Related papers: Solving Diophantine Equations
Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and…
In this paper, we present new techniques for solving a large variety of partial differential equations. The proposed method reduces the PDEs to first order differential equations known as classical equations such as Bernoulli, Ricatti and…
Methods for the computation of invariants and symmetries of nonlinear evolution, wave, and lattice equations are presented. The algorithms are based on dimensional analysis, and can be implemented in any symbolic language, such as…
Based on the MRDP theorem, we introduce the ideas of the proof equation of a formula and universal proof equation of Peano Arithmetic (PA); and then, combining universal proof equation and G\"odel's Second Incompleteness Theorem, it is…
A code for the numerical evaluation of hyperelliptic theta-functions is presented. Characteristic quantities of the underlying Riemann surface such as its periods are determined with the help of spectral methods. The code is optimized for…
In this paper, we use a variety of classical and new research methods for ternary exponential Diophantine equations and extensive use of computer calculations to study the conjecture of R. Scott and R. Styer which asserts that for any fixed…
We present an algorithm (CoDeFi) which overcomes the curse of dimensionality (CoD) in scientific computations and, especially, in mathematical finance (Fi). Our method applies a broad class of partial differential equations such as…
Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are…
Our goal is to finally settle the persistent problem in Diophantine Approximation of finding best linear approximates. Classical results from the theory of continued fractions provide the solution for the special homogeneous case in the…
The application of the approximation-operational approach to solving linear differential equations of fractional order with variable coefficients is considered. It is shown that the method can also be applied to solving differential…
By finding all integral points on certain elliptic and hyperelliptic curves we completely solve the Diophantine equation $\binom{n}{k}=\binom{m}{l}+d$ for $-3\leq d\leq 3$ and $(k,l)\in\{(2,3),\; (2,4),\;(2,5),\; (2,6),\; (2,8),\; (3,4),\;…
Stochastic differential equations (SDEs) offer powerful and accessible mathematical models for capturing both deterministic and probabilistic aspects of dynamic behavior across a wide range of physical, financial, and social systems.…
We utilize a combination of integral transforms, including the Laplace transform, with some classical results in analytic number theory concerning the Riemann $\xi$-function, to obtain a new integral equation. We also provide a new proof of…
In this paper we analyze computational properties of the Diophantine problem (and its search variant) for spherical equations $\prod_{i=1}^m z_i^{-1} c_i z_i = 1$ (and its variants) over the class of finite metabelian groups…
We are interested in solving decision problem $\exists? t \in \mathbb{N}, \cos t \theta = c$ where $\cos \theta$ and $c$ are algebraic numbers. We call this the $\cos t \theta$ problem. This is an exploration of Diophantine equations with…
A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It…
Two fractional two-phase Stefan-like problems are considered by using Riemann-Liouville and Caputo derivatives of order $\alpha \in (0, 1)$ verifying that they coincide with the same classical Stefan problem at the limit case when…
This paper reports on the current status of the project in which we order all polynomial Diophantine equations by an appropriate version of "size", and then solve the equations in that order. We list the "smallest" equations that are…
We report the fourth known primitive solution to the Diophantine equation $a^5 + b^5 + c^5 + d^5 = e^5$, extending the list of solutions from 1966, 1996, and 2004. This result was obtained via a large-scale computational search based on an…
An original approach to solving rather difficult probabilistic problems arising in studying the readout of random discrete fields and having no exact analytical solutions at the moment is proposed. Several algorithms for direct, iterative,…