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.     

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.     

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.     

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 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).     

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.     

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 + .     

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.     

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 .     

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 .     

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).     

Existence of Periodic Solutions in Nonlinear Asymmetric Oscillations

The existence of periodic solutions for the nonlinear asymmetricoscillator x' + x⁺ – rßx^– = h(t),(' =d/dt (is discussed, where , rß are positive constantssatisfying for some positive integer n N and h(t) L (0,2) is 2-periodic with x^±= max {±x,0}. 2000 Mathematics Subject Classification34C10, 34C25.     

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.     

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.     

The Dirichlet problem by variational methods

Let ^N be a bounded open set and C( ). Assume that has an extensionC() such that H^–1().Then by the Riesz representation theorem there exists a unique

We show that u+ coincides with the Perron solutionof the Dirichlet problem

This extends recent results by Hildebrandt [Math. Nachr. 278(2005), 141–144] and Simader [Math. Nachr. 279 (2006),415–430], and also gives a possible answer to Hadamard'sobjection against Dirichlet's principle.     

Application to Dirichlet Series of Maximal Large Sieve Inequality

In this note, the maximal Large Sieve inequality has been usedto obtain following estimate: where D(s)=, s=+it, is a Dirichletseries, dµ is a nonnegative Borel measure on R and U =[–Clog_–1N] (say T1). In addition, the higher-dimensionalversion of (*) has been obtained.     

Counterexamples to Tischler's Strong Form of Smale's Mean Value Conjecture

Smale's mean value conjecture asserts that for every polynomial P of degree d satisfying P(0)=0,where K = (d–1)/d and the minimum is taken over all criticalpoints of P. A stronger conjecture due to Tischler assertsthat with . Tischler's conjecture is known to be true: (i) for local perturbations of the extremumP₀(z)=z^d – dz, and (ii) for all polynomials of degreed 4. In this paper, Tischler's conjecture is verified for alllocal perturbations of the extremum P₁(z)=(z – 1)^d –(–1)^d, but counterexamples to the conjecture are givenin each degree d 5. In addition, estimates for certain weightedL¹- and L²-averages of the quantities are established, which lead to the best currentlyknown value for K₁ in the case d=5. 2000 Mathematics SubjectClassification 30C15.     

Global Branch of Solutions for Non-Linear Schrodinger Equations with Deepening Potential Well

We consider the stationary non-linear Schrödinger equation where > 0 and the functionsf and g are such that and for some bounded open set R^N. We use topological methods to establish the existenceof two connected sets D^± of positive/negative solutionsin R x W², ^p R^N where that cover the interval (, ()) in the sense that and furthermore, The number () is characterized as the unique value of in theinterval (, ) for which the asymptotic linearization has a positiveeigenfunction. Our work uses a degree for Fredholm maps of indexzero. 2000 Mathematics Subject Classification 35J60, 35B32,58J55.     

Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces

The shape-space whose points represent the shapes of not totally degenerate k-ads in R^mis introduced as a quotient space carrying the quotient metric.When m = 1, we find that when m 3, the shape-space contains singularities. This paper dealsmainly with the case m = 2, when the shape-space can be identified with a version of CP^k–2.Of special importance are the shape-measures induced on CP^k–2by any assigned diffuse law of distribution for the k vertices.We determine several such shape-measures, we resolve some ofthe technical problems associated with the graphic presentationand statistical analysis of empirical shape distributions, andamong applications we discuss the relevance of these ideas totesting for the presence of non-accidental multiple alignmentsin collections of (i) neolithic stone monuments and (ii) quasars.Finally the recently introduced Ambartzumian density is examinedfrom the present point of view, its norming constant is found,and its connexion with random Crofton polygons is established.     

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