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.     

Convergence of Solutions of General Dispersive Equations Along Curve

In this paper, the authors give the local L~2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R~+→R satisfies some suitable conditions and φ((-?)~(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L~2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.     

Exponential Convergence to Time-Periodic Viscosity Solutions in Time-Periodic Hamilton-Jacobi Equations

Consider the Cauchy problem of a time-periodic Hamilton-Jacobi equation on a closed manifold, where the Hamiltonian satisfies the condition: The Aubry set of the corresponding Hamiltonian system consists of one hyperbolic 1-periodic orbit. It is proved that the unique viscosity solution of Cauchy problem converges exponentially fast to a 1-periodic viscosity solution of the Hamilton-Jacobi equation as the time tends to infinity.     

含极限次临界增长项p-Laplace方程的无穷多解

讨论了有界光滑区域上一类p-Laplace方程,非线性项具奇对称性且在无穷远为极限次临界增长．证明了变分泛函在大范围内满足推广的Palais-Smale条件,构造了变分泛函的一列临界值,进而得到了无穷多个弱解的存在性,对应泛函的能量趋于正无穷．所得到的结果推广了次临界增长的情形．     

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 -     

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.”     

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.     

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.     

二阶非线性微分方程组的正规解及极限边值问题

本文对微分方程组其中a(t)>0,r(t)>0,t≥t₀;f(x)对x>0正值递减;g(y)>0,给出存在正规解,有界正规解或两类无穷区间[c,∞)(c≥t₀)上边值问题解的充要条件.同时给出若干例子说明结果的条件.     

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 ∞.     

Low Dimensional Born-Infeld Equations Coupled with a Collisionless Matter Model

We consider the Born-Infeld nonlinear electromagnetic field equations and study its Cauchy problem in the case that the Vlasov equation is considered as a matter model. In the present paper, the Vlasov equation is considered on the so-called one and one-half dimensional phase space, and in consequence the Born-Infeld equations are reduced to a quasilinear hyperbolic system with two unknowns. A transformation is introduced in order to make the field equations easy to handle, and suitable assumptions are made on initial data so that the nonlinearity of the field is controlled.