Correlation functions of gauged linear σ-model

This is the second paper in a series following Tian and Xu(2015), on the construction of a mathematical theory of the gauged linear σ-model(GLSM). In this paper, assuming the existence of virtual moduli cycles and their certain properties, we define the correlation function of GLSM for a fixed smooth rigidified r-spin curve.     

On deformations of linear systems of differential equations and the Painlevé property

We consider systems of linear differential equations discussing some classical and modern results in the Riemann problem, isomonodromic deformations, and other related topics. Against this background, we illustrate the relations between such phenomena as the integrability, the isomonodromy, and the Painlevé property. The recent advances in the theory of isomonodromic deformations presented show perfect agreement with that approach.     

On solving systems of linear algebraic equations by Gibbs’s method

The paper presents an algorithm of approximate solution of a system of linear algebraic equations by the Monte Carlo method superimposed with ideas of simulating Gibbs and Metropolis fields. A solution in the form of a Neumann series is evaluated, the whole vector of solutions is obtained. The dimension of a system may be quite large. Formulas for evaluating the covariance matrix of a single simulation run are given. The method of solution is conceptually linked to the method put forward in a 2009 paper by Ermakov and Rukavishnikova. Examples of 3 × 3 and 100 × 100 systems are considered to compare the accuracy of approximation for the method proposed, for Ermakov and Rukavishnikova’s method and for the classical Monte Carlo method, which consists in consecutive estimation of the components of an unknown vector.     

Some properties of Malgrange isomonodromic deformations of linear 2 × 2 systems

We study movable singularities of the Malgrange isomonodromic deformation of a linear differential 2 × 2 system with two irregular singularities of Poincaré rank 1 and with an arbitrary number of Fuchsian singular points.     

On exact convergence rates for solutions of linear systems of Volterra difference equations†

The asymptotic behaviour of the solution of general linear Volterra non-convolution difference equations on a finite dimensional space, is investigated. It is proved under appropriate assumptions that the solution converges to a limit, which is in general non-trivial. These results are then used to obtain the exact rate of decay of solutions of a class of convolution Volterra difference equations, which have no characteristic roots. In particular, we obtain the exact rate of convergence of the solution of equations whose kernel does not converge exponentially. A useful formula for the weighted limit of a discrete convolution is also obtained.     

Ont ∞ quasi-similarity of linear systems

Summary Conti defined t similarity of systems (1)x=tA(t)xand (2)y=B(t)y, and showed that it is an equivalence relation which preserves uniform and strict stability. Here the definition is weakened by imposing less stringent integrability conditions, some in terms of perhaps conditionally convergent improper integrals, on the matrix function relating A and B. The extended relation, t quasi-similarity, is not symmetric or transitive; however, it is shown that if (2) is t quasi-similar to (1)and (1)is uniformly, uniformly asymptotically, or strictly stable, then so is (2).Results are also given concerning linear asymptotic equilibrium of (2)in the case where (1)is stricly stable or has linear asymptotic equilibrium.     

Existence of complex ℓ2 solutions of linear delay systems of difference equations

We give sufficient conditions for the existence of complex ?₂ solutions of a non-homogeneous system of linear difference equations and of two general classes of delay systems of linear difference equations. In some cases, bounds of the established solutions are also given. As a consequence of the space ?₂ where we work, information can be obtained about the asymptotic behavior of the established solutions and, the asymptotic stability of the zero equilibrium point of the systems under consideration. The method we use is a functional-analytic one.     

Perturbation of systems of linear algebraic equations ∗

A classical perturbation result for nonsingular systems of linear algebraic equations is extended to general consistent systems under any norm. An optimal perturbation result is also obtained for general linear least squares problems under a Euclidean norm.     

On the number of limit cycles in general planar piecewise linear systems of node–node types

We investigate the existence and number of limit cycles in a class of general planar piecewise linear systems constituted by two linear subsystems with node–node dynamics. Using the Liénard-like canonical form with seven parameters, some sufficient and necessary conditions for the existence of limit cycles are given by studying the fixed points of proper Poincaré maps. In particular, we prove the existence of at least two nested limit cycles and describe some parameter regions where two limit cycles exist. The main results are applied to the PWL Morris–Lecar neural model to determine the existence and stability of the limit cycles.     

Asymptotic equivalence of linear stochastic Itô systems and oscillation of solutions of linear second-order equations

We obtain conditions for the asymptotic equivalence of linear stochastic and deterministic systems and analyze the oscillation of solutions of the Itô stochastic equation of the second order of the form \(\ddot x + (p(t) + q(t)\dot W(t))x = 0\) on the half-line.     

On Bayes linear unbiased estimation of estimable functions for the singular linear model

The unique Bayes linear unbiased estimator (Bayes LUE) of estimable functions is derived for the singular linear model. The superiority of Bayes LUE over ordinary best linear unbiased estimator is investigated under mean square error matrix (MSEM) criterion.     

Several splittings for non-Hermitian linear systems

For large sparse non-Hermitian positive definite system of linear equations,we present several variants of the Hermitian and skew-Hermitian splitting(HSS)about the coefficient matrix and establish correspondingly several HSS-based iterative schemes.Theoretical analyses show that these methods are convergent unconditionally to the exact solution of the referred system of linear equations,and they may show advantages on problems that the HSS method is ineffiective.     

On subnormal solutions of second order linear periodic differential equations

In this paper, we investigate the existence and the form of subnormal solution for a class of second order periodic linear differential equations, estimate the growth properties of all solutions, and answer the question raised by Gundersen and Steinbart.     

Symmetric modified AOR method to solve systems of linear equations

We propose a class of symmetric modified accelerated overrelaxation (SMAOR) methods for solving large sparse linear systems. The convergence region of the method has been investigated. Numerical examples indicate that the SMAOR method is better than other methods such as accelerated overrelaxation(AOR) and modified accelerated overrelaxation(MAOR) methods, since the spectral radius of iteration matrix in SMAOR method is less than that of the other methods. Also, we apply the method to solve a real boundary value problem.     

Continuity in weak topology:higher order linear systems of ODE

We will introduce a type of Fredholm operators which are shown to have a certain con- tinuity in weak topologies.From this,we will prove that the fundamental matrix solutions of k-th, k≥2,order linear systems of ordinary differential equations are continuous in coefficient matrixes with weak topologies.Consequently,Floquet multipliers and Lyapunov exponents for periodic systems are continuous in weak topologies.Moreover,for the scalar Hill's equations,Sturm-Liouville eigenvalues, periodic and anti-periodic eigenvalues,and rotation numbers are all continuous in potentials with weak topologies.These results will lead to many interesting variational problems.