The complete sets of solutions of the equation (n k ) = (m ` ) are determined for the cases (k, `) = (2, 3), (2, 4), (2, 6), (2, 8), (3, 4), (3, 6), (4, 6), (4, 8). In each of these cases the equation is reduced to an elliptic equation, which is solved by using linear forms in elliptic logarithms. In all but one case this is more or less routine, but in the remaining case ((k, `) = (3, 6)) we h...

A rational elliptic balance method is introduced to obtain exact and approximate solutions of nonlinear oscillators by using Jacobi elliptic functions. To illustrate the applicability of the proposed rational elliptic forms in the solution of nonlinear oscillators, we first investigate the exact solution of the non-homogenous, undamped Duffing equation. Then, we introduce first and second order...

The paper is largely expository. The first part is devoted to studying integer solutions to Pell’s Equation: u2 − dv2 = 1. The authors present the classic construction of a fundamental solution via continued fractions, from which all solutions can be derived. The primary focus of the second part is on rational solutions to the Thue’s equation, u3−dv3 = 1. The authors explain why these rational ...

This paper concerns the homogenization of a one-dimensional elliptic equation with oscillatory random coefficients. It is well-known that the random solution to the elliptic equation converges to the solution of an effective medium elliptic equation in the limit of a vanishing correlation length in the random medium. It is also well-known that the corrector to homogenization, i.e., the differen...

In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on R-matrix description which provides Lax pairs in terms of quantum and classical R-matrices. First, we prove that for relativistic (and non-relativistic) tops such Lax pairs with spectral parameter follow from the associative Yang-Baxter equation and its degenerations. T...

A study of the diophantine equation v2 = 2u4 − 1 led the authors to consider elliptic curves specifically over Q(i) and to examine the parallels and differences with the classical theory over Q. In this paper we present some extensions of the classical theory along with some examples illustrating the results. The well-known diophantine equation v2 = 2u4 − 1 , has, ignoring signs, only two integ...

‎in this paper‎, ‎we obtain exact solutions involving parameters of some nonlinear pdes in mathmatical physics; namely the one-‎dimensional modified complex ginzburg-landau equation by using the $ (g^{'}/g) $ expansion method‎, homogeneous balance method, extended f-expansion method‎. ‎by ‎using homogeneous balance principle and the extended f-expansion, more periodic wave solutions expres...

We consider a semilinear elliptic equation ∆u+ λf(u) = 0, x ∈ Ω, ∂u

Some oscillation criteria for the second order semilinear elliptic differential equation

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, regularized twolevel schemes are used. The numerical implementation is based on solving the equation with the square root...