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In this paper, the following problem is studied. Let Ω 1 and Ω 2 be two domains in the complex plane with Ω 1 n Ω 2 ¬= O. Suppose that f j : Ω j → f j (Ω j ) (j = 1, 2) are two quasiconformal mappings satisfying f 1 | Ω1 ∩ Ω2 = f 2 | Ω1 ∩ Ω2 . Let F be the mapping in Ω 1 U Ω 2 defined by F| Ωj = fj (j = 1,2). If both f 1 and f 2 are uniquely extremal, is F always uniquely extremal? It is shown in this paper that the answer to this problem is no.

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.

Let U n ( q ) U_{n}(q) denote the unitriangular group of degree n n over the finite field with q q elements. In a previous paper we obtained a decomposition of the regular character of U n ( q ) U_{n}(q) as an orthogonal sum of basic characters. In this paper, we study the irreducible constituents of an arbitrary basic character ξ D ( φ ) \xi _{{\mathcal {D}}}(\varphi ) of U n ( q ) U_{n}(q) . We prove that ξ D ( φ ) \xi _{ {\mathcal {D}}}(\varphi ) is induced from a linear character of an algebra subgroup of U n ( q ) U_{n}(q) , and we use the Hecke algebra associated with this linear character to describe the irreducible constituents of ξ D ( φ ) \xi _{{\mathcal {D}}}(\varphi ) as characters induced from an algebra subgroup of U n ( q ) U_{n}(q) . Finally, we identify a special irreducible constituent of ξ D ( φ ) \xi _{{\mathcal {D}}}(\varphi ) , which is also induced from a linear character of an algebra subgroup. In particular, we extend a previous result (proved under the assumption p ≥ n p \geq n where p p is the characteristic of the field) that gives a necessary and sufficient condition for ξ D ( φ ) \xi _{{\mathcal {D}}}(\varphi ) to have a unique irreducible constituent.

We address the problem of interrelations between the properties of an action of a discrete group Γ \Gamma on a compact Hausdorff space X X and the algebraic and analytical properties of the module of all continuous functions C ( X ) C(X) over the algebra of invariant continuous functions C Γ ( X ) C_\Gamma (X) . The present paper is a continuation of our joint paper with M. Frank and V. Manuilov. Here we prove some statements inverse to the ones obtained in that paper: we deduce properties of actions from properties of modules. In particular, it is proved that if for a uniformly continuous action the module C ( X ) C(X) is finitely generated projective over C Γ ( X ) C_\Gamma (X) , then the cardinality of orbits of the action is finite and fixed. Sufficient conditions for existence of natural conditional expectations C ( X ) → C Γ ( X ) C(X)\to C_\Gamma (X) are obtained.

It is a well-known result of M. Brodmann that if a is an ideal of a commutative Noetherian ring A, then the set of associated primes Ass(A/α n ) of the n-th power of a is constant for all large n. This paper is concerned with the following question: given a prime ideal p of A which is known to be in Ass(A/a n ) for all large integers n, can one identify a term of the sequence (Ass(A/a n )) n ∈ N beyond which p will subsequently be an ever-present? This paper presents some results about convergence of sequences of sets of associated primes of graded components of finitely generated graded modules over a standard positively graded commutative Noetherian ring; those results are then applied to the above question.

This paper answers a question of Jan J. Dijkstra by giving a proof that all three-point sets are zero dimensional. It is known that all two-point sets are zero dimensional, and it is known that for all n > 3, there are n-point sets which are not zero dimensional, so this paper answers the question for the last remaining case.

Trudinger-Moser inequality with logarithmic weight was first established by Calanchi and Ruf [J. Differential Equations 258 (2015), pp. 1967–1989]. The aim of this paper is to address the higher order version; more precisely, we show the following inequality sup u ∈ W 0 , r a d 2 , 2 ( B , ω ) , ‖ Δ u ‖ ω ≤ 1 ∫ B exp ⁡ ( α | u | 2 1 − β ) d x > + ∞ \begin{equation*} \sup _{u \in W_{0,rad}^{2,2}(B,\omega ),{{\left \| {\Delta u} \right \|}_\omega } \le 1} \int _B {\exp \left ( {\alpha {{\left | u \right |}^{\frac {2}{{1 - \beta }}}}} \right )} dx > + \infty \end{equation*} holds if and only if \[ α ≤ α β = 4 [ 8 π 2 ( 1 − β ) ] 1 1 − β , \alpha \le {\alpha _\beta } = 4{\left [ {8{\pi ^2}\left ( {1 - \beta } \right )} \right ]^{\frac {1}{{1 - \beta }}}}, \] where B B denotes the unit ball in R 4 \mathbb {R}^{4} , β ∈ ( 0 , 1 ) \beta \in \left ( {0,1} \right ) , ω ( x ) = ( log ⁡ 1 | x | ) β \omega \left ( x \right ) = {\left ( {\log \frac {1}{{\left | x \right |}}} \right )^\beta } or ( log ⁡ e | x | ) β {\left ( {\log \frac {e}{{\left | x \right |}}} \right )^\beta } , and W 0 , r a d 2 , 2 ( B , ω ) W_{0,rad}^{2,2}(B,\omega ) is the weighted Sobolev spaces. Our proof is based on a suitable change of variable that allows us to represent the laplacian of u u in terms of the second derivatives with respect to the new variable; this method was first used by Tarsi [Potential Anal. 37 (2012), pp. 353–385].

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 establish an integral inequality on centroaffine hyperovaloids in R n + 1 \mathbb {R}^{n+1} , in terms of the Ricci curvature in direction of the Tchebychev vector field and the norm of the covariant differentiation of the difference tensor with respect to the Levi-Civita connection of the centroaffine metric. This integral inequality is optimal, and its equality case provides a new centroaffine characterization of the ellipsoids.

In this paper we shall deduce a Bailey type formula as a consequence of the residual identity of a q q -series transformation due to Gasper. Our formula leads to a variety of q q -series identities which are related to the arithmetic function counting integer representations of the form \[ n ( A n + B ) 2 + r ( C r + D ) 2 + E n r . \frac {n(An+B)}{2}+\frac {r(Cr+D)}{2} + Enr. \]

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].

When ordered by discriminant, it is known that about 83% of quartic fields over Q have associated Galois group S_4, while the remaining 17% have Galois group D_4. We study these proportions over a general number field F. We find that asymptotically 100% of quadratic number fields have more D_4 extensions than S_4 and that the ratio between the number of D_4 and S_4 quartic extensions is biased arbitrarily in favor of D_4 extensions. Under GRH, we give a lower bound that holds for general number fields., Fixed a typo with Theorem 1.3 that is present in the published version of the paper. The main results remain unchanged

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the Zeilberger--Bressoud $q$-Dyson theorem or the $q$-Dyson constant term identity. This conjecture was proved by Karolyi, Lascoux and Warnaar in 2015. In this paper, by slightly changing the variables of Kadell's conjecture, we obtain another symmetric function generalization of the $q$-Dyson constant term identity. This new generalized constant term admits a simple product-form expression.

Consider the affine space consisting of pairs of matrices ( A , B ) (A,B) of fixed size, and its closed subvariety given by the rank conditions rank ⁡ A ≤ a , rank ⁡ B ≤ b \operatorname {rank} A \leq a, \, \operatorname {rank} B \leq b , and rank ⁡ ( A ⋅ B ) ≤ c \operatorname {rank} (A\cdot B) \leq c , for three non-negative integers a , b , c a,b,c . These varieties are precisely the orbit closures of representations for the equioriented A 3 \mathbb {A}_3 quiver. In this paper we construct the (equivariant) minimal free resolutions of the defining ideals of such varieties. We show how this problem is equivalent to determining the cohomology groups of the tensor product of two Schur functors of tautological bundles on a 2-step flag variety. We provide several techniques for the determination of these groups, which is of independent interest.

In 1974, Askey and Steinig proved that for all n ≥ 0 n\geq 0 and x ∈ ( 0 , 2 π ) x\in (0,2\pi ) the trigonometric sums sin ⁡ ( x / 4 ) 1 + sin ⁡ ( 5 x / 4 ) 2 + ⋯ + sin ⁡ ( ( 4 n + 1 ) x / 4 ) n + 1 \begin{equation*} \frac {\sin (x/4)}{1}+\frac {\sin (5x/4)}{2}+\cdots + \frac {\sin ((4n+1)x/4)}{n+1} \end{equation*} and cos ⁡ ( x / 4 ) 1 + cos ⁡ ( 5 x / 4 ) 2 + ⋯ + cos ⁡ ( ( 4 n + 1 ) x / 4 ) n + 1 \begin{equation*} \frac {\cos (x/4)}{1}+\frac {\cos (5x/4)}{2}+\cdots + \frac {\cos ((4n+1)x/4)}{n+1} \end{equation*} are positive. Recently, the Askey-Steinig inequalities were improved by the present authors. In this paper, we further improve these inequalities and provide new sharp upper and lower bounds for the two sums given above.

This paper investigates connections between the long-time asymptotics of heat distribution on a body Ω in Z n , and various geometric properties of Ω, starting from an initially constant heat distribution supported on Ω. We use combinatorial and differential geometric methods. We begin the paper with a result in R.

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

A topological space X is said to belong to the class of Stegall (weakly Stegall) spaces if for every Baire (complete metric) space B and minimal usco φ: B → 2 X , φ is single-valued at some point of B. In this paper we show that under some additional set-theoretic assumptions that are equiconsistent with the existence of a measurable cardinal there is a Banach space X whose dual, equipped with the weak* topology, is in the class of weakly Stegall spaces but not in the class of Stegall spaces. This paper also contains an example of a compact space K such that K belongs to the class of weakly Stegall spaces but (C(K)*, weak*) does not.

In this paper, the authors consider the boundary value problem (E) x (2m) (t) + (-1) m+1 f(x(t)) = 0, 0 < t < 1, (B) x (2i) (0) = x (2i) (1) = 0, i = 0, 1, 2, …, m - 1, and give sufficient conditions for the existence of any number of symmetric positive solutions of (E)-(B). The relationships between the results in this paper and some recent work by Henderson and Thompson (Proc. Amer. Math. Soc. 128 (2000), 2373-2379) are discussed.

In a previous paper, the author proved that in characteristic zero the jacobian J(C) of a hyperelliptic curve C: y 2 = f(x) has only trivial endomorphisms over an algebraic closure K a of the ground field K if the Galois group Gal(f) of the irreducible polynomial f(x) ∈ K[x] is either the symmetric group S n or the alternating group An. Here n > 4 is the degree of f. In another paper by the author this result was extended to the case of certain smaller Galois groups. In particular, the infinite series n = 2 r + 1, Gal(f) = L 2 (2 r ):= PSL 2 (F 2 r) and n = 2 4r+2 +1, Gal(f) = Sz(2 2r+1 ) were treated. In this paper the case of Gal(f) = U 3 (2 m ):= PSU 3 (F 2 m) and n = 2 3m + I is treated.

In his famous paper The image of a derivation is contained in the radical, Marc Thomas establishes the (commutative) Singer-Wermer conjecture, showing that derivations from a commutative Banach algebra A to itself must map into the radical. The proof goes via first showing that the separating subspace of a derivation on A must lie in the radical of A. In this paper, we exhibit discontinuous derivations on a commutative unital Frechet algebra A such that the separating subspace is the whole of A. Thus, the situation on Frechet algebras is markedly different from that on Banach algebras.

Let G be a finite group of Lie type in characteristic p. This paper addresses the problem of describing the irreducible complex (or p-adic) representations of G that remain absolutely irreducible under the Brauer reduction modulo p. An efficient approach to solve this problem for p > 3 has been elaborated in earlier papers by the authors. In this paper, we use arithmetical properties of character degrees to solve this problem for the groups G ∈ { 2 B 2 (q), 2 G 2 (q), G 2 (q), 2 F 4 (q), F 4 (q), 3 D 4 (q)} provided that p < 3. We also prove an asymptotical result, which solves the problem for all finite groups of Lie type over F q with q large enough.

We define and study the universal norm distribution in this paper, which generalizes the well studied universal ordinary distribution by Kubert (1979). We display a resolution of Anderson type for the universal norm distribution. Furthermore, we prove a general index formula between different. universal norm distributions. As a special case, this general index formula recovers the hard calculation in Sinnott's Annals paper (1978).

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.

This paper deals with the spectra of the Laplace and stability operators of a constant mean curvature surface in the hyperbolic space. In a preceding work, the author described the essential spectra of these operators, assuming that the surface is of finite total curvature. In this paper, we prove that these two operators have a finite number of eigenvalues below their essential spectra.

In this paper, we first characterize the rank one operators in the preannihilator U⊥ of a weakly closed T(N)-module U. Using this characterization for the rank one operators in U⊥, a complete description of the extreme points of the unit ball U 1 is given. Finally, we show how to apply the techniques of the present paper to other operator systems and characterize their extreme points.

In this paper, we generalize Chirka's theorem on the extension of functions holomorphic in a neighbourhood of S U (oD x D) where D is the open unit disc in C and S is the graph of a continuous D-valued function on D to higher dimensions, for certain classes of graphs S C D x Dn, n > 1. In particular, we show that Chirka's extension theorem generalizes to configurations in Cn+1, n > 1, involving graphs of (non-holomorphic) polynomial maps with small coefficients. 1. THE MAIN THEOREM This paper is motivated by an article by Chirka [1] (also see [2]) in which he proves the following result (in what follows, D will denote the open unit disc in C, while Dr will denote the open disc of radius r, centered at 0 C C): Theorem 1.1 (Chirka). Let 0: D -> C be a continuous function having sup cIq5(z)I 1. Rosay [3] showed that the theorem fails in general for higher dimensions. A natural question that arises is whether holomorphic extension to Dn+1r n > 1, occurs when the component functions of the Dn -valued map defining our graph are small in some appropriate sense (for instance, when the graph is a sufficiently small perturbation of a holomorphic graph). We are able to answer Chirka's question in the affirmative for the class of graphs described in Theorem 1.3 below. Before stating that theorem, however, we state the following proposition, which is a special case of Theorem 1.3. We highlight this as a separate proposition because of the clarity of its statement. Received by the editors May 1, 2000. 2000 Mathematics Subject Classification. Primary 32D15.

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.

We consider real spaces only. Definition. An operator T : X → Y between Banach spaces X and Y is called a Hahn-Banach operator if for every isometric embedding of the space X into a Banach space Z there exists a norm-preserving extension T of T to Z. A geometric property of Hahn-Banach operators of finite rank acting between finite-dimensional normed spaces is found. This property is used to characterize pairs of finite-dimensional normed spaces (X, Y ) such that there exists a Hahn-Banach operator T : X → Y of rank k. The latter result is a generalization of a recent result due to B. L. Chalmers and B. Shekhtman. Everywhere in this paper we consider only real linear spaces. Our starting point is the classical Hahn-Banach theorem ([H], [B1]). The form of the Hahn-Banach theorem we are interested in can be stated in the following way. Hahn-Banach Theorem. Let X and Y be Banach spaces, T : X → Y a bounded linear operator of rank 1 and Z a Banach space containing X as a subspace. Then there exists a bounded linear operator T : Z → Y satisfying (a) ||T || = ||T ||; (b) T x = Tx for every x ∈ X. Definition 1. An operator T : Z → Y satisfying (a) and (b) for a bounded linear operator T : X → Y is called a norm-preserving extension of T to Z. The Hahn-Banach theorem is one of the basic principles of linear analysis. It is quite natural that there exists a vast literature on generalizations of the HahnBanach theorem for operators of higher rank. See papers by G. P. Akilov [A], J. M. Borwein [Bor], B. L. Chalmers and B. Shekhtman [CS], G. Elliott and I. Halperin [EH], D. B. Goodner [Go], A. D. Ioffe [I], S. Kakutani [Kak], J. L. Kelley [Kel], J. Lindenstrauss [L1], [L2], L. Nachbin [N1] and M. I. Ostrovskii [O], representing different directions of such generalizations, and references therein. There exist two interesting surveys devoted to the Hahn-Banach theorem and its generalizations; see G. Buskes [Bus] and L. Nachbin [N2]. We shall use the following natural definition. Definition 2. An operator T : X → Y between Banach spaces X and Y is called a Hahn-Banach operator if for every isometric embedding of the space X into a Banach space Z there exists a norm-preserving extension T of T to Z. Received by the editors February 9, 2000. 2000 Mathematics Subject Classification. Primary 46B20, 47A20.

We give a reduction of the irregular case for the effective non vanishing conjecture by virtue of the Fourier-Mukai transform. As a conse quence, we reprove that the effective non-vanishing conjecture holds on alge braic surfaces. In this paper we consider the following so-called effective non-vanishing conjec ture, which has been put forward by Ambro and Kawamata [Am99, KaOO]. Conjecture 1 (EN,). Let X be a proper normal variety of dimension n, B an effective R-divisor on X such that the pair (X, B) is Kawamata log terminal, and D a Cartier divisor on X. Assume that D is nef and that D (Kx + B) is nef and big. Then Ho (X, D) 74 0. This conjecture is closely related to the minimal model program and plays an important role in the classification theory of Fano varieties. For a detailed intro duction to this conjecture, we refer the reader to [XieO6]. By the Kawamata-Viehweg vanishing theorem, we have HL(X, D) = 0 for any positive integer i. Thus H?(X, D) =& 0 is equivalent to X(X, D) /& 0. Under the same assumptions as in Conjecture 1, the Kawamata-Shokurov non-vanishing theorem says that Ho (X, mD) =A 0 for all m > 0. Thus the effective non-vanishing conjecture is an improvement of the non-vanishing theorem in some sense. Note that EN1 is trivial by the Riemann-Roch theorem and that EN2 was settled by Kawamata [KaOO, Theorem 3.1] by virtue of the logarithmic semipositivity theo rem. For n > 3, only a few results are known. For instance, ENn holds trivially for toric varieties [Mu02], EN3 (X, 0) holds for all canonical projective minimal three folds X [KaOO, Proposition 4.1], and EN3 (X, 0) also holds for almost all canonical projective threefolds X with -Kx nef [XieO5, Corollary 4.5]. In this paper, we shall prove that, in the irregular case, the effective non vanishing conjecture can be reduced to lower-dimensional cases by means of the Fourier-Mukai transform. As consequences, EN2 is reproved after Kawamata, and ENn holds for all varieties of maximal Albanese dimension. Throughout this paper, we work over the complex number field C. For the definition of the Kawamata log terminal (KLT, for short) and the other notions, we refer the reader to [KMM87, KM98]. For irregular varieties, the study of the Albanese map provides enough infor mation to understand their birational structure. Therefore, through the Albanese Received by the editors January 17, 2007, and, in revised form, October 18, 2007, November 16, 2007, and December 27, 2007. 2000 Mathematics Subject Classification. Primary 14E30. (@)2008 American Mathematical Society Reverts to public domain 28 years from publication 61 This content downloaded from 157.55.39.35 on Thu, 01 Sep 2016 04:38:29 UTC All use subject to http://about.jstor.org/terms

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.

This paper extends the result from Amnon Neeman regarding strong generators in D p e r f ( X ) \mathbf {D}_{perf}(X) , from X X being a quasicompact, separated scheme to X X being quasicompact, quasiseparated scheme that admits a separator with some conditions. Neeman’s result states a necessary and sufficient condition for D p e r f ( X ) \mathbf {D}_{perf}(X) being regular. Together with being proper over a noetherian commutative ring, those conditions give an interesting description for when an R R -linear functor H H is representable.

In this paper, we follow an idea of Lusztig to define the Fourier transform on the Ringel–Hall algebra of a valued quiver (given by a quiver with automorphism). As an application, this provides a direct proof of the fact that the Ringel–Hall algebra of a valued quiver is independent of its orientation. Furthermore, by combining the BGP-reflection operators defined on double Ringel–Hall algebras of valued quivers with Fourier transforms, we obtain an alternative construction of Lusztig’s symmetries of the associated quantum enveloping algebras. This generalizes a result of Sevenhant and Van den Bergh in the quiver case.

The goal of this short paper is to show that the hyperbolic metric is uniquely determined by the induced metric on the strictly convex smooth boundary for dimension ≥ 4 \geq 4 .

This paper presents a combinatorial approximation of the monoidal topological complexity T C M \mathrm {TC}^M of a simplicial complex K K that controls reserved robot motions in K K . We introduce an upper bound S C r M \mathrm {SC}^M_r of T C M \mathrm {TC}^M using the r r -iterated barycentric subdivision of K × K K \times K modulo the diagonal and consider the refinement of the approximation. We show that T C M \mathrm {TC}^M can be described as S C r M \mathrm {SC}^M_r for sufficiently large r ≥ 0 r \geq 0 . As an example, we consider a simplicial model S n S_n of an n n -sphere and demonstrate that S C r M ( S n ) \mathrm {SC}^M_r(S_n) presents the best estimate of the monoidal topological complexity of an n n -sphere for r ≥ 1 r \geq 1 .

In this paper we establish a lower bound for the distance induced by the Kähler-Einstein metric on pseudoconvex domains with positive hyperconvexity index (e.g. positive Diederich-Fornæss index). A key step is proving an analog of the Hopf lemma for Riemannian manifolds with Ricci curvature bounded from below.

In this paper, we establish some conformable fractional Hermite-Hadamard type integral inequalities via harmonic s s -convexity, and the estimates of the products of two harmonic s s -convex functions are also considered.

In this paper, we extend the work of Cao-Chen [9] on Bach-flat gradient Ricci solitons to classify $n$-dimensional ($n\ge 5$) complete $D$-flat gradient steady Ricci solitons. More precisely, we prove that any $n$-dimensional complete noncompact gradient steady Ricci soliton with vanishing $D$-tensor is either Ricci-flat, or isometric to the Bryant soliton. Furthermore, the proof extends to the shrinking case and the expanding case as well., 10 pages; final version to appear in Proc. Amer. Math. Soc. arXiv admin note: text overlap with arXiv:1105.3163

This paper deals with the existence of periodic solutions and attractiveness for some partial functional differential equations in Banach spaces. We assume that the first linear part generates a strongly continuous semigroup, while the delayed part is periodic with respect to the first argument. We prove that the existence of a bounded solution implies the existence of a periodic solution. Several results regarding uniqueness and global attractiveness of periodic solutions are also established. The analysis relies on a fixed point theorem of Chow and Hale’s type and uses some arguments of weak topology. Our theorems extend in a broad sense some new and classical related results. An application to a transport equation with delay is also presented.

In this paper, we study gradient estimates for a parabolic p p -Laplace equation with logarithmic nonlinearity, which is related to the L p L^p -log-Sobolev constant on Riemannian manifolds. We prove a global Li-Yau type gradient estimate and a Hamilton type gradient estimate for positive solutions to a parabolic p p -Laplace equation with logarithmic nonlinearity on compact Riemannian manifolds with nonnegative Ricci curvature. As applications, the corresponding Harnack inequalities are derived.

Let A be the disk algebra. In this paper we address the following question: Under what conditions on the points Zk,, E D do there exist operators Ln : A -W A such that Mn Lnf = f f (Zk,n)lk,n, fL 1k,n E A, k=1 and Lnf -+ f, n -* no, for every f E A? Here the convergence is understood in the sense of sup norm in A. Our first result shows that if Zk,n satisfy Carleson condition, then there exists a function f E A such that Lnf 74 f, n -4 oc. This is a non-trivial generalization of results of Somorjai (1980) and Partington (1997). It also provides a partial converse to a result of Totik (1984). The second result of this paper shows that if Ln are required to be projections, then for any choice of Zk,, the operators Ln do not converge to the identity operator. This theorem generalizes the famous theorem of Faber and implies that the disk algebra does not have an interpolating basis.

The Steenrod problem asks: given a G-module, when does there exist a Moore space realizing the module? By using the equivariant Postnikov Tower, it is shown that a ZG-module is ZG-realizable if and only if it is ZHrealizable for all p-Sylow subgroups H, for all primes pl GI. Let G be a finite group. Let M be a finitely generated ZG-module. We say that M is a Steenrod representation if there exists a Moore space X with G-action such that the homology of X is isomorphic to M as a ZG-module. Recall that a Moore space is a topological space whose reduced homology vanishes in all dimensions except one. Without loss of generality, M will be assumed 2-free [1] from now on. In this paper, the following statement will be proved: M is a Steenrod representation as a ZG-module if and only if M is a Steenrod representation as a 7H-module for all p-Sylow subgroups H of G, for all primes pII GIC. This statement is inspired by papers by J. Arnold [1, 2] and P. Vogel [18]. Arnold [2] showed that if G is cyclic, then every finitely generated ZG-module is a Steenrod representation. And Vogel showed that if for any ZG-module M, we can always find a G-Moore space X realizes M, then G has only cyclic Sylow subgroups. Let us first consider a Moore space X whose homology is isomorphic to M as a 2-module. Let G(X) denote the space of self homotopy equivalences of X and let BG(X) [9] denote the classifying space of the H-space G(X) for a certain fibration defined by Dold and Lashof [10]; see also Stasheff [15]. Since a homotopy equivalence of X induces an automorphism on M, there is a map G(X) -* Aut(M). And the map G(X) -* Aut(M) induces another map a : BG(X) -' BAut(M). On the other hand, since M is a ZG-module, there is a map G -* Aut(M) (= GL(n, 2) if M = Zn), which induces .p : BG -) BAut(M). Following Cooke's theory [9], as pointed out in [12], M is a Steenrod representation as a ZG-module if and only if the lifting f exists in

We estimate double exponential sums of the form Sa(X, Y) = E E exp(27riaI0xY/p), xGE X yG Y where V is of multiplicative order t modulo the prime p and X and Y are arbitrary subsets of the residue ring modulo t. In the special case t = p 1, our bound is nontrivial for I XI > I Yl > p15/16+6 with any fixed 6 > 0, while if in addition we have Xl > p1-6/4 it is nontrivial for I YI ? p3/4+6. Let p be a prime and let Fp be a finite field of p elements. For an integer m > 1 we denote by Z,m = {0, . .. , m 1} the residue ring modulo m. We also identify Fp with the set {o,... ,p-1}. Finally we define e(z) = exp(27ri/p) and use log z for the natural logarithm of z. Throughout the paper the implied constants in symbols 'O', ' ' may occasionally, where obvious, depend on the small positive parameter E and are absolute otherwise (we recall that A A are equivalent to A 0 (B)). We fix an element ) E IFp of multiplicative order t, that is, 79s 7 1, 1 pl5/16+6 with any fixed 6 > 0. Further examples are given below. Our results rely on the following estimate for certain double exponential sums from [1]; see the proof of Theorem 8 of that paper. Let A E Fp be of multiplicative order T. For any a, b (E IF we have the bound 4 (1) E| e(a\v +bAuv) 1 satisfies (2) T(m) < m(1). 2 Our main estimate is the following. Theorem. For any sets X, Y C Zt, the bound max ISa( X, Y)| << I x 1/21 yI5/6tl/2pl/8+E aEF* p holds. Proof. For a divisor dlt we denote by Y(d) the subset of y Y Y with gcd(y, t) d. Then ISa(X, Y)I ? Z ud| dlt

We prove that the homeomorphism group of each one of a collection of continua constructed in a paper by the first author (Trans. Amer. Math. Soc. 121 (1966), 516–548) is one dimensional. This answers a question posed in that paper.

This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form ϕ ( x ) = g ( x ) + ∑ α ∈ Z s a ( α ) ϕ ( M x − α ) , x ∈ R s , \begin{equation*} \phi (x) = g(x) + \sum _{\alpha \in \mathbb {Z}^s} a(\alpha ) \phi (Mx-\alpha ), \qquad x \in \mathbb {R}^s, \end{equation*} where ϕ = ( ϕ 1 , … , ϕ r ) T \phi = (\phi _1,\ldots ,\phi _r)^T is the unknown, g = ( g 1 , … , g r ) T g = (g_1,\ldots ,g_r)^T is a given vector of functions on R s \mathbb {R}^s , M M is an s × s s \times s dilation matrix, and a a is a finitely supported refinement mask such that each a ( α ) a(\alpha ) is an r × r r \times r (complex) matrix. Let ϕ 0 \phi _0 be an initial vector in ( L 2 ( R s ) ) r (L_2(\mathbb {R}^s))^r . The corresponding cascade algorithm is given by ϕ k := g + ∑ α ∈ Z s a ( α ) ϕ k − 1 ( M ⋅ − α ) , k = 1 , 2 , … . \begin{equation*} \phi _k := g + \sum _{\alpha \in \mathbb {Z}^s} a(\alpha ) \phi _{k-1}({M \cdot } - \alpha ), \qquad k=1,2,\ldots . \end{equation*} In this paper we give a complete characterization for the L 2 L_2 -convergence of the cascade algorithm in terms of the refinement mask a a , the nonhomogeneous term g g , and the initial vector of functions ϕ 0 \phi _0 .

We prove the following theorems: (1) Suppose that f; 2W 2W is a continuous function and X is a Sierpiiiski set. Then (A) for any strongly measure zero set Y, the image f[X + Y] is an so-set, (B) f [X] is a perfectly meager set in the transitive sense. (2) Every strongly meager set is completely Ramsey null. This paper is a continuation of earlier works by the authors and by M. Scheepers (see [N], [NSW], [S]) in which properties (mainly, the algebraic sum) of certain singular subsets of the real line R and of the Cantor set 2' were investigated. Throughout the paper, by a set of real numbers we mean a subset of 2' and by "+" we denote the standard modulo 2 coordinatewise addition in 2W. Let us also assume that a "measure zero" (or "negligible") set always denotes a Lebesgue measure zero set. We apply the following definition of sets of real numbers. Definition 1. An uncountable set X is said to be a Luzin (respectively, Sierpin'ski) set iff for each meager (respectively, measure zero) set Y, XnY is at most countable. We say that a set X is of strong measure zero (respectively, strongly meager) iff for each meager (respectively, measure zero) set Y, X + Y 7& 2'. Remark 1. It is well known (see [M] for example) that every Luzin set is strongly measure zero. Quite recently J. Pawlikowski proved that each Sierpin'ski set must be strongly meager as well (see [P]). Let us recall that a set X is called an so-set (or Marczewski set) iff for each perfect set P one can find a perfect set QCP that is disjoint from X. M. Scheepers showed in [S] that for a Sierpin'ski set X and a strong measure zero set Y, X + Y is an so-set. Later, in [NSW] it was proven that this also holds when X is strongly meager. We have the following functional version of the M. Scheepers' result. Theorem 1. Let X be a Sierpin'ski set and let Y be a strong measure zero set. Assume also that f: 2' -* 2W is a continuous function. Then the image f[X + Y] is an so-set. Received by the editors July 16, 1998 and, in revised form, September 9, 1998 and March 10, 1999. 2000 Mathematics Subject Classification. Primary 03E15, 03E20, 28E15.

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.

We formulate and prove a topological Paley-Wiener theorem for the normalized spherical Laplace transform defined on the rank 1 causal symmetric spaces M = SO0(1,n)/SO(1,n -1), for n > 2. INTRODUCTION The spherical Laplace transform on causal symmetric spaces was introduced in [FHO, ?8] as a generalization of the spherical Fourier transform on Riemannian symmetric spaces defined by Helgason (see [Hi, Chapter 4]). Both transforms can be expressed in terms of (integrating against) spherical functions. It was furthermore shown in [01, ?5] that the spherical functions on the Riemannian dual of a causal symmetric space can be written as an expansion in spherical functions on the causal symmetric space. The inversion formula for the spherical Laplace transform easily follows (see [01, ?6]). One of the most important results on the spherical Fourier transform is the (topological) Paley-Wiener theorem (see [Hi, Chapter 4, ?7] and [H2, Chapter 3, ?5] for details) generalizing the classical Paley-Wiener theorem on Euclidean spaces. In this paper we generalize these results to the normalized spherical Laplace transform on causal symmetric spaces M of rank 1, thereby partially solving an open problem posed by the second author in [02, ?5]. The paper is divided into two sections: in the first section we recall some results on the spherical Fourier transform on the Riemannian dual Md of M, and in the second we consider the spherical Laplace transform defined on M. We define the Paley-Wiener space, the supposed image space of spherical Laplace transform, according to the growth and symmetry conditions satisfied by the spherical functions on M. The Paley-Wiener theorem for the normalized spherical Laplace transform follows by using Cauchy's theorem to change the path of integration in the inversion formula and from the Paley-Wiener theorem for the spherical Fourier transform on Md. Received by the editors December 9, 1998 and, in revised form, March 22, 1999. 2000 Mathematics Subject Classification. Primary 43A85, 22E30; Secondary 43A90, 33C60. The first author was supported by a postdoc fellowship from the European Commission within the European TMR Network "Harmonic Analysis" 1998-2001 (Contract ERBFMRX-CT97-0159). The second author was supported by LEQSF grant (1996-99)-RD-A-12. (?2000 American Mathematical Society

This paper proves the existence of solutions to the initial value problem \[ ( I V P ) { x ′ ( t ) = f ( t , x ( t ) ) ( 0 ≤ t ≤ 1 ) , x ( 0 ) = 0 , (\mathrm {IVP})\qquad \qquad \left \{\begin {array}{l} x’(t)=f(t,x(t))\qquad \quad (0\le t\le 1), x(0)=0,\end {array} \right . \] where f : [ 0 , 1 ] × R M → R M f:[0,1]\times \mathbb {R}^M\to \mathbb {R}^M may be discontinuous but is assumed to satisfy conditions of superposition-measurability, quasimonotonicity, quasisemicontinuity, and integrability. The set M M can be arbitrarily large (finite or infinite); our theorem is new even for card ( M ) = 2 \mbox {card}(M)=2 . The proof is based partly on measure-theoretic techniques used in one dimension under slightly stronger hypotheses by Rzymowski and Walachowski. Further generalizations are mentioned at the end of the paper.