Related papers: A development of Lagrange interpolation, Part I: T…
Let $f$ be a real polynomial of $x = (x_1,\dots,x_n)$ and $\varphi$ be a locally integrable function of $x$ which satisfies a holonomic system of linear differential equations. We study the distribution $f_+^\lambda\varphi$ with a…
A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be…
A group classification of first-order delay ordinary differential equation (DODE) accompanied by an equation for delay parameter (delay relation) is presented. A subset of such systems (delay ordinary differential systems or DODSs) which…
For linear and fully non-linear diffusion equations of Bellman-Isaacs type, we introduce a class of approximation schemes based on differencing and interpolation. As opposed to classical numerical methods, these schemes work for general…
Identifying integrable coupled nonlinear ordinary differential equations (ODEs) of dissipative type and deducing their general solutions are some of the challenging tasks in nonlinear dynamics. In this paper we undertake these problems and…
A previous article was devoted to an analysis of the symmetry properties of a class of first-order delay ordinary differential systems (DODSs). Here we concentrate on linear DODSs. They have infinite-dimensional Lie point symmetry groups…
A procedure to obtain differentiation matrices is extended straightforwardly to yield new differentiation matrices useful to obtain derivatives of complex rational functions. Such matrices can be used to obtain numerical solutions of some…
We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations…
In recent years many efforts have been devoted to finding bidiagonal factorizations of nonsingular totally positive matrices, since their accurate computation allows to numerically solve several important algebraic problems with great…
Deep Reinforcement Learning (DRL) is emerging as a promising approach to generate adaptive behaviors for robotic platforms. However, a major drawback of using DRL is the data-hungry training regime that requires millions of trial and error…
Diffusion-based inverse algorithms have shown remarkable performance across various inverse problems, yet their reliance on numerous denoising steps incurs high computational costs. While recent developments of fast diffusion ODE solvers…
This paper introduces a new functional expansion framework that extends classical ideas beyond the Taylor series. Unlike traditional Taylor expansions based on local polynomial approximations, the proposed approach arises from exact…
In deep learning, classification tasks are formalized as optimization problems often solved via the minimization of the cross-entropy. However, recent advancements in the design of objective functions allow the usage of the $f$-divergence…
Linear algebraic expressions are the essence of many computationally intensive problems, including scientific simulations and machine learning applications. However, translating high-level formulations of these expressions to efficient…
Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We…
We introduce the Schur class of functions, discrete analytic on the integer lattice in the complex plane. As a special case, we derive the explicit form of discrete analytic Blaschke factors and solve the related basic interpolation…
We present several methods, which utilize symplectic integration techniques based on two and three part operator splitting, for numerically solving the equations of motion of the disordered, discrete nonlinear Schr\"odinger (DDNLS)…
Recent advancements in the realm of deep learning, particularly in the development of large language models (LLMs), have demonstrated AI's ability to tackle complex mathematical problems or solving programming challenges. However, the…
A numerical method ADER-DG with a local DG predictor for solving a DAE system has been developed, which was based on the formulation of ADER-DG methods using a local DG predictor for solving ODE and PDE systems. The basis functions were…
We prove a useful formula and new properties for the recently introduced power fractional calculus with non-local and non-singular kernels. In particular, we prove a new version of Gronwall's inequality involving the power fractional…