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.     

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).     

Binomial Sums and Functions of Exponential Type

Let (a_n)_n0 be a sequence of complex numbers, and, for n0, let A number of results are proved relating the growth of the sequences(b_n) and (c_n) to that of (a_n). For example, given p0, if b_n= O(n^p and for all > 0,then a_n=0 for all n > p. Also, given 0 < p < 1, then for all > 0 if and onlyif . It is further shown that, given rß > 1, if b_n,c_n=O(rßⁿ), then a_n=O(ⁿ),where , thereby proving a conjecture of Chalendar, Kellay and Ransford. The principal ingredientsof the proogs are a Phragmén-Lindelöf theorem forentire functions of exponential type zero, and an estimate forthe expected value of e^(X), where X is a Poisson random variable.2000 Mathematics Subject Classification 05A10 (primary), 30D15,46H05, 60E15 (secondary).     

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     

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.     

On the Best Constant in Weighted Inequalities for Riemann-Liouville Integrals

For 1 k < and 1 p q , the problem of finding the bestconstant C_pq in the weighted inequality involving the Riemann-Liouville integrals of theform is considered.     

Generalized Catalan Numbers, Weyl Groups and Arrangements of Hyperplanes

For an irreducible, crystallographic root system in a Euclideanspace V and a positive integer m, the arrangement of hyperplanesin V given by the affine equations (, x) = k, for and k =0, 1, ..., m, is denoted here by . The characteristic polynomial of is related in the paper to that of the Coxeter arrangement A(corresponding to m = 0), and the number of regions into whichthe fundamental chamber of A is dissected by the hyperplanesof is deduced to be equal to the product , where e₁,e₂, ..., e_l are the exponents of and h is the Coxeter number.A similar formula for the number of bounded regions follows.Applications to the enumeration of antichains in the root posetof are included. 2000 Mathematics Subject Classification 20F55(primary), 05A15, 52C35 (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.     

Uniqueness of a Nonharmonic Trigonometric Series under an Exponential Growth Condition

We prove uniqueness for the nonharmonic trigonometric series under the weaker condition (*) where , for some 0 < < 1. In other words, if satisfies the above condition (*), and if , then a_k = 0 for all k = 0, 1,.... Finally, we statean improvement of Zygmund's uniqueness result as a corollary.     

On Prime Divisors of Binomial Coefficients

In 1985, Sárközy proved a conjecture of Erdösby showing that ) is never square-free for sufficiently large n. By applying a new estimateon exponential sums, we prove that this also holds for if dis not ‘too big’. Let 0 < < 1, p₀ N. For m m₀ and 1 k m satisfying |m– 2k| < m^{1 –} , there is a prime p > p₀ suchthat .     

Estimates of Eigenfunctions for One Class of Boundary Conditions

Requirements for boundary conditions are found which imply sharpby order estimates in L-metrics of eigenfunctions of the ordinarydifferential operator It is shown that the inequalities for the eigenfunctions {y} yield certain restrictions on thespectrum. 1991 Mathematics Subject Classification 34L05.     

Outer Automorphisms on Cuntz Algebras

We prove that if an automorphism on the Cuntz algebra O_n satisfies(S₁) = S₁ for one isometry S₁ of generators S₁, ..., S_n suchthat , , l i n and a complex number with modulus one,then it is outer. In particular, any non trivial automorphismon O_n leaving one generator fixed is outer.     

A Relation between Hausdorff Dimension and a Condition on Time Sets for the Image by the Brownian Sheet to Possess Interior Points

We prove that if W_{N, d} is a Brownian sheet mapping to R^d and E is a set in (0, )^N of Hausdorff dimensiongreater than , then for almost every rotation about a point x and translation _x such that _x(E) (0, )^N, the set _x(E) is such that almost surely W(E) containsinterior points. The techniques are adapted from Kahane andRosen and generalize to higher dimensional time and range.     

Approximation of Ground State Eigenvalues and Eigenfunctions of Dirichlet Laplacians

Let be a bounded connected open set in R^N, N 2, and let –0be the Dirichlet Laplacian defined in L²(). Let > 0 be thesmallest eigenvalue of –, and let > 0 be its correspondingeigenfunction, normalized by ||||₂ = 1. For sufficiently small>0 we let R() be a connected open subset of satisfying Let – 0 be the Dirichlet Laplacian on R(), and let >0and >0 be its ground state eigenvalue and ground state eigenfunction,respectively, normalized by ||||₂=1. For functions f definedon , we let Sf denote the restriction of f to R(). For functionsg defined on R(), we let Tg be the extension of g to satisfying 1991 Mathematics SubjectClassification 47F05.     

Reducing Subspaces for a Class of Multiplication Operators

Let D be the open unit disk in the complex plane C. The Bergmanspace is the Hilbert space of analytic functions f in D such that where dA is the normalized area measure on D. If are two functions in , then the inner product of f and g is given by We study multiplication operators on induced by analytic functions. Thus for H (D), the space ofbounded analytic functions in D, we define by It is easy to check that M is a bounded linear operator on with ||M||=||||=sup{|(z)|:zD}.     

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).     

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 + .     

The {ell}1-norm of the Fourier transform on compact vector spaces

Suppose that G is a compact Abelian group. If A G, then howsmall can ||_A||_A(G) be? In general, there is no non-triviallower bound. In a recent preprint, Green and Konyagin show thatif has sparse small subgroup structure and A has density with (1 – ) >> 1, then||_A||_A(G) does admit a non-trivial lower bound. In this paperwe address the complementary case of groups with duals havingrich small subgroup structure, specifically the case when Gis a compact vector space over ₂. The results themselves arerather technical to state, but the following consequence capturestheir essence: if is a set of density as close to 1/3 as possible, then we show that . We include a number of examples andconjectures which suggest that what we have shown is very farfrom a complete picture.     

On Banach Spaces Verifying Grothendieck's Theorem

The projective tensor product ₂ with any Banach space X sits inside the space Rad(X) of allalmost unconditionally summable sequences in X. If X is of cotype2 and u : XY is 2-summing, then u takes Rad(X) into . Consequently, if X is of cotype 2,then every operator from X to ₂ is 1-summing if and only if. In this case, each 2-summing operator from ₂ to X is nuclear, and X does not have non-trivialtype provided that dim X = . 2000 Mathematics Subject Classification46B28, 46M05.     

Corrigendum: Elliptic Curves and Quadratic Recurrence Sequences

The statement of the numerical values and z₀ on page 167 of[1, Section 3] contained an error. The values that were actuallyused were (to nine decimal places): thesebeing shifted, by the periods 2₁ and 2₃ respectively, comparedwith the values given in [1] (with ₁ = 1.496729323 and ₃ = 1.225694691i).With ₀ = ₁ and (z) denoting the sigma function (z; g2, g3) withinvariants g2 = 4, g3 = –1 associated with the ellipticcurve given by equation (3.2), these values of and z₀ yield and the latter three values all agreewith those stated in the paper (apart from rounding down thelast digit in the imaginary part of A). 2000 Mathematics SubjectClassification 11B37 (primary), 33E05, 37J35 (secondary).