The Bicanonical Map of Surfaces with pg = 0 and K2 >= 7

A minimal surface of general type with p_g(S) = 0 satisfies 1 K² 9, and it is known that the image of the bicanonical map is a surface for , whilst for , the bicanonical map is always a morphism. In this paper it is shown that is birationalif , and that the degree of is at most 2 if or By presenting two examples of surfaces S with and 8 and bicanonical map of degree 2, it is alsoshown that this result is sharp. The example with is, to our knowledge, a new example of a surfaceof general type with p_g = 0. The degree of is also calculated for two other known surfacesof general type with p_g = 0 and . In both cases, the bicanonical map turns out to be birational.     

The Complex Plank Problem

It is shown that if is a sequence of norm 1 vectors in a complex Hilbert space and is a sequence of non-negative numbers satisfying then there is a unit vector z for which for every j. The result is a strong,complex analogue of the author's real plank theorem.     

Some Relations Between Packing Premeasure and Packing Measure

Let K be a compact subset of Rⁿ, 0 s n. Let , P^s denote s-dimensional packing premeasure andmeasure, respectively. We discuss in this paper the relationbetween and P^s. We prove:if , then ; and if , then for any > 0, there exists a compact subset F of K such that and P^s(F) P^s(K) – .1991 Mathematics Subject Classification 28A80, 28A78.     

On the Discrete Hardy Inequality

Necessary and sufficient conditions for the boundedness of thediscrete Hardy operator of the form , from to when 0 < q < 1 <p , is given.     

A Radial Uniqueness Theorem for Sobolev Functions

We show that continuous functions u in the Sobolev space , 1 < p n, which have the limitzero in a certain weak sense in a set of positive p-capacityon B with where B is the open unit ball of Rⁿ and for 0 > > , are identically zero. Conversely, we produce for each 1 > p n and each positive a non-constant function u in , continuous in , and a compact set EB of positive p-capacity such that u = 0 in E and the aboveinequality holds with exponent p – l + .     

Oscillation Criteria for First-Order Delay Equations

This paper is concerned with the oscillatory behaviour of first-orderdelay differential equations of the form (1) where is non-decreasing, (t)< t for t t₀ and . Let the numbers k andL be defined by It is proved here that when L < 1 and 0 < k 1/e all solutionsof equation (1) oscillate in several cases in which the condition holds, where ₁ is the smaller root of the equation = e^k. 2000Mathematics Subject Classification 34K11 (primary); 34C10 (secondary).     

Calculs De Facteurs Epsilon De Paires Pour GLn Sur Un Corps Local, I

Soient F un corps commutatif localement compact non archimédienet un caractère additif non trivial de F. Soient unereprésentation du groupe de Weil–Deligne de F,et sa contragrédiente. Nous calculons le facteur (, , ). De manière analogue, nous calculons le facteur (x, , ) pour toute représentationadmissible irréductible de GL_n(F). En conséquence,si F est de caractéristique nulle et si et se correspondentpar la correspondance de Langlands construite par M. Harris,ou celle construite par les auteurs, alors les facteurs (, , s) et (x, , s) sont égaux pour tout nombre complexe s. Let F be a non-Archimedean local field and a non-trivial additivecharacter of F. Let be a representation of the Weil–Delignegroup of F and its contragredient representation. We compute (, , ). Analogously, we compute (x, , ) for all irreducible admissible representations of GL_n(F).Consequently, if F has characteristic zero, and , correspondvia the Langlands correspondence established by M. Harris orthe correspondence constructed by the authors, then we have(, , s) = (x, , s) for all sC. 1991 Mathematics Subject Classification22E50.     

A Local Limit Theorem for Moderate Deviations

The purpose of this note is to establish a uniform estimatefor the mass function P(S_m = y) of an integer-valued randomwalk when y and where µ isthe mean of the step distribution. (The local central limittheorem provides such an estimate when ( is bounded.) The assumptions are that the mass functionp of the step distribution is regularly varying at with index–, where > 3, and that for some ' > 2. From this result, a ratio limit theorem isderived, and this in turn is applied to yield some new informationabout the space–time Martin boundary of certain randomwalks.     

Determinants of Lattices induced by Rational Subspaces

Let L and be orthogonal complementary rational linear subspaces of Eⁿ, and let = L Zⁿ and $$\stackrel{\&macr;}{\Lambda}$$ = Zⁿ be the sublatticesof the usual integer lattice Zⁿ induced by L and . Then the determinants of and are equal. The samerelationship holds between the determinants of the lattices and obtained by orthogonal projection of Zⁿ on to L and .     

Paley's Inequality for the Jacobi Expansions

Let be an analytic function in the unit disc satisfying Then, which is familiar as Paley's inequality. In this paper, an analogue of this inequality withrespect to the Jacobi expansions is established.     

Property (T) for C*-Algebras

A notion of Property (T) is defined for an arbitrary unitalC*-algebra A admitting a tracial state. This is extended toa notion of Property (T) for a pair (A, B) where B is a C*-subalgebraof A. Let be a discrete group and its reduced algebra. We show that has Property (T) if and only if the group has Property (T).More generally, given a subgroup of , the pair has Property (T) if and only if the pair of groups(, ) has Property (T). 2000 Mathematics Subject Classification46L05, 22D25.     

Intersections of the Left and Right Essential Spectra of 2 x 2 Upper Triangular Operator Matrices

In this paper, perturbations of the left and right essentialspectra of 2 x 2 upper triangular operator matrix M_C are studied,where is an operator acting on the Hilbert space H K. For given operators A and B, thesets and are determined, where _le(T) and _re(T) denote, respectively,the left essential spectrum and the right essential spectrumof an operator T. 2000 Mathematics Subject Classification 47A10,47A55.     

Small Exotic 4-Manifolds with Formula

We construct an infinite family of simply connected, pairwisenondiffeomorphic 4-manifolds, all homeomorphic to . Similar ideas provide examples of 4-manifolds with, , vanishing first homology and nontrivial Seiberg–Witteninvariants. 2000 Mathematics Subject Classification 57R55, 57R57.     

Grothendieck Spaces and Duals of Injective Tensor Products

Let E and F be Fréchet spaces. We prove that if E isreflexive, then the strong bidual is a topological subspace of . We also prove that if, moreover, E is Montel and F has the Grothendieckproperty, then has the Grothendieck property whenever either E or has the approximation property. A similar result is obtainedfor the property of containing no complemented copy of c₀.     

A Combinatorial Application of the Maximal Ergodic Theorem

Let X be a compact space,µ a Borel probability measureon X, T: X X a measure preserving continuous transformationand g: X R a continuous function. Then for some yX, This Lemma is used to give an alternative proof of a resultby Ruzsa [6], which implies the following extension of a resultof Bergelson [1]. If E N satisfies then there exists a set N such that n–1|[1,n]| (E) for all, n 1, and any finite subset{₁, ... _k} satisfies Ø. 7 Moria St., Ramat Hasharon, Israel     

A Schwarz Lemma for the Symmetrized Bidisc

Let be an analytic function from D to the symmetrized bidisc We show that if (0) = (0,0) and () = (s, p) in the interiorof , then Moreover, the inequality is sharp: we give an explicit formulafor a suitable in the event that the inequality holds withequality. We show further that the inverse hyperbolic tangentof the left-hand side of the inequality is equal to both theCaratheodory distance and the Kobayashi distance from (0,0)to (s, p) in int      

On Storer Difference Sets

Let p, q be distinct odd primes, and let a, b be positive integers.In this paper we prove that if S(p^a, q^b) is a Storer differenceset with the parameters = p^aq^b, k = (–1)/4 and =(–5)/16,then we have a = b = 1, and , where , and r is a positiveinteger. 1991 Mathematics Subject Classification 05B10.     

Bernstein-Type Inequalities for Linear Combinations of Shifted Gaussians

Let P_n be the collection of all polynomials of degree at mostn with real coefficients. A subtle Bernstein-type extremal problemis solved by establishing the inequality for all , and m= 1, 2, ..., where c is an absolute constant and Some related inequalities and direct and inversetheorems about the approximation by elements of in L_q (R) are also discussed. 2000 Mathematics SubjectClassification 41A17 (primary).     

Differential Operators on some Singular Surfaces

Let X be an irreducible affine algebraic variety of dimension2 defined over an algebraically closed field of characteristiczero. Suppose that the normalisation of X is non-singular, and the natural projection : X is injective. Further, suppose that X is Cohen–Macaulay.Then the rings of differential operators D(X) and D() are Morita equivalent. Current address of S. P. Smith: Department of Mathematics, Universityof Washington, Seattle, WA 98195. USA     

Multipliers and Isometries in Formula

Let E be a complex Banach space and the space of bounded analytic functions on the unit disc toE. By means of a study of the multipliers on it is shown that, if Mult(E) = C, then every isometryT of is of the form (TF)(z)= TF((z)), where is a conformal map of the disc and T is aconstant isometry of E.