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In this paper, we continue to discuss the properties of iterates generated by a strict pseudocontraction or a finite family of strict pseudo-contractions in a real q-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [Marino, G., Xu, H. K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl., 324, 336–349 (2007)]. In order to get a strong convergence theorem, we modify the normal Mann’s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C. This result extends a recent result of Kim and Xu [Kim, T. H., Xu, H. K.: Strong convergence of modified Mann iterations. Nonl. Anal., 61, 51–60 (2005)] both from nonexpansive mappings to λ-strict pseudo-contractions and from Hilbert spaces to q-uniformly smooth Banach spaces.

In this paper we classify, up to analytic isomorphism, the family of almost stretched Artinian complete intersection A=R/I with a given Hilbert function, in the case R is a power series ring with an arbitrary number of variables., 20 pages. This paper generalizes a previous version where the result was proven for a power series ring in two variables

This paper complements a recent result of Dilworth, Ferenczi, Kutzarova and Odell regarding the ergodicity of strongly asymptotic ℓ p \ell _p spaces. We state this result in a more general form, involving domination relations, and we show that every asymptotically Hilbertian space which is not isomorphic to ℓ 2 \ell _2 is ergodic. In particular, every weak Hilbert space which is not isomorphic to ℓ 2 \ell _2 must be ergodic. Throughout the paper we construct explicitly the maps which establish the fact that the relation E 0 E_0 is Borel reducible to isomorphism between subspaces of the Banach spaces involved.

Let T be an operator on a Hilbert space \( \mathcal{H} \). The problem of computing of the norm of T, norm of selfcommutators of T, and the numerical radius of T are discussed in many papers and a number of textbooks. In this paper we determine the relationships between these values for self inverse operators and explain how we can determine any three of these (‖T‖, ‖[T*,T]‖, ‖{T*,T}‖, and the numerical radius of T) by knowing any one of them. Also, we find the spectrum of T*T,[T*,T] and {T*,T} in the case that T is self inverse and the spectrum of T*T is an interval. Finally, by giving some examples on automorphic composition operators, we show that these results make it possible to replace lengthy computation with quick ones.

A composite dilation wavelet is a collection of functions generating an orthonormal basis for L2(ℝn) under the actions of translations from a full rank lattice and dilations by products of elements of non-commuting groups A and B. A minimally supported frequency composite dilation wavelet has generating functions whose Fourier transforms are characteristic functions of a lattice tiling set. In this paper, we study the case where A is the group of integer powers of some expanding matrix while B is a finite subgroup of the invertible n×n matrices. This paper establishes that with any finite group B together with almost any full rank lattice, one can generate a minimally supported frequency composite dilation wavelet system. The paper proceeds by demonstrating the ability to find such minimally supported frequency composite dilation wavelets with a single generator.

This paper deals with the regularity properties for a class of semilinear integrodierentia l functional dierential equations. It is shown the relation between the reachable set of the semilinear system and that of its corresponding linear system. We also show that the Lipschitz continuity and the uniform boundedness of the nonlinear term can be considerably weakened. Finally, a simple example to which our main result can be applied is given. Let H and V be two complex Hilbert spaces such that V is a dense subspace of H. Identifying the antidual of H with H we may consider V ‰ H ‰ V ⁄ . In this paper we deal with the approximate controllability for the semilinear equation in H as follows. (SE) ( d dt x(t) = Ax(t) + R t 0 k(t i s)g(s,x(s),u(s))ds + Bu(t)

Let H be a Hilbert space which is the direct sum of five closed subspaces X0, X1, X2, X3 and X4 with X1, X2, X3 of finite dimension. Let J be a C 1,1 functional defined on H with J(0) = 0. We show the existence of at least four nontrivial critical points when the sublevels of J (the torus with three holes and sphere) link and the functional J satisfies sup-inf variational inequality on the linking subspaces, and the functional J satisfies (P.S.)⁄ condition and f|X0'X4 has no critical point with level c. For the proof of main theorem we use the nonsmooth version of the classical deformation lemma and the limit relative category theory. 1. Introduction and statement of main result Let H be a Hilbert space which is a direct sum of five closed subspaces X0, X1, X2, X3 and X4 with X1, X2, X3 of finite dimension. Let J be a C 1,1 functional defined on H with J(0) = 0. In this paper we investigate the number of nontrivial critical points of the C 1,1 functional J under some conditions on the sublevels of J and the shape of J. We show the existence of at least four nontrivial critical points when the sublevels of J (the torus with three holes and sphere) link and the functional J satisfies the sup-inf variational inequality on the linking subspaces, and the functional J satisfies (P.S.) ⁄ condition and J|X0'X4 has no critical point with level c. Micheletti and Saccon prove in (13) that the functional J has at least two nontrivial critical points under the same conditions on J except the condition that the sublevel sets are the torus with one hole and the sphere. In this paper we improve this result to the case that the sublevel sets are the torus with three holes and sphere. Now, we state the main result

In this paper we consider a family of random Cantor sets on the line. We give some sufficient conditions when the Lebesgue measure of the arithmetic difference is positive. Combining this with the main result of a recent joint paper of the second author with M. Dekking we construct random Cantor sets F1, F2 such that the arithmetic difference set F2 − F1 does not contain any intervals but ℒeb(F2 − F1)> 0 almost surely, conditioned on non-extinction.

Regular polynomials with quaternionic coefficients admit only isolated zeroes and spherical zeroes. In this paper we prove a factorization theorem for such polynomials. Specifically, we show that every regular polynomial can be written as a product of degree one binomials and special second degree polynomials with real coefficients. The degree one binomials are determined (but not uniquely) by the knowledge of the isolated zeroes of the original polynomial, while the second degree factors are uniquely determined by the spherical zeroes. We also show that the number of zeroes of a polynomial, counted with their multiplicity as defined in this paper, equals the degree of the polynomial. While some of these results are known in the general setting of an arbitrary division ring, our proofs are based on the theory of regular functions of a quaternionic variable, and as such they are elementary in nature and offer explicit constructions in the quaternionic setting.

In this paper we generate a fuzzy dynamical matrix Lyapunov system andobtain a sufficient condition for the observability of this system. 1. IntroductionThe importance of control theory in Applied mathematics and its occurrencein several problems such as mechanics, electromagnetic theory, thermodynamics,artificial satellites etc., are well known. The observability condition assures theconstruction of the state from the output. This property is intrensic for systemsand play an important role in the theory of linear systems.The objective of this paper is to provide sufficient condition for observability offirst order matrix Lyapunov system described by(1.1) X 0 (t) = A(t)X(t)+X(t)B(t)+F(t)U(t), X(0) = X 0 , t>0,(1.2) Y(t) = C(t)X(t)+D(t)U(t),where U(t) is a n× nfuzzy input matrix called fuzzy control and Y(t) is a n× nfuzzy output matrix. Here A(t),B(t),F(t),C(t), and D(t) are matrices of ordern×n, whose elements are continuous functions of t on J= [0,T] ⊂ R(T>0).The problem of controllability and observability for systems of ordinary differ-ential equations has been studied by Barnett [2] and for matrix Lyapunov systemsby Murty, Rao and Suresh Kumar [8]. Recently the observability criteria for fuzzydynamical control systems was studied by Ding and Kandel [5].In section 2 we present some basic definitions and results relating to fuzzy setsand also some properties of Kronecker products and obtain general solution of thesystem (1.1), when U(t) is a crisp continuous matrix.In section 3 we generate a fuzzy dynamical matrix Lyapunov system and alsoobtain its solution set.

In this paper, we deal with the problem of meromorphic functions that havethree weighted sharing values, and obtain some uniqueness theorems which improve thosegiven by N. Terglane, Hong-Xun Yi & Xiao-Min Li, and others. Some examples are pro-vided to show that the results in this paper are best possible. 1. Introduction and main resultsIn this paper, by meromorphic functions we will always mean meromorphicfunctions in the complex plane. We adopt the standard notations in the Nevan-linna theory of meromorphic functions as explained in [3]. It will be convenient tolet E denote any set of positive real numbers of finite linear measure, not necessarilythe same at each occurrence. For any nonconstant meromorphic function h(z), wedenote by S(r,h) any quantity satisfying S(r,h) = o(T(r,h)) (r → ∞,r 6∈E).Let f(z) and g(z) be two nonconstant meromorphic functions, and let a ∈C ∪ {∞}, where C ∪ {∞} denotes the extended complex plane. We denote byN 0 (r,a,f,g) the counting function of the common zeros of f(z) − a and g(z) − a,and each point is counted only once, where f(z)−∞ means 1/f(z) (see [10]). We saythat f and g share the value a CM, provided that f and g have the same a−pointswith the same multiplicities. Similarly, we say that f and g share the value a IM,provided that f and g have the same a−points ignoring multiplicities (see [12]).Throughout this paper, we denote by N

In this paper we provide a solution of the functional equation unsolved in the paper, by the second author, “On functional equations arising from map enumerations” that appeared in Discrete Math, 123: 93–109 (1993). It is also the number of combinatorial distinct rooted general eulerian planar maps with the valency of root-vertex, the number of non-root vertices and non-root faces of the maps as three parameters. In particular, a result in the paper, by the same author, “On the number of eulerian planar maps” that appeared in Acta Math Sinica, 12: 418–423 (1992) is simplified.

In this paper we give simple, sufficient conditions for the existence of the stochastic integral for vector-valued processes X with values in a Banach space E; namely, X is of class (LD), and the stochastic measure I X is bounded and strongly additive in L p E (in particular, if I X is bounded in L p E and c 0 ⊈ E) and has bounded semivariation. The result is then applied to martingales and processes with integrable variation or semivariation. For martingales the condition of being of class (LD) is superfluous. For a square-integrable martingale with values in a Hilbert space, all the conditions are superfluous. For processes with p-integrable semivariation or p-integrable variation, the conditions of I X to be bounded and have bounded semivariation are superfluous. For processes with 1-integrable variation, all conditions are superfluous. In a forthcoming paper, we shall extend these results to local summability. The extension needs additional nontrivial work.

Euler made many contributions to what is now called combinatorics. Some of these arose from recreational mathematics, such as magic squares and knight's tours on a chessboard; others from his study of lotteries; and, perhaps his most important work, from the study of partitions of an integer. In what follows I shall attempt to show the breadth of his work. Euler's papers will be referred to by their Eneström numbers, such as E338; this cataloguing was carried out in the early twentieth century by the Swedish Mathematician Gustav Eneström, who was to Euler what Köchel has been to Mozart. Euler's papers can all be studied in detail on the Euler Archive, details of which are given at the end of this article.

application/pdf, Synthetic di¤erential geometry occupies a unique position in topos-theoretic physics. Nevertheless it has appeared somewhat too conceptual to physicists in general, partly because it has appeared to lack computational aspects. Its computational facets are really concerned with computation of the quasi-colimit of a …nite diagram of in…nitesimal spaces, or equivalently, with computation of the limit of a …nite diagram of Weil algebras. Indeed we have been forced to do a highly invovled computation of the above kind by hand in our previous papers ([International Journal of Theoretical Physics, 36 (1997) , 1099-1131] and [International Journal of Theoretical Physics, 38 (1999) , 2163-2174]). The principal objective in this paper is to show that Gröbner bases techniques pro- vide us with means that relegate such computations to computers.

The paper is concerned with bounds for integrals of the type $$ \int\limits_{U_k (x)} {\left| {p_n^{(\alpha , \beta )} (t)} \right|^p w^{(a, b)} } (t) dt, p \geqq 0 $$ , involving Jacobi polynomials pn(α,β) and Jacobi weights w(a,b) depending on α,β, a, b > −1, where the subsets Uk(x) ⊂ [−1, 1] located around x and are given by \( U_k (x) = \left[ {x - \tfrac{{\phi _k (x)}} {k}, x + \tfrac{{\phi _k (x)}} {k}} \right] \cap [ - 1, 1] \) with \( \phi _k (x) = \sqrt {1 - x^2 } + \tfrac{1} {k} \). The functions to be integrated will also be of the type \( \left| {\tfrac{{p_n^{(\alpha , \beta )} (t)}} {{x - t}}} \right| \) on the domain [−1,1] t/ Uk(x). This approach uses estimates of Jacobi polynomials modified Jacobi weights initiated by Totik and Lubinsky in [1]. Various bounds for integrals involving Jacobi weights will be derived. The results of the present paper form the basis of the proof of the uniform boundedness of (C, 1) means of Jacobi expansions in weighted sup norms in [3].

The purpose of this paper is to discuss a first-return integration process which yields the Lebesgue integral of a bounded measurable function f: I → R defined on a compact interval I. The process itself, which has a Riemann flavor, uses the given function f and a sequence of partitions whose norms tend to 0. The “first-return” of a given sequence \( \bar x \) is used to tag the intervals from the partitions. The main result of the paper is that under rather general circumstances this first return integration process yields the Lebesgue integral of the given function f for almost every sequence \( \bar x \).

We compare program controls and strategies continuous in the phase vector in evasion problems with disturbance. The target set is specified in the function space of trajectories. In particular, this setting covers the case of a terminal set in a finite-dimensional space at the terminal time. The target set is not required to be convex. The problem is formalized as a differential game with a nonlinear equation. The players’ controls are chosen in given sets of integrable functions. We show that if evasion can be guaranteed with the help of a continuous strategy, then it can be guaranteed with the help of a program control as well. The continuous strategies that we consider can contain a deviation of the argument. We assume that some sets constructed in the problem in the space of continuous functions are acyclic (their homology groups are degenerate). The present paper deals with the theory of positional differential games [1–5] and is motivated by the results in [6, 7], where the case of linear dynamics and a convex terminal set was comprehensively studied. It is known that the specific properties of a continuous feedback also manifest themselves in the stabilization of a nonlinear control system (see [8–10] and the bibliography therein). Our proofs are based on the Eilenberg–Montgomery fixed point theorem [11, 12]. One can assume that homology groups are obtained, for example, with the help of the Vietoris construction. In the present paper, we impose no conditions on the growth rate of the right-hand side of the differential equation. Instead, we assume that the closure in the space of continuous functions of the set of all solutions of the problem for all possible pairs of the players’ admissible controls is a compact set. This condition holds in many typical situations. In what follows, we present also some considerations as to what the set of strategies on which the strategy is defined must be so as to ensure that a theorem of this type can be proved. The generalized Tietze continuation theorem proves useful in this connection. If the independent variable can be treated as time, then one can consider a fixed initial point. Below, we consider the more general case of boundary conditions. Similar relations appear, for example, in some conflict control problems for rod heating [13]. In this case, a solution can be treated as a stationary temperature distribution along the rod. Boundary value problems with similar conditions were considered in [14, 15]. The dynamics of a control system is not necessarily linear. We also assume that there may be a nonlinear dependence on the control of the evader. However, for simplicity, we assume that the differential equations are linear in the pursuer’s control. This permits one to show (with the help of weak topology by analogy with [6]) that the graph of the corresponding multimapping is closed. However, the linearity assumption can be avoided. In this case, one should prove closedness by imposing some conditions, as in [16]. At the end of the present paper, we use the results to study a specific control system. We use the following notation: R (where n ≥ 1 is an integer) is the space of n-vectors (columns) with given norm |·|n; Rn×n is the space of n×nmatrices with real entries whose norm |·|n×n is coordinated with the vector norm (i.e., |Ab|n ≤ |A|n×n|b|n, where A ∈ Rn×n and b ∈ R are arbitrary); C is the space of continuous functions; Lr (where r ≥ 1 is a real number) is the space of Lebesgue measurable functions with r-integrable absolute value [|x(t)|n or |x(t)|n×n is integrable for functions x(t) ranging in the space of n-vectors or n× n matrices]; AC is the space of absolutely continuous functions. The symbol cl co stands for the closed convex hull, and cl stands for the closure.

The central purpose of this paper is to illustrate that combining the recently developed theory of random conjugate spaces and the deep theory of Banach spaces can, indeed, solve some difficult measurability problems which occur in the recent study of the Lebesgue (or more general, Orlicz)-Bochner function spaces as well as in a slightly different way in the study of the random functional analysis but for which the measurable selection theorems currently available are not applicable. It is important that this paper provides a new method of studying a large class of the measurability problems, namely first converting the measurability problems to the abstract existence problems in the random metric theory and then combining the random metric theory and the relative theory of classical spaces so that the measurability problems can be eventually solved. The new method is based on the deep development of the random metric theory as well as on the subtle combination of the random metric theory with classical space theory.

0 K(t, s)x(s)ds, t ∈ I ≡ [0, 1], p1, p2 ∈ R+, tj ∈ (0, 1), mj ∈ N (j = 1, q); K and y are known continuous functions with certain properties of pointwise “smoothness“, and x is the desired function. These equations are connected with several problems of the theory of elasticity, scattering of particles, neutron transfer (see, e. g., [1] and references therein; [2]). As a rule, the intrinsic classes of solutions of the ETK are special spaces of distributions. The equations under consideration allow explicit solution in rare special cases. Therefore both the theory and the applications require approximative solution methods with the corresponding theoretical support. Several relevant results are obtained in papers [3], [4]. In [3] we propose and prove special direct solution methods for ETK (1) in the space of distributionsD{p1, p2;m, τ}. Paper [4] is dedicated to the construction of the complete solvability theory for the equations under consideration in a certain space of distributions V {p1, p2;m, τ}. In this paper, following the works [3]–[6], we propose and substantiate in the sense of [7] (Chap. 1) one special direct method for the approximate solution of ETK (1) in the space V {p1, p2;m, τ}. We prove that the constructed method is optimal by the order of exactness on a certain class Φ generated by the classHr ω among all projection solution methods for ETK (1). 1. The space of trial functions. Let C ≡ C(I) be the space of continuous on I functions with the customary max-norm and m ∈ N . According to [8], we denote by C t0 ≡ C{m; t0} the class of functions g ∈ C such that at a point t0 ∈ (0, 1) they have the Taylor derivative g{m}(t0) of the orderm. Let t1, t2, . . . , tq be arbitrarily fixed pairwise distinct points of the interval (0, 1). We associate each point tj with a certain numbermj ∈ N (j = 1, q). Let us introduce the vector space

In this paper we study the b -chromatic number of the cartesian product of two graphs. The b -chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G , such that we obtain a proper coloring and each color i has at least one representative χi adjacent to a vertex of every color j , 1 ≦ j ≠ i ≦ k . In this paper we get ρ( G□H ) ≦ ρ( G )( nH + 1) + Δ( H ) + 1, when the girth of G is assumed to be greater than or equal to 7.

For the Poisson line process the empirical polygon is defined thanks to ergodicity and laws of large numbers for some characteristics, such as the number of edges, the perimeter, the area, etc... One also has, still thanks to the ergodicity of the Poisson line process, a canonical relation between this empirical polygon and the polygon containing a given point. The aim of this paper is to provide numerical simulations for both methods: in a previous paper (Paroux, Advances in Applied Probability, 30:640–656, 1998) one of the authors proved central limit theorems for some geometrical quantities associated with this empirical Poisson polygon, we provide numerical simulations for this phenomenon which generates billions of polygons at a small computational cost. We also give another direct simulation of the polygon containing the origin, which enables us to give further values for empirical moments of some geometrical quantities than the ones that were known or computed in the litterature.

This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u(x, t)) in the quasi-linear parabolic equation u t (x, t) = (k(u(x, t))u x (x, t)) x , with Dirichlet boundary conditions u(0, t) = ψ 0 , u(1,t) = ψ 1 . The main purpose of this paper is to investigate the distinguishability of the input-output mappings Φ[·]: K →C 1 [0, T], Ψ[·]: Κ → C 1 [0, T] via semigroup theory. In this paper, it is shown that if the null space of the semigroup T(t) consists of only zero function, then the input-output mappings Φ[·] and Ψ[·] have the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of these mappings. Moreover, under the light of measured output data (boundary observations) f(t):=k(u(0, t))u x (0, t) or/and h(t):=k (u(1,t))u x (1, t), the values k(ψ 0 ) and k(ψ 1 ) of the unknown diffusion coefficient k(u(x,t)) at (x, t) = (0,0) and (x,t) = (1,0), respectively, can be determined explicitly. In addition to these, the values k u (ψ 0 ) and k u (ψ 1 ) of the unknown coefficient k(u(x, t)) at (x,t)=(0,0) and (x, t) = (1, 0), respectively, are also determined via the input data. Furthermore, it is shown that measured output data f(t) and h(t) can be determined analytically by an integral representation. Hence the input-output mappings Φ[·]:Κ→ C 1 [0, T], Ψ[·]:K→ C 1 [0, T] are given explicitly in terms of the semigroup.

We consider partially hyperbolic maps which are close to the direct product of a hyperbolic map and an identity map and prove that their central leaves depend Holder continuously on the base point in the -metric. We use this result to construct an open set of diffeomorphisms with rather unusual properties (they have transitive sets with periodic points of different indices and orbits with zero Lyapunov exponent). This paper concludes a series of joint papers with Yu. S. Ilyashenko.

This is a paper in mathematics, specifically in set theory. On the example of the measure recognition problem (MRP), the paper highlights the phenomenon of the utility of a multidisciplinary mathematical approach to a single mathematical problem, in particular, the value of a set-theoretic analysis. MRP asks if for a given Boolean algebra, , and a property, Φ , of measures, one can recognize by purely combinatorial means if supports a strictly positive measure with property Φ . The most famous instance of this problem is MRP (countable additivity), and in the first part of the paper, we survey the known results on this and some other problems. We show how these results naturally lead to asking about two other specific instances of the problem MRP, namely MRP (non-atomic) and MRP (separable). Then, we show how our recent work gives an easy solution to the former of these problems and some partial information about the latter. The long-term goal of this line of research is to obtain a structure theory of Boolean algebras that support a finitely additive strictly positive measure, along the lines of Maharam theorem, which gives such a structure theorem for measure algebras.

The principal problem considered in this paper can be formulated as follows: given an L-function satisfying the Riemann Hypothesis or at least a non-trivial density estimate, is it true that it has an Euler product expansion? A positive answer would mean that arithmetic is necessary for proving the Riemann Hypothesis or a non-trivial density estimate, respectively. The paper contains a solution for degree one L-functions from the extended Selberg class. The main tools in the proof are: explicit description of the structure of the extended Selberg class in degree one due to A. Perelli and the first named author (see [7]) and a “hybrid” type joint universality theorem for Dirichlet L-functions. The latter result seems to be of an independent interest, embodying in one statement a special case of the classical Kronecker-Weyl theorem on diophantine approximations and Voronin’s joint universality theorem for Dirichlet L-functions.

In a previous paper, Auslander–Reiten triangles and quivers were introduced into algebraic topology. This paper shows that over a Poincare duality space, each component of the Auslander–Reiten quiver is isomorphic to \(\mathbb{Z}A_{\infty }\).

Andrei Andreevich Markov Jr. (born September 22, 1903, in St. Petersburg; died Oc tober 11, 1979, in Moscow) was the late and only child of the great Russian mathe matician Andrei Andreevich Markov Sr. (born June 14, 1856, in Ryazan7; died July 20, 1922, in Petrograd), universally recognized, in particular, for his contributions to the theory of probability (e.g., Markov chains and Markov processes). At his father's suggestion, young Andrei entered the chemistry section of the School of Physics and Mathematics at the University of Petrograd (formerly St. Petersburg, then Leningrad, and now again St. Petersburg). The young man was fascinated by chemistry and al ready by 1920 had taken part in chemical research. The results of his and his coau thor's work were published in 1924. Thus Markov's first paper dealt with chemistry. In his sophomore year he became interested in theoretical physics, and he graduated in 1924 with a physics degree. Markov's publications in chemistry were followed by a series of papers on the three body problem and dynamical systems (1926-1937), and a paper on Schr?dinger's quantum mechanics. The latter was one of the first papers on quantum mechanics published in the U.S.S.R., appearing less than a year after Schr?dinger's own ground breaking series of publications. In this connection it should be noted that it was Markov who, in 1931, introduced the concept of an abstract (topological) dynamical system. In 1932 Markov published an intriguing paper (in German) on relativity titled "Deriving a World Metric from the Relation 'Earlier Than.' " His interest in abstract mathematics is represented by series of papers on topology, algebra, analysis, and geometry. After World War II Markov's interests turned to axiomatic set theory, mathematical logic, and the foundations of mathematics. He founded the Russian school of con structive mathematics in the late 1940s and early 1950s. But in private conversations Markov often said that he had nurtured constructivist convictions for a very long time, in fact, long before the war. The Moscow mathematical school had been interested in constructivism, especially intuitionism, since its inception in the 1920s. It is enough to mention the 1925 work of Kolmogorov on intuitionistic logic [10]. It may be that this interest was due, at

In 1941 S. N. Bernstein proved the following characterization theorem. Let ξ1 and ξ2 be independent random variables. If ξ1 + ξ2 and ξ1 - ξ2 are independent, then ξ1 and ξ2 are Gaussian with equal dispersion. This paper is devoted to a group analogue of the Bernstein theorem. Let X be a locally compact separable abelian group and ξ1, ξ2 be independent random X-valued variables with distributions μ1, μ2. Denote by $$ \user1{\mathbb{T}} $$ and by ∑ a the group of rotations of the circle and the a-adic solenoid, respectively. In this paper we solve the following problem: In the case where either $$ X = \user1{\mathbb{R}} \times \user1{\mathbb{T}} $$ or X = ∑ a we describe all possible distributions μ1 and μ2 under the assumption that ξ1 + ξ2 and ξ1 - ξ2 are independent.

We develop the theory of multiresolutions in the context of Hausdorff measure of fractional dimension between 0 and 1. While our fractal wavelet theory has points of similarity that it shares with the standard case of Lebesgue measure on the line, there are also sharp contrasts. These are stated in our main result, a dichotomy theorem. The first section is the case of the middle-third Cantor set. This is followed by a review of the essentials on Hausdorff measure. The remaining sections of the paper cover multiresolutions in the general context of affine iterated function systems., We have followed all the suggestions and corrections made by the referee. In the introduction, we added an overview of the three themes in the paper, and added some motivation. This was motivated by the comments and suggestions of the referee. We also rearranged the material as suggested by the referee. This makes the three themes in our paper more clear, we believe

Let X be a random variable with survival function G(x)=ℙ{X≥x}. Let α > 0. Consider a transform, say T α, of survival functions G↦T α G defined by $$(T_\alpha G)(x)\mathop = \limits^{def} \mathop {\inf }\limits_{h:h < x} (x - h)^{ - \alpha } E(X - h)^\alpha \mathbb{I}\{ X \geqslant h\} $$ . In this paper, we examine the properties of the transform as well as provide explicit expressions for T α G in some special cases. Our motivation to study the properties of the transform comes from the theory of inequalities for tail probabilities of sums of independent random variables, martingales, super-martingales, etc. The transform is a commonly used tool in this field, and up to date it has led to most advanced or new-type inequalities. For statistical applications, such as construction of upper confidence bounds, particularly, in problems related to the auditing mathematics, computable and as precise as possible inequalities are required. This is the main motivation to write this paper. We examine general properties of the transform and consider special cases of normal, exponential, Bernoulli, binomial, uniform, and Poisson tails. These types and other survival functions appear in known applications. Until now only the values of α = 1, 2, 3 were actually used, and we concentrate our attention to these three special cases.

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that any graph is close to being the disjoint union of expanders. Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. Three examples of our results are the following: A classical result of Lipton, Rose and Tarjan from 1979 states that if F is a hereditary family of graphs and every graph in F has a vertex separator of size n/(log⁡n)1+o(1), then every graph in F has O(n) edges. We construct a hereditary family of graphs with vertex separators of size n/(log⁡n)1−o(1) such that not all graphs in the family have O(n) edges. Trevisan and Arora-Barak-Steurer have recently shown that given a graph G, one can remove only 1% of its edges to obtain a graph in which each connected component has good expansion properties. We show that in both of these decomposition results, the expansion properties they guarantee are essentially best possible, even when one is allowed to remove 99% of G's edges. Sudakov and the second author have recently shown that every graph with average degree d contains an n-vertex subgraph with average degree at least (1−o(1))d and vertex expansion 1/(log⁡n)1+o(1). We show that one cannot guarantee a better vertex expansion even if allowing the average degree to be O(1). The above results are obtained as corollaries of a new family of graphs which we construct in this paper. These graphs have a super-linear number of edges and nearly logarithmic girth, yet each of their subgraphs has (optimally) poor expansion properties.

Each Cantor measure $\mu$ with scaling factor $\frac{1}{2n}$ has at least one associated orthonormal basis of exponential functions (ONB) for $L^2(\mu)$. In the particular case where the scaling constant for the Cantor measure is $\frac14$ and two specific ONBs are selected for $L^2(\mu_{1/4})$, there is a unitary operator $U$ defined by mapping one ONB to the other. This paper focuses on the case in which one ONB $\Gamma$ is the original Jorgensen-Pedersen ONB for the Cantor measure $\mu_{{1}/{4}}$ and the other ONB is $5\Gamma$. The main theorem of the paper states that the corresponding operator $U$ is ergodic in the sense that only the constant functions are fixed by $U$.

This paper studies series of independent random variables in rearrangement invariant spaces X on [0, 1]. Principal results of the paper concern such series in Orlicz spaces exp(Lp), 1 ~ p ~ c~ and Lorentz spaces A¢. One by-product of our methods is a new (and simpler) proof of a result due to W. B. Johnson and G. Schechtman that the assumption Lp C X, p < oc is sufficient to guarantee that convergence of such series in X (under the side condition that the sum of the measures of the supports of all individual terms does not exceed 1) is equivalent to convergence in X of the series of disjoint copies of individual terms. Furthermore, we prove the converse (in a certain sense) to that result.

Let φ be an analytic self-map of the complex unit disk and X a Banach space. This paper studies the action of composition operator C φ : f → f ˆ φ on the vector-valued Nevanlinna classes N(X) and Na(X). Certain criteria for such operators to be weakly compact are given. As a consequence, this paper shows that the composition operator Cφ is weakly compact on N(X) and Na(X) if and only if it is weakly compact on the vector-valued Hardy space H1(X) and Bergman space B1(X) respectively.

The aim of this paper is to introduce a notion of L-homology in the context of von Neumann algebras. Finding a suitable (co)homology theory for von Neumann algebras has been a dream for several generations (see [KR71a, KR71b, JKR72, SS95] and references therein). One’s hope is to have a powerful invariant to distinguish von Neumann algebras. Unfortunately, little positive is known about the Kadison-Ringrose cohomology H∗ b (M,M), except that it vanishes in many cases. Furthermore, there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von Neumann algebra. Our interest in developing an L-cohomology theory was revived by the introduction of Lcohomology invariants in the field of ergodic equivalence relations in the paper of Gaboriau [Gab02]. His results in particular imply that L-Betti numbers β (2) i (Γ) of a discrete group are the same for measure-equivalent groups (i.e., for groups that can generate isomorphic ergodic measure-preserving equivalence relations). Parallels between the “worlds” of von Neumann algebras and measurable equivalence relations have been noted for a long time (starting with the parallel between the work of Murray and von Neumann [MvN] and that of H. Dye [Dy]). Thus there is hope that an invariant of a group that “survives” measure equivalence will survive also “von Neumann algebra equivalence”, i.e., will be an invariant of the von Neumann algebra of the group. The original motivation for our construction comes from the well understood analogy between the theory of II1-factors and that of discrete groups, based on the theory of correspondences [Con, Con94]. This analogy has been remarkably efficient to transpose analytic properties such as “amenability” or “property T” from the group context to the factor context [Con80] [CJ] and more recently in the breakthrough work of Popa [Popa] [Con03]. We use the theory of correspondences together with the algebraic description of L-Betti numbers given by Luck. His definition involves the computation of the algebraic group homology with coefficients in the group von Neumann algebra. Following the guiding idea that the category of bimodules over a von Neumann algebra is the analogue of the category of modules over a group, we are led to the following algebraic definition of L-homology of a von Neumann algebra M

In the present paper, we prove a fixed-point theorem for completely continuous multivalued mappings defined on a bounded convex closed subset X of the Hilbert space H which satisfies the tangential condition \(F(x) \cap (x + T_X (x)) \ne \emptyset\), where TX(x) is the cone tangent to the set X at a point x. The proof of this theorem is based on the method of single-valued approximations to multivalued mappings. In this paper, we consider a simple approach for constructing single-valued approximations to multivalued mappings. This approach allows us not only to simplify the proofs of already-known theorems, but also to obtain new statements needed to prove the main theorem in this paper.

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic s r a n k ( π ) = O ( π ) − O ( π ′ ) , \begin{equation*} \mathrm {srank}(\pi ) = {\mathcal O}(\pi ) - {\mathcal O}(\pi ’), \end{equation*} where O ( π ) {\mathcal O}(\pi ) denotes the number of odd parts of the partition π \pi and π ′ \pi ’ is the conjugate of π \pi . In a forthcoming paper, Andrews proved the following refinement of Ramanujan’s partition congruence mod 5 5 : p 0 ( 5 n + 4 ) a m p ; ≡ p 2 ( 5 n + 4 ) ≡ 0 ( mod 5 ) , p ( n ) a m p ; = p 0 ( n ) + p 2 ( n ) , \begin{align*} p_0(5n+4) &\equiv p_2(5n+4) \equiv 0 \pmod {5}, p(n) &= p_0(n) + p_2(n), \end{align*} where p i ( n ) p_i(n) ( i = 0 , 2 i=0,2 ) denotes the number of partitions of n n with s r a n k ≡ i ( mod 4 ) \mathrm {srank}\equiv i\pmod {4} and p ( n ) p(n) is the number of unrestricted partitions of n n . Andrews asked for a partition statistic that would divide the partitions enumerated by p i ( 5 n + 4 ) p_i(5n+4) ( i = 0 , 2 i=0,2 ) into five equinumerous classes. In this paper we discuss three such statistics: the ST-crank, the 2 2 -quotient-rank and the 5 5 -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the 2 2 -quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan’s congruence mod 5 5 . This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo 5 5 . Finally, we discuss some new formulas for partitions that are 5 5 -cores and discuss an intriguing relation between 3 3 -cores and the Andrews-Garvan crank.

In this paper I consider a property of sets in the real line such that every non-empty union of finitely many sets with the property does not contain a set with a positive Lebesgue measure. Selectors of the real numbers $$\mathbb R$$ related to any proper dense subgroup of the additive group $$(\mathbb R, +)$$ as well as cosets of any proper dense subgroup of $$(\mathbb R, +)$$ possess this property.

In this paper, we obtain inequalities for trigonometric and algebraic polynomials supplementing and strengthening the classical results going back to papers of S. N. Bernstein and I. I. Privalov. The method of proof is based on the construction of the conformal and univalent mapping from a given trigonometric polynomial and on the application of results of the geometric theory of functions of a complex variable to this mapping.

Let (\Pi,\Sigma) be a Coxeter system. An ordered list of elements in \Sigma and an element in \Pi determine a {\em subword complex}, as introduced in our paper on Gr\"obner geometry of Schubert polynomials (math.AG/0110058). Subword complexes are demonstrated here to be homeomorphic to balls or spheres, and their Hilbert series are shown to reflect combinatorial properties of reduced expressions in Coxeter groups. Two formulae for double Grothendieck polynomials, one of which is due to Fomin and Kirillov, are recovered in the context of simplicial topology for subword complexes. Some open questions related to subword complexes are presented., Comment: 14 pages. Final version, to appear in Advances in Mathematics. This paper was split off from math.AG/0110058v2, whose version 3 is now shorter

In this paper, we study the classification of matrix GI/M/1-type Markov chains with a tree structure. We show that the Perron–Frobenius eigenvalue of a Jacobian matrix provides information for classifying these Markov chains. A fixed-point approach is utilized. A queueing application is presented to show the usefulness of the classification method developed in this paper.

In this paper we consider two-person zero-sum stochastic games with unbounded payoffs and the average reward criterion. State and action spaces are assumed to be Borel spaces. Under conditions which are more general than the ergodicity assumptions in related works [5], [7] and [12], we show that the optimality equation has a solution and that e-optimal stationary strategies exist. Our proofs use a completely different approach compared to the above mentioned papers. We prove that operators of a parametrized class have fixed points, and then we use continuity and monotonicity properties of these fixed points with respect to the class parameter to show that the optimality equation has a solution.

Let (G, ρ, X) be a prehomogeneous vector space with singular set S over an algebraic number field F. The main result of this paper is a proof for the convergence of the zeta fucntions Z(Φ, s) associated with (G, ρ, X) for large Re s under the assumption that S is a hypersurface. This condition is satisfied if G is reductive and (G, ρ, X) is regular. When the connected component of the stabilizer of a generic point x is semisimple and the group Πx of connected components of Gx is abelian, a clear estimate of the domain of convergence is given.Moreover when S is a hypersurface and the Hasse principle holds for G, it is shown that the zeta fucntions are sums of (usually infinite) Euler products, the local components of which are orbital local zeta functions. This result has been proved in a previous paper by the author under the more restrictive condition that (G, ρ, X) is irreducible, regular, and reduced, and the zeta function is absolutely convergent.

The first part of this paper is concerned with the proof of a discrete analog of the Poisson summation formula. In the second part, we describe an elementary proof of a functional equation for the function $$\theta (t)$$ , based on the summation formula derived in the paper.

The paper is concerned with an eectiv e formula for the Euler charac- teristic of the link of a weighted homogeneous mapping F : R n ! R k with an isolated singularity. The formula is based on Szafraniec's method for calculating the Euler charac- teristic of a real algebraic manifold (as the signature of an appropriate bilinear form). It is shown by examples that in the case of a weighted homogeneous mapping it is possible to make the computer calculations of the Euler characteristics much more eectiv e. Let F : R n ! R k , where n > k + 1, be a polynomial mapping and let L =fx2 S n 1 j F (x) = 0g. Let (L) denote the Euler characteristic of L. There is an algebraic formula, due to Szafraniec (7), which expresses (L) in terms of the signature of an appropriate bilinear form (provided that L is smooth). The aim of this paper is to show that in the weighted homogeneous case the general formula for the Euler characteristic of L can be replaced by a simpler one. It makes practical (computer) calculations of the Euler characteristic more eectiv e. Actually, in some cases the general method is ineectiv e because of huge computations. The basic idea is to consider the intersection of F 1 (0) with the hyper- planesfxn = 0g andfxn = 1g. We prove that the Euler characteristic of L is determined by the behavior of F on these spaces, so the formula for (L) becomes simpler (we consider only the case when the manifold L is of even dimension, for otherwise its Euler characteristic is zero). We also consider the simpler invariant (L)=2 (mod 2); we give a simple formula which requires only calculating the dimension of an appropriate algebra (instead of computing a signature). There is a computer program by Andrzej aƒcki to compute the dimen- sions of quotient algebras and signatures of bilinear forms on these algebras

We prove that the symmetric convexified Tsirelson space is of weak cotype 2 but not of cotype 2. Introduction Weak type 2 and weak cotype 2 spaces were originally introduced and investigated by V.D. Milman and G. Pisier in [10] and weak Hilbert spaces by Pisier in [13]. A further detailed investigation can be found in Pisier’s book [14]. The first example of a weak Hilbert space which is not isomorphic to a Hilbert space is the 2-convexified Tsirelson space (called the convexified Tsirelson space in this paper). This follows from the results of W.B. Johnson in [4]. For a detailed study of the original Tsirelson space we refer to [3]. Let X be a Banach space with a symmetric basis. It was proved in [14] that if X is a weak Hilbert space, then it is isomorphic to a Hilbert space and this has lead to the belief that if X is just of weak cotype 2, then it is of cotype 2. However, this turns out not necessarily to be the case. The main result of this paper states that the symmetric convexified Tsirelson space is of weak cotype 2 but not of cotype 2. We now wish to discuss the arrangement of this paper in greater detail. In Section 1 we give some basic facts on properties related to weak type 2 and weak cotype 2 while Section 2 is devoted to a review of some results on the convexified Tsirelson space which we need for our main result. Most of these results are stated without proofs since they can be proved in a similar manner as the corresponding results for the original Tsirelson space. In Section 3 we make the construction of the symmetric convexified Tsirelson space, investigate its basic properties and prove our main result stated above. 02000 Mathematics Subject Classification: 46B03, 46B07. ∗Supported by NSF grants DMS 9706108 and 0102686. †Supported by the Danish Natural Science Research Council, grant 9801867.

In a recent paper on semidefinite linear complementarity problems, Gowda and Song (2000) introduced and studied the P-property, P2-property, GUS-property, and strong monotonicity property for linear transformation L: Sn → Sn, where Sn is the space of all symmetric and real n × n matrices. In an attempt to characterize the P2-property, they raised the following two questions: (i) Does the strong monotonicity imply the P2-property? (ii) Does the GUS-property imply the P2-property? In this paper, we show that the strong monotonicity property implies the P2-property for any linear transformation and describe an equivalence between these two properties for Lyapunov and other transformations. We show by means of an example that the GUS-property need not imply the P2-property, even for Lyapunov transformations.

This paper answers an open question in the design of complimentary metal-oxide semiconductor VLSI circuits. The question asks whether a polynomial-time algorithm can decide if a given planar graph has a plane embedding ${\cal E}$ such that ${\cal E}$ has an Euler trail P = e1 e2 ... em and its dual graph has an Euler trail $P^*=e^*_1 e^*_2 \ldots e^*_m$, where $e^*_i$ is the dual edge of ei for i=1,2,...,m. This paper answers this question in the affirmative by presenting a linear-time algorithm.

In this paper we give a new characterization of the closure of the set of the real parts of the zeros of a particular class of Dirichlet polynomials that is associated with the set of dimensions of fractality of certain fractal strings. We show, for some representative cases of nonlattice Dirichlet polynomials, that the real parts of their zeros are dense in their associated critical intervals, confirming the conjecture and the numerical experiments made by M. Lapidus and M. van Frankenhuysen in several papers. The second author was partially supported by Vicerrectorado de Investigación, Desarrollo e Innovación de la Universidad de Alicante under project GRE11-23.