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.     

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.     

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.     

Estimates for the Norm of the nth Indefinite Integral

Let T be the Volterra operator on L²[0, 1] where f L²[0, 1], 0 x 1. It is well known that||n!Tⁿ|| = O(1/n!). In a recent paper [1], D. Kershaw has provedthat a result which wasfirst conjectured by Lao and Whitley in [2]. It is easy to provethat For completeness, wegive the proof using the familiar Schmidt norm estimate forthe norm of an integral operator (see Section 2 below). Kershawproves that by analysingthe special positivity preserving properties of T*T. He usesone of the many abstract theorems on eigenvalues and eigenfunctionsof compact operators which preserve a cone. In this paper weshall reprove (1), giving a short and direct proof of (2). 1991Mathematics Subject Classification 47G10, 45-04.     

On Approximately Midconvex Functions

A real-valued function f defined on an open, convex set D ofa real normed space is called (, )-midconvex if it satisfies The main result of the paper states that if f is locally boundedfrom above at a point of D and is (, )-midconvex, then it satisfiesthe convexity-type inequality where : [0, 1] R is a continuous function satisfying The particular case = 0 of this result is due to Ng and Nikodem(Proc. Amer. Math. Soc. 118 (1993) 103–108), while thespecialization = = 0 yields the theorem of Bernstein and Doetsch(Math. Ann. 76 (1915) 514–526). 2000 Mathematics SubjectClassification 26A51, 26B25.     

Determinants of Lattices induced by Rational Subspaces

Let L and be orthogonal complementary rational linear subspaces of Eⁿ, and let = L Zⁿ and $$\stackrel{\¯}{\Lambda}$$ = 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 .     

Some Sharp Bounds for the Cone Multiplier of Negative Order in R3

This paper considers the cone multiplier operator which is definedby where and . For –3/2 < µ < –3/14, sharp L^p – L^q estimatesand endpoint estimates for S^µ are obtained. 2000 MathematicsSubject Classification 42B15 (primary).     

Remarks on Maximal Operators Over Arbitrary Sets of Directions

Throughout this paper, we shall let be a subset of [0, 1] havingcardinality N. We shall consider to be a set of slopes, andfor any s , we shall let e_s be the unit vector of slope s inR². Then, following [7], we define the maximal operator on R²associated with the set by The history of the bounds obtained on is quite curious. The earliest study of relatedoperators was carried out by Cordoba [2]. He obtained a boundof C(1 + log N) on the L² operator norm of the Kakeya maximaloperator over rectangles of length 1 and eccentricity N. Thisoperator is analogous to with However, for arbitrary sets, the best known result seems to be C(1 + log N). This followsfrom Lemma 5.1 in [1], but a point of view which produces aproof appears already in [8]. However, in this paper, we provethe following.     

On Ramanujan's Double Inequality for the Gamma Function

Ramanujan presented (without proof) the following double inequalityfor the gamma function: .Recently, Karatsuba established that these inequalities holdfor x 1. We show that this can be slightly improved: the inequalitieshold for all x 0, even if we replace 1/100 by where f(x) = (1/)³[(x + 1)(e/x)^x]⁶–8x³–4x²–x.Moreover, and 1/30 are the best possible constant terms. 2000Mathematics Subject Classification 33B15 (primary); 26D15, 26D20(secondary).     

Relation Modules of Infinite Groups

Let F_n be the free group of rank n with basis x₁, x₂, ..., x_n,and let d(G) denote the minimal number of generators of thefinitely generated group G. Suppose that n d(G). There existsan exact sequence and wemay view the free abelian group as a right ZG-module by defining (rR')^g = r^g–1R' for allg G, where g^–1 is any preimage of g under , and = (g^–1)^–1 r(g^–1),the conjugate of r by g^–1. We call the relation module of G associated with the presentation(1), and say that has ambient rank n. Furthermore, we call the group F_n/R' the free abelianizedextension of G associated with (1). 1991 Mathematics SubjectClassification 20F05, 20C07.     

On the 'pits effect' of Littlewood and Offord

Asymptotic behaviour of the entire functions , with real _n is studied. It turns out that the Phragmén–Lindelöfindicator of such a function is always non-negative, unlessf(z)=e^az. For a special choice of _n= n² with irrational , theindicator is constant and f has completely regular growth inthe sense of Levin and Pfluger. Similar functions of arbitraryorder are also considered.     

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.     

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.     

Choquet Integrals, Hausdorff Content and the Hardy-Littlewood Maximal Operator

Using the BMO-H¹ duality (among other things), D. R. Adams provedin [1] the strong type inequality whereC is some positive constant independent of f. Here M is theHardy–Littlewood maximal operator in Rⁿ, H is the -dimensionalHausdorff content, and the integrals are taken in the Choquetsense. The Choquet integral of 0 with respect to a set functionC is defined by Precise definitionsof M and H will be given below. For an application of (1) tothe Sobolev space W^{1, 1} (Rⁿ), see [1, p. 114]. The purpose of this note is to provide a self-contained, directproof of a result more general than (1). 1991 Mathematics SubjectClassification 28A12, 28A25, 42B25.     

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).     

On Weighted Distortion in Conformal Mapping II

Some years ago, Blatter [1] gave a result of the form for any function f regular and univalentin D: |z| < 1, where is the hyperbolic distance betweenz₁ and z₂. Kim and Minda [5] pointed out that the multiplieron the right is incorrect. They say that Blatter's proof givesthe correct multiplier, but Blatter's formula for in termsof z₁, z₂ is wrong. Kim and Minda proved the generalized formula where D₁(f) = f'(z) (1 – |z|²),valid for p P with some P, . In each case there was an appropriate equality statement. Kimand Minda made the important and easily verified remark thatthese problems are linearly invariant in the sense that if theresult is proved for f, then it follows for , where U is a linear transformation of the planeonto itself and T is a linear transformation of D onto itself.This means that we need to prove such results only in an appropriatelynormalized context. 1991 Mathematics Subject Classification30C75, 30F30.     

The Set of {omega}-Limit Points of A Map of the Circle is Closed

Let f be a continuous self-map of the unit circle, S¹. The -limitpoints (x) of a point x are the set of all limit points of thesequence of iterates of f acting on x. We shall show that theset of all -limit points _xS¹(x) a closed set in S¹.     

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 + .     

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.     

Generalized Logarithmic Derivative Estimates of Gol'dberg-Grinshtein Type

For f meromorphic in the complex plane and meromorphic in theunit disc, sharp upper bounds are obtained for and where k andj are integers satisfying k > j 0. The results generalizethe logarithmic derivative estimate due to Gol'dberg and Grinshteinto derivatives of higher order. 2000 Mathematics Subject Classification30D35.