Searcharxiv⌕ Search

arXiv subjects

Igor Kondrashuk

Publications and source records attributed to Igor Kondrashuk.

At least 19 recordsLinked to original sources
arXiv2026

Inverse Laplace and Mellin integral transforms modified for use in quantum communications

Integral transformations are useful mathematical tool to work out signals and wave-packets in electronic devices. They may be used in software protocols. Necessary knowledge may come from quantum field theory, in particular from quantum chromodynamics, in which the optic theorem and the renormalization group equation can be solved by a unique contour integral written in two different "dual" ways related between themselves by a complex map in the complex plane of Mellin variable. The inverse integral transformation should be modified to be applied for these contour integral solutions. These modified inverse transformations may be used in security protocols for quantum computers. Here we do a brief review of the basic integral transforms and propose their modification for the extended domains.

quant-ph↗
arXiv2025

DGLAP-BFKL duality from QCD to quantum computers

DGLAP integro-differential equation can be solved by applying certain map in the complex plane of Mellin moments. It may be re-written as Schrödinger equation for $n$ particles. By applying another map in the complex plane of Mellin moment we may re-write the DGLAP equation as a dual DGLAP equation. Regge limit of the dual DGLAP equation coincides with BFKL integro-differential equation which in turn is the Regge limit of optic theorem in quantum field theory and may be re-written as another Schrödinger equation. This means that the BFKL equation and the corresponding Schrödinger equation may be solved by the proposed method of complex mapping in the complex plane of Mellin moments. This approach may be useful in solving tasks related to quantum communication processes.

quant-ph↗
arXiv2025

A simple way to reduce the number of contours in the multi-fold Mellin-Barnes integrals

Mellin-Barnes integral representation of one-loop off-shell box massless diagram is five-fold by construction. On the other hand, it is known from the year 1992 that it may be reduced to certain two-fold Mellin-Barnes integral. We propose a way to reduce the number of the Mellin-Barnes integration contours from five to two by using the Mellin-Barnes integral representation only in combination with basic methods of mathematical analysis such as analytical regularization. We do not use any Barnes lemma to prove the reduction but we use the integral Cauchy formula instead. We recover first the well-known two-fold Mellin-Barnes representation for the one-loop triangle massless diagram and then show how the five-fold Mellin-Barnes integral representation of one-loop box diagram with all the indices 1 in four spacetime dimensions may be reduced to the two-fold Mellin-Barnes representation for one-loop triangle diagram. Singular integrals over Feynman parameters appear in the integrand of the five-fold Mellin-Barnes integral representation at the intermediate step. Such integrals should be treated as distributions with respect to certain linear combinations of the initial Mellin-Barnes integration variables in the Mellin-Barnes integrands. These distributions may be integrated out with a finite number of residues in the limit of removing the analytical regularization. We explain how to apply this strategy to an arbitrary Feynman diagram in order to reduce the number of Mellin-Barnes integration contours. On the practical side, we analyze connections between the obtained results and the knot theory, Trotter integrals, quantum computing.

hep-ph↗
arXiv2023

Finiteness of N =4 super-Yang-Mills effective action in terms of dressed N =1 superfields

We argue in favor of the independence on any scale, ultraviolet or infrared, in kernels of the effective action expressed in terms of dressed ${\cal N} =1$ superfields for the case of ${\cal N} =4$ super-Yang--Mills theory. Under ``scale independence '' of the effective action of dressed mean superfields we mean its `` finiteness in the off-shell limit of removing all the regularizations.'' This off-shell limit is scale independent because no scale remains inside these kernels after removing the regularizations. We use two types of regularization: regularization by dimensional reduction and regularization by higher derivatives in its supersymmetric form. Based on the Slavnov-Taylor identity we show that dressed fields of matter and of vector multiplets can be introduced to express the effective action in terms of them. Kernels of the effective action expressed in terms of such dressed effective fields do not depend on the ultraviolet scale. In the case of dimensional reduction, by using the developed technique we show how the problem of inconsistency of the dimensional reduction can be solved. Using Piguet and Sibold formalism, we indicate that the dependence on the infrared scale disappears off shell in both the regularizations.

hep-th↗
arXiv2023

Evaluation of the second virial coefficient for the Mie potential using the method of brackets

The second virial coefficient for the Mie potential is evaluated using the method of brackets. This method converts a definite integral into a series in the parameters of the problem, in this case this is the temperature $T$. The results obtained here are consistent with some known special cases, such as the Lenard-Jones potential. The asymptotic properties of the second virial coefficient in molecular thermodynamic systems and complex fluid modeling are described in the limiting cases of $T \rightarrow 0$ and $T \rightarrow \infty$.

math-ph↗
arXiv2023

Analytical solution to DGLAP integro-differential equation in a simple toy-model with a fixed gauge coupling

We consider a simple model for QCD dynamics in which DGLAP integro-differential equation may be solved analytically. This is a gauge model which possesses dominant evolution of gauge boson (gluon) distribution and in which the gauge coupling does not run. This may be ${\cal N} =4$ supersymmetric gauge theory with softly broken supersymmetry, other finite supersymmetric gauge theory with lower level of supersymmetry, or topological Chern-Simons field theories. We maintain only one term in the splitting function of unintegrated gluon distribution and solve DGLAP analytically for this simplified splitting function. The solution is found by use of the Cauchy integral formula. The solution restricts form of the unintegrated gluon distribution as function of momentum transfer and of Bjorken $x$. Then we consider an almost realistic splitting function of unintegrated gluon distribution as an input to DGLAP equation and solve it by the same method which we have developed to solve DGLAP equation for the toy-model. We study a result obtained for the realistic gluon distribution and find a singular Bessel-like behaviour in the vicinity of the point $x=0$ and a smooth behaviour in the vicinity of the point

hep-ph↗
arXiv2022

Convergence Arguments to Bridge Cauchy and Matérn Covariance Functions

The Matérn and the Generalized Cauchy families of covariance functions have a prominent role in spatial statistics as well as in a wealth of statistical applications. The Matérn family is crucial to index mean-square differentiability of the associated Gaussian random field; the Cauchy family is a decoupler of the fractal dimension and Hurst effect for Gaussian random fields that are not self-similar. Our effort is devoted to prove that a scale-dependent family of covariance functions, obtained as a reparameterization of the Generalized Cauchy family, converges to a particular case of the Matérn family, providing a somewhat surprising bridge between covariance models with light tails and covariance models that allow for long memory effect.

math.ST↗
arXiv2021

Mellin-Barnes and the method of brackets

The method of brackets is a method for the evaluation of definite integrals based on a small number of rules. This is employed here for the evaluation of Mellin-Barnes integral. The fundamental idea is to transform these integral representations into a bracket series to obtain their values. The expansion of the gamma function in such a series constitute the main part of this new application. The power and flexibility of this procedure is illustrated with a variety of examples.

math.CV↗
arXiv2021

A deep look into the Dagum family of isotropic covariance functions

The Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and Hurst effect for Gaussian random fields that are stationary and isotropic over Euclidean spaces. Sufficient conditions that allow for positive definiteness in Rd of the Dagum family have been proposed on the basis of the fact that the Dagum family allows for complete monotonicity under some parameter restrictions. The spectral properties of the Dagum family have been inspected to a very limited extent only, and this paper gives insight into this direction. Specifically, we study finite and asymptotic properties of the isotropic spectral density (intended as the Hankel transform) of the Dagum model. Also, we establish some closed forms expressions for the Dagum spectral density in terms of the Fox{Wright functions. Finally, we provide asymptotic properties for such a class of spectral densities.

math.ST↗
arXiv2020

Algorithm to find an all-order in the running coupling solution to an equation of the DGLAP type

We propose an algorithm to find a solution to an integro-differential equation of the DGLAP type for all the orders in the running coupling $α$ with splitting functions given at a fixed order in $α.$ Complex analysis is significantly used in the construction of the algorithm, we found a simpler way to calculate the involved integrals over contours in the complex planes than by any of the methods known at present.

hep-ph↗
arXiv2020

Analytical solution to DGLAP integro-differential equation via complex maps in domains of contour integrals

A simple model for QCD dynamics in which the DGLAP integro-differential equation may be solved analytically has been considered in our previous papers arXiv:1611.08787 [hep-ph] and arXiv:1906.07924 [hep-ph]. When such a model contains only one term in the splitting function of the dominant parton distribution, then Bessel function appears to be the solution to this simplified DGLAP equation. To our knowledge, this model with only one term in the splitting function for the first time has been proposed by Blumlein in hep-ph/9506403. In arXiv:1906.07924 [hep-ph] we have shown that a dual integro-differential equation obtained from the DGLAP equation by a complex map in the plane of the Mellin moment in this model may be considered as the BFKL equation. Then, in arXiv:1906.07924 we have applied a complex diffeomorphism to obtain a standard integral from Gradshteyn and Ryzhik tables starting from the contour integral for parton distribution functions that is usually taken by calculus of residues. This standard integral from these tables appears to be the Laplace transformation of Jacobian for this complex diffeomorphism. Here we write up all the formulae behind this trick in detail and find out certain important points for further development of this strategy. We verify that the inverse Laplace transformation of the Laplace image of the Bessel function may be represented in a form of Barnes contour integral.

hep-th↗
arXiv2019

Periodic solution and asymptotic stability for the magnetohydrodynamic equations with inhomogeneous boundary condition

We show, using the spectral Galerkin method together with compactness arguments, existence and uniqueness of the periodic strong solutions for the magnetohydrodynamics's type equations with inhomogeneous boundary conditions. Also, we study the asymptotic stability for time periodic solution for this system. In particular, when the magnetic field h(x,t) is zero, we obtain existence, uniqueness and asymptotic behavior of the strong solutions to the Navier-Stokes equations with inhomogeneous boundary conditions.

math.AP↗
arXiv2018

Multi-fold contour integrals of certain ratios of Euler gamma functions from Feynman diagrams: orthogonality of triangles

We observe a property of orthogonality of the Mellin-Barnes transformation of the triangle one-loop diagrams, which follows from our previous papers [JHEP {\bf 0808} (2008) 106, JHEP {\bf 1003} (2010) 051, JMP {\bf 51} (2010) 052304]. In those papers it has been established that Usyukina-Davydychev functions are invariant with respect to Fourier transformation. This has been proved at the level of graphs and also via the Mellin-Barnes transformation. We partially apply to one-loop massless scalar diagram the same trick in which the Mellin-Barnes transformation was involved and obtain the property of orthogonality of the corresponding MB transforms under integration over contours in two complex planes with certain weight. This property is valid in an arbitrary number of dimensions.

math-ph↗
arXiv2018

On the algorithm to find S-related Lie algebras

In this article we describe the Java library that we have recently constructed to automatize the S-expansion method, a powerful mathematical technique allowing to relate different Lie algebras. An important input in this procedure is the use of abelian semigroups and thus, we start with a brief review about the classification of non-isomorphic semigroups made in the literature during the last decades, and explain how the lists of non-isomorphic semigroups up to order 6 can be used as inputs in many of the methods of our library. After describing the main features of the classes that compose our library we present a new method called fillTemplate which tuns out to be very useful to answer whether two given algebras can be S-related.

physics.comp-ph↗
arXiv2018

On a Java library to perform S-expansions of Lie algebras

The S-expansion method is a generalization of the Inönü-Wigner (IW) contraction that allows to study new non-trivial relations between different Lie algebras. Basically, this method combines a Lie algebra $\mathcal{G}$ with a finite abelian semigroup $S$ in such a way that a new S-expanded algebra $\mathcal{G}_{S}$ can be defined. When the semigroup has a zero-element and/or a specific decomposition, which is said to be resonant with the subspace structure of the original algebra, then it is possible to extract smaller algebras from $\mathcal{G}_{S}$ which have interesting properties. Here we give a brief description of the S-expansion, its applications and the main motivations that lead us to elaborate a Java library, which automatizes this method and allows us to represent and to classify all possible S-expansions of a given Lie algebra.

math-ph↗
arXiv2017

Nearly perturbative lattice-motivated QCD coupling with zero IR limit

The product of the gluon dressing function and the square of the ghost dressing function in the Landau gauge can be regarded to represent, apart from the inverse power corrections 1/Q^{2n}, a nonperturbative generalization A(Q^2) of the perturbative QCD running coupling a(Q^2)=alpha_s(Q^2)/pi. Recent large volume lattice calculations for these dressing functions indicate that the coupling defined in such a way goes to zero as A(Q^2)~Q^2 when the squared momenta Q^2 go to zero (Q^2<<1 GeV^2). In this work we construct such a QCD coupling A(Q^2) which fulfills also various other physically motivated conditions. At high momenta it becomes the underlying perturbative coupling a(Q^2) to a very high precision. And at intermediate low squared momenta Q^2~1 GeV^2 it gives results consistent with the data of the semihadronic tau lepton decays as measured by OPAL and ALEPH. The coupling is constructed in a dispersive way, resulting as a byproduct in the holomorphic behavior of A(Q^2) in the complex Q^2-plane which reflects the holomorphic behavior of the spacelike QCD observables. Application of the Borel sum rules to tau-decay V+A spectral functions allows us to obtain values for the gluon (dimension-4) condensate and the dimension-6 condensate, which reproduce the measured OPAL and ALEPH data to a significantly better precision than the perturbative MSbar coupling approach.

hep-ph↗
arXiv2017

Explicit calculation of multi-fold contour integrals of certain ratios of Euler gamma functions. Part 1

In this paper we proceed to study properties of Mellin-Barnes (MB) transforms of Usyukina-Davydychev (UD) functions. In our previous papers [Nuclear Physics B 870 (2013) 243], [Nuclear Physics B 876 (2013) 322] we showed that multi-fold Mellin-Barnes (MB) transforms of Usyukina-Davydychev (UD) functions may be reduced to two-fold MB transforms and that the higher-order UD functions were obtained in terms of a differential operator by applying it to a slightly modified first UD function. The result is valid in $d=4$ dimensions and its analog in $d=4-2\varepsilon$ dimensions exits too [Theoretical and Mathematical Physics 177 (2013) 1515]. In [Nuclear Physics B 870 (2013) 243] the chain of recurrent relations for analytically regularized UD functions was obtained implicitly by comparing the left hand side and the right hand side of the diagrammatic relations between the diagrams with different loop orders. In turn, these diagrammatic relations were obtained due to the method of loop reduction for the triangle ladder diagrams proposed in 1983 by Belokurov and Usyukina. Here we reproduce these recurrent relations by calculating explicitly via Barnes lemmas the contour integrals produced by the left hand sides of the diagrammatic relations. In such a way we explicitly calculate a family of multi-fold contour integrals of certain ratios of Euler gamma functions. We make a conjecture that similar results for the contour integrals are valid for a wider family of smooth functions which includes the MB transforms of UD functions.

math-ph↗