Primes dividing both 2 n –3 and 3 n –2 are rare

Let S(x) denote the number of primes p < x which divide both 2ⁿ–3 and 3ⁿ–2 for some We prove that

On the mixed problem for hyperbolic partial differential-functional equations of the first order

We consider the mixed problem for the hyperbolic partial differential-functional equation of the first order where is a function defined by z _(x,y)(t, s) = z(x + t, y + s), (t, s) [–, 0] × [0, h]. Using the method of bicharacteristics and the method of successive approximations for a certain integral-functional system we prove, under suitable assumptions, a theorem of the local existence of generalized solutions of this problem.     

Exponentiating a Bundle of Linear Operators

This paper deals with the concept of exponentiability for a special class of multivalued maps. To be more precise, we discuss the exponentiability of a multivalued map F: X⇉X expressible in the form F(x) = {Ax:A ∈ Ξ}, with Ξ denoting a collection of linear continuous operators defined on a Banach space X. Among other results, we prove that, under suitable assumptions on Ξ, the Painlevé–Kuratowski limit

An Ω-theorem for an error term related to the sum-of-divisors function

Let denote the sum-of-divisors function, and set . Gronwall and Wigert proved (independently) in 1913 and 1914, respectively, thatE ₁ (x)= (x log logx). In this paper we obtain the more preciseE ₁ (x)=_–(x log logx). The method consists in averaging over suitable arithmetic progressions, and was suggested by the work ofP. Erdös andH. N. Shapiro [Canad. J. Math. 3–4, 375–385 (1951)] on the error term corresponding to Euler's functions, .     

An existence theorem for generalized quasi-variational inequalities

In this paper, we deal with the following generalized quasi-variational inequality problem: given a closed convex subsetX ⁿ , a multifunction :X 2 ⁿ and a multifunction :X 2 ^X , find a point ( ) X × ⁿ such that We prove an existence theorem in which, in particular, the multifunction is not supposed to be upper semicontinuous.     

Embeddings of graphs in euclidean spaces

The dimension of a graphG=(V, E) is the minimum numberd such that there exists a representation and a thresholdt such thatxy E iff . We prove that d(G)n–x(G) and wheren=|V| andx(G) is the chromatic number ofG; we present upper bounds for the dimension of graphs with a large girth and we show that the complement of a forest can be represented by unit vectors inR ⁶. We prove that d(G)1/15n for most graphs and that there are 3-regular graphs with d(G)c logn/log logn.     

The greatest prime factor of the integers in a short interval (IV)

LetP(x) denote the greatest prime factor of n. In this paper, we shall prove thatP(x)>x ^0.728 holds true for sufficiently largex.Project supported by the Tian Yuan Item in the National Natural Science Foundation of China.     

On the Asymptotic Behavior of Solutions of Differential Systems

There are many studies on the asymptotic behavior of solutions of differential equations. In the present paper, we consider another aspect of this problem, namely, the rate of the asymptotic convergence of solutions. Let ϕ (t) be a scalar continuous monotonically increasing positive function tending to ∞ as t → ∞. It is established that if all solutions of a differential system satisfy the inequality

On a class of functions with equal infima over a domain and its boundary

In this paper, we present a class of functions:f:X such that inf _xX f(x)= , whereX is a nonempty, finitely compact and convex set in a vector space andB _x ={xX: y aff(X){x:[x, y]X={x}. Our main tool is a recent minimax theorem by Ricceri (Ref. 1).     

Limit theorems for transient diffusions on the line

Summary LetX be a diffusion in natural scale on (0,1], with 1 reflecting, and letc(x)(H _x ) andv(x)var (H _x ), whereH _x =inf{t: X _t =x}. Let _x =sup{t:X _t =x}. The main results of this paper are firstly that (i)c is slowly varying; (ii) are all equivalent: and secondly that (v) are all equivalent, and are implied by the condition . Other partial results for more general limit theorems are proved, and new results on regular variation are established.     

A vector-sum theorem in two-dimensional space

Given a finite setX of vectors from the unit ball of the max norm in the twodimensional space whose sum is zero, it is always possible to writeX = {x₁, , x_n} in such a way that the first coordinates of each partial sum lie in [–1, 1] and the second coordinates lie in [–C, C] whereC is a universal constant.     

An application of Shoenfield's absoluteness theorem to the theory of uniform distribution

IfC is a Polish probability space, a Borel set whose sectionsW _x ( have measure one and are decreasing , then we show that the set _x W _x has measure one. We give two proofs of this theorem—one in the language of set theory, the other in the language of probability theory, and we apply the theorem to a question on completely uniformly distributed sequences.Supported by DFG grant Ko 490/7-1.     

Generalized Quasi-Variational Inequalities in Infinite-Dimensional Normed Spaces

In this paper, we deal with the following problem: given a real normed space E with topological dual E*, a closed convex set XE, two multifunctions :X2^X and , find such that We extend to the above problem a result established by Ricceri for the case (x)X, where in particular the multifunction is required only to satisfy the following very general assumption: each set (x) is nonempty, convex, and weakly-star compact, and for each yX–:X the set is compactly closed. Our result also gives a partial affirmative answer to a conjecture raised by Ricceri himself.     

On the Jump Number of Lexicographic Sums of Ordered Sets

Let Q be the lexicographic sum of finite ordered sets Q _x over a finite ordered set P. For some P we can give a formula for the jump number of Q in terms of the jump numbers of Q _x and P, that is, , where s(X) denotes the jump number of an ordered set X. We first show that where w(X) denotes the width of an ordered set X. Consequently, if P is a Dilworth ordered set, that is, s(P) = w(P)–1, then the formula holds. We also show that it holds again if P is bipartite. Finally, we prove that the lexicographic sum of certain jump-critical ordered sets is also jump-critical.     

On a conditional Gołab-Schinzel equation

Let We show that for every function satisfying the conditional equation

Multiple recurrence theorem for nilpotent group actions

We prove that ifX is a compact space andT ₁, ...,T _t are homeomorphisms ofX generating a nilpotent group, then there existx X andn ₁,n ₂, ... N such that for eachj=1, ...t.Supported by British Technion Society.     

Microlocal energy methods and pseudo-differential operators

A functionk(x, u) of the(n+n)-variables is said to be a positive Hermite kernel if , and the matrix(k(x _i, x_j))_i,j is positive semi-definite for every integerN and everyx ₁, ..., x_N. In this paper, we prove that this positive structure can be microlocalized in the category of microfunctions. Further we obtain a useful theorem concerning the positivity of pseudo-differential operators. This theory will play important roles in the study of analytic singularities of solutions of boundary value problems.     

Tensor ideals in the category of tilting modules

We study the tensor category of tilting modules over a quantum groupU _q with divided powers. The setX ₊ of dominant weights is a union of closed alcoves numbered by the elementswW ^f of a certain subset of affine Weyl groupW. G. Lusztig and N. Xi defined a partition ofW ^f into canonical right cells and the right order _R on the set of cells. For a cellAW ^f we consider a full subcategory formed by direct sums of tilting modulesQ() with highest weights . We prove that is a tensor ideal in , generalizing H. Andersen's theorem about the ideal of negligible modules which in our notations is nothing else then . The proof is an application of a recent result by W. Soergel who has computed the characters of tilting modules.This material is based upon work supported by the U.S. Civilian Research and Development Foundation under Award No. RM1-265.     

Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems

Summary This paper deals with polynomial approximations ø(x) to the exponential function exp(x) related to numerical procedures for solving initial value problems. Motivated by positivity and contractivity requirements imposed on these numerical procedures we study the smallest negative argument, denoted by –R(ø), at which ø is absolutely monotonic. For given integersp1,m1 we determine the maximum ofR(ø) when ø varies over the class of all polynomials of a degree m with (forx0).     

Un critère sur les petites boules dans le théorème limite central

We show that a Banach space valued random variableX such that t} \right\} = 0$$ " align="middle" border="0"> satisfies the central limit theorem if and only if the following criterion on small balls is fulfilled: