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1.
We consider the set ?? of nonhomogeneous Markov fields on T = N or T = Z with finite state spaces En, 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 ≦ kN, there are local characteristics with k phases, whereas for T = Z every number l · k, 1 ≦ lN-∞, 1 ≦ kN 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.  相似文献   

2.
Let LΨ and EΨ 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}.  相似文献   

3.
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/nc < 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/nc > 1 the largest component of $ \mathop {\rm D}\limits^ \to $(n, M) is of the order n with probability 1 - o(1).  相似文献   

4.
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 Xd being projective limits of (discrete) finite decomposition systems for each topological measure space X = (X, O, A, P) and each net (Aα) 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). Xd 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 $\end{document} Aα, hence a Boolean measure space with regular Borel measure. Some measure theoretical and topological relations between X, X(\documentclass{article}\pagestyle{empty}\begin{document}$ \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.).  相似文献   

5.
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.  相似文献   

6.
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 LL 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}.  相似文献   

7.
Let k be an arbitrary field, X1,….,Xn indeterminates over k and F1…, F3 ε ∈ k[X1…,Xn] polynomials of maximal degree $ d: = \mathop {\max }\limits_{1 \le i \le a} \deg $ (Fi). 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…,Xn] such that $ 1: = \mathop \Sigma \limits_{1 \le i \le a} $ PiFi 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.  相似文献   

8.
The Radon transform R(p, θ), θ∈Sn?1, p∈?1, of a compactly supported function f(x) with support in a ball Ba 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 Sn?1, and all p∈(? ∞, ∞), n≥2. An approximate formula is given to calculate f(x) from the given data.  相似文献   

9.
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
$\begin{gathered} \alpha (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\sup }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\}, \hfill \\ \beta (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\inf }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\} \hfill \\ \end{gathered}$\begin{gathered} \alpha (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\sup }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\}, \hfill \\ \beta (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\inf }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\} \hfill \\ \end{gathered}  相似文献   

10.
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.  相似文献   

11.
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.  相似文献   

12.
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 tα 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.  相似文献   

13.
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) .  相似文献   

14.
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 < ∞.  相似文献   

15.
It is shown that the following three limits
  相似文献   

16.
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 ?.  相似文献   

17.
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 всехfH 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 $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } = 0(f \in H^\omega ,x \in [0;2\pi ]).$$   相似文献   

18.
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.  相似文献   

19.
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.  相似文献   

20.
Uniform Approximation of Nonperiodic Functions Defined on the Entire Axis   总被引:1,自引:1,他引:0  
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:
is summable on each finite interval, we establish some assertions similar to the following theorem: Let 0$$ " align="middle" border="0"> ,
Then for fV the series
uniformly converges with respect to and the following equality holds:
This theorem develops some results obtained by Zubov relative to the approximation of probability distributions. Bibliography: 4 titles.  相似文献   

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