Related papers: On the Solution to a Countable System of Equations…
The inverse power method is a numerical algorithm to obtain the eigenvectors of a matrix. In this work, we develop an iteration algorithm, based on the inverse power method, to numerically solve the Schr\"odinger equation that couples an…
This paper aims at the efficient numerical solution of stochastic eigenvalue problems. Such problems often lead to prohibitively high dimensional systems with tensor product structure when discretized with the stochastic Galerkin method.…
An infinite system of point particles placed in $\mathds{R}^d$ is studied. The particles are of two types; they perform random walks in the course of which those of distinct types repel each other. The interaction of this kind induces an…
The chemical birth-death process, whose chemical master equation (CME) is exactly solvable, is a paradigmatic toy problem often used to get intuition for how stochasticity affects chemical kinetics. In a certain limit, it can be…
In this paper, we first show the existence of solutions to the following system of nonlinear equations \begin{eqnarray*}\left\{\begin{array}{l} a_{11}x_1+a_{12}x_2+a_{13}x_3+\cdots+a_{1n}x_{n} =…
For non-equilibrium systems described by finite Markov processes, we consider the number of times that a system traverses a cyclic sequence of states (a cycle). The joint distribution of the number of forward and backward instances of any…
As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a {\em stochastic maximal inequality} derived by using the formula for…
Recent advances in optimization theory have shown that smooth strongly convex finite sums can be minimized faster than by treating them as a black box "batch" problem. In this work we introduce a new method in this class with a theoretical…
The problem of iterated partial summations is solved for some discrete distributions defined on discrete supports. The power method, usually used as a computational approach to finding matrix eigenvalues and eigenvectors, is in some cases…
We study orbit-finite systems of linear equations, in the setting of sets with atoms. Our principal contribution is a decision procedure for solvability of such systems. The procedure works for every field (and even commutative ring) under…
The time evolution of complex systems usually can be described through stochastic processes. These processes are measured at finite resolution, what necessarily reduces them to finite sequences of real numbers. In order to relate these data…
This paper presents a novel approach to constructing finite generating sets for infinitely generated ideals. By integrating algebraic and computational techniques, we provide a method to identify finite generators, demonstrated through…
In this paper, we develop algorithms for computing the recurrence coefficients corresponding to multiple orthogonal polynomials on the step-line. We reformulate the problem as an inverse eigenvalue problem, which can be solved using…
To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…
Leveraging recent work on data-driven methods for constructing a finite state space Markov process from dynamical systems, we address two problems for obtaining further reduced statistical representations. The first problem is to extract…
We investigate invertible matrices over finite additively idempotent semirings. The main result provides a criterion for the invertibility of such matrices. We also give a construction of the inverse matrix and a formula for the number of…
Many problems in quantum information theory can be formulated as optimizations over the sequential outcomes of dynamical systems subject to unpredictable external influences. Such problems include many-body entanglement detection through…
This article generalizes a recently introduced procedure to solve nonlinear systems of equations, radically departing from the conventional Newton-Raphson scheme. The original nonlinear system is first unfolded into three simpler…
Poisson's equation plays a fundamental role as a tool for performance evaluation and optimization of Markov chains. For continuous-time birth-death chains with possibly unbounded transition and cost rates as addressed herein, when…
Markov matrices have an important role in the filed of stochastic processes. In this paper, we will show and prove a series of conclusions on Markov matrices and transformations rather than pay attention to stochastic processes although…