Related papers: Effective Methods for Norm-Form Equations

This article is concerned with the almost sure existence of global solutions for initial value problems of the form $\dot{\gamma}(t)= v(t,\gamma(t))$ on separable dual Banach spaces. We prove a general result stating that whenever there…

We derive solvability conditions and closed-form solution for the Weber type integral equation, related to the familiar Weber-Orr integral transforms and the old Weber-Titchmarsh problem (posed in Proc. Lond. Math. Soc. 22 (2) (1924),…

Using increasing sequences of real numbers, we generalize the idea of formal moment differentiation first introduced by W. Balser and M. Yoshino. Slight departure from the concept of Gevrey sequences enables us to include a wide variety of…

We find all solutions to the parametrized family of norm-form equations $x^3-(t^3-1)y^3+3(t^3-1)xy+(t^3-1)^2 = \pm 1$ studied by Amoroso, Masser and Zannier. Our proof relies upon an appeal to lower bounds for linear forms in logarithms and…

We give a new completion for the quasi-uniform spaces. We call the whole procedure {\it $\tau$-completion} and the new space {\it $\tau$-complement of the given}. The basic result is that every $T_{_0}$ quasi-uniform space has a…

New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, it gives an explicit solution to the Ditkin-Prudnikov problem (1966). The 3-term recurrence relations, explicit representations,…

This paper is concerned with the numerical solution of the third kind Volterra integral equations with non-smooth solutions based on the recursive approach of the spectral Tau method. To this end, a new set of the fractional version of…

For arbitrarily large times $T>0$, we prove the uniform-in-$\hbar$ propagation of semiclassical regularity for the solutions to the Hartree$\unicode{x2013}$Fock equation with singular interactions of the form $V(x)=\pm\,|x|^{-a}$ where…

In this paper we find new solutions for the so called Einstein-Chern-Simons Friedmann-Robertson-Walker field equations studied in refs. (Phys. Rev. D 84 (2011) 063506, Eur. Phys. J. C 74 (2014) 3087). We consider three cases:(i) in the…

In this work we consider a simple, approximate, tending toward exact, solution of the system of two usual Lotka-Volterra differential equations. Given solution is obtained by an iterative method. In any finite approximation order of this…

We solve the Fermat-type equation \[ x^{13} + y^{13} = 3 z^7, \qquad \gcd(x,y,z) = 1 \] combining a unit sieve, the multi-Frey modular method, level raising, computations of systems of eigenvalues modulo 7 over a totally real field, and…

We examine several of the normal-form multivariate polynomial rootfinding methods of Telen, Mourrain, and Van Barel and some variants of those methods. We analyze the performance of these variants in terms of their asymptotic temporal…

In this paper, two numerical approaches based on the Newton iteration method with spectral algorithms are introduced to solve the Thomas-Fermi equation. That Thomas-Fermi equation is a nonlinear singular ordinary differential equation (ODE)…

We resolve two problems pertaining to indefinite integral ternary quadratic forms, one highlighted by Margulis and the other initiated by Serre, both from 1990. To do so we develop tools for dealing with high ramification in problems…

A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical…

This paper provides a thorough exploration of the absolute value equations $Ax-|x|=b$, a seemingly straightforward concept that has gained heightened attention in recent years. It is an NP-hard and nondifferentiable problem and equivalent…

We develop a theory connecting the following three areas: (a) the mean field equation (MFE) $\triangle u + e^u = \rho\, \delta_0$, $\rho \in \mathbb R_{>0}$ on flat tori $E_\tau = \mathbb C/(\mathbb Z + \mathbb Z\tau)$, (b) the classical…

Following the famous proof of Fermat's Last Theorem by Andrew Wiles using the modularity of elliptic curves over $\mathbb{Q}$, significant developments have been made in the study of Diophantine equations using the modularity method. This…

Form methods give a very efficient tool to solve evolutionary problems on Hilbert space. They were developed by T. Kato [Kat] and, in slightly different language by J.L. Lions. In this expository article we give an introduction based on…

We study the \emph{complex-valued} solutions to the Cauchy problem of the modified Korteweg-de Vries equation on the real line. To study the low-regularity problems, we employ a generalized Fourier-Lebesgue space…