Related papers: Effective Methods for Norm-Form Equations
For every number field $\mathbb{K}$, with $[\mathbb{K}:\mathbb{Q}] \geq 3$, we show that the number of non-associates of the same norm in a full module in $\mathbb{K}$ does not depend only on $\mathbb{K}$, but can also depend on the module…
Finding integer solutions to norm form equations is a classical Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It is known that these solutions can be…
In this paper we present a new method of solving certain quartic and higher degree homogeneous polynomial diophantine equations in four variables. The method can also be extended to solve simultaneous homogeneous polynomial diophantine…
We complete the solution of the relative class number one problem for function fields of curves over finite fields. Using work from two earlier papers, this reduces to finding all function fields of genus 6 or 7 over $\mathbb{F}_2$ with one…
In the present paper, a Nystrom-type method for second kind Volterra integral equations is introduced and studied. The method makes use of generalized Bernstein polynomials, defined for continuous functions and based on equally spaced…
We propose a two-dimensional generalization of Constantin-Lax-Majda model [2]. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line…
We analyze the inverse problem to reconstruct the shape of a three dimensional homogeneous dielectric obstacle from the knowledge of noisy far field data. The forward problem is solved by a system of second kind boundary integral equations.…
We introduce an effective method to solve the $\bar\partial$-harmonic forms on the Kodaira-Thurston manifold endowed with an almost complex structure and an Hermitian metric. Using the Weil-Brezin transform, we reduce the elliptic PDE…
We solve the local equivalence problem for second order (smooth or analytic) ordinary differential equations. We do so by presenting a {\em complete convergent normal form} for this class of ODEs. The normal form is optimal in the sense…
In this paper we develop an adaptive procedure for the numerical solution of general, semilinear elliptic problems with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton method and an adaptive…
In this paper, we consider the numerical solution of highly-oscillatory Vlasov and Vlasov-Poisson equations with non-homogeneous magnetic field. Designed in the spirit of recent uniformly accurate methods, our schemes remain insensitive to…
It is well known that the ultra-relativistic Thomas-Fermi equation, amply adopted in the study of heavy nuclei, admits an exact solution for a constant proton distribution within a spherical core of radius Rc. Here exact solutions of a…
We provide several extensions of the modular method which were motivated by the problem of completing previous work to prove that, for any integer $n \geq 2$, the equation \[ x^{13} + y^{13} = 3 z^n \] has no non-trivial solutions. In…
The method of separation of variables is significant, it has been applied to physics, engineering , chemistry and other fields. It allows to reduce the diffculity of problems by separating the variables from partial differential equation…
We establish Ecalle's mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to…
The aim of the research presented in this paper is to derive the systems of ordinary differential equations (ODEs) satisfied by modular forms of level six and to construct extensions of the differential field of the cubic theta functions,…
The stable and efficient numerical simulation of viscoelastic flows has been a constant struggle due to the High Weissenberg Number Problem. While the stability for macroscopic descriptions could be greatly enhanced by the log-conformation…
The objective of this paper is to investigate the existence and the forms of the pair of finite order entire and meromorphic solutions of some certain systems of Fermat-type partial differential-difference equations of several complex…
We classify all totally real number fields of degree at most 5 that admit a universal quadratic form with rational integer coefficients; in fact, there are none over the previously unsolved cases of quartic and quintic fields. This fully…
In this note we develop a numerical method for partial differential equations with changing type. Our method is based on a unified solution theory found by Rainer Picard for several linear equations from mathematical physics. Parallel to…