The Stable Periodic Solutions of a Discrete Neural Network System

In this paper, a nonlinear difference system {xn=βxn-1＋f（yn-κ）,yn=βyn-1＋f（xn-κ）,n∈N is considered a,nd sufficient conditions for the existe~lce of the stable 2κ ＋ 1 periodic solution are obtained.     

An Inertial Proximal Algorithm with Dry Friction: Finite Convergence Results

Let H be a Hilbert space and A, B: H ⇉ H two maximal monotone operators. In this paper, we investigate the properties of the following proximal type algorithm:

Approximate <Emphasis Type="Italic">C</Emphasis>*-ternary ring homomorphisms

In this paper, we establish the generalized Hyers–Ulam–Rassias stability of C*-ternary ring homomorphisms associated to the Trif functional equation

Principles of the theory of two-dimensional infinite systems of algebraic equations

We investigate infinite systems of algebraic equations of the form

Oscillation of Second Order Nonlinear Delay Damped Difference Equations

Some new oscillation criteria for nonlinear delay difference equation with damping △^2xn＋pn△xn＋F（n,x（n-τ,△x（n-σ）=0,n=0,1,2,…,（＊）are given. Our results partially solve the open problem posed in [Math. Bohemica, 125 （2000）, 421- 430]. Also, we will establish some new oscillation criteria for special cases of （＊）, which improve some of the well-known results in the literature.     

On certain polylogarithmic double series

In this paper, we study certain polylogarithmic double series

Small deviations of modified sums of independent random variables

Let S_n = X₁ + · · · + X _n , n ≥ 1, and S ₀ = 0, where X ₁, X ₂, . . . are independent, identically distributed random variables such that the distribution of S _n/B _n converges weakly to a nondeoenerate distribution F _α as n → ∞ for some positive B _n . We study asymptotic behavior of sums of the form

Some generalizations of Knopp's identity*

For integers a, b and n > 0, define

Multiple rational trigonometric sums and multiple integrals

We obtain an estimate of the modulus of a complete multiple rational trigonometric sum: $$\left| {\sum {_{x_{1, \ldots ,} x_r = 1^{\exp \left( {{{2\pi if\left( {x_{1, \ldots ,} x_r } \right)} \mathord{\left/ {\vphantom {{2\pi if\left( {x_{1, \ldots ,} x_r } \right)} q}} \right. \kern-\nulldelimiterspace} q}} \right)} }^q } } \right| \ll q^{{{r - 1} \mathord{\left/ {\vphantom {{r - 1} {n + \varepsilon }}} \right. \kern-\nulldelimiterspace} {n + \varepsilon }}} ,$$ where $$\begin{gathered} f\left( {x_{1, \ldots ,} x_r } \right) = \sum {_{0 \leqslant t_1 , \ldots ,t_r \leqslant n^a t_1 , \ldots ,t_r x_1^{t_1 } \ldots x_r^{t_r } ,} } \hfill \\ a_{0, \ldots ,0} = 0,\left( {a_{0, \ldots ,0,1} , \ldots ,a_{n, \ldots ,n,} q} \right) = 1 \hfill \\ \end{gathered} $$ , and an estimate of the modulus of a multiple trigonometric integral.     

On a periodic boundary value problem for cyclic feedback type linear functional differential systems

Nonimprovable effective sufficient conditions are established for the unique solvability of the periodic problem

On the Relationship Between the Baum-Katz-Spitzer Complete Convergence Theorem and the Law of the Iterated Logarithm

For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which （i） lim sup n→∞ ｜｜Sn｜｜/an〈∞ a.s.and ∞ ∑n=1（1/n）P（｜｜Sn｜｜/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ [0, ∞） are equivalent; （ii） For all constants λ ∈ [0, ∞）, lim sup ｜｜Sn｜｜/an =λ a.s.and ^∞∑ n=1（1/n） P（｜｜Sn｜｜/an ≥ε）{〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.     

On Approximation by Reciprocals of Spherical Harmonics in L p Norm

Let S^1-1,q≥2,be the surface of the unit sphere in the Euclidean space R^1,f（x）∈L^p（S^q-1）,f（x）≥0,f absohutely unegual to 0,1≤p≤＋∞,Then,it is proved in the present paper that there is a spherical harmonics PN（x） of order≤N and a constant C〉0 such that where ω（f,δ）L^p=sup 0〈t≤δ‖St（f）-f‖L^p is a kind of moduli of continuity and ^‖f-1/PN‖L^p≤Cω（f,N^-1）L^p,St（f,μ）=1/｜S^q-2｜Sin^2λt ∫-μμ’=t f（μ＇）dμ＇ is a translation operator.     

Multiple Wick Product Chaos Processes

Let u(x) xR ^q be a symmetric nonnegative definite function which is bounded outside of all neighborhoods of zero but which may have u(0)=. Let p _x, (·) be the density of an R ^q valued canonical normal random variable with mean x and variance and let {G _x, ; (x, )R ^q ×[0,1 ]} be the mean zero Gaussian process with covariance

A Useful Extension of Ito＇s Formula with Applications to Optimal Stopping

Given a continuous semimartingale M = （Mt）t≥〉0 and a d-dimensional continuous process of locally bounded variation V = （V^1,……, V^d）, the multidimensional Ito Formula states that f（Mt, Vt） - f（M0, V0） = ∫[0, t] Dx0f（Ms, Vs）dMs＋∑i=1^d∫[0, t] Dxi F（Ms, Vs）dVs^i＋1/2∫[0, t] Dx0^2 f（Ms, Vs）d 〈M〉s if f（x0,……,xd） is of C^2-type with respect to x0 and of C^1-type with respect to the other arguments This formula is very useful when solving various optimal stopping problems based on Brownian motion. However, in such application the function f typically fails to satisfy the stated conditions in that its first partial derivative with respect to x0 is only absolutely continuous. We prove that the formula remains true for such functions and demonstrate its use with two examples from Mathematical Finance.     

Periodic Solutions for a Class of Forced Liénard-type Equations

Abstract   By applying the topological degree theory, we establish some sufficient conditions for the existence on T-periodic solutions for the Liénard-type equation

Sequential Dynamical Systems Over Words

In this paper we study sequential dynamical systems (SDS) over words. An SDS is a triple consisting of: (a) a graph Y with vertex set {v₁, ..., v_n}, (b) a family of Y-local functions , where K is a finite field and (c) a word w, i.e., a family (w₁, ..., w_k), where w_i is a Y-vertex. A map is called Y-local if and only if it fixes all variables and depends exclusively on the variables , for . An SDS induces an SDS- map, , obtained by the map-composition of the functions according to w. We show that an SDS induces in addition the graph G(w,Y) having vertex set {1, ..., k} where r, s are adjacent if and only if w_s, w_r are adjacent in Y. G(w, Y) is acted upon by S_k via and Fix(w) is the group of G(w, Y) graph automorphisms which fix w. We analyze G(w, Y)-automorphisms via an exact sequence involving the normalizer of Fix(w) in Aut(G(w, Y)), Fix(w) and Aut(Y). Furthermore we introduce an equivalence relation over words and prove a bijection between word equivalence classes and certain orbits of acyclic orientations of G(w, Y). Received September 12, 2004     

Bloch-Bergman Pullbacks with Logarithmic Weights

We characterize the composition operators mapping Blochs boundedly into the weighted Bergman spaces of logarithmic weight. For 0 < p < ∞, 1 < α < ∞, let A^{p, log α} denote the space of holomorphic functions F in the unit disc D for which

Estimate of a multiple sum involving the legendre symbol

Supposef(x₁,..., x_n) is a polynomial of even degree d having coefficients in the finite field k=[q] and satisfying certain natural conditions, and let χ be the quadratic character of k. Then $$\left| {\sum {x_1 , \ldots ,} x_n \in k\chi (f(x_1 , \ldots ,x_n ))} \right| \leqslant Cq^{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} $$ where the constant C depends only on d and n.     

Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.