Matrix-Fractional Invariance of Diophantine Systems of Linear Forms

Metric Diophantine Approximation for Systems of Linear Forms via Dynamics

The goal of this paper is to generalize the main results of [KM1] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish ‘joint strong extremality’ of arbitrary finite collection of smooth nondegenerate submanifolds of R. The proofs are based on quantitative nondivergence estimates for quasi-polyn...

Diophantine Approximation of Ternary Linear Forms . II

Let 6 denote the positive root of the equation xs + x2 — 2x — 1 = 0; that is, 8 = 2 cos(27r/7). The main result of the paper is the evaluation of the constant lim supm-co min M2\x + By + 02z|, where the min is taken over all integers x, y, z satisfying 1 g max (\y\, |z|) g M. Its value is (29 + 3),/7 = .78485. The same method can be applied to other constants of the same type.

On Solving Linear Diophantine Systems Using Generalized Rosser's Algorithm

Exact and numerical solutions of linear and non-linear systems of fractional partial differential equations

The present study introduces a new technique of homotopy perturbation method for the solution of systems of fractional partial differential equations. The proposed scheme is based on Laplace transform and new homotopy perturbation methods. The fractional derivatives are considered in Caputo sense. To illustrate the ability and reliability of the method some examples are provided. The results ob...

On Systems of Linear Diophantine Equations

Introduction Something happened to me recently I would wager has happened to many who read this note. Teaching a new topic, you cannot understand one of the proofs. Your first attempt to fill the gap fails. You look through your books for an answer. Next, you ask colleagues, go to the library, maybe even use the interlibrary loan. All in vain. Then it strikes you that, in fact, you cannot answe...