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A classical result of K. L. Chung and W. Feller deals with the partial sums S k S_k arising in a fair coin-tossing game. If N n N_n is the number of “positive” terms among S 1 S_1 , S 2 S_2 , …, S n S_n then the quantity P ( N 2 n = 2 r ) P(N_{2n} = 2r) takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for P ( N 2 n + 1 = r ) P(N_{2n+1} = r) , r = 0 r = 0 , 1 1 , 2 2 , …, 2 n + 1 2n+1 . We get to this ansatz by adaptating the Feynman–Kac methodology.

Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. In [Di], DiPerna found a global entropy solution of the isentropic Euler equations for the following exponents in the equation of state for the pressure: γ = 1 + 2/(2m+ 1), m ≥ 2 integer. (1) He divided his arguments into the following two steps. 1. Compactness framework Assume that a sequence of approximate solutions (ρ (x, t),m (x, t)), 0 ≤ t ≤ T , satisfies: (i). There exists a constant C(T ) > 0, independent of > 0, such that 0 ≤ ρ (x, t) ≤ C, |m (x, t)/ρ (x, t)| ≤ C; (ii). For all weak entropy pairs (η, q) of the isentropic Euler equations, the measure sequence η(ρ ,m )t + q(ρ ,m )x is contained in a compact subset of H −1 loc (R× [0, T ]). If γ satisfies (1), then the sequence (ρ (x, t),m (x, t)) is compact in Lloc(R× [0, T ]). The reason for the restriction on the number γ is that, in such a case, any weak entropy function is a polynomial function of the Riemann invariants (w, z). This is the key step in DiPerna’s arguments and is also his main contribution to the compensated compactness method in this aspect. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06.

Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.

M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.

PROOF. By a theorem of Rudin a function gEH' in U whose boundary values are real a.e. on I can be analytically continued to D [3, p. 59]. The lemma follows on applying Rudin's result to gi= (1/2) (fl+f2) and g2=(i/2) (fi-f2). PROOF OF THEOREM 1. By a well-known decomposition theorem [2, p. 87], f(z)=B(z)F2(Z), where B(z) is a Blaschke product and F(z) EH1. Since the boundary values of B (z) have absolute value one a.e. on K, we have a.e. on I, f(ei0)= |f(eio)I, or B(ei0)F2(ei0) = F2(ei0) |, and hence

In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and 83 83 powers of 2 2 .

Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-27T11:25:28Z No. of bitstreams: 0Bitstream added on 2014-05-27T14:41:41Z : No. of bitstreams: 1 2-s2.0-79951846250.pdf: 494002 bytes, checksum: 56b6ee8beddda3e7dae971355d44a19f (MD5) Made available in DSpace on 2014-05-27T11:25:28Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-03-01 Item merged in doublecheck by Felipe Arakaki (arakaki@reitoria.unesp.br) on 2015-12-11T17:28:11Z Item was identical to item(s): 71803, 21370 at handle(s): http://hdl.handle.net/11449/72321, http://hdl.handle.net/11449/21804 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) The simplest necessary conditions for an entire function ψ(x) =∞ ∑ k=0 γk xk/k! to be in the Laguerre-Pólya class are the Turán inequalities γ2 k- γk+1γk-1 ≥ 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with ψ to be hyperbolic. The higher order Turán inequalities 4(γ2 n - γn-1γn+1)(γ2n +1 - γnγn+2) - (γnγn+1 - γn-1γn+2) 2 ≥ 0 are also necessary conditions for a function of the above form to belong to the Laguerre-Pólya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic. Pólya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turán inequalities hold for the coefficients of the Riemann ψ-function. In this short paper, we prove that the higher order Turán inequalities also hold for the ψ-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three. © 2010 American Mathematical Society. Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP Departamento de matemática Aplicada IMECC UNICAMP, 13083-859 Campinas, SP Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP FAPESP: 03/01874-2 FAPESP: 06/60420-0 CNPq: 305622/2009-9 CAPES: DGU-160

We are concerned with the study of the Lane-Emden equation with variable exponent and Dirichlet boundary condition. The feature of this paper is that the analysis that we develop does not assume any subcritical hypotheses and the reaction can fulfill a mixed regime (subcritical, critical and supercritical). We consider the radial and the nonradial cases, as well as a singular setting. The proofs combine variational and analytic methods with a version of the Palais principle of symmetric criticality., The final version this paper will be published in Proc. AMS

In this paper, we investigate asymptotic behaviors of Racah polynomials with fixed parameters and scaled variable as the polynomial degree tends to infinity. We start from the difference equation satisfied by the polynomials and derive an asymptotic formula in the outer region via ratio asymptotics. Next, we find the asymptotic formulas in the oscillatory region via a simple matching principle. Unlike the varying parameter case considered in a previous paper, the zeros of Racah polynomials with fixed parameters may not always be real. For this unusual case, we also provide a standard method to determine the oscillatory curve which attracts the zeros of Racah polynomials when the degree becomes large.

In this paper we show that any Bohr-zero non-periodic set B B of traceless integer-valued matrices, denoted by Λ \Lambda , intersects non-trivially the conjugacy class of any matrix from Λ \Lambda . As a corollary, we obtain that the family of characteristic polynomials of B B contains all characteristic polynomials of matrices from Λ \Lambda . The main ingredient used in this paper is an equidistribution result for an S L d ( Z ) SL_d(\mathbb {Z}) random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work [J. Amer. Math. Soc. 24 (2011), 231–280].

In his paper "A Construction for Coisotropic Subalgebras of Lie Bialgebras", Marco Zambon gave a way to use a long root of a complex semisimple Lie biaglebra $\mathfrak{g}$ to construct a coisotropic subalgebra of $\mathfrak{g}$. In this paper, we generalize Zambon's construction. Our construction is based on the theory of Lagrangian subalgebras of the double $\mathfrak{g}\oplus\mathfrak{g}$ of $\mathfrak{g}$, and our coisotropic subalgebras correspond to torus fixed points in the variety $\mathcal{L}(\mathfrak{g}\oplus\mathfrak{g})$ of Lagrangian subalgebras of $\mathfrak{g}\oplus\mathfrak{g}$.

We will show that two functions in the extended Selberg class satisfying the same functional equation must be identically equal if they have sufficiently many common zeros. This paper concerns the question of how L-functions are determined by their zeros. L-functions are Dirichlet series with the Riemann zeta function ζ(s) = ∑∞ n=1 1 ns as the prototype and are important objects in number theory. The Selberg class S of L-functions is the set of all Dirichlet series L(s) = ∑∞ n=1 a(n) ns of a complex variable s = σ + it with a(1) = 1, satisfying the following axioms (see [7]): (i) (Dirichlet series) For σ > 1, L(s) is an absolutely convergent Dirichlet series. (ii) (Analytic continuation) There is a nonnegative integer k such that (s − 1)L(s) is an entire function of finite order. (iii) (Functional equation) L satisfies a functional equation of type ΛL(s) = ωΛL(1− s), where ΛL(s) = L(s)Q s ∏K j=1 Γ(λjs+μj) with positive real numbers Q, λj and with complex numbers μj , ω with Reμj ≥ 0 and |ω| = 1. (iv) (Ramanujan hypothesis) a(n) n for every e > 0; (v) (Euler product) logL(s) = ∑∞ n=1 b(n) ns , where b(n) = 0 unless n is a positive power of a prime and b(n) n for some θ < 12 . The degree dL of an L-function L is defined to be dL = 2 ∑K j=1 λj , where K, λj are the numbers in axiom (iii). The Selberg class includes the Riemann zeta-function ζ and essentially those Dirichlet series where one might expect the analogue of the Riemann hypothesis. At the same time, there are a whole host of interesting Dirichlet series not possessing a Euler product (see e.g. [3], [8]). Throughout the paper, all L-functions are assumed to be functions from the extended Selberg class of those only satisfying the axioms (i)-(iii) (see [3]). Thus, the results obtained in the present paper particularly apply to L-functions in the Selberg class. Received by the editors October 5, 2011 and, in revised form, February 12, 2012. 2010 Mathematics Subject Classification. Primary 11M36, 30D30.

The object of this short note is to give some observations on Bernoulli numbers and their function field analogs and point out ‘known’ counter-examples to a conjecture of Chowla. Bernoulli numbers Bn defined (for integer n > 1) by z/(e z − 1) = ∑ Bnz /n!, and their important cousins Bn/n, play interesting roles in many areas of mathematics. (Below we only restrict to these for n even, precisely the case when they are non-zero.) We mention some key words by which the reader can search: Power sums, Zeta special values, Eisenstein series, measures, p-adic L-functions, finite differences, combinatorics, Euler-Maclaurin formula, Todd classes in topology, Grothendieck-Hirzebruch-Riemann-Roch formula, K-theory of integers, Stable homotopy, Bhargava factorial associated to the set of primes, Kummer-HerbrandRibet theorems in cyclotomic theory, Kervaire-Milnor formula for diffeomorphism classes of exotic spheres. Their factorization is of interest, the denominators (which show up explicitly in the third-fourth items from the end) are well understood via theorems of von-Staudt, but the numerators (which show up explicitly in the last two items above) are mysterious and connected to many interesting phenomena. In one of the rare lapses, Ramanujan, in his very first paper [R1911, (14), (18) and Sec. 12], claimed to have proved (editors downgrade it to a conjecture) that the numerator Nn of Bn/n is always a prime, when it was already known since Kummer (in Fermat’s last theorem connection) that ‘irregular’ prime 37 is a proper divisor of N32, and even N20 is composite. In [C1930], Chowla showed that Ramanujan’s claim had infinity of counter-examples. Note that this also follows from one counterexample and the Kummer congruences (recalled below) for that prime! Interestingly, in his last paper [CC1986], Chowla (jointly with his daughter) asks as unsolved problem whether the numerator is always square-free. (This is also mentioned in the nice survey article by Murtys and Williams on Chowla’s work in Vol. 1 of [C1999], where the author learned about it.) Theorem 1. Chowla’s conjecture stated above has infinity of counter-examples. In fact, for any given irregular prime p less than 163 million, and given arbitrarily large k, there is n such that p divides Nn. Proof. Using the tables (or the reader can try to check directly!) giving factorizations of Bn/n, for example the table by Wagstaff at the Bernoulli web page www.bernoulli.org, we see that 37 divides N284. Now recall the well-known Kummer congruences that the value of (1 − p)Bn/n modulo p depends only on (even) n modulo pk−1(p − 1), for n not divisible by p − 1. The first claim follows by taking p = 37. Supported in part by NSA grant H98230-10-1-0200.

In this paper we consider continuous, SL ( 2 , R ) \text {SL}(2,\mathbb {R}) -valued, Schrödinger cocycles over irrational rotations. We prove two generic results on the Lyapunov exponents which improve the corresponding ones contained in a paper by Bjerklöv, Damanik and Johnson.

In this paper we present two-sided estimates of Chernoff type for the weighted L w 2 L_{w}^{2} -distance of a smooth function to the k k -dimensional space of all polynomials of degree less than k k , whenever the weight function w w solves the Pearson differential equation and generates a finite or infinite sequence of classical orthogonal polynomials. These inequalities are simple corollaries of a unified general theorem, which is the main result of the paper.

This paper complements a recent result of Dilworth, Ferenczi, Kutzarova and Odell regarding the ergodicity of strongly asymptotic ℓ p \ell _p spaces. We state this result in a more general form, involving domination relations, and we show that every asymptotically Hilbertian space which is not isomorphic to ℓ 2 \ell _2 is ergodic. In particular, every weak Hilbert space which is not isomorphic to ℓ 2 \ell _2 must be ergodic. Throughout the paper we construct explicitly the maps which establish the fact that the relation E 0 E_0 is Borel reducible to isomorphism between subspaces of the Banach spaces involved.

In this paper we give simple, sufficient conditions for the existence of the stochastic integral for vector-valued processes X with values in a Banach space E; namely, X is of class (LD), and the stochastic measure I X is bounded and strongly additive in L p E (in particular, if I X is bounded in L p E and c 0 ⊈ E) and has bounded semivariation. The result is then applied to martingales and processes with integrable variation or semivariation. For martingales the condition of being of class (LD) is superfluous. For a square-integrable martingale with values in a Hilbert space, all the conditions are superfluous. For processes with p-integrable semivariation or p-integrable variation, the conditions of I X to be bounded and have bounded semivariation are superfluous. For processes with 1-integrable variation, all conditions are superfluous. In a forthcoming paper, we shall extend these results to local summability. The extension needs additional nontrivial work.

In a former paper we studied transformations on the set of all n n -dimensional subspaces of a Hilbert space H H which preserve the principal angles. In the case when dim ⁡ H ≠ 2 n \dim H\neq 2n , we could determine the general form of all such maps. The aim of this paper is to complete our result by considering the problem in the remaining case dim ⁡ H = 2 n \dim H=2n .

Let ρ: G → GL(n, F) be a faithful representation of a finite group G. In this paper we proceed with the study of the image of the associated Noether map η G G : F[V(G)] G → F[V] G . In our 2005 paper it has been shown that the Noether map is surjective if V is a projective FG-module. This paper deals with the converse. The converse is in general not true: we illustrate this with an example. However, for p-groups (where p is the characteristic of the ground field F) as well as for permutation representations of any group the surjectivity of the Noether map implies the projectivity of V.

An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results complement a recent work by Ismail and Stanton. Quite surprisingly, in some cases the Taylor polynomials converge to a function which differs from the original one. We provide explicit expressions for the integral remainder. As application, we obtain some summation formulas for basic hypergeometric series. As far as we know, one of them is new. We conclude by studying the different forms of the binomial theorem in this context. 1. Introduction and definitions The problem of expanding a function with respect to a given polynomial basis has many implications in analysis. The simplest example of this kind is the Taylor's expansion theorem. In this paper, we replace the classical derivative by a difference operator of Askey-Wilson type. Our results complement the paper (3) of Ismail and Stanton and are a natural continuation of the point of view presented in (5), where a new approach to the theory of classical hypergeometric polynomials is given. In contrast with (3), our aim is to find sufficient conditions for t he Taylor series to converge, but not necessarily to the original function. In this more general setting, we may consider non-necessarily entire functions and we give an explicit expression for the limit of the remainders in terms of a contour integral. Using this and a new estimate for the q-shifted factorials, which might be of independent interest, we obtain a summation formula which is new as far as we know. As we explain below, it can be regarded as a non-symmetrized version of the non-terminating q-Saalschutz sum. As applications, we also provide a new proof of the q-Gauss summation formula and a list of binomial type summation formulas in the same line than Ismail's paper (2). Now we give some definitions which will be used in what follows. The notions we are presenting were already introduced in (5) with the aim of studying some aspects of the theory of hypergeometric polynomials. The relevance of this approach is justified in (5), where a more detailed exposition is given.

We consider the initial value problem for the fourth order nonlinear Schrodinger type equation (4NLS) related to the theory of vortex filament. In this paper we prove the time local well-posedness for (4NLS) in the Sobolev space, which is an improvement of our previous paper.

Weighted Opial-type inequalities are shown to be equivalent to weighted norm inequalities for sublinear operators and for nearly positive operators. Examples involving the Hardy-Littlewood maximal function and the non-increasing rearrange- ment are presented. Opial-type inequalities are related to norm inequalities much as quadratic forms are related to bilinear forms. A linear operator T on Hilbert space gives rise to the bilinear form (f,g) 7! hTf,gi and the quadratic form f 7! hTf,fi. Duality shows that the norm of T and the norm of the bilinear form coincide and a standard polarization argument shows that this norm is equivalent to but not necessarily equal to the norm of the quadratic form, called the numerical radius of T. In this paper, far from the luxuries of Hilbert spaces and linear operators, we show that the equivalence of operator norm and numerical radius persists. The work is in response to Richard Brown's suggestion that Steven Bloom's result (2, The- orem 1) which gives the equivalence for positive operators should apply in greater generality. Opial-type inequalities have been much studied since Opial's original paper in 1960 and the papers (2), (3) and (4) include many references. After the main theorem showing equivalence of Opial-type and norm inequali- ties, an example involving the Hardy-Littlewood maximal function is included to illustrate that the equivalence cannot be taken in a pointwise sense. To show that the method can be readily applied to generate non-trivial inequal- ities from known norm inequalities we give a simple weight characterization of an Opial-type inequality for the non-increasing rearrangement.

In this paper, we reinvestigate an old problem of prescribing Gaussian curvature in the negative case. In 1974, Kazdan and Warner considered the equation -Δu+α=R(x)e u , x ∈ M, on any compact two dimensional manifold M with a α > α o and it is not solvable for α < α o . Then one may naturally ask: Is the equation solvable for a = α o ? In this paper, we answer the question affirmatively. We show that there exists at least one solution for α = α o .

We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada. The spectral density function of the DML is defined using the von Neumann trace associated with the free action of a discrete group on a graph. The main result in this paper states that when the group is amenable, the spectral density function is equal to the integrated density of states of the DML that is defined using either Dirichlet or Neumann boundary conditions. This establishes the main conjecture in a paper by Mathai and Yates. The result is generalized to other self adjoint operators with finite propagation., LaTeX2e, 7 pages

In this paper, we study the sign-changing Kazdan-Warner's problem on two dimensional closed Riemannian manifold with negative Euler number $\chi(M)

In this paper, we extend the classical Bohr's inequality to the setting of the non-commutative Hardy space $H^1$ associated with a semifinite von Neumann algebra. As a consequence, we obtain Bohr's inequality for operators in the von Neumann-Schatten class $\cl C_1$ and square matrices of any finite order. Interestingly, we establish that the optimal bound for $r$ in the above mentioned Bohr's inequality concerning von Neumann-Shcatten class is 1/3 whereas it is 1/2 in the case of $2\times 2$ matrices and reduces to $\sqrt{2}-1$ for the case of $3\times 3$ matrices. We also obtain a generalization of our above-mentioned Bohr's inequality for finite matrices where we show that the optimal bound for $r$, unlike above, remains 1/3 for every fixed order $n\times n,\ n\ge 2$.

The first section of this paper provides an improvement upon known finiteness theorems for Riemannian submersions; that is, theorems which conclude that there are only finitely many isomorphism types of fiber bundles among Riemannian submersions whose total spaces and base spaces both satisfy certain geometric bounds. The second section of this paper provides a sharpening of some recent theorems which conclude that, for an open manifold of nonnegative curvature satisfying certain geometric bounds near its soul, there are only finitely many possibilities for the isomorphism class of a normal bundle of the soul. A common theme to both sections is a reliance on basic facts about Riemannian submersions whose A A and T T tensors are both bounded in norm.

Let B denote the unit ball in RI. This paper characterizes the subsets E of B with the property that supE h = SUPB h for all harmonic functions h on B which have finite Dirichlet integral. It also examines, in the spirit of a celebrated paper of Brelot and Doob, the associated question of the connection between non-tangential and fine cluster sets of functions on B at points of the boundary.

In this paper it is shown that if MC = ( iS a 2 x 2 upper triangular operator matrix acting on the Hilbert space 7H E IC and if W(.) denotes the "Weyl spectrum", then the passage from w(A) U w(B) to w(MC) is accomplished by removing certain open subsets of w(A) n w(B) from the former, that is, there is equality w(A) U w(B) = w(MC) U G, where G is the union of certain of the holes in w(Mc) which happen to be subsets of w (A) n w (B). Let At and IC be Hilbert spaces, let 1C(7-, 1C) denote the set of bounded linear operators from 7t to IC, and abbreviate LC(QH,7) to C(H). When A E LC(t) and B E L(/C) are given we denote by Mc an operator acting on 1t E 1C of the form M A C) Mc:=( 0 ), where C E iC(!, 7-). The invertibility and spectra of Mc were considered by Du and Jin [5]. In this paper we give some conditions for operators A and B to exist an operator C such that Mc is Weyl, and describe the Weyl spectra of Mc. Recall ([7], [8]) that an operator A E 12(X, Y) for Banach spaces X and Y is called regular if there is an operator A' E L(Y, X) for which A = AA'A; then A' is called a generalized inverse for A. In this case, X and Y can be decomposed as follows (cf. [8, Theorem 3.8.2]): A-1(0) e A'A(X) = X and A(X) e (AA')-'(0) = Y. It is familiar ([6], [8]) that A E LC((, IC) is regular if and only if A has closed range. An operator A E LC(7-, IC) is called relatively Weyl if there is an invertible operator A' E 12(IC,7t) for which A = AA'A. It is known ([8, Theorem 3.8.6]) that A is relatively Weyl if and only if A is regular and A-1(0) A(H)', where means a topological isomorphism between spaces. An operator A E LQ(H, IC) is called left-Fredholm if it is regular with finite dimensional null space and rightFredholm if it is regular with its range of finite co-dimension. If A is both leftand right-Fredholm, we call it Fredholm. The index, ind A, of a leftor right-Fredholm Received by the editors November 21, 1997 and, in revised form, May 1, 1998 and March 10, 1999. 1991 Mathematics Subject Classification. Primary 47A53, 47A55.

This paper introduces two main concepts, called a generalized Watson transform and a generalized skewWatson transform, which extend the notion of a Watson transform from its classical setting in one variable to higher dimensional and noncommutative situations. Several construction theorems are proved which provide necessary and sufficient conditions for an operator on a Hilbert space to be a generalized Watson transform or a generalized skew-Watson transform. Later papers in this series will treat applications of the theory to infinite-dimensional representation theory and integral operators on higher dimensional spaces.

In his 1986 paper in the Rev. Mat. Iberoamericana, A. Carbery proved that a singular integral operator is of weak type (p, p) on LP(lRTn) if its lacunary pieces satisfy a certain regularity condition. In this paper we prove that Carbery's result is sharp in a certain sense. We also obtain a weighted analogue of Carbery's result. Some applications of our results are also given.

In this paper, we prove that if S is an n-dimensional subspace of L(H), then S is ([i ] + 1)-reflexive, where [n ] denotes the greatest integer not n larger than '. By the result, we show that if 1?( ) = E Ai( )Bi is an 2=1 elementary operator on a C*-algebra A, then 'D is completely positive if and only if 'D is ([n1 ] + 1)-positive. In this paper, let H denote a complex Hilbert space. Let H(') denote the direct sum of n copies of H. For T E L(H), we write T(') for the operator on H(') which is the direct sum of n copies of T; the notation is extended to a subset of L(H) by S(n) = {T(n) E L(H(n)): T E S}. If S is a subspace of L(H), S is called n-reflexive if S(n) = ref (S(n)) =_ {T(n) E L(H(n)): T(n)X E [S(n)X], for all x E H (n)}, where [] denotes norm closed linear span. By the definiton, we have that if S is m-reflexive, then S is n-reflexive for n > m. A separating vector for a subspace S of L(H) is a vector x E H such that T 4 Tx, T E S, is an invective map. For x, y E H, let x 0 y denote the rank-one operator u | 4 (u, x)y. Let A denote a C*-algebra. Then A is called primitive, if A has a faithful irreducible representation on some Hilbert space. An elementary operator AP on A n is a linear mapping of the form AP: T F4 AiTBi, where {Ai},nL1 and {Bi},nU1 are i=l subsets of A. In this paper, we assume that all elementary operators are nonzero. A linear map 4J on A is positive (resp. hermitian-preserving) if 4)(T) is positive (resp. hermitian) for all positive (resp. hermitian) T in A. We define 4'n = 4{I 0 In: Mn(A) -4 Mn(A) by 4) 0 In((Tij)nxn) = (4)(Ti ))nxn. 4) is said to be n-positive if 4J 0 In is positive. If 4J is n-positive for all n, then 4J is said to be completely positive. In [4], Magajna states the following problem: For each positive integer r determine the smallest k = k(r) such that all rdimensional subspaces of L(H) are k-reflexive. In [4], Magajna proves k < r. In this paper, we prove that if S is an n-dimensional subspace of L(H), then S is ([n] + 1)-reflexive. Also by this result, we study complete positivity of elementary operators on a C*-algebra A. We prove that if n 4 ) = Ai( )Bi is an elementary operator on a C*-algebra A, then 4) is i=l1 Received by the editors July 8, 1996 and, in revised form, May 14, 1997. 1991 Mathematics Subject Classification. Primary 47B47, 47B49; Secondary 46L05.

In 2012, Chan proved a number of congruences for the coefficients of Ramanujan’s ϕ \phi function. In this paper, we prove some new congruences modulo powers of 2 and 3 for Ramanujan’s ϕ \phi function by employing Newman’s identities and theta function identities.

In this paper, the authors establish a multiplier theorem for Herz type Hardy spaces. Let Tm be a multiplier operator defined in terms of Fourier transforms by Tmf Q) = M(W for suitable functions f. It is well-known that there is a multiplier theorem for H1 (Rn) (see [FS]): if a > n/2 and (1) J ID3m(e)12d d 0, let us denote m6QW) = M(6077(0) It is easy to check that (1) is equivalent to (2) sup |m6||Ka 2(Rn) < 0, 6 2 where K2'2 2(R) is a non-homogeneous Herz space (see [BS]). By using some embedding relations on Herz spaces, A. Baernstein II and E. T. Sawyer [BS] weakened (2) into (3) sup iM6i||Kf1l (Rn) < 0, 6 where 0 < E < an2. In fact, this is just a special case of their theorem. In [BS], Baernstein and Sawyer showed that m is a bounded multiplier of H1 (Rn) under an even weaker condition than (3); see Theorem 3b in [BS, page 21]. By using the technique of Herz type Hardy spaces developed by the authors in [LY1]-[LY3] and [Y], in this paper, we shall first establish a multiplier theorem for the homogeneous Herz type Hardy space Hk n(1-1/q), 1(IRn) which is introduced by y ~~~q the authors of this paper in [LY1]. Then as simple consequences of this theorem, a multiplier theorem for the corresponding non-homogeneous version of the space Received by the editors April 13, 1995 and, in revised form, April 5, 1997. 1991 Mathematics Subject Classification. Primary 42B15; Secondary 42B30.

We present a new proof for Morita equivalence of twisted crossed products by coactions within the abstract context of crossed products of Hilbert C∗-modules. In this context we are free from representing all C∗-algebras and Hilbert C∗-modules on Hilbert spaces. The notion of Morita equivalence of twisted coactions was introduced in [B]. In [B, Theorem 3.3] we established conditions on twisted coactions which are sufficient to ensure Morita equivalence of the corresponding crossed product C∗-algebras. Later [ER] gave a shorter proof for this result using their results on multipliers of imprimitivity bimodules. However in the proofs of both [B] and [ER], all C∗algebras and Hilbert C∗-modules need to be represented on Hilbert spaces. In this paper we present a new proof for [B, Theorem 3.3] based on the notion of crossed products of Hilbert C∗-modules introduced in [B2]. Crossed products of Hilbert C∗-modules in [B2] were defined as subspaces of adjointable operators between Hilbert C∗-modules. In this abstract context, we are free from representing all C∗-algebras and Hilbert C∗-modules on Hilbert spaces as in [B] and [ER]. As a consequence, the proof here is shorter and more elegant than that of [B]. Our approach is close to the spirit of [BS], and different from [ER]. Throughout this paper G is a locally compact group and N is a closed normal amenable subgroup of G. Recall from [M, Lemma 3] that there is a surjective homomorphism Ψ from C∗ r (G) into C ∗ r (G/N) such that Ψ(λ (r)) = λ (qN (r)), where q N : G → G/N is the quotient map, λ and λ are the left regular representations ofG andG/N . We denote by WG the unitary operator on L 2(G×G) defined by [WGξ](r, s) = ξ(r, r −1s). If f is an element of the Fourier algebra A(G), then Sf (WG) = Mf . Here Sf denotes the slice map, see [LPRS, §1]. To apply [B2, Theorem 1.6] to this paper, we need to show that WG is a regular multiplicative unitary. For any ξ, η ∈ L(G), we define ωη,ξ = 〈Tξ|η〉, ∀T ∈ B(L(G)). Then for any ω = ωη,ξ, we have 〈(id⊗ ω)(WG)ξ′|η′〉 = 〈Mω◦λGξ′|η′〉, ∀ξ′, η′ ∈ L(G). It then follows that ŜWG = C0(G), and the crossed product of [B2, Proposition 1.5] is just the crossed product of [LPRS, Definition 2.4]. The unitary operator Received by the editors October 23, 1995 and, in revised form, February 6, 1996. 1991 Mathematics Subject Classification. Primary 46L05, 22D25. c ©1997 American Mathematical Society 2109 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

If a smooth manifold has a Riemannian metric with positive scalar curvature, it follows immediately that the universal covering also has such a metric. The paper establishes a converse if the manifold in question is closed of dimension at least 5 and the fundamental group is an elementary abelian p-group of rank 2, where p is an odd prime. A conjecture of J. Rosenberg [Rs3] states that a closed smooth manifold with odd order fundamental group and dimension ≥ 5 admits a Riemannian metric with positive scalar curvature if and only if its universal covering admits such a metric. Standard considerations involving transfers (cf. [KS1]) reduce the conjecture to the special case of p-groups (where p is an odd prime). Results of Rosenberg [Rs1]– [Rs3] and S. Kwasik and the author [KS1] prove the conjecture if the fundamental group G is a cyclic p-group. In this work we study the conjecture when G is a finite abelian p-group. The following interim conclusion reflects many of the basic ideas and disposes of the first examples not covered by [KS1], [Rs1]–[Rs3]. Theorem. Let p be an odd prime, and let M be a closed smooth manifold of dimension n ≥ 5 with fundamental group Zp × Zp. Then M admits a Riemannian metric with positive scalar curvature if and only if its universal covering M does. In [RsS] J. Rosenberg and S. Stolz prove a stable result that yields a weaker conclusion for G ∼= Zp × Zp but applies to all finite groups. In particular, if G is a finite group of odd order and M is a closed smooth manifold with fundamental group G, their result states that the universal covering M has a Riemannian metric with positive scalar curvature if and only if some product M ×X × · · · ×X does, where X is an 8-dimensional Bott manifold; i.e., it is a spin manifold whose Â-genus is equal to 1. Although the methods should yield quantitative information on the number of factors of X that are needed, it is not clear how precise such estimates would be. Our result implies that if G ≈ Zp×Zp, then no copies of X are needed if n ≥ 5 and at most one copy of X is needed if n < 5. A more geometric proof of this result in dimensions ≤ 2p− 2 was obtained independently by Stolz (unpublished). It seems likely that the methods of this paper can be combined with the multiple Kunneth formula decomposition of [Hn] and finite induction to prove at least a semistable version of Rosenberg’s conjecture for all elementary abelian p-groups Received by the editors February 13, 1995 and, in revised form, September 13, 1995. 1991 Mathematics Subject Classification. Primary 53C21, 55N15, 57R75; Secondary 53C20, 57R85. c ©1997 American Mathematical Society

In this paper, we investigate the probabilistic pointwise convergence problem of Schrödinger equation on the manifolds. We prove probabilistic pointwise convergence of the solutions to Schrödinger equations with the initial data in L 2 ( T n ) L^{2}(\mathrm {\mathbf {T}}^{n}) , where T = [ 0 , 2 π ) \mathrm {\mathbf {T}}=[0,2\pi ) , which require much less regularity for the initial data than the rough data case. We also prove probabilistic pointwise convergence of the solutions to Schrödinger equation with Dirichlet boundary condition for a large set of random initial data in ∩ s > 1 2 H s ( Θ ) \cap _{s>\frac {1}{2}}H^{s}(\Theta ) , where Θ \Theta is three dimensional unit ball, which require much less regularity for the initial data than the rough data case.

In this paper, we investigate the strict inclusion relation associated with intersections and unions of a general family of function spaces. We answer partially a question left open in Korhonen and Rättyä [Comput. Methods Funct. Theory 5 (2005), pp. 459–469].

Let F, G E C[x, y]. If the Jacobian determinant of F and G is 1, then G is said to be a Jacobian mate of F. If, in addition, G has degree less than that of F, then G is said to be a younger mate of F . In this paper, a necessary and sufficient condition is given for a polynomial to have a younger mate. This also gives rise to a formula for the younger mate if it exists. Furthermore, a conjecture concerning the existence of a younger mate is shown to be equivalent to the Jacobian conjecture. Throughout this paper, F and G will be polynomials in C[x, y] where C denotes the field of complex numbers. We say that F and G satisfy the Jacobian hypothesis if their Jacobian determinant is one, i.e., Fx Gy Fy Gx = 1 . In this case, we also say that G is a Jacobian mate of F. Furthermore, if the x-degree (resp. y-degree, total degree) of G is less than that of F, then G is said to be a younger mate of F relative to the x-degree (resp. y-degree, total degree). For instance, x + y has younger mates y and -x relative to the x-degree and the y-degree, respectively, but has no younger mate relative to the total degree. This paper was motivated by the Jacobian conjecture which asserts that if F has a Jacobian mate G, then (F, G) is an automorphism pair. In Section 1, it is shown that a younger mate is unique (up to an additive constant) and universal, i.e., if a Jacobian mate G of F exists, then any other mate of F can be expressed as G plus a polynomial in F. In Section 2, the problem of existence of a younger mate of F is reduced to the case where F is monic in both variables. In Section 3, a necessary and sufficient condition for the existence of a younger mate and a formula for a younger mate provided one exists are given. Finally, in Section 4, a conjecture concerning the existence of younger mates is formulated and shown to be equivalent to the Jacobian conjecture. Received by the editors July 6, 1993 and, in revised form, January 18, 1994. 1991 Mathematics Subject Classification. Primary 13B25, 13F20, 14E09, 16W20.

By using Deodhar's combinatorial setting and Bernstein-Gelfand projective functors, this paper provides some necessary and sufficient conditions for a highest weight category to have a Kazhdan-Lusztig theory. A consequence of these conditions is that in the semisimple Lie algebra case, the Kazhdan-Lusztig conjecture on the multiplicities of a Verma module implies the nonnegativity conjecture on the coefficients of Kazhdan-Lusztig polynomials. One of the central topics in representation theory in recent years is the socalled Kazhdan-Lusztig theory. The Kazhdan-Lusztig polynomials play a key role in this theory. These polynomials can be defined by using a distinguished basis of the Hecke algebra associated to a Coxeter group. In [KL1], there are two conjectures about these polynomials: (a) For any Coxeter group, the coefficients of these polynomials are nonnegative integers; (b) If the Coxeter group is the Weyl group of a complex semisimple Lie algebra, then the multiplicities of the composition series of a Verma module are given by the values of these polynomials at 1. Conjecture (b) is usually referred to as the Kazhdan-Lusztig conjecture and was proven in [BB] and [BK] shortly thereafter. Conjecture (a) is now known to be true for all crystallographic Coxeter groups (for a more upto-date reference on recent developments of Kazhdan-Lusztig theory, we refer to [DS]). It was shown in [D] that if the coefficients of the Kazhdan-Lusztig polynomials of a Coxeter group are nonnegative, then these polynomials can be defined by using certain sets derived from the elements of the Coxeter group. In fact, these sets give a closed formula for the Kazhdan-Lusztig polynomials under the nonnegativity assumption (see [D]). Since the Kazhdan-Lusztig polynomials are not easy to get at in general, the results in [D] give strong evidence for the importance of the nonnegativitiness. In an attempt to understand the results of [D], we observed that in the semisimple Lie algebra case, conjecture (b) implies conjecture (a). The connection is provided by some tensor functors called projective functors defined in [BG]. In this paper, we will give some necessary and sufficient conditions for the validity of the Kazhdan-Lusztig conjecture in certain special cases of the highest weight categories defined by CPS (see [CPS1] Received by the editors June 2, 1993; the contents of this paper have been presented to the Nineteenth Holiday Symposium held in December 1992 at New Mexico State University. 1991 Mathematics Subject Classification. Primary 22E47, 17B10; Secondary 22E46, 17B35.

Let p p be a rational prime and D D a positive rational integer coprime with p p . Denote by N ( D , 1 , p ) N(D, 1,p) the number of solutions ( x , n ) (x, n) of the equation D x 2 + 1 = p n D x^2 + 1 = p^n in rational integers x ≥ 1 x \geq 1 and n ≥ 1 n \geq 1 . In a paper of Le, he claimed that N ( D , 1 , p ) ≤ 2 N(D, 1, p) \leq 2 without giving a proof. Furthermore, the statement N ( D , 1 , p ) ≤ 2 N(D, 1, p) \leq 2 has been used by Le, Bugeaud and Shorey in their papers to derive results on certain Diophantine equations. In this paper we point out that the statement N ( D , 1 , p ) ≤ 2 N(D, 1, p) \leq 2 is incorrect by proving that N ( 2 , 1 , 3 ) = 3 N(2, 1, 3)=3 .

It is well known that a generic compact Riemann surface of genus greater than two admits only the identity automorphism; however, examples of such Riemann surfaces with their defining algebraic equations have not appeared in the literature. In this paper we give the defining equations of a doubly infinite, two-parameter family of projective curves (Riemann surfaces if defined over the complex numbers), whose members admit only the identity automorphism. It is well known that a generic curve of genus greater than two admits only the identity automorphism. Although this result was probably known by the turn of the century, the first published proof was given by Bailey in 1961 [1]. To obtain the strongest results, Bailey's method is necessarily nonconstructive; it does not yield an example of a defining algebraic equation for a curve with no nontrivial automorphisms. Similarly a proof by Greenberg [4], using techniques of Teichmuller theory, does not yield an explicit example of a Riemann surface with a trivial automorphism group. Much of the subsequent work on automorphisms of Riemann surfaces, including the author's, has relied on the representation of a given Riemann surface as the upper half plane under the action of a Fuchsian group. This again has the disadvantage of rarely yielding a defining algebraic equation for the given Riemann surface. Indeed, in the preface to his book, The complex analytic theory of Teichmiiller spaces, Subhashis Nag exclaimed, "Almost every compact Riemann surface of genus g > 3 allows only the identity automorphism. (I don't know, though, of even a single explicit such algebraic curve whose automorphism group is demonstrably trivial!)." The author believes that examples pertinent to famous theorems should be readily at hand. Therefore, in this paper we give the defining equations of a doubly infinite, two-parameter family of curves which admit only the identity automorphism (see equation (1) below). The curves in this family have genus (n 1)(m 1)/2 for relatively prime integers m and n which satisfy n > m+ 1 > 3. Let C be a curve defined by (1) and let C' be a nonsingular projective model for C. The proof that C' admits only the trivial automorphism depends on the Received by the editors February 5, 1993. 1991 Mathematics Subject Classification. Primary 14E09, 14H55, 30F99. @ 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

We show that extension problems for generalized Toeplitz kernels may be completely reduced to extension problems for positive definite functions, where the solution is well known. These considerations in particular imply that generalized Toeplitz kernels may be represented as Fourier transforms of positive operator-valued measures. The notion of generalized Toeplitz kernels (g.T.k.) was introduced in [5], where also a generalized Bochner theorem was proved in the discrete case. Since then many papers concerning this topic appeared, for example, [2, 3]. The extension problem for the discrete case was discussed in [ 1 ] and for the continuous case, for example, in [4]. The proofs for the extendibility and for the generalized Bochner theorem use mainly lifting theorems for families of operators. In this paper we want to show that these questions can be reduced directly to appropriate questions for positive definite operator-valued functions. Fix Hilbert spaces H1, H2, and let 0 0 a, ,B=1,2 sEI4(a),tEIp(a) for all pairs of functions (oc: I(a(a) -+ Ho, a = 1, 2, with finite support. Recall that an operator-valued function F: (-2a, 2a) -. L(H) with some Hilbert space H is called positive definite if (1) Z (F(s t)4(s), 'k(t))H > 0 s, tE(-a, a) Received by the editors May 29, 1992. 1991 Mathematics Subject Classification. Primary 43A35, 42A82, 47A20; Secondary 46L30. ? 1994 American Mathematical Society

In this paper we demonstrate various reducible examples of the scheme I d , g , r ′ \mathcal {I}{’ _{d,g,r}} of smooth curves of degee d and genus g in P r {\mathbb {P}^r} with positive Brill-Noether number. An example of a reducible I d , g , r ′ \mathcal {I}{’ _{d,g,r}} with positive ρ ( d , g , r ) \rho (d,g,r) , namely, the example I 2 g − 8 , g , g − 8 ′ , \mathcal {I}{’ _{2g - 8,g,g - 8}}, , has been known to some people and seems to have first appeared in the literature in Eisenbud and Harris, Irreducibility of some families of linear series with Brill-Noether number − 1 -1 , Ann. Sci. École Norm. Sup. (4) 22 (1989), 33-53. The purpose of this paper is to add a wider class of examples to the list of such reducible examples by using general k-gonal curves. We also show that I d , g , r ′ \mathcal {I}{’ _{d,g,r}} is irreducible for the range of d ≥ 2 g − 7 d \geq 2g - 7 and g − d + r ≤ 0 g - d + r \leq 0 .

In this paper we shall consider the factorization of positive cones of order n of a von Neumann algebra. Namely, we shall show the existence of a *-subalgebra inducing the positive cone of order n of a von Neumann algebra. Let M be a von Neumann algebra on a Hilbert space H. It is known that a positive cone M,(M)+ (= (M 0 Mn)+), where Mn denotes an algebra of all n x n matrices, coincides with the convex hull of all elements [xlx;] for xi E M. We shall find a *-subalgebra N of M such that a positive cone of order n is generated by all elements [alc*caj] for ai E N, c E M. We may then say that the cone is factorized by N. The purpose of this paper is to consider when a positive cone of order n can be factorized or not. The factorization of a self-dual cone in the Hilbert space associated with a standard form of a von Neumann algebra was already shown in [4]. We shall use the books of Pedersen [5] and Takesaki [8] as references of concepts and results of operator algebras. We begin with the following definition, in which we consider two convex cones of order n . Definition 1. Let M be a von Neumann algebra on a Hilbert space H and N a subspace of M. For a natural number n we put Cn(N) = cos{[a c*caj] E Mn(M)jai E N, c E M}

The maximal nilpotent and supersoluble automorphism groups of Riemann surfaces were given in earlier papers by this author. In this note the author wishes to correct the necessity of the condition given in Theorem (4.3) of Bounds for the order of supersoluble automorphism groups of Riemann surfaces (Proc. Amer. Math. Soc. 108 (1990), 587-600), which was left out at the time of writing the paper. The author also wishes to apologize to the readers for that.

In this paper we prove that if an atriodic and weakly chainable continuum is the inverse limit of trees with simplicial bonding maps, then it is chainable. By a graph we understand a one-dimensional polyhedron G with finite and fixed collections of vertices %"F(G) and edges o'(G) . We will assume that every graph is a subset of three-dimensional Euclidean space and every edge is a straight linear closed segment between its vertices. Two vertices belonging to the same edge are called adjacent. A connected graph without a simple closed curve is called a tree. A tree in which every vertex belongs to at most two edges is called an arc. If u and v are two adjacent vertices of a graph, by (u, v) we will denote the edge between u and v. Additionally, if u and v are two vertices of a tree, by (u, v) we will denote the arc between u and v. A map ( of a graph G1 into a graph Go is simplicial provided that 9(Y(G1)) c Y(Go)X and if u and v are two adjacent vertices of GI , then f I (u, v) is a linear map either onto an edge or onto a single vertex. All topological spaces considered in this paper are metric. A continuum is a connected and compact space. A continuum is tree-like (chainable) provided that it can be expressed as the inverse limit of a sequence of trees (arcs) with continuous (not necessarily simplicial) bonding maps. A continuum is weakly chainable if it is a continuous image of a chainable continuum (see [3, 2]). We will say that three continua A, B, and C form a triod if none of them is contained in the union of the remaining two and 0 0 A n B n C = A n B = A n C = B n C. A continuum is atriodic if it does not contain a triod. In 1981 Mohler asked whether every weakly chainable atriodic tree-like continuum X is chainable (see [16, Problem 171; 6, Problem 16]). This problem is closely related to the question of whether span (see [4, 5]) zero implies chainability (see [1, Problem #8]). Both problems were extensively studied by Oversteegen and Tymchatyn in [11-15] and by Oversteegen in [10]. Among other results Oversteegen and Tymchatyn proved in [1 1, 14] that any tree-like Received by the editors January 26, 1991. 1991 Mathematics Subject Classification. Primary 54F 1 5.

In this paper, we have obtained two main results by using probabilistic methods: (i) For a domain, we obtained a representation formula of the bounded solution to the first boundary value problem for Schrödinger equation; (ii) For α ∈ R 1 \alpha \in {R^1} , under certain conditions, we proved that the bounded solution having limit α \alpha at infinity to the generalized first boundary value problem for Schrödinger equation exists and is unique, and it is represented in explicit formula. The results of this paper are generalizations of Chung and Rao.

In this paper, we first establish a version of multidimensional analogues of the refined Bohr’s inequality. Then we establish two versions of multidimensional analogues of improved Bohr’s inequality with initial coefficient being zero. Finally we establish two versions of multidimensional analogues of improved Bohr’s inequality with the initial coefficient being replaced by absolute value of the function, and to prove that most of the results are sharp.