On Large Values of the Divisor Function

Let d(n) denote the divisor function, and let D(X) denote the maximal value of d(n) for n X. For 0 < z 1, both lower and upper bounds are given for the number of integersn with n X, zD(X) d(n).     

Optimization of pressure control in a gas pipeline

We consider the synthesis of an approximate optimal control for a gas transport system when state information is acquired at the points X₁,...,x_n of the interval 0 x during the entire control period 0 t . This means that the functions (t, x₁),...,(t, x_n) characterizing the state of the controlled system at the points X₁,...,x_n are known. The approximate optimal control is determined using this information. We solve the control optimization problem which, for a given deviation of the approximate optimal control from the optimum, produces a minimum-information approximate optimal control, i.e., a control determined by a formula with the least number of points x_i, and establishes the location of these points.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 118–123, 1987.     

Rational chebyshev approximations of elementary functions

Explicit formulæ which should be useful in practical computations, are given for the functions e^–x, log x, (1 + x), arctg x/x, sin x/x, cos x and (2/x) arcsin (x/2) in the interval 0x1 except for log x, where the interval is 1/2x1 instead.     

Configuration of subgroups in the special linear group over a skew field with infinite center

Let T be a skew field with infinite center, let be the special linear group over T of degree 3, and let be the subgroup of diagonal matrices with unit Dieudonee determinant. It is proved that for each intermediate subgroup H, H , there exists a net of order n such that ( H N().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 5–12, 1989.In conclusion, the author would like to thank his instructor Z. I. Borevich, as well as N. A. Vavilov, for their assistance.     

Efficient Regular Polygon Dissections

We study the minimum number g(m,n) (respectively, p(m,n)) of pieces needed to dissect a regular m-gon into a regular n-gon of the same area using glass-cuts (respectively, polygonal cuts). First we study regular polygon-square dissections and show that n/2 -2 g(4,n) (n/2) + o(n) and n/4 g(n,4) (n/2) + o(n) hold for sufficiently large n. We also consider polygonal cuts, i.e., the minimum number p(4,n) of pieces needed to dissect a square into a regular n-gon of the same area using polygonal cuts and show that n/4 p(4,n) (n/2) + o(n) holds for sufficiently large n. We also consider regular polygon-polygon dissections and obtain similar bounds for g(m,n) and p(m,n).     

On the effective thermal conductivity of multiphase composites

While several micromechanical models have been developed today in the literature for two-phase media, the extent of their applicability to multiphase materials need yet to be investigated. The present paper studies the effective thermal conductivity of multicomponent composites, and concentrates on two methods: (a) the Mori-Tanaka model, (b) The generalized self-consistent scheme. The Mori-Tanaka method of back-stress previously developed in the context of elasticity of composites is applied here to the conduction problem. The generalized self-consistent scheme, based on a particle-matrix embedding in the effective medium, is studied in this paper in the context of multicomponent media and two variations of this method distinctly different in their imbedding procedure are proposed.Numerical results are given for three-phase composites illustrating and comparing the proposed methods.

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Resumé Plusieurs modéles de micromécanique existent aujoud'hui dans la litérature des matériaux á deux phases. L'applicabilité de ces modéles aux matériaux á multi-phases nécessite d'autre part d'être examinée. Ce travail étudie la conductivité thermale des matériaux á multi-phases et se concentre sur deux méthodes: a) le modéle de Mori-Tanaka b) la méthode auto-consistante généralisée. La méthode de Mori-Tanaka, développée auparavant dans le cadre des propriétés mécaniques des composites est appliquée ici au probléme de conduction. La méthode autoconsistante généralisée est étudiée dans le cadre des matériaux á multi-phases et deux alternatives de cette méthode distinctement différentes dans leur formalisme sont proposées. Pour des matériaux á trois phases on donne des resultats numériques qui illustrent et comparent les méthodes proposées.

     

Function Spaces of Posets with Projections

This paper investigates function spaces of structures consisting of a partially ordered set together with some directed family of projections.More precisely, given a fixed directed index set (I,), we consider triples (D,,(p _i ) _iI ) with (D,) a poset and (p _i ) _iI a monotone net of projections of D. We call them (I,)-pop's (posets with projections). Our main purpose is to study structure preserving maps between (I,)-pop's. Such homomorphisms respect both order and projections.Any (I,)-pop is known to induce a uniformity and thus a topology. The set of all homomorphisms between two (I,)-pop's turns out to form an (I,)-pop itself. We show that its uniformity is the uniformity of uniform convergence. This enables us to prove that properties such as completeness and compactness transfer to function pop's.Concerning categorical properties of (I,)-pop's, we will see that we are in a lucky situation from a computer scientist's point of view: we obtain Cartesian closed categories. Moreover, by a D -construction we get (I,)-pop's that are isomorphic to their own exponent. This yields new models for the untyped -calculus.     

Convergence of Series of Fourier Coefficients of p-Absolutely Continuous Functions

In the paper, we generalize the well-known criteria of Bernstein and Stechkin on the absolute convergence in terms of best approximations and moduli of smoothness of continuous functions. We give conditions for the convergence of the series of Fourier coefficients raised to the power in terms of best approximations in the space of p-absolutely continuous functions and in terms of fractional moduli of continuity with respect to this space. We also prove the sharpness of our conditions for 0 < 1 with no restriction and for 1 < 2 under some restrictions.     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

On elementary equivalence and isomorphism of clone segments

The principal application of a general theorem proved here shows that for any choice 1mnp of integers there exist metric spacesX andY such that the initialk-segments of their clones of continuous maps coincide exactly whenkm, are isomorphic exactly whenkn, and are elementarily equivalent exactly whenkp.Dedicated to Prof. László Fuchs on the occasion of his 70th birthday     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

Tensor products and Banach ideals of p-compact operators

A study is made of the norm w_p (1 p ) on the tensor product of two Banach spaces E and F. It is shown that w_p is a tensor norm, and a representation is deduced for the elements in the completion of E F equipped with w_p. Finally it is shown that the w_p-nuclear operators in the sense of Grothendieck [3] coincide with those operators factoring compactly through^p (if 1 p ) or Co (if p=), with related equalities concerning the idea1 norms.     

Quasilinear elliptic equations with small boundary data

It is shown that if the order of non-uniformity of a quasilinear elliptic equation is h, 1<h2, then the critical norm separating existence and non-existence of a solution to the Dirichlet problem with small boundary data is the norm. For 0h1, existence of a solution is guaranteed without any smallness assumption on the given boundary data, provided that the usual a priori interior gradient bound for solution is available.     

On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.     

Inclusion statements for operator equations by a continuity principle

The paper is concerned with Range-Domain Implications MvCvK, where M is a given operator and C,K denote given sets. Sufficient conditions are derived by a very general continuity principle. Various special cases are considered such as inverse-positivity, MvMwvw, and a generalization H(,[,])MvH(,[,]) v, where Mu=H(u,u) and [,] denotes an order interval. These results are applied to differential operators related to boundary or initial value problems. The goal is to furnish a simple uniform approach, to explain its application, and to provide a kind of survey on what problems have been treated in this way.     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

Montgomery's weighted sieve for dimension two

The most well-known application of Montgomery's weighted sieve is to the so-called Brun-Titchmarsh inequality, which was proved byH. L. Montgomery andR. C. Vaughan in the form (x, k, l)2x((k)log(x/k))^–1 for 1k<x, (k, l)=1, (x, k, l) being the number of primespx andpl modk, (k) being Euler's function. In this paper an upper estimate is given for a certain class of two-dimensional sieve problems, among them bounds for the number of twin primes and the number of Goldbach representations.     

Vector Valued Differentiation Theorems for Multiparameter Additive Processes in Lp Spaces

Let X be a Banach space and (,,µ) be a -finite measure space. We consider a strongly continuous d-dimensional semigroup T={T(u):u=(u₁,..., u_d, u_i >0, 1 i d} of linear contractions on L_p((,,µ); X), with 1 p<. In this paper differentiation theorems are proved for d-dimensional bounded processes in L_p((,,µ); X) which are additive with respect to T. In the theorems below we assume that each T(u) possesses a contraction majorant P(u) defined on L_p((,,µ); R), that is, P(u) is a positive linear contraction on L_p((,,µ); R) such that T(u)f(w) P(u)f(·)() almost everywhere on for all f L_p((,,µ); X).     

Bases in the spaces C and Lp

In this paper it is proved that for any numbers A and B, 0k(x), k=1, 2, ..., whose graphs lie in the strip 0x1, AyB. It is shown that for the space L_p, p>1, there is no analogous basis in a strip theorem.Translated from Matematicheskie Zametki, Vol. 10, No. 6, pp. 635–640, December, 1971.     

Approximation of the Distribution of the Supremum of a Centered Random Walk. Application to the Local Score

Let (X _n ) _{n 0} be a real random walk starting at 0, with centered increments bounded by a constant K. The main result of this study is: |P(S _n n x)–P( sup_{0 u 1} B _u x)| C(n,K) n/n, where x 0, ² is the variance of the increments, S _n is the supremum at time n of the random walk, (B _u ,u 0) is a standard linear Brownian motion and C(n,K) is an explicit constant. We also prove that in the previous inequality S _n can be replaced by the local score and sup₀ u 1 B _u by sup₀ u 1|B _u |.