Related papers: Some results on number theory and differential equ…

We prove some results on the border of Ramsey theory (finite partition calculus) and model theory. Also a beginning of classification theory of finite models in undertaken.

Differential equations with constant and variable coefficients over octonions are investigated. It is found that different types of differential equations over octonions can be resolved. For this purpose non-commutative line integration is…

Using, as main tool, the convergence theorem for discrete martingales and the mean value property of harmonic functions we solve, a particular case of, Dirichlet problem.

We discuss various forms of the classical van der Corput's difference theorem and explore applications to and connections with the theory of uniform distribution, ergodic theory, topological dynamics and combinatorics.

We study completeness in partial differential varieties. We generalize many results from ordinary differential fields to the partial differential setting. In particular, we establish a valuative criterion for differential completeness and…

This article gives a brief survey of the theory and applications of anomalies.

We study first-order concatenation theory with bounded quantifiers. We give axiomatizations with interesting properties, and we prove some normal-form results. Finally, we prove a number of decidability and undecidability results.

A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation…

We discuss a version of the fundamental theorem of calculus in several variables and some applications, of potential interest as a teaching material in undergraduate courses.

This short communication develops a new numerical procedure suitable for a large class of ordinary differential equation systems found in models in physics and engineering. The main numerical procedure is analogous to those concerning the…

Partially or totally unsolved questions in number theory and geometry especially, such as coloration problems, elementary geometric conjectures, partitions, generalized periods of a number, length of a generalized period, arithmetic and…

We prove a convergence theorem for partial sums of sectorial forms with vertex zero and a common semi-angle. As an example we prove an absorption theorem for sectorial forms.

By using a perturbation technique in critical point theory, we prove the existence of solutions for two types of nonlinear equations involving fractional differential operators.

We continue the study of the center problem for the ordinary differential equation $v'=\sum_{i=1}^{\infty}a_{i}(x)v^{i+1}$ started in our earlier papers. In this paper we present the highlights of the algebraic theory of centers.

In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…

Many theorems of mathematics have the form that for a certain problem, e.g. a differential equation or polynomial (in)equality, there exists a solution. The sequential version then states that for a sequence of problems, there is a sequence…

There is no general existence theorem for solutions for nonlinear difference equations, so we must prove the existence of solutions in accordance with models one by one. In our work, we found theorems for the existence of analytic solutions…

We prove some new results on existence of solutions to first--order ordinary differential equations with deviating arguments. Delay differential equations are included in our general framework, which even allows deviations to depend on the…

In this paper, we use differential forms to prove a number of theorems of integral vector calculus that are rarely found in textbooks. Two of them, as far as the author knows, have not been published before. Some possible applications to…

In this paper, we present some applications of a difference equation of degree k in Cryptography and Coding Theory.