Related papers: Quadratic First Integrals of Kinetic Differential …
Four classes of three dimensional quadratic algebras of the type $\lsb Q_0 , Q_\pm \rsb$ $=$ $\pm Q_\pm$, $\lsb Q_+ , Q_- \rsb$ $=$ $aQ_0^2 + bQ_0 + c$, where $(a,b,c)$ are constants or central elements of the algebra, are constructed using…
The existence of solutions to Cauchy type problems of linear Riemann-Liouville fractional differential equations with variable coefficients is considered in a space of integrable functions. First, we consider the existence and uniqueness of…
The role of differential equations in the process of calculating Feynman integrals is reviewed. An example of a diagram is given for which the method of differential equations was introduced, the properties of the inverse-mass-expansion…
In this work we wish to highlight some consequences of a recent result proved in [N. D. Cong and H. T. Tuan, Generation of nonlocal fractional dynamical systems by fractional differential equations, J. Integral Equations Appl. 29 (2017),…
It is shown that the squared operation of the Dirac equation which is widely applied may create new solutions and moreover may change the inner nature of original equation. Some illustrating examples are considered as well.
We present here the explicit parametric solutions of second order differential equations invariant under time translation and rescaling and third order differential equations invariant under time translation and the two homogeneity…
We consider the family of dynamical modular curves associated to quadratic polynomial maps and determine precisely which of these curves have infinitely many cubic points. We use this to prove a classification statement on preperiodic…
A quantum phase space version of the continuity equation for systems with internal degrees of freedom is derived. The $1$ -- D Dirac equation is introduced and its phase space counterpart is found. The phase space representation of free…
We present several constructions of paths and processes with finite quadratic variation along a refining sequence of partitions, extending previous constructions to the non-uniform case. We study in particular the dependence of quadratic…
Differential constraints compatible with the linearized equations of partial differential equations are examined. Recursion operators are obtained by integrating the differential constraints.
The notion of quadratic maps between arbitrary groups appeared at several places in the literature on quadratic algebra. Here a unified extensive treatment of their properties is given; the relation with a relative version of Passi's…
The integrable time-dependent central potentials that admit linear and quadratic first integrals other than those constructed from the angular momentum are determined. It is shown explicitly that previous answers to this problem are…
A sketch of a recent approach to quantum gravity is presented which involves several unconventional aspects. The basic ingredients include: (1) Affine kinematical variables; (2) Affine coherent states; (3) Projection operator approach for…
We present a set of Feynman integrals appearing in calculations of different QED processes to the one-loop accuracy. We consider scalar, vector, and tensor integrals with two, three, four and five denominators. The cases of equal and…
Partial differential equations (PDEs) are at the heart of many mathematical and scientific advances. While great progress has been made on the theory of PDEs of standard types during the last eight decades, the analysis of nonlinear PDEs of…
We study the autonomous systems of quadratic differential equations of the form $\dot{x}_i(t)=\mathbf{x}(t)^T \mathbf{A}_i \mathbf{x}(t) + \mathbf{v}_i^T \mathbf{x}(t)$ with $\mathbf{x}(t) = (x_1(t),x_2(t),\dots,x_i(t),\dots)$ which, in…
We investigate a special class of quadratic Hamiltonians on so(4) and so(3,1) and describe Hamiltonians that have additional polynomial integrals. One of the main results is a new integrable case with an integral of sixth degree.
The Fundamental Theorem of Integral Calculus links the integrand and its antiderivative via a simple first order differential equation. A numerical solution of this ode yields the antiderivative and hence the required integral. This…
This paper is devoted to the description of the evolution of states of quantum many-particle systems within the framework of a one-particle density operator, which enables to construct the kinetic equations in scaling limits in the presence…
In this paper we establish some results about the existence and precise forms of finite order entire solutions of some systems of quadratic trinomial functional equations one of which in $\mathbb{C}^n$, $n\in\mathbb{N}$ and other two in…