Local and Global LIPSCHITZian Mappings on Ordered Metric Spaces

We consider the set ?? of nonhomogeneous Markov fields on T = N or T = Z with finite state spaces E_n, n ? T , with fixed local characteristics. For T = N we show that ?? has at most \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop N\nolimits_\infty = \mathop {\lim \inf}\limits_{n \to \infty} \left| {\mathop E\nolimits_n} \right| $\end{document} phases. If T = Z , ?? has at most N_-∞ · N_∞; phases, where \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop N\nolimits_{-\infty} = \mathop {\lim \inf}\limits_{n \to -\infty} \left| {\mathop E\nolimits_n} \right| $\end{document}. We give examples, that for T = N for any number k, 1 ≦ k ≦ N_∞, there are local characteristics with k phases, whereas for T = Z every number l · k, 1 ≦ l ≦ N_-∞, 1 ≦ k ≦ N_∞ occurs. We describe the inner structure of ??, the behaviour at infinity and the connection between the one-sided and the two-sided tail-fields. Simple examples of Markov fields which are no Markov processes are given.     

Unions and Intersections of Families of Lp-Spaces

Let L^Ψ and E^Ψ be the ORLICZ space and the space of finite elements respectively, on a measure space (Ω, Σ, μ), and let T ? (0, ∞). It is proved that if inf {p: p ? T} ? T or sup {p: p ? T} ? T and μ is an infinite atomless measure, then there is no ORLICZ function Ψ such that: \documentclass{article}\pagestyle{empty}\begin{document}$ L^\varphi = Lin\mathop { \cup L^p }\limits_{p\varepsilon T} $\end{document} or \documentclass{article}\pagestyle{empty}\begin{document}$ E^\varphi = Lin\mathop { \cup L^p }\limits_{p\varepsilon T} $\end{document} and moreover, there is no ORLICZ function Ψ such that: \documentclass{article}\pagestyle{empty}\begin{document}$ L^\varphi = Lin\mathop { \cap L^p }\limits_{p\varepsilon T} $\end{document} or \documentclass{article}\pagestyle{empty}\begin{document}$ E^\varphi = Lin\mathop { \cap L^p }\limits_{p\varepsilon T} $\end{document}.     

The phase transition in the evolution of random digraphs

Let $ \mathop {\rm D}\limits^ \to $(n, M) denote a digraph chosen at random from the family of all digraphs on n vertices with M arcs. We shall prove that if M/n ≤ c < 1 and ω(n) → ∞, then with probability tending to 1 as n → ∞ all components of $ \mathop {\rm D}\limits^ \to $(n, M) are smaller than ω(n), whereas when M/n ≥ c > 1 the largest component of $ \mathop {\rm D}\limits^ \to $(n, M) is of the order n with probability 1 - o(1).     

Projective limits of finite decomposition systems

Special finite topological decomposition systems were used to get compactifications of topological spaces in [6]. In this paper the notion of finite decomposition systems is applied for topological measure spaces. We get two canonical topological measure spaces X_∞ and X_∞^d being projective limits of (discrete) finite decomposition systems for each topological measure space X = (X, O, A, P) and each net (A_α) α ? I of upward filtering finite σ-algebras in A. X_∞ is a compact topological measure space and the idea to construct is the same as used in [6]. The compactifications of [6] are cases of some special X_∞. Further on we obtain that each measurable set of the remainder of X_∞ has measure zero with respect to the limit measure P_∞ (Theorem 1). X_∞^d is the STONE representation space X(\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document}) of \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document} A_α, hence a Boolean measure space with regular Borel measure. Some measure theoretical and topological relations between X, X(\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document}) and x(A) where x(A) is the Stone representation space of A, are given in Theorem 2. and 4. As a corollary from Theorem 2. we get a measure theoretical-topological version to the Theorem of Alexandroff Hausdorff for compact T₂ measure spaces x with regular Borel measure (Theorem 3.).     

A Note on Bounds for the Supremum Metric for Discrete Random Variables

For positive or negative integer-valued random variables X and Y with finite second moments the inequality sup \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop {\sup }\limits_n |\Pr \{ X \le n\} - \Pr \{ Y \le n\} |\, \le \,|EX - EY| + \frac{1}{2}(EX(EX - 1) + (EY(Y - 1)) $\end{document} is established by elementary manipulation, and shown to be tight. Use of generating functions and an inversion formula yields the larger bound with 1/2 replaced by 2/π.     

The PENROSE Transform for Bundles Non-Trivial on the General Line

Let L₀ be a fixed projective line in CP ³ and let M ? C ⁴ be the complexified MINKOWSKI space interpreted as the manifold of all projective lines L ? CP ³ with L ∩ L ₀ ?? Ø. Let D ? M , D ′ ? CP ³/ L ₀ be open sets such that \documentclass{article}\pagestyle{empty}\begin{document}$ D' = \mathop \cup \limits_{L \in D} $\end{document}. Under certain topological conditions on D, R. S. WARD'S PENROSE transform sets up an 1–1 correspondence between holomorphic vector bundles over D ′ trivial over each L ? D and holomorphic connections with anti-self-dual curvature over D (anti-self-dual YANG-MILLIS fields). In the present paper WARD'S construction is generalized to holomorphic vector bundles E over D′ satisfying the condition that \documentclass{article}\pagestyle{empty}\begin{document}$ E|_L \cong E|_{\tilde L} $\end{document} for all \documentclass{article}\pagestyle{empty}\begin{document}$ L,\tilde L \in D $\end{document}.     

Nullstellensatz effectif et Conjecture de Serre (Théorème de Quillen-Suslin) pour le Calcul Formel

Let k be an arbitrary field, X₁,….,X_n indeterminates over k and F₁…, F₃ ε ∈ k[X₁…,X_n] polynomials of maximal degree $ d: = \mathop {\max }\limits_{1 \le i \le a} \deg $ (F_i). We give an elementary proof of the following effective Nullstellensatz: Assume that F₁,…,F have no common zero in the algebraic closure of k. Then there exist polynomials P₁…, P₃ ε ∈ k[X₁…,X_n] such that $ 1: = \mathop \Sigma \limits_{1 \le i \le a} $ P_iF_i and This result has many applications in Computer Algebra. To exemplify this, we give an effective quantitative and algorithmic version of the Quillen-Suslin Theorem baaed on our effective Nullstellensatz.     

Inversion of the Radon transform with incomplete data

The Radon transform R(p, θ), θ∈S^n?1, p∈?¹, of a compactly supported function f(x) with support in a ball B_a of radius a centred at the origin is given for all $ \theta \in \mathop {S^{n - 1} }\limits^\tilde $, where $ \mathop {S^{n - 1} }\limits^\tilde $ is an open set on S^n?1, and all p∈(? ∞, ∞), n≥2. An approximate formula is given to calculate f(x) from the given data.     

Exponents for three-dimensional simultaneous Diophantine approximations

Let Θ = (θ ₁,θ ₂,θ ₃) ∈ ℝ³. Suppose that 1, θ ₁, θ ₂, θ ₃ are linearly independent over ℤ. For Diophantine exponents

Compactness Conditions for the Green Operator in Lp(ℝ) Corresponding to a General Sturm–Liouville Operator

We consider the equation ℝ, where , for ℝ, (ℝ), (ℝ), (ℝ), (ℝ) := C(ℝ)). We give necessary and sufficient conditions under which, regardless of , the following statements hold simultaneously: I) For any (ℝ) Equation (0.1) has a unique solution (ℝ) where $\int ^{\infty}_{-\infty}$ ℝ. II) The operator (ℝ) → (ℝ) is compact. Here is the Green function corresponding to (0.1). This result is applied to study some properties of the spectrum of the Sturm–Liouville operator.     

What is the smallest number of dicycles in a dicycle decomposition of an eulerian digraph?

A digraph D with n vertices is said to be decomposable into a set S of dicycles if every arc of D is contained in exactly one member of S. Counterexamples are given to the following conjectures which are generalizations of three well-known conjectures by G. Hajós, P. Erd?s, and P.J. Kelly: (1) [B. Jackson] Every eulerian-oriented graph is decomposable into at most \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{n}{2} $\end{document} dicycles. (2) [W. Bienia & H. Meyniel] Every eulerian digraph is decomposable into at most n dicycles. Certain observations lead us to make three other conjectures: (a) Every eulerian-oriented graph is decomposable into at most \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{2n}}{3} $\end{document} dicycles. (b) Every symmetric digraph with n > 1 is decomposable into at most 2n – 3 dicycles. (c) Every eulerian digraph with n > 1 is decomposable into at most \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{8n}}{3} $\end{document} – 3 dicycles.     

A uniqueness condition for the polyharmonic equation in free space

Consider the polyharmonic wave equation ?u + (? Δ)^mu = f in ?ⁿ × (0, ∞) with time-independent right-hand side. We study the asymptotic behaviour of u ( x , t) as t → ∞ and show that u( x , t) either converges or increases with order t^α or In t as t → ∞. In the first case we study the limit $ u_0 \left({\bf x} \right) \colone \mathop {\lim }\limits_{t \to \infty } \,u\left({{\bf x},t} \right) $ and give a uniqueness condition that characterizes u₀ among the solutions of the polyharmonic equation ( ? Δ)^mu = f in ?ⁿ. Furthermore we prove in the case 2m ? n that the polyharmonic equation has a solution satisfying the uniqueness condition if and only if f is orthogonal to certain solutions of the homogeneous polyharmonic equation.     

A Mean-Value Theorem for Multiplicative Functions on the Set of Shifted Primes

Let f be a complex-valued multiplicative function, letp denote a prime and let π(x) be the number of primes not exceeding x. Further put $$m_p (f): = \mathop {\lim }\limits_{x \to \infty } \frac{1}{{\pi (x)}}\sum\limits_{p \leqslant x} {f(p + 1)} {\text{, }}M(f): = \mathop {\lim }\limits_{x \to \infty } \frac{1}{x}\sum\limits_{n \leqslant x} {f(n)}$$ and suppose that $$\mathop {\lim \sup }\limits_{x \to \infty } \frac{1}{x}\sum\limits_{n \leqslant x} {\left| {f\left( n \right)} \right|^2 } < \infty ,\sum\limits_{p \leqslant x} {\left| {f\left( n \right)} \right|^2 } \ll x\left( {\ln x} \right)^{ - \varrho } ,$$ with some \varrho > 0. For such functions we prove: If there is a Dirichlet character χ_d such that the mean-value M(f χ_d) exists and is different from zero,then the mean-value m_p(f) exists. If the mean-value M(f) exists, then the same is true for the mean-valuem_p(f) .     

Weighted Inequalities for Monotone Functions

Weighted norm inequalities are investigated by giving an extension of the Riesz convexity theorem to semi-linear operators on monotone functions. Several properties of the classes B(p, n) and C(p, n) introduced by Neugebauer in [13] are given. In particular, we characterize the weight pairs w, v for which $ \int\limits_0^\infty {f(x)^p w(x)dx \le C \int\limits_0^\infty {({\textstyle{1 \over x}}\int\limits_0^x f)^p } v(x) dx,} $ for nondecreasing functions f and 1 ≦ p < ∞.     

Uniformly maldistributed sequences in a strict sense

It is shown that the following three limits

A conjecture of Erdős and Reddy and the rational approximation of Mittag-Leffler type functions

Пусть \(f(z) = \mathop \sum \limits_{k = 0}^\infty a_k z^k ,a_0 \ne 0, a_k \geqq 0 (k \geqq 0)\) — целая функци я,π _n — класс обыкновен ных алгебраических мног очленов степени не вы ше \(n,a \lambda _n (f) = \mathop {\inf }\limits_{p \in \pi _n } \mathop {\sup }\limits_{x \geqq 0} |1/f(x) - 1/p(x)|\) . П. Эрдеш и А. Редди высказали пр едположение, что еслиf(z) имеет порядок ?ε(0, ∞) и $$\mathop {\lim sup}\limits_{n \to \infty } \lambda _n^{1/n} (f)< 1, TO \mathop {\lim inf}\limits_{n \to \infty } \lambda _n^{1/n} (f) > 0$$ В данной статье показ ано, что для целой функ ции $$E_\omega (z) = \mathop \sum \limits_{n = 0}^\infty \frac{{z^n }}{{\Gamma (1 + n\omega (n))}}$$ , где выполняется $$\lambda _n^{1/n} (E_\omega ) \leqq \exp \left\{ { - \frac{{\omega (n)}}{{e + 1}}} \right\}$$ , т.е. $$\mathop {\lim sup}\limits_{n \to \infty } \lambda _n^{1/n} (E_\omega ) \leqq \exp \left\{ { - \frac{1}{{\rho (e + 1)}}} \right\}< 1, a \mathop {\lim inf}\limits_{n \to \infty } \lambda _n^{1/n} (E_\omega ) = 0$$ . ФункцияE _ω (z) имеет порядок ?.     

On the strong summation of Fourier series with variable exponents

Пустьf 2π-периодическ ая суммируемая функц ия, as _k (x) еë сумма Фурье порядк аk. В связи с известным ре зультатом Зигмунда о сильной суммируемости мы уст анавливаем, что если λn→∞, то сущес твует такая функцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _{2n} } } \right\}^{1/\lambda _{2n} } = \infty .$$ Отсюда, в частности, вы текает, что если λn?∞, т о существует такая фун кцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } } \right\}^{1/\lambda _n } = \infty .$$ Пусть, далее, ω-модуль н епрерывности и $$H^\omega = \{ f:\parallel f(x + h) - f(x)\parallel _c \leqq K_f \omega (h)\} .$$ . Мы доказываем, что есл и λ _n ?∞, то необходимым и достаточным условие м для того, чтобы для всехf∈H ^ω выполнялос ь соотношение $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _n } } \right\}^{1/\lambda _n } = 0(x \in [0;2\pi ])$$ является условие $$\omega \left( {\frac{1}{n}} \right) = o\left( {\frac{1}{{\log n}} + \frac{1}{{\lambda _n }}} \right).$$ Это же условие необхо димо и достаточно для того, чтобы выполнялось соотнош ение $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } = 0(f \in H^\omega ,x \in [0;2\pi ]).$$      

Limit theorems in approximation theory

В РАБОтЕ ОпРЕДЕлЕНы У слОВИь, ОБЕспЕЧИВАУЩ ИЕ ВыпОлНЕНИЕ сООтНОшЕ НИИ ВИДА $$\begin{array}{*{20}c} {\mathop {\lim }\limits_{n \to \infty } E(h_n ,B_n ,F_n ) = E(f,B,F),} \\ {\mathop {\lim }\limits_{n \to \infty } \mathop {\sup }\limits_{h_n \in K_n } E(h_n ,B_n ,F_n ) = \mathop {\sup }\limits_{f \in K} E(f,B,F),} \\ \end{array}$$ гДЕE(g, H, M) — НАИлУЧшЕЕ пРИ БлИжЕНИЕ ЁлЕМЕНтАg ЁлЕМЕНтАМИ ИжH В МЕтР ИкЕ НОРМИРОВАННОгО пРОстРАНстВАM, K _n ,K — НЕк ОтОРыЕ МНОжЕстВА ЁлЕ МЕНтОВ. с пОМОЩьУ ЁтИх РАВЕНс тВ пОлУЧЕНы ВсЕ ИжВЕс тНыЕ пРЕДЕльНыЕ тЕОРЕМы Д ль НАИлУЧшИх пРИБлИжЕНИИ ФУНкцИИ АлгЕБРАИЧЕскИМИ И тРИгОНОМЕтРИЧЕскИМ И МНОгОЧлЕНАМИ И сплА ИНАМИ, А тАкжЕ ДОкАжАН РьД НО Вых РЕжУльтАтОВ. В ЧАс тНОстИ, пОлУЧЕНы пРЕДЕльНыЕ сООтНОшЕНИь МЕжДУ тОЧНыМИ ВЕРхНИ МИ гРАНьМИ НАИлУЧшИх пРИБлИжЕНИИ НА МНОгОМЕРНых клАсс Ах гЕльДЕРА.     

A Self‐Dual Polar Factorization for Vector Fields

We show that any nondegenerate vector field u in \begin{align*}L^{\infty}(\Omega, \mathbb{R}^N)\end{align*}, where Ω is a bounded domain in \begin{align*}\mathbb{R}^N\end{align*}, can be written as \begin{align*}u(x)= \nabla_1 H(S(x), x)\quad {\text for a.e.\ x \in \Omega}\end{align*}}, where S is a measure‐preserving point transformation on Ω such that \begin{align*}S^2=I\end{align*} a.e. (an involution), and \begin{align*}H: \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}\end{align*} is a globally Lipschitz antisymmetric convex‐concave Hamiltonian. Moreover, u is a monotone map if and only if S can be taken to be the identity, which suggests that our result is a self‐dual version of Brenier's polar decomposition for the vector field as \begin{align*}u(x)=\nabla \phi (S(x))\end{align*}, where ? is convex and S is a measure‐preserving transformation. We also describe how our polar decomposition can be reformulated as a (self‐dual) mass transport problem. © 2012 Wiley Periodicals, Inc.     

Uniform Approximation of Nonperiodic Functions Defined on the Entire Axis

Using the following notation: C is the space of continuous bounded functions f equipped with the norm , V is the set of functions f such that , the set E consists of fCV and possesses the following property: