Absolute stability of a stochastic integro-differential system

A stochastic integro-differential system (1) $\text{x}\text{?}\text{(t, ω) = A(ω) X(t, ω) + ∝}{}_0{}^{\mathrm{<mtext>t</mtext>}}\text{B(t ? s, ω) X(s, ω) ds + b(ω) Φ(σ(t, ω))}\text{subject to}\text{σ(t, ω) = 〈c(ω), X(t, ω)〉, t ? 0}$ is considered where the integrals are interpreted as Bochner integrals. Existence and absolute stability of random solutions of (1) are studied.     

Asymptotic behavior of nonlinear contraction semigroups

The asymptotic behavior of weak solutions of the equation $\text{{u(t) + Au(t) ? ?(t)}$ (A maximal monotone in Hilbert space) is determined via a characterization of ω-limit sets of the contraction semigroup generated by ?A.     

Homogenization of Hamilton‐Jacobi‐Bellman equations with respect to time‐space shifts in a stationary ergodic medium

We consider a family {u_? (t, x, ω)}, ? < 0, of solutions to the equation ?u_?/?t + ?Δu_?/2 + H (t/?, x/?, ?u_?, ω) = 0 with the terminal data u_?(T, x, ω) = U(x). Assuming that the dependence of the Hamiltonian H(t, x, p, ω) on time and space is realized through shifts in a stationary ergodic random medium, and that H is convex in p and satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, of u_?(t, x, ω) as ? → 0 to the solution u(t, x) of a deterministic averaged equation ?u/?t + H?(?u) = 0, u(T, x) = U(x). The “effective” Hamiltonian H? is given by a variational formula. © 2007 Wiley Periodicals, Inc.     

Explicit solution family for the equation of the resistively shunted Josephson junction model

We obtain and study a family of solutions of the equation $\dot \varphi $ + sin ? = B + A cos ωt, which is applicable to several problems in physics, mechanics, and geometry. We use polynomial solutions of double confluent Heun equations associated with this equation to construct the family. We describe the manifold M _P of parameters (A,B, ω) of these solutions and obtain explicit formulas for the rotation number and Poincaré map of the dynamical system on a torus corresponding to this equation with parameters (A,B, ω) ∈ M _P .     

广义时滞Hopfield型神经网络系统的平稳周期振荡

本文利用重合度理论和 V-泛函研究了一类具有周期输入的广义时滞 Hopfield型连续神经网络系统的平稳周期振荡问题 ,得到了其周期解存在、唯一和全局吸引的充分条件 .     

Description of the closed ideals of the algebra λω (n)

If on the semiaxis [0, + ∞) a continuous positive function ω does not decrease and ω(t)/t does not increase, then the space λ⁽ⁿ⁾ _ω(n≥1) has only standard ideals. For n =0 one has to require for this that ω(t)=0(t^α) for some number α, α >0.     

Continuation of separately analytic functions defined on part of the domain boundary

Let D ? ? ⁿ be a domain with smooth boundary ?D, let E??D be a subset of positive Lebesgue measure mes(E) > 0, and let F ? G be a nonpluripolar compact set in a strongly pseudoconvex domain D ? ? ^m . We prove that, under an additional condition, each function separately analytic on the set X = (D × F) ∪ (E × G) has a holomorphic contination to the domain $\rlap{--} X = \{ (z,w) \in D \times G:\omega _{in}^ * (z,E,D) + \omega ^ * (w,F,D) < 1\} $ , where ω* is the P-measure and ω*_in is the interior P-measure.     

Doubly stochastic graph matrices,II

If G is a graph on n vertices, its Laplacian matrix L(G) = D(G) - A(G) is the difference of the diagonal matrix of vertex degrees and the adjacency matrix. The main purpose of this note is to continue the study of the positive definite, doubly stochastic graph matrix (I_n + L(G))?¹= ω(G) = (w_ij). If, for example, w(G) = min w_ij, then w(G)≥0 with equality if and only if G is disconnected and w(G) ≤ l/(n + 1) with equality if and only if G = K_n. If i¦j, then w_ii ≥2wij, with equality if and only if the ith vertex has degree n - 1. In a sense made precise in the note, max w,, identifies most remote vertices of G. Relations between these new graph invariants and the algebraic connectivity emerge naturally from the fact that the second largest eigenvalue of ω(G) is 1/(1 + a(G)).     

Generalized Characteristics of Smoothness and Some Extreme Problems of the Approximation Theory of Functions in the Space L2(ℝ). I

Ukrainian Mathematical Journal - We consider the generalized characteristics of smoothness of the functions ωw(f, t) and Λw(f, t), t &gt; 0, in the space L2(?) and on the...     

含半正非线性项的三阶三点边值问题的正解存在性

研究了半正非线性三阶三点边值问题ω′″(t) - λf(t,w(t)) = 0, 0 ≤ t ≤ 1, ω(0) = ω′(η) = ω"(1) = 0的正解存在性。     

POSITIVE PERIODIC SOLUTIONS OF FIRST AND SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper the existence results of positive ω-periodic solutions are obtained forsecond order ordinary differential equation-u"(t)=f(t,u(t)) (t∈R), and also for firstorder ordinary differential equation u"(f)=f(t,u(t)) (t∈R), where f: R×R~ →Ris a continuous function which is ω-periodic in t. The discussion is based on the fixedpoint index theory in cones.     

t函數的

<正> §1.引言 在1938年華羅庚首先研究零維模函數的富氏係數,特別是求出把一整數n分為不等部分的数目q(n)的表式。這結果,由於抗日戰爭關係直到1942年才發表。華又曾建立正維模向量的理论,閔嗣鶴推廣之並求出正維模向量     

Equivalent theorems on simultaneous approximation by combinations of Gamma operators

In this paper,some equivalent theorems on simultaneous approximation for combinations of Gamma operators by weighted moduli of smoothness ωr(4)λ(f,t)w(4)s(O≤λ≤1)are given.The relation between derivatives of combinations of Gamma operators and smoothness of derivatives of functions is also investigated.     

具有拟周期系数的schrodinger方程的可约化与平衡点的线性稳定性

对Schrodinger方程(A):iut-uxx+c(t)u=0u(t,0)=u(t,2π)=0,u(t,x)=∑∞n=1qn(t)n(x)进行讨论.n(x)是特征方程y″+λy=0y(0)=y(2π)=0中特征值对应的特征函数,c(t)=a+εc1(t),其中a是常数,c1(t)是以ω为频率的拟周期函数.直接判断方程的稳定性十分困难,把方程中的c(t)约化为常数,然后利用约化后的结果来判断方程(A)的平衡点的线性稳定性,方法简单实用.     

積分∫F(x,y,y′,…,y~((n)))dx的等價問题

<正> 命g為x,y平面上由所有保切變換所構成的羣。在本文內我們將定義一類廣義空間使這空間與積分∫F(x,y,y′,…y~((n)))dx對於羣g而言有不變的聯繫。所謂一空間對於羣g而言與積分∫Fdx有不變的聯繫,意義是:如我們施用羣g     

The optimal solution to the problem of complex extrapolation of a first-order scheme

Given the linear stationary first-order iterative scheme $\text{x(}{}^{\mathrm{m+1}}\text{= Tx(}{}^{\mathrm{m}}\text{+ c}$ for the solution of the linear complex system (I ? T)x = c, its extrapolated complex scheme $\text{x(}{}^{\mathrm{m+1}}\text{) = T}{}_{\mathrm{\omega }}\text{x(}{}^{\mathrm{m}}\text{) + ωc [T}{}_{\mathrm{\omega }}\text{≡ (1 ? ω)I + ωT]}$ is considered. The problem which is studied and solved is that of determining an optimum value for ω, over the set of complex numbers, such that the extrapolated scheme considered converges asymptotically as fast as possible.     

一类非线性连续分布时滞系统的周期正解

研究一类非线性周期连续时滞传染病模型ny′i(t)=-αi(t)yi(t)+(ci(t)-yi(t))∑j=1βij(t∫)0-TKj(s)yj(t+s)ds,i=1,2,…,n作者主要讨论了该传染病模型的周期正解的全局存在性,运用重合度延拓理论证明了该模型至少存在一个满足容许值的ω-周期正解.     

GABOR ANALYSIS OF THE SPACES M（p,q,w）（R~d） AND S（p,q,r,w,ω）（R~d）

Let g be a non-zero rapidly decreasing function and w be a weight function. In this article in analog to modulation space, we define the space M(p, q, w)(Rd) to be the subspace of tempered distributions f ∈ S′(Rd) such that the Gabor transform Vg(f) of f is in the weighted Lorentz space L(p, q, wdμ) (R2d). We endow this space with a suitable norm and show that it becomes a Banach space and invariant under time frequence shifts for 1 ≤ p, q ≤∞. We also investigate the embeddings between these spaces and the dual space of M(p, q, w)(Rd). Later we define the space S(p, q, r, w, ω)(Rd) for 1 < p < ∞, 1 ≤ q ≤∞. We endow it with a sum norm and show that it becomes a Banach convolution algebra. We also discuss some properties of S(p, q, r, w, ω)(Rd). At the end of this article, we characterize the multipliers of the spaces M(p, q, w)(Rd) and S(p, q, r, w, ω)(Rd).     

On the Interval Number of a Triangulated Graph

The interval number of a simple undirected graph G, denoted i(G), is the least nonnegative integer r for which we can assign to each vertex in G a collection of at most r intervals on the real line such that two distinct vertices v and w of G are adjacent if and only if some interval for v intersects some interval for w. For triangulated graphs G, we consider the problem of finding a sharp upper bound for the interval number of G in terms of its clique number ω(G). The following partial results are proved. (1) For each triangulated graph G, i(G) ? ?ω(G)/2? + 1, and this is best possible for 2 ? ω(G) ? 6. (2) For each integer m ? 2, there exists a triangulated graph G with ω(G) = m and i(G) > m^1/2.     

Some consequences of MA + ¬wKH

Let wKH denote the following statement: There exists a ω₁-tree of power ω₁ with >ω₁ ω₁-branches. First, using methods of [6] and [13], we shall prove theconsistency of MA + ¬wKH. Then we shall prove that MA + ¬wKH implies the following: (a) There is no θ-dense (splitting) poset of power ω₁; (b) Every LOTS of density ω₁ has a θ-disjoint π-base; (c) There is no Baire LOTS of power ω₁ without isolated points; (d) Every perfectly normal non-Archimedian space of weight ω₁ is metrizable. These results are connected to problems from [4], [10], [7] and [15], respectively. A part of these results was announced in [17].