Mathematical and Computational Methods
69 papers on this topic
AI Mathematician as a Partner in Advancing Mathematical Discovery -- A Case Study in Homogenization Theory
Artificial intelligence (AI) has demonstrated impressive progress in mathematical reasoning, yet its integration into the practice of mathematical research remains limited. In this study, we...
Automated Discovery of Integral with Deep Learning
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...
Fast Algorithms for Discrete Differential Equations
Discrete Differential Equations (DDEs) are functional equations that relate polynomially a power series $F(t,u)$ in $t$ with polynomial coefficients in a "catalytic" variable $u$ and the...
The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants
Fundamental mathematical constants like $e$ and $\pi$ are ubiquitous in diverse fields of science, from abstract mathematics to physics, biology and chemistry. For centuries, new formulas relating...
Analyzing the Nuances of Transformers' Polynomial Simplification Abilities
Symbolic Mathematical tasks such as integration often require multiple well-defined steps and understanding of sub-tasks to reach a solution. To understand Transformers' abilities in such tasks in...
An algorithm for real and complex rational minimax approximation
Rational minimax approximation of real functions on real intervals is an established topic, but when it comes to complex functions or domains, there appear to be no algorithms currently in use....
Computation of parabolic cylinder functions having complex argument
Numerical methods for the computation of the parabolic cylinder $U(a,z)$ for real $a$ and complex $z$ are presented. The main tools are recent asymptotic expansions involving exponential and Airy...
The Mathematical Foundations of Physical Systems Modeling Languages
Modern modeling languages for general physical systems, such as Modelica, Amesim, or Simscape, rely on Differential Algebraic Equations (DAE), i.e., constraints of the form f(dot{x},x,u)=0. This...
Quantifying the ill-conditioning of analytic continuation
Analytic continuation is ill-posed, but becomes merely ill-conditioned (although with an infinite condition number) if it is known that the function in question is bounded in a given region of the...
Choosing the Variable Ordering for Cylindrical Algebraic Decomposition via Exploiting Chordal Structure
Cylindrical algebraic decomposition (CAD) plays an important role in the field of real algebraic geometry and many other areas. As is well-known, the choice of variable ordering while computing...
Similarity-Based Equational Inference in Physics
Automating the derivation of published results is a challenge, in part due to the informal use of mathematics by physicists, compared to that of mathematicians. Following demand, we describe a...
Quantum algebra in R: the weyl package
Weyl algebra is a simple noncommutative system used in quantum mechanics. Here I introduce the weyl package, written in the R computing language, which furnishes functionality for working with...
Solving linear difference equations with coefficients in rings with idempotent representations
We introduce a general reduction strategy that enables one to search for solutions of parameterized linear difference equations in difference rings. Here we assume that the ring itself can be...
An in-place truncated Fourier transform
We show that simple modifications to van der Hoeven's forward and inverse truncated Fourier transforms allow the algorithms to be performed in-place, and with only a linear overhead in complexity.
Optimized numerical inverse Laplace transformation
Among the numerical inverse Laplace transformation (NILT) methods, those that belong to the Abate-Whitt framework (AWF) are considered to be the most efficient ones currently. It is a...
A Flawed Dataset for Symbolic Equation Verification
Arabshahi, Singh, and Anandkumar (2018) propose a method for creating a dataset of symbolic mathematical equations for the tasks of symbolic equation verification and equation completion....
qFunctions -- A Mathematica package for $q$-series and partition theory applications
We describe the qFunctions Mathematica package for $q$-series and partition theory applications. This package includes both experimental and symbolic tools. The experimental set of elements...
Finding polynomial roots by dynamical systems -- a case study
We investigate two well known dynamical systems that are designed to find roots of univariate polynomials by iteration: the methods known by Newton and by Ehrlich-Aberth. Both are known to have...