Convergence of Solutions to Equations Arising in Neural Networks

We study polynomial ordinary differential systems whereQ0 is an n×n matrix and M(t) is an n×k matrix. It is proven that, as t grows to infinity, the solution M(t) tends to a limit BU, where U is a k×k orthogonal matrix and B is an n×k matrix whose columns are k pairwise orthogonal, normalized eigenvectors of Q. Moreover, for almost every M ₀, these eigenvectors correspond to the k maximal eigenvalues of Q; for an arbitrary Q with independent columns, we provide a procedure of computing B by employing elementary matrix operations on M ₀. This result is significant for the study of certain neural network systems, and in this context it shows that M() provides a principal component analyzer.     

具无限时滞泛函微分方程解的收敛性

本文分别就紧性条件和耗散型条件给出了具无限时滞泛函微分方程的解的收敛性定理，推广了现有文献中的某些结果．     

Speed of Convergence of Chernoff Approximations to Solutions of Evolution Equations

Mathematical Notes -     

具有马尔可夫调制的随机微分方程数值解的收敛性

本文在无穷维Hilbert空间中研究了一类具有马尔可夫调制的随机微分方程(SDEwMSs).在一般情况下SDEwMSs没有解析解.因此合适的数值逼近法,例如欧拉法,就是在研究它们性质时所采用的重要工具.本文在较弱的条件下不仅证明了欧拉近似解收敛于SDEwMSs的精确解(分析解),而且给出了欧拉近似阶的界.     

带有Poisson随机测度的比例微分方程数值解的收敛性

主要研究了带跳的随机比例微分方程dX(t)=f((X(t),X(qt))dt+g(X(t),X(qt))dW(t)+∫nh(X(t),X(qt),u)N(dt,du),0≤t≤T,X(0)=X0,给出了此方程的Euler数值解,并在局部Lipschitzs条件下,证明了数值解依均方和概率测度意义下收敛于精确解.     

On the Convergence of Formal Solutions of a System of Partial Differential Equations

We study a version of the classical problem on the convergence of formal solutions of systems of partial differential equations. A necessary and sufficient condition for the convergence of a given formal solution (found by any method) is proved. This convergence criterion applies to systems of partial differential equations (possibly, nonlinear) solved for the highest-order derivatives or, which is most important, “almost solved for the highest-order derivatives.”     

两参数Ito型随机微分方程解的收敛定理

设ωｚ是Ｒ＾２＋上的布朗单,考虑两参数Ｉｔｏ型随机微分方程：ｄｘｚ＝ａ（ｚ,ｘｚ）ｄωｚ＋ｂ（ｚ,ｘｚ）ｄｚ（１）ｄｘ＾＊ｚ＝ａｚ（ｚ,ｘ＾＊ｚ）ｄωｚ＋ｂｚ（ｚ,ｘ＾＊ｚ）ｄｚ（２）则在方程系数满足一定条件下,本证明了方程（２）的解向方程（１）的解收敛。     

Limit Theorems for the Solutions of Linear Boundary-Value Problems for Systems of Differential Equations

Ukrainian Mathematical Journal - We establish sufficient conditions for the convergence of a sequence of solutions of general boundary value problems for systems of linear ordinary differential...     

带跳的时滞随机微分方程近似解的收敛性

本文研究了一类具有Possion跳的时滞随机微分方程（SDDEwJPs）．在一般情况下SDDEwJPs没有解析解．因此合适的数值逼近法，例如欧拉法，就是在研究它们性质时所采用的重要工具．本文在局部李普希兹条件下证明了欧拉近似解强收敛于SDDEwJPs的精确解（分析解）．     

On Asymptotic Stability and Sweeping of Collisionless Kinetic Equations

We study a collisionless kinetic equation describing density distribution function of the position and velocity of particles moving in a slab with finite thickness and with a partly diffusive boundary reflection. In particular, we deal with existence of an invariant density and with the convergence to the equilibrium. We also study the long time behavior of densities when the equilibrium does not exists.     

On Convergence of One-Step Schemes for Weak Solutions of Quantum Stochastic Differential Equations

Several one-step schemes for computing weak solutions of Lipschitzian quantum stochastic differential equations (QSDE) driven by certain operator-valued stochastic processes associated with creation, annihilation and gauge operators of quantum field theory are introduced and studied. This is accomplished within the framework of the Hudson–Parthasarathy formulation of quantum stochastic calculus and subject to the matrix elements of solution being sufficiently differentiable. Results concerning convergence of these schemes in the topology of the locally convex space of solution are presented. It is shown that the Euler–Maruyama scheme,with respect to weak convergence criteria for Itô stochastic differential equation is a special case of Euler schemes in this framework. Numerical examples are given.     

On the Convergence of a Trust-Region Method for Solving Constrained Nonlinear Equations with Degenerate Solutions

This paper presents a trust-region method for solving the constrained nonlinear equation F(x) = 0, x , where R ⁿ is a nonempty and closed convex set, F is defined on the open set containing and is continuously differentiable. The iterates generated by the method are feasible. The method is globally and quadratically convergent under local error bounded assumption on F. The results obtained are extensions of the work of Yamashita Fukushima (Ref. 1) and Fan Yuan (Ref. 2) for unconstrained nonlinear equations. Numerical results show that the new algorithm works quite well.     

一类非单调二元算子方程解的Mann迭代序列的收敛性

利用Mann迭代技巧,讨论了不具有连续性和紧性条件的非单调二元算子方程解的存在唯一性,并给出了迭代序列收敛于解的误差估计,所得结果是某些已知结果本质改进和推广.     

Fourier-Integral-Operator Approximation of Solutions to First-Order Hyperbolic Pseudodifferential Equations I: Convergence in Sobolev Spaces

An approximation Ansatz for the operator solution, U(z′,z), of a hyperbolic first-order pseudodifferential equation, ? _z  + a(z,x,D _x ) with Re (a) ≥ 0, is constructed as the composition of global Fourier integral operators with complex phases. An estimate of the operator norm in L(H ^(s),H ^(s)) of these operators is provided, which yields a convergence result for the Ansatz to U(z′,z) in some Sobolev space as the number of operators in the composition goes to ∞.     

Convergence of Weighted Polynomial Approximations to Solutions of Partial Differential Equations with Quasianalytical Coefficients

Sequences of polynomial functions which converge to solutions of partial differential equations with quasianalytical coefficients are constructed. Estimates of the degree of convergence are also given.     

The Infinite-Genus Limit of the Whitham Equations

We derive the infinite-genus limit of the KdV–Whitham equations based on the special scaling of the spectral curve introduced by Venakides in the study of the continuum limit of theta functions. The limit describes evolution of the integrated density of states in a one-dimensional soliton gas.     

Limit Sets of Free Convergence Groups

We study the limit sets of convergence groups and prove that the limit set of a free discontinuous cocompact convergence group with an invariant component of the discontinuity domain is a discontinuum.