Related papers: Shnirelman's Theorem Applications

This paper investigates a generalized version of Diophantine tuples in finite fields. Applying Shparlinski's method, we obtain power-saving results on the number of such tuples.

This work determine the entire family of positive integer solutions of the diophantine equation. The solution is described in terms of $\frac{(m-1)(m+n-2)}{2} $ or $\frac{(m-1)(m+n-1)}{2}$ positive parameters depending on $n$ even or odd.…

By using the degree theory and the $\tau-$topology of Kryszewski and Szulkin, we establish a version of the Fountain Theorem for strongly indefinite functionals. The abstract result will be applied for studying the existence of infinitely…

We study metric Diophantine approximation in local fields of positive characteristic. Specifically, we study the problem of improving Dirichlet's theorem in Diophantine approximation and prove very general results in this context.

A combinatorial methods are used to investigate some properties of certain generalized Stirling numbers, including explicit formula and recurrence relations. Furthermore, an expression of these numbers with symmetric function is deduced.

Schr\"{o}dinger equations in \emph{functional derivatives} are solved via quantized Galerkin limit of antinormal functional Feynman integrals for Schr\"{o}dinger equations in \emph{partial derivatives

In this paper we study random iterated function systems. Our main result gives sufficient conditions for an analogue of a well known theorem due to Khintchine from Diophantine approximation to hold almost surely for stochastically…

A novel method for finding the eigenvalues of a Sturm-Liouville problem is developed. Following the minimalist approach the problem is transformed to a single first-order differential equation with appropriate boundary conditions. Although…

We study connections between linear equations over various semigroups and recursively enumerable sets of positive integers. We give variants of the universal Diophantine representation of recursively enumerable sets of positive integers…

An infinite dimensional algebra, which is useful for deriving exact solutions of the generalized pairing problem, is introduced. A formalism for diagonalizing the corresponding Hamiltonian is also proposed. The theory is illustrated with…

We employ Schauder fixed-point Theorem to prove the existence of at least one positive continuous solution of the quadratic integral equation Moreover, the maximal and the minimal solutions of the last equation are also proved.

In the present paper we generalize the Eulerian numbers (also of the second and third orders). The generalization is connected with an autonomous first-order differential equation, solutions of which are used to obtain integral…

Sturm theory for second order differential equations was generalized to systems and higher order equations with positive leading coefficient by several authors. Here we propose a Sturm oscillation theorem for indefinite systems of even…

The main point of this paper is to present a class of equations over integers that one can check if they have a solution by checking a set of inequalities. The prototype of such equations is the equations appearing in the well-known…

The theory of Schur functors provides a powerful and elegant approach to the representation theory of GL_n - at least to the so-called polynomial representations - especially to questions about how the theory varies with n. We develop…

We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system…

In this note we recall the definition of the digital root, and apply the notion of the digital root to searching solutions of Diophantine equations. A table of arithmetic operations with digital roots is given. This method is incapable of…

We prove multidimensional integration by parts formulas for generalized fractional derivatives and integrals. The new results allow us to obtain optimality conditions for multidimensional fractional variational problems with Lagrangians…

We prove a version of the Khinchine--Groshev theorem for Diophantine approximation of matrices subject to a congruence condition. The proof relies on an extension of the Dani correspondence to the quotient by a congruence subgroup. This…

Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei-Norman method is applied to obtain the associated…