Related papers: Some results on number theory and differential equ…
We take some first steps in providing a synthetic theory of distributions. In particular, we are interested in the use of distribution theory as foundation, not just as tool, in the study of the wave equation.
Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last…
In this paper we give simple extension and uniqueness theorems for restricted additive and logarithmic functional equations.
In view of the usefulness and importance of the kinetic equation in certain physical problems, the authors derive the explicit solution of a fractional kinetic equation of general character, that unifies and extends earlier results.…
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…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…
In this paper, we study the consequences of the fundamental theorem of calculus from an algebraic point of view. For functions with singularities, this leads to a generalized notion of evaluation. We investigate properties of such…
We study a generalization of the classical Pentagonal Number Theorem and its applications. We derive new identities for certain infinite series, recurrence relations and convolution sums for certain restricted partitions and divisor sums.…
The book gives a detailed exposition of basic concepts and results of a theory of processes. The presentation of theoretical concepts and results is accompanied with illustrations of their application to solving various problems of…
The general conditions under which the quadratic, uniform and monotonic convergence in the quasilinearization method of solving nonlinear ordinary differential equations could be proved are formulated and elaborated. The generalization of…
Here we develop a technique of computing the invariants of $n-$ary forms and systems of forms using the discriminants of corresponding multilinear forms built of their partial derivatives, which should be cosidered as analogues of classical…
The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application…
We give new proofs of some well-known results from Invariant Theorey using the Kempf-Ness theorem.
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
In this short note, we establish an operator theoretic version of the Wiener-Ikehara tauberian theorem, and point out how this leads to a new proof of the Prime number theorem that should be accessible to anyone with a basic knowledge of…
The theory of Lie remarkable equations, i.e. differential equations characterized by their Lie point symmetries, is reviewed and applied to ordinary differential equations. In particular, we consider some relevant Lie algebras of vector…
We study function-valued solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable parabolicity hypotheses. We provide…
The present paper provides a method for finding partial differential equations satisfied by the Feynman integrals for diagrams of various types, using the Griffiths theorem on the reduction of poles of rational differential forms. As an…
In this work we establish a theory of Calculus based on the new concept of displacement. We develop all the concepts and results necessary to go from the definition to differential equations, starting with topology and measure and moving on…
In this article we determine several theorems and methods for solving linear congruences and systems of linear congruences, and we find the number of distinct solutions. Many examples of solving congruences are given.