arXiv subjects

I. Kondrashuk

Publications and source records attributed to I. Kondrashuk.

4 recordsLinked to original sources

arXiv2019

Zastavnyi Operators and Positive Definite Radial Functions

Positive definite functions are fundamental to many areas of applied mathematics, probability theory, spatial statistics and machine learning, amogst others. Motivated by a problem coming from the maximum likelihood estimation under fixed domain asymptotics, we consider a new operator acting on rescaled weighted differences between two members of the class $Φ_d$ of positive definite radial functions,. In particular, we study the positive definiteness of the operator for the Matérn, Generalized Cauchy and Generalized Wendland families. It turns out that proposed operator allows to govern differentiability at the origin, and to attain negative correlations.

math.SP↗

arXiv2017

A Java library to perform S-expansions of Lie algebras

The contraction method is a procedure that allows to establish non-trivial relations between Lie algebras and has had succesful applications in both mathematics and theoretical physics. This work deals with generalizations of the contraction procedure with a main focus in the so called S-expansion method as it includes most of the other generalized contractions. Basically, the S-exansion combines a Lie algebra $\mathcal{G}$ with a finite abelian semigroup $S$ in order to define new S-expanded algebras. After giving a description of the main ingredients used in this paper, we present a Java library that automatizes the S-expansion procedure. With this computational tool we are able to represent Lie algebras and semigroups, so we can perform S-expansions of Lie algebras using arbitrary semigroups. We explain how the library methods has been constructed and how they work; then we give a set of example programs aimed to solve different problems. They are presented so that any user can easily modify them to perform his own calculations, without being necessarily an expert in Java. Finally, some comments about further developements and possible new applications are made.

cs.MS↗

arXiv2015

A Clifford Bundle Approach to the Wave Equation of a Spin 1/2 Fermion in the de Sitter Manifold

In this paper we give a Clifford bundle motivated approach to the wave equation of a free spin $1/2$ fermion in the de Sitter manifold, a brane with topology $M=\mathrm{S0}(4,1)/\mathrm{S0}(3,1)$ living in the bulk spacetime $\mathbb{R}^{4,1}=(\mathring{M}=\mathbb{R}^{5},\boldsymbol{\mathring{g}})$ and equipped with a metric field $\boldsymbol{g:=-i}^{\ast}\boldsymbol{\mathring{g}%}$ with $\boldsymbol{i}:M\rightarrow\mathring{M}$ being the inclusion map. To obtain the analog of Dirac equation in Minkowski spacetime in the structure $\mathring{M}$ we appropriately factorize the two Casimir invariants $C_{1}$ and $C_{2}$ of the Lie algebra of the de Sitter group using the constraint given in the linearization of $C_{2}$ as input to linearize $C_{1}$. In this way we obtain an equation that we called \textbf{DHESS1,}which in previous studies by other authors was simply postulated.$.$Next we derive a wave equation (called \textbf{DHESS2}) for a free spin $1/2$ fermion in the de Sitter manifold using a heuristic argument which is an obvious generalization of a heuristic argument (described in detail in Appendix D) permitting a derivation of the Dirac equation in Minkowski spacetime and which shows that such famous equation express nothing more than the fact that the momentum of a free particle is a constant vector field over timelike \ integral curves of a given velocity field. It is a remarkable fact that \textbf{DHESS1}and \textbf{DHESS2}\ coincide. One of the main ingredients in our paper is the use of the concept of Dirac-Hestenes spinor fields. Appendices B and C recall this concept and its relation with covariant Dirac spinor fields usualy used by physicists.

math-ph↗

arXiv2002

QCD effective action with dressing functions - consistency checks in the perturbative regime

In a previous paper, we presented solution to the Slavnov--Taylor identity for the QCD effective action, and argued that the action terms containing (anti)ghost fields are unique. These terms have the same form as those in the classical action, but the gluon and (anti)ghost effective fields are convoluted with gluon and ghost dressing functions G_A and G_c, the latter containing perturbative and nonperturbative effects (but not including the soliton-like vacuum effects). In the present work we show how the perturbative QCD (pQCD) can be incorporated into the framework of this action, and we present explicit one-loop pQCD expressions for G_A and G_c. We then go on to check the consistency of the obtained results by considering an antighost Dyson--Schwinger equation (DSE). By solving the relations that result from the Legendre transformation leading to the effective action, we obtain the effective fields as power expansions of sources. We check explicitly that the aforementioned one-loop functions G_A and G_c fulfil the antighost DSE at the linear source level. We further explicitly check that these one-loop G_A and G_c have the regularization-scale and momentum dependence consistent with the antighost DSE at the quadratic source level. These checks suggest that the the effective action with dressing functions represents a consistent framework for treating QCD, at least at the one-loop level.

hep-ph↗