, this is the case if co-*\F(co)\ sup{ |i 2 1| :zeF(co)}e L+ . Such multifunctions are called "integrably bounded". Next let 7,Z be Hausdorff topological spaces and G: Y-*2 Z ,{9} a multifunction. We will say that G(-) is upper semicontinuous (u.s.c.) (resp. lower sernicontinuous (Ls.c.)), if ^ for every 17 c z open G+ (7) = {ye Y: G(y) s 17} (resp. G'(U) = {ye Y: G(y)n U is open in 7. A multifunction G(-) which is both u.s.c. arid Ls.c. is said to be continuous. So a continuous multifunction G(-), is one that is continuous from Y into 2 Z ,{0} equipped with the Vietoris topology (see Klein-Thompson [12]). If Z is a metric space, then on P f (Z) we can define a (generalized) metric, known in the literature as the Hausdorff metric, by setting h(A 9 B) = max [sup a6A d(0, B\ sup beB d(b, A}}, for every A,BeP f (Z). If Z is complete, then so is the metric space (P s (Z\h). A multifunction F:Q~>P / (Z) is said to be Hausdorff continuous (/i-continuous), if it is continuous from Y into the metric space (P/(Z), h). If the multifunction has nonempty compact values, then continuity and /z-continuity coincide. This follows from the fact that on the collection of nonempty compact sets of a metric space, the Vietoris and Hausdorff topologies coincide (see Klein-Thompson [12], corollary 4.2.3, p. 41). Let V be a Banach space and {A n , A} n ^ 1 c 2 v {ty}. Denote by s- the strong topology on V and by w- the weak topology. We define: x n A n ,n^ 1} = {xe V:\im d(x,A n ) = s-lim A n = {xe V:x = s-lim x Bjk , x^eA^, n 1 < n 2 < - - - < n k < - } and _ = {xe F:x = ^-limx^x^eA^n^ <n 2 Maximal monotone inclusions 61 It is clear from the above definitions, that we always have s-]\mA n ^ s-lim ^4 n e v-limA n . If s-timA n = w-limy4 rt = A, then we say that the A K *s converge to A in the <Curatowski-Mosco sense, denoted by A n K ~ M > A (see Mosco [15]). If dim V < oo, ;hen the weak and strong topologies coincide, and so we recover the well-known iCuratowski mode of set convergence (see Kuratowski [13]). If s-lim A n A = s-lim A n9 :hen we write A n -^ A. Let us also recall some basic facts about maximal monotone operators. So let H 3e a Hilbert space. An operator A:D(A)^H-+2 H is said to be "monotone", if and mly if (x x', y / ) ^ for all [x, y] 9 [x', / ] eGr A (here (, ) denotes the inner product n H). It is said to be maximal monotone if and only if (x v, y w) ^ for all ~x,y]eGr,4, which implies that weAv (i.e., the graph of A is not properly included in my other monotone subset of H x H ). From a well-known theorem of Minty, we have :hat A(-) is maximal monotone if and only if from some /I > 0, R(I -j- /L4) = H. Then br every A > 0, J^ = (/ + AA) ~ * : #(/ H- &A) = H - />(/4), and is called the "resolvent of 4". The resolvent J A (-) is nonexpansive and J^xAx as /l-^O"*" for each x6D(/4). Let Jt be the set of all maximal monotone operators in H. The topology of R-convergence on Jt^ is the weakest topology, that makes continuous the maps J^ x :^ -+H for every /I > and xeH, where J^ X (A) = (/ 4- A^4)~ 1 x. We will denote 3y ^^ (or Jt R (H)\ the set ujf equipped with the topology of R-convergence. If H is separable, then Jt R is a Polish space (i.e., a separable, metrizable, complete space). Furthermore, we know that A n ^ A if and only if GrA n K ~~ M GrA For further letails, we refer to Attouch [1]. Finally, note that if A() is maximal monotone, for ;very xeD(A\ Ax is closed and convex. Hence for every xeD(A\ Ax contains an dement of minimum norm (the projection of the origin on Ax). This unique element s denoted by Ax. Thus we have AxeAx and \\Ax\\ =inf{||3;||:j;6Xx}. The jingle-valued operator A:D(A)-+H, is called the "minimal section" of A. 3. Existence theorem Let fc,r>0 and set f = [-r,fe], T = [~r,0] and T= [0,i]. We will be studying :he following maximal monotone differential inclusion with memory: - x(t)eAx(t) H- F(t, x t ) a.e. on T] Here x f eC(T ,lR JV ) and is defined by x t (s) = x(t + s). So x r (-) gives us the history 3f state x(-) from t r up to the present time t. In this section we present two existence results concerning (*). The first assumes ;hat the multivalued perturbation F(t 9 x t ) is convex valued, while the second that it s nonconvex valued. For the first existence theorem, we will need the following hypotheses on the data. H(A): A:D(A)^ IR N ~>2 R " is a maximal monotone operator. F: T x C(T , R N ) -> P fe (R N ) is a multifunction s.t. (1) t^F(t, y) is measurable, (2) j;->F(i,j;).isiLs.a, (3) \F(t,y)\ =mp{M:vF(t 9 y)} <aWH-iWl^lloo a.e. with 62 Nikolaos S Papageorgiou By a solution of (*), we understand a function xeC(f, R N ) s.t. x(v) = q>(v) for t?e T ( and x| r is a solution of the initial value problem x(t)eAx(t) + F(t, x,) a.e., x(0) = (i.e., x: T-- R N is absolutely continuous and there exists feSp ( . tXf )S.t. x(t)eAx(t) a.e., x(0) = o Theorem 3.1. // hypotheses H(A\H(F) and H(cp) hold, then (*) admits a solution and the solution set is compact in C(T 9 R N ). Proof. First we will obtain an a priori, uniform bound for the solutions of (*). So let x()eC(f,R N ) be such a solution. Then by definition we can find f(-)eL l (T 9 R N ) 9 f(t)eF(t, x t ) a.e. s.t. x(t)eAx(t) + f(t) a.e. From Benilan's inequality (see for example Vrabie [19], corollary 1.7.1, p. 35), we have that T Jo where {S(t)} tT is the nonlinear semigroup of contractions generated by the maximal operator A(\ Recalling that r-* ||5(r)<p(0)|| is continuous on T, we can find M>0 s.t. || S(t)cp(Q)\\ ^M for all te T. Hence using growth hypothesis #(F)(3), we get ||x(t)KM+ o Let h(t) = ||x, H^,. Then clearly h(-)eC(T, U N ) and we have h(t) ^M+\ (a(s) + b(s)h(s))ds, te T, Jo with M = max[M, || <p IL]. Invoking Gronwall's inequality we get M x > s.t. for all te T we have Then consider the following, modified orientor field F &y)v \\y\\* Note that F(t,y) = F(t,p Ml (y)\ where p Ml (-) is the M r radial retraction on the Banach space C(T , R N ). Hence, t -> F(t, y) is measurable, while since p Ml () is Lipschitz continuous, y->f(t 9 y) is u.s.c. (see Klein-Thompson [12], theorem V.3. 11, p. 87). Furthermore we have: \F(t,y)\ = sup{ ||v|| :veF(t,y)} ^ a(t) + b(t)M 1 - \l/(t) a.e. with vH')e+. Then we consider problem (*) with F(t 9 y) instead of F(t,y). Set V={g<=L 1 (T,R N ):\\g(t)\\^il/(t) a.e.}. From the Dunford-Pettis theorem, we know that V is sequentially weakly compact. In what follows V will be equipped with the relative weak-L^T, ^ N ) topology. Let R: V-*P fc (V) be the multifunction defined by Maximal monotone inclusions 63 R (0) = s k P (9).)> where prL^R^HC^R") is the map that to each geL^T^R") assigns the unique solution of x(t)eAx(t) -4-0(f)a.e., x(0) = (p(0) (its existence is guaranteed by the Benilan-Brezis theorem; see Vrabie [19], theorem 1.9.1, p. 41). We claim that R(-) is an u.s.c. multifunction. Given that V equipped with the relative weak-L 1 ^ 1R N ) topology is compact, metrizable (see Dunford-Schwartz [8], theorem 3, p. 434), it is enough to show that GrR is sequentially closed in V x V (see Klein- Thompson [12], theorem 7.1.16, p. 78). To this end, let [#,/] eGrKn^ 1 and assume that [0 n ,/ n ]->[0,/3 in V x V. From corollary 2.3.1, p. 67 of Vrabie [19], we know that p(-) is sequentially continuous from L^TJffP) equipped with the weak topology into C(T,R N ) with the strong topology. So if we define p(g)(-)eC(f,R N ) by setting P(ff)(t) = p(g)(t) for teT and p(g)(v) = q>(v) for ve T , then we have p(g n ) l ^p(g\ in C(T , R N ) for all te T. Hence applying theorem 4.2. of [16] we have =>R(-) is indeed u.s.c. as claimed. Since R(-) is closed, convex valued we can apply the Kakutani-KyFan fixed point theorem to get geR(g). Let p(g)() = x(-)eC(f, U N ). Then this function solves (*) with F(t, y) instead of F(t,y). But as in the beginning of the proof, via GronwalFs inequality (see Vrabie [19], p. 3) we can get that || x J *, < M t . Hence F(t, x t ) = F(t, x,). Thus x(-) solves (*). Since the solution set of (*) lies in p(V) and the latter is compact in C(T, R N ), to establish the compactness of the solution set, it suffices to show that it is closed. So let {x B } n>1 c C(f, R N ) be solutions of (*) and- assume that x n -*x in C(f, R N ). Then x n = p(f n ) for some/ n Sf (%( w . Because of hypothesis H(F)(3) and the Dunford-Pettis theorem, we have that (f n } n>l is relatively sequentially w-compact in L^T, IR N ). So by passing to a subsequence if necessary, we may assume that f n ^+f in L 1 (T, R N ) and so p(f n )-+p(f) = x in C(f,R N ). Also from theorem 4.2 of [16], we have that /eSj? ( . x >. Hence x(-) = (/)(') is.also a solution of (*), establishing the compactness in C(7JR*)of(*). Q.E.D. We can also have an existence result for the case where the orientor field F(% ) is not necessarily con vex- valued. For this we will need the following hypotheses on the F(t, y). H(F) 1 : F: T x C( T , R N ) -> P f (R N ) is a multifunction s.t (1) (t,y)^>F(t,y) is graph measurable, (2) y~>F(t,j;)isl.s.c., (3) \F(t,y)\ <a(t)-h 6||j>IL a.e., with a(-), 6(-)6U- Theorem 3.2. // hypotheses H(A\ H(F) l , and H(<p) hold, then (*) admits a solution. Proo/ Let V^ L l (T, R N ) and F(t, j) be defined as in the proof of theorem 3.1. Let K = p( V) (i.e., K = p(V)on T and = {<p} on T ). From the continuity property of #() (see the proof of theorem 3.1), we have that K is compact in C(T,(R Ar ). Hence by Mazur's theorem so is X = c6nvJK. Let jRiK-^Py^^^IR^)) be defined by Sj? ( . iy>) . Using theorem 4.1 of [16], we get that R(-) is l.s.c. Applying Fryszkowski's w\ 64 Nikolaos S Papageorgiou selection theorem [9], we get r.K- L^T,^') continuous s.t. r(y)eR(y) for all yeK. Then let q:K-*K be defined by q = por. Clearly q(-) is continuous. So applying Schaudefs fixed point theorem, we get xeK s.t. x = q(x). Again through the definition of F(t, y) and GronwalTs inequality, we can check that || xj < M 1 =>F(t, x,) = F(t, x t ) => x(-)eC(f, R") solves (*). Q.EJX Remark. Theorem 3.2 above, improves the result of Cellina-Marchi [7], who considered memoryless systems and assumed that the orientor field was /z-continuous in both variables, also extends the existence result of Bressan [5], who also considered memoryless systems and assumed that A == 0. 4. A continuous dependence result In this section, we investigate the dependence of the solution set of (*) on the data that determine it; namely the maximal monotone operator, the orientor field and the function cp. So let be a metric space. We consider the following family of problems parametrized by elements in E: (r)x(t) + F(r,x r ,r)a.e. on T We denote the solution set of (*) r by P(r) c C(T 9 M N ). We want to examine the dependence of P(-) on re. To this end we will need the following two auxiliary results. Recall that if A:D(A) R Ar -2 R " is a maximal monotone operator, then we can define the realization of A on L 2 (T,U N \A:D(A)^L 2 (T,R N )-+2 L2( * RN] by Ax = {ye L 2 (T > R N ):y(t)eAx(t) a.e. on T} for each xeD(A) = {veL 2 (T,U N ):v(t)eD(A) a.e. and there exists coeL 2 (T 9 R N ) s.t. co(t)eAv(t)a.Q.}. It is well-known and easy to prove that the realization A(-) is maximal monotone too. Lemma 4.1. // A:E-+J( R (R N ) is continuous, then so is A:E-+J R (L 2 (T, Proof. Let r n -+ r in E and let s(t) = 1^= 1 Xc k ^^ w ^h C fc e, co fc e R N (a simple function). Then since by hypothesis A(r n )-^A(r) in Jt r (U N \ we have forallfce{l,...,m}, A>0. Thus m m Z %c k (t)Ji (r " )(Jt) k-* Z Xc k (t)J* (r)o} k as n->oo for all re T. From this we deduce that as n-*oo in L 2 (TJ Maximal monotone inclusions 65 for all A > 0. Since s(-) was an arbitrary simple function, simple functions are dense in L 2 (T, R N ) and the resolvent operator is nonexpansive, we get jf (rj (x) A jfW(x) as n-> oo in L 2 (T, R N ) for all xeL 2 (7JR N ) and all l>Q=>A(r n )~+A(r) in ^(L 2 ^^)) (see 2)=>^:-> Jf R (L 2 (T 9 U N )) is indeed continuous. QJLD. The second auxiliary result that we will need is the following: Lemma 4.2. // X is a Banach space, {A n } >l ^ P f (X\ A n ^K for alln^l with K^X compact, and A n ^*A as n -* oo, then A n -+ A as n -> oo. Proof. Let a n eA n , n ^ 1 s.t. d(a n , A) = sup bAn d(b, A). It exists since A n , n ^ 1 is compact (being a closed subset of the compact set K). Note that {a n } n ^ l K. So by passing to a subsequence if necessary, we may assume that a n -*a. Because A n -^A, we have aeA. Then d(a n ,A)-^d(a,A) = 0=>sup hAn d(b^A)-^Q as n-*oo. Similarly, let a n e/l n^l s.t. sup fre/4 rf(i,>l w ) = d ( (a w ,y4 n ). Since /I is compact and (a n } i ^1, we may assume that a n -+aeA. Then we have But since A n ^ A, we have d(a, A n ) -> 0. So d(a M , ^ B ) ~> as n -> oo =>sup &e/l d(b, A n ) -> A Q.E.D. To prove our continuous dependence result, we will need the following hypotheses on the data. H(A) l : A:E~+Jt R (R N ) is continuous, for every re D(A(r)) is closed and A(r) is bounded on compact subsets of D(A(r)\ uniformly in reB E nonempty, compact. Remark. This hypothesis is clearly satisfied if A(r)-d5 K{r} with K:E-+P fc (U N ) continuous, or if for every reE,D(A(r)) = U N and r -> ^4(r) is bounded on compact sets. fl(F) 2 : F: T x C( T , 1R N ) x -* P /e (R N ) is a multifunctions s.t. (1) t -> F(t, y, r) is measurable. (2) (y, r) -> F(t, v, r) is continuous, (3) fc(F(r, y, r), F(t, /, r)) < ij (t) || y - / 1| a.e. with ^)e Li , (4) |F(r,y,r)|<fl B (0 + MOWco with fl B (-), & B C)e Li and for all nonempty, compact. (p:E-C(T , U N ) is continuous and for all reE q>(r)(0)eD(A(r)). Theorem 4.3. // hypotheses H(A) l ,H(F) 2 and H((p) l hold, then r->P(r) from E into ?' the nonempty compact subsets of C(T, R N ) is continuous and h-continuous. * I/ Remark. The hypotheses of this theorem and theorem 3.1 guarantee that for every ij! re, P(r) is nonempty and compact in C(f,U N ). f 66 Nikolaos S Papageorgiou Proof. Let r n ->r in E and let xes-limP(r n ). Then by denoting subsequences with the same index as the original sequences for economy in the notation, we know that there exist x n eP(r n ) n ^ 1 s.t. x n ->x in C(f, (R N ). Then by definition )a.Q. on T x n (v) = (p(r n )(v) veT , where / n e L 1 ( T, (R N ) and f n (t)eF(t, (x n ) r , r n ) a.e. . Then from the Benilan-Brezis theorem (see Vrabie [19], theorem 1.9.1, p. 41), we have But from hypothesis H(A) l9 we know that there exists M 2 > s.t. for all n ^ 1 and all t<= T, we have \\A(r n )x n (t)\\ ^M 2 . Hence we have: ^ a B (t) + b B (t)M i + M 2 = fc B (t) a.e. with JijsQeL 2 and J5 = {r n ,r} B>1 ^E (note that the bound M^O derived in the proof of theorem 3.1 holds for all *() n ^ 1, because of hypothesis H(F) 2 ). From this last inequality, we deduce that {x n } n ^ 1 L 2 (I^ DR^) is relatively sequentially w-compact. Also as in the proof of theorem 3.1, we can get that {x n } n ^ { ^p(V\ where F={0eL 2 (T,(R N ): \\g(t)\\ ^\// B (t) = a B (t) + b B (t)M 1 a.e.}. Since p(V) is compact in C(T, (R N ) (see the proof of theorem 3.1), we get that {x n } n>1 C(T, R N ) is relatively compact. Finally note that for n^ 1, || /(*)!! <^(0 a.e. Hence {/ n } n>1 is relatively sequentially w-compact in L 2 (T, IR N ). Thus by passing to an appropriate subsequence if necessary, we may assume that x n -^x in C(f,R N ), x n ^*v in L 2 (T,R N ) and f n ^f in L 2 (T, R N ). Clearly i? = x on T. Then note that for all n > 1 and Also from lemma 4.1, we have that A(r n )-+A(r) in ^ jR (L 2 (T,R N ))=>Gr ^(O^-^GT^M (see Attouch [1], theorem 3.6.2, p. 365). Hence in the limit as n->oo, we get ^x(t) + f(t) a.e. on T and x(v) = (p(r)(v) veT Q (because of hypothesis Furthermore because of hypothesis H(F) 2 (1) and theorem 4.4 of [16], we have that f(t)eF(t,x t ,r) a.e. Therefore xeP(r) and so we have proved that ^P(r). (I) Next let xeP(r). Then by definition, we have - x(t)eA(r)x(t) + f(t) a.e. on T Maximal monotone inclusions 67 with /s L l (T,R"), f(t)eF(t,x t ,r) a.e. Let m,(t) = proj [/(r); F(t, x,, rj] and (t, >>, r_) = proj im n (t); F(t, y, rj], yeC(T , R"), where for every CePf C (K N ), proj(-;C) denotes the metric projection function. From theorem 4.2 of [1 1], we know that ?() is measurable, while from theorem 3.33, p. 322 of Attouch [1], we have that (t, y, r) - u(t, y, r) is measurable on t, continuous in (y, r) (i.e. a Caratheodory function), hence jointly measurable. Consider the following differential inclusions n > 1: x n (v) = <p(r n )(v) veT . For each n > 1, we can find a solution x n (-)eC(f, R") of the above problem. From the bounds obtained in the first half of the proof, we know that by passing to a subsequence if necessary, we may assume that x B -^x in C(f, R N ) and x n ^x in L 2 (T,M N ). Let ?(), y(-)eL 2 (7;R N )s.t. y a (t)eA(r n )x n (t)*.e., y(t)e,4(r)x(t)a.e. and yn(t)=-x n (t)-u(t,(x n ),,r a ),y(t)=-x(t)-f(t)a.e. Note that y^y in L 2 (T,R") with y(t) = - x(t) - u(t,^,,r)a.e. Then we have: (- x n (t) + x(), x(t) - x n (t)) = (y n (t) - y(t), x(t) - x n (t)) + (u(t, (xj r , r n ) - /(t), x(t) - x n (t))a.e. Recalling that GiA(r n ) ^GrA(r) in L 2 (T, R*) x L 2 (i; R N ) (since A(r n )->A(r) in ^r K (L 2 (T,R N ))), we can find ft,eL 2 (r,R"),ft,We,4(r B )x(Oa.e. s.t. j8 B ^-y in L 2 (T,R JV ). Then we have the last inequality, being a consequence of the monotonicity of the operator A(r H )(-).. Thus we get: II x,(t) - x(t) || 2 < 2 f ' (/?(*) - y (s), x(s) - x n (s))d5 Jo + 2 f ' ((s, (X B ) S , r.) - /(s), x(s) - x M (s))ds Jo f || (s, (x n ) s , r J - Jo + 2 || (s, (x n ) s , r J - /(s) || || x(s) - x n (s) || ds. Note that || u(s, (x n ) s ', r.) -f(s) |1 S || u(s, (x n ) s , r,) - u(s, x s) rj || + || u(s, x s , r n ) - /(s) || < h(F(s, (x n ) s , r.X F(s, x., rj) + fc(/(s, x,, r n ), F(s, x., r)). 68 Nikolaos S Papageorgiou Hence we have: fr 2 (&(*)- y(s),x(s)-x n (s))ds Jo + 2 f ' h(F(s 9 (xj s , rj, F(s, x s , r n ))- 1| x(s) - x n (s) \\ ds Jo + 2 r/z(F(5,x s ,rJ,F(s 5 x s ,r))-||x(s)-x n (5)||d5. Jo Recalling that p n ^y in L 2 (T,R N ) and x n Ajc in C(f,R N ), and using hypotheses /f(F) 2 (2) and (3), in the limit as n-+ oo, we get r ||*W-x(t)|| 2 <2 Ks) II *,-*. II co J *()-*(*) lids ' i Jo Set 0(s) = ||x s -x s ||^. Then we have Invoking Gronwall's inequality, we deduce that 0(t) = for all re T=> x = xe C(t (R N ). But note that x w ^?(rjn ^ 1 and x n -4x in C(T, R N ). Therefore . (2) From (1) and (2) above we get that P(r n )2>P(r)asn-KX>. But P(r n ) c jj(K) n ^ 1 and the latter is compact in C(f, R*). So lemma 4.2 tells us that P(r n )-P(r). Therefore P(-) is continuous for both the Vietoris and Hausdorff metric topologies as claimed by the theorem. Q.E.D. 5. Relaxation In 3 we established the existence of solutions for both the "convex" and "nonconvex" problems. In this section, we show that under some additional regularity hypotheses on the orientor field F(t,y\ we can in fact show that the solutions of the nonconvex problem are dense in the C(t IR*)-topology to those of the "convex" problem. Such a result is usually called in the literature "relaxation theorem". In optimal control, the "relaxed" (i.e. convexified) problem, plays an important role, because on the one hand captures the asymptotic behavior of the minimizing sequences of the original problem and on the other hand, thanks to its convex structure, always has a solution under very general hypotheses on the data (see Avgerinos-Papageorgiou [3], [4], Warga [21] and references therein). Maximal monotone inclusions 69 So we consider problem (*) and its "convexified" version: r,x.)a.e. on T} , , ' |. (*)c Denote the solution set of (*) by P(<p) and that of (*) c by P c ((p). We will need the following hypothesis on the orientor field F(t 9 y). H(F) 3 : F: TxC(T Q , R N )-+P f (R N ) is a multifunction s.t. (1) t -> F(t, j>) is measurable, (2) h(F(t 9 y) 9 F(t 9 z))^ri(t)\\y z\\ 00 a..e. with ?/() L+, (3) \F(t,y)\^a(t) + b(t) \\y\\ad~S' with a(-), i?(*)e L+. Theorem 5.1. // hypotheses H(A\ H(F) 3 and H(<p) hold, then P(cp) = P c (<p) in C(t K N ). Proof. From theorem 3.2 we know that P((p) ^ and so P c ((p) ^ 0. Furthermore, from theorem 3.1 we have that P c ((p) is compact in C(T, IR N ). Let x(-)P c (<p). Then by definition, we have .e. on T with /6L 1 (r,R N ),/(r)eF(t,x f )a.e. Recall that the map p:L l (T 9 R N )-+C(T 9 R N ) 9 which to each geL l (T 9 R N ) assigns the unique solution of the initial value problem x(t)e Ax(t) + gr(t)a.e., x(0) = <p(0), is sequentially continuous from L^T, R N ) with the weak topology into C(T 9 (R N ) with the strong topology. As before p\ L l (T, R N )-> C(f, R N ) is defined by (/)W = P(/)W re T and 0(/)(i?) = <p(v) ve T . Let F^ L J (r, 1R N ) be as in the proof of theorem 3.1. Then V equipped with the relative weak-L x (T, U N ) topology, is compact, metrizable. So p: F->C(f, R N ) is "weak-to-strong" continuous. Thus given e > 0, we can find a symmetric, weak neighborhood of the origin in C(T, IR N )s.t. if / ^el/n F, then II x - p(f l ) || = || x - x 1 1| , < a (here we have set x 1 = p(f v )). From theorem 4.2 of [17], we know that we can choose /ieSj- ( . >x) . Next, via a straightforward application of Aumann's selection theorem (see Wagner [20], theorem 5.10), we can find / 2 eS]r ( . Xi) s.t. r -(xJJ^ Suppose /!,...,/ eL^T,^) have been chosen ) r )a.c.fc = > l,...,n-l(x x) (3) and ||/ t (r)-/ + (OII ^ ' Again through Aumann's selection theorem, we can choose /,, + iGS ( . t(JCn)0 s.t. 70 Nikolaos S Papageorgiou But from corollary, 1.7.1, p. 35 of Vrabie [19] (Benilan's inequality), we have that rt rt g ( s \ r r s "I" -2 j^ * S *1-1 Thus by induction, we get a sequence {/ t } fc>1 L^r.R*) satisfying (3) and (4) above. Clearly then {/}>! L^T.R") is Cauchy. So / n A/ in L l (T,R N ). Then x n = p(/ n )^>p(/) = x in C(f,FP), and from theorem 4.5 of [16] we have that f(t)eF(t, i r )a.e. Thus x = p(f)eP(<p). Also exploiting the monotonicity of A(\ we have -x Jk (t))<(/ fc + 1 (t)-/ fc (f),x k+1 (0 Invoking lemma A.5, p. 157 of Brezis [6], we get ;rr( I l& ds Jo Hence the triangle inequality gives us Since xeP(<p) and > is arbitrary, we will conclude that P(q>) = P c ((p\ the closure taken in C(T 5 IR N ). Q.E.D. Acknowledgement The author would like to thank the referee for his helpful comments. This work is supported by a NSF grant. Maximal monotone inclusions 71 References [1] Attouch H, Variational Convergence for Functionals and Operators (London: Pitman) (1984) [2] Aubin J -P and Cellina A, Differential Inclusions (Berlin: Springer) (1983) [3] Avgerinos E and Papageorgiou N S, On the sensitivity and relaxability of optimal control problems governed by nonlinear evolution equations with state constraints, Monatsh. Math. 109 (1990) pp. 1-23 [4] Avgerinos E and Papageorgiou N S, Optimal control and relaxation for a class of nonlinear distributed parameter systems, Osaka J. Math. 21 (1990) pp. 745-767 [5] Bressan A, On differential relations with lower semicontinuous right hand side, J. Differ. Equ. 37 (1980) pp. 89-97 [6] Brezis H, Operateurs Maximaux Monotones (Amsterdam: North Holland) (1973) [7] Cellina A and Marchi M, Nonconvex perturbations of maximal monotone differential inclusions, Israel J. Math. 46 (1983) pp. 1-11 [8] Dunford N and Schwartz J, Linear Operators I (New York: Wiley) (1958) [9] Fryszkowski A, Continuous selections for a class of nonconvex multivalued maps, Stud. Math. 76 (1983) pp. 163-174 [10] Henry C, Differential equations with discontinuous right hand side for planning procedures, J. Econ. Theory 4 (1972) pp. 545-551 [11] Kandilakis D and Papageorgiou N S, Nonsmooth analysis and approximation, J. Appro*. Theory 52 (1988) pp. 58-81 [12] Klein E and Thompson A, Theory of Correspondences (New York: Wiley) (1984) [13] Kuratowski K, Topology I (New York: Academic Press) (1966) [14] Moreau J -J, Evolution problem associated with a moving convex set in a Hilbert space, J. Differ. Equ. 26 (1977) pp. 347-374 [15] Mosco U, Convergence of convex sets and solutions of variational inequalities, Adv. Math. 3 (1969) pp. 510-585 [16] Papageorgiou N S, Convergence theorems for Banach space valued integrable multifunctions, Int. J. Math. Math. Sci. 10 (1987) pp. 433-442 [17] Papageorgiou N S, Measurable multifunctions and their applications to convex integral functionals, Int. J. Math. Math. Sci. 12 (1989) pp. 175-192 [18] Stacchetti E, Analysis of a dynamic, decentralized exchange economy, J. Math. Econ. 14 (1985) pp. 241-259 [19] Vrabie I, Compactness Methods in Nonlinear Evolutions (Essex: Longman) (1987) [20] Wagner D, Survey of measurable selection theorems, SI AM J. Control Optim. 15 (1977) pp. 859-903 [21] Warga J, Optimal Control of Differential and Functional Equations (New York: Academic Press) (1970) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 1, April 1992, pp. 73-81. Printed in India. Remark on GronwalPs inequality J POPENDA Institute of Mathematics, Technical University, Potrowo 3a, 60-965 Poznan, Poland MS received 20 November 1990; revised 26 June 1991 Abstract. Gronwall's inequality has many extensions and analogues among them the discrete one. In this paper we present theorems which look like Gronwall's lemma in the classical prepositional calculus. Keywords. Gronwall's inequality; classical prepositional calculus. 1. Introduction One of the most famous inequalities in the theory of differential equations is Gronwall's inequality. Extended and generalized in many directions (see e.g. [1-3,5]) this inequality has also discrete version embodied in the following theorem. Theorem, (discrete analogue of Gronwall's lemma). Let be any real sequences with a - non-negative, c - any real constant. If n h. x n+1 ^c + fyXj j=i holds for every n = 1, 2, . . . , then n c. x rt + 1 ^(c + a 1 x 1 )[]( 1 - f a j) J = 2 for n= 1,2,... For understanding the meaning of "look like" Gronwall's lemma, let us see that in the hypothesis h. terms of the unknown sequence x appear on both sides of the inequality. In the thesis (consequence) c. the terms of x are estimated, bounded by the terms of a,c, and (generally not necessary) the first element x 1 . The theorems presented below look similar. The main result of this note is to show that the theorems have their analogues in many branches of mathematics. We construct our theorems in the classical prepositional calculus 32- However, some of them can be considered in other languages or in metalanguage. Furthermore it seems that the results can be used as a manner of proving theorems as well, direct and indirect. 74 J Popenda 2. Preliminaries We denote by P 9 q 9 s,t,a l9 a 29 ... 9 b 1 ,b 2 ,... 9 c 1 ,c 29 ... 9 x l9 x 29 ... 9 the infinite set of statement variables and by ~, A, V, ID, = connectives in 32- Furthermore we shall use the symbols A" =1 aj and V" =1 flj to denote generalized conjunction and disjunction i.e. (a n l\(a n __ l A (.. A (a 2 A a,)..))), (a n V(a n -, V(.. V (a 2 V a, )..))) respec- tively. We suppose A J^o,-, Vj-^o,- means a k . Since we use the axioms in the proofs we recall them a.l (p ID q) ID \_(q ID s) ID (p ID s)] a.2 [p ID (p ID g)] ID (p ID q) a.3 p ID (q ID p) a.4 p A 4 ID p a.5 p A <? ID q a.6 (p ID g) =) [(p ID s) =5 (p ID q A s)] a.7 p ID p V q a.8 4 z> p V q a.9 (p ID 5) ID [(g ID 5) :D (p V q =D s)] a.10 (p = g) z> (p z> q) a.ll (p = 4)iD(4iDp) a.12 (p ID q) ID [(4 ID p) ID (p = q)~] a. 1 3 ( ~ q ID ~ p) ID (p ID q) The rules of inference are substitution, modus ponens, and the derivable rule hypothetical syllogism use of which is denoted by /, C, r syl respectively (see e.g. [4 pp. 170]). For making the proofs shorter we apply different statement forms which are known or can be easily inferred from the axioms. In the sequel, whenever we use some known formula of the classical prepositional calculus, we note this f.x. The formulae we use in the presented proofs are f.2 (p ID q) ID (s A p ID s A q) f.3 p ID p f.4 p A q V p A s ID p A (q V 5) f.5 (p A q) A 5 ID p A (q A s) f.7 (p ID q) : f.8 (p V q) V s f.9 p =5 p A p f.10 (p ID g) ID [(s ID t) ID (p A s ID q A t)] AT denotes the set of positive integers. 3. Main results We start with inequality which is easy to prove and is embodied in the following Vfc.V VbjAxj^x^V |h V^V^Ax^^^. Remark on Gronwall's inequality We prove this by applying n-times 0.8 and r syl and get 1- b^x^ \/^bjf\Xj. Now by f.l using the rule of substitution we obtain Hence using modus ponens 2., 1. we have 75 q ^bjhxj, s/a n -2. 3. fl ll V6 1 Ajc 1 =>o ll V V 1 b j /\x j . By premises (pr.), 3., and the rule of syllogism we obtain r syl 3.-pr.-c. Remark. If we introduce the connective <= defined by the truth table T T T F F T F F T T F T then p c: q is logically equivalent to q ^> p, or by writing q z> p we denote this statement by p <= q- Then the theorem we have proved above can be expressed in the Torm more familiar for specialists of differential equations and in fact similiar to the inequality considered in the introduction i.e. If n h.n. holds (has logical value truth) for every neAT then c.n. ^ +i c:a B Vfr 1 Axj is also true for every neN. In such a meaning our theorem ought to be understood. Notice that the statement considered is true if we replace both in the premise (hypothesis) and conclusion neN by neNm:= {l,2,...,m}. It is evident that by f.l and 76 J Popenda we can obtain many similar statements; e.g. or using f.2 instead of f.l Regarding axioms a.7 and a.8 we state that the conclusion of the proved statement will be better if many disjuncts appear in the conclusion's antecedent. This leads to the problem of finding the conclusion. The question that whether such a conclusion exists is left open. In the differential equations theory this problem reduces to solving instead inequality respective equation. Theorem 1. // n j - 1 J J has logical value truth for every rceJV (for some interpretation) then C.l ai V^ f\X^X 2 c.n a n V V b i+ A a, V I A b, ) A x, I => X H for n = 2,3,... is also true. Proof. r syj Op/^ V b i /\x l -h.l c.l Therefore c.l holds. We shall prove that c.l is true f.2p/aj V fcj A x l9 q/x 29 s/b 2 C h.l - 1. 1. b 2 A(a l V b 1 /\x 1 )^>b 2 A x 2 r syl f.4/>/6 2 , g/fl l9 s/fci Ax x - 1. - 2. 2. fc 2 Afli Vb 2 Afbj A x 1 )=)fc 2 Ax 2 f-lp/^AbJAXi, q/b 2 /\(b 1 /\x l \ s/fr 2 A a 1 Cf.5p/& 2 , 9/6^ S/XJL - 3. 3. fe 2 A aj V (& 2 A &J A x x ID fo 2 A a V fc 2 A (b x A xj r syl 3.~2.^4. 4. fo 2 A a x V (& 2 A bj A Xj z) ft 2 A x 2 r syl 4. - a.7 p/b 2 A x 2 , q/b l /\x i ~ 5. Remark on Gronwall's inequality 77 5. b 2 f\a l V(b 2 Ab 1 )Ax 1 =>b 2 Ax 2 Vb 1 A x t f.lp/^Aa! V(6 2 AbJAXi, <?/6 2 A x 2 V b 1 A x l5 s/a 2 C5.-6. 6. a 2 V [6 2 A a t V (b 2 A bj A xj => a 2 V [i> 2 A x 2 V ^ A xj r syl 6. - h.2 - c.2 This means that the theorem holds for n = 1. Suppose c.m is satisfied for m = 1, 2, . . . , k. We prove that c.fe + 1 holds. r syl a.7p/a l5 q/^ Ax x -c.l - 7. 7. a t r>x 2 r syl a.7 p/a m , q / f V b j+ 1 A al V [ .A ^ J A x x - c.m - 8. 8. a m =>x m+l m = 2,3,...,/c f.2p/a m , ?/x m+1 , s/fo m+1 C8.-9. m = 2,3,...,fe f.2p/a 1; g/x 2 , s/b 2 Cl.-9. 9. i) m+1 Aa m 3b m+1 Ax m+1 m=l,2,...,/c r syl f.6p/a k , 10. a k vf .A atVfA^JAXi 11. fr* ^, s/b k+1 f\a k CL5p/b k+1 , q jbj, s/Xj-12. 12. ^^Aa.V^A r sy ,1.2-f.4p/b t+1 , q/a k , s.A^-J Ax t - 11.- 13. 13. b k+1 /\a k v(''K i bjJ/\x 1 =>b k+ ^x k+l r syl 9.(m = 1) - a.7p/b 2 A x 2 , q/b^ A x x - 14. 14. fc 2 Aa 1 =>b 2 Ax 2 Vb 1 Ax! f.7p/6 2 Aa lt q/b 2 *x 2 Vb 1 *x 1 , s/b 3 /\a 2 , t/b 3 Ax 3 C 14. -C9.(m = 2)-15. 78 J Popenda 15. b 3 A fl 2 -V 6 2 A fl x 3 fo 3 A x 3 V V fe; A x, Lip I .V & 7+ i A a,, 4 / .Vfe; A Xj , 5/fc 4 A 3 , t/b 4 A x 4 C 15. - -C9.(m = 3)-16. , 16. V b j+l Aa,=> VfcyAx., j=i ./-n j 7=1 J ^ Repeating the above reasoning we get 17. V^Aa^.y^.Ax,, Hence /*-! / k /fc+1 f.7p/ Vb ;+1 A flj , q VjjAxj, s/fr^iAa fc-i fc+i fc* + 1 Afl* VA fc, Ax x V .V 6 J+1 A flj .=> .V q V^.Aaj, s I Ajj Ax, - 18.- 19. Jk+1 Xi, q 20. a k+ 1 V ^ .V b j+l A a ; J V ^A bj J A x t J ^ a k+ x V .V 6, A x, r syl 20.-/i./c+l-c./c+l. We obtain c.fe + 1 holds so the proof is complete by induction. Remark. Note that by a.5, a.8, f.l, and f.2 Kfc .V Therefore the statement we have proved at the beginning of this chapter follows from Theorem 1. In the next theorem we consider a statement wherein the premise instead generalized wedge stands generalized inverted wedge. We start with similar remarks as before. Theorem 1. It is evident by a.7 that Remark on Gronwall's inequality ' 79 and e.g. We suppose that the set of premises is semantically consistent. By a.4 we see that if P ^ x n + 1 then b y the r ^e of syllogism p A 4 ID x n+ 1 . Therefore the conclusion would be better if less conjunction appears in the antecedent of consequence. Theorem 2. // / n h.n a n V J A i (b ; Vx ; )=>x,, +1 has logical value truth for every neN, then c.n a n V(bi ^x 1 )^x n + 1 n = l,2,.. is a/so true. Proof. r syl f.3p/a 1 V(b 1 Vx 1 )-h.l-c.l. The theorem holds for n = 1. Suppose c.n is true for n = 1, 2, . . . , k then the proof of this yields c.k + 1. x -h.l -a.8p/b 2 , q/x 2 -l. jVx!, q/b 2 Vx 2 , s/b.Vx,, t/b^ x 1 e Cl.-Cl3p/b l \/ x\-2. 2. (b, V x t ) A (ft t V xj 3 (b 2 V x 2 ) A (fci V x t ) 3. iiVXi^&aVxJAfoVXi) tlp/^VXi, q/(b 2 Vx 2 )h(b 1 Vx l \ s/a 2 C3.-4. 4. c 2 V(b 1 Vx 1 )=5fl 2 V(6 2 Vx 2 )A(fr 1 Vx 1 ) r sy ,4.-h.2-c.2 r syl a.8p/a 2 , /&! Vx t -4.-h.2-a.8p/Z> 3 , <?/x 3 -5. 5. b 1 Vx 1 ^b 3 Vx 3 C 5. - C 3. - 6. 6. (ft t V x t ) A (ft t V Xl ) = (6 3 V x 3 ) A [(& 2 V x 2 ) A (&! V 80 J Popenda 3 Following this way we obtain and similarly as we obtain 3. and 7. applying f.9, f.10 we get Q A>\/v \ A fi \/ v 8- btVx^frbjVxj. Now Ik+i Hence r syl 9. - h.k + 1 - c.k + 1 proves that the theorem holds for k + 1. By the induction argument theorem is proved. Remark. By a.7 p/a n9 q/b 1 V x l -a n ID a n V (b l V xj. Hence by Theorem 2 we get a n =>x n+l what we have noticed before this theorem. Applying f.10 we have 10p/b 2 , q/b 2 yx 2 , s/b l9 t/b^x^Ca.lp/b^ q/x 2 Ca.lp/b l9 ql*i~ 1. 1. b 2 A &! ID (fc 2 V x 2 ) A (fti V xj. Continuing this reasoning we obtain 2. b j=> hbjVxj. From there Lip I ^b j9 q l^bjVxj, s/a n C2.-3. 3. ^V.Ab^a.V.A^Vx! r syl 3. h.n 4. 4. <*nV j^ The conclusion we have just proved is different from the one given in Theorem 2. In a similar way we can proye Theorem 3. // n h.n abVxiDX Remark on Gronwall's inequality 81 has logical value truth for every neN then c - n a- A (b, Vx for n = 2, 3, ... r -i i (fejVxJA A (b, + l Va,) 3x nH L J- 1 J f.s 1 a /so Some of the theorems we can construct in such a manner have their premises which look like hypothesis in the Gronwall inequality while the conclusion has more different forms. We present as an example one such statement Theorem 4. V A b:V ~x,=>x n+l Ih Vx^ V - A ft,. neNj^i J J n+l \\ neN n+ 1 J - =1 j We omit the proof of this theorem. References [ 1 j Ahramowich J, On Gronwall and Wendroff type inequalities, Proc. Am. Math. Soc. 87 (1983) 481-486 [2] Agarwal R P and Thandapani E, On discrete generalizations of GronwalFs inequality, Bull. Inst. Math. Acad. Sinica 9 (1981) 235-248 [3 j Bainov D D, Myshkis D and Zahariev A I, On an abstract analog of the Bellman-Gronwall inequality, Puhl. Res. Inst. Math. Sci. Kyoto Univ. 20 (1984) 903-911 [4] Georgacarakos G N and Smith R, Elementary formal logic (McGraw-Hill) (1979) [5J Popenda J, Finite difference inequalities, Comm. Math. 26 (1986) 89-96 Proc. Indian Acad. Sci. (Math. ScL), Vol. 102, No. 2, August 1992, pp. 83-91. Printed in India. Proof of some conjectures on the mean-value of Titchmarsh series - III .R BALASUBRAMANIAN and K RAMACHANDRA* The Institute of Mathematical Sciences, Madras 600 113, India *School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India MS received 17 June 1992 Abstract. With some applications in view, the following problem is solved in some special case which is not too special. Let F(s) = I^ =1 a n ^ n " s be a generalized Dirichlet series with l = A 1 <A 2 <...,A n ^)n, and A n + 1 - ^D~ l ^ l where a>0 and 0(^1) are constants. Then subject to analytic continuation and some growth conditions, a lower bound is obtained for (l/H)Jo \F(it)\ 2 dt. These results will be applied in other papers to appear later. Keywords. Titchmarsh series; mean value; lower bounds. 1. Introduction In the previous papers [1] and [2] with the same title (as the present one) we proved some conjectures made by the second author [4]. In this paper we formulate a new conjecture (which we believe to be true at least in some modified form) and indicate a slight progress towards it. Conjecture. Let 1 = ^ < u 2 < ...be any sequence of real numbers with 1/C fj. n ^ C where C(> 1) is an integer constant and n = 1, 2, 3,. . . . Let us form the sequence l"-^! <A 2 <...0/ all possible (distinct) finite power products of l=/ii,^ 2 ,.-. w ' th non-negative integral exponents. Lets = a + it, H(> 10) be a real parameter, and {a n } (n = 1,2,3,...) with a i = l be any sequence of complex numbers (possibly depending on H)such 'that F(s) = S = l a n ^ s is absolutely convergent ats^B where B^lis an integer constant. Suppose that F(s) can be continued analytically in (OO,0<*^H) and that there exist T l9 T 2 withO< T^H 3 /* H-H.^< T 2 <H such that for some KO30) there holds max(|F(cr + iT^I + \F(a + iT 2 )|) ^ K. FinaHy tet E 00 , t |a |^" B < H A where A(& 1) w an integer constant. Then there exists a constant 5 >Q (depending only on A, B and C) such that for all H > H Q (A 9 B, C) there holds provided that ff' 1 loglog K does not exceed a small positive constant. * to whom correspondence should be addressed (2) 84 R Balasubramanian and K Ramachandra Remark L We can strengthen the Conjecture (1) by replacing i by a more specific function of H which is asymptotic to 1 as H-+ oo. Remark 2. By the method of [1] we can prove that 1 f H 1 -- \F(it)\dt^-. H J o * The Remark 1 is also applicable. Remark 3. Under the condition kl < D (logX) R (R = H\ D any constant, X ^ 30), (3) *n**X we can prove (2). Remark 1 is also applicable. For both these results the conditions involving K are unnecessary. For the results mentioned in Remarks 2 and 3 we refer the reader to [1] and [5]. Remark 4. Actually the proof of (1) in [1] goes through without serious problems until we come to a lower bound for 1 2 dt. To apply Montgomery- Vaughan theorem we need good lower bounds for A M+ 1 - X n . These are not available in general. But we can work with \i n = (n -f n l)/n where n (^2) is any integer constant (of course using Montgomery- Vaughan Theorem). Thus in this special case we can prove Conjecture (1). We can also handle /i n = (1 -f P)~ l (n + /?) where /?(> 0) is any real algebraic constant. Remark 5. We can formulate Conjecture (1) with no conditions involving K, but instead we have to assume condition (3). Remark 1 is also applicable. Before closing this section we like to make two important remarks. First X n ^ f.i n ^ Cn which is obvious because {A n } contains the subsequence {JK B }. Secondly for x ^ 1 and r\ ^ 2C + 1, we have, -1 00 ) du =x"l-C oV c) J V Jo (i + ")< c Hence in (1) the condition A n =$ /f^ is equivalent to a condition of the type with a different constant <5 > 0. Mean-value of Titchmarsh series 85 2. Main lemma Let rbea positive integer, H^(r + 5)U where U > 2 70 (16B) 2 and M and N are positive integers subject to N > M ^ 1, and B(^3) an integer constant. Let {b m } (1 ^ m ^ M) and {c n } (n^ N) be two sequences of complex numbers, 1 = A 1 <A 2 <... be any increasing sequence of real numbers and let A(s) = S m ^ M b m A~ s . Let B(s) = S n>N c n X~ s he absolutely convergent for s = B and continuable analytically in ((7^0,0^ t ^H). Write g(s) = A(-s)B(s), G(s)=U~ r o where (here and elsewhere) A = w t -f ... + u r . Assume that there exist real numbers T l and T 2 with ^ 7\ ^ (7, H - V ^ T 2 ^ H, such that ( U |ff(a + ITJI + \g(<r + iT 2 )\ ^ expexp I uniformly in < a ^ J5. Let and X Then, we have, <*H r \g(it)\dt G(it)dt ( U + 64B 2 )S 1 + 16J3 2 S 2 exp - \ O/5 \ Remark. This lemma is borrowed from [1] (see pages 2 to 8). 3. Progress towards the conjecture From now on we assume that 1 = a l5 a 2 ,a 3r ..is any sequence of complex numbers. We set b m = a m and c n = a n and assume that I^i kl^ * is convergent Lemma 1. We have, with |a m | 2 , where A(A M ) = max |A M A v /4*V 86 R Balasubramanian and K Ramachandra Proof. Follows from Montgomery-Vaughan Theorem (see [3]). Lemma 2. We have, / n=l and Proof. The first inequality is trivial and the second follows from log^=_log(l-(l-A We now make the following. Hypothesis. {/!} is any increasing sequence of real numbers satisfying 1 1 = 1, l n ^ Dn, A rt+1 X n ^ A~+i D" 1 , where D(^ 1) is an integer constant and a a positive constant. Also we assume that n= 1 where 0<e^l/[2(a + 1)] and r ^ [(200J? + 200)e~ 1 ] is any integer. Also F(s) = E^ =1 a n A~ s shall be as in the introduction except that the {A n } are not related to the {Hn}- {^n} will now be an independent sequence. From now on we set N = M + 1, M = [H (1/(a+ ^^^l, 17 = H 1 ~ (fi/2) + 50B loglog K t where K 1 =H r K. Note that if H ^(r 4- 5)17 is not satisfied, our main theorem (to follow) asserts that a positive quantity is non-negative. Also note that min max |F (<r + ft) [ 0<r=$H 3/4 <T>0 ^ min max |F (0-4- it) I 0^<I7 r>0 and a similar result holds for the intervals (H - H 3/4 , H) and (H - 17, H). Lemma 3. and Proof. We have A M < DM ^ DH and this proves the first inequality. Also Mean-value of Titchmarsh series 87 and this proves the second inequality. The third follows from The lemma is completely proved. Now we apply the main lemma (we closely follow the proof of the first main theorem in [1]). Let A(S)= Mm* m;$M and Then, we have, in a > B, F(s) = A(s) + B(s) and so where g(s) = A( s)jB(s). Hence |F(it)| 2 df r dti r ... O JO 2U + A = J 1 + 2J 2 say. By Lemmas 1 and 3, we have, Again, we have, for < a ^ B, \g(s)\ = \A(- s)B(s)| = M(- s)(F(s) < the last two inequalities being true for instance if H > 10D. Observe that / U \ / 5 , 1 r Wr ' expexp ( ig P expexp^loglogKi J J^ and hence the condition on g and 17 required by the main lemma is satisfied. Hence by the main lemma, we have, Balasubramanian and K Ramachandra \J 2 1 < U JJ provided H ^ (r + 5)U and U > 2 10 (16B) 2 . As remarked already we can ignore the condition H ^ (r + 5) [7. Also we will satisfy H > (50rBl) B4 " a + 2 ) 8/ and we will show later that this implies U ^ 2 70 (16) 2 . We can assume that ft\F(it)\ 2 dt ^ #2 nJSM \a n \ 2 (otherwise the result asserted by the main theorem to follow, is trivially true). Hence \g(it)\dt= o o \A(-it)\ 2 dt+ \B(it)\ 2 dt o J o < 3 J \A(- it)\ 2 dt + 2 f \F(it)\ 2 dt Jo Jo (on noting that B(it) = F(it) - A(it)) by Montgomery- Vaughan Theorem and the third part of Lemma 3. Hence _ , 4B 2 108B + (2H + 32J5 2 S 2 exp -- 108J3 f Thus where 5 > and a n \\ Here we have used exp ( + (2H + 1285 2 )2 r (I J ff) 2B (2D) r(a+2) H~ r/4 Note that U Mean-value of Titchmarsh series 89 loglogKi ^loglog(H r X)<loglogK -f log(rlogH) ^ loglog K + log r 4- log H and that i(logH) 2 ^ H and so log H < 2H 1 ' 2 and so Hence (r + 5)17 < lOOBrfloglogK + logr + logH) + (r + 5)^ ~* 12 . Thus S 3 < lOOflr loglog K + r(log r)D^H l ~ W*> (r + 5) 10 4B 2 108jB(15> a+2 )# I /l \ r^.fvJ.') e/A I ft \ wrTLalA I* < ^ r(logr ; Denote the expression in the last curly bracket by S 4 . Then we have c 200J? 2r 10 4B 2 1620B $ 4 ^ . | 1 1 f. yyg/A .}- jLp Let H E/s ^4D a+2 . We have H rBl *>H ((2mB+m& ~ l ~ l} ^^H 2B + 2 . Let H l/ *^8Br. We have H r/4 > H 2J5+ x . Now both H /8 > 4D a+2 and H 1/4 ^ 8Br are satisfied if Hence under this only condition, we have, S 4 ^ (200B + 2r + 10 -f 4B 2 4- 1620B + 130B 2 D B + 32 2 D 2B )#- /4 < rB 2 D 2B H~ /4 (200 -f 2 -f 10 + 4 + 1620 H- 130 + 32) provided H ^ (2000r 2 D 2B ) 4/fi . Now this last condition and H ^ (32BrD* +2 ) 8/e are both satisfied if H ^ (50rJ5Z) B+a+2 ) 8/ . Finally 17 > H 1 ~* 12 ^ H 3/4 ^ (50 x 200) (8/fi)(3/4) = i (10,OOOJ5-B) 6/ ^2 13(6/) (16J5) 2 (B 24 /(16B) 2 )^2 70 (16B) 2 since J5^3. Collecting, we have proved the following \ Main Theorem. Let {A n }(rc = 1,2,^3,...) with Aj_ = 1 be any increasing sequence of real I numbers with the properties l n ^ Dn and A n+1 A, n >D~~ 1 A ~/i where a(>Q)isa constant and D(^ 1) is an integer constant. Let {a n }(n = 1,2,3,...) wit/i a x = 1 be any sequence 90 R Balasubramanian and K Ramachandra of complex numbers such that F(s) = ZL t a B A~ s is absolutely convergent ats~ B, where B(^ 3) is an integer constant. LetO<e< (2(1 + a))" 1 and let r(^ [(200J3 + 200)8^]) be any integer constant. Let Assume that F(s) possesses an analytic continuation in (a > 0,0 ^ t ^ H) and that there exist T 15 T 2 with < T\ < H 3/4 , H - H 314 ^ T 2 < H such that for some K(^ 30) there holds uniformly in ^ a ^ B. Let Then, there holds, 1 r\F(it) " JO |a n | 2 , where M=H "' 0= rb- e ' and In view of the two closing remarks at the end of 1 we can be now deduce some corollaries. COROLLARY 1. Let u n = n. Then the conjecture is true. Proof. We can take C == 1, a = & and D=L COROLLARY 2. Letn (^2) be an integer constant and jj, n = (n + n l)/n . Then the conjecture is true. Proof. First, since {u n } is a subsequence of {A ff } is follows that A rt ^ /* ^ n. To apply the main theorem we have to verify that A n+1 -^^D'U".^ holds with some constant a > and DO 1) an integer constant. To prove this we observe that we can assume that A n+ 1 l n ^ 1. In this case where; = max(fc, /). Now (1 + (l/n )) k < A n + 1 and (1 + (l/n ))' < l n and so j = max(k, 1) < (log A n+ i) (log (( + IVno)- 1 . But log (n + 1)/ = - log(l - (1/ + 1)) > Wo + 1)- Mean-value of Titchmarsh series 91 Thus jf <; (n 4- l)(log k n+1 ) and so HO j ^ >C+i where a = (n + I)logn . Plainly we can take D = 1. COROLLARY 3. Let /? > be an algebraic constant and ^ n = (n + /?)/(! + j?). T/zen the conjecture is true. (The conjecture is also true for the choice //i = 1, // = n -f /? 1 /or n > 1). Pr00/. As before >l n < // ^ (n/? + 2)(/? 4- 1)" *. Also considering the norm of A n+ t A n (in case it is ^ 0) we can prove that A rt+1 A n ^ Dk~$ l . The latter assertion follows similarly. Post-script. The results of this paper were necessitated by a lot of applications to the zeros of generalized Dirichlet series. All these applications will form the subject matter of our forthcoming paper "On the zeros of a class of generalized Dirichlet series-XI". References [1] Balasubramanian R and Ramachandra K, Proof of some conjectures on the mean-value of Titchmarsh series-I, Hardy-Ramanujan J. 13 (1990) 1-20 [2] Balasubramanian R and Ramachandra K, Proof of some conjectures on the mean-value of Titchmarsh series-II, Hardy-Ramanujan J. 14 (1991) 1-20 [3] Montgomery H L and Vaughan R C, Hilbert's inequality, J, London Math. Soc. (2) 8 (1974) 73-82 [4] Ramachandra K, Progress towards a conjecture on the mean-value of Titchmarsh series-I, Recent Progress in Analytic Number Theory (eds) H Halberstam and C Hooley (Academic Press) (1981) 303-318 [5] Ramachandra K, Proof of some conjectures on the mean-value of Titchmarsh series with applications to Titchmarsh's phenomenon, Hardy-Ramanujan J. 13 (1990) 21-27 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 2, August 1992, pp. 93-123. Printed in India. Combinatorial meaning of the coefficients of a Hilbert polynomial M R MODAK Department of Mathematics, S. P. College, Pune 411030, India MS received 8 July 1991; revised 10 May 1992 Abstract. In [1] Abhyankar defines an ideal I(p,a) generated by certain minors of a matrix X, the entries of X being independent indeterminates, and proves that the Hilbert function of I(p,a) coincides with its Hilbert polynomial F(V) and obtains it in the form He also proves that F( V] is the number of certain "indexed" monomials of degree V in the entries of X and that the coefficients F D (m, p, a) are non-negative integers and asks for their combinatorial meaning. In this paper we characterize the indexed monomials in terms of certain sets of lattice paths, called frames, and prove that the coefficients F D (m,p 9 a) count certain families of such frames. Keywords. Determinantal ideals; Hilbert polynomial; lattice paths; binomial determinants. 1. Introduction Let X be an m(l) by m(2) matrix whose entries Xy's are ^determinates over some field X, and let X[X] be the ring of polynomials in these m(l)m(2) variables over K. Let I(p + 1) denote the ideal, in K[X], generated by all (p + 1) by (p + 1) minors of X. A part of the second fundamental theorem for vector invariants says that I(p + 1) is a prime ideal in K[X]. In [1], Abhyankar proves two generalizations of this result, the first of which is as follows. Let a be a bivector of length p and bounded by m = (m(l), m(2)), i.e. let a be a bisequence (a(fc, i)} of positive integers such that a(fc, 1) < a(/c, 2) < . . . < a(fc, p) ^ m(/e), for k = 1,2. Let /(p, a) denote the ideal, in K[X], generated by all (p + 1) by (p + 1) minors of X, all i by z minors of X whose row numbers are less than a(l, i), 1 ^ U P and all ; by ; minors of X whose column numbers are less than a(2J), 1 ^j<p. Clearly, in the case when a(l, i) = i = a(2, i) for all U < i < P, we have J(p, a) = /(p 4-1). Abhyankar proves that J(p,a) is a homogeneous prime ideal in K[X]. He also considers, for each non-negative integer V, a certain finite set, denoted by mon(m, p, a F), of "indexed" monomials in the variables X y 's determined by a and K He proves that the cardinality, F(V) 9 of this set is a polynomial in V and obtains it in the form C-D 93 94 MRModak where the positive integer C is the "degree" of F(V) in V and the coefficients F D are non-negative integers given by where the sum is over all integers E and H E is a sum of determinants whose entries are products of binomial coefficients^ He also proves that the Hilbert function and also the Hilbert polynomial of I(p>a) is this polynomial F(V). That is, if K[X"] V is the F-th homogeneous component of the homogeneous ring K[X~\ and /(p,fl) w = J(p, a)r\K[X] V9 then the dimension of A v = K[X~\JI(p 9 a) v - as a K-vector space - is F(V\ for every non-negative integer V. Thus he shows that the set mon(m,p,a, V\ of indexed monomials, forms a K-basis of A v . He also gives a determinantal basis for A v using certain standard Young bitableaux. See also [4]. In [1], page 309, Remark (9.14), Abhyankar asks whether the non-negative integer F D is the cardinality of some "natural" family of finite sets parametrized by m,p,a,D. In this paper we determine one such family. Thus in 2 below we define, for each bivector a of length p and bounded by m, certain p-tuples of lattice paths - called frames - and prove the relation between indexed monomials and frames in Theorem (2.5), Using this relation we deduce an alternative expression for the cardinality of mon(m,p,a, V) and obtain, in (48), a new expression for the coefficients F D , by comparison with Abhyankar's formula F(V). This shows, in particular, that F is the total number of frames corresponding to the bivector a. This latter result is essentially the same as Theorem 1 of Gessel and Viennot [3]. We also note that lattice paths, different from ours, have been associated to the indexed monomials by Abhyankar and Kulkarni [2]. In 3 we consider two of the many equivalent expressions for F D given in [1], and these involve certain integer-valued functions ff^ 12) (m,p,a) and H^ 22) (m,p,a) defined in (42a,c) below. We prove that these also have combinatorial interpretation. In proving these results, we follow the method used in [1]: to find the cardinality, |S|, of a finite set S, first discover a recursive relation, together with initial conditions, satisfied by | S \. Then prove that this recursive relation, with initial conditions, uniquely determines a function. Finally, discover a formula which satisfies this recursive relation and initial conditions. Then |S| is given by that formula. Thus Theorem (3.3.2) says that H ( E 2} (m, p, a) is the number of frames having E "antinodes" each and then Theorem (3.3.3) says that F D is the number of frames in each of which D of the antinodes have been "marked". From this combinatorial meaning of F D we easily deduce the following result conjectured by Abhyankar [1] and proved by Udpikar [5]: if for some positive integer E, F E = 0, then F D = for all D ^ E. Also, we characterize the integer C such that (see (45) (iii) below) Finally, in Theorem (3.3.4) we obtain H^ 12) (m,p,a) in terms of H^ 22) (m,p,a) and show that H ( E 2} (m,p',a) is the number of frames in each of which S E of the "intermediate" points have been "labelled", where S is the integer C[2] in (33) below. A Hilbert polynomial 95 2. Paths, frames and mon We denote by Z, N, N* and Q the set of all integers, non-negative integers, positive integers and rational numbers respectively. For any set S we put \S\ = the cardinal number of S. For any A,BeZ we put [4,B] = {DeZ\A^D^B}. Given peN and Me{Z, N, N*}, let M(p) denote the set of all maps from [l,p] to M. Given any peAf, by a bivector a of length p we mean a bisequence a(k 9 l)<a(k,2)<...<a(k 9 p) . (1) of positive integers for k = 1,2, and we write len(a) = p. Given meN*(2) we say that a bivector a of length p is bounded by m and we write a ^ m, if a(fc,p) ^ m(k), V/ee[l,2]. Given peN" and we AT* (2), we put vec[p] = the set of all bi vectors of length p and vec(m,p) = Given me N(2\ we denote by cub(m) the positive integral rectangle bounded by m i.e., cub(m) = (yeZ(2)| 1 < y(k) < m(k), fe = 1,2}. 2.1 Paths Given weJV*(2) and neN, a directed lattice path (briefly, a path) in cub(m) is a sequence of points in cub(m) such that for i e[0, n - 1], if P(i) = (x, y\ then P(i -f 1) is either (x + 1, y) or (x,y 1). That is, P(i)P(i + 1) is either a 'vertical' or a 'horizontal' unit segment. See figure 1. Also then P(i) are called the vertices of w and w is called a path joining P(0) to P(n). Thus along w the ^-coordinates of the vertices steadily increase and the y-coordinates steadily decrease and we may describe this by saying that w is a non-increasing path in the direction from top right to bottom left in cub(m). Let P(x,y) be a vertex of a path w. Then P is called a node of w if (x,y+ l)ew and (x 4- 1, y)ew, and P is called a right-end point in w if and P is called an antinode of w if (x l,y)ew and (x,y l)ew ^ and P is called an intermediate point of w if 1 (x,y l)ew and (x l,y)w. 96 M R Modak m(2) (1,1) P(2) P(1) P(0) i , P(5) P(n) Figure 1. Thus in figure 1,'P(2) is a node, P(5) is an antinode and P(0), P(l) are intermediate points. It follows from these definitions that (2) if P(x, y) is a right-end point in a path w, then P is either an antinode or an intermediate point. Given points A(a,b) and B(c,d) in cub(m) such that a < c and b ^ d, (3) let w be the unique path from A to B such that (e,b)ew. We say that this non-empty path w joins A to B minimally or that w is the minimal join of A and B. (4) Suppose points A(a, b) and B(c, d) satisfy (3) and let w(l) be the path joining A to B minimally. Suppose points C(a + 1, b + 1) and D(c + 1, d + 1) are in cub(m) and are joined minimally by the path w(2). Then clearly (x, y)Gw(l)o(x + 1, y + l)ew(2). In particular, w(l) and w(2) are non-intersecting. (5) Given points A(a,m(2)) and B(m(l\b) in cub(m), let w be a path from .4 to B. A point (u,t;) in cub(m) is said to lie above the path w if for each vertex (x,y) of w we have either x > u or y > v. Also (w, t;) is said to lie below w if for each vertex (x, y) of w we have either x < u or y < v. (6) Let the points A, B and path w be as in (5). Then (i) w has r = (m(l) - a) vertical unit segments, 5 = (m(2) - b) horizontal unit segments and r 4- s + 1 vertices. (ii) Suppose a point (11,1?) of cub(m) lies below w. Choose vertices C(c,i?) and D(u,d) of w. Then the part w' of w, joining C to D, has at least one node (x,y) such that x < u and y < v. A Hilbert polynomial 97 Proof, (i) is obvious. For (ii), by data we have c < u and d < v. Now since (w ? i;) we see that if x is the largest integer such that (x, v)e w', then c < x < u and (x -f 1, t Hence (x. v l)ew'. Let y be the smallest integer such that (x, y)e w'. Then d ^ y < v 1, and (x, y !)< w'. Hence (x + 1, y)ew'. Thus (x, y) is a node of w' as required. D (7) Suppose paths w and w' in cub(m) respectively join A(a,m(2)) to J3(m(l),ib) and C(<s', m(2)) to D(m(l), fe') where a < a' and fc < V. Suppose every node of w' lies below w. Then every vertex of w' lies below w and w, w' are non-intersecting. Proof. Follows by noting first that if points P and P' are below w then every vertex of the minimal join of P and P' lies below w and secondly the join of successive nodes is always minimal. D (8) Let A, B and w be as in (5). Then for each ce\b m(l), m(2) a], there is a unique point (x, j) on w such that y x = c. Proof. Obvious if A = B. So let A ^ B and n = m(l) - + m(2) - b and Now P(l) = (x 1 ,); 1 ) is such that either *i = ^ ^i = w(2) 1 or Xj = a -f 1, y x = m(2). In either case, Similarly, for each fe[0,n] 5 P(0 = (x f ,j i ) satisfies y. x. = m(2) a i. D 2.2 Minimal paths Given meN*(2\ peN* and aevec(m,p), consider the points ^ - ( fl (l, i), m(2)), B t - = (m(l), a(2, OX for ie[l,p]. (9) The bivector a satisfies, by definition, the inequalities 1 < a(/c, < a(k, i + 1) < m(/c), , (10) for all ie[l,p- 1] and for all fce[l,2] and hence also ^i, (11) for all ie[l,p] and for all fce[l,2]. As a consequence of these we now prove the existence of a particular non-empty path, denoted by mw(a, 1), which joins the point A 1 to B^. The path mw(a, 1) is obtained by joining minimally the successive points of the sequence {P(U)Ue[0,2p-l]} 98 M R Modak where V/e andV;e[p,2p-l], P(l J) = Ml) - 2p + 1 +7, a(2, 2p -7-) - 2p + 1 +7). It is easy to see using (10) and (11) that V/e[0, 2p - 2], the points P(lj') and P(1J + 1) belong to cub(m) and can be joined minimally. Hence the path mw(a, 1) is well-defined. Note that if p = 1, the path mw(a, 1) joins A^ to B 1 minimally. The path ww(a, 1) will be called the first minimal path belonging to the bi vector a. See figure 2. We define the other minimal paths belonging to a inductively as follows. Having defined the path mw(a, i) for 1 ^ f < p, we define the path mw(a, i 4- 1) to be the first minimal path belonging to the bivector b[f|evec(w,p i), where bli](kj) = a(kj + i), VMl,p - i] and Vfce[l,2]. (12) Thus corresponding to aevec(m,p), we have in cub(m), the p-tuple of paths mW\a\ = (mw(a, 1), . . . , mw(a, p)\ (13) where mw(a,i) is a particular path joining A t to B,-Vz6[l,p], as defined above. For example, the path mw(a,2) is obtained by joining minimally the successive points of the sequence {P(2,7)|MO,2p-3]} (1,1: P(1,< P(1,1) B Figure 2. A Hilbert polynomial 99 where for;e[0,p 2], and for JG[P 1, 2p 3], P(2J) = (m(l) - 2p 4- 3 +7, a(2, 2p - 1 - j) - 2p + 3 +7)- 2.2.1 The above p-tuple mW[a~\ has the following two properties: (i) The paths in mW[a] are pairwise non-intersecting. (ii) Every point of cub(m) lying below the path mw(a, 1), lies on a path mw(a, i) for some ze[2,p]. Proo/. Suppose p > 1 and consider the paths mw(a,), i = 1,2. In view of (10), we see that V/e[l, 2p - 3], the pairs P(l,7), P(U+ 1) and P(2,7- 1), P(2J) satisfy the assumptions of (4). Hence by (4) it follows that (14) If(x,j;)ecub(m) with x^a(l,2)-l and y or x < m(l) and y ^ a (2, 2) - 1 then (x,j;)6mw(a, !)<*>(x + 1, y + l)eww(a,2). From this and (10) it follows first that the paths ww(a, 1) and mw(a,2) are non-intersecting and so (i) follows by a simple induction on p. Secondly, (14) can be used to prove (ii). For this note that if p = 1 then there is no point of cub(m) lying below mw(a, 1) and so there is nothing to prove. Let p > 1. It is enough to show that there is no point of cub(m) which lies below mw(a, 1) and above mw(a, 2). For then it will follow that if a point P lies below mw(a, 1) then P lies on or below mw(a, 2) and so (ii) will follow by an easy induction on p. So let, if possible, (w,i;)ecub(w) which lies below mw(a, 1) and above ww(a,2). Then by (8) there is a point (x, >?)emw(a, 2) such that y x = v u. Hence y-v = x u=j, say. Now (u,v) is above ww(0,2) and (x,y) = (u-f;', u+7*)6mw(a,2). Hence either u +7 > u or v +7 > v, so that 7 ^ 1. Thus by (14), (u +j l,v +7 l)emw(a, 1). But (M, i?) lies below mw(<a, 1). Hence either w 4-7 1 < w or v -f 7 1 < i?, that is, j ^ 0. This is a contradiction. D 2.3 Frames Given weAT*(2), peN* and aevec(m,p), consider the points A t and B t as in (9). By a frame W[a\ in cub(m), belonging to the bivector a, we mean an ordered p-tuple where Vie[l, p], w(a, z) is a path in cub(m) joining A t to B and these paths are pairwise non-intersecting. Sometimes we will regard W\_d] as a subset of cub(m) containing the vertices of the paths in the frame. This will be clear from the context. Also fr(m, p, a) will denote the set of all frames in cub(m) belonging to a. 100 M R Modak Note that, by (2.2.1) (i), the p-tuple mW[a] in (13) is a frame belonging to a and so the set fr(m,p,a) is non-empty. We call wW[a] the minimal frame belonging to a. (15) Further, VeZ, let fr(m,p,a;) = { W[X]efr(m,p,a)|the p paths in W[a] have E antinodes in all}. (16) Note that obviously, |fr(m,p, a;E)\ = 0, VE < 0. 2.4 Index and mon Let S cub(m) and ee Af. By a chain T of length e in 5 we mean a sequence of points ^eS, ie[l,e], such that ^(/c)<^ +1 (fe), Vie[l,e-l] and Vfce[l,2]. Also, if T is non-empty, we call the point y 1 the initial point on T. (17) For every finite subset S of Z(2), we define the index of S, written ind(S), as follows ind(S) = the largest non-negative integer e such that there is a chain of length e in S, and we note that then ind(S)eN and ind(S) = 0, if and only if S is empty. Note that for A, B cub(m), A B=>ind(A) ^ ind(). Let there be given peN, weJV*(2) and aevec(m,p). Then for every ie[l,p] we put T (1, - truc(m, p, a, 1, i) = cub(*(l, i) - 1, m(2)), , = truc(m, p, a, 2, = cub(w(l), a(2, i) - 1). T(k, i) will be called the (fc, i)-th truncation of cub(m) with respect to a. Note that by (1) we have Vfe[l,p-l]andV/ce[l,2]. By a monomial on cub(m) we mean a map of cub(m) into N. We put mon(m) = the set of all monomials on cub(m), and for temon(m) we define the support of t, written supp(r), and the degree of t, written deg(t), as follows supp(t) = {j;ecub(m)| t(y) * 0}, deg(t)= S t(y). 3?ecub(m) Also for VeN we define mon[[m, F]] = {temon(m)|deg(t) = V] and mon(m, p) = { t emon(m) | ind (supp(t)) < p} t A Hilbert polynomial 101 and ] and ind(supp()n truc(m,p, fl, fc, i)) < i 1}. (18) Finally, for FeN, we define mon(m, p, a, V) mon(m, p, a) nmon [[m, F]]. Let X be an m(l) by m(2) matrix whose entries X^s are independent indeterminates over some field K. For every temon(m), denoted by X\ the ordinary monomial in X t jS where x =n^ ( j i )3 , (2) where the product is over yecub(m). Note that t-+X* is a bijection of mon(m) onto the set of all ordinary monomials in X tj \ (i J)ecub(m), ana t is thus the "exponent system" of an ordinary monomial. The elements of either of the sets mon(m,p,<2, F) and {X^temon^p.a, F)} are called indexed monomials. Given Secub(m) and temon(m), we say that S is occupied by t if supp(t)nS is non-empty and that S is free of t otherwise. Given a point P cub(m), we say that P is occupied by t if Pesupp(t) and that P is free of t otherwise. The following two results follow easily from these definitions. (i) Given temon(m) and eeN* such that ind (supp()) = e, let (x 9 y) be the initial point of any chain of length e in supp(t). Then cub ((x I 9 y 1)) is free of t. (ii) For every emon(m,p,<2), by definition (18) V/ce[l,2], ind(supp(t)n and hence T(fc, 1) is free of t, Vfee[l,2]. 2.5 Relation between mon and frames Theorem. Let a bivector aevec(m,p) and a monomial temon(m) be given. Then temon (m,p,a) if and only if there is a frame W[a]Efr(m,p,a) such that supp(t)^ Proof. First consider any frame JF[a]fr(w,p, a) and let temon (m) be any monomial such that supp(t) s W[a]. Suppose Given de[l,p], let X(d) = {Pecub(m)|Pew(a, i) for some ie[l,<f|}. Then (i) 102 Af R Modak To see this, let S be any (non-empty) chain in X(d) of length e. Then 1 ^ e < d, because, by the definition of a path, each of the d paths w(a 9 1), .. . , w(a, d) can contain at most one point of the chain S. Hence (i) follows. Now, w(a,i) is a non-increasing path joining the point A t to B i9 Ve[l,p] and so it follows that for each ze[2, p], w(<2, 1), . . . , \v(a, i 1) are exactly the paths from W[a\ which meet the truncations T(k, i\ k = 1, 2. Therefore, since supp(r) W[a], we see by (i) thatVfce[l,2] and*e[l,p], ind(supp(t) n T(fe, i}) i - 1 and also that Hence emon(?M,p,<3). Conversely v/e prove that 2.5.1: For every temon(w,p,a), there is a frame W[a]efr(m,p,a) such that supp(f) JF[a]. For this we start by constructing for each temon(m, p, a) a particular path, wm(a, 1, t) say, joining point y4 x to B x and call this path the first minimal path determined by t. For this construction we shall make use of the minimal frame, m W[a\, defined in (13). Given aevecfm, p), for each te[0, m(l) a(l, 1)], let d(i) = smallest integer such that (a(l, I) -f f,rf(i))eww(a, 1). Clearly then, d(i) > 1 and d(i + 1) < d(i). Given for each ie[0, m(l) - a(l, 1)], let ron(i,a, t) be the integer defined thus: m(2), if the point (a(l, 1) + fj), is free of t V/e[l,m(2)], ron(f,a,0 = 7, if; is the smallest integer in [l,m(2)] such that the point l, 1) H- ij) is occupied by t. Consider the sequence of points where and for i y(i) = min {y(i - 1), d(i), ronft a, t)}. Then each pair of successive points of the sequence can be joined minimally; let wm(a, 1, t) be the path so obtained. We note the following five properties of this path wm(a, 1, t). Put n = m(l) - 0(1, 1). (19) Every point of cub(m), which lies above wm(a, 1, t), is free of t Proof. This happens because for each point P = (fl(l,l) + i,y), ie[0,n], lying above wm(a, 1, f), we have A Hilbert polynomial 103 (20) Every vertex of ww(a, 1, t) lies on or above mw(a, 1). Proof. Follows from the definition of wm(a, 1, t). D (21) The path wm(a,l,t) coincides with the path ww(a,l) if and only if no point above mw(a, 1) is occupied by t. Proof. If wm(a, 1, t) coincides with mw(a, 1), the result follows from (19). Conversely, suppose every point of cub(m), lying above mw(a, 1) is free of t. Then y(O) = d(0), and Vie[0, n], d(i) < ron(z, a, r), and hence y(i) = d(z). D (22) Every node of wm(a, 1, t), which is not a vertex of ww(0, 1), is necessarily occupied by t. Proof. Suppose P = (0(1, 1) + j, y), je[0, n - 1], is a node of wm(a, 1, t) which is not a vertex of ww(a, 1). Then by (20), P lies above mw(a, 1), and so y < d(j). Now, by the definition of wm(a, 1, t), P belongs to the minimal join of the points P(j - 1) and P(j)- But P is a node and so the points (a(l, 1) +7, y + 1) and (a(l, 1) +;' + l,y) belong to wm(a, 1, t). Hence we see that P = P(J) and y = y(i) < y(j - 1). Therefore, y = y(j) = ron(j, a, t), so that P is occupied by t. D (23) Ifforanyie[2,p], ind(supp(t) n T(k, i)) = i - 1 for some ke[l, 2], (i) then the initial point of every chain in supp(t)n T(fc,i), of length i- 1, belongs to wm(a, 1,0- Further, if ind(supp(t)) = p, (ii) then the initial point of every chain in supp(t), of length p, belongs to wm(a, 1, t). Proof. Suppose S is any chain in supp(f)n T(M of length f-1, and let P = (a(l, 1) + jf , y) be the initial point of S, where je[0, n]. If; = 0, we see that because P is occupied by t. Hence Pewm(a, 1, t). If j > 1, then we will show that X/Ky^O-i). P) To prove this, note first that by (i) and (3) we have Secondly, P must lie on or above the path mw(a> 1). For suppose that P lies below mw(a, 1). Then every point of S lies below mw(a, 1). Therefore, by (2.2.1) (ii) it follows that every point of S lies on some path mw(o,) for *e[2,p]. Also the points of S must belong to distinct paths. But this is impossible because the length of S is i - 1, while S must be contained in the union of only the i - 2 paths mw(a, e\ ee[2, i - 1], since the paths mw(a,/), /e[i,p], do not meet the truncations T(/c,i), fc= 1,2. Hence 104 M R Modak P lies on or above mw(a, 1), and so y ^ d(j). Therefore, since the sequence d is non-increasing, we also have (v) Thirdly, we show by induction that,Vee[OJ 1] For e = 0, (vi) follows from (iv), and assuming (vi) for some ee[0,fc 2], we easily obtain from (iv) and (v) that y ^y(e+ 1). Therefore y^y(j 1). Finally, note that y(j) < ronO' &> t) < y, because P is occupied. Hence (iii) holds and so Pewmte 1, t). Thus (i) is proved, (ii) can be proved similarly. Hence (23) follows. We define the other minimal paths determined by t inductively, using the bivectors b [i] defined in (12). For example, in order to define the second minimal path wm(a, 2, t) for , consider t'emon(w) thus: Vyecub(m), if 3>e su PP(0\ wrote 1, t) =0, otherwise. Then by (23) we see that We define wrote 2, t) to be wm(fc[l], 1, t'). These paths wm(a, z,t), i= 1,2, are non-intersecting. To see this note that by (19) ^ and by the definition oft', every point of cub(m), lying on or above wm(<3, 1, t), is free oft'. Hence every point of cub(m), occupied by t', lies below wm(a, l,t). Now as in (22), every node P of wrote 2, t), which is not a vertex of mw(a,2), is necessarily occupied by t'. Hence P lies below wwtel,t). Hence by (7), wrote 2, t) an d wro(a, l,t) are non-intersecting. Repeating the above procedure, we obtain the p-tuple, Wm [a] = (wrote 1 > 0, , wro(a, P ? 0), where Vie[l, p], wm(a, i, t) is the i-th minimal path for t. Also these paths are pair- wise non-intersecting and hence FFm[a]efr(ro,p,a) and we call Wro[a] the minimal frame for t. Now we prove the following result which certainly includes (2.5.1). (2.5.1)' for every temon(ro,p,a), supp(t) c Wm{a\, where Wm[a\ is the minimal frame for t. Proof. Induction on p. Let p = 1. Then a = (a(l, 1), a(2, 1)) and the minimal frame ro W[a\ has only one path mw(a, 1), which joins A 1 and B^ minimally. Let temon(ro, p, a) be given. If supp(t) row(a, 1)> then mW[a\ is the required frame by (21). Suppose that some point lying above row(a, 1) is occupied by t. Construct the A Hilbert polynomial 105 path ww(a, l,t). By (19), no point of supp(r) lies above wm(a, l,t). Let, if possible, P(w, t>) be a point lying below wm(a, 1, f) and occupied by t. Then, by (6) (ii), wm(a, 1, t) has a node #(x, y) (which clearly lies above mw(a, 1)) such that x < u and y<v. Hence by (22), R is occupied by t and so {K,P} is a chain of length 2 in supp(t). Thus ind(supp(f)) ^ 2, which contradicts the assumption that emon(m, 1, d). Hence supp(t) wm(0, 1, t) and so Wm\_d] is the required frame. Next let p > 1 and assume the result for all bivectors a'evec(m, p f ) with p'e[l, p 1], Let aevec(m,p) be given and let emon(m,p,a) be also given. Now construct the first minimal path for t, namely wm(a, 1, t) and consider the monomial t' as in (*) above. Then, as seen above, and so by inductive assumption, supp(t')^ Wm [&[!]], where is the minimal frame for t'. Now it is easy to see that the frame (wm(a, I,0,w 2 ,...,w p ) is the same as Wm[a] and supp() W>w[a]. This proves (2.5.1)' and completes the proof of Theorem (2.5). D (2.6) For an element V in an overring of Q and AeZ we define the usual binomial coefficient V A \~\ Al ' ' if^<0 r*n and we define the twisted binomial coefficient .by putting We note that then / y\ VV\ (24) if A^O, each of I land can be regarded as a polynomial of degree^ in an indeterminate V with coefficients in Q and (25) for V, AeN, (26) for all A, FeZ, I = 0,if^l<0orif^> F^Oand A 106 M R Modak (27) for all VeN and AeZ, v V-A For every S cub (m) and VeN, put mon(S) = (temon(m)|supp(t) c S}, mon(S, 7) = mon(S)nmon[[m, 7]]. Given S, T^ cub(m) and 7eAf, it is clear that mon(S)nmon(T) = mon(Sn T), mon(S, F)nmon(r,F) = mon(5n T, 7), (28) (29) (30) (31) (32) Let there be given peN*, weN* (2) and aevec(m,p). Let Wi,... 9 W F , where F = |fr(m, p, a)|, be all the frames in fr(m, p, a) labelled in some order. Then by (6) (i) and (2.3) we see that | W | = C + 1, Vie[l,F], where C is defined as follows. (33) For every fce[l,2] let C[k] = Sf =1 (m(k)-a(/c,i)) and C = C[l] + C[2] +p- 1. Hence by (32) we have (34) CJ (35) NextVr,seJV*,r>2,let los(r, s) = family of all sets { W tl , . . . , W ir }, 1 < ij < i 2 < . . < i r < F, such that n ^- Then by (31) and (32) we get (36) V VeN and for every set { W h , . . . , W ir } elos(r, s), r \| T V 1 n^y)- c . 5 ' j=i /I L c 5 J In view of (28), by (2.5) we have F mon(w,p,fl)= (J mon(W^) i=l and so VFeN, by (29), F mon(m,p,a, F)= (J mon(W^ F). A Hilbert polynomial Hence by the inclusion-exclusion principle, we get VFeN, F |mon(ro,p,a,JO| = + \mon(W i ,V)nmon(W j ,V)nmon(W h V)\ 107 p V)\ =1 Hence by (34) and (31), ,n W J9 V)\ i <j < I ^ F / F mon fj W t , V \i=i Hence by (35) and (36) we get ]~nos(Mir v i J seAf* L^ ~ 5 J where each sum is clearly essentially finite. Finally, regrouping the terms in (37) we obtain: (38) For every FeN, |mon(m,p,a, V)\ = % (- DeN where F, if D = Z r>2 (-l)r |los(r,D)|,ifD>0. (37) (39) Note that the summation in (39) is essentially finite. Hence (38) shows that |mon(m,p,0, V)\ is a polynomial in V of degree C (since G = F > 0) with coefficients in Q. Now we describe Abhyankar's formula for |mon(w,p,, V)\. (40) Given peN*, DeZ and Me{Z,N} 9 let (41) Given fee [1,2], we put 2, if k = 1 1, iffc = 2' 108 M R Modak Let fcs[l, 2]. First, for eeZ(p) let G (U) (m, p, a, e) denote pby p matrix whose (i,j )-th element is and let and , p, a, e) = G^fo p, a, e)det G (U) (m, p, a, e), where det ^4 denotes the determinant of the matrix A. Second, for eeZ(p) let G (1 " k >(m,p,a,e) denote p by p matrix whose (tj')-th element is and let HW(m, p, a, e) = det G (1 * fc) (m, p, a, e). Third, for every Le{l, 1*} and EeZ we define (42a) eeZ(p,E) and for DeZ we define ) (m, p, a) (42b) eZ where Fourth, for eEZ(p), let G (2/c) (m, p, a, e) and G (2 * fc) (m, p, a, e) denote p by p matrices whose (ij )-th elements are e(0 p a e) = - ~ ' ~ )P " } respectively and let VLe{2,2*} H (Lk) (m,p, a, e) = det G (Lk >(m, p, a, e). Fifth, VLe{2,2*} and EeZ we define ,p,a, e ) (42c) eeZ(p,) A Hilbert polynomial and VLe{2,2*} and DeZ we define (42d) Sixth, VLe{l, 1*,2,2*} and V J/eZ we define )[% P l. (42) We now quote some results proved by Abhyankar. (43) ([1], pages 60,61). For Le{l,l*,2,2*} and fce[l,2], the summations in the definitions (42a) (42d) and (42) are essentially finite. (44) ([!]. page 171). For Lejl, 1*,2,2*} and /ee[l,2], ,a) = Fi> u >(m,p,a), VDeZ. (45) ([1], pages 309, 310). For Le{l, 1*,2,2*} and ke[l,2] (i) F ( Lk) (m,p,a) is a positive integer (ii) VjDeZ,F! k) (m,p,a) is a non-negative integer (iii) (DeZ|F<f >(m,/>, a) * 0} [Q, C], where and C[fc] are as in (33). We see, by (43), that the function F (Lk) (m,p,a, V) defined in (42) is a polynomial in V and, by (44), that this polynomial is independent of L and k and so we denote it by F( V). By (45) (i) it follows that the degree of F( V\ in F, is the integer C defined in (33). Finally, we note that ([1], Theorem (9-8), page 306) |mon(m,p,a, V)\=F(V). (46) In view of (42) and (25), it follows from (46) and (38) above that VLe{l, 1*,2,2*}, V/ce{l,2} and VFeJV, we have De. Now since j \\AeN I is a Q-vector space basis of the polynomial ring Q{V}, we may compare coefficients of ^ D , DeN, and obtain from (47) that ) = G D , ( 4g ) '}andVfc6{l,2}. In particular, (49) 110 M R Modak 3. Combinatorial interpretation 3.1 Notation Given peN and AeZ, we put J(p) = the set of all subsets of [l,p] and J(p,A)={aeJ(p)\\a\=A}. Given peN*, bevec[p + 1] and /ce[l, 2], with k as in (41), first for every ue J(p) we let and a(fc',z and a(KJ and second, Vl/e[0,p] we let ,k,u]. (50) Note that the union is disjoint. Also Vl/e[0,p], M(p,6,fc,l7)c=vec(m,p). (51) Let there be given fee[l,2] and meN*(2), m*eN*(2) such that 3.1.1: m*(/c x ) = m(/c r ) and m*(/c) = m(fe) -h 1. Further let peJV*, p*eN*, 0*evec (m*,p*) and fcevec[p + 1] be such that either 3.1.2: p* = p and m*(fc) - a*(fc, p*) 5^ and a*(k, i) = b(k, z)V/ce[l, 2] and ie[l, p] and holds or 3.1.3: p* = p+l ) m*(fe)~a*(/c,p*) = and a* = b holds. We note that if (3.1.1) holds and if either (3.1.2) holds or (3.1.3) holds, then V U e[0, p], we have M(p,b,fc,l/)c=vec(m,p). (52) 3.2 Theorem. Let there be given any m*eA/ r *(2), p*e7V*, a*evec(m*,p*) andmeN*(2\ peJV*,fce[l,2] and fcei;ec[p4-l] such that (3.1.1) holds and either (3.1.2) holds or (3.1.3) fcoMs. Tfeen VEeZ, |fr(m*,p*,a*;)|= ^ ^ fr(m,p,a; -/) (53) "[/6[0,p] fleM(p,fc,fc,l/) and when p = 1, m(k) a(k, 1) = 0, (54) A Hilbert polynomial 111 Proof. Clearly, by (16), both sides of (53) vanish if E <0. Also (54) is obvious. Let k = 2 so that k f = 1; the other case is exactly similar. Fix EeN and consider any frame W[a*] = (w(fl*, 1), . . . , w(a*, p*))efr(m*, a* 9 p* ; E) where Vze[l,p*], w(<3*, z) denotes a path joining the points 4 = (fl*(l, 0, m*(2)), B f = (m*(l), a*(2, 0). (55) Let (3.1.2) hold. Then each path in PF[a*] meets the line y = m*(2)~l i.e. Y = m(2) and Vze[l,p*] we let XJ to be the vertex with the smallest x co-ordinate which is common to the path w(a*,z) and the line 7= m(2). (56) Let (3.1.3) hold. Then each path, except w(<2*,p*), in W[a*] meets the line 7 = ra(2) and we define the point A\ as before for every z'e[l,p*- 1]. Now let either of (3.1.2) and (3.1.3) hold and define Vze[l,p], ffl(l, = x co-ordinate of A\ and ( / and consider the bisequence Let (3.1.2) hold. Then since the paths in W[a*] are non-increasing, it follows (Vz'e[l,p*]) from the definition of the point A\(a(lJ), m(2)) that the point (fl(l, z), w*(2))ew(fl*,0 and a(l, z) ^ fl*(l, z) (58) and the portion of w(<3*, z) joining y4 f (a*(l,z), m*(2)) to -4J is in fact a minimal join. Further, since the paths in P^[a*] are pair-wise non-intersecting, it follows from (58) that fl(l,0e[**(l,i), **(U+ 1)- 1], , .andfl(l,p*)e[fl*(l,p*),m(l)]. l j Finally^ it is clear that Vze[l,p*], the path w(fl* 9 z') has an antinode on the line F = m*(2)iff fl(l,i)>a*(l,0. (60) If (3.1.3) holds, we see similarly that fl(l,i)6[a*(l,i), fl*(l,i+ 1)- 1], Vze[l,p*-~ 1] (59)' and Vze[l, p* 1], the path w(a*, z) has an antinode on the line Y = m*(2) iff (60) holds. Now let either of (3.1.2) or (3.1.3) hold. TJien it follows from (59) or (59)' that for the bisequence a defined by (57), we actually have aevec(m,p). In fact, we see that 112 MRModak there exists a unique integer /e[0,p] and a unique subset ueJ(p, U) such that .2, ], ^ and if.w(a, denotes the portion of w(a*, f) joining AJ to B i9 Vie[l,p], then the p-tuple T/r/r ~i / / 1\ ^ \\ //^ i \ is, by (60), a frame in fr (m, p, a; E - I/). We shall call W[a], the contraction of J>F[a*]. Now consider the set of frames T= U U fr(m,p,a;E-(7), (62) C/6[0,p] a6M(p,6,2,[7) and note that the unions are disjoint. We now show that the map W[a*] -+ W[a] = the contraction of W\_a*~\ (63) is a bijection of fr(m*,p*,a*,E) onto the set T. To see that the map (63) is surjective, take any I7e[0,p], weJ(p, U), c and consider a frame W[<2]efr(m,p,a,E U). Let where Vze[l,p], w(a,z) is a path joining the point I i (fl(l,0,m(2))toB (m(l),fl(2,0). If (3.1.2) holds, then p = p* and a (2, = a*(2, so that JB f = B i9 Vie[l, p*], with A i9 B t as in (55). Join, Vi'e[l,p*], A t and A t minimally and let w(a*,f) be the extension of w(a,i) thus obtained, and let _ If (3.1.3) holds, then p = p* - 1 and a* (2,p*) = m*(2) andji(2, = a*(2, so that Bi = B f - } Vfe[l,p* - 1]. For every ie[l,p* - 1] join A t and A t minimally. Also join A p *(a*(l 9 p*),m*(2)) and J5 p *(m*(l),a*(2,p*)) and let w(a*,p*) be the path obtained. Also for every ie[l,p*], let w(a*,z) be the extension of w(a,i) obtained above and let Then, if either of (3.1.2), (3.1.3) holds, it is clear that and that the contraction of W[a*] is W[a]. Hence the map (63) is surjective. Next, to show that the map (63) is injective, consider any two distinct frames W^ta*], j= 1,2, in fr(m*,p*,a*;E). Then there exists ze[l,p*] such that the paths w ; (a*,i) in W[^*l J= 1 9 2, are different. A Hilbert polynomial 1 i 3 Consider the points A^fJ = 1, 2) as in (56) and let W/[0/] 0" = *> 2) be the contraction 'of Wj^a*]. We now show that ^[fliD^WilAl (64) If A'uitAM, then clearly a l ^a 2 and so (64) holds. Hence suppose that A' u = ,4' 2l .. Then the portions Wj(a j9 i), of Wj(a*,i) 9 joining A$ to B u (j= 1,2) must be different. Hence again (64) holds. Thus the map (63) is injective. Hence Theorem (3.2) is proved. D (65) Remark. It can be easily shown after the manner of [1], lemma (5.6"), page 167, by double induction on p and s(p) = m(k) a(k,p\ that the recurrence relation (53) and the condition (54) uniquely determine a function. (3.3) In Theorem (3.3.1) below we shall obtain the recurrence relation satisfied by the function H% k \m,p 9 a\ defined in (42c). We start by stating the notation and lemmas needed in the proof. (66) By convention, the sum over an empty family is taken to be zero. (67) ([1], page 61) For the functions H% k \m, p, a) we also have VLe{2, 2*} and VeZ, eeN(p,E) where the summation is essentially finite. Also, by definition, VLe{2,2*} and VDeZ, .4 (68) We note that the summation in (68) is essentially finite. (69) ([1], page 211) Given K y (d*)sfiViJ = [l,p] and d*eZ. Then VDeZ we have X det[J^)]= deN(p,D) (70) It follows from (69) that VeZ, (71) Let peN* be given. For ueJ (p) and V*, yeZ(p) such that x t < y t , Vie[l, p], we put X[u] = {rZ(p)|r c e[x l + l,yi- 11 Viet* and o and Vl/e[0,p] we put and note that if u' 9 u*eJ(p) and M' ^ w* then 114 MRModak so that (72) the union in (i) is disjoint. Hence we see that / (73) Given any l/e[0,p], for every map /:Z(p)-Q, we have = Z Z W ueJ(p.U) reX[u] (74) For the rest of this section we assume that there are given m*eN*(2\ p*eW*> a*evec(m*,p*) and meiV*, peJV*, fee[l,2] and &evec[p + 1] such that (3.1.1) holds and either (3.1.2) holds or (3.1.3) holds. We shall deal with the case k = 2 so that kf = 1; the other case is exactly similar. Define sequences r*,s*eZ(p*) by putting Vje[l,p*], 5J 8 = m* (k) - a* (kj ), rj = m* (kf) - a* (k J ). (75) Then since a*evec(m*,p*), it is clear that r* and 5* are strictly decreasing sequences of non-negative integers. LetVie[l,p], Xi = m(kf)-b(kr 9 i+l) 9 yi ~m(k')~-b(k',i) (76) and Si = m(k)-b(kJ). (77) Now by assumption a*evec(m*,/?*) and (3.1.1) holds. Hence by (76) and (77) we see that (78) If (3.1.2) holds then first p = p* and secondly, and * = - 1 > = r = * and thirdly, s** > and sf-l-p* + and s f = s?-l, (79) If (3.1.3) holds then first p = p* - 1, and secondly, and and thirdly, s** = and Next let x p = r** >0 9 y l = rf (80) A Hilbert polynomial 115 and (81) VaeM, let r[a]eZ(p) be defined by putting, Then, by the definition of M in (3.1) we see that (82) given ueJ(p) and aeM, a](;) = y j5 Vje[l,p]\u and equivalently (83) given we J (p) and aeM, aeM[p,6,/c,u] =>Vjeu, r [a](j) = r* - 1 - 1 ; for some tj6[0, r* - r}V x - 2] By (82) we see that VueJ(p) a->r[a] gives a bijection of M[p,fc,k,u] onto X[w] when x,yeZ(p) are chosen as in (76). Hence, in view of (50) and (72), we see that (84) Vt/6[0,p], c^r[a] gives a bijection of M(p,b,fc, 17) onto X(U). 3.3.1 Theorem. Suppose (74) fcoWs. Tfcew VEeZ W6: have Ht 2fc) (m*,p*,a*)= E Z p=l,m(2)-a(2,l) = 0, Hg k) (m, p, a) = 1, if E = and zero otherwise. Proof. By definition and (67), we have for EeZ, /fi?* ) (m*,p*,fl*)= Z det[G y ()] eeiV(p*,) where Vi,;s[l,p*] and eeN(p*,E\ Fix eeN(p*,E) and consider the matrix G(e) = [G y (e)]. We use the following two basic results 116 MRModak and given any A, V, WeZ with W\ f V\ w-v-i (W 1 sf- ei -l and hence by (86), Also,Vi6[l,p*], (II) Now, if ^ > p* - f then ( P " ) = by (26), since p* - i ^ 0. \ e i Next let gj < p* - i. Then V (86) A) \A ' ^ * A * ' V } First let (3.1.2) hold. Then s** > 0. For every j"e[l, p* 1], in G(e) subtract (j -h l)-th column from;-th column to obtain the matrix H(e) whose (i,j)-th element is, say H tj , l^i^p* where H ip * = G ip *(e) and for 1 < j < p* 9 " . i Then by (85) we have for 1 ^7 < p* }* A Hilbert polynomial 111 and so s?-l-p* + i = by (26), since sf - 1 - p* 4- i ^ by (78). Hence the last term in (II) above is always zero. Now let (3.1.3) hold. Then s** = 0. Hence Vje[l,p*], Now e p * ^ and p* -; ^ V/e[l,p*]. Hence,by(26),V/e[l,p*], Therefore det G(e) = if e p * > and each element of the last row of G(e) is 1 if e p * = 0. Hence let e p * = 0. Then V;[l,p* - 1] we have Expanding det H (e) by its last row we obtain a determinant of order p* 1 = p whose (, j)-th element is given by (I) above. Hence, in view of (66), we see by (78) and (79) that when either of (3.1.2) and (3.1.3) holds, (HI) Hk 2k) (m*,p*,fl*) = where Vfje[l,p], H tj = JT Now for every weJ(p) and ie[l,p], let ' ~ (. *\ / i *\ VT' r j Then using the fact that (87) det is a multilinear function of its columns we get (IV) ueJ(p) For every ueJ(p), reX[u] and eeN(p,E), let -(- i, if;eu 118 MR Modak Note that, by (26), (88) if for any ie[l,p], e ; <0, then k u (r,e t ) = 0, Vje[l,pJ. Then again by (87) we get VweJ(p), det[H<f] = Z det[fc lV M,-)]. reX[u] Hence by (III) and (IV) we get H<?*>(m*,p*,a*)= Z Z Z detlTcyfce,)]. eeN(p,E)u./(p)reX[u] = Z Z Z det[fcy(r,e,)]. ueJ(p) reX[u] eeN(p,E) = Z Z Z det[k i; .(r, e; .)]. (by (69)) ueJ(p) reX[] eeN(p,E) = Z Z Z Z 0,p] ueJ(p,U) reX[u] eeN(p,E) Z Z det[fc y (r, ej )]. (by (73)) X(iy)eeN(p,E) = Z Z Z det[ky(rM,^)]- (^ (84)) I7e[0,p] aeM(p,b,k,U) eeN(p.E) = Z . Z Z det[ky(rM.j)]- ( b J7E[0,p] fleM(p,5,fc,l7) eeZ(p,E} Now VUe[0,p], weJ(p, 17) and eeZ(p,) consider d(u,e)eZ(p) where so that (*) e -> d(u, e) is a bijection of Z(p, E) onto Z(p, 17). Then Vl7e[0,p], weJ(p, 17), aeM [p, 6, fc, w] and eeZ(p,E) we see, by (51) that det [kyO-lXUj)] = H< 2 * fc) (m 9 p 5 ^d(w^)). Also we then have ^ X det [k,./r [a], <>;)]= I eeZ(p,E) eeZ(p,E) eeZ(p,F - U) = ffg^(m j p,a) = Hg D (m,p,4 (by (70)) Hence by the above {/e[0 t p] A Hilbert polynomial 119 Finally, if p = 1 and m(fc) a(k,p) = 0, we have [0, ifeZ\{0}. Thus Theorem (3.3.1) is proved. D (3.3.2) Theorem. Let there be given peN*, meJV*(2) and aevec(m,p). Then VkE[l,2] and VEeZ, = |fr(m,p,a;E)| = number of frames in fr(w,p, a) each having exactly E antinodes. Proof. Follows from (3.2) and (3.3.1) in view of (65) and (70). D Let p, w and a be as in the above theorem. Fix DeAT. For EeN, consider any frame /E\ W[a\ in fr(w,p,<3;E). Now D of the E antinodes in W[a] can be chosen in I J different ways. For each such choice, mark the D chosen antinodes by red colour and call the frame obtained a D-marked frame. Let A(E, D) denote the set of all D-marked frames obtained from various frames in fr(w,p,a;E). Then the set (D)= (j A(E,D\ EeN where the union is disjoint, is the set of all D-marked frames obtainable from various frames in fr(m,p,a). Hence by the above theorem, (3.3.3) Theorem. Let there be given peN*, meN*(2) and aEvec(m,p). Then Vke[l,2] and VDeZ, F ( D k) (m,p, a) = |B(D)| = number of D-marked frames obtainable from those in fr(m, p, a). Proof. Follows from (*), since by definition, VDeZ, 120 MR Modak COROLLARY (i) F D ^ 0, for all DeJV, where F D F D (m, p, a). (ii) // for some EeN*, F E = then F D = for all D^E. (iii) Let C be the maximum number of antinodes that a frame in fr(w, p, a) can have. Then (iv) Let, for ze[l,p], r / = m(l)-a(l,0,s f = m(2)-a(2,z), C, = min{r,,sj Then, Proof, (i) (iii) are obvious. For (iv) note that in any frame Wefr(w,p,0), the path ^ w() has at most r, x-steps and s t y-steps and hence at most C t antinodes. Hence the frame W has atmost SC f antinodes. D Let the assumptions of (3.3.3) hold. For all EeZ, write = H E , Hg 2) (m, p, a) - H E and x and s t = m(2) - fl (2, 0, ie[l,p] and S = 5| . (89) Then, by definition, VDeZ, EeZ \D + <Sy and (91) We see from (2.1), (6) (i) and (89) above that in any frame the p paths have in all S right-end points in them. Hence, by (2), it follows that if in a frame j of these S right-end points are antinodes then the remaining S ; are intermediate points. Now S E of these S j intermediate points can be chosen in ( J different ways. For \S-EJ each such choice, mark the S E chosen intermediate points by blue colour and call the frame so obtained an ^-labelled frame. (3.3.4) Theorem. Under the assumptions of (3.3.3), VEe[0,S], ---<- "(Jlf A Hilbert polynomial 121 (in) H E = number of E-labelled frames obtainable from those in fr(m,p, a). Proof. We need the following lemmas (iv) ([1], Lemma 4.2', page 96). For all 17, T, FeZ, Uj\TJ \TJ\U-T (v) Let there be given neAT and x t eQ and y.-eQ, Vz'e[0,n]. IfVje[0,n],. then Vfce[0,n], (vi) Let there be given S, EJeN such that ; < S and < S. Then E J\iJ VS-B Using (iv), (v) follows easily. For (vi) it is enough to show that for all R Here, the generating function of left side is 1+x 122 MR Modak ReN = the generating function of right side. Now VDeW, by (27), (90) is Hence applying (v), we obtain from that Hence and so (i) follows. Again, - (by(91)) *' Hence (ii) follows. Finally, from (3.3.2) and (ii) above we obtain (iii). Acknowledgements The author thanks Professor S S Abhyankar for his valuable guidance throughout this work. He also thanks the referee for helpful comments, suggestions about rearranging the material and reference [4] and wishes to acknowledge the Department of Mathematics, University of Poona, Pune and Bhaskaracharya Pratishthana, Pune for providing necessary facilities. A Hilbert polynomial 123 References [1] Abhyankar S S, Enumerative Combinatorics of Young Tableaux (New York: Marcel Dekker) (1988) [2] Abhyankar S S and Kulkarni D M, On Hilbertian ideals, Linear Algebra 116 (1'989) 53-79 [3] Gessel I and Viennot G, Binomial Determinants, Paths and Hook Length Formulae, Adv. Math. 58 (1985) 300-321 [4] Ghorpade Sudhir R, Abhyankar's work on Young Tableaux and Some Recent Developments, to appear in Proc. Conf. on Algebraic Geometry and applications (Purdue Univ. June 1990), (New York: Springer- Verlag) (1992?) [5] Udpikar S G, On Hilbert polynomial of certain determinantal ideals, Int. Math. Math. Sci. 14 (1991) 155-162 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 2, August 1992, pp. 125-128. Printed in India. A note on two absolute summability methods HOSEYIN BOR* Department of Mathematics, Erciyes University, Kayseri 38039, Turkey *Mailing Address: P K 213, Kayseri 38002, Turkey MS received 15 December 1991 Abstract. In this paper we have established a relation between \R,p n ;5\ k and \R,q n ;5\ k summability methods which generalizes the results of Bor [1] and Bosanquet [2], Keywords. Absolute summability; summability methods. 1. Introduction Let a n be a given infinite series with the sequence of partial sums (s n ). By u n we denote the nth (C, 1) mean of the sequence (sj. The series ^La n is said to be summable |C,l;<5| fc , where k^ 1 and <5>0, if (see [3]) 1 !*, --!!*< oo. (1) n=l Let (p n ) be a sequence of positive numbers such that Pn = p,-> oo as n-* oo, (P., = ?-! = <), f^l). (2) 17=0 The sequence-to-sequence transformation. (3) defines the sequence (t n ) of the (R,p n ) means of the sequence (sj, generated by the sequence of coefficients (p n ) (see [4]). We say that the series Ea n is said to be summable \R*Pni <5|/t> where k ^ 1 and 5 ^ 0, if In the special case when p n = 1 for all values of n (resp. <5 == 0), \R,p n ;d\ k summability is the same as |C, l;5| fc (resp. \R,p n \k) summability. 2. Regarding the relation between two absolute summability methods the following theorem due to Sunouchi [5] is given. 125 126 Huseyin Bor Theorem A. Let (p n ) and (q n ) be positive sequences (where Q n = ^ = q v ). In order that every \R,p n \ summable series should be \R,q n \ summable it is sufficient that (5) PnQn While reviewing this paper, Bosanquet [2] observed that the condition (5) is also necessary for conclusion and so completed Theorem A in necessary and sufficient form, 3. The aim of this paper is to generalize Bosanquet's result for |jR, p n ; S\ k and \R, q n ; 5\ k summability methods. Now, we shall prove the following theorem. Theorem. Let k^l and 5 ^ 0. In order that every \R,p n ; d\ k summable series should be \R,q n ;5\ k summable, the condition (5) is necessary. If we suppose that then the condition (5) is also sufficient. Remark. If we take k = 1 and <5 = 0, in this theorem, then we get the observation of Bosanquet [2]. Also if we take only 6 = in this theorem, then we get a result due to Bor [1]. 4. We need the following lemma for the proof of our theorem. Lemma. Let fe^l and A (a nv ) be an infinite matrix. In order that Ae(l k j k ) it is necessary that a nv = O(l) for all n,v^Q. Proof. This follows since I 1 c l k c= / (see [1] for details). 5. Proof of the theorem. Necessity. We consider the series-to-series version of (3), i.e., for n ^ 1, let A simple calculation shows that for n ^ 1 c -"Z 1 &-i-fi-^-i) + - (9) nn-l v = l From this we can write down at once the matrix A transforms {n (6k+k ~ l}/k b n } into {n ( * k+k ~ m c n }. Thus every |jR,p n ;<5| fc summable series \R,q n ;8\ k summable if and only if Ae(P, l k ). By the lemma, it is necessary that the diagonal terms of A must be bounded, which gives that (5) must hold. A note on two absolute summability methods 127 Sufficiency. Let c n>1 denote the sum of the right of (9) and let c n ^ 2 denote the second term on the right of (9). Suppose the condition is satisfied, then, it is enough to show that, if n* k+k - 1 \b n \ k x>, (10) n=l we have I H* + *-Mc,,/< oo,(i=l,2). (11) n=l For i = 2 this is an immediate corollary of (5). Thus, to complete the proof of the sufficiency, it is enough to show that we have P 9 Q v -i-Q,P 9 -i Now, applying Holder's inequality, with k> 1, we have f a**"*/" n=l n-1 = 0(1) n^^^ \ \b,\ k q.(Q,/q.n\ * n=l oo /1/1\ V M 5/c + fc 1 = O(i) 2L, n=l oo ^ua^)" n = 0(1) Ib/^/^V^- 1 -, by (6) v=i y y Thus This completes the proof of the theorem. If we take p n = 1 for all values of n, then \R, p n \6\ k summability reduces to |C, 1; <5| k summability and the condition (5) reduces to Now, we obtain the following corollary from our theorem. COROLLARY JT Let k^ I and 6^ 0. / ord^r t/iat et;^ry |C, 1; 5| fc summable series should be \R, q n \ <5| k summable, (i) is necessary. If the condition (6) is satisfied, then the condition (i) is also sufficient. 128 Hilseyin Bor Acknowledgement This paper was written during the author's visit to Birmingham University, England. The author wishes to express his thanks to Professor B Kuttner and Dr B Thorpe in the Mathematics Department for thier hospitality. References [1] Bor H, On the relative strength of two absolute summability methods, Proc. Am. Math. Soc. 113 (1991) j 1009-1012 / [2] Bosanquet L S, Math. Rev. 11 (1950) 654 / [3] Flett'T M, Some more theorems concerning the absolute summability of Fourier series and power ( series, Proc. London Math. Soc. 8 (1958) 357-387 ' [4] Hardy G H, Divergent series (Oxford University) (1949) W [5] Sunouchi G, Notes on Fourier Analysis XVIII, Absolute summability of a series with constant terms, ] Tdhoku Math. J. 2 (1949) 57-65 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 2, August 1992, pp. 129-139. Printed in India. On deficiencies of differential polynomials ANAND PRAKASH SINGH and RAJ SHREE DHAR Department of Mathematics, University of Jammu, Jammu 180004, India MS received 8 July 1992 Abstract. Bounds for deficiencies of differential polynomials have been given and some relations between Nevanlinna characteristic of differential polynomials in / and Nevanlinna characteristic function of / have been obtained. Keywords. Meromorphic functions; differential polynomials; Nevanlinna theory and deficiencies. 1. Introduction Let / be a non-constant meromorphic function in the complex plane. We use the usual notations m(rj\ n(rj\ N(rJ\ T(r,f), N(rJ) and S(r,/) etc of the Nevanlinna theory (see [2]). Let P , P , . . . , P k be non-negative integers. Following Doeringer [1], we call a monomial in/ with d M = p -f p^ 4- . . .p fc , its degree and F M = p -f- 2p x + . . . 4- (fc -f l)p fc , its weight. Further let M 1 [/],M 2 [/],..., M n [/] denote monomials in / and a l . , a 2 , . . . , a n meromorphic functions satisfying T(r, a^) = S(r,/), 1 ^j ^ n, then QW= I, ajMjUl, 7=1 is called diJBferential polynomial in / of degree d Q = Max" = 1 d Mj and weight T Q = Max"= i T Mj with coefficients a,-. For simplicity, we will denote d Mj by dj and T Mj by Tj for all;. Throughout this paper we shall denote the quantity l(d l +d 2 + ...d n ) (n l)d^\ by A. Then clearly A^ d Q . If all the terms in g[/l have the same degree, then Q[/] is called a homogeneous differential polynomial. As usual the Nevanlinna deficiency 6(a,f) and the Valiron deficiency A (a,/) are defined by 129 130 Anand Prakash Singh and Raj Shree Dhar Also as usual In this paper we find bounds for deficiencies of differential polynomials and some relations between Nevanlinna characteristic of differential polynomials in / and Nevanlinna characteristic of /. 2. Statements of the results We shall prove the following theorems: | Theorem 1. Let f be a meromorphic function of finite order. \ (i) // Q is a differential polynomial in f of degree d Q and weight T Q , then \ to ' =^ T Q (T Q d Q )(<x),f). (ii) // Q is a differential polynomial of degree d Q and if Q does not involve f, then A / &(b,f ) ^ #(0, Q) lim ~ 6,* oo r-*oo * (T )] } and A 6(b 9 f) < A(0, G) lim ^7^ ^ b^co r-*oo * ViJ ) The following theorem gives the relation between <5(0, Q) and <5(a,/), a ^ oo: Theorem 2. Let f be a meromorphic function with N(r,f) = S(r,/) (m particular for entire function f) and let Q[f~] be a differential polynomial of degree d Q which does not contain the factor /, then A ^ Remark 2.1. If Q =/' where / is an entire function then we obtain Theorem 4.6 of "p Hayman [2]. A consequence of Theorems 1 and 2 is the following corollary. COROLLARY 1 // / is a meromorphic function of finite order with N(rJ) -f N(r, I//) = S(r,/) and if Q is a differential polynomial in f which does not contain the factor f with On deficiencies of differential polynomials 131 then a Q and T(r,Q)~d Q T(r,f). Remark 2.2. We note that the above corollary is not true if lim ^' i 7^, T(r,0 d Q as shown by the following example: Let f(z) = e z and Q [/] = (/"") 2 + /' - (/"' ) 2 , then Q [/] = e\ In this <5(0, Q) = 1 but A/d Q =l/2. Theorem 3. Let fbea meromorphic function of finite order. Let \ a f | < oo (i = 1 , 2, . . . ). // | is a differential polynomial of degree d Q which does not contain f then for A > [1-<5(0,0 + A(0,0] f (5(a f ,/K F Q ~ (F Q " d Q )&(coJ) 1 A( ^ g) The following theorem gives us the bounds for deficiencies of differential polynomials in terms of its corresponding degree and weight. Theorem 4. Let fbea meromorphic function of finite order with % aiX> <5(a,/) = 2. Then for any positive integer /c, we have 2A T(r, Q) ~ T c T(r,f), S(co t Q) = 0, d(0, Q) ^ =-, 1 where poles of M 7 -[/], 1 ^j ^ n are different from zeros of a } (z) in Q [/] = I? j= i aj(z)Mj[f]. Remark 4.1. For Q =f (k \ we get Theorem 1 of Xiao-Mao and Chong Ji [4]. Remark 4.2. Let f(z) be a meromorphic function of finite order and let a 1 ^ a 2 be two finite complex numbers. Let then T(r, Q) ~ T Q T(rJ) where poles of M;[/] are not zeros of a^z). This follows since lim r _ 00 (A/ r (r,a i )/T(r,/)) = implies (5(a )= 1 for i- 1,2 so that = 2 and 5(oo,/) = 0. 132 Anand Prakash Singh and Raj Shree Dhar 3. Some lemmas For the proofs of the above theorems we shall need the following lemmas. Lemma 1 (see [1]). Let f be a non-constant meromorphic function. If Q\_f~\ is a differential polynomial in f with arbitrary meromorphic coefficients q h l^j ^n then m(r, 2[J]) *S d Q m(r,f) + f ( r > 9j) + S(r,f). j=i Lemma 2. Let Q be a differential polynomial in f of degree d Q having n terms and suppose that Q does not involve f. If Q is not a constant and bi,b 29 ... 9 b q are distinct complex numbers (where q is any positive integer) then Proof of lemma 2. Let where k is a positive integer. Following Theorem 2.1 of Hayman [2] it is easy to see that m(r, F) + 0(1) ^ k t m(r, fc l9 /). (2) i=l Taking k as d Q , we obtain by using Theorem 3.1 of Hayman ([2], p. 55) that m(r, 1/0 m(r, 1/S) + I (d c - A)m(r, b h f) + S(r,f). i=l This gives that which proves the desired result. Lemma 3 ([3], Lemma 3). Let Q[f] be a non-constant differential polynomial Let z be a pole of f of order p and neither a zero nor a pole of coefficients of <2[/]. Then z , ~|f is a pole of Q [f] of order almost pd Q + (T Q d Q ). On deficiencies of differential polynomials 133 Lemma 4. Let f be a meromorphic function of finite order. Then a?*oo 1 Q 1 Q Froo/ o/ /emma 4. We have, on replacing / by Q and & = 1 in (1) and (2), that and Therefore m(r, 1/QO + m(r, FQ') + AT r, + 0(1) i = T(r, Q) - JV(r, + AT(r, Q') + S(r, Q), (3) which gives ,-+ mbb^ , QJ i=l 1 Q < T(r, 2) + N(r, Q') - N(r, Q) + ^N(r, Q) + S(r, Q). 1 Q Since it follows that 'Q' Hence we obtain / l_ S~fra -t Now from (3) we note that r T(r 9 ( lnn-^7 .^y. I - 134 Anand Prakash Singh and Raj Shree Dhar Therefore, we have (M = [l-5(0,Q')]. -i Combining the above with (4), we have [1 - 5(0, Q')] Z ^ 6) ^ 2 - 5 ( 2) ~ Z ^ 2) i=l i = 1 _lZi(l_5(oo,Q)), r <2 which gives [2 - 3(0, Q')] Z *&> G) + F" 5 <' 2) < -4 - i=l 1 Q 1 Q This proves Lemma 4. Lemma 5. Let f be a meromorphic function of finite order. If I fl?6oo <5(fl,/) = 2, then N(r,/)~r(r,/)asr-*oo. Proof. Replacing Q by /' and b t by a f in Lemma 2, we get Z m(r, fl b /) ^ 7W') - N(r, I//') 4- S(r,/). 1=1 Adding I, q i=l N(r,a i ,f) to both the sides, it follows that q T(rJ) ^ T(r,f) + J N(r, fl ,/) - N(r, l/f) + S(rJ). Dividing by T(r,/) and taking limit inferior as r-> oo, it follows that i=lr-*oo ^ U J J And so Making q -> oo, it follows that lim ^i >2. (5) On deficiencies of differential polynomials Also T(r,f) ^ lim < 2. r-cc T(r,/) Thus we have from (5), 135 <-\m(r-} + m(r,/) + AT(rJ) + N(r,/) | 2 V f ) J Thus -7WK-JV or equivalently T(rJ) < N(r,/) 4- S(r,f). The lemma now follows since N(r,/) ^ T(r,/) for all r. 4. Proofs of the theorems Proof of theorem 1. By Lemmas 1 and 3, we get T(r, Q) < d Q T(rJ) + (F Q - d Q )N(rJ) + S(r,/). (6) Thus we obtain This proves part (i) of Theorem 1. Next from Lemma 2 and noting that S(r, = S(r,/), we obtain m(r,l/Q) Using the property that lim. /(r)flf(r) < Bm/(r) lim g(r) for /(r) 5* r-*co r-*oo 136 Anand Prakash Singh and Raj Shree Dhar and g(r) ^ 0, it follows on making q -> oo that A d(bj) ^ 6(0, Q) to ^fj 6*0, r->oc 1 (/,./ J and This proves the required result. Froq/ of theorem 2. Since by the hypothesis N(r 9 f) = S(rJ), we have by (6) that T(r,Q)^d Q r(r,/) + S(rJ). (7) From Lemma 2 and using (7) we obtain A v r ^4 2, li Thus by making # -> oo we obtain Hence the result. Proof of corollary 1. By Theorems I and 2, we obtain >A 6(b,f) r -oo J (r,J ) ft^cc' and " d Q Also by hypothesis, we have Thus we have On deficiencies of differential polynomials Since 0(oo,/) = 1 and > 8(b,f) = 1, we have from the above Thus we have T(r,C)~d Q T(r,/). This proves Corollary 1. Proof of theorem 3. Let B = Hm ^ and C = Hm ~ ^ ^ r -oo J l r J) r ~" GC ' 1 ^' J By Lemma 2 we have ^ ,40 T(r, f ) 4- N(r, 1/2) < ^(r, Q) + A N(r, a f ,/) + S(r ,/), (8) a \ ./ / i= 1 which gives Thus we have which reduces to Also from (8) we have (r, :r- T(r, Q) A So, we obtain Aq + (\- o(0, Q))B ^ Therefore on rearranging we get - 5(0,0) r==1 From (10) and (11), we get . (12) 138 Anand Prakash Singh and Raj Shree Dhar But by our supposition and Theorem 1, we have From (12) and (13) we get A(l - S(0 9 Q)) t 5(a f ,/) < [T Q - (T Q - d Q )0(oo,/) i- 1 Making q -> oo it follows that This proves the desired result. Proof of theorem 4. For the proof of Theorem 4, we put By using (6) we have = r Q -(r fl -d Q )0(oo,/). (H) On the other hand, by using Lemma 3 we have b(l - <5(oo, 0) = Hm ? lim ? r * oo * \J)J ) r~~* <x ^ \' s xJ/ /v r?* D i ^ lim-rr~^- ) (15) where Since by hypothesis S a9tOD <5(a 5 /) = 2, by using Lemma 5 it can be easily seen that @(oo,/) = and A(oo,/) = 0. Hence, it follows from (14) and (15) that B^T Q and bl-d(ao,Q))^r. Thus i.e. B = b = Y n . On deficiencies of differential polynomials 139 Hence we have T Q T(r,/) and <5(oo, Q) = 0. Therefore, by Lemma 4 we have (2-^(0,Q')) Y 6( a ,Q)^I&l .e. Finally by Lemma 2, we have ft n . Consequently on making <7 -> oo we obtain 2,4 . 1 Q 5(0,Q; Hence the result. References [1] Doeringer W, Exceptional values of differential polynomials, Pacific J. Math. 98 (1982) 55-62 [2] Hayman W K, Meromorphic functions (Oxford: Clarendon Press) (1964) [3] Hong Xun Yi, On the theorem of Tumura-Clunie, Kodai Math. J. 12 (1989) 49-55 [4] Wang Xiao-Mao and Dai Chong Ji, On meromorphic functions with maximum defect, Bull. Cal. Math. Soc. 80 (1988) 373-376 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 2, August 1992, pp. 141-146. Printed in India. A note on the identity operators of fractional calculus R K RAINA Department of Mathematics, College of Technology and Agricultural Engineering, Udaipur313001, India MS received 1 February 1990; revised 10 June 1992 Abstract. The application of an identity operator for Saigo's fractional calculus operators is shown by evaluating the limit of an indeterminate form. Its special case yields the result which has been used as an infinitesimal generator in the semigroup theory. Also, an identity operator for the recently introduced multi-dimensional fractional operators (due to Srivastava and Raina [8]) is discussed. Keywords. Fractional calculus operators; identity operator; Riemann-Liouville operators; Gauss hypergeometric function I. Introduction and main result In the theory of fractional calculus, a fractional integral operator Jf '* due to Saigo [6] (see also [9] and [10]) is defined by where a > 0, ft and r\ are real, f(x) is any real continuous function defined on (a, oo), with the order that where e Here F is the Gauss hypergeometric function defined by ([7]) in \z\ < 1 (and its analytic continuation into iarg(l z)\ < n\ with (a\ denoting the usual Pochhammer symbol (a) r = T(a + r)/Y(d). As stated in [6], there exists the relationship j 0> V(x)-/(x) (3) which shows that I^ '* serves as an identity operator. A logistically permissive proof of (3) can be given by following Ross [4, pp. 13-15], and we skip here these details. 141 142 R K Raina We examine the following limit of an indeterminate form involving the operator (1): A r (*-g) 2 'JffV(x) -/(*)' A = lim - - a ~ ' _o+a|_r(a) J V x-a (4) where /(x) is continuous and differentiate on (a, x) and a > 0. In view of (3), (4) takes the indeterminate form (0/0) when a = 0. Integration by parts leads (4) to 4 - 1 1 (x - a)T(l - a (5) by virtue of the summation formula [7, p. 28, (1.7.6)]. Clearly (5) is again of the indeterminate form (0/0) when a = 0, and by L' Hospital's rule, we find that o+ aF(l + a) + F(l + a) L T(l - a)F(l + a + ly) x ty(l + a) - \l/(l - a + 1;) + ^(1 - a) - \l/(l 4- a + //) + log(x - a)} (6) UU. J where P=\ (x - t)"F 2o, - if; 1 + a; /' (t) dt, (7) Ja \ x-a/ and \l/(z) = r ; (z)/T(z), is the psi-function. By differentiating under the integral sign with Leibnitz rule, we get ,->7;l+a; -r)-f \l/(l 4- a) - trf'(t)\ Z ~~r^ L r > i 1 + a )r r! a + r) 2^(2a)} dt. (8) On the identity operators ,*-> Evidently when a^0+, the last term in (8) above vanishes, and we get .. dP C* Sfe-J. 10 **-').*- (9) Finally, in view of (4), (6) and (9) give - " J J - 2^(1 + iy) + log(jc - a)} -f log(x-t)f f (t)dt. (10) A special case of (10) when a = r\ = 0, is the result obtained recently by Ross and Sachdeva [5, p. 205]. This special case of eq. (10) giving the limit of the difference quotient was described as an infinitesimal generator in semi-group theory in [1], and obtained in an L p setting by means of functional analysis. 2. Multidimensional operator In a recent paper, Srivastava and Raina [8] have introduced a multidimensional extension of the familiar differential operator c D q z (see [4] for its definition). This extension is defined by ([8, p. 357, (1.6)]; see also [2] and [3]) where m t > Re(^), m t eN = {0,1,2,...}, for i = l,...,n; and for convenience c = (c 19 ..., c n ). It must be pointed out here that different subscripts on D in (11) require different classes of functions to have a convergent integral. The multidimensional extensions of the Riemann-Liouville and Weyl operators of the fractional calculus are defined in [8]. If the function f(x l , . . . , x n ) is continuous on each of the interval (c,, x t ) for i = 1, . . . , n; then where m,eAT (i =!,...,). To prove the assertion (12), we write (11) as 144 R K Raina f/V __fV. <?i-l ) ami + .-.+m,, [ Xl fx n rfa a) dti Kx- a*- - ^'->*) ( il/Wi <?iJ J 0*1 ..0x n j Ci j CB _ f Vi-i-l *} / *,. (13) Introducing points d t > (z = 1, . . . , n) in the interval of integrations then / t can be expressed as the sum of two rc-tuple integrals d H n ( Y _ "TT J l_i__lii Ml d ' = A! + A 2 (suppose). (14) For the multiple integral A 2 , let M = max then \f(t 1 , . . . , tj /(xj , . . . , x n )|< M, where M depends on x f and <5 f , and for fixed x f , M~>-0 as <5i->0, for all i= l,...,n (because / is continuous). We now have Similarly for the first multiple integral A A in (14), let m = max \f(t 1 ,...,*) /(x x , . . . , x n ) |, c,<sr,<x,-, then US* I . . . fA M i = as qi-^m h for fixed <5 f (z' = 1, . . . , n). The value of J 2 in (3) is easily given by ' (x -i Adding On the identity operators to both the sides of (13) and using (14) and (11), we find that 3mi + . i , , X.) - 145 m + .-. + mr, r n C /^.. __ . ^.V"*""^ "} (18) If fi is an arbitrary positive number, then choosing each <5 f (f = l,...,n) such that | A 2 | < e, we have (in view of (16) and (18)) SU P (19) By letting s->0 (since e is arbitrary), the assertion (12) follows. For m = (i = 1, . . . , n), (12) gives the identity operator (20) for the multidimensional fractional operators. A direct proof of (12) can be given by first integrating by parts the multiple integral which defines (11), and then resorting to the process of limits when each #i-+m f (i = !,...,). We leave to the interested reader further details concerning this alternative proof of the assertion (12). Acknowledgement The author is thankful to the referee for pointing out a few additional references relevant to Saigo's operator. References [1] Hille E and Phillips R S, Functional analysis and semigroups, Am. Math. Soc., Colloq. Publ. Volume 31 (Revised edition), (1957) 663-678 [2] Raina R K, A note on the multidimensional Weyl fractional operator, Proc. Indian Acad. Sci. (Math. Sci.) 101 (1991) 179-181 [3] Raina R K, On the multidimensional fractional differintegrals of Riemann-Liouville and Weyl types Boll. Un. Mat. Ital. (7) 6-A (1992) (to appear) [4] Ross B, A brief history and exposition of the fundamental theory of fractional calculus, in Fractional calculus and its applications, (ed.) B Ross, pp. 1-36, 1975 (Berlin, Heidelberg, New York: Springer- Verlag) [5] Ross B and Sachdeva B K, The continuity property D/(x) = /(x), its proof, usefulness and application, in Fractional calculus and its applications, pp. 201-206, Int. Con/. Proc., College of Engineering, Nihon Univ., 1990 [6] Saigo M, A certain boundary value problem for the Euler-Darboux equation, Math. Jpn. 24 (1979) 377-385 [7] Slater L J, Generalized Hyper geometric Functions, (Cambridge: University Press) (1966) 146 R K Raina [8] Srivastava H M and Raina R K, The multidimensional Holmgren-Riesz transformation and fractional differintegrals of Riemann-Liouville and Weyl types, in Univalent Functions, Fractional Calculus, and Their Applications, (eds) H M Srivastava and S Owa pp. 355-370, (New York: John Wiley) (1989) [9] Srivastava H M and Saigo M, Multiplication of fractional calculus operators and boundary value problems involving the Euler-Darboux equation, J. Math. Anal Appl 121 (1987) 325-369 [10] Srivastava H M, Saigo M and Owa S, A class of distortion theorems involving certain operators of fractional calculus, J. Math. Anal. Appl 131 (1988) 412-420 s^ 1 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 2, August 1992, pp. 147-153. Printed in India. An RSA based public-key cryptosystem for secure communication V CH VENKAIAH Central Research Laboratory, Bharat Electronics Limited, 25, M.G. Road, Bangalore 560001, India MS received 20 March 1991; revised 10 May 1992 Abstract. A new cryptosystem that uses modulo arithmetic operations is proposed. It is based on Rivest-Shamir-Adleman's public key cryptosystem. A feature of the proposed system is that the encryption and decryption procedures are computationally less intensive, and hence the system is amenable for high data bit rate communications. Keywords. Cryptanalysis; high data bit rate communication; encryption; decryption. 1. Introduction In his paper, Shannon [6] indicated the need to develop new cryptosystems as they would improve the level of security of the existing cryptosystems. In this article, we propose a cryptosystem [8] that uses Rivest-Shamir-Adleman's system (RSA) to bootstrap into the new encryption scheme designed here. It is known that the much celebrated RSA public-key cryptosystem [4] cannot be used in high data bit rate communication systems as the encryption and decryption procedures of RSA are computationally involved. Even the less known ElGamaFs public-key cryptosystem [1] is computationally prohibitive for high data bit rate situations. Also, the size of the ciphertext of this system is double the size of the message. The encryption scheme designed here is computationally less intensive and hence can be used in high data bit rate communication systems. The proposed system is explained in 2 and 3 discusses some issues in system design. An example to illustrate the system is given in 4 and analysis of the attacks is considered in 5. Finally, an algorithm to compute the multiplicative inverse of an integer is explained in Appendix. 2. Proposed system In this system, the sender chooses the set of integers p t and g j9 1 ^ i ^ 3, 1 ^j ^ 6, that satisfy the conditions given in (1), uses the encryption procedure of RSA with the public-key, given in the public directory or obtained from the receiver to encrypt the chosen set of integers and obtains the corresponding ciphertext. This ciphertext is then transmitted to the intended receiver. The receiver applies the decryption scheme of RSA with the secret key known only to him, to the received ciphertext and obtains 147 148 V Ch Venkaiah the set of integers p t and g jf After transferring the chosen set of integers, the sender then uses (3) that involves certain modulo arithmetic operations on the chosen set of integers and the message blocks m l9 i ^ 1, obtains the corresponding ciphertext c 9 + i , and transmits the ciphertext c 9 + i to the receiver. By performing the reverse modulo arithmetic operations specified in (4), the receiver recovers the message blocks. Formally, the system is as follows. Public Key N : integer that has large prime factors P and Q, e : multiplicative inverse of d modulo $(#), Euler's totient function of N Secret Key d : integer less than (/>(N) and is relatively prime to </>(N), P and Q : prime factors of JV, (j)(N) : Euler's totient function of N and is equal to (P 1) *(Q 1) Encryption Choose the set of integers p , 1 ^ i ^ 3, (not necessarily be prime) and g j9 1 <j < 6, such that they satisfy the following requirements. Pi < PJ whenever i <j, gcd(p i ,g i )=l for all i. (1) Use RSA encryption procedure with the public-key e and N, to encrypt the chosen set of integers and obtain the corresponding ciphertext c h i.e. compute c t = pfmodN for all i c 3 + j = g e j mod N for all j (2) where p f and g j are assumed to be less than N. Otherwise, each integer should be split into a set of small integers which are less than N. Transmit the ciphertext to the intended receiver. Partition the message into blocks so that the numeric value m t of each block is less than p 1 . Now encrypt each message block w f , using the relation c 9+i = (((m t + g 4 )g 1 modp 1 + g 5 )g 2 modp 2 + 6 ) 3 modp 3 , (3) obtain the ciphertext c 9+i , and transmit c 9+i . Decryption Receive c i9 i< 10, and compute, using RSA decryption procedure with secret key d, ctmodN^pfmodN^pi for all z = l,2,3, c d 3 +j mod N = gfmod N = g } for all j = 1, . . . , 6 to obtain the set of integers p { and g jf A new cryptosystem 149 Use the following relation 2 ~ 1 (P 2 ) (4) to recover the message blocks m t from the ciphertext blocks c 9+i . gr l (pt) is the mutliplicative inverse of g t with respect to p f . 3. Issues in system design For the choice of prime numbers P and g, and the secret key d in RSA, refer the book by Salomaa [5]. d should be large and relatively prime to (f>(N). If d is small, direct search with reasonable amount of computation will reveal the key. To check whether the chosen integer d is relatively prime to $(N) or not, generate (rj as explained in the appendix by taking a = 4>(N) and b = d.lf {r n } = 1, then d is relatively prime to <p(N\ else not. Since d and &(N) are relatively prime, the multiplicative inverse e of d modulo (f)(N) exists, e can be computed using Euclid's algorithm given in the appendix. The sender can choose the integers p t and #,- randomly. These integers should be from the set of large integers. Otherwise, the frequency of the plaintext characters may be preserved in the ciphertext. Also, direct search may reveal the key (integers Pi and gj) of the message encryption scheme. Coming to the computational aspect of encryption and decryption procedures, generation of ciphertext c i9 i^ 10, in the encryption requires three multiplications, three additions, and three divisions. To compute c , 1 ^ i < 9, exponentiation by repeated squaring and multiplication [4] may be adopted. This procedure requires atmost 2 Iog 2 e multiplications, 2 Iog 2 e divisions. For more efficient procedures, see [3]. Exponentiation by repeated squaring and multiplication method can also be employed in decryption to obtain p and g jf Note that, since d is known only to the receiver, a cryptanalyst cannot use this procedure. To recover the message from c , i^ 10, the receiver should first compute g^ l (Pi) for each i= 1,2,3, say, by Euclid's algorithm which will terminate in atmost log/p iterations [7], where / = (1 -f *j5)/2 is the golden ratio. Each iteration requires one division with remainder operation. Before the start of communication, the sender may compute gr i (Pi) and transmit g{ l (pi) instead of # . The receiver can then recover the message in just three multiplications, three subtractions and three divisions. If need arises, the system may be modified to amortize the number of arithmetic operations, by dropping # 4 , # 5 , # 6 , without sacrificing the level of security of the system. The proposed system is explained in the following example. 4. Example Let m l = 301, m 2 = 250 and m 3 = 276 be three blocks of message to be transmitted. Public key 150 V Ch Venkaiah Secret Key P = 17 and Q = 23 (AT) = 352 Encryption Choose Pl = 317, p 2 = 323, p 3 = 371 and g, = 41, 2 = 47, 3 = 1 1, g 4 = 10, # 5 = 13, g 6 = 8. Note that p f < p 7 - whenever i <;' and gcd(p,.,0 ) = 1 for all i = 1,2, 3. Use RSA with public-key e = 3 and N = 391 to encrypt p l = 317 and obtain the ciphertext Similarly, encrypt p 2 >P3>0i>--->#6 and obtain the ciphertext c 2 = 323, c 3 = 211, c 4 = 205, c 5 = 208, c 6 = 158, c 7 = 218, c 8 = 242, c 9 = 121. II Transmit the ciphertext c^, l</<9, to the intended receiver. This is one time transmission and the integers p and g j become the key for the message encryption scheme. Encrypt the first message block m l = 301 and obtain the ciphertext c 10 =(((m 1 +# 4 )0 1 modp 1 + 5 )# 2 modp 2 4-# 6 )# 3 modp 3 = (((301 + 10)41 mod 317 + 13)47 mod 323 + 8) 1 1 mod 371 = 138. Repeat this for the remaining message blocks m 2 = 250, m 3 = 276 and obtain the ciphertext c n = 134, c 12 = 301. Decryption Use RSA with secret key d = 235 to decrypt c l = 243 and obtain Similarly, decrypt C 2 ,c 3 ,...,c 9 and-obtain p 2 ,p 3 ,^i,...,/ 6 - Receive the ciphertext c 10 and recover the message m-^ by computing = (((((138*135~8)mod371)55-13)mod323)116~10)mod317 = 301, ' where ^p(Pi) = 4rH317) = 116, ^Mp 2 ) = 47-M323)== 55, ^ 1 (/> 3 )=11 135. Similarly, compute the messages m 2 = 250 and ra 3 = 276 from the ciphertexts c l 1 and c 12 respectively. 5. Cryptanalysis of the proposed system The proposed system is analyzed against possible cryptanalytic approaches and found that the system is secure. A new cryptosystem 151 It is easy to see that computing p t and g } from c h 1 =^ i ^ 9, is equivalent to breaking RSA. Hence it is assumed that p- t and g j cannot be derived from c t , 1 < z < 9. Therefore, security of the proposed system is equivalent to security of the proposed encryption scheme. Ciphertext only attack Upon receiving the ciphertext, an intruder may be able to partition the ciphertext into blocks that reveal the size of p 3 and hence p 3 . Also, an intruder, though almost impossible, may find maximum number of plaintext characters corresponding to blocks of ciphertext; knowing the size of p t and hence P!. To prevent this leakage of information on the size of p l9 partition the message so that each block m t is much smaller than p l . But p 2 and g j9 1 ^j < 6, can be obtained only by exhaustive search, which is computationally not feasible. Known and chosen plaintext attacks To analyze the system under known plaintext attack and chosen plaintext attack, write c h i > 10, in terms of the message and the key, i.e. write C 9 + i = ^010203 +01020304 + 029335 + 0306 ~ where fc^m,-) are the quotients in the mod operation with p jf This may conveniently be written as where fl &K')= 01020304 + 020305 + 0306 ~ Let J be a subset of positive integers. In the known plaintext attack, cryptanalyst is assumed to have knowledge of (m , c 9+i ) for all ieJ, but not a and b(m t ) even when p 3 is assumed to be known. He then wants to solve for m /5 i$ J. Since there are | J| -f 1, |J| is the cardinality of J, equations in |J| -f- 3 variables, the system is under determined and has large number of solutions. Hence the system is secure against the known plaintext attack. Similarly, the system can be proved to be secure against the chosen plaintext attack. The scheme also seems to be secure against the threat of solving for an equivalent key. Acknowledgement The author wishes to acknowledge Mr M Sethuraman, of Central Research Laboratory, Bharat Electronics Limited, for suggestions in cryptanalysis. 152 V Ch Venkaiah Appendix (Multiplicative inverse computation) Let a > b> be two integers. The following procedure (Euclid's algorithm) computes the greatest common divisor of a and b, gcd(a, b). If gcd(a, b) = I, a and b are relatively prime, then the procedure also computes the multiplicative inverse of b modulo a. By division property r 9 0< r l <b, 7 0<r 3 <r 2 ;* where, r n ^ (but r n + 1 = 0). The greatest common divisor of a and b is then r n . Now to compute the multiplicative inverse of b modulo a, define If r n = 1, the multiplicative inverse of b modulo a is then the positive residue of y n modulo a. If r n > 1, then gcd(a, b) = r n >\ implying a and b are not relatively prime to each other and hence the multiplicative inverse of b modulo a does not exist. {r }, {y t } can be computed as in the following table [2]: 42 a b 1 3^2 y n As an example, let a be 352 and b be 235. Then the computation of the multiplicative inverse of b modulo a is 7 > 352 235 1 1 2 117 1 1 3 117 -352 A new cryptosystem 153 Since r 2 = 1,235 ~ 1 (352) = 3. As a check, consider 235* 3 mod 352 = 705 mod 352 = 1. References [1] ElGamal T, A public key cryptosystem and a signature scheme based on discrete logarithms, IEEE Trans. Inform. Theory, IT-31 (1985) 469-472 [2] Gregory R T and Krishnamurthy E V, Methods and applications of error-free computation, (New York: Springer- Verlag), (1984) [3] Knuth D E, The art of computer programming, 2: Seminumerical algorithms (Reading Mass: Addison- Wesley) (1969) [4] Rivest R L, Shamir A and Adleman L, A method for obtaining digital signatures and public-key cryptosystems, Comm. ACM, 21 (1978) 120-126 [5] Salomaa A, Public - Key Cryptography, EATCS monographs on Theoretical Computer Science, Vol. 23 (Berlin, Heidelberg: Springer- Verlag) (1990) [6] Shannon C E, Communication theory of secrecy systems, Bell Syst. Tech. J. 28 (1949) 656-715 [7] Van Tilberg H C A, An introduction to cryptology (Hingham Mass: Kluwer Academic) (1988) [8] Venkaiah V Ch, Is this a secure cryptosystem?, Labtalk 1 (1991) (Labtalk is an house journal of Central Research Laboratory, Bharat Electronics, Bangalore, India) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 2, August 1,992, pp. 155-158. Printed in India. A note to the paper "An efficient algorithm for linear programming" of V Ch Venkaiah JOACHIM KASCHEL Technische Universitat Chemnitz, Fakultat fur Wirtschafts- und Rechtswissenschaften, Reichenhainer Str. 39, D O-9022 Chemnitz, Germany MS received 26 June 1991; revised 20 April 1992 Abstract. In Venkaiah [1] an algorithm for solving linear optimization problems based on the idea of the projective algorithm of Karmarkar, is proposed. The essential simplification in the new algorithm is the use of a fixed projection operator. In this way the algorithm requires only 0(n 2 ) operations to obtain a sufficient exact solution. In this note it is shown that in some special cases the algorithm of Venkaiah yields a feasible solution that is far from the optimal one. 1. The algorithm of Venkaiah Let us consider the linear optimization problem, which is denoted by (P2) in [1]: CX^min AX = b X^Q (P2) with C, XeR n , beR m and A as a [m, n] -matrix. Let be the generalized inverse to A. The algorithm proposed by Venkaiah [1] is the following: Algorithm A2 Step 1: Compute an initial feasible solution X > 0. Step 2: Compute the projection operator P = / A + A. Step 3: Compute C P = PC and 7= C P /\\C P \\. Step 4: Set k = 0. Step 5: Compute The problem (P2) is unbounded if all Y| < and at least one Y t < 0. If all YI = then X k is a solution. 155 156 Joachim Kdschel Step 6: Compute X k + 1 = X k - el Y where < e < 1 . Step 7: If where L is defined by Karmarkar, then stop. Step 8: Set k = k 4- 1, D k = diag(<4) with yk Step 9: Compute go to step 5. 2. The convergence of the algorithm In [1] in theorem 1 it is formulated that this algorithm A2 converges to an (optimal) solution of problem (P2). The proof of this theorem contains the following steps: (i) V ^ for every k. (ii) AX k = b for every k. (iii) CX l = CX-BJi\\C p \\<CX and CX k + 1 = CX k -Bl(C p D k C p /\\PD k C p \\< CAT* for every /c=l,2,... Since in [1] (P2) is assumed to have a bounded solution it follows that CX k is bounded below. Hence OX k converges: In [1] without any proof it is assumed that this value C is the optimal value C* of objective function in (P2). But C must not be equal to the optimal value C*. The following example shows that the case C > C* is possible. Y- 3. Counter example I Consider the following problem: f 2x l x 2 + x 3 =4 The optimal solution is X* = (4; 4; 0; 0)' with the optimal value C* = - 8. Now we use the proposed algorithm A2. p-1 10 An efficient algorithm for linear programming 157 .Step 1: * = (0-1; 2-0; 5-8; 0-1)' Step 2: We compute / 3 2 -4 - 2 3-1-4 -4 -1 7 -2 % -1 -4 -2 . 7, Step 3: C p = PC = ^(-5; - 5; 5; 5)' = (- 1; - 1; 1; 1)' Step 4: k = Step 5: A = min{ 5-8/0-5; 0-1/0-5} = 0-2 Step 6: With e = 0-5 we obtain X l = (0-1; 2-0; 5-8; 0-l)'-0-5*0-2*0-5*(- 1; -1; 1; 1)' = (0-1 5; 2-05; 5-75; 0-05)' Step 7: Step 8: D! = diag(0-27386; 3-74277; 6-87256; 0-05976) Step 9: r = (- 0-53832; -0-28356; 0-79308; 0-02879)' Step 5: A = 1-7363 Step 6: JT 2 = (0-61734; 2-29617; 5-06149; 0-025)' Step 7: Step 8: Z> 2 = diag(H2711; 4-19221; 6-04964; 0-02988) Step 9: 7 = (-0-54434; -0-31758; 0-77109; 0-09087)' Step 5: A = 0-275 Step 6: ^T 3 = (0-69226; 2-33988; 4-95537; 0-0125)' Step 6: X 4 = (0-72633; 2-36004; 4-90738; 0-00625)' Step 6: * 5 =(0-74270; 2-36979; 4-88438; 0-003125)' Step 6: X 6 = (0-75074; 2-37459; 4-87310; 0-00156)' Step 6: X = (0-75856; 2-37927; 4-86214; 0-00002) r . We see that the algorithm converges, but the solution is far from the optimal. With another value < a < 1 we have the same effect. If we start the algorithm with another point * = (2;2;2;2)' after some iterations the algorithm yields a solution X = (3-99988; 3-99988; 0-00012; 0-00012)' which is near to the optimal solution. 4. Conclusion The algorithm A2 converges to a feasible solution. The quality of this solution depends on the starting point. Other conditions as described in [1] are necessary to guarantee 158 Joachim Kdschel the convergence to an optimal solution. Then the algorithm will, however, require more than 0(n 2 ) operations. Reference [1] Venkaiah V Ch, An efficient algorithm for linear programming Proc. Indian Acad. Sci. -Math. Sci. 100 (1990) 295-301 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 3, December 1992, pp. 159-173. Printed in India. Stochastic dilation of minimal quantum dynamical semigroup A MOHARI and K B SINHA Indian Statistical Institute, 7 SJ.S. Sansanwal Marg, New Delhi 110016, India MS received 23 February 1992; revised 19 August 1992 Abstract. A necessary and sufficient condition is formulated for minimal quantum dynamical semigroups to be conservative. The paper also provides a Markovian dilation of the minimal semigroups, as a contractive solution of an associated quantum stochastic differential equation in Boson-Fock space, which is isometric if and only if the minimal semigroup is conservative. Using the reflection principle of Brownian motion a necessary and sufficient condition for the contractive solution to be co-isometric is also obtained. Keywords. Quantum dynamical semigroup; Markovian cocycle; quantum stochastic differential equation. 1. Introduction Feller [8] proved the existence of a unique minimal semigroup P r , t ^ on / x associated with the Fokker- Planck equation: j*. t > 0,p tt (0) = 6 ik (1) subject to the Markov condition: and 0.^= -Q.^ oo. (2) Exploiting the special nature of J t , Kato [14] constructed the minimal semigroup in the framework of semigroup theory. It was also shown in [8, 14] that the minimal semigroup is conservative i.e. !|P t )>|li = l|j>lli for all yel* if and only if k = tej} = {>}> forsomeA>0. (3) In this paper we consider the quantum mechanical Fokker-Planck equation the Banach space of trace class operators in j^: in P(O)-P, p(t)'=yp(t)+p(t)y*+lzp(oz* (4> fceS subject to = 0, (5) fceS 159 160 A Mohari and K B Sinha where 7, Z k ,keSaI. + are densely defined operators in J^ and pe^ h , the real Banach space of self-adjoint elements in ?T. Davies [4], following essentially Kato's method, constructed the minimal dynamical semigroup of in (t ^ 0) in <y h as a solution to (4)-(5). In this context, we formulate a condition similar to (3) as the necessary and sufficient one for the preservation of trace under the action of <7 in . We also provide a Markovian dilation of <r min in the sense of Accardi [1,2] as a contractive solution of an associated Hudson- Parthasarathy equation [11, 12, 19]. The solution is isometric if and only if <r min is trace preserving. Finally using Journe's reflection principle [13, 17] we also obtain a necessary and sufficient condition for the contractive solution to be co-isometric. Some results on the related dilation problem may be found in Chebotarev [3] and Fagnola [6, 7]. The method employed here is different from that in [3,7]. The paper is organized as follows: In 2 we describe the framework of quantum stochastic calculus and a class of contractive cocycles satisfying quantum stochastic differential equation (qsde) with bounded coefficients and also recall [11, 17, 20] the necessary and sufficient condition for the solution to be isometric, co-isometric or unitary. Section 3 is devoted to exactly the same "questions as in 2, this time with unbounded coefficients subject to some conditions. Many of the results in this section are quoted without proof since they are published elsewhere [19]. In 4 we consider the problem mentioned at the beginning. 2. Contractive bar-cocycles All the Hilbert spaces that appear here are assumed to be complex and separable with inner product <, > linear in the second variable. For any Hilbert space H, we denote by r(H) the symmetric Fock space over H and B(H) the C* algebra of all bounded linear operators in H. For any ueH, we denote by e(u) the exponential vector in T(H) associated with u. The family {e(u):ueJt} is total for any dense linear manifold Jt in H and linearly independent in T(H). We fix two Hilbert spaces 34f Q and k and write It is clear that for any pair of linear manifolds 3f and Jt dense in #e and L 2 (IR+ , k) respectively, the algebraic tensor product ^ e(^) is dense in j/P 9 where t(Jf) is the linear manifold generated by the vectors e(u):ueJf. We also denote the vacuum conditional expectation on ^f by E . For the basic notions in boson stochastic calculus such as adapted, regular, bounded, contractive, isometric, co-isometric and unitary process, we refer to [1 1, 21]. The notion of Markovian cocycle was first introduced in [1]. However in this paper we follow the definition introduced in [13] and call it bar-cocycle to avoid confusion. We fix an orthonormal basis {e t :ieS} in k and set F. = I^X^IiiJeS. With respect Jo this basis we ^define the basic quantum stochastic processes {Aj:f,7S: = Su{0}} as in [18, 23]. Then quantum Ito's formula [11] can be expressed as: dAjdAj^dA* (6) Quantum dynamical semigroup 161 for all iJ,k,leS where if/ <5j, otherwise. We denote by u j (s) = <> j5 w(s)>, Uj(s) = u j (s) for jeS and u (s) = w(s) = 1. Choose Jf = {ueH:u j (") = Q for all but finitely manyyeS} and set N(u) = {j;w j '(-)^0}. So #N(u) < oo for ueJ?. We also denote by 2f R the class of elements L=(L(e&(34f \ijS) such that for each jeS there exists a non-negative constant c ; - (depending on L) satisfying ieS for all fe^f . For any Le^ R define the family of bounded linear operators {J2?j., zj eS} on 34? o by where the necessary convergence follows from (7). Fix Le& R . Then there exists a unique adapted process X = {X(t\ t ^ 0} satisfying the following qsde: AX = LJ.dAf (t)X(t\ X(Q) = / (8) on ^ e(^). Moreover X is isometric whenever Le/ R , where ^ R = jSf j = 0, for all iJeS}. For a complete account of these facts the reader is referred to [11,16,17,18,20,21]. Observe that for all i, jeS, J*?j. = (JS?/)* and jSf 5 , = ((^fj)) jtes , is a self adjoint operator on the Hilbert space Jf ^(^) f r an Y fi n i te subset S ; of S. We set &' = {L, ^f s , ^ 0, for all S' c S,#S' < oo}. Hence S R c &~ d& R . The following proposition gives a necessary and sufficient condition for X to be contractive. PROPOSITION 2.1. Fix LG& R . Consider the family X = (X(t), *$ t < 00} of operators satisfying (8). The following statements are valid: (i) X has a contractive extension if and only if LE$? R ', (ii) X has an isometric extension if and only if Le</ R . Proof. By (6) and (8) we have <X(t)fe(u),X(t)ge(v)y - <fe(u) 9 ge(v 1 . . \ , X(r)fe(u) 9 E UittWr) ^]X(i)ge(v) ) dr, ^ t (9) -f Jo 162 A Mohari and K B Sinha for all /,#eJf , w, vsJl. For finitely many vectors / a e^ a "e<Jf let ^: = I a / a e( a u)' Ue^tOir 1 . It is convenient to introduce the Hilbert space H = a H a with H a = yewo ^, the vector in H:(t) = ffi a ^(t) with ^(0 = ; - 6]V(u) ^(O^W/e( W > || e(w) || ~ \ and the bounded operator I in H: L* = 'JS? for all a, /?. Then from (9) we have (10) dt Also observe that < is negative semi-definite if and only if i_ is negative semi-definite. Hence from (10) it is clear that the map t-+\\X(t)\l/\\,t^Q is decreasing whenever Le^f ~ . This completes the proof of the sufficiency part of (i). Conversely, let X be contractive s.o that (d/dt)\\X(t)\//\\f tssl Q^O. Fix any finite set of vectors # a eJf > ae ^> where S' c S,#S' < oo. Taking continuous functions a we^ so that V(0) = 6* and - Z in (10) we have a,/!eS' Hence Le^"^ . This completes the proof of (i). The proof of (ii) is very similar to that For any L = (Lj.:iJeS) with L l . densely defined closed operators in 3tf Q we define Is{Lj:UeS}by and set ^ = (L, Le^}, SI = {L s Ze^; } and S R = {L, As a consequence of Proposition 2.1 and 'time reversal principle' [17], we have the following theorem, which we state without proof. The proof can be found in [17, 20]. Theorem 2.2. Suppose Ze^ R r\^ R . Then there exists a unique regular adapted process V= {V(t\Q^ t < 00} satisfying I (11) UeS ^o e(^). Moreover the following hold: (i) 7/ze following statements are equivalent: (a) V has a contractive extension', - In such a case V is a strongly continuous bar-cocycle. (ii) V has an isometric extension if and only if Quantum dynamical semigroup 163 (iii) V has a co-isometric extension if and only if (iv) V has a unitary extension if and only if ZeJ> R r\JP R . 3. A class of qsde with unbounded coefficients In this section we recall some results from [19] which will enable us to deal with more general quantum evolutions satisfying (1 1) on 2 &(Jt\ where 2 is a common dense domain of the family Z == {Z(, iJeS} of operators in the initial Hilbert space J^ . We denote by 2C"(QI) the class of elements Zs{Zj,i,jeS} such that Z is the generator of a strongly continuous contractive semigroup with 2 as a core and assume furthermore that j); (iJeS); (12) (b) there exists a sequence Z(n)e2 R r\&~,n^ 1 so that for all /e, i,jfeS s =limZ;(n)/-Z;./. (13) n-*oo Let Z&~(@). From Lemma 3.1 in [19] we observe that for each feSJeS there exists a constant Cj(f) > such that <c//) (14) n>l ief and IJZJ/II 2 <<://). (15) feS For any Xe^(^f ) we define the bilinear forms jS?j.pQ(ijeS) on ^ by fceS where the necessary convergence follows from (15) and Cauchy-Schwarz inequality. We set for A > and denote by / the class of elements Ze3f~(3i) such that JSf(/) = foraUiJeS. Fix a sequence Z^eJT'nJ'" satisfying (12) and (13). We denote by F (n) = (V (n \t)\t ^ 0} the unique regular (3f ,^) adapted contractive process satisfying (11) with Z(n) as its coefficients. We state the following propositions without proof, referring to [19, 20] for the proofs. PROPOSITION 3.1. Let Ze&-(@) and V (n} be as above. Then (i) w - lim^ V (n) (t) = V(t) exists for all t ^ 0, 164 A Mohari and K B Sinha (ii) V= [V(t):t^ 0} is the unique strongly continuous contractive barcocycle satisfying (iii) V is isometric only if Ze,/; (iv) if ZeJ^ and /? A = {0} for some X > then V is isometric. Remark 3.2. Suppose for each n^ 1, V (n) is a regular contractive (^f , ^-adapted process satisfying (11) on @s(J) where Z(n) are densely defined operators on 2. Then Proposition 3.1 holds as well for the associated sequence V (n) provided (12)-(14) are valid for Z. We omit the proof since it follows by the method employed for the proof of Proposition 3.3 in [19]. Let ZeT~(JO, for some a dense linear manifold J> in Jf . We denote by 3 and /? A the classes / and /? A respectively, with Z replaced by Z. COROLLARY 3.3. Consider the contractive cocycle V defined as in Proposition 3.1. Let in addition Z e&-(&). Then the following hold: (i) if V is co-isometric then Ze./; (ii) if ZeJ> and fl^ = {0} for some X > then V is co-isometric. 4. Minimal quantum dynamical semigroup and its dilation We consider the quantum mechanical Fokker-Planck equation written formally as p(0) = p, p(t)' = Yp(t) + p(t) 7* + Z Z k p(t)Z* (16) keS subject to y -|_ y* _j_ Y 2* Z ^ (17) fceS for pe^,, where Y,Z fc , fceS are densely defined operators in J^ and ^ is the real Banach space of all self-adjoint trace class operators in jf? Q . When Y is a bounded operator, (17) implies that {Z k , keS} is a family of bounded operators and the series kgS Z*Z fc converges in strong operator topology. In such a case, for each p (16) admits a unique ^-valued solution p(t\ t^O and the map p-+a t (p) = p(t\ t^O is a one parameter contraction semigroup in the Banach space (F^ ||-|| tr ). On the other hand by Theorem 2.2 (i) there exists a unique regular (^f , ^-adapted contractive operator valued process 7= {V(t) 9 t^Q} satisfying / (18) keS on ^f (^) where ^ Z,, Quantum dynamical semigroup 165 and S = ((Sj)) is a contractive operator in ^f / 2 (S). The contractive one parameter semigroup r f := E [F(t)*(x(g)/) 7(t)], t^O of completely positive maps [1] and tr t , t ^ satisfy the relation whenever t^O, Here our aim is to deal with the dilation problem associated with the Fokker-Planck equations (16)-(17) when the operators Y, Z k , keS are not necessarily bounded operators. DEFINITION 4.1. [9, 15] A one parameter family of completely positive maps T = {T,, t > 0} on is said to be a quantum dynamical semigroup if the following hold: (ii) ||T,KU>0; (iii) The map t->tr(pr t (x)) is continuous for any fixed xe^(c?f ) and pe^, the trace class operators in 2? Q . (iv) For each t^O the map x-+r t (x) is continuous in the ultra- weak operator topology. Given a dynamical semigroup T we define the predual semigroup <r = {a t ,t^0} on ^"as tr(x<j t (p)) = tr(pT,(x)) (20) wherever t ^ 0, peT, xe8(jf ). Note that the family tr is uniquely determined if (20) holds for p: = |/> <g\, f,geJ4? . It is also evident that a is a strongly continuous one parameter semigroup in the Banach space (^", ||'|| tr ). Conversely, for a strongly continuous one parameter semigroup a on ^", (20) determines a unique dynamical semigroup T. Moreover for any t ^ 0, tr<r t (p) = tr(p), pe^ h if and only if t t (/) = /. The central aim of this section is to exploit the theory developed in 3 and the construction of the minimal quantum dynamical semigroup, as outlined in Davies [4], in dilating the minimal semigroup in a boson-Fock space. Before we proceed to the next result we state the following simple but useful lemmas without proof. Lemma 4.2. Let s.lim n ^^A n = A and s.lim n ^ (X) B n = B. Then lim^^AnpB* = ApB* in ll'lltr topology whenever Lemma 4.3. Let A k ,k^l and B k ,k^l be two families of bounded operators such that both the series Z fe>1 ^4*^4 fc and ^L k ^ l B*B k converge in strong operator topology. Then for each pe&" h the series 2^^ B k pA* converges in ||-|| tr norm topology. As in Davies [4], let Y be the generator of a strongly continuous contractive semigroup in J^Q and let Z k9 keS be a family of densely defined operators on <tff such that keS (21) 166 A Mohari and K B Sinha and </, y/> + < y/,/> + 1 <zj,z k f> < o (22) /ceS for all fe9(Y). In view of Lemma 4.2 the following relation defines a strongly continuous, positive, one parameter, contraction semigroup on P^ whose generator G is given formally by G(p)= Yp + pY*. (23) We introduce the positive one-to-one map n on y h defined by As in [4] we set n (P k ) = {n(p) 9 P^^ h } and define the positive linear map /iTrC^H^by ZZ* (24) keS where the convergence follows from (22) and Lemma 4.3. PROPOSITION 4.4. Consider the family Y, Z k , keS of operators satisfying (21) and (22). Then the following hold: (i) n(y h ) is a core for G and (23) is valid for all pe7t(^ h ); (ii) The map / has a positive extension /' on ^(G) such that (25) wherever pe^(G). Moreover equality holds in (25) if and only if equality holds in (22); (iii) For each fixed 1 > 0, /'(A - G)"" x is a map /rom 7t(^" fc ) into P h and has a unique bounded positive extension A x in 3~ h such that \\A^ \\ ^ 1 and /'(p) = A 1 [1 - G](p) for all pe^(G); (iv) For any fixed ^ r < 1, n(3T h ) is a core for the operator W (r) = G + r/ f defined on @(G). Moreover W (r) is the generator of a strongly continuous positive one parameter contraction semigroup <r| r) , whose resolvent at /l>0 is given by l = (A - G)- x X rM*> ( 26 ) t/ze series converges in trace norm; (v) For eac/z p ^ 0, t ^ tfce map r-+a ( ' } (p\ re[0, 1) is increasing and continuous; (vi) T/iere exists a positive one parameter strongly continuous contraction semigroup (j in on 3~ h such that /or a// Quantum dynamical semigroup 167 (vii) For each J,>0, R ( ->(A): = (A- G)~ l ****** A\-* R(X) strongly as N~CO, wfcer* JR(/) = (/ - W) \W is the generator of a n . Proof. For (i)-(vi) see Davies [4]. Now for (vii) we follow Kato [14] (Lemma 7). For each / > 0, < r < 1 we have Letting rjl we get R (n} (X)^R(X). But as R (n] (X) is increasing with n, sJim^K^Ol) = U'(A) exists and K'M < #(4 We also have R<?\X) ^ R M (Z) ^ R'(Z). Hence R r (X) = lim^/?^) ^ R'(l), R(X) = lim^,*,^) < R'W by (vi). This completes the proof. Now our aim is to obtain a necessary and sufficient condition for a to be trace preserving. It is evident that equality in (22) is necessary. We have the following theorem giving sufficient conditions. Theorem 4.5. Consider the semigroup <7 in , t^Q defined as in Proposition 4.4. Let W Q = G + /'with domain n(T h ) and let W* be the adjoint of W . Assume furthermore the equality in (22). Then the following statements are equivalent: (i) tr(v n (p)) = tr(p) for all t ^ 0, pe^,; (ii) for each fixed A > 0, A\ -+0 strongly as n-^coi (iii) for each fixed A > 0, (A - W )(n(^ h )) is dense in 9~ h \ (iv) for each fixed A>0, the characteristic equation W*(x) = Ax has no non-zero solution in 38(3? ); (v) for any fixed /I > 0, fa s {x ^ 0,XG^(^ ):</,x7^> + < r/,x^> + X <Z k f,xZ k gy = A</ 5 ^> fceS (27) toW /or fl// /, 06=0(7)} = {0}. Proo/. The proof is exactly along the lines of Theorem 3 in [14]. We write a = cr min . As in [14] in this context we note that \\RW(p)\\ ir = exp(-^)||(7 r (p)|| tr df (28) Jo for all p ^ 0, which follows from the resolvent formula R(X) = ^ exp( Ar)or f dr, A > 0. As a simple consequence of the following identity / + fR (n) (X) = (U - G)R W (A) + 4J + ' (29) and (22) and (25), we get trG9) = Atr(jR (li) (A)(p)) + tr(^ +1 (p)) for pe<T. Since (n --+ we have (30) 168 A Mohari and K B Sinha for all p ^ 0. Now taking limit as n-> co in (30) we get by Proposition 4.4(vii) lim \\Al +1 m*= \\P\\ - (31) for all p ^ 0, where we have used (28) in the second equality. Since for each fixed pe$~ the map t->\\<r t (p)\\ ir is continuous and |k,(p)|| tr < ||p|| tr , t^O from (31) we conclude that (i) and (ii) are equivalent. Our next aim is to show that (ii) and (iii) are equivalent for any fixed A > 0. From (29) we note that (ii) is equivalent to for all pe&~. Since R w (X)(p)e(G) we conclude that [A- G -/'](0(G)) is dense in 9~. Since n(^ h ) is a core for G, for any fixed pe^(G) we choose a sequence p n such that p w -p and G(pJ-G(p) as n->oo. By Proposition 4.4(iii) we have for all p in 3t(G\ hence /'(p) = lirn^^/^pj. Thus it is evident that (X - G - /')(P) = lim^A - G - /')(pj Hence [A - G - /'](7r(^" h )) is dense in ^. Conversely let (iii) be valid. Since [/- [A-G-/'][7i(^(G)]=D[A-G-/ / ][7r(^)], [/-^](^) is dense in ^. Set C ( ? = (l/n + l)Z 0<Jk<B X*, which is a uniformly bounded by 1. That slirn^C^ = is now an easy consequence of C ( A n) [/ - A^] = (l/n 4- !)[/ - A n * *]. On the other hand, A x being a contractive positive map, || A \\ ^ || A\ || whenever m ^ n, hence n 4- whenever p ^0. Thus we have A n (p)-*Q as n-> oo. This shows that (ii) and (iii) are equivalent. That (iii) and (iv) are equivalent follows by the definition of adjoint of a densely defined operator and Hahn-Banach theorem. Finally we need to show that an element x@(W*) satisfies W*(x) = Jbc if and only if x satisfies (27). For any fixed f, and xe<$(W*) we have 0l) W*(x))= I <Z k (l - keS (32) Quantum dynamical semigroup 169 Since 0t(\ - Y)~ 1 ) = (Y) 9 that (iv) and (v) are equivalent is a simple consequence of (32). & We use the same symbol for the linear canonical extension of a bounded map that appeared in Proposition 4.4 to the Banach space of all trace class operators. In the case of an unbounded operator, say G we extend it to 9(G) + i@(G) by linearity. The family of maps i n : = (<j in )* on the dual space d?pP ) * s called the minimal dynamical semigroup. For further details we refer to [4]. Our next aim is to deal with the dilation problem associated with the Fokker-Planck equation (16) whenever the operators Y 9 Z fc , keS satisfy the following assumption. Assumption A. Y is the generator of a strongly continuous semigroup on Jtf Q and Z k , keS is a family of densely defined operators satisfying (21) and (22). There exists . a dense linear manifold @> in Jf so that it is a core for Y and > where S = ((Sj, fcjeS)) is a contractive operator on Jt? 1 2 (S). Furthermore for any fixed jeS, Sj^O for finitely many i'eS. The last hypothesis ensures that the third expression (19) is meaningful and is indeed verified for most applications [19]. For any A > we define bounded operators 7 A , Z, keS by where boundedness of ZjJ, fceS follows from (22). Moreover for each A>0, Y X ,Z*, /eeS satisfies (17), hence the series E^ZjJ^ZjJ converges in strong operator topology. On the other hand for each ge@>(Y) we have Y^g-tYg as A-oo. Taking /= (/ - A(A - 7)- x )^ in (22) we get -A(A- Y)- l g\\ \\ y(/-A(A- Y)' 1 )^!! Hence Z*g-+Z k g as A->- oo for all #e^(F) [5]. For any 1 > 0, ^ r ^ 1 we define bounded operators Z(A, r) s {Z^(A, r), z, jeS} as in (19) with y, Z fe , feeS. replaced by 7 A) r i/2 Z^ fce5 respectively. So for each ^ r < 1 and X > 0, Z(A, r)e^~ n &~ . We denote by K (A ' r) = { K (W (t), t ^ 0} the unique regular (JP ,Jf) - adapted contractive process satisfying (18) with Z(A,r) as its coefficients. We also define operators Z(r) on as in (19) with Z k replaced by r 1/2 Z k , keS and write Z(A, 1) = Z(A), Z(l) == Z. For each ^ r ^ 1 it is evident that for all ij'eS, PROPOSITION 4.6. Consider the operators Y, Z ki keS satisfying Assumption A. Then the following hold: (i) For each < r < 1, w./wn^ K (A ' r) (t) = F (r) (t) exists /or allt&Q and V (r) = (F (r) (t), t^O} is the unique regular (J^ ,J^)-adapted contractive operator valued process 170 A Mohari and K B Sinha satisfying (18) on @&(Jt) with Z(r) = [Z\(r\ iJeS} as its coefficients. Moreover V (r} is a strongly continuous contractive bar-cocycle; (ii) For each t^Q the map r -> ! /(r) (t), ^ r < 1 is continuous is weak operator topology. Proof. For each ^ r ^ 1, A > 0; Z(A, r)e^~ n % ~ and the triad Z(r), Z(A, r), satisfy (12) and (13) with n replaced by L By our hypothesis 2 is also a core for 7. Hence we conclude (i) by Proposition 3.1 (i)-(ii). Choose ^ r, r n < 1 (n ^ 1) such that r n -> r as n -* oo. Since the triad (Z(r),Z(r B ),n> 1,) satisfies (12) and (13) on S and Z f (rJ/^Z^(r)/, as r n ->oo for any /e, remark 3.2 implies that wlim^^ V (rn \t) = 7 (r '(t),0 < r < oo. This completes the proof. M For each A,^ > 0, < r,s < 1, we define the semigroup t u ' Ai ' r ' s) on 8(Jtf Q ) by T (A, M ,r,s) (;x) ^ E [7 (A ' r) (t)*x7 ( ^ s) (t)], t ^ 0, where the semigroup property follows from the cocycle property of the contractive processes V M . The associated pre-dual semigroup a ( ^' r>s) on ST is defined as in (20) whose bounded generator ? i A " u ' l '' s) is given by keS For each ^ r < 1 we also have * W^(p) = Yp + pY ke5 where VF (r) is described in Proposition 4.4. We write r (A ' r) , T (A ^ r) , cr (A ' r) and cr (A '^ r) for T (A ' A ' r ' r) , t (A '^ r ' r) , (j (A * A ' r ' r) and (T (A ^' r ' r) respectively. When r = 1 we omit the symbol r. For each < r, s ^ 1 we also define the one parameter semigroup T <^>;= E [F (r) (0*xK (s) (t)], t ^ on ^(Jf ). Again when r = s = 1 we omit the symbol r. Our aim is to show that crj min) is the pre-dual map of r r for all t ^ 0, where cr min is . defined as in Proposition 4.4. For this we need the following lemma. Lemma 4.7. Let A k ,k^l and B k9 k^lbe two families of bounded operators such that the series E n> 1 A*A k converges in strong operator topology and slim^^Bn = B. Then for each pef h , lim m ^^C(m,n) = Im^^I.^A^pB^A* = X keS A k BpB*A* = C in \\'\\ tr norm. Proof. Lemma 4.3 implies that C, C(m 9 n) 9 m,n^l are elements in ^". For any fixed m, n ^ 1 and peJ~ we have )-C|| tr ^ * {\\A k (B m -B)p(A k B n )*\\ ir k^\ + \\A k Bp(A k (B n -B))*\\ ir .}. y 7 Quantum dynamical semigroup 171 Hence for p = |/)<#| we have { II A k (B n - B)f || || A k B n g \\ + || A k Bf \\ || A k (B n -B)g\\] 2 y/Y 2 \i/ 1/2 / \ 1/2 Z M*/ll 2 Z M*(B where a, /? are some positive constants independent of /, g. Hence the result follows for p = |/><0|, /,0eJf . For a general p = I.c,!/,)^!, ||/ || = ||^|| - 1, S.|c,| < oo, we use dominated convergence theorem to conclude the required result. H PROPOSITION 4.8. Consider the family of operators {Y,Z k ,keS} satisfying (21) and (22). Then for each fixed ^ r < 1 the following hold: (i) For each 1, fi > 0, ^ r, 5 < 1, (ii) For pen(r), Um M ^ \\ tf^(p) - W (r \p) \\ tr = where the limit is independent of the order of L & (iii) Hm ( , M)r w I! of >(/>) - ^(P) t = f r aU P^> where <r (r ' is the map defined as in Proposition 4.4; (iv) The pre-dual map of ij r) is of , t ^ 0; (v) For ^c/z ^ s < 1, a[ r ' s) = a ( t (rs}i/2} for all t ^ 0. Proof. Since for each fixed A, ^ > 0, jSfJ,^^^) = ^(^^frsj^^rs)!/^ we conclude $ by the fact that a bounded generator uniquely determines the semigroup [5]. Now for (ii) first observe that keS and Yn(p) + TC(P) F* + r 2 JkeS for all pe^r and 7 M 7c(p) = // 2 (^- F*)" 1 ^- Y)~ l (Y(l- Y)~ l p(l- Y"*)" 1 ), Now (ii) is immediate from Lemma 4.7. Since n(&~] is a core for W (r} which is the generator of a strongly continuous contraction semigroup, (iii) is evident from (ii) and a standard result (Corollary 3.18 [5]) in the theory of semigroups. 172 A Mohan and K B Sinha 'For any fixed f t geJ^ , hu > we have Hence (iv) follows from Proposition 4.6 (i) and (iii). Finally, we arrive at (v) from (i) and (iii). The following theorem establishes the main result. Theorem 49. Let Y,Z k ,keSbe a family of operators satisfying Assumption A. Consider the family Z = {Z , iJeS} defined as in (19) on Q). Then there exists a unique regular (@,J?)-adapted contractive process 7s {V(t) 9 t^0} satisfying (18) on @&(Jt\ Moreover the following hold: (i) Tj nin (x) = E [7(t)*x7(t)], where r min is the minimal dynamical semigroup on ^(J^ Q ) associated with (16) and (17) (ii) Assume furthermore that S is an isometry and the equality in (22) holds. Then ZeJf. In such a case V is isometric if and only if /? A = for some A > 0, where /? A is defined as in Theorem 4.5 (v). Proof. The first part is a restatement of Proposition 4.6 (i) for r = 1. In view of Proposition 4.8 it is evident that for all 0< r < 1, = lim for any /, geJf . Now taking limit as rf 1 in the above identity we get the required identity for (i) by Proposition 4.4(vi). That Ze*/ is simple to verify. The 'only if part of (ii) follows from (i) and Theorem 4.5. For the converse we appeal to Proposition 3.1 (iv). This completes the proof. Now combining Corollary 3.3 and Theorem 4.9 we arrive at necessary and sufficient conditions for V to be co-isometric. Theorem 4.10. Consider the family V={V(t\ t^O} of operators defined as in Theorem 4.9. Suppose the family {7*, Z k , k<=S] of operators also satisfy (21), (22) and D is a core for 7* so that <=.2(Z%\ feeS. Assume further the equality in (22) and S is a co-isometry then ZeJ. In such a case V is co-isometric if and only if /^ = for some A > 0, where /? A defined in Corolary 3.3 is modified as /? A was in the statement of Theorem 4.5 (v). Proof. S being a contractive operator we observe that keS keS for each /e<(Z fe ), feeS, where L k = E (gS (S[)*Z,. Hence the family {Y*, L k ,keS} also Quantum dynamical semigroup 173 satisfy (21) and (22). Thus Ze3T(). The proof is complete once we appeal to Corollary 3.3. Example 4.11. Let L fc , /ceS be a family of closed operators in ^ and Y i?e the generator of a contractive C semigroup satisfying (21) and (22). For each /ceS consider the polar decomposition L k = S k \L k \ 9 where S k is the partial isometry with initial subspace as #(|L k |), hence S*L, = |L fc |. Now with Z k = L k ,S k = 8 k .S k , define the family of operators Z == {zj, UeS} as in (19) on 9(Y). It is evident that Assumption A is valid. In general it is difficult to verify if /^ or j8 A or both are trivial. However, when | L fc |, fceS is a family of commuting self-adjoint operators then /? A = for some (hence for all) A > 0. For more explicit examples refer [19]. References [1] Accardi L, On the quantum Feynman-Kac formula. Rend. Semin. Mot. Fis. Milano Vol. XLVIII (1978) [2] Accardi L, Frigerio A and Lewis J T, Quantum stochastic processes. Proc. Res. Inst. Math. ScL, Kyoto 18(1982)94-133 [3] Chebotarev A M, Conservative dynamical semigroups and quantum stochastic differential equations, Moscow Institute for Electronic Engineering, (Preprint 1991) [4] Da vies E B, Quantum dynamical semigroups and neutron diffusion equation. ( Rep. Math. Phys. 11 (1977) 169-189 [5] Davies E B, One parameter semigroups. (London: Academic Press) (1980) [6] Fagnola F, Pure birth and pure death processes as quantum flows in Fock space. Sankkya, A53 (1991) 288-297 [7] Fagnola F, Unitarity of solution to quantum stochastic differential equations and conservativity of the associated semigroups, University of Trento (preprint, 1991) [8] Feller W, An introduction to probability theory and its applications, vol 2, (New York: John Wiley) (1966) [9] Gorini V, Kossakowski A and Sudarshan E C G, Completely posititive dynamical semigroups of n-level systems. J. Math. Phys. 17 (1976) 821-825 [10] Hudson R L and Lindsay J M, On characterizing quantum stochastic evolutions. Math. Proc. Cambridge. Philos. Soc. 102 (1987) 263-269 [11] Hudson R L and Parthasarathy K R, Quantum Ito's formula and stochastic evolutions Commun. Math. Phys. 93 (1984) 301-323 [12] Hudson R L and Parthasarathy K R, Stochastic dilations of uniformly continuous completely positive semigroups. Acta. Appl. Math. 2 (1988) 353-398 [13] Journe J L, Structure des cocycles markoviens sur Fespace de Fock. Probab. Th. Rel Fields 75 (1987) 291-316 [14] Kato T, On the semi-groups generated by Kolmogoroffs differential equations. J. Math. Soc. Jpn: 6 (1954) 1-15 [15] Lindblad G, On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48 (1976) 119-130 ' [16] Meyer P A, Fock Spaces in Classical and Noncommutative Probability, Chapters I-IV, Publication de 1'IRMA, Strasbourg (1989) [17] Mohari A and Parthasarathy, K R, A quantum probabilistic analogue of Feller's condition for the existence of unitary Markovian cocycles in Fock space. Indian Statistical Institute (preprint 1991). To appear in Bahadur Festschrift [18] Mohari A and Sinha K B, Quantum stochastic flows with infinite degrees of freedom and countable state Markov processes, Sankhva, A52 (1990) 43-57 [19] Mohari A, Quantum stochastic differential equations with unbounded coefficients and dilations of Feller's minimal solution, Sankhya, A53 (1991) 255-287 [20] Mohari A. Quantum stochastic differential equations with infinite degrees of freedom and its applications, thesis submitted to Indian Statistical Institute, Delhi centre, 15 Nov 1991 [21] Parthasarathy K R, An introduction to quantum stochastic calculus, (Basel: Birkhauser Verlag)(1992) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 3, December 1992, pp. 175-188. Printed in India. Almost periodicity of some Jacobi matrices ANAND J ANTONY and M KRISHNA* School of Mathematics, SPIC Science Foundation, Madras 600017, India "Institute of Mathematical Sciences, Madras 600013, India MS received 24 June 1992; revised 8 November 1992 Abstract. We show that random Jacobi matrices are almost periodic whenever they have purely absolutely continuous spectrum having finitely many bands. Keywords. Almost periodicity; random potential; Jacobi inversion. 1. Introduction Studies on random Schrodinger operators acquired importance in view of their usefulness in understanding the properties of condensed matter systems. The theory is well developed both in the continuous and in the discrete settings and there are excellent reviews by Simon [14], Spencer [15], Carmona [2] on this subject, in addition the theory also appears in the book of Cycone et al [5]. In one dimension the theory is much sharper as the spectral properties of such operators are shown to have consequences on the nature of the random potential, see for example the Kotani theory on the determinicity of potentials having some absolutely continuous spectrum ([7] for the continuous case and Simon [13] for the discrete case.) Further it was shown by Kotani and Krishna [8] and Craig [4] that some random Schrodinger operators with purely absolutely continuous spectrum of a certain type are almost periodic of necessity. The route for showing such a result was through inverse spectral theory. Showing the almost periodicity of such random potentials involved setting up and solving the Dubrovin equation for some spectral parameters, Jacobi inversion on a Riemann surface etc., All this was done for the continuous case by Levitan [9] McKean and Moerbeke [11], McKean and Trubowitz [10]. In the discrete case (i.e. for Jacobi matrices) the inverse spectral theory exists, see Kac, Moerbeke [6], Moerbeke [17], Toda [16] for the periodic case and Carmona, Kotani [3] for the random case and the references therein. However while the existence of solutions for the inverse spectral problem is given in the above works it was only in [6] and [17] that the nature of the matrices so constructed is presented. Of necessity these turn out to be periodic matrices. In this paper we show that in the discrete setting given a random potential, with finite band absolutely continuous spectrum, it is almost periodic, in the sense that the support of the random potential consists only of almost periodic sequences. The theory extends to the case of infinitely many bands and will appear in [1]. We have organized this paper in four sections. In the first we set up the direct spectral theory and identify the spectral functions that will play a role in the subsequent 175 176 Anand J Antony and M Krishna sections. In the second section we do the inverse theory and finally 3 the Jacobi inversion is presented. 2. Direct-theory We consider Q = /J(Z, 1/(1 -f \n\ 2 )) and B the associated Borel cr-algebra. We consider a bimeasurable invertibte transformation T on Q whose action is Tco(n) = a)(n + 1) and consider a probability measure P on (Q, B) such that it is invariant and ergodic with respect to T. Given CD Supp P we consider the operator q* of multiplication by o) on / 2 (Z) and consider the family of operators H " = A + g w , A the discrete Laplacian. We assume further that CD e Supp P implies that CD is bounded as a sequence in which case CD-^H * will be a measurable self-adjoint operator valued map. Then the general theory [2] of such operators shows that there exist constant sets, E, ! ac , E sc and S pp such that the spectrum, the absolutely continuous, singularly continuous and pure point spectra are the above sets respectively for the operators H a.e. CD. The main theorem of the paper is the following. Theorem 2.1. Suppose (Q B, P) z's as above with P ergodic with respect to the shift T acting on Q. Suppose that for the associated self-adjoint operators H 03 the spectrum is purely absolutely continuous and consists of finitely many bands. Then, any CD in Supp P is an almost periodic sequence. We present the proof of this theorem at the end of 3. Before we proceed further let us outline the strategy employed in proving the theorem. The information on the absolutely continuous spectrum, together with the zeros of the Green function for a given potential q gives us an expression for the Green function. From this we obtain an expression for the sum of the Weyl m-functions. Using the additional property that the imaginary parts of the m-functions agree on the absolutely continuous spectrum, we obtain expressions for the difference of the Weyl m-functions. The Weyl m-functions thus obtained are identified as the values, on the two sheets of a Riemann surface, of a meromorphic function and the poles of this meromorphic function are related, via the trace formula, to the potential q. These poles are then shown to be related to theta functions, from which almost periodicity follows. We would like to emphasize here that unlike the continuous case, [8, 4], where the reflectionless property of the potentials is sufficient [4], it is not sufficient here. The reason is that the Dubrovin equation, that came for free in the continuous case, was the heart of the matter there and its analog is missing in the discrete case. Henceforth we fix a CD and consider H 03 on / 2 (Z) and drop the superscript also, referring to q 03 as q and the associated sequence CD(n) as q(n). We shall also write the limits lim 8 _ g A+te as simply # A for ease of writing and the appropriate definition should be clear from the context. Then from Weyl theory it is known that for AeC + , the difference equation, (H - A)u A = 0, has two independent solutions u i , w 2 such that Wi(0) = 1, 1^(1) = 0, M 2 (0) = 0, u 2 (l) = 1. One also has unique solutions u+ A in 1 2 (Z ) and there exist holomorphic functions m (1) such that u + A (n) = u^ri) +~m+ (X)u 2 (n). We also have MO) Almost periodicity of some Jacobi matrices 177 which can equivalently be taken as their definition. Weyl theory also gives the expression for the green kernel of H, for m ^ n in terms of the Wronskian [u + , u _ ], as / \ /rr ix i/ N u + (ri)u_(m) g A (n, m) = (H - A) " 1 (n, m) = -il^ L2 (9 ) and the definition extended by symmetry tom^n. An evaluation of the Wronskian leads to the expression where m n (A) are defined as in (1) by taking T"*? instead of q. Let us recall the following relations for m from [13]. u (n) and ..W = ()-*- ( . o, T i,)- 1 (5) and in terms of the operators H + n of H restricted to the subspaces {uel 2 (Z ): (u(n) = 0), we have m , n = (H , n -A)- l (5 n1 ^ n1 ). (6) From now on we retain the superscript CD and consider the set Then we have the following theorem from the theory of random Jacobi matrices. See Simon [14] (proofs of Theorems 1 and 2) for the proofs. PROPOSITION 2.2. Suppose A is as above with \A\>0, then Ef c (A) *Q. If further A is an open interval, Theorem 2.3. Suppose the spectrum of H" contains the set A, then limRe^ +fa (n,n) = Oa.e co a.e., For a.e. pair {(co,A)} we have determines {q(n),neZ} uniquely a.e. co. In fact we can also deduce as in [4] or [8] that COROLLORY 2.4. Suppose the spectrum of the Jacobi matrix H is purely absolutely continuous and is 178 Anand J Antony and M Krishna the union of closed intervals (finitely or infinitely many), then we have and also and 3. Inverse theory In this section we specify the spectral data that are uniquely associated to a given potential q. The necessary spectral data are the band edges /!,- of the absolutely continuous spectrum, the Dirichlet eigenvalues f of an appropriate half space problem and the + valued variables a { specifying as to which half space problem the Dirichlet eigen values f correspond. DEFINITION 3.1. A set A is called reflectionless whenever, Re0 A+f0 (n,n) = 0, VneZ and a.e. /leA Then we have, Lemma 3.2. Suppose the spectrum of H is a reflectionless set and is also the union of closed sets [^p^zi+i^ * = 0,...,A7", then, the following are valid for every neZ. i0 (n,n) = on S and Img Ji+i0 (n 9 n) = Q on R\I. There is a unique zero ^(n) of g^(n, n) in each of the intervals [A 2 ._ 1 , A 2 ], i = 1, . . . , N. Proof. The first part of lemma follows from the assumption that the spectrum is a reflectionless set and the vanishing of the imaginary part on the complement of the spectrum in R is easy to verify. As for the zeros of Green functions we note that if a zero exists it is unique, since the Green function is real analytic and is strictly increasing as a function of A in each connected component of R\L. If there is a zero in (A 2 _ 1 ,A 2 .) call it { (n). Otherwise if g^ is positive there then set ,-(n) to be ^ 2 i-i and if it is negative set f (n) to be A 2 . This proves the lemma. DEFINITION 3.3. We consider 1 as in the previous section. We call a point q of 1 a reflectionless potential if the spectrum of A + q is a reflectionless set. DEFINITION 3.4. We define the hull H(q) of a reflectionless potential as the closure in the topology of Q, of {q(- + n):neZ and <?eQ}. PROPOSITION 3.5. If q is a reflectionless potential with spectrum S of the type mentioned in the above lemma. Then each q in the Hull H(q) is also a reflectionless potential with the same spectrum. / Almost periodicity of some Jacobi matrices 179 Proof. It is known that the functions m^ converge uniformly in C + whenever a sequence of potentials q k converges to q in the topology of Q, see [7]. The rest of the proof follows as in [8]. Remark 3.6. We note that recalling the equations (2, 3, 4, 5) for g^(n,n\ m + ^ n (X\ w_ w (A), any zero ^(ri) of g^(n,n) in D^-i'^aJ corresponds to a pole of m + n (X) (or m_ n (X)). Equivalently to an eigen value of H + n (or H_ n ). It is possible to reconstruct the Green function of the Jacobi matrix satisfying the conditions of Lemma (3.2) from the spectrum and the zeros of the Green function. Theorem 3.7. Suppose the spectrum of H satisfies the conditions of Lemma 3.2. Then, the data {A J and {^(n)} are sufficient to recover the Green function g z (n, n) uniquely. Proof. Since g z (n,ri) is Herglotz, its logarithm lng z (n,ri) is also Herglotz. Further < Im lng z < n in C" 1 " and the Herglotz representation theorem gives the expression for lng z (n.,n) as - 7lJ R -2) 1+X Since we have that lmlng x+iQ (n,n)dx. (7) and Imlnflf je+i0 (n,n) = or n for g x+i0 >0 or <Q. Equation (7) becomes, after dropping the zero contribution from the integrand, Now using the values of the imaginary part of the logarithm from (8) and performing the integration we have that ^ f Azi + i 1 f) +- Z JA 21 2i=i J A2( _ (9) where C(A 2JV+1 ) is a constant. Now the asymptotic expansion for (^ z (n,n)) 2 requires that it behaves like 1/z 2 at oo since the spectrum of H is bounded. Hence, we see that collecting the constant terms in the above equation together, the expression reduces to ,n)^ Y ln^^ + I In ^. (10) z\ 5 / ^ JL*I / *\ A~I /i i\ ' 2 = (A-A) i=i (-""^) 180 Anand J Antony and M Krishna PROPOSITION 3.8. Suppose H satisfies the conditions of Lemma 3.2, then the potential has the following expression in terms of {/l t -, ^(rc)} N <?(") = i^o + 1 1=1 Proof. Using the asymptotic expansion of g^(n,n) as a power series and comparing the (I/A 2 ) terms from equation (2) on one hand and equation (4) on the other we get the above formula. Now we define two functions R and P n as follows 2N+1 N Then in terms of R and P n the expression for g^(n,n) becomes in view of the equation (10) / N *nW (1?\ gi(n,n) = = . (13) The sign of the square root is determined and fixed according to the requirement that A (n,n) is positive in ( oo,A ). At this stage we see that the absolutely continuous spectrum together with the Dirichlet eigenvalues gives the Green function uniquely. The knowledge of the Green function 0^(0, 0) is actually sufficient to recover the potential when it is symmetric about zero. However when the potential is not symmetric we need to know either of m (A) and #(0) is sufficient to recover the potential uniquely. We proceed to find the parameters that will determine m + (A) uniquely. We already know from equation (4) that the Green function is written in terms of m (A). We observe that the function F(A) = -flf^O)" 1 = m + (A) + m_(A) + A - q(0) (14) is Herglotz in C + and has purely imaginary boundary values of Z and is analytic except for poles in the complement of Z. The poles are located precisely at the zeros of A (0,0). Since the function m+(A),m_(/i) are also Herglotz as can be seen from equation (6), it is clear that if we can find an expression for m+(A) w_(/l) we can find m + and m_ . To this end we have PROPOSITION 3.9. Suppose q is a reflectionless potential with spectrum of the type given in Lemma 3.2, suppose further that m+ and m_ have the same imaginary parts of the spectrum Zofq and % t is a pole of m according as a t is 1. Then m can be uniquely recovered from the knowledge of 2, {,(0) and crj. Proof. From equation (14) it is clear that m + (1) + m_ (A) H- A - <?(0) is Herglotz and has boundary values a.e. on [R and poles at (0). Also m +) m^ both are analytic in / Almost periodicity of some Jacobi matrices 181 C + and have non zero imaginary parts on E and zero imaginary parts on R\E. Therefore the function G(X) = (m+ w_)(A) is analytic in C . Also G has zero imaginary part on (R, Further from equation (14) it is clear that the poles of F(l) are simple and come precisely from those of w+ or w_. Therefore G(l) has the same poles on IR. Therefore it is a meromorpic function in C with simple poles at f . Then we use the relation (14) to compute the residues of the function G at the poles we find that Z:-Z; (15) where / = {i:^ is a pole of m } and C t are the residues of F(l) at the points Explicitly Q's turn out to be prime denoting the derivative with respect to A. Now defining the function a from /+u/_to{ + , }we can write the expression for G(>i) as (16) since the difference of the left and right hand sides of the above equation is an entire function bounded on C and has the value at infinity, hence vanishes identically by Liouville's theorem. From equations (15) and (14) we can write the expressions for m+ as Clearly since the poles of m in (R are the eigenvalues of H , we can collect all the above results into the following Theorem 3.10. Suppose we have the operator H = A + q with spectrum S, a reflectionles set of the type assumed in Lemma 3.2. Further suppose on , Im m + = Im m_ a.e. T/ien corresponding to each set of points {^-(0), cr (0)}, t/rere is a unique potential q such that the spectrum of the associated H is purely absolutely continuous and equals S and further {^(0):cr l .(0)= } are precisely the eigen values of H . In this case the values of the potential at is given by equation (5). Remark 3.1 1. The above theorem is valid if q, <^(0), ^.(0), m are replaced by T n q, t (n\ m + n for each fixed neZ. use in the next section we note that equations (17) and (6) imply that the following relations are valid as. 182 Anand J Antony and M Krishna and We also note the relation (A) (20) -i' P (A) where 4> + .W- "fl m + ,(*) and 4>_ JA)= nV-tfCO + m^A)). (21) i = ' i = Clearly that at oo the behaviour of + n (X) is given by ^(A)*^". (22) 4. Almost-periodicity In the last section we showed that given the operator H = A + q with purely absolutely continuous spectrum we can recover the operator H uniquely under certain additional data given by theorem 3.10. Apriori it is not clear that there is any relation between the zeros (n) and <^(0) of the Green functions A (n,n) and A (0,0). In this section however we show that actually these zeros cannot vary independently for each n but infact need to move in a way to make the function <X>(n) = f (n) almost periodic as n varies. As a consequence the potential which necessarily satisfies the 'trace' formula equation (11) also becomes almost periodic in n. We recall the relations (18, 19 and 20), of the last section, for the functions </>+ . It is clear that these can be thought of as the same function </> on the Riemann surface ^associated with the function ^/R(ty. We recall that $ is constructed by taking two copies of the A sphere slit along [A 2l .,A 2i+r ], i=l,...,N and joined appropriately to form a two sheeted branched cover, with branch points the A/s, i = 2, . . ., 2N + 1. The points on the spheres corresponding to the points at infinity of the plane are denoted as p^ and p^, respectively and fall on either sheet of &. On $ we consider closed paths a and &, i = 1,...,N forming crosscuts in such a way that (x.i lies on the upper sheet and encloses [A 2P A 2 . +1 ], i = 1,...,JV and j8 4 starts at A! goes to A 2 ., i= 1,...,JV on the upper sheet crosses over to the lower sheet and returns to A L . These paths form a set of generators for the group of closed paths on $?. We also choose a point P on &t, away from the branch points, the points p^ and p^, and the paths a f and &, as a reference point where the value of the function l/V/i*(4) is chosen and fixed. Then the family {A m dA/ v /5(J)} m = 0, . . . , N - 1 forms a collection of N holomorphic differentials in terms of which we choose a basis, (23) of holomorphic differentials for the vector space of holomorphic differentials on ^, Almost periodicity of some Jacobi matrices 183 which necessarily has dimension N. The bais {dco m } is normalized so that the matrix T with entries. takes the form [/, T], where / is the identity matrix and IT, of necessity, a symmetric nonpositive definite matrix. T is called the period matrix associated with the basis a x -, j8,. and dco t -. Now we consider the single valued function n (A) on Si which takes values </> n (ty defined in equation (21) on the upper and lower sheets of ^?. Clearly a point ^ in (^2;-i> ^2f) ^ as un i ( l ue point p f over it in & for each value 1 of <j . Therefore the single valued function W (A) has poles at p^, of order n and zero of order n at p^ (by equation 22) and also has exactly N poles of order 1 at the points p f (0) and N zeros j of order 1 at the points p t (n) by (20). The exact location of these points in terms of which sheet they belong to is unimportant in the subsequent discussion. Having identified the function <j> n (X) as having the appropriate behaviour we shall use it to construct the differential A (24 ) d/l with the property that it has poles at p^p^ with residues n and n simple poles at Pi(n\ pi(Q)i=l,... 9 N with residues + 1 and 1 respectively. This differential will be crucial for us to obtain a relation between the points p (n) and p (0). To start with we have the relations, which are consequences of Cauchy's integral formula da)(n) = 2itikj and da>(n) = 27riw / . (25) Jftj Now do)(n) being a differential with poles we can write it in terms of the differentials of first, second and normalized differentials of the third kind as N N ,p 00 ,)+ Z do)(p l .(),p,(0))+ X Cjdcoj (26) where da)(a,b) is a normalized differential of the third kind with residues -h 1 and 1 respectively at a and b. D is a differential of the second kind and dco,- are defined in (23). Since it is always possible to add differentials dco,- to any dco(a, b) to make the integral over the a/ vanish we obtain the relation that in view of the normalization (25) Cj = 2nikj for some integer kj for each; by integrating the above equation over a,-. Lemma 4.1. We have the following relation among the points p f (n), Pi(0), p^ and p^ ,, (27) N fPj(n) N rPj(O) N r do,- da>- fc J .T i . + m i + J = ijp ;=iJp J =1 J 184 Anand J Antony and M Krishna Proof. Integrating of equation (26) on both sides with respect to fa we have (28) By the comment before the lemma we know that the sum of integrals w.r.t. fa of da)j gives the term, 27r/Ef =1 /c J T. ; and the integral of the left hand side provides us with the term Inim^ The remaining sum is computed as follows. Suppose da}(a,b) is a normalized differential of the third kind with residues -f 1 and 1 at p and q respectively. Then, we claim that f f dco(p,4) = 27ri JBi Ja da>i. To show the claim we note first that by addition of differentials of first kind we can always make J a .dco(p,#) vanish for each L Now consider the normalized differentials do) fc and compute the integral J c o)^dco(p, q), where co k is the integral of the differential dco fc . Then a> k da)(p,q) is an Abelian differential, regular everywhere except for poles p, q. Therefore by Cauchy integral theorem the integral evaluates on the one hand to I; (D k dco(p, q) = 2ni(co k (p) - co k (q)) (29) since the residues of dco(p,<?) at p and q are respectively 4- 1 and 1. On the other hand going down to the polygonal region S corresponding to the normal form of the canonically dissected Riemann surface we obtain the relation (see Siegal [12], Chapter 4, 7) L' co k d(o(p, q) = X {co t (X,)eo(p, q)(B,) - (a k (B t )(o(p, q)(A t ) (30) where A lt Bi are the sides of the polygon corresponding to the closed paths Now, we use the normalizations and their implications f J aj d ik . (31) J aj Hence equation (29), and (31) together yield, a) k dco(p 9 q)=:2ni(cD k (p)-co k (q)). (32) = Jc But the integral co k (p) co k (q) is precisely J^dco^ J^dco fc from which the claim follows. Equation (27) can be inverted to get a relation for P;(n)'s in terms of the right hand side of (27). The existence of a unique inverse is assured by the Jacobi inversion. Then an explicit formula will be obtained for the inverse through functions. Therefore we define the necessary quantities here. Almost periodicity of some Jacobi matrices 185 DEFINITION 4.2. A divisor P is a formula product P^ . . . PJQ l . . . Q k of points in 9t, An integral divisor is the product of the type P 1 ...P n . An integral divisor P 1 ...P N is called general if the matrix with entries A { . = (d<y /cU)(Pj), ij = 1, . . . , N is nonsingular. In the following theorem we identify the divisors which are general. Lemma 4.3. An integral divisor Pi.-.p^ is general provided the factors p t are distinct and no two of the p L is in {Pa^p^,}. Proof. We note that the result follows by checking that the matrix J. . with entries (d/dA)(A'~ 1} /^/R(X)) at Pj is nonsingular. Using the local parameters t = z at ordinary points and t = ^/(z ~- P ) at branch points the verification is easy. DEFINITION 4.4. The function on C N is defined by 0(z) = X exp(27ri<w,zexp(7rz<w,Tm (33) meZ N where T is as defined after equation (23). The function satisfies the periodicity relations as follows (z + e k ) = 0(z) and 0(z + r fc ) = exp [ - 27112* - nh kk ]&(z). (34) The Jacobi's imaginary transformation is given by :N/2 1 ^-!- 1 ) (35) where 0(w,r) is the theta function at u with the period matrix given by T. Next we consider a divisor P = P . . . P N and define the Abel-Jacobi function A(P) as the function >4(P) = If == J pl dco, taking the divisor to a point in C N /IL We recall the following theorems from Siegel [12], 10. y Theorem 4.5. Consider A(P) defined above for an integral divisor P. Whenever P is general, then it is the unique integral divisor in the preimage of A(P). While the above theorem guarantees a solution for the Jacobi inversion, the components of the point P are obtained as the zeros of an appropriate theta function acting on ^?. Explicitly we consider on 3% the integrals co(P) = j do>. Then we consider the function c/>(P) = (a>(P)-~s + c) where c is a vector of Riemann constants depending upon t and <w,seC N /II. It is also known that there are exactly N zeros for the function <(P) in ^. We have the following theorem from [12] ( 10). Theorem 4.6. Suppose (f>(P) does not vanish identically for a fixed s. Then, the zeros q k $ of </>(P) satisfy the relation s = EL x J^ dco, the equation is to be understood component wise. 186 Anand J Antony and M Krishna The upshot of the above theorems is that if we rewrite equation (27) as N dG> = nc f + X f (36) to obtain Pj(n) in terms of nc + K, one needs only to check that the required inverse is a general point and consider the function / N rPi(n) \ &(co(P)- X dco + c ) = (a)(P)-nC'-K + c) (37) \ f-lJpo / and find its zero. The Riemann constant c is chosen so that the function 0(P) does not vanish identically. Till now we are still on the Riemann surface and identified the function whose zeros are precisely the points above ^(n). Now we shall obtain an expression for the sum of <^-(n) in terms of the theta function mentioned above. To this end we have the following Lemmas. We consider the map A from & to C. The map 1 is a meromorphic function on & with poles precisely at the points p^ and p^,. Lemma 4.7. We have the following relation between the theta function and the (n). ) = Const + C..W ( " C + M (38) t=i i = i \((n+l)c + d)y where the constants {C iN } are the same as those which appear in equation (1), D t is the partial derivative in the ith direction, c is a vector with purely imaginary entries and d some constant vector independent of n. Proof. We consider the meromorphic differential M In (</>(P)). It has poles at p^,p^, and at the zeros P- t (n) or 4>(P). Let C be a closed contour issuing from the reference point P and enclosing all the poles and zeros of the function Adln(<(/l)). Therefore a computation using the Cauchy's integral theorem gives that [ Ad In <t> = *>(Pi(n)) + Residue at p^ + Residue at p^,. (39) The computation of the residues is done using the local parameter as follows. The local parameter at p^ is X = C"" 1 . Therefore we have dl (40) where co is the integral of the differential dco and D t refers to the derivative of the argument of <f> in the ith direction. The summand of the above equation is evaluated as Since p^ belongs to the upper sheet, the square root has a positive sign and also since R(A) goes like /( N+1 at oo, we have that the above equation evaluates to - C iN . Almost periodicity of some Jacobi matrices 187 Therefore we have N Residue at p^ = - C lW Z) I .ln(0(o>(p 00 )) - nc + rf). (42) Similarly noting that at p^,, the square root has a negative sign since p^, belongs to the lower sheet of ^, we have N Residue at p^, = C w l) ln(0(e0(p 00 ,)) nc + d). (43) 1 = 1 Now, since c = J p * dco, we have coQ?^) = c + ^(p^,)- Putting these equations together and absorbing the value o^p^) into d, ' + d} ' (44) This equation provides us with one expression for the left hand side. On the other hand going down to the Riemann region corresponding to the canonically dissected surface and taking the polygonal path {a l b l a^ l b~ l ...b^ 1 }, we can integrate the function explicitly using the relations, for each /c, "(; l ) = te)- 2nidco k and (b~ 1 ) = (b k ). (45) The equations are understood to mean equality of evaluation of the differentials at a point on the ark a k , b k and the corresponding point on a" 1 ,^"" 1 etc., Then we obtain that I J (46) c t=ij a . Putting equations (39) and (37) together we obtain the lemma. Lemma 4.8. The sum Sf =1 ^-(n) of the above lemma are almost periodic in n. Proof. The Jacobi-imaginary transformation equation (35) shows that the right hand side of the equation (38) for the sum Ef =1 ^(n) is indeed real and further the vector T~ X C is a vector with real entries. Therefore the required almost periodicity is now immediate from that of the theta function with real argument. Now we are ready to prove the Theorem 1 of 1. Proof. The assumptions of Theorem 3.10 are satisfied for each fixed co in support of P. Therefore for each fixed o> the potential q 03 is almost periodic as a sequence in view of Lemma 4.8 and the trace formula. We would like to make a few comments regarding the theorems of this paper. Firstly if we replace A in H by non constant off diagonal entries (i.e. coming from a positive sequence a n \ the theorems of the last two sections will go through with some modifications for the expressions for the m-functions and the trace formula. Secondly we have stated that the meromorphic function </> has exactly N poles. It might happen that if for some n, some of the zeros ^ t (n) coincide with ^-(0), then this is not 188 Anand J Antony and M Krishna exactly correct. However the proof still go through as this phenomenon does not persist for the neighbouring values n 1 and n + 1. Such coincidence will show only the periodicity of the appropriate () in n. A forteriori using the first order equations (4) for m+(/i), we can write a difference equations for the t .(n) in terms of ^(n 1) as follows. This equation shows clearly that the poles of m + n are determined by the zeros of m + n _ 1 and similarly for m_. Therefore by looking at the appropriate single valued function on the Riemann surface which agrees with m + on the upper and /I q + m_ on the lower sheets, we can write the following equation for the p(n) above t -(rc) as p(n)=F(p(tt),p(rc-l)). (47) The explicit form for V can be obtained by combining equations (4) and (18, 19). This equation will provide the analogue of the Dubrovin equation of the continuous case. Acknowledgement The authors would like to thank Prof. S Nag for some useful discussions on the Jacobi-inversion and related concepts. References [1] Antony Anand J, Thesis (in preparation) [2] Carmona R, Springer Lecture Notes in Mathematics, Vol. 1180 (1986) [3] Carmona R and Kotani S, Inverse spectral theory for random Jacobi matrices J. Stat. Phys. 46 (1989) 1091-1114 [4] Craig W, Trace formulas for Schrodinger operators, Commun. Math. Phys. 126 (1989) 379-407 [5] Cycone H L, Froese R, Kirsh W and Simon B, Schrodinger operators, (New York: Springer) (1987) [6] Kac M and Van Moerbecke P, On some periodic Toda lattice, Proc. Natl. Acad. Sci. USA 72 (1975) 1627-1629; The solution of the periodic Toda lattice, Proc. Natl. Acad. Sci. USA 72 (1975) 2879-2880 [7] Kotani S, One dimensional random Schrodinger operators and Herglotz functions, Proc. Taniguchi Symposium (ed.) S A Katata (1987) 443-452 [8] Kotani S and Krishna M, Almost periodicity of some random potentials, J. Fund. Anal 78 (1989) 390-405 [9] Levitan B M, On the closure of finite zone potentials, Math. USSR-Sbornik 51 (1985) 67-89; An inverse problem for the Strum-Liouville operator in the case of finite-zone and infinite-zone potentials, Trudy Moskov Mat. Obsch. 45 (1982) 3-36; Engl. Transl Moscow Math. Soc. 1 (1984) [10] McKean H P and Trubowitz E, Hill's operator and hyper elliptic function theory in the presence of infinitely many branch points, Commun. Pure Appl. Math. 29 (1976) 143-226 [1 1] McKean H P and Van Moerbecke P, The spectrum of Hill's operator, Invent. Math. 30 (1975) 21 7-274 [12] Siegel Carl L, Topics in complex function theory, Vol. II, Inter science tracts in pure and applied mathematics, Vol. 25 (New York: John Wiley) (1971) [13] Simon B, Kotani theory for one dimensional random Jacobi matrices, Commun. Math. Phys. 89 (1983) 227 [14] Simon B, Almost periodic Schrodinger operators: a review, Adv. Appl. Math. 3 (1985) 463-490 [15] Spencer T, The Schrodinger equation with a random potential -a review in Proc. Les H ouches Summer School (1984) [16] Toda M, Theory of non-linear lattices, Solid State Sciences Series (New York: Springer- Verlag) 20 (1989) [17] Van Moerbecke P, The spectrum of Jacobi matrices, Invent. Math. 37 (1976) 45-81 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 3, December 1992, pp 189-200 Printed in India. Three-dimensional diffraction of congressional waves by a rigid cylinder in an ^homogeneous medium B K RAJHANS and S K SAMAL Department of Applied Mathematics, Indian School of Mines, Dhanbad 826004, India MS received 27 February 1991; revised 12 September 1992 Abstract. In this paper we consider the diffraction of compressional waves by a rigid cylinder embedded in an unbounded inhomogeneous elastic medium. The point source, generating the incident pulse, is situated at a finite distance from the obstacle. It is assumed that the velocities of P and S waves are given by a = a r, /? = p r q respectively, q < 1. The formal solutions of displacement field are obtained in the integral form. These integrals are evaluated asymptotically by the Residue Cagniard method to obtain the short-time estimate of the motion near the wave front in the shadow zone of the elastic medium. Numerical computations are done to investigate the behaviour of diffracted P and S waves. Keywords. Diffraction; elastic mhomogeneity; residue-Cagniard method. 1. Introduction Friedlander [3] applied the modified form of Fermat's principle to the problem of diffraction of two-dimensional pulses by a rigid cylinder in a homogeneous medium. Using Friedlander's method Jha and Mishra [5] solved the problem of diffraction of sound pulses by a rigid cylinder in an inhomogeneous medium. Kesari and Rajhans [6] studied the scattering of shear (SV) waves by a rigid cylinder in an inhomogeneous medium using Friedlander's method. Hwang et al [4] applied a similar method and discussed the case of three-dimensional elastic wave scattering by a rigid cylinder in an elastic medium. In this paper we investigate the diffraction of compressional waves by a rigid cylinder in an inhomogeneous medium using the technique of Hwang et al [4]. The point source generating the incident pulse is supposed to be situated in the surrounding elastic medium at a finite distance from the obstacle. It is assumed that the velocities of P and S waves are functions of r only and are given by a = a r* and ft = far* respectively, where q is the inhomogenity factor and < q < 1. This law of variation of velocity has been found to be prevalent in actual earth ([1], [8]) and is more general as it includes the homogeneous case when q = 0. The results show that in the case of a compressional point source there exist in the shadow zone (1) a diffracted PP d P wave, that is, the P wave incident on the surface of the cylinder travels along the cylindrical surface as the P wave and finally emerges as the diffracted P wave, (2) a diffracted PS d S wave, that is, the P wave incident on the surface is converted into the S wave which finally emerges as the diffracted S wave, (3) a diffracted PP d S wave, which denotes that the P wave incident on the 190 B K Rajhans and S K Samal surface travels on the cylindrical surface as the P wave and it emerges as the diffracted S wave after mode conversion ([1], [4]). 2. Formulation of the problem and formal solution Let the axis of the cylinder be taken as the z axis and let the point source be located at r = r , 9 = 0, z = (figure 1). The velocity potentials <p and *F are functions of r, 0, z and t which satisfy the equations 271 d(r~r )5(z)S(t) (r>a) where and The notations used in (1), (2), and (3) are defined as follows V 2 : Laplacian operator 5: Dirac delta function /l,/z: Lame's parameters p: Density of the medium a, j8: Velocities. of P and S waves respectively. Equation (2) can be divided into two parts namely [4] (1) V 2 G--^e = RECEIVER P (r,6,z) (1) (2) (3) COMPRESS! ON AL POINT SOURCE RECEIVER Figure 1. The geometry of the problem. (4) Three-dimensional diffraction 191 (2) V 2 #~~H = 0, (5) where G and H are scalar wave functions. The boundary conditions are where C/ r , U Q and U z are components of displacement. We define Laplace transform pairs with respect to t as f *(r,0,z,s) = <X>(r,0,z,t)exp(-st)dt J oo (7) _ (D(r, 0, z, t) = <&(r, 0, z, s)exp(st)ds, (8) 27"J Vl -;oo where ^ lies to the right of all singularities in the s-plane. We define bilateral Laplace transform with respect to z as f 00 - (r, 0, p, s) = 0(r, 0, z, s)exp( - pzs)dz J -oo (9) 00 <f>(r,0,z,s) = : O(r,0,p,s)exp(psz)d(ps). (10) We further, define bilateral Laplace transform pairs with respect to as f 00 - O*(r,v,p,s)= <E>(r,0 > p,s)exp( v0)d0 (11) J-oo 1/*y3"fioO I ifi*/ \ / /3\r! C12^ 27CI Jy 3 _ io0 Using these transformations in (1), (4) and (5) for m = we get ^ 2 <T>* 1 ^db* / v 2 \ 2n ._ f2 )<!)*= <5(r r n ) \ ^ i ~2 / v v 3r 2 dr 2 where dr 2 r 3r I (14) Also the transformed boundary conditions are 192 B K Rajhans and S K Samal 6 r dr r The solutions of the ordinary differential equations (13), (14) and (15) are modified Bessel functions [9]. A particular solution of (13) is (r<r ) (19) d>* (r, v, p, s) = 2nK iv/1 _ f ^L ) / w _ f ( I**- ) (r > r o) . (20) While the complementary solution is where / v/1 _ 4 and K iv/1 _ q are modified Bessel functions of imaginary order fv/1 <? of the first and second kind respectively, and C t (v) is a constant coefficient to be determined by the boundary conditions. Similarly solutions of (14) and (15) are G*(r, v, p, s) = C 2 (v)X ( -&- } (r ^ a) (21) H*(r, v, p, s) = C 3 (v)X iv/1 _^ ( ^- 1 (r ^ a) (22) where C 2 and C 3 are constants. The constants C l9 C 2 and C 3 can be expressed as C u (v) C 22 (v) ' 2 ~' where Cn(") = Three-dimensional diffraction 193 K K> Finally the exact solutions are x exp (v0 + psz 4- st) dv d (ps) ds (27) r* x exp(v0 4- psz + st)dvd(ps)ds ( 28 ) * x exp(v0 + psz + st)dvd(ps)ds ( 29 ) /..\ / / ^ \ (30) (31) (32) A(v) ' v/1 /Q __3 - * I/ o/C ^^yi-ioo, L 2 = y 2 - foo 3. Evaluation of integrals 194 B K Rajhans and S K Samal s SHADOW I ZONE \ \ \ J^***^^ COMPRESSIONAL POINT SOURCE Figure 2. The Projection of the z = plane. For this we have assumed s, the Laplace transform variable to be large, real and positive. We know that this corresponds to the short-time approximations of pulses and diffraction of short-harmonic waves [3]. For an early time response corresponding to the high frequency approximation, let the radial part of the displacement be (33) For the region of v~f^a, v>f t a as s becomes large, the asymptotic expansion of the modified Bessel function can be expressed as (34) I iv (ti ~ [2 1/2 /(3z 1/3 )]exp(7cv/2)[ I /(0 + 0(03 (35) ) (36) (Q + ff'(0] (37) where (38) 2 1 / 3 (39) A t and B t are Airy functions, and is defined as = 2- i [3* (f cosh $ - sinh ^)] 2/3 cos = v/*. (40) Using the above expressions, we write (23) in the form -q V(0exp(- Jiv/2(l - )). (41) / Three-dimensional diffraction 195 Thus the zeros of A(v) for large s correspond to those of /(), we denote the latter by , which can be related to v n as v n ~/ia + Cn(/i0) 1/3 , n = l,2,3.... (42) The contour in (33) is chosen to be closed in the right half plane for 9 < and in the left half plane for 9 > 0. Here the Jordan's Lemma is applicable. Hence there are no contributions to the integrals of (33) other than the ones arising from the poles. Then the series of residues of (33) can be written as I- n=l xexp(|0|v,,) (43) The motion can be divided into compressional and shear motions. We consider the P-wave, which corresponds to the first term of (43) and denote it as t7 rfPPdP (r, 0, p, s). For further evaluation we have U r PPdP (r, 6>, z, s) = -L : f ^ l/ r>PPd p(r, 0, p, 5) exp(p5z)d( P 5). (44) ^ ni JL 2 Because of the complexity of evaluation of (44) we use a simple formula for the Bessel functions which are ([3]) (45) V 2 ^^ (46) Substituting the above expressions of (45) and (46) into (44) we get x exp[- s(MD p - zp) - s 113 M l/3 E a p ]dp (47) where A 1 = i2 2 a 1/3 r- 1 (r 2 -a 2 r 1/4 (r 2 -a 2 ) 11 * (48) (49) - 2 - 1) 1/2 + (rg<r 2 - 1) 1/2 + A p ] (50) os-^-cos- 1 - (51) o 196 B K Rajhans and S K Samal n ,, = ; 1/2 (52) In (51), A p , the angle in the shadow zone measured from the shadow zone boundary must be positive such that Jordan's Lemma can be applied and (47) is valid in the region of the geometric shadow. Analysing (47), we find that there are four branch points namely p = a ~ 1 r ~ q and p ^Q 1 r~~ q . We choose the branch cuts along the real axis given by | real (p) | < oo and <|real(p)| < oo Change the variables as i p = MD p ~zp. (53) So that path of integration is modified such that F p , which can be represented by a hyperbola in the p-plane as with -2 r 2q x r (54) (55) The point of intersection of the hyperbola with the real axis in the p plane is between the branch point l/a r 4 and the origin (figure 3). Now (47) becomes (56) (57) IMA6 P- PLANE Figure 3. The paths of integration of the P wave (F p ), S wave (F s ) and PP d S wave (T ). Three-dimensional diffraction 197 T! is the first arrival time of the diffracted PP d P waves. Then the result for the calculation of the inverse Laplace transform of the P-wave motion is obtained as M. *>') = : I Imag f" \ l F 2m n = i J T1 J Ll x exp[s(-T p )~s 1/3 M 1/3 E np ]dT p ds, (58) where Imag J is the imaginary part of the integration. Equation (58) can be evaluated by means of the following formula (Ragab [7]). 1 3(6m (59) \:T'~ l A '~/ Here we have m = -, T= t - i p , Q = M ll3 E np H(T) = the Heaviside function. Finally the diffracted PP d P wave can be obtained in the domain as T p )dt p (60) where Now, we propose to obtain the diffracted S waves which correspond to the second term (M-motion) and the third term (Af-motion) of (43). If we follow the technique used above we find, after a little calculation, that these diffracted events are obtained as follows: T,)dt s , (62) where the path of integration for these motions is along T s (figure 3). Similarly the diffracted PP d S waves are obtained as xH(r-t )dz DS (63) 198 B K Rajhans and S K Samal t ps )d V (64) where the path of integrations is along T PPaS (figure 3), (65) (66) = ", A s = |0|(l-, ? )-cos- 1 --cos- 1 - + cos- 1 (d) (67) : = p 3 , T2 and T 3 = the first arrival time of the PS d S and PP d S respectively. s> " (l-q) ' r r ? = j8 /ao ( 68 ) (70) ( ?1 ) j = aA * + ( r o~ a2 ) 1/2 9 J = (r 2 -a 2 d 2 )V 2 -a(l-d 2 ) (?2) (1-^) (1-^) For the path of integration F PPdS (figure 3) (Lr 2 + (/ + J)^(L)- 2 4-i<5 G = (73) T ps with W = (L 2 M^ 2 z 2 ) 1/2 . (74) 2 V c n't' * \- ' 4. Numerical results and discussion The numerical results as shown here are for the case in which all quantities are normalised with respect to the radius a of the cylinder and the wave velocity a . For convenience the material properties A, JLI and p are chosen such that a /j3 = 3 and the Poisson's ratio is assumed to be 1/4. Figures 4, 5 and 6 show the variations of the radial displacement of the diffracted PP d P wave, the diffracted M-type PS d S wave and the diffracted N-type PS d S waves, a function of time. Four cases are plotted with the source position r = 3a, z = 0, = and the observational positions r = 2a, z = a, = 180, 170, 160 and 150. The figures also show that the magnitude of the diffracted PP d P waves changes more rapidly than that of the diffracted M and N type PS d S waves, when 9 changes from 1 50 to 1 80. However, our result shows that PP d P waves do not exist when 9 < 1 50. Three-dimensional diffraction sr 199 r = 3a, z = 0, 6 =0 r = 2a, z = a, 6 = Varied q =-1 Figure 4. Variation of the radial displacement of the diffracted PP d P wave as a function of time. -1-0- J- i,- -2-0 - -3-0- -005 -01 -015 -02 -025 =150 =160 ie< = 170 iei =180 = 3a, z =0,9o = r =2a, z =a, 9 = varied q=-1 Figure 5. Variation of the radial displacement of the diffracted M-type PS d S wave as a function of time. 200 B K Rajhans and S K Samal -0-5 -VO -1-5 r =3a, 6 = 0,z = r = 2a, z = a, 8 = varied q =-1 Figure 6. Variation of the radial displacement of the diffracted JV-type PS d S wave as a function of time. The short time approximations for the diffracted pulses by a rigid cylinder in a homogeneous medium can be obtained by putting q = in our result. The results obtained agree with those obtained by Hwang et al [4]. References [1] Bullen K E, Theory of Seismology, (Cambridge: University Press) (1963) [2] Cagniard L, (New York: McGraw Hill) (1962) [3] Fricdlander F G, Commun. Pure Appl. Math. 1 1954 705-732 [4] Hwang L, Kuo J T and Teng Y, Three dimensional elastic wave scattering and diffractions due to a rigid cylinder embedded in an elastic medium by a point source, Pure Appl Geophys. 120 (1982) 548-576 [5] Jha R N and Mishra S K, Proc. Indian Acad. Sci. A62 (1965) 271-292 [6] Kesari P and Rajhans B K, Int. J. Eng. Sci. 25 (1987) 797-806 [7] Ragab F M, Commun. Pure Appl. Math. 11 (1958) 115-127 [8] Roy A, Bull. Seism. Soc. Am. 59 (1969) 1989 [9] Watson G N, A treatise on the theory of Bessel functions 2nd Edn. Cambridge (1944) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 3, December 1992, pp. 201-215. Printed in India. Topological algebras with C* -enveloping algebras SUBHASH J BHATT and DINESH J KARIA Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar 388 120, India MS received 28 January 1992 Abstract. Let A be a complete topological *algebra which is an inverse limit of Banach *algebras. The (unique) enveloping algebra $ (A) of A, providing a solution of the universal problem for continuous representations of A into bounded Hilbert space operators, is known to be an inverse limit of C*-algebras. It is shown that $(A) is a C*-algebra iff A admits greatest continuous C*-seminorm iff the continuous states (respectively, continuous extreme states) constitute an equicontinuous set. A Q-algebra (i.e., one whose quasiregular elements form an open set) A has C*-enveloping algebra. There exists (i) a Frechet algebra with C*-enveloping algebra that is not a g-algebra under any topology and (ii) a non-<2 spectrally bounded algebra with C*-enveloping algebra. A hermitian algebra with C*-enveloping algebra turns out to be a g-algebra. The property of having C*-enveloping algebra is preserved by projective tensor products and completed quotients, but not by taking closed subalgebras. Several examples of topological algebras with C*-enveloping algebras are discussed. These include several pointwise algebras of functions including well-known test function spaces of distribution theory, abstract Segal algebras and concrete convolution \ algebras of harmonic analysis, certain algebras of analytic functions (with Hadamard product) f and Kothe sequence algebras of infinite type. The enveloping C*-algebra of a hermitian topological algebra with an orthogonal basis is isomorphic to the C*-algebra c of all null sequences. Keywords. C*-enveloping algebra; g-algebra; hermitian algebra; Segal algebra; Kothe sequence space. 1. Introduction A complete locally m-convex * algebra A is a topological * algebra that is an inverse limit of Banach* algebras. In representation theory of such algebras, the enveloping algebra (S(A\ T) of A has been introduced in [10], [21], [16] which provides a solution of the universal problem for continuous Representations of A into bounded Hilbert space operators. This corresponds to the construction of the enveloping C*-algebra of a Banach* algebra [13, 2.7.2, p. 48]. The algebra (S(A\i) is a pro-C*-algebra [27] in the sense that it is an inverse limit of C*-algebras. This paper is concerned with those A for which ((A\i) is a C*-algebra. In fact, in [16], [17], A is called a bQ-algebra if ($(A\ i) is a barreled space that is a Q-algebra (a topological algebra A is a Q-algebra [26] if the set A _ l of all quasiregular elements of A is an open set). The barreled assumption turned out to be redundant; for a pro-C*-algebra, which is a g-algebra, is a C*-algebra [18, Corollary 2.2], [27, Proposition 1.14]. / Topological *algebras with C*-enveloping algebras are important for a couple of reasons. Though non-normed, they are well-behaved. In the literature, bQ-condition 201 202 Subhash J Bhatt and Dinesh J Karia has been assumed in several aspects like tensor products [17], hermitian K-theory [24] and representation theory [16]. In fact, the representation theory of such algebras is quite similar to that of Banach* algebras. Further, as exhibited in the present paper, there are several classes of examples of such algebras arising in function theory, Courier series, abstract harmonic analysis, complex analysis and nuclear spaces, in particular, sequence spaces. In what follows, we briefly describe the contents of the present paper. In [17], the question of completely specifying the class of bQ-algebras was discussed. We show that a complete lmc-*algebra A has C*-enveloping algebra (i.e., A is a bQ-algebra) iff A admits greatest continuous C*-seminorm p x (*) iff the continuous states (respectively, continuous extreme states) constitute an equicontinuous set. This is used to show that the enveloping algebra of a Q lmc-*algebra is a C*-algebra, but the converse does not hold. An lmc-*algebra A is spectrally bounded (sb) (respectively, *spectrally bounded (*sb)) if the spectrum of each xeA (respectively, the spectrum of each element of the form x*x) is bounded. We discuss the examples exhibiting: (i) a Frechet algebra with C*-enveloping algebra, which is not sb, and which fails to be a Q-algebra under any topology and (ii) a non-Q sb algebra with C*-enveloping algebra. However, if A is hermitian and having C*-enveloping algebra, then A is a 2-algebra. Further, it is also shown that if A is *sb, then A admits greatest C*-seminorm -| (not necessarily continuous); and such an A is hermitian iff |-| = s(-) iff s(-) is a C*-seminorm. Here s(x) = r(x*x) 1/2 (xe>4), r(-) denoting the spectral radius. Thus if A is *sb, then A is hermitian and has C*-enveloping algebra iff $() is a continuous C*-seminorm (in which case, sO^PooO)- We also show that if A is Frechet, then (i) A is sb iff A is a g-algebra and (ii) if A is *sb, then A has C*-enveloping algebra. Projective tensor products and complete quotients of algebras with C*-enveloping algebras are algebras with C*-enveloping algebras; but the enveloping algebra of a closed *subalgebra of an algebra with C*-enveloping algebra need not be a C*-algebra. We have also discussed several classes of algebras with C*-enveloping algebras. Notable among these, besides pointwise algebras of functions (including the algebra C ao (X) of smooth functions on a compact manifold) are the various test function spaces of distribution theory, topological Segal algebras [11] of harmonic analysis (in particular, certain convolution group algebras of locally compact groups) and Kothe G^ -sequence algebras [22] (of significance in the theory of nuclear and Schwartz spaces). This also incorporates certain topological algebras with orthogonal bases [15], [20]; and via Fourier expansion and Taylor expansion, algebras of smooth periodic functions (convolution product) and of analytic functions (Hadamard product). The enveloping algebra, of an lmc-*algebra with hermitian orthogonal basis and having C*-enveloping algebra, is *isomorphic to the C*-algebra c of all null scalar sequences. Let us note that the class of topological *algebras with C*-enveloping algebras also include the Frechet algebra of C-elements of automorphic action of Lie group on a C*-algebra, certain x F*-algebras of pseudo- differential operators and the algebra of local observables of quantum field theory. These will be discussed in a subsequent paper. Preliminaries and notations A locally m-convex *algebra (lmc-*algebra) [25], [26], [9], [10] is a linear associative involutive algebra A with complex scalars and with a Hausdorff locally convex Topological algebras with C* -enveloping algebras 203 topology t on it which is determined by a separating directed family P (p a :aeA) of seminorms satisfying, for all a and for all x 9 y 9 p a (xy) ^ p a (x)p a (y) (submultiplicativity) and p a (x*) = p a (x) (*-invariance). Let (e y ) be a bounded approximate identity (bai) for A, i.e. (e y )c:A is a net such that (a) for each xeA, e y x-*x, xe y -+x and (b) for each a, p a (e v ) ^ 1 for all 7. One can take P to be the collection K(A) of all continuous invariant submultiplicative seminorms p satisfying p(e y ) ^ 1 for all y. A pro-C*-algebra [27], [28], [6], [7] (also called an Lm.c. C*-algebra [16] or a locally C*-algebra [21]) is a complete lmc-*algebra A in which each p a is a C*-seminorm, i.e., each p a additionally satisfies p a (x*x) = p a (x) 2 for all xe A Given an lmc-*algebra A and peK(A), let N p = {xeA:p(x)-Q}, and ^4 p be the Banach *algebra obtained by completing the quotient *algebra A/N p in the norm ||x p || p = p(x), x p = x + N p . Forap a eP,letv4 a = yl pa .Ify4iscomplete,thenv4 =ljm a6A v4 a = ]jm p6j5:u) X p , an inverse limit of Banach* algebras [26, Theorem 5.1]. Similarly, a pro-C*-algebra is an inverse limit of C*-algebras. An lmc-*algebra A is hermitian if for each h = ft* in A, the spectrum sp(ft) c R. Let ^4 be a complete lmc-*algebra with a bai (e y ). In representation theory of such algebras [10], [21], [18], the enveloping algebra (^(A\i) of a has been introduced as follows. Let R(,4) (respectively, R'(A)) be the set of all continuous (respectively, continuous topologically irreducible) *representations ii:A-+B(H n ) of A into the C*-algebras B(H n ) of all bounded linear operators on Hilbert spaces H n . For a peK(A) 9 let R p (A) = (neR(A): there exists k > such that || n(x) \\ < kp(x) for all x}, R' p (A) = R p (A)nR'(A) 9 R a (A) = R P .(A) 9 R' x (A) = R' p (A). Then >4)=u a l?;(^) = u{U;(A):p6JC(4)}. For = sup{||7c(x)||:7E6jR^)} [16, Lemma 4.1] (xeA) defines a continuous C*-seminorm on A. Let r a (-) = r pa (-). The *radical of ^1 is the *ideal srzdA = n a N(r x ) = n{N(r p ):peK(A)} 9 where N(r p )= {xe>l:r j ,(x) = 0}. The algebra (f(A) 9 -c) is the Hausdorff completion of (A, {r a }) (equivalently, of (A,{r p :peK(A)})), i.e. the completion of ,4/srad^ in the topology T defined by C*-seminorms q a (x + srad^) = inf {r a (x + 0:*'esrad,4} = r a (x) (xeA). The pro-C*-algebra ((A) 9 T) = Ijm^(y4 p ) = l^m(v4 a ), where $(A P ) is the enveloping C*-algebra of the Banach *algebra A p [13, 2.7.2, p. 48]. Let <t>\A-+g(A) be </>(x) = x-hsradA The algebra ((A) 9 t) satisfies the universal property that, given neR(A) (respectively, iteR'(A)) there exists a unique aeR(S(A)) (respectively, aeR'(S(A))} such that n~(?o<t> [16, p. 69-70]. Further, it is easily seen that (S(A) 9 i) is a unique (up to a homeomorphic isomorphism) pro-C*-algebra satisfying this universal property. Thus the following unambiguously makes sense. DEFINITION A complete lmc-*algebra A has C*-enveloping algebra if((A),i;) is a C*-algebra. 2. Basic theory of algebras with C* -enveloping algebras Throughout the section, A denotes a complete lmc-*algebra with a bai. The following corresponds to the fact that a Banach* algebra admits greatest C*-seminorm (automatically continuous) viz the Gelfand-Naimark pseudonorm [8, 39]. Theorem 2.1. The algebra A has C*-enveloping algebra iff A admits greatest continuous C*-seminorm. In this case, if p^O dentoes the greatest continuous C*-seminorm on A, 204 Subhash J Bhatt and Dinesh J Karia then p^O = sp a r a (x) = sup { \\ n(x) \\ :neR(A)} = sup{ \\ n(x) \\ :rcER f (A)} (xeA); and (S(A\x) is the C*-algebra (A/N(p (X) ))^ the completion of A/N(p x ) in the norm Proof. Observe that on A/smdA, q a (x + srad^l) = r a (x) for each aeA. Indeed, for any xeA, q a (x + smdA) = inf {r a (x + i):iesradA} = inf fe/ [sup{/((x + z)*(x + i)) 1/2 : feP a (A)}-] using r a (z) = sup{/(z*z) 1/2 :/eP a (,4)} [16, Lemma 4.1], where P a (A) denotes the set of all continuous positive linear functional / on A such that I/Ml < PM for a11 ueA - Since tesradA, r a (i) = 0; and so f(i*i) = for all feP a (A). Further, for all such /, by the Cauchy-Schwarz inequality, f(i*x) = =/(x*z) for all xe/4. Hence q a (x + srad^) = inf [sup{/(x*x) +/(i*x) + /(x*0 + f(i*i):ieI} ll2 :feP tt (A)'] = sup{/(x*x) 1 ' 2 :/eP a (,4)} = r a (x). Now suppose that A has C*-enveloping algebra, so that ($(A\i) is a C*-algebra, the topology t being determined by a C*-norm ||-||. By [27, p. 165], for any sup a ^f a (z) < oo, and ||z|| =sup a g a (z). Thus p^x) = ||x + 5radA|| =sup a r a (x) defines a C*-seminorm on A; and there exists k > and aeA such that for all xeA, PooW = II* + sradA \\ ^ kq a (x 4- srad^) = fcr a (x) < /cp a (x) using [16, p. 69]. Let p be any continuous C*-seminorm on A, so that, for some />0 and some /JeA, P(x) ^ lp ft (x) (xeA). Then R p (A) a R^(A) and for all x, r p (x) ^ r p (x). Identifying R p (A) and R(A p ) canonically [16, Proposition 3.5] and using that A p is a C*-algebra; it follows that for each x, p(x)= ||x-f A/ r (p)|| p = sup{||7r(x4-N(p))||:7re J R(^ p )} = sup{||7u(x)||:7rejR p (A)}=r p (x)<r / ,(xXp 00 (x). Thus p^Q is the greatest continuous C*-seminorm on A. Conversely, let A admit greatest continuous C*-seminorm, say p^O- There exist jSeA, k> such that for all xeA, p (X (x) ^ fcp y3 (x). Hence, as above p^x) ^ r^(x) and so Poo(x) = r^(x) for all x. Since each r a (-) satisfies ^ a (x)^p a (x) (xeA) [16, p. 69], r aW ^Poo W for all a, for all x. Thus p^x) = sup a r a (x) (xeA), st&dA = N(p 00 ) ) and for any x, ||x + sradX|| px = p 00 (x) = sup a q a (x + sradA) = r^(x) = ^(x + sradA). It follows that the topology T on g(A) is determined by ||-||^; and then (<f(A) 9 r) is a C*-algebra. This completes the proof. COROLLARY 2.2. // A is a Q-algebra, then A has C* '--enveloping algebra. Proof. Let A be a Q-algebra. By [26, Lemma E.3] A is sb; and [26, Proposition 13.5] implies that there exists an a eA and k> such that r(x) ^ fcp ao (x) for all x. Let q be any continuous C*-seminorm on A. There exists peK(A) and M >0 such that q(x) < Mp(x) for all x. Then for any h = h*mA and for n = 1, 2, 3, . . . , q(h) = q(h 2 ") 1/2n < M 1/2 >(/i 2ri ) 1/2M . By the spectral radius formula [26, p. 22], ^(ftXlim^^supM 172 "- p a (ft 2M ) 1 / 2n ^sup p6 ^ ) lim^ w supp(/i") 1/ " = r(/i)^/cp ao (/2). Hence for any xeA, q(x) = ^(x*x) 1/2 ^/c 1/2 p ao (x). Thus Poo(x) = sup{^(x):qf is a continuous C*-seminorm on A} ^fc 1/2 p ao (x)(xeX), and p^Q is the greatest continuous C*-seminorm. Let A be commutative. Let Jf(A) be the Gelfand space (with weak* topology) of A consisting of all continuous multiplicative linear functionals on A. For cj)eJf(A), Topological algebras with C* -enveloping algebras 205 let 0*G^(>1) be defined as 0*(x) = </>(x*). The hermitian Gelfand space of A is ^*(A) = {(/)eJ?(A):<j) = </>*}. The following can be shown, as in [8, Theorem 40.2, p. 220] using the machinery in [16]. Lemma 2.3. Let A be commutative. Let neR'(A). Then n is one dimensional, and there exists (f)eJP*(A) such that n(x) = </>*(x)l for all xeA. Example 2.4. There exists a unital commutative Frechet *algebra B with C*-enveloping algebra such that B is not sb and B fails to be a Q-algebra (under any topology). Let U = {zeC: 1 < Rez < 1}. Let C(U) be the a]gebra, with pointwise operations, of all continuous complex valued functions on U with compact open topology t. Let B = {feC(U):f is analytic in U}. Then (B,t) is a Frechet *algebra with involution /->/*, /*(z) = /(z) (zeU). The topology t is defined by the family of seminorms P = (p n :n = 0,1, 2,...), where /?(/) = sup {\f(z)\:zeK n }, K n = {zeU:n^ Imz<n + 1}. It is easily seen that </>e^(B) iff <t> = <j> z for some zet/, ^(/)=/(^); and c^*(B) = {0 r :z is real; 1 ^z< 1}. In view of Theorem 2.1 and Lemma 2.3, Poo(/) = su Pn r p(/) = SU P{I/( Z )I : ~~ l^z<l}<oo (feB) defines greatest continuous C*-seminorm; and ((B\t) is the supnorm C*-algebra C[ 1,1] of all continuous functions on [1,1]. By 26, Corollary 5.6], for any /e#, the spectrum sp(/) = {f(z):z<=U}. Thus B is not sb; hence it fails to be a g-algebra under any topology making it a topological algebra [26, Appendix E]. For xeA, let the hermitian spectral radius of x be defined as r h (x) = sup{r(7r(x)): neR'(A)}, r(n(x)) being the spectral radius of the operator n(x) in the C*-algebra COROLLARY 2.5. The algebra A has C* -enveloping algebra iff there exists peK(A) and k >0 such that r h (x) ^ kp(x) for all xeA. Proof. If A has C*-enveloping algebra, then there exists peK(A) and k > such that Poo W < kp(x) for all x. It follows that r^x) ^ fcp(x) for all x. Conversely, suppose that there exists peK(A) and k> such that r h (x) ^ fcp(x) for all x. Let q be any continuous C*-seminorm on A. Let aeK q (A). Then, for any xeA, \\ a(x) || 2 = || a(x*x) || < r(<r(x*x)) < r*(x*x) < kp(x*x) ^ kp(x) 2 giving <reR' p (A). Thus R' q (A) <=: R' p (A), with the result, for each xeA, q(x) = r q (x)^r p (x). It follows that r p (-) is the greatest continuous C*-semmorm on A; and A has C*-enveloping algebra. COROLLARY 2.6. Let A be a hermitian algebra with C* -enveloping algebra. Then A is a Q-algebra. Proof. By [26, Theorem 5.2], the hermiticity of A implies that, for each qeK(A\ the Banach* algebra (A q , \\ \\ q ) is hermitian. Hence, by [8, Lemma 41.2], for each zeA q , the spectral radius in A q , r A(i (z) < r Aq (z*z) 1 ' 2 = \z q \ q , where \-\ q denotes the Gelfand-Naimark pseudonorm on A q . Then m q (x) = \x q \ q (xeA) defines a continuous C*-seminorm on A. By Theorem 2.1, there exists greatest continuous C*-seminorm p^O on A. By [26, Corollary 5.3], for each xeA 9 the spectral radius in A, r(x) = sup{r Aq (x q ):qK(A)} < 206 Subhash J Bhatt and Dinesh J Karia sup{m q (x):qeK(A)} ^p ao (x). By continuity of p^O, there exists peK(A) and k>Q such that r(x)^p ao (x)^kp(x) (xeA). It follows from [26, Proposition 13.5] (or [3, Theorem 14]) that A is a Q-algebra. Recall that P a (/4) is the set of all continuous positive linear functionals / on A such that \f(x)\ ^ p a (x) for all x. As in [16, Theorem 3.1], the bijective correspondence /-/ a :x a -+/ a (x a ) =/(x) (xeA) 9 identifies P a (/4) with the set P(A X ) of all positive linear functionals (automatically continuous) on the Banach* algebra A x (having bai ((e y )J). Let P C (A) = u a P a G4).'The following identifies P C (A) intrinsically. Lemma 2.7. Let f be a continuous positive linear functional on A. Then feP c (A) iff I/Ml 2 **f(x*x) for all xeA. Proof. Let there be an aeA such that feP a (A\ so that |/(x)| ^ p a (x) (xeA). Then, for any xeA, by the continuity of/ and Cauchy-Schwarz inequality, \f(x)\ 2 = lim \f(e y x)\ 2 ^ (lhn~ y /(e y **))/(x*x) ^ (Hmp a (e y e*))/(x*x) ^ (]tap a (e y ) 2 )/(x*x) < /(x*x). Conversely, assume that |/(x)| 2 </(x*x) (xeA). Since /is continuous and P = (p a :aeA) is directed, there exists /c>0 and aeA such that \f(x)\^kp x (x) (xeA). Thus for any x, \f(x) 2 \^f(x*x)^k Pa (x*x)^kp a (x) 2 i hence by iterations, |/(x)| </c 1/2 > a (x) (xeA, neN). It follows that /eP a (A) c P C (A). For each a, let B X (A) be the set of all nonzero extreme points of P X (A). Let B C (A) be the nonzero extreme points of P C (A). As in [16], B C (A) = u a B a (/4). Let S c (-4) denote the set of all continuous C*-seminorms on A. The following is immediate in view of [16, Lemma 4.1] and Theorem 2.1. COROLLARY 2.8. Let A have C*-enveloping algebra. ThenforeachxeA^p^x) sup{f(x*x) 1/2 :feP c (A)} = sup{f(x*x)V 2 :feB c (A)} = su P {p(x):peS c (A)}. COROLLARY 2.9. The algebra A has C*-enveloping algebra iff P C (A) is equicontinuous iff B C (A) is equicontinuous. Proof. Let A have C*-enveloping algebra, so that by Theorem 2.1, the topology T on* A/sradA is determined by the C*-norm ||x + 5rad^i|| pro = p 00 (x) (xeA); and there is c>Q and peK(A) such that p 00 (x)^cp(x) (xeA). Since" the quotient topology t q on A/ s rad A induced by the topology t of A is finer than T, it follows that for a given bai (e y ) of A, there exists k> such that p^(e y ) = \\ e y -f srad^ \\ ^ k for all y. Let feP c (A). By Corollary 2.8 and the Cauchy-Schwarz inequality, it follows that for any xeA, \f(x)\ = lim y \f(e y x)\ ^ (lim y sup/( e *e y ) 1/2 )(/(x*x)) 1 / 2 < (lim y sup Poo - (e*e y ) l/2 )p x> (x)^kp 00 (x)^kcp(x). Thus P C (A) (hence B C (A)) is equicontinuous. Conversely, let B C (A) (or P C (A)) be equicontinuous, so that, there exists peK(A) and fc>0 such that |/(x)| ^fcp(x) for all xeA, feB c (A). Then the quantity q( x ) = sup{f(x*x) l/2 :feB c (A)} < oo (xeA) defines greatest continuous C*-seminorm on A. Thus Theorem 2.1 applies. COROLLARY 2. 10. Let A be commutative. Then A has C*-enveloping algebra iff the hermitian Gelfand space W- Topological algebras with C*-enveloping algebras 207 is equicontinuous. In this case, for each xeA, Pao ( x ) = sup{\f(x)\-feJt*(A)\ = lemark 2.11. There is an analogy between Q Imc algebras and lmc-*algebras with :*-enveloping algebras. Corollaries 2.5 and 2.9 correspond to the fact that B is a )-algebra iff there is a continuous submultiplicative seminorm p on B and k > such hat r(x)^fcp(x) (xsX) iff (in commutative case) ^T(A) is equicontinuous [3]. Analogous to Theorem 2.1, it holds that a commutative B is a Q-algebra iff B admits greatest continuous submultiplicative seminorm q with square property q(x 2 ) = l(x) 2 (xeA). The details will appear elsewhere. Remark 2.12. It follows from [17, Theorem 4.1] that if A and B have C*-enveloping ilgebras, then so^ does the completed projective tensor product 40J3; and f(A n B) = (A) Tm (B) 9 the maximal C*-tensor product. Also, if / is a closed egular *ideal of A, then A has C*-enveloping algebra iff /and the completion of 4/J have C*-enveloping algebras. Example 2.13. The purpose of this example is to exhibit that (a) there exists a commutative Frechet *algebra B and a closed *subalgebra D such that B has ^-enveloping algebra but D fails to have C*-enveloping algebra, (b) there exists a commutative non-Q-algebra B with C*-enveloping algebra such that B is strongly ipectrally bounded (ssb) [17], i.e. for some family Q = (q$)^K(B) determining the :opology of B, sup^ q 6 (x) < oo for all x eB. The example is a modification of [8, Example 16, p. 202]. Let C be a complete lmc-*algebra having identity with a P = (p^ c K(C) ietermining its topology. Let B = C C with the product topology defined by the jeminorms g a ((x,30) = max(p a (x),p a (y)). The involution (x,j;)* = (y* 9 x*) makes B a complete unital lmc-*algebra on which /(z*z) = (zeB) for any positive linear 'unctional / on B. Thus JR(B) = {0}, B = srad B and f(B) = (0), the trivial C*-algebra. i) Let D = {(x,x)eB:xeC}, a closed *subalgebra of B, homeomorphically *isomorphic to A. Take C to be the Frechet *algebra C(R) of all continuous functions DH R with pointwise operations, complex conjugation and with the compact open topology k. The resulting algebra D does not have C*-enveloping algebra. li) Note that A is ssb (respectively, g-algebra) iff B is ssb (respectively, Q-algebra). Take C to be the *algebra C[0,l] of all continuous functions of [0,1] with the topology of uniform convergence on all countable compact subsets of [0, 1]. The resulting algebra B is non-Q, ssb and having C*-enveloping algebra. Let / be a positive linear functional, not necessarily continuous, on A. The GNS construction (n f9 D(n f ),H f ) defines a *homomorphism n f of A into linear operators (not necessarily bounded) all defined on a dense invariant subspace D(itj) of a Hilbert space H f as follows: Let N f = {x 6 4:/(x*x) = 0}, D(n f ) = A/N f with inner product <x + ]Y / ,j;-h]V / >=/(y*x),H / = the Hilbert space obtained by completing D(it f ), and n f (x)(y 4- N f ) = xy + N f . Further, each ^(x) is a bounded operator (so that, by extension, L(x)eB(H f )\ iff / is admissible i.e., for each x, sup {/0>*x*xy)//(y*y): fly* y ) ^ o, y eA} 1/2 ( = || n f (x) || ) < oo . Also, / is extendible if / can be extended as a positive linear functional on the *algebra A e obtained by adjoining the identity to A. As a consequence of the presence of a bai on A, every continuous positive functional f on A is extendible and hence admissible by Propositions 3.2 and 3.3 of [2]. Let 208 Subhash J Bhatt and Dinesh J Karia P(A) be the set of all positive linear functional / on A satisfying |/(x)| 2 ^ /(x*x) (xeA). Let AP(A) = {feP(A)\f is admissible}, S(A) be the set of all C*-seminorms on A (not necessarily continuous). For feAP(A), p f (x) = || n f (x) \\ gives a p f eS(A). For xeA, define l(x) = mp{p(x):peS(A)} 9 u(x) = sup{p f (x):feAP(A)}, m(x) = sup{/(x*x) 1/2 : Theorem 2.14. (1) // the algebra A is *sb, then A admits greatest C*-seminorm \-\ (say). In this case, \x\ = l(x) = u(x) = m(x) /or a// xeA (2) Let >4 6e *sb. The following are equivalent. (i) /4 is hermitian. (ii) s(x) = |x| /or a// x. (iii) x-*s(x) is a C*-seminorm on A. If, moreover, A is hermitian, then A is sb and r(x) ^ s(x) (xeA). (3) Let A be *sfe and have C* -enveloping algebra. The following are equivalent. (i) A is hermitian. (ii) s(x) = p ao (x)forallx. (iii) x >s(x) is a continuous C*-seminorm on A. If A is commutative and hermitian, then r(x) = p^(x) for all x. Lemma 2.15. (1) The algebra A is hermitian iff A is symmetric. (2) Let a = a* in A with r(a)< 1. There exists x = x* in A with r(x)< 1 such that 2x - x 2 = a. (3) Let feP(A\ be A, f b (x) = f(b*xb) (xeA). Then the following hold. (i) '\f b (k)\ ^ r(k)f(b*b) for all k = k* in A. (ii) |/ & (a)| < s(a)f(b*b) for all a in A. (4) Given peS(A\ be A, there exists feAP(A) such that |/(x)|<p(x) for all xeA and /(&*&) = p(b*fc). (5) Let A be Frechet. Each peS(A) is continuous. (6) Let A be *sb. For all peS(A), p(x)^ s(x) for all xeA. (7) Let A be *sb and hermitian. The following hold. (i) r(x)^s(x) for all xeA. (ii) x-s(x) is a C*-seminorm on A. Proof of the lemma. We prove (2). Let 0* m (A) denote the collection of all families Q = (q d )aK(A) such that Q determines the topology of A. Given such a Q, the *subalgebra B Q = {xeA:sup d q 6 (x)<ao} is a Banach* algebra with the norm qf(x) = sup (5 ^(x), xeB Q . Now let a = a*eA and r(a)< 1. By [19, Theorem 4], there exists a Qe0> m (A) such that aeB Q and the spectral radius of a in B Q , r BQ (a) < 1. Ford's square root lemma [8, Proposition 12.11] applied to (B Q ,q) gives x = x* in B Q with 2x x 2 = a and r(x) = r Bg (x) < 1. This gives (2). The assertion (5) is a consequence of the automatic continuity of the homomorphism n:A-+A p from a Frechet *algebra with a bai to a C*-algebra. The remaining assertions can be proved by using inverse limit decomposition of A and using corresponding results for Banach* algebras from [8]. Proof of Theorem 2.14. Given peS(A\ beA, there exists feAP(A) as in Lemma 2.15(4). Then p f (b) 2 = p f (b*b)= \\n f (b*b)\\ =sup{/(x*b*bx):/(x*x) = 1} ^ showing l(b)^u(b). Thus u(x) = l(x) for all xeA. Similarly, Topological algebras with C* -enveloping algebras 209 p(b) 2 =f(b*b)^m(b)\ so that l(x)^m(x) for all x. Let feAP(A) and f eH f be a topologically cyclic vector of norm 1 for n f so that f(x) = <7t / (x)iJ / ,^ / > (xe/1). Then /(x*x) = || ^(x) ^ || 2 ^ py-(x) 2 ^ w(x) 2 implies m(x) ^ w(x). Thus, for all x, l(x) = m(x) = w(x) = x| (say), which is the greatest C*-serninorm, if it exists. Assuming A to be *sb, one gets |x| ^ s(x) for all x, as follows: For feAP(A), xeA, the boundedness of n f (x) implies that p f (x) = \\n f ( x )\\ = /Yy*x*xy) 1/2 by Lemma 2.15(3). (2) follows from Lemma 2.15 and arguments similar to those in Banach* algebras [18, Theorem 41. 11]. (3) Let A be *sb and have a C*-envel oping algebra. Then (ii) implies (i) follows from above (2). Conversely, if A is hermitian, then each A x is hermitian, hence symmetric by the Shirali-Ford Theorem, and so by the Raikov symmetry criterion [29, Theorem 4.7.21], for all xeA, r(x*x) = sup^^x^xj = sup a [sup{/(x*x a ):/eJ3(;4 a )}] = sup a [sup- {f(x*x):fEB(A)}] = sup{f(x*x):feB c (A)}==p ao (x) 2 by Corollary 2.8. This, with Lemma 2.15(6), gives s(x) = Poo(x) for all x. That (ii) implies (iii) is immediate and (iii) implies (i) follows from (2). If A is commutative, then using [26, 5], it is easily seen, as in [8, Theorem 35.3], that A is hermitian iff Jit (A) = Jf*(A); and if A is *sfc, then this holds iff r(x*x) = r(x) 2 for all x. Thus x ->r(x) = s(x) is a C*-seminorm dominated by a peK(A) 9 as A is also g-algebra by Corollary 2.6. Thus r(x) = p^x) for all x. Remarks 2.16. (1) The pro-C*-algebra C[0, 1] of continuous functions on [0, 1] with the topology of uniform convergence on all countable compact subsets of [0, 1] admits greatest C*-seminorm, but fails to admit greatest continuous C*-seminorm. (2) It follows from Theorem 2.15 that a *sb Frechet *algebra has C*-enveloping -. algebra. This is analogous to the result that a sb Frechet algebra is a <2-algebra [3, Theorem 1]. (3) The Frechet *algebra B of Example 2.4 has C*-enveloping algebra, it admits greatest C*-seminorm, but is not *sb (as is exhibited by the function /(z) = z in B). 3. Examples: function algebras Throughout this section, we consider *algebras of functions with pointwise operations and complex conjugation as the involution (except in Example 3.5). 3. 1 Let X be a compact, second countable C 00 -manifold. Let C (X) be the *algebra of all C -functions on X with the topology of uniform convergence on X of functions and all their derivatives. It is a Frechet sb hermitian g-algebra, and &(C*(X)) = C(X\ the 210 Subhash J Bhatt and Dinesh J Karia supnorm C*-algebra of all continuous functions on X. If A is a complete lmc-*algebra with C*-enveloping algebra, then by [17, Corollary 4.3] (C* > (X) rt A) = S > (C^(X))\,S > (A) = C (X) $ (A) = C(X,S(A)), ^(A)-valued continuous functions on*. 3.2 Let CB fc ) =1 be a sequence of commutative Frechet lmc-*algebras with identities. Let A n = B l @"-B n with the product topology and the natural involution. Then (A n )* is an m-compatible sequence, and A = v* =1 A n is an involutive LF-algebra. It is a Q-algebra (hence has C*-enveloping algebra), if each A n is a g-algebra (in particular, each B k is a Banach* algebra) [26, Proposition 15.8]. In particular, the algebra C C ([R") of continuous function on [R" with compact supports and with the measure topology [26, Example 3.5], as well as the test function algebras C*((R n ), 1 ^ k ^ oo of (^-functions with compact supports and. with the Schwartz topologies are Q-algebras. One has $ (C c (R n )) = g (C*(R")) = <f(C?(R")) = C (R n ), the C*-algebra of continuous functions on R n vanishing at infinity. The Frechet *algebra s(R n ) of rapidly decreasing C 00 -functions on R n -with the Schwartz topology in an AE-algebra [12, p. 89], hence is Imc [12, Proposition 1], and <?(s(R")) = C (R n ). 3.3 The constructions, analogous to the one due to Arens [1], also lead to several algebras with C*-enveloping algebras. For l^p<oo, let AC P [Q, 1] = {/eC[0, 1]: the derivative /' exists a.e. and /'eL p [0, 1]}, a Banach* algebra with norm I/I P = 11/11 co +(JJI/ / Wi p dt) 1/p , || -|| co bein S the supnorm on C[0,l]. Let AC"[Q 9 1] = 3, 1], a Frechet *algebra, with topology defined by /->|/| p , 1 ^p < oo; ^m p ^C p [0,l]. Since J?(AC P [Q, 1]) = [0, 1] by [29, p. 303]. Jt(AC<\, 1]) = [0, 1] by [26, Proposition 7.5]. The algebra AC W [0, 1] is hermitian Q-algebra, and (AC*[Q, 1]) = C[0, 1]. One can also consider the Sobolev spaces W p , fc [0, 1] = {/eC fc-1 [0, l]:/ (fc " 1} e^C[0, 1] and / (fc) eL p [0, 1]}, which are Banach* algebras with norms A i fW(t\\ / r 1 \ i /p ll/llp. k = and analogously construct the Sobolev-Arens algebras W mJl ,[Q 9 1] = n Up<00 ^^[0, 1]. 3.4 Let C b (R) be the C*-algebra of all bounded continuous functions on R. Let BV loc C b (U) = {feC b (R):f is of bounded variation on [~n,n] for all n= 1,2,3,...}, a Frechet Imc *algebra having seminorms p n (f) =11/11^+ V n (f), V n (f) denoting the total variation of / on [-n,n]. One has S'(BV loc C b (U)) = C b (R). 3.5 Let U = {zeC\\z\ < 1}, H(U) be the algebra, with pointwise operations, of all holomorphic functions on U. Let A n (U)= {/e#(C/):/ (/c) has continuous extension on U for all k, ^ k ^ n}, a Banach* algebra with involution /*(z) =/() and norm ||/|| n = sup re ^SJ| =0 (l/fc!)|/ w (z)|. The Frechet *algebra A^(U) = n^ =0 /l n (L r ), with the topology defined by /-Hi/IL = 0,1,2,..., is a non-hermitian Q-algebra with 4. Segal Algebras The following is a modification of the definition in [1 1] tailored for the present set up. Topological algebras with C* -enveloping algebras 211 DEFINITION 4.1. Let (A, \\-\\) be a Banach* algebra with a bai. A *subalgebra B of A is an ,4-Segal *algebra, if there exists a topology T on B satisfying the following: (a) B is a dense *ideal in A. (b) (B, T) is a complete Imc *algebra with a bai. (c) The inclusion (B 9 t)-+(A, \\-\\) is continuous. (d) The multiplication (A 9 1|-||) x (J3, T) -> (B, T) is continuous. PROPOSITION 4.2. // B is an A-Segal *algebra, then B is a Q-algebra\ and $(B) = S(A\ the enveloping C* -algebra of A. Proof. B being an ideal, B_ l = A _ 1 r\B\ which is open in (B, T) by above (c). We show that for continuous (with respective topologies) topologically irreducible ^presenta- tions, R'(A) = R'(B) via restriction map. By (c), there exists p eK(B)(= K(B 9 r)) such that Hxll ^ kpo(x)(xeB\ with the result, neR'(A) implies ||TT(;C)|| ^ ||x|| ^ kp Q (x) (xeB) and 7r| B eR'(J3) in view of (a). Let neR'(B). Let ^eH^ be any topologically cyclic vector for n so that H n = closed span of {n(B)}. Let (e y ) be a bai for B. Let xeA, yeB. Then xyeBi and e y y ~> y in (B, T). By (d), xe v y -+ xy in (B, t); hence || n (xe y y) n(xy) \\ -> 0. Since (e y ) is i-bounded, || n(xe y ) \\^M X < oo. Thus || n(xy)\\ = lim y || 7r(xe r )7r(y)(^ || ^ ^xll^M^II- Thus the bounded linear operator n(x):n(y)->n(xy), defines neR'(A\ n\ B = TC. Thus jR'^4) = R'(B). Let P = (p a ) c K(B,r) determine T on B. Then, for any yeB, p co () ; ) = su P a r a (3 ; ) = su Pa{ su P^ J R'(B)ll 7I WII} =sup{||7i(}?)||:7TejR / (#)} =sup{||7c(y)H: neR'(A)} < ||j;|| < kp (y); and the greatest continuous C*-seminorm p^-) on (J5,t) is the restriction of the Gelfand-Naimark pseudonorm (denoted by p^*) only). Finally, we show that t(B) = [B/(N (pj nB), II'I! P J^= (A/(N(pJ) 9 \\'\\J~= *(A). The map <5f>(x -f (N(p X) )r^B)) = x H- N(p^)is a well defined *isomorphism of B/(N(p (Xi )r^B) into A/N(p 00 ). Thus ^(B) is a C*-subalgebra of <f (/4). Let zeS > (A). There exist sequences (x n ) in y4, (};) in B such that Pir ' B Then, 2 n showing that <$(B) is dense in g(A). Examples: Convolution algebras. For various group algebras on locally compact groups, we take convolution multiplication and the involution /* (g) = A(g ~ x )f(g ~ 1 ), A being the modular function. Throughout, || || p denotes the usual norm on I/-space. 212 Subhash J Bhatt and Dinesh J Karia 4.3 For a locally compact abelian group G, take A = L l (G\ B = L W (G): = {feL l (G)\ /G L P (G) for all p, 1 < p < oo}, the topology T on is determined by submultiplicative seminorms /?*(/)= ll/lii + II/IU, /c = 2,3,4,.... Then <?(J3) = C*(G), the group C*-algebra of G. 4.4 For A = L l (R), B^{feL 1 (K):feC co (R) and the derivative f^eL^R) for all n= 1,2,...}. The topology T on B is defined by p k (/) = \\fh + ||/ fc || l5 fc = 0, 1,2,.... That p k (f*g)^p k (f)Pk(g) is a consequence of the identity (f*g) (k) =f (k) *g=f*g (k) . Then B is an ,4-Segal *algebra. 4.5 Let G be a compact group. Let (S, |-|) be a Banach* algebra with a bai. For K p < oo, B P (G> S) be the Banach* algebra of functions /:G-*S with |/| p = [f |/(0)| p d/x] 1/p < oo. Then B P (G,S) can be realized as a suitable completed tensor product L P (G) P S, with the norm r\ p () defined by taking a finite tensor /= Sx^^, as r\ p (f) = ^\Lx i (g)y i \ p dfi] i/p . By [23, Proposition 7.10], for all p, ff (B P (G,S)) = C*(G) mil /(S). Taking A = B^G^S), B = B (0 (G,S)=n l<p<ao B p (G,S) = lim p B p (G,S) with the topology of ||-|| ^convergence for each p 9 (B(G, S)) = C*(G) (g> 4.6 Let (^4, ||-||) be a commutative hermitian Banach* algebra with a bai. Let \JL be a positive regular Borel measure on Jt(A) = ^*(^). For 1 ^p < oo, let A P (JJ) = {xeA: xeL p (Jt(A),iJi)}, x denoting the Gelfand transform of x. It is a Banach* algebra with norm \\x\\ Ap = ||x|| + ||x|| p . For the 4-Segal *algebra B = n 1<p<00 ^ p (^), (B) = ^(A). In particular, for a locally compact abelian group G with dual group G and Haar measure \JL on G, consider A = L l (G), B = {/e L 2 (G): Fourier transform / is in L P (G, /x), oo}. 5. Topological algebras with bases and Kothe sequence algebras Let co denote the *algebra of all scalar sequences (a n ) with pointwise operations and complex conjugation. Let Q a co be a Kothe power set, i.e. Q satisfies (i) for each aeQ, a n ^Q for all n; (ii) for each a eg, beQ, there exists ceg such that a n ^ c n , b n ^ c n for all n; and (iii) for each n, there exists aeQ satisfying a n > 0. We assume a l ^ for all aeQ. Further, let Q satisfy G^-property, i.e. (iv) for each aeQ, a n ^a n+l for all n; (v) for each aeQ, there exists deQ such that a 2 n ^ d n for all n. Kothe space of infinite type ([22], [32, p. 203]) is the complete locally convex space A 00 (Q) = (x = (x M )eco: p a (x) = Z|x n |<2 n < oo for all aeQ} with the locally convex Kothe topology t defined by the family T (p a :aeQ) of seminorms. It so turns out [4] that (A 00 (Q),t) is a complete lmc-*algebra which is a *subalgebra of (/\ || || l ) and which is a g-algebra. (Note that a complete commutative continuous inverse Q-algebra is Imc [31].) Further, it is hermitian. For the present purpose, we note the following. PROPOSITION 5.1. (i) A 00 (0 is an t l -Segal * algebra. (ii) <?(A 00 (Q)) = c , t/i^ C*-algebra of all null sequences. (i) can be easily checked; whereas -(ii) follows from a more general result to follow. The following important particular case of A 00 (Q) we shall need. Topological algebras with C* ''-enveloping algebras 213 5.2 (A a, [#]): For an increasing sequence (9 n ) of positive numbers, Q = {(fc a ")^L x :/c = 1,2,...} gives the algebra A^Q) denoted by A^flJ called power series space of infinite type [32, p. 204]. The algebra 5 = {x:a):!,* =l n k \x n \ < oo for all k = 1,2,...} of rapidly decreasing sequences is A^^J taking n = logw, n= 1,2,... [32, p. 205]. Recall [5, 1] that an orthogonal basis on a topological algebra A is a basis (f n ) for A such that f n f m = (5 nm / w for all n, m in N, <5 nm being the Kronecker delta. The algebra A 00 (Q) admits f n = (^ nm )^ =1 as an orthogonal basis. PROPOSITION 5.3. Let A be a complete hermitian lmc-*algebra with C*-enveloping algebra. Let A admit an orthogonal basis consisting of hermitian elements. Then A is a Q-algebra, and is *isomorphic to the C*-algebra c . Proof. Let (/) be an orthogonal basis for A 9 f* = f n for all n. Then (/) is a Schauder basis and A is commutative [5], [20]. Let (/) be the coefficient functional defined by / viz expanding xeA as x = E w x w / w , x n eC, <j) n (x) = x n . Then the Gelfand space ^(/4) = {</>} = JP*(A) by hermiticity; and 4>:A-+co, </>(x) = (x n ) is a isomorphism of A onto a *subalgebra K of CD; which is continuous for the topology of pointwise convergence on K [20, Corollary 1.3]. We identify A with K algebraically. By [26, Corollary 5.6], for all xeA, the spectrum sp(x) = {(j) n (x)} = {x n }; and by Corollary 2.10 and Thorem 2.1, p 00 (x) = sup n |x ll |= ||x|| OC) = r(x)<oo for all x. Thus Ac:/ 00 . By Corollary 2.6, A is a Q-algebra. Further, A contains the set of all finitely many nonzero sequences. Also, A cannot have identity, otherwise A has to be co with pointwise convergence [20, Theorem 2.1], which is not an algebra with C*-enveloping algebra. It follows that S(A\ which is the completion of (A 9 IHU), contains c . On the other hand, p^(-) being a continuous C*-seminorm on A (in the topology of A), x (n) = " 1 ;c fc / /c -x = If x fc / fc in .4 implies that |x w + 1 | ^sup fc> Jx fc | =p 00 (x- x (rt) )-^0 as n - CXD. Thus x = (xjec , (^(A) c c with the result, <^(yl) = c . Example 5.4. Let A = C(r), the convolution algebra of all C -functions on the circle F with involution u*(z) = u(z~ l ). By [14, p. 48], for any weC(r), the Fourier series expansion u = I* (x> u(ri)e ml gives a sequence (w(n))es(Z), = two sided rapidly decreasing sequences. The map $:C co (r)-+s(Z), <(u) = (tf(n)) establishes a *isomorphism of C(r) onto 5, which is a homeomorphism for the (usual) Frechet C-topology on C(r) and Frechet Kothe topology on s(Z) [30, Theorem 5.1]. Now, s(T) is a g-algebra and (s(Z)) = c Q . Thus, via fa C(r) is a complete Q lmc-*algebra with ^(C^OT)) = {^ePM(r):(/i(n))ec }, where PM(F) is the convolution algebra of all pseudo measures on F, isomorphic to f via Fourier expansion [14, 12.11]. Example 5.5. For the open unit disc (7, let H P (U) be the Hardy space, for 1 < p < oo. The Banach space (H P (U), \\-\\ p ) is a Banach* algebra with Hadamard product having involution f*(z) f(z). The sequence e n (z) = z n is an orthogonal basis for H P (U) [20, Example 3]. Thus, the Hardy-Arens algebra H 0) (l/)= n l<p<(X H p (U) is a 214 Subhash J Bhatt and Dinesh J Karia Frechet lmc-*algebra with basis (e n \ the topology being the topology of || -lip-convergence for each p, l<p<oo. The coefficient functional n are B (/)=/ <B) (0)/n!, the nth Taylor coefficient of /(exactly as in [15, Example- 3.2(ii)] for the *algebra H(L/)); and for any feH"(U), sp(/) = {<(/)}. It is easily seen that H(V) is hermitian, and Pao (/) = sup n (l/ (ri) (0)l/n!)^ ||/|| p (p>l) is the greatest continuous C*-seminorm. Example 5.6. Let E be the Frechet space of all entire functions of one complex variable with compact open topology. It is a topological *algebra with Hadamard product and the involution /*(z) = /(), admitting orthogonal basis e n - z", neN. The mapping (/>:-> co, 0(Sx n e B ) = (xj is a "isomorphism of onto the sequence algebra A^O]. Also, </> is a homeomorphism for the respective topologies on E and A^n] [32, p. 206]. Thus ef () is *isomorphic to c . References [1] Arens R, The space L w and convex topological rings, w//. Xm. Mar/z. Soc. 52 (1946) 931-935 [2] Bhatt S J, Representability of positive functionals on abstract star algebras without identity with application to locally convex *algebras, Yokohama Math. J. 29 (1981) 7-16 [3] Bhatt S J, On spectra and numerical ranges in locally m-convex algebras, Indian J. Pure Appl Math. 14(1983)596-603 [4] Bhatt S J, Kothe sequence algebras I, J. Ramanujan Math. Soc. 6 (1991) 167-183 [5] Bhatt S J and Deheri G M, Kothe spaces and topological algebras with basis, Proc. Indian Acad. Sci. (Math. 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Lecture Note No. 135 (1988) [29] Rickart C E, General Theory of Banach Algebras (Princeton: Van Nostrand Publ. Co.) (1960) [30] Ruckle W, Sequence Spaces (London Melbourne: Pitman Adv. Publ. Program, Boston) (1981) [31] Turpin Ph., Un remarque sur la algebra a inverse continue, C. R. Acad. Sci. Paris. 270 (1970) 1686-1689 [32] Wong Y C, Schwartz spaces, Nuclear spaces and Tensor products, Lecture Notes in Maths. No. 726, Springer- Verlag (1979) Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 3, December 1992, pp. 217-224. Printed in India. C(K,X) as an M-ideal in WC(K, X) T S S R K RAO Indian Statistical Institute, R. V. College Post, Bangalore 560059, India MS received 12 May 1992; revised 4 November 1992 Abstract In this paper we study the classes of Banach spaces X for which the space of continuous X -valued functions forms an M-ideal in the space of weakly continuous functions. We also study a lifting problem for weakly continuous functions. Keywords. M-ideals; weakly continuous functions; Schur property. 1. Introduction For an infinite dimensional Banach space X and an infinite compact Hausdroff space K let C(K,X) denote the Banach space of X- valued continuous functions on K equipped with the supremum norm and let WC(K, X) denote the space of functions that are continuous when X has the weak topology, equipped with the supremum norm. In a recent work Diestel et al [4] show that any element of WC(K,X) is Bochner /Mntegrable w.r.t. each regular Borel probability measure / on K and thus identify C(K,X)* as a subspace of WC(K,X)* and obtain the decomposition WC(K, X)* = C(K, X)* C(K 9 X)^ via the restriction map The question raised by them is "when is the sum above an / x sum?" Let us recall that a subspace JcrJ^ is said to be an M-deal if J 1 1 AT = X*(/ 1 -sum) for some closed subspace N. Also if JaX is such that J x is the kernel of a norm one projection P in X* and if J is an M-ideal then N = Range P. Hence, in terms of M-structure theory (see [1] for all relevant definitions) an equivalent formulation is that "When is C(K 9 X) an M-ideal in WC(K,X)T In this paper we look at this question and obtain some positive and negative results. Our first theorem disposes off the trivial situation and more. Theorem 1. The following statements are equivalent. 1. X has the Schur property 2. C(K 9 X) = WC(K, X) for any K 3. For any K 9 every element of WC(K,X) attains its norm on K 4. For some K, every element of WC(K 9 X) attains its norm Proof. 1=>2: Let feWC(K,X). Since X has the Schur property, f(K) is a norm compact subset of X and hence on f(K) weak and norm topologies coincide. Therefore / is norm continuous. 217 218 T S S R K Rao 2=>3=>4 are clear. 4=>1: Suppose X fails the Schur property. Assume w.l.o.g3 a y n eX, \\y n \\ = 1 and y n -> weakly. Fix any ae/ with Let x n ~oL(n)y n then x n -+Q weakly. Fix a distinct sequence k n eK and a pairwise disjoint sequence of open sets U n with k n eU n . Choose f n eC(K\ 0^/ n ^l and Define g:K-+Xbyg(k) = I,f n (k)x n . Clearly g is well defined and || g \\ = 1. To see that g is weakly continuous, note that for any x*eX*, x*g = I,x*(x n )f n and since x*(x n )-+0 ^e RHS is a continuous function. To obtain the required contradiction we now show that g fails to attain its norm on K. Suppose for some /c , ||#(/c )|| = 1. Let n be such that fc 6L/ Mo , then g(k )=f no (k )x tto =f no (k )a n0 y n0 . Since \\y n0 \\ = 1 and / W0 (fe ) < 1, | a w0 1 < 1, we get a contradiction. Therefore X has the Schur property. In spite of the decomposition of WC(K, X)* the precise nature of its elements is far from being clear, hence the following corollary is of some interest. Let 8eX* denote the extreme points of the dual unit ball and for any keK, let <5(fe) denote the Dirac measure at fc. It is well known that deC(K 9 X)* = {6(k)x*:keK, x*edeX*}. Note that for any function f:K-+X, (5(fc)0**M = **(/"(*))- COROLLARY. X has the Schur property iff deWC(K,X)* = {<5(fc)0;x*:keK, x*edeX*}. Proof. Suppose X fails the Schur property and Let g be the function constructed during the proof of 4 => 1 above. By the Hahn-Banach theorem, 1 = II 9 II = Afe) for some AedeWC(K,X)*. By our assumption, A = 5(k)x* for some keK and x*edeX*. Now 1 = \(g) = x*(g(k))^ \\g(k)\\ ^ 1. Hence \\g(k)\\ = 1 contradicting the fact that g fails to attain its norm. From now on we assume that C(K,X) is a proper subspace of WC(K,X). For Banach spaces X, Y let us denote by XT (X, 7) = space of compact operators, ^(X, Y) = space of weakly compact operators and &(X, Y) = space of bounded operators. For any index set T, let Jjf denote X- valued functions defined on F and C(K, X) as an M-ideal in WC(K, X) 219 vanishing at oo and let r ^X denote the space of X-valued bounded functions defined on P. Both these spaces are equipped with the supremum norm. It is well-knowflr that o^ is an M-ideal in Q^X for any Banach space X and index set F. Our first result is based on the following easy observation about M-ideals. Observation. For Banach spaces X, Y, Z with Z c Yd X, if Z is an M-ideal in X then Z is an M-ideal in Y. PROPOSITION 1. For any discrete set F, and for any compact K, C(K,c (F)) is an M-ideal in WC(K 9 c (T)). Proof. Let us note the canonical identification and via evaluation at elements of F. Since C(K) is an M-ideal in . C(K) 9 using the observations mentioned above we get that C(K, c (F)) is an M-ideal in WC(K 9 c (F)). In [10] the authors study a class of Banach spaces Y (the so called M^-spaces) with the property, Jf(J, 7) is an M-ideal in (X, Y) for any Banach spaces X, our next result involves subspaces of such a space Y. Theorem 2. Let K be any compact extremally disconnected space and let X be a closed subspace of an M^-space then C(K,X) is an M-ideal in WC(K,X\ Proof. Case i: Suppose K is the Stone-Cech compactification /?(F) of some discrete space F. It is easy to identify C(/?(F), X) as jf (^(F), X) by restricting the functions to F and the same mapping allows one to identify WC(fi(r\ X) as a closed subspace of &(l l (T), X). In view of our observation, the result is proved once we note that jf (I 1 (T), X) is an M-ideal in &(P(T\ X). That this is indeed the case can be proved by using arguments identical to the ones given in the proof of Proposition 2.9 in [8]. Case ii: Let K be any compact extremally disconnected space. By well known results in topology (see [6]), there exist a discrete set F, an into homeomorphism ^:K-+/?(F) and a continuous onto map <p:[l(r)-+K such that </>i^ = identity on K. Let $ denote the canonical isometry /->/</> taking function spaces on K isometrically into function spaces on /?(F) and let P denote the norm one projection g^><b(g(j>) on the appropriate spaces. By Case (i) C(jS(F), X) is an M-ideal in WC($(T\ X). We shall verify the restricted 3 ball property for C(K,X) c WC(K,X) to conclude that it is an M-ideal (see [1]). Let f t eC(K 9 X) l9 1 <i<3, geWC^X), and e>0. Since C(j3(F), X) is an M-ideal in WC(p(T) 9 X), applying the restricted 3 ball property to ^(/ f ), $>(g) we get a 220 T S S R K Rao h'eC(P(r), X) such that || 9(g) + <(/;) - h' K 1 + eVz. Since P is a projection of norm one .e. Now h'\l/eC(K,X). Hence C(K, X) is an M-ideal in WC(K 9 X). Remark. Whether the above theorem is valid for any compact space K is not known. The properties of M^-spaces and their subspaces seem to indicate that this should be so. Related to the above ideas is a question of lifting weakly compact sets. We are interested in the following two situations. (a) X is a Banach space, Fez X is a closed subspace. and n:X-+X/Yis the quotient map. Given a weakly compact set K in X/Y and e > there is a weakly compact set K in X such that n(K) = K and Sup^ || ||< (1 + e) sup x || ||. (b) For a compact K and feWC(K,X/Y) 9 e>0 there is a geWC(K,X) such that 7t0=/ and Let us note that this is trivial when X/Y has the Schur property and a quotient map from a l l (T) onto a Banach space X does the lifting in (a) only when X has the Schur property (ie in general there is no weakly continuous cross-section map for n). Examples 1) The authours of [13] show that if Y is a reflexive subspace of a Banach space X, n has lifting as in (a). 2) Let T denote the unit circle and H 1 Q the Hardy space in L l (T\ then a classical theorem in analysis (see [11]) says that n has lifting as in (a). Note that the norm-restrictions are valid since K is the weak closure of image of K under the nearest point cross-section map in this case. 3) X a Banach space, Y c X be an L 1 -predual. Consider n:X*-*X* /"*". Let K be any compact set and let feWC(K,Y*). Define T:Y-+C(K) by T(y)(k) =f(k)(y). It is well known that T is a weakly compact operator and || T || = \\f \\ . Since Y c:X and Y is an L x -predual by Theorem 6.1 of [9], 3 a weakly compact operator T:X-+C(K), extending T and such that || f \\ = || T|| = ||/||. Now g = (f)*<5 (where <5:K->C(K)* the Dirac map) is the necessary weakly continuous lifting. PROPOSITION 2. Let X be a Banach space and let Y cX be a closed subspace. B^>A, and A=>B for compact extremally disconnected spaces. When B holds and if C(K,X) is an M-ideal in WC(K,X) then the same is true of C(K,X/Y) in WC(K,X/Y). Proof. B=>A is clear. Let K be compact, extremally disconnected. As in the proof of case [ii] of Theorem 2, get a discrete set T and mappings \l/:K-+fi(r\ <t>:f}(r)-+K with </>^ = identity. C(K 9 X) as an M-ideal in WC(K,X) 221 3iven feWC(K 9 X/Y) 9 e > since f<l>eWC(P(r), X/Y) by property (A) (f(t>)(P(T)) ;an be lifted and hence we can define a g'eWC(f!(r) 9 X)B\\g'\\ <(1 +e)||/|| and ind for io that ng=f. ^roof of the rest of the proposition can be completed as in Theorem 2 using the 'restricted 3-ball" characterization of M-ideals. ^roblem. Does A => F? Bven though we do not have a complete description of situations when C(K,X) is in M-ideal in WC(K,X\ our last proposition shows that they exhibit properties imilar to c -spaces. PROPOSITION 3. / C(K, X) is an M-ideal in WC(K, X) then QedeX* (Closure taken in the w*-topology). roof. Let us observe that *:/ceX, x*edeX*) w*-closed convex hull), since the functional on the RHS determine the norm. If WC(K, X)* = C(K 9 X)* 1 C(K, X) 1 - then choose By Milman's converse to the Krein-Milman theorem ([3])Ae{5(fc)x*: JceK, <*edeX*}-\ Let A = lim a ^(fc a )x*, k a eX, x*edeX*. For any xeX considered as constant function in C(K 9 X) Iherefore QedeX* Negative results. \s before these observations are based on the corresponding facts known for operator spaces. An observation due to Saatkamp [12] in operator theory says that when , Y) * &(X 9 Y\ 3T(X, Y) is not an M-summand in &(X 9 Y). Similar argument 222 T S S R K Rao works to show that C(K,X) is not an M-summand in WC(K,X). Hence, a standard procedure now to show that C(K,X) is not an M-ideal is to notice when C(K,X) has the intersection property (I.P, See [2], [5]) and then appeal to Theorem 4.3 of [2] to conclude that C(K,X) is hot an M-ideal in WC(K,X). It has been observed in [5] that when X has the I.P, C(K,X) has the I.P and examples of Banach spaces X with I.P include C(K) spaces, reflexive Banach spaces and more generally spaces with the Radon-Nikodym property, spaces with a non-trivial / p -summand for p < oo (see [2]). In all these situations C(K,X) is not an M-ideal in WC(K,X\ For a dual space X*, identifying C(K, X*) = tf(X, C(K)\ WC(K, X*) = P(X 9 'C(K)) when X* fails the I.P, since X* has a copy of c (see [2])., arguments given during the Proof of Proposition 2.2 [8] work to show that if jf(X, C(K)) is an M-ideal in ^(X 9 C(K)) then tf(l l ,C(K}) is an M-ideal in &(P,C(K)). But as we have noted before jf(/ 1 ,C(K)) = C(0(N), C(K)) and &(l l , C(K)) = WC(ft(N), C(K)) and since C(K) has the I.P this cannot happen. So for no dual space X*, C(K,X*) can be an M-ideal in WC(K,X*). Acknowledgement Part of this work was done during 1991 at the Freie Universitat Berlin under a bursary from the Commission of the European Communities. References [I] Behrends E, M-structure and the Banach-Stone Theorem, Lecture Notes in Math-136, Berlin-Heidelberg-New York: Springer (1979) [2] Behrends E and Harmand P, Banach spaces which are proper M-ideals, Studia Math., 81 (1985) 159-169 [3] Diestel J, Sequences and series in Banach spaces, Berlin-Heidelberg-New York: Springer) (1984) [4] de Reyna J A, Diestel J, Lomonosov V and Rodriguez L Piazza (Preprint) (1991) [5] Harmand P and Rao T S S R K, An intersection property of balls and relations with M-ideals, Math. Z. 197 (1988) 277-290 [6] Lacey H E, The isometric theory of classical Banach spaces (Berlin-Heidelberg-New York: Springer) (1974) [7] Lima A, Intersection properties of balls and subspaces in Banach spaces, Trans. Am. Math. Soc. 227 (1977) 1-62 [8] Lima A, Rao T S S R K and Werner D, Geometry of operator spaces, Preprint Nr A 92-18, Fachbereich Mathematik, Freie Universitat Berlin [9] Lindenstrauss J, Extension of compact operators, Mem. Am. Math. Soc. 48 (1964) 1-112 [10] Pay a R and Werner W, An approximation property related to M-ideals of compact operators, Proc. Am. Math. Soc. Ill (1991) 993-1001 [II] Pelczynski A, Banach spaces of analytic functions and absolutely summing operators, CBMS 30 (1977) 1-91 [12] Saatkamp K, Best approximation in the space of bounded operators and its applications, Math. Ann. 250 (1980) 35-54 [13] Schliichtermann G and Wheeler R F, On strongly WCG Banach spaces, Math. 2. 199 (1988) 387-398 [14] de Reyna J A, Diestel J, Lomonosov V and Rodriguez L Piazza: Some observations about the space of weakly continuous functions, Questiones Mathematicae (to appear) C(K, X) as an M-ideal in WC(K, X) 223 [15] Cambern M, Jarosz K and Wodinski G, Almost ZAprojections and ^-isomorphisms, Proc. R. Soc. Edinburgh, 113A (1989) 13-25 [16] Harmand P, Werner D and Werner W, M-ideals in Banach spaces and Banach algebras (Monograph in preparation). [17] Werner D, Contributions to the theory of M-ideals in Banach spaces (Habilitationsschrift, Freie Universitat Berlin) (1991) [18] Domanski P and Drewnowski L, The uncomplemenlability of the space of continuous functions in the space of weakly continuous functions, (to appear). Note After submitting this paper for publication, I have received an expanded version of [4] from Professor J. Diestel. The authors of [14] now have also some answers to the M-ideal question. Their approach is different from the M-structure theoretic approach that I have taken. The purpose of this note is to illustrate this point. The following characterization of the class M^ appears in [16]. XeM^ iff there is a net K a in the unit ball of jf(X) such that and K* and for any a > there is an a 9Va > a . || K a x + (/ - KJy K (1 + e)max { || x ||, || y \\ } for all x, Note that if a K x satisfying (|) is a projection then one has and and Projections satisfying these conditions are called almost L -projections (see [15]). Now let us recall from [14] the definition of Schur approximation property. A Banach space X has the Schur approximation property (SAP for shor) if for any compact set K c X and e > there is a projection P with range (?) having the Schur property such that ||x-Px||<sVxe || ? IK 1 + fi, || / - P IK 1 + e and ||x|K(H-e)max{||Px||,||x-Px||}. PROPOSITION 4. Let Kbea compact Hausdorff space and let XeM^ . C(K, X) is an U-ideal in WC(K, X). Proof. As before we shall verify the restricted 3-ball property. Let feWC(K,X) 9 f t C(K 9 X) be in their respective unit balls and let e>0. Put 224 T S S R K Rao K~ = uf =1 / (jK) and use (t) to get a compact operator K a such that \\K^ + (I-K a )y\\^(l and Put = K a o/. Clearly geC(K 9 X). For any /ceK \\fi(k) +/(*) Remark 1. It follows from the results in Chapter VI of [16] that for any YeM^ of infinite dimension, every infinite dimensional subspace has an isomorphic copy of C Q . Consequently only finite dimensional subspaces here have the Schur property. So for a X c= 7, YeM^ the SAP for X already implies the bounded approximation property. However in Theorem 2 above we have made no assumptions about approximation property and such spaces X without the bounded approximation property are known to exist. Remark 2. An argument similar to the one above gives an M-structure theoretic proof of "C(K,X) is an M-ideal in WC(K, X) when X has the SAP," which is Theorem 7 of [14]. PROPOSITION 5. // a Banach space X has the Schur property then every M-ideal in X is an M-summand and there are only finitely many M-summands. Proof. Key fact is that X and none of its subspaces have an isomorphic copy of c . So if M cr X is an M-ideal and infinite dimensional then since M has no copy of c , M must be an M-summand see [2]. Of course when M is finite dimensional it is already an M-summand, see [1]. An example of a space with the SAP mention in [4] is a c direct sum of spaces with the Schur property. We now show PROPOSITION 6. // X has the SAP with M -projections then X is isometric to a C Q direct sum of spaces with the Schur property. Proof. Here we consider the maximal function module representation of X( [1] ). Then the base spaces have the Schur property and in view of Proposition 5 M-projections correspond to multiplication operator by indicator functions of finite sets, one concludes that X is isometric to a c direct sum of spaces with the Schur property. Remark. The above formulation and proof are inspired by Proposition 6.5 and its proof in [17]. Proc. Indian Acad. Sci. (Math. Sri.), Vol. 102, No. 3, December 1992, pp. 225-233. Printed in India. On the zeros of a class of generalised Dirichlet series XI R BALASUBRAMANIAN and K RAMACHANDRA* Institute of Mathematical Sciences, Tharamani PO, Madras 600113, India * School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India MS received 17 June 1992 Abstract. A sufficiently large class of generalised Dirichlet series is shown to have lots of zeros in cr>l/2. Some examples are (i) '(s) a (a any complex constant) (ii) a-C(s)-* =0 ((n + N /2)~ s -(rc-f l)~ s ) (where a is 'any positive constant) and (iii) a + *__ j( lyxiogn^n" 5 (where A is any real constant > 1/2 and a any complex constant). Here as is usual we have written s = a + it. Keywords. Zeros; generalised Dirichlet series; Riemann zeta-function. 1. Introduction In paper [1] of this series we considered zeros of G(s)==^ =1 a n n~ s (under fairly general conditions. We have changed the notation for F(s) to G(s) to avoid a clash of notation later) in the rectangle (1) where <5 = <5(T)-*0 as T-> oo, and as usual s = o + it. The only restrictive condition was something like I|a p | 2 x/logx, (the sum being over all primes p subject to x<p^2x) for all large x and what was irksome was the condition a 1 ^ 0. The main object of the present paper is to relax the condition a t ^ to a 1 = 0, . . . , a no = and a n +1 7^0 where n (^0) is an integer constant. Of course we can (as we do) assume n ^ 1 since the case rc = is considered in the paper X [1] of this series. Also the condition involving a p was designed to include C(s); but if we strengthen the lower bound to say S \a p \ 2 x(logx) 2 then we can prove that G(s) has at least one zero in {<T>-~, T^t^2T (2) provided only that |G(s)| does not exceed a fixed power of T (assuming T to be sufficiently large). Also by using ideas of this paper and those of [7] it is possible to prove that Riemann hypothesis implies that if q = [a(loglog T) 1/2 ] (where a > is a constant) then lim hiM \(j + ii)\ 2lq dt >^exp(a~ 2 ). (3) 225 226 R Balasubramanian and K Ramachandra (We may also formulate a result for \/Hft +H (--)dt where 7>/f loglog T). The first of these results follows from a routine application of the method of X [1] (except when a 1 = in which case the method of the present paper succeeds) while the second follows from the following observation. Consider G(s) where the a n are multiplicative over square-free integers n. Then the coefficient of (p l p/ c )~ s (pi , ~',p k distinct primes) in (G(s)) l/q is the same as in a V/4/ a \ llq ( a \ llq +5*1) (1 + ^2 ... I+^E P S J \ P S 2 J \ PlJ i.e. q~ k a pl a p2 '" a pk' We ^ ave t ^ en to use tfle Hardy-Ramanujan theorem as in [7]. We do not give further details of the proof of these results. Instead we define a property P q of a Dirichlet series G(s) = S^ =1 ^ju~ s where {b n } is any sequence of complex numbers and {^} is any sequence of real numbers with b^ =/4 1 = l, Mi < A<2 < Ms < ---and 1/C ^ // B + 1 ju n ^ C where C(> 1) is an integer constant. We assume that the series for G(s) converges absolutely for some complex number s. DEFINITION Let q(^2) be an integer. We say that G(s) has the property P q if there exists a constant 5>Q and a positive integer n* = n*(<5) (n* not divisible by q) both depending on G(s) such that G(s) can be continued analytically in (4) and has T zeros all of order n* in this rectangle. Remarks. Also we consider functions like log ((s) a where a is any complex constant. These have singularities but continuable in <r> 1/2. We prove that logC(s) a has the property P 2 (if we allow analytic continuation except on horizontal lines which contain singularities). In what follows n* may depend on T; but n* will be bounded above by a constant depending only on 6. Accordingly our theorems which illustrate our method are Theorem 1. The function ('(s) a has the property P 2 for every complex constant a. Theorem 2. The function log C(s) a has the property P 2 (in the sense explained in the remark above) for every complex constant a. Theorem 3. The function G(s) = a 2* aBO (n + v^)"' has the property P 2 for every positive real constant a. Theorem 4. Let 1(> 1/2) be any constant. Then G(s) = a 4- 2* =1 (- l)"(logn) A n~ s has the property P 2 for every complex constant a. Theorem 5. The function G(s) = a + Z^ = 1000 (- l) n (loglogn) 3/4 rc~ s has the property P q (for some integer q = q(8)) for every complex constant a. Generalised Dirichlet series 227 Remarks. More general results will be found in the later sections of this paper. It is possible to deal with the zeros in {a^\6, T^t^lT} in a somewhat general setting. These questions will be taken up elsewhere. We would like to remark that our results hold good for zeros of Dirichlet polynomials like T, n ^ T a n ^ s and ,, r oooflj;~ s (with conditions on {a n } of a fairly general nature and somewhat restrictive conditions on {^}). The previous history of Theorems 1 and 2 is well-known and due to many authors. (For references see [8]. Of great relevance here is the work of Bohr and Jessen [4, 5]. But both our methods and results seem to be new). 2. A conjecture and its proof in special cases We believe that the following conjecture is true (at least in a modified form). In [2] we have proved it in some special cases and these will be used in the present paper. (We stipulate that certain constants shall be integers only for a technical reason which is not serious). We quote from the paper just cited. Conjecture. Let I = u^ < u 2 < be any sequence of real numbers with l/C H n ^C where C( ^ 1) is an integer constant and n = 1, 2, 3, . Let us form the sequence 1 = A t < A 2 < of all possible (distinct) finite power products of 1 = A*i> A*2 "* w * tn non-negative integral exponents. Let s = cr + it, H(^IQ) a real parameter, and {a n } (n = 1, 2, 3, ) with a 1 = lbe any sequence of complex numbers (possibly depending on H) such that F(s) = J L^ > =l a n X~ s is absolutely convergent at s = B where B(^3) is an integer constant. Suppose that F(s) can be continued analytically in (<r ^ 0, < t ^ H) and that there exist T l9 T 2 with 0^7^ # 3/4 , H - H 3 ' 4 ^T 2 ^H such that for some K(30), there holds max(|F(<7 + iT 1 )| + \F(a + iT 2 )\)^K. (5) ff^Q Finally let 'L* Sil \a n \/.~ B ^H A where A(^ 1) is an integer constant. Then there exists a 6 1 ( > 0) (depending only on A, B, C) such that for all H ^ H (A, B, C) there holds ~f H \F(it)\ 2 dt^ 1 - |aj 2 , (6) provided that H~ 1 loglogK does not exceed a small positive constant. Remark. We have used the symbol d^ (in place of d) so that it should not clash with the (5 already introduced. Also we recall that 1/2 can be replaced by a quantity ~ 1 (as jF/->oo) and whenever we have succeeded in proving this conjecture we have proved it in this stronger form. We now quote the corollaries to the main theorem of [2]. COROLLARY 1. Let i n = n. Then the conjecture is true. 228 R Balasubramanian and K Ramachandra COROLLARY 2. Let n ( ^ 2) be an integer constant, and u n = (n 4- n l)/(n ) ^n ^ conjecture is true. COROLLARY 3. Let J8>0 be aw algebraic constant, and u n = ((n + p)/(l + /?)). T/ien t/ie conjecture is true. ( The conjecture is also true for the choice u 1 = 1, ^ w = n 4- j8 1 /or w > 1). Remark. It is possible to state a slightly more general corollary than Corollary 3. But we do not state it since our ambition is to prove a sufficiently general result. 3. Two important observations We record the observations as two lemmas. Lemma 1. Let u n = (H O + n- l)/(n ) and G(s)==S^ =1 b n ^~ s be absolutely convergent for some complex s. Then, we have, for any integer q>Q and a large enough, = a n /- (7) where the / are formed as in the conjecture, a 1 = 1, and further whenever is prime \a n = q" 1 \b n \, and so the RHS of (6) is >A I where the sum is restricted to those n for which n= 1, and also to those n for which n -f n 1 is prime. Proof. It is sufficient to check that if p is a prime > n 4- 1, the equality ^i-'^k P where ^ l ,--,^ k are integers ^ n -f 1, is not possible except when fe = 1 and ^ = p. This is trivial since p has to divide at least one ^ say ^ . Now which is impossible unless fe= L Lemma 2. Let G(s) = 1 ^ 2 b n A*~ 5 where b n are reaJ and non-negative and the series involved converges for some complex s. Then for any integer q(> 0) and a large enough, we have, n-l Generalised Dirichlet series 229 where the / are as in the conjecture, a t = 1 and further for rc^2, a n ^Q and -^n^b n q~ l wherever / = /v 4. Proof of theorems 1, 3, 4 and 5 We sketch the proof in a general setting. Note that after an easy normalisation the functions in question look like G(s) = E^L 1 6 fJJ a~ >s , where b 1 = 1, {/;} as in any of the Corollaries 1,2 or 3 (of 2), which converges absolutely for some complex number s and is analytically continuable in cr > 1/2. It is easy to see that, for a = 1/2 -f <5, f l&.fn- 1 - 2 ^ K(2<5), say. (9) From this and the fact that the absolute value of an analytic function at the centre of a circle is majorised by its mean- value over the disc enclosed by it, it follows that max |G(s)| 2 (T 2 K(<5)r (10) ,co)x/) where / runs over all disjoint intervals of length H into which [T,2T] can be divided with a suitable meaning at the end points. We assume that H ^ T 1/2 and that H is a large enough function of <5. From (10) it follows that :<r 3 V(5)H}dT/H. (11) Let q ^ 2 be an integer. In order to obtain the lower bound |G(s)| 2 "dt Z \a n \ 2 n-*- 2 *, ( Sa =I + 5 + fc), (12) we have to check the condition that H' 1 loglogK shall not exceed a small positive constant. In (12) {a n } are defined by If we assume that in [^ -f 6, oo] x /, F(s) is regular (i.e. G(s) has no zeros of order not divisible by q) then (12) holds if H exceeds a large constant depending on <5 since we can take K = <5~ 3 V(8)H provided we omit the intervals counted in (11). Also HI. (13) Jrf where r\ > is a small constant. Hence we have TH' 1 intervals / (with |/| = H) for which (12) holds and also If - |G( 5 )| 2 dt<^ 1 F(2<5). (14) H J j We now show that each of the rectangles [| H- 5, oo] x / (for these J) must contain 230 R Balasubramanian and K Ramachandra a zero of G(s) of order not divisible by q (if we impose a suitable condition on V(2S) and V(S)). Otherwise from (12) and (14) we must have T \b n \ 2 n-*- 2 ><D 2 f,-iV(2S) (15) q n*H*i ) where D l > 0, D 2 > 0, and rj are independent of T, H, q and 8. Also the accent restricts the sum as in (8). If the {/*} are as in Corollary 3 we end up with ^T I \b H \ 2 n-*- 2 '}*<D 2 f,-iV(2S). (16) q H'i / Since we are interested in finding some H = H(5) contradicting (15) and (16) we can as well contradict (17) for proving Theorems 1,4 and 5. To prove Theorem 3 we have to contradict (18) It is a trivial matter to check that (17) and (18) are false for the particular cases in question. This completes the proofs of Theorems 1,3,4 and 5 except for the remark concerning n* (for this see 7). 5. Some generalisations It is plain that we can prove analogues of Theorem 1 (also Theorem 2 as will be seen) to "(s), '"(s), -, derivatives of L-functions and also to derivatives of the zeta and L-functions of any quadratic field. We can also prove the analogues of Theorems 3, 4 and 5 to more general Dirichlet series. We are particularly interested in (stating the analogue for) a class of functions in which we were interested in [3]. We proceed to recall their definition. Let %(ri)(n = 1,2, 3, ) be a periodic sequence of complex numbers not all zero (if the period is k we require that there is at least one integer n with (n, k) = 1 and X(n) ^ 0) such that the sum L#(n) extended over a period is zero. Let /(x) be a positive real valued function of x defined for x ^ 1 such that for every fixed s > 0, /(x)x is increasing and /(x)x~ is decreasing for all x^x (e). Let {d n } (n= 1,2,3,--) be a sequence of complex numbers satisfying /(n) \d n \f(n) and for all X ^ 1 we should have x< Z <2x \d n+1 -d n \f(x). The functions that we wish to consider are Generalised Dirichlet series 231 Let us suppose that the expression l/2/oo \ -1 Z f(n)(log(n +!))' 1 n" i ~ 2S } (19) V = i / tends to zero as S - 0. Then, we have Theorem 6. The function G(s) a has the property P 2 for every complex constant a. Proof. This follows from the arguments of 5 and 7. We have only to observe that f(x)f(2x)f(x) and that n(x)xx/\ogx. Remark. We can also state a similar theorem for the property P q (q = q(d)). t 6. Proof of theorem 2 The proof is not very much different from the one sketched in 4. Note that we have the density theorem that N(a, T) defined by is O(T V(1 " ff) (log T) 5 ) where v = 3/(2 - <r) due to Ingham [6] (see also page 236 of [8]). The 0-constant is independent of a and T. In view of this theorem the number of t-intervals / of constant length H = H(8), satisfying T^t^2T such that \_j + (5/2, oo) x / is zero free is ~ T/H. This and the remark in 7 are enough for the proof of Theorem 2. Remark 1. We may also remark that the analogue of Theorem 2 is true for the logarithm of a finite power product (with complex exponents not all zero) of ordinary L-functions or L-functions of a fixed quadratic field since for these L-functions the function N(<r, T) is 0(r v ' (1 ' ff) (log T) c ) where v' = 4/(3 -2<r) and C is an absolute constant. The 0-constant depends on the modulii of the characters. Remark 2. Starting from Theorem 2 one may deduce easily the following. f Theorem 7. The function f (s) e" has the property P 2 for every complex constant a. 7. Completion of proofs We have proved that for the functions in question the number of distinct zeros in {cr ^ f + d, 7Xt< 2T} whose orders are not divisible by q is T. But by a slight variant of the considerations of the proof we can secure that the TH" 1 intervals / selected for the contradiction have the property that in the rectangles [+ .5/2, oo] x I the functions are bounded by a function of <5. By Jensen's theorem it follows that the number of zeros (in these rectangles) counted with multiplicity is bounded. Thus the orders of the T zeros as proved already in 4, 5 and 6 are bounded by a function of <5 alone. Hence (by classifying these zeros according to their 232 R Balasubramanian and K Ramachandra orders) we see that T zeros (in at least one class) have the same order (a fixed integer not divisible by q). This completes the proof of all our assertions. References [1] Balasubramanian R and Ramachandra K, On the zeros of a class of generalised Dirichlet series-X Indag. Math, (to appear) [2] Balasubramanian R and Ramachandra K, Proof of some conjectures on the mean-value of Titchmarsh series-Ill Proc. Indian Acad. ScL (Math. Sci.) 102 (1992) 83-91 [3] Balasubramanian R and Ramachandra K, On the zeros of a class of generalised Dirichlet series-Ill, J. Indian Math. Soc. 41 (1977) 301-315 [4] Bohr H and lessen B, Ober die Werteverteilung der Riemannschen Zetafunktion. Acta Math. 54 (1930) 1-35 [5] Bohr H and lessen B, Uber die Werteverteilung der Riemannschen Zetafunktion. Acta Math. 58 (1932) 1-55 [6] Ingham A E, On the estimation of N(cr, T). Quart. J. Oxford. 11 (1940) 291-292 [7] Ramachandra K, Application of a theorem of Montgomery and Vaughan to the zeta-function-II, J. Indian Math. Soc. (to appear) [8] Titchmarsh E C and Heath-Brown D R, The Theory of the Riemann zeta- function, Second edition (Revised by D R Heath-Brown), (Oxford: Clarendon Press) (1986) POST-SCRIPT. The condition E(5) -> as d -> see ((9)) can be proved under various choices of /(). For example let (log rif ^ f(n) ^ exp((log n)* 1 ). Then E(8) -> as 6 -> 0. To see this we begin with a Lemma. Let f(n)(n= 1,2,3, ) be any sequence of positive real numbers such that (log/(n))(log n)~ 1 ->0 as n - oo. For any d > put 2i= Z (/(a)) 2 *- 1 -", Q 2 = Z (/(nM'Oogfo + l))- 1 !*- 1 - 2 ', n=l n=l and 63= Z (f(n)) 2 n-i- 2 . l<exp2|/ 4 ) // Qi - 63 <|Ci and Q, ^(1/s) 2 , (0<6<f), then Q, eQ 2 2 . Proof. We have Q 2 ^ Z (/wpaogoi+i))- 1 *- 1 -" KexpCQJ' 4 ) Gr 1/4 6i( with an implied absolute constant, since Q 3 ^|d) i-e- Q 2 2Ql l2 (l/z)Qi since Q^tt/e) 2 . This completes the proof of the lemma. COROLLARY. Let (logn) 2 </(n) < exp((logn) 0>1 ). Then E(8)^0 as <5->0. Generalised Dirichlet series 233 Proof. In this case Q 1 ^Z* =1 (logn) 4 'n~ 1 ~ 2S xd^ 5 ^(l/e) 2 if 6 is sufficiently small. We have only to prove that fii 63^261- Let d be any positive } constant. We will show that Q 4 = n>cxp (,/$-i-2S)(/(n)) 2 n~ I ~" 2a tends to zero as <5-*0. For Oexp^" 1 " 25 ), we have n 28 n 2d - - -^exp{2 < 51ogn-2(logn)' 1 } Fl ; (/(n)) 2 exp(2(logn)- 1 ) (log n) 2 (for all n exceeding an absolute constant if 8 is small enough). Thus (24.-+0 as <5->0. This proves the corollary completely since S* =2 n(logn) is convergent. (For the validity of E(<5)-~-0 clearly we can impose (logn)* 1 </(n) exp((logn)* 2 ) where R^> 3/2) and R 2 ( < 1 - 4(21?! + I)" 1 ) are constants). -2 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 3, December 1992, pp. 235-238 (G) Printed in India. On L l -convergence of a modified cosine sum HUSEYIN BOR* Department of Mathematics, Erciyes University, Kay sen 38039, Turkey * Mailing Address: P.K. 213, Kayseri 38002, Turkey MS received 31 March 1992 Abstract. In this paper a generalization of a theorem of Kumari and Ram [4] has been proved by considering the class S(<5) instead of the class S. Keywords. L 1 -convergence; convex sequence; modified cosine sum. 1. Introduction A sequence (a k ) is said to be convex if A 2 a k ^0, where A 2 a fc = A(Aa fc ) and Aa k = a k a k+ 1 , and quasi-convex if i(/c+l)|A 2 a fe |<oo. (1) fc=i A sequence (a k ) of positive numbers is said to be quasi-monotone if kka k ^ ota k for some positive number a and <5-quasi-monotone if a fc -0, a k >Q ultimately and Aa k ^ <5 k , where (d k ) is a sequence of positive numbers [2]. The concept of quasi-convex sequence was generalized by Sidon [5] in the following manner: A sequence (a k ) is said to belong to class S, or a k eS, if a k -+0 as fc- oo, and there exists a sequence of numbers (A k ) such that (a) A JO, fc->aD, (b) Z^<oo, (2) k=l (c) lAaJ^/ifc, for all fc. This class S of sequences has been further generalized to the class S* and S(d) by Singh and Sharma [6] and Zaini and Hasan [7], respectively: <2 k eS* if (2) holds with the condition (a) replaced by, (a') (A k ) is quasi-monotone. a k eS(S) if (2) holds with the condition (a) replaced by, (a") (A k ) is (5-quasi-monotone and Efe fc < oo. 235 236 Huseyin Bor Thus, in view of the above definitions it is obvious that S c S* c S(d). Let + Y a fc cos fcx (3) 2 k = i be a cosine series and satisfy a k 0(1), k-+co. Let the partial sum of (3) be denoted by S n (x) and /(x) = lim n _ >ool S n (x). Recently Kumari and Ram [4] introduced the modified cosine sum as (*) = y+ I lA(V7)fccosfcx, (4) and obtained its L 1 -convergence by proving the following theorem. Theorem A. Let (a k ) in (3) belong to the class S. If ti n -*< x> \a n + i\ logn = Q, then The aim of this paper is to generalize Theorem A by considering the class S(S) instead of the class S. Now, we shall prove the following theorem. Theorem. Let (a k ) occurring in (3) belong to class S(5). If lim n ^ ^ \ a n + 1 1 log n = 0, then 3. We need the following lemmas for the proof of our theorem. Lemma 1. ([3]). // |c fc | < 1, then I sin(fc+l/2)x ^ . X 2sm- 2 dx ^ C(n + 1), where C is a positive absolute constant. Lemma 2. ([7]). Let (a k ) be a 5-quasi-monotone sequence with Efc<5 k < oo. // Za k < oo, then S(fc+ l)Aa k < oo. Proof of the theorem We have + Z ^kCosfcx -- ^^^ Z kcos/cx 2 ^=1 n+ljt=i ), say. On L l -convergence of a modified cosine sum Now, making use of Abel's transformation and Lemma 1, we get 237 \f(x)-g,( -i: =i: *-n+i dx + f Jo -n n + l i f Jo n+l dx X AaD/ dx dx (5) Since S(k+ l)AA k < oo, by Lemma 2, the first term in (5) tends to zero as n-oo. On the other hand, by Zygmund's Theorem ([1], p. 458) n+l n> i: |D;(x)|dx~|a n+1 |logn (6) since | D' n (x)|dx behaves like log n. j -n; The conclusion of the theorem now follows from (5) and (6). COROLLARY. V ( a n) occurring in (3) belongs to the class S(S) and lim n ^ OQ \a n ^ 1 \logn = then Proof. We notice that |/(x)-S w (x)|dx- f * J -7C f * J I I/W - 9.(x)\dx + \g n (x) - S.(x)| dx l/(x)-flf.Mldx+ f* J -1C n+l " dx. 238 Huseyin Bor -* oo J _ Since lim |/(x) g n (x)\dx = 0, by our theorem and _ v , dx behaves n+1 like |<3 n+1 |logn by Zygmund's Theorem cited above, for large values of n, the conclusion of the corollary follows. Acknowledgement! This research was supported by Tiibitak (TBAG-C2). References [1] Barf N K, A treatise on trigonometric series (Londonij Pergamon Press) Vol. II (1964) [2] Boas R P, Quasi-positive sequences and trigonometric series, Proc. London Math. Soc. A14 (1965) 38-46 [3] Fomin G A, On linear methods for summing Fourier series, Math. Sbornik 66 (1964) 144-152 [4] Kumari S and Ram B, L 1 -convergence of a modified cosine sum, Indian J. Pure AppL Math. 19 (1988) 1101-1104 [5] Sidon S, Hinreichende Bedingungen fur den Fourier-Charekter einer trigonometrischen Reihe, J. London Math. Soc. 14 (1939) 158-160 [6] Singh N and Sharma K N, Convergence of certain sums in a metric space L 1 , Proc. Am. Math. Soc. 72(1978) 117-120 [7] Zaini S Z A and Hasan S, Integrability of Rees-Stanojevic sums, Math. Seminar Notes 10 (1982) 637-641 Proc. Indian Acad. Sci. (Math. Sci.), Vol. 102, No. 3, December 1992 p 239 Printed in India. ' Addendum to the paper "generalised parabolic sheaves on an integral projective curve" USHA N BHOSLE School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India MS received 21 August 1992 In [1] we constructed moduli spaces of generalised parabolic sheaves (GPS in short) of rank n on an integral projective curve, the generalised parabolic structure being at finitely many (smooth) points of the curve. The method of construction was a natural generalisation of the method of Simpson [2] which uses the well-known embedding of Quot scheme into a Grassmannian. The method of [2] was generalised by M S Narasimhan and T R Ramadas to construct moduli of (generalised) parabolic sheaves of rank 2 (on an integral projective curve) with generalised parabolic structure at a point y and usual parabolic structure at finitely many points y^ different from y [Appendix, [3]). By oversight the reference to this interesting paper was not given in [1]. We would also like to point out the following correction to 2, [1]. In (2-1, [1]) one should have "a point qeQ gives for each;, a ^-dimensional quotient of C n H (^), ^ is given by the composite map H(V n D ) -+ H(g fl D ) -> Fffi q )/F{ (g,). Hence we get an embedding Q - Z = Grass p(w) (C" W) x ( x j Grass ^(C 11 H (0 D )). .... We denote a point of Z by (P, (?,)) where P: C" W-+ 17, P/.C" <g> H((9 Dr ) -+ 'u j are surjective maps, . . . ". Also in the expression for CT H in Proposition 2-2 and in the proofs of Lemmas 2-5 and 2-6, one should replace "?/#)" by " References [1] Bhosle Usha N, Generalised parabolic sheaves on an integral projective curve, Proc. Indian Acad. Sci. (Math. Sci.), 102(1992)13-22 [2] Simpson C T, Moduli of representations of the fundamental group of a smooth projective variety, (Preprint, 1990) [3] Narasimhan M S and Ramadas T R, Factorisation of generalised theta functions I, T.I.F.R. (preprints 1991-1992) Proceedings (Mathematical Sciences) Volume 102, 1992 SUBJECT INDEX Absolute summability Factors for |N, p n \ 6\ k summability of Fourier series 53 A note on two absolute summability methods 125 Almost periodicity Almost periodicity of some Jacobi matrices 175 Binomial determinants Combinatorial meaning of the coefficients of a Hilbert polynomial 93 Boundary value problem Solution of convex conservation laws in a strip 29 C*-enveloping algebra Topological algebras with C*-enveloping algebras 201 Classical propositional calculus Remark on Gronwall's inequality 73 Conservation laws Solution of convex conservation laws in a strip 29 Convex sequence On L 1 convergence of a modified cosine sum 235 Critical exponent Non-existence of nodal solution for m-Laplace equation involving critical Sobolev exponents 23 Cryptanalysis An RS A based public-key cryptosystem for secure communication 147 Decryption An RSA based public-key cryptosystem for secure communication 147 Determinantal ideals Combinatorial meaning of the coefficients of a Hilbert polynomial 93 Differential polynomials On deficiencies of differential polynomials 129 Diffraction Three dimensional diffraction of compressional waves by a rigid cylinder in an inhomogeneous medium 189 Eigenvalue Non-existence of nodal solution for m-Laplace equation involving critical Sobolev exponents 23 Elastic inhomogeneity Three dimensional diffraction of compressional waves by a rigid cylinder in an inhomogeneous medium 189 Encryption An RSA based public-key cryptosystem for secure communication 147 Enveloping algebra Topological algebras with C*-enveloping algebras 201 Explicit formula Solution of convex conservation laws in a strip 29 Fourier series Factors for \N 9 p n ; d\ k summability of Fourier series 53 Fractional calculus operators A note on the identity operators of fractional calculus 141 Frequency On the frequency of Titchmarsh's phenomenon for CM VIII 1 Gauss hypergeometric function A note on the identity operators of fractional calculus 141 Generalised Dirichlet series On the zeros of a class of generalised Dirichlet series XI 225 Generalized parabolic sheaf Generalized parabolic sheaves on an integral protective curve 13 Gronwall's inequality Remark on GronwalFs inequality 73 Hermitian algebra Topological algebras with C*-enveloping algebras 201 High data bit rate communication An RSA based public-key cryptosystem for secure communication 147 Hilbert polynomial Combinatorial meaning of the coefficients of a Hilbert polynomial 93 241 242 Subject index Identity operator A note on the identity operators of fractional calculus 141 Jacobi inversion Almost periodicity of some Jacobi matrices 175 Kothe sequence space Topological algebras with C*-enveloping algebras 201 Random potential Almost periodicity of some Jacobi matrices L 1 convergence On L 1 convergence of a modified cosine sum 235 Lattice paths Combinatorial meaning of the coefficients of a Hilbert polynomial 93 Lower bounds Proof of some conjectures on the mean-value of Titchmarsh series-Ill 83 M-ideals C(X, X) as an M-ideal in WC(K t X) 217 m-Laplacian Non-existence of nodal solution for m-Laplace equation involving critical Sobolev exponents 23 Markovian cocycle Stochastic dilation of minimal quantum dynamical semigroup 159 Maximal monotone operator Maximal monotone differential inclusions with memory 59 Mean value Proof of some conjectures on the mean-value of Titchmarsh series-Ill 83 Meromorphic functions On deficiencies of differential polynomials 129 Modified cosine sum On L 1 convergence of a modified cosine sum 235 Nevanlinna theory and deficiencies On deficiencies of differential polynomials 129 Projective curve Generalized parabolic sheaves on the integral projective curve 13 Q-algebra Topological algebras with C*-enveloping algebras 201 Quantum dynamical semigroup Stochastic dilation of minimal quantum dynami- cal semigroup 159 Quantum stochastic differential equation Stochastic dilation of minimal quantum dynami- cal semigroup 159 175 Regular space An example of a regular space that is not completely regular 49 Relaxation Maximal monotone differential inclusions with memory 59 Residue-Cagniard method Three dimensional diffraction of compressional waves by a rigid cylinder in an inhomogeneous medium 139 Resolvent Maximal monotone differential inclusions with memory 59 Resolvent convergence topology Maximal monotone differential inclusions with memory 59 Riemann zeta-function On the frequency of Titchmarsh's phenomenon for C(s) VIII 1 On the zeros of a class generalized Dirichlet series XI 225 Riemann-Liouville operators A note on the identity operators of fractional calculus 141 Schur property C(K,X) as an M-ideal in WC(K, X) 217 Segal algebra Topological algebras with C*-enveloping algebras 201 Selection theorem Maximal monotone differential inclusions with memory 59 Summability factors Factors for |JV, p w ; <5| fc summability of Fourier series 53 Summability methods A note on two absolute summability methods 125 Titchmarsh phenomenon On the frequency of Titchmarsh's phenomenon for C(s) VIII i Titchmarsh series Proof of some conjecture on the mean-value of Titchmarsh series-Ill $3 Weakly continuous functions C(K,X) as an M-ideal in WC(K,X) 217 Zeros On the zeros of a class of generalized Dirichlet series XI 225 AUTHOR INDEX i Adimurthi Non-existence of nodal solution for m-Laplace equation involving critical Sobolev exponents 23 Antony Anand J Almost periodicity of some Jocobi matrices 175 Balasubramanian R On the frequency of Titchmarsh's phenomenon for C(s) VIII 1 Proof of some conjectures on the mean-value of Titchmarsh series III 83 On the zeros of a class of generalised Dirichlet series XI 225 Bhatt Subhash J Topological algebras with C*-enveloping algebras 201 Bhosle Usha N Generalized parabolic sheaves on an integral projective curve 13 Addendum to the paper "Generalized parabolic sheaves on an integral projective curve" 239 Bor Htiseyin Factors for |N, p w ; 6\ k summability of Fourier series 53 A note on two absolute summability methods 125 On L 1 -convergence of a modified cosine sum 235 Dhar Raj Shree see Singh Anand Prakash 129 Joseph K T Solution of convex conservation laws in a strip 29 Karia Dinesh J see Bhatt Subhash J 201 Kaschel Joachim A note to the paper "An efficient algorithm for linear programming" of V Ch Venkaiah 155 Krishna M see Antony Anand J 175 Modak M R Combinatorial meaning of the coefficients of a Hilbert polynomial 93 Mohari A Stochastic dilation of minimal quantum dynamical semigroup 1 59 Papageorgiou Nikolaos S Maximal monotone differential inclusions with memory 59 Popenda J Remark on Gronwall's inequality 73 Raha A B An example of a regular space that is not completely regular 49 Raina R K A note on the identity operators of fractional calculus 141 Rajhans B K Three-dimensional diffraction of compressional waves by a rigid cylinder in an inhomogeneous medium 189 Ramachandra K see Balasubramanian R 1, 83, 225 Rao T S S R K C(K, X) as an M-ideal in WC(K, X) 217 Samal S K see Rajhans B.K 189 Sankaranarayanan A see Balasubramanian R 1 Singh Anand Prakash On deficiencies of differential polynomials 129 Sinha K B see Mohari A 159 Veerappa Gowda G D see Joseph K T 29 Venkaiah V Ch An RS A based public-key cryptosystem for secure communication 147 Yadava S L see Adimurthi 23 243