Related papers: A reduction algorithm for Volterra integral equati…
In this paper we explicitly prove that Integrable System solved by Quantum Inverse Scattering Method can be described with the pure algebraic object (Universal R-matrix) and proper algebraic representations. Namely, on the example of the…
In this article, numerical integration is formulated as evaluation of a matrix function of a matrix that is obtained as a projection of the multiplication operator on a finite-dimensional basis. The idea is to approximate the continuous…
A lattice of integers is the collection of all linear combinations of a set of vectors for which all entries of the vectors are integers and all coefficients in the linear combinations are also integers. Lattice reduction refers to the…
Runge-Kutta methods have an irreplaceable position among numerical methods designed to solve ordinary differential equations. Especially, implicit ones are suitable for approximating solutions of stiff initial value problems. We propose a…
We consider optimization problems that are formulated and solved in the framework of tropical mathematics. The problems consist in minimizing or maximizing functionals defined on vectors of finite-dimensional semimodules over idempotent…
A simple iteration methodology for the solution of a set of a linear algebraic equations is presented. The explanation of this method is based on a pure geometrical interpretation and pictorial representation. Convergence using this method…
We construct the basis of a stochastic calculus for so-called Volterra processes, i.e., processes which are defined as the stochastic integral of a time-dependent kernel with respect to a standard Brownian motion. For these processes which…
The Integral Image algorithm is often applied in tasks that require efficient integration over images, such as object detection. In this paper we discuss theoretical aspects of the algorithm's continuous version. We suggest to define the…
Learned image reconstruction has become a pillar in computational imaging and inverse problems. Among the most successful approaches are learned iterative networks, which are formulated by unrolling classical iterative optimisation…
We introduce a new class of computationally tractable scattering problems in unbounded domains, which we call decomposable problems. In these decomposable problems, the computational domain can be split into a finite collection of…
A Rota-Baxter algebra, also known as a Baxter algebra, is an algebra with a linear operator satisfying a relation, called the Rota-Baxter relation, that generalizes the integration by parts formula. Most of the studies on Rota-Baxter…
Solving systems of non-autonomous ordinary differential equations (ODE) is a crucial and often challenging problem. Recently a new approach was introduced based on a generalization of the Volterra composition. In this work, we explain the…
A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly…
Numerical integration (NI) packages commonly used in scientific research are limited to returning the value of a definite integral at the upper integration limit, also commonly referred to as numerical quadrature. These quadrature…
Complex systems are composed of a large number of simple components connected to each other in the form of a network. It is shown that, for some network configurations, the equivalent dynamic behavior of the system is governed by an…
Machine learning approaches relying on such criteria as adversarial robustness or multi-agent settings have raised the need for solving game-theoretic equilibrium problems. Of particular relevance to these applications are methods targeting…
An expression for the coefficients of a linear iterative equation in terms of the parameters of the source equation is given both for equations in standard form and for equations in reduced normal form. The operator generating an iterative…
We introduce and analyse a sparse spectral method for the solution of Volterra integral equations using bivariate orthogonal polynomials on a triangle domain. The sparsity of the Volterra operator on a weighted Jacobi basis is used to…
We propose a quantum algorithm to solve systems of nonlinear algebraic equations. In the ideal case the complexity of the algorithm is linear in the number of variables $n$, which means our algorithm's complexity is less than $O(n^{3})$ of…
We present a historiographical review of algorithms and computer codes developed for solving integration-by-parts relations for Feynman integrals. This procedure is one of the key steps in the evaluation of Feynman integrals, since it…