Related papers: Ordinary Differential Equations and Integrable Mod…
The search for new integrable (3+1)-dimensional partial differential systems is among the most important challenges in the modern integrability theory. It turns out that such a system can be associated to any pair of rational functions of…
A brief description of the relations between the factorization method in quantum mechanics, self-similar potentials, integrable systems and the theory of special functions is given. New coherent states of the harmonic oscillator related to…
Recent progress in quantum field theory and quantum gravity relies on mixed boundary conditions involving both normal and tangential derivatives of the quantized field. In particular, the occurrence of tangential derivatives in the boundary…
An elliptic version of quantum groups is proposed. It comes form the quantization of the Knizhnik-Zamolodchikov- Bernard equation on the torus. The relation with elliptic IRF models is explained.
We review very briefly the main mathematical structures and results in some important areas of Quantum Mechanics involving PDEs and formulate open problems.
Quantum field theory unifies concepts from quantum theory and from special relativity. Its mathematically rigorous description is quite intricate and is only partially understood; this is particularly true for the construction of operators…
A previously established correspondence between definite-parity real functions and inner analytic functions is generalized to real functions without definite parity properties. The set of inner analytic functions that corresponds to the set…
Let $k$ be a differential field of characteristic zero with an algebraically closed field of constants. In this article, we provide a classification of first order differential equations over $k$ and study the algebraic dependence of…
Necessary and sufficient conditions for the solvability of boundary value problems for a family of functional differential equations with a non-integrable singularity are obtained.
Starting from space-discretisation of Maxwell's equations, various classical formulations are proposed for the simulation of electromagnetic fields. They differ in the phenomena considered as well as in the variables chosen for…
The concept of unified field theory is discussed. Two nonlinear field models with world volume type action are considered, namely extremal space-time film model and Born -- Infeld nonlinear electrodynamics. The natural appearance of two…
In this article, we will showcase some analytical concepts that can be used to tackle Functional Equations (FE) in the positive real numbers domain. Such concepts and related techniques have occasionally appeared in recent High School Math…
In a recent paper, Cohl and Costas-Santos derived a number of interesting multi-derivative and multi-integral relations for associated Legendre and Ferrers functions in which the orders of those functions are changed in integral steps.…
In quantum mechanics, one can express the evolution operator and other quantities in terms of functional integrals. The main goal of this paper is to prove corresponding results in the geometric approach to quantum theory. We apply these…
In this paper are examined general classes of linear and non-linear analytical systems of partial differential equations. Indeed the integrability conditions are found and if they are satisfied, the solutions are given as functional series…
We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a…
The status of multifractional theories is reviewed using comparative tables. Theoretical foundations, classical matter and gravity dynamics, cosmology and experimental constraints are summarized and the application of the multifractional…
The exact solutions to quantum string and gauge field theories are discussed and their formulation in the framework of integrable systems is presented. In particular I consider in detail several examples of appearence of solutions to the…
Ordinary differential equations have an arithmetic analogue in which functions are replaced by numbers and the derivation operator is replaced by a Fermat quotient operator. In this survey we explain the main motivations, constructions,…
Based on previous work we consturct an equation (Lagrange equation) and relate it with a system of generalized integrals and differential equations in such a way to provide useful evaluations and connections between them.