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).     

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.     

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.     

Traces of Singular Values and Borcherds Products

Let p be a prime for which the congruence group ₀(p)* is ofgenus zero, and let be the corresponding Hauptmodul. Let f be a nearly holomorphic modularform of weight 1/2 on ₀(4p) which satisfies some congruencecondition on its Fourier coefficients. We interpret f as a vectorvalued modular form. Applying Borcherds lifting of vector valuedmodular forms we construct infinite products associated to and relate them to Zagier's traceformula for the singular values of . 2000 Mathematics Subject Classification 11F03, 11F30 (primary);11F22, 11F37, 11F50 (secondary).     

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.     

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     

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).     

Localization of Closed (or Periodic) Solutions of a Differential System with Concave Nonlinearities

Consider a scalar differential equation , where I is an open interval containing [0,T]. Assumethat f(t, x) is continuous with a continuous derivative , and weakly concave (or weakly convex)in x for all t I, though strictly concave (or strictly convex)for some t [0, T]. It is well known that in this case therecan be either no, one or two closed solutions; that is, solutions(t) for which (0) = (T) If there are two closed solutions, thenthe greater has a negative characteristic exponent and the smallerhas a positive one. It is easily seen that this is equivalentto a statement on localization of closed solutions. It is shownhow this statement can be generalized to systems of differentialequations . The requirements are that the coordinate functions ) be continuous with continuous derivatives with respect to x₁,x₂, ...,x_n, that the f_j are weakly concave (or weakly convex)in , and that a certain condition pertaining to strict concavity (or strict convexity) is fulfilled.2000 Mathematics Subject Classification 34C25, 34C12.     

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.     

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.     

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.     

Multiple Positive Solutions of Semi-Positone Sturm-Liouville Boundary Value Problems

This paper treats the existence of multiple positive solutionsof the semi-positone Sturm–Liouville boundary value problem. almost everywhere on [R₀, R₁], where and f is allowed to take negative values (that is, f is semi-positone).When = 1, new results on the existence of one or two nonzeropositive solutions are obtained. These results generalize previousresults for positone cases (that is, f 0) to the semi-positonecases. We illustrate our results with an explicit example whichhas two nonzero positive solutions. These results are used todeduce results on intervals of eigenvalues for which there existone or two nonzero positive eigenfunctions. Applications ofthese eigenvalue results are provided. 2000 Mathematics SubjectClassification 34B18 (primary), 34B15, 34B16, 47H10, 47H30 (secondary).     

Commutative Subalgebras and Sphericity in Z2-Graded Lie Algebras

Let g = g₀g₁ be a Z₂-graded simple Lie algebra. Fix a Borelsubalgebra b₀g₀. Let ag₁ be a b₀-stable subalgebra. Then a isautomatically commutative. It is known that if is a spherical G-variety, then G₀.a is a sphericalG₀-variety. We describe all Z₂-gradings having the propertythat is a spherical G-varietyfor any a. 2000 Mathematics Subject Classification 14L30, 17B20,20E46.     

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 .     

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.     

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 + .     

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).     

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.     

Spectral Multipliers for Hardy Spaces Associated with Schrodinger Operators with Polynomial Potentials

Let T_t be the semigroup of linear operators generated by a Schrödingeroperator – A = – V, where V is a non-negative polynomial,and let be the spectral resolution of A. We say that f is an element of if the maximal function Mf(x) = sup_t>0|T_tf(x)| belongs toL_p. We prove a criterion of Mihlin type on functions F whichimplies boundedness of the operators on , 0 < p 1. 1991 MathematicsSubject Classification 42B30, 35J10.