Related papers: Generalization of the hypervirial and Feynman-Hell…
The great innovation of the Generalized Theorem is that it gives us the philosophy to work out the knowledge that the number of roots of an equation depends on the subfields of the functional terms of the equation they generate. Thus, the…
The present thesis represents developments in two main directions related to the simple Lie algebras. The first one is devoted to the representation theory of the simple Lie algebras. Specifically, we present recent results, which include…
Starting from the Mellin-Barnes integral representation of a Feynman integral depending on set of kinematic variables $z_i$, we derive a system of partial differential equations w.r.t.\ new variables $x_j$, which parameterize the…
A generalization of the Einstein equation is considered for complex line elements. Several second order semilinear partial differential equations are derived from it as semilinear field equations in uniform and isotropic spaces. The…
Generalizations of the Korteweg-de Vries equation are considered, and some explicit solutions are presented. There are situations where solutions engender the interesting property of being chiral, that is, of having velocity determined in…
We introduce a hypergoemetirc series with two complex variables, which generalizes Appell's, Lauricella's and Kemp\'e de F\'eriet's hypergeometric series, and study the system of differential equations that it satisfies. We determine the…
This dissertation investigates three main topics, all of which dealing with alternative, higher-order gravity theories in four dimensions. Firstly, we study the variational and conformal structure of those theories. Next, we analyse their…
Considered is the Schr\"odinger equation in a finite-dimensional space as an equation of mathematical physics derivable from the variational principle and treatable in terms of the Lagrange-Hamilton formalism. It provides an interesting…
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…
Seminar held at JINR, Dubna, May 15, 2012. In General Relativity, spacetime singularities raise a number of problems, both mathematical and physical. One can identify a class of singularities - with smooth but degenerate metric - which,…
In this review paper we present some basic notions about f(R) theories of gravity and some simple cosmological models derived from it. We first make an introduction to General Relativity (GR), followed by the discussion of…
We elaborate on the proposed general boundary formulation as an extension of standard quantum mechanics to arbitrary (or no) backgrounds. Temporal transition amplitudes are generalized to amplitudes for arbitrary spacetime regions. State…
It was observed that hyperlogarithms provide a tool to carry out Feynman integrals. So far, this method has been applied successfully to finite single-scale processes. However, it can be employed in more general situations. We give examples…
We shall present and analyze two examples of extended theories of gravitation in Palatini formalism with matter that couples to the connection. This will show that the class of Further Extended Theories of Gravitation introduced in ref. [1]…
The Schr\" odinger equations which are exactly solvable in terms of associated special functions are directly related to some self-adjoint operators defined in the theory of hypergeometric type equations. The fundamental formulae occurring…
We extend the classical Landesman-Lazer results to the setting of second order Hamilton-Jacobi-Bellman equations. A number of new phenomena appear.
This thesis studies general relativity (GR) using chiral formulations, which take advantage of the decomposition of the four-dimensional Lorentz group into self-dual and anti-self-dual sectors. Within this framework, GR can be expressed…
Development of quantum engineering put forward new theoretical problems. Behavior of a single mesoscopic cell (device) we may usually describe by equations of quantum mechanics. However if experimentators gather hundreds of thousands of…
The Thirring model and various generalizations of it are analyzed in detail. The four-Fermi interaction modifies the equation of state. Chemical potentials and twisted boundary conditions both result in complex fermionic determinants which…
Quantum nonrelativistic systems with $2\times2$ matrix potentials are investigated. Physically, they simulate charged or neutral fermions with non-trivial dipole momenta, interacting with an external electric field. Assuming rotationally…