Related papers: On an elementary method for solving $Ax^4-By^2=1$
Some interesting (periodic!) solutions of certain systems of $4$ nonlinear Ordinary Differential Equations $dx_{n}\left( t\right) /dt=P_{2}^{\left( n\right) }\left[ x_{m}\left( t\right) \right] /\left[ x_{1}\left( t\right) +x_{2}\left(…
In this work, a novel quaternary algebra has been proposed that can be used to implement an arbitrary quaternary logic function in more than one systematic ways. The proposed logic has evolved from and is closely related to the Boolean…
Asymptotic symmetries of the Einstein-Yang-Mills system with or without cosmological constant are explicitly worked out in a unified manner. In agreement with a recent conjecture, one finds a Virasoro-Kac-Moody type algebra not only in…
Suppose that A and B are real stable matrices, and that their difference A-B is rank one. Then A and B have a common quadratic Lyapunov function if and only if the product AB has no real negative eigenvalue. This result is due to Shorten…
Methods of solving big Boolean equations can be broadly classified as algebraic, tabular, numerical and map methods. The most prominent among these classes are the algebraic and map methods. This paper surveys and compares these two types…
Using an elementary identity, we prove that for infinitely many polynomials $P(x)\in \mathbb{Z}[X]$ of fourth degree, the equation $\prod\limits_{k=1}^{n}P(k)=y^2$ has finitely many solutions in $\mathbb{Z}$. We also give an example of a…
A general method for the construction of solutions of the d'Alambertian and double d'Alambertian (harmonic and bi-harmonic) equations with local dependence of arbitrary functions upon two independent arguments is proposed. The connection…
This article proves that there are infinitely many primes of the form a^2 + b^4, in fact getting the asymptotic formula. The main result is that \sum_{a^2 + b^4\le x} \Lambda(a^2 + b^4) = 4\pi^{-1}\kappa x^{3/4} (1 + O(\log\log x / \log x))…
In this note we find all the solutions of the Diophantine equation $x^4\pm y^4=iz^2$ using elliptic curves over $\mathbb Q(i)$. Also, using the same method we give a new proof of Hilbert's result that the equation $x^4\pm y^4=z^2$ has only…
The Painlev\'e equations possess transcendental solutions $y(t)$ with special initial values that are symmetric under rotation or reflection in the complex $t$-plane. They correspond to monodromy problems that are explicitly solvable in…
We prove a generalization of the Li-Yau estimate for a board class of second order linear parabolic equations. As a consequence, we obtain a new Cheeger-Yau inequality and a new Harnack inequality for these equations. We also prove a…
New modifications of the methods for simultaneous extraction of all roots of polynomials over an arbitrary Chebyshev system are elaborated. A cubic convergence of iterations is proved. The method presented is a generalisation of the…
A unified approach, for solving a wide class of single and many-body quantum problems, commonly encountered in literature is developed based on a recently proposed method for finding solutions of linear differential equations. Apart from…
Several exact, cylindrically symmetric solutions to Einstein's vacuum equations are given. These solutions were found using the connection between Yang-Mills theory and general relativity. Taking known solutions of the Yang-Mills equations…
The solution of equations from the title is well known since the Euler's time. However, its proof in the case of multiple roots of the characteristic polynomial is rather long and technical and even appearance of the factors $x^m$ looks…
We propose a protocol for solving systems of linear algebraic equations via quantum mechanical methods using the minimal number of qubits. We show that $(M+1)$-qubit system is enough to solve a system of $M$ equations for one of the…
In various areas of applied numerics, the problem of calculating the logarithm of a matrix A emerges. Since series expansions of the logarithm usually do not converge well for matrices far away from the identity, the standard numerical…
We consider a dominance order on positive vectors induced by the elementary symmetric polynomials. Under this dominance order we provide conditions that yield simple proofs of several monotonicity questions. Notably, our approach yields a…
We give a specific method to solve with quadratic complexity the linear systems arising in known algorithms to deal with the sign determination problem. In particular, this enable us to improve the complexity bound for sign determination in…
This paper presents the classification of a general quadric into an axisymmetric quadric (AQ) and the solution to the problem of the proximity of a given point to an AQ. The problem of proximity in $R^3$ is reduced to the same in $R^2$,…