On Some Rings of Arithmetical Functions

In this paper, we consider several constructions which from a given B-product * _B lead to another one We shall be interested in finding what algebraic properties of the ring are shared also by the ring . In particular, for some constructions the rings R _B and will be isomorphic and therefore have the same algebraic properties.     

Equivariant K-Homology and Restriction to Finite Cyclic Subgroups

For a discrete group G, we prove that a G-map between proper G–CW-complexes induces an isomorphism in G-equivariant K-homology if it induces an isomorphism in C-equivariant K-homology for every finite cyclic subgroup C of G. As an application, we show that the source of the Baum–Connes assembly map, namely K _* ^G (E(G, in)), is isomorphic to K _* ^G (E(G, )), where E(G, ) denotes the classifying space for the family of finite cyclic subgroups of G. Letting be the family of virtually cyclic subgroups of G, we also establish that and related results.     

Time-Localization of Random Distributions on Wiener Space

A nuclear space of distributions on Wiener space was constructed by Gorostiza and Nualart [10] as a framework for studying weak convergence of trajectorial fluctuations of particle systems. A basic problem in recovering the usual time-evolution results from the trajectorial ones consists in associating in a unique way an -valued process to a random distribution on by localizing it at each time t [0,1]. In this paper we solve this problem for a large class of random distributions which includes trajectorial fluctuation limits of some systems of diffusions.     

Convergence Analysis of Some Methods for Minimizing a Nonsmooth Convex Function

In this paper, we analyze a class of methods for minimizing a proper lower semicontinuous extended-valued convex function . Instead of the original objective function f, we employ a convex approximation f _{k + 1} at the kth iteration. Some global convergence rate estimates are obtained. We illustrate our approach by proposing (i) a new family of proximal point algorithms which possesses the global convergence rate estimate even it the iteration points are calculated approximately, where are the proximal parameters, and (ii) a variant proximal bundle method. Applications to stochastic programs are discussed.     

The Ingham Divisor Problem on the Set of Numbers without kth Powers

Suppose that k and l are integers such that and , M _k is a set of numbers without kth powers, and . In this paper, we obtain asymptotic estimates of the sums over      

On the Structure of a Certain Class of Mixed Tsirelson Spaces

We study Banach spaces of the form We call such a space a p-space, p[1,), if for every k the space is isomorphic to _pk and the sequence (p_k) strictly decreases to p. We examine the finite block representability of the spaces _r in a p-space proving that it depends not only on p but also on the sequences (p_k) and (n_k). Assuming that _i n_i ^1/q decreases to 0, where q is the conjugate exponent of p, we prove the existence of an asymptotic biorthogonal system in X and also that c ₀ is finitely representable in X. Moreover we investigate the modified versions of p-spaces proving that, if n_km1/pkm-1/pk_m-1 increases to infinity for a subsequence (n_km) , then ₁ embeds into X. We also investigate complemented minimality for the class of spaces where is either a subsequence of the sequence of Schreier classes ( _n)_{n N} or a subsequence of ( _n)_{n N}.     

An Intersection Theorem on an Unbounded Set and Its Application to the Fair Allocation Problem

We prove the following theorem. Let m and n be any positive integers with mn, and let be a subset of the n-dimensional Euclidean space ⁿ . For each i=1, . . . , m, there is a class of subsets M _i ^j of ^Tn . Assume that for each i=1, . . . , m, that M _i ^j is nonempty and closed for all i, j, and that there exists a real number B(i, j) such that and its jth component _{xjB(i, j)} imply . Then, there exists a partition of {1, . . . , n} such that for all i and We prove this theorem based upon a generalization of a well-known theorem of Birkhoff and von Neumann. Moreover, we apply this theorem to the fair allocation problem of indivisible objects with money and obtain an existence theorem.     

On Singular Phenomena in Certain Time-Optimal Problem

The purpose of this paper is to present the solution of time-optimal problem ofthe controlled object the dynamics of which is given by: , , where and motion resistance function if ,f(x)=-A if x > 0 where . That model describes dynamicsof a very important class of industrial installations. As the time-optimalproblem will be understood a transfer of the initial state to the target state in a minimumtime . There has been shown that in the formula defining resistancefunction f(x)there exists a value that plays an essentialrole in time-optimal structure formation. Namely, if then thetime-optimal control process is typical, analogous as in classical case , i.e. there exists a switching curve formed by thetrajectories of time-optimal solutions reaching the target state and thetime-optimal process is formed by at most one switching operation. For the caseA>A_bwe will examine two following singular phenomena.(a) If the target state z₁=(0, 0) then there exists theswitching curve, dividing the state plane into two sets, however only one itsbranch is formed by the time-optimal solution reaching the target z₁=(0, 0) and generated by the control u=-1. None of solution formsthe second branch of switching curve. It is formed by a state-locus dependingon the value of Aonly. In dependency of the starting state z₀ thetime-optimal control process is generated by bang-bang control with none,one or two switching operations. This is the first singular phenomenon,because any small decrease of the value Aover A _b requires to change thestructure which would be able to generate the time-optimal process.(b) The paper shows, that if the target state z ₁(x_1, 0), x₁>0then there exists a set of the starting states from which there start twotrajectories reaching the target in the same minimum time. This is thesecond phenomenon.Finally, some suggestions as to practical applications have been given too.     

On posets with isomorphic interval posets

Let be a partially ordered set, Int the system of all (nonempty) intervals of partially ordered by the set-theoretical inclusion . We are interested in partially ordered sets with Int isomorphic to Int . We are going to show that they correspond to couples of binary relations on A satisfying some conditions. If is a directed partially ordered set, the only with Int isomorphic to Int are corresponding to direct decompositions of ( denotes the dual of . The present results include those presented in the paper [11] by V. Slavík. Systems of intervals, particularly of lattices, have been investigated by many authors, cf. [1]–[11].     

Auxiliary Problem Principle and Proximal Point Methods

An extension of the auxiliary problem principle to variational inequalities with non-symmetric multi-valued operators in Hilbert spaces is studied. This extension concerns the case that the operator is split into the sum of a single-valued operator , possessing a kind of pseudo Dunn property, and a maximal monotone operator . The current auxiliary problem is k constructed by fixing at the previous iterate, whereas (or its single-valued approximation ^k) k is considered at a variable point. Using auxiliary operators of the form ^k+ , with _k>0, the standard for the auxiliary problem principle assumption of the strong convexity of the function h can be weakened exploiting mutual properties of and h. Convergence of the general scheme is analyzed and some applications are sketched briefly.     

On a General Projection Algorithm for Variational Inequalities

Let H be a real Hilbert space with norm and inner product denoted by and . Let K be a nonempty closed convex set of H, and let f be a linear continuous functional on H. Let A, T, g be nonlinear operators from H into itself, and let be a point-to-set mapping. We deal with the problem of finding uK such that g(u)K(u) and the following relation is satisfied: , where >0 is a constant, which is called a general strong quasi-variational inequality. We give a general and unified iterative algorithm for finding the approximate solution to this problem by exploiting the projection method, and prove the existence of the solution to this problem and the convergence of the iterative sequence generated by this algorithm.     

Quotient Groups of Groups of Certain Classes

For an arbitrary variety of groups and an arbitrary class of groups that is closed on quotient groups, we prove that a quotient group G/N of the group G possesses an invariant system with - and -factors (respectively, is a residually -group) if G possesses an invariant system with - and -factors (respectively, is a residually -group) and N (respectively, N is a maximal invariant -subgroup of the group G).     

Strictly Positive Definite Functions on a Real Inner Product Space

If converges for all with all coefficients , then the function is positive definite on H×H for any inner product space H. Set K={k: a _k >0}. We show that is strictly positive definite if and only if K contains the index 0 plus an infinite number of even integers and an infinite number of odd integers.     

A Higher-Order Analog of the Linking Number for a Pair of Divergence-Free Vector Fields

Pairs B, of divergence-free vector fields with compact support in are considered higher-order analog M(B, c (of order 3) of the Gauss helicity number H(B, )= , curl(A)=B; (of order 1) is constructed, which is invariant under volume-preserving diffeomorphisms. An integral expression for M is given. A degree-four polynomial m(B(x₁), B(x₂), ( ₁), ( ₂)), x₁, x₂, _{1 2} , is defined, which is symmetric in the first and second pairs of variables separately. M is the average value of m over arbitrary configurations of points. Several conjectures clarifying the geometric meaning of the invariant and relating it to invariants of knots and links are stated. Bibliography: 11 titles.     

Imprimitive Q-polynomial Association Schemes

It is well known that imprimitive P-polynomial association schemes with are either bipartite or antipodal, i.e., intersection numbers satisfy either for all for all . In this paper, we show that imprimitive -polynomial association schemes with are either dual bipartite or dual antipodal, i.e., dual intersection numbers satisfy either .     

Theorems of the Alternative and Duality

This paper investigates the relations between theorems of the alternative and the minimum norm duality theorem. A typical theorem of the alternative is associated with two systems of linear inequalities and/or equalities, a primal system and a dual one, asserting that either the primal system has a solution, or the dual system has a solution, but never both. On the other hand, the minimum norm duality theorem says that the minimum distance from a given point z to a convex set is equal to the maximum of the distances from z to the hyperplanes separating z and . We consider the theorems of Farkas, Gale, Gordan, and Motzkin, as well as new theorems that characterize the optimality conditions of discrete l ₁-approximation problems and multifacility location problems. It is shown that, with proper choices of , each of these theorems can be recast as a pair of dual problems: a primal steepest descent problem that resembles the original primal system, and a dual least–norm problem that resembles the original dual system. The norm that defines the least-norm problem is the dual norm with respect to that which defines the steepest descent problem. Moreover, let y solve the least norm problem and let r denote the corresponding residual vector. If r=0, which means that z , then y solves the dual system. Otherwise, when r0 and z , any dual vector of r solves both the steepest descent problem and the primal system. In other words, let x solve the steepest descent problem; then, r and x are aligned. These results hold for any norm on . If the norm is smooth and strictly convex, then there are explicit rules for retrieving x from r and vice versa.     

Limiting Laws for Entrance Times of Critical Mappings of a Circle

A renormalization group transformation R ₁ has a single stable point in the space of the analytic circle homeomorphisms with a single cubic critical point and with the rotation number (the golden mean). Let a homeomorphism T be the C ¹-conjugate of . We let denote the sequence of distribution functions of the time of the kth entrance to the nth renormalization interval for the homeomorphism T. We prove that for any , the sequence has a finite limiting distribution function , which is continuous in , and singular on the interval [0,1]. We also study the sequence for k>1.     

λ-Divergence of the Fourier Series of Continuous Functions of Several Variables

In this paper, we consider the behavior of rectangular partial sums of the Fourier series of continuous functions of several variables with respect to the trigonometric system. The Fourier series is called -convergent if the limit of rectangular partial sums over all indices for which for all j and k exists. In the space of arbitrary even dimension 2m we construct an example of a continuous function with an estimate of the modulus of continuity such that its Fourier series is -divergent everywhere for any .     

Zeros of H ∞-Functions on Hyperplanes in \mathbb{B} n

Let ⁿ be the unit ball in ℂⁿ, n ≥ 2. Let T_α = {z ∈ ⁿ : (z, a) = |a|²} for a ∈ ⁿ and denote for a discrete set A in ⁿ. We find a sharp necessary condition for a set A to be a part of the zero-set for a function in H^∞( ⁿ). Bibliography 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 272–278.