arXiv subjects

Vladimir P. Gerdt

Publications and source records attributed to Vladimir P. Gerdt.

At least 19 recordsLinked to original sources

arXiv2020

Quantum Information Scrambling and Entanglement in Bipartite Quantum States

Investigating the influence of quantum information (QI) scrambling on quantum correlations in a physical system is an interesting problem. In this article we establish the mathematical connections among the quantifiers known as quantum information scrambling, Uhlmann fidelity, Bures metric and bipartite concurrence. We study these relations via four point out-of-time-order correlation (OTOC) function used for quantum information scrambling. Further we study the dynamics of all the quantifiers and investigate the influence of QI scrambling on entanglement in two qubits prepared in Bell states. We also determine the related QI scrambling and entanglement balancing points and investigate that they are periodic in nature for the Ising Hamiltonian.

quant-ph↗

arXiv2020

Milestone developments in quantum Information and No-Go theorems

In this article we present milestone developments in the theory and applications of quantum information from historical perspectives. The domain of quantum information is very promising to develop quantum computer, quantum communication and varieties of other applications of quantum technologies. We also give the light on experimental manifestations of major theoretical developments. In addition, we present important no-go theorems frequently used in quantum information along with ideas of their respective mathematical proofs.

physics.hist-ph↗

arXiv2020

Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations

For a wide class of polynomially nonlinear systems of partial differential equations we suggest an algorithmic approach that combines differential and difference algebra to analyze s(trong)-consistency of finite difference approximations. Our approach is applicable to regular solution grids. For the grids of this type we give a new definition of s-consistency for finite difference approximations which generalizes our definition given earlier for Cartesian grids. The algorithmic verification of s-consistency presented in the paper is based on the use of both differential and difference Thomas decomposition. First, we apply the differential decomposition to the input system, resulting in a partition of its solution space. Then, to the output subsystem that contains a solution of interest we apply a difference analogue of the differential Thomas decomposition which allows to check the s-consistency. For linear and some quasi-linear differential systems one can also apply difference \Gr bases for the s-consistency analysis. We illustrate our methods and algorithms by a number of examples, which include Navier-Stokes equations for viscous incompressible flow.

cs.SC↗

arXiv2019

Entanglement sudden death and birth effects in two qubits maximally entangled mixed states under quantum channels

In the present article, the robustness of entanglement in two qubits maximally entangled mixed states (MEME) have been studied under quantum decoherence channels. Here we consider bit flip, phase flip, bit-phase-flip, amplitude damping, phase damping and depolarization channels. To quantify the entanglement, the concurrence has been used as an entanglement measure. During this study interesting results have been found for sudden death and birth of entanglement under bit flip and bit-phase-flip channels. While amplitude damping channel produces entanglement sudden death and does not allow re-birth of entanglement. On the other hand, two qubits MEMS exhibit the robust character against the phase flip, phase damping and depolarization channels. The elegant behavior of all the quantum channels have been investigated with varying parameter of quantum state MEMS in different cases.

quant-ph↗

arXiv2019

Algorithmic approach to strong consistency analysis of finite difference approximations to PDE systems

For a wide class of polynomially nonlinear systems of partial differential equations we suggest an algorithmic approach to the s(trong)-consistency analysis of their finite difference approximations on Cartesian grids. First we apply the differential Thomas decomposition to the input system, resulting in a partition of the solution set. We consider the output simple subsystem that contains a solution of interest. Then, for this subsystem, we suggest an algorithm for verification of s-consistency for its finite difference approximation. For this purpose we develop a difference analogue of the differential Thomas decomposition, both of which jointly allow to verify the s-consistency of the approximation. As an application of our approach, we show how to produce s-consistent difference approximations to the incompressible Navier-Stokes equations including the pressure Poisson equation.

cs.SC↗

arXiv2018

A Strongly Consistent Finite Difference Scheme for Steady Stokes Flow and its Modified Equations

We construct and analyze a strongly consistent second-order finite difference scheme for the steady two-dimensional Stokes flow. The pressure Poisson equation is explicitly incorporated into the scheme. Our approach suggested by the first two authors is based on a combination of the finite volume method, difference elimination, and numerical integration. We make use of the techniques of the differential and difference Janet/Groebner bases. In order to prove strong consistency of the generated scheme we correlate the differential ideal generated by the polynomials in the Stokes equations with the difference ideal generated by the polynomials in the constructed difference scheme. Additionally, we compute the modified differential system of the obtained scheme and analyze the scheme's accuracy and strong consistency by considering this system. An evaluation of our scheme against the established marker-and-cell method is carried out.

math.NA↗

arXiv2018

The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs

We present the Maple package TDDS (Thomas Decomposition of Differential Systems). Given a polynomially nonlinear differential system, which in addition to equations may contain inequations, this package computes a decomposition of it into a finite set of differentially triangular and algebraically simple subsystems whose subsets of equations are involutive. Usually the decomposed system is substantially easier to investigate and solve both analytically and numerically. The distinctive property of a Thomas decomposition is disjointness of the solution sets of the output subsystems. Thereby, a solution of a well-posed initial problem belongs to one and only one output subsystem. The Thomas decomposition is fully algorithmic. It allows to perform important elements of algebraic analysis of an input differential system such as: verifying consistency, i.e., the existence of solutions; detecting the arbitrariness in the general analytic solution; given an additional equation, checking whether this equation is satisfied by all common solutions of the input system; eliminating a part of dependent variables from the system if such elimination is possible; revealing hidden constraints on dependent variables, etc. Examples illustrating the use of the package are given.

physics.comp-ph↗

arXiv2015

On the Partial Analytical Solution of the Kirchhoff Equation

We derive a combined analytical and numerical scheme to solve the (1+1)-dimensional differential Kirchhoff system. Here the object is to obtain an accurate as well as an efficient solution process. Purely numerical algorithms typically have the disadvantage that the quality of solutions decreases enormously with increasing temporal step sizes, which results from the numerical stiffness of the underlying partial differential equations. To prevent that, we apply a differential Thomas decomposition and a Lie symmetry analysis to derive explicit analytical solutions to specific parts of the Kirchhoff system. These solutions are general and depend on arbitrary functions, which we set up according to the numerical solution of the remaining parts. In contrast to a purely numerical handling, this reduces the numerical solution space and prevents the system from becoming unstable. The differential Kirchhoff equation describes the dynamic equilibrium of one-dimensional continua, i.e. slender structures like fibers. We evaluate the advantage of our method by simulating a cilia carpet.

math.AP↗

arXiv2015

Lagrangian Constraints and Differential Thomas Decomposition

In this paper we show how to compute algorithmically the full set of algebraically independent constraints for singular mechanical and field-theoretical models with polynomial Lagrangians. If a model under consideration is not singular as a whole but has domains of dynamical (field) variables where its Lagrangian becomes singular, then our approach allows to detect such domains and compute the relevant constraints. In doing so, we assume that the Lagrangian of a model is a differential polynomial and apply the differential Thomas decomposition algorithm to the Euler-Lagrange equations.

math.DS↗

arXiv2014

Lie Symmetry Analysis for Cosserat Rods

We consider a subsystem of the Special Cosserat Theory of Rods and construct an explicit form of its solution that depends on three arbitrary functions in (s,t) and three arbitrary functions in t. Assuming analyticity of the arbitrary functions in a domain under consideration, we prove that the obtained solution is analytic and general. The Special Cosserat Theory of Rods describes the dynamic equilibrium of 1-dimensional continua, i.e. slender structures like fibers, by means of a system of partial differential equations.

math.AP↗

arXiv2014

Noetherian quotients of the algebra of partial difference polynomials and Grobner bases of symmetric ideals

In this paper we develop a Grobner bases theory for ideals of partial difference polynomials with constant or non-constant coefficients. In particular, we introduce a criterion providing the finiteness of such bases when a difference ideal contains elements with suitable linear leading monomials. This can be explained in terms of Noetherianity of the corresponding quotient algebra. Among these Noetherian quotients we find finitely generated polynomial algebras where the action of suitable finite dimensional commutative algebras and in particular finite abelian groups is defined. We obtain therefore a consistent Grobner bases theory for ideals that possess such symmetries.

math.AC↗

arXiv2013

Involutive Bases Algorithm Incorporating F5 Criterion

Faugere's F5 algorithm is the fastest known algorithm to compute Groebner bases. It has a signature-based and an incremental structure that allow to apply the F5 criterion for deletion of unnecessary reductions. In this paper, we present an involutive completion algorithm which outputs a minimal involutive basis. Our completion algorithm has a nonincremental structure and in addition to the involutive form of Buchberger's criteria it applies the F5 criterion whenever this criterion is applicable in the course of completion to involution. In doing so, we use the G2V form of the F5 criterion developed by Gao, Guan and Volny IV. To compare the proposed algorithm, via a set of benchmarks, with the Gerdt-Blinkov involutive algorithm (which does not apply the F5 criterion) we use implementations of both algorithms done on the same platform in Maple.

math.AC↗

arXiv2012

Computation of Difference Groebner Bases

To compute difference Groebner bases of ideals generated by linear polynomials we adopt to difference polynomial rings the involutive algorithm based on Janet-like division. The algorithm has been implemented in Maple in the form of the package LDA (Linear Difference Algebra) and we describe the main features of the package. Its applications are illustrated by generation of finite difference approximations to linear partial differential equations and by reduction of Feynman integrals. We also present the algorithm for an ideal generated by a finite set of nonlinear difference polynomials. If the algorithm terminates, then it constructs a Groebner basis of the ideal.

cs.SC↗

arXiv2011

Consistency Analysis of Finite Difference Approximations to PDE Systems

In the given paper we consider finite difference approximations to systems of polynomially-nonlinear partial differential equations whose coefficients are rational functions over rationals in the independent variables. The notion of strong consistency which we introduced earlier for linear systems is extended to nonlinear ones. For orthogonal and uniform grids we describe an algorithmic procedure for verification of strong consistency based on computation of difference standard bases. The concepts and algorithmic methods of the present paper are illustrated by two finite difference approximations to the two-dimensional Navier-Stokes equations. One of these approximations is strongly consistent and another is not.

math.AP↗

arXiv2011

A Variant of Gerdt's Algorithm for Computing Involutive Bases

Ihe first author presented an efficient algorithm for computing involutive (and reduced Groebner) bases. In this paper, we consider a modification of this algorithm which simplifies matters to understand it and to implement. We prove correctness and termination of the modified algorithm and also correctness of the used criteria. The proposed algorithm has been implemented in Maple. We present experimental comparison, via some examples, of performance of the modified algorithm with its original form which has been implemented in Maple too. In doing so, we have taken care to provide uniform implementation details for the both algorithms.

math.RA↗

arXiv2006

Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations

In this paper we present an algorithmic approach to the generation of fully conservative difference schemes for linear partial differential equations. The approach is based on enlargement of the equations in their integral conservation law form by extra integral relations between unknown functions and their derivatives, and on discretization of the obtained system. The structure of the discrete system depends on numerical approximation methods for the integrals occurring in the enlarged system. As a result of the discretization, a system of linear polynomial difference equations is derived for the unknown functions and their partial derivatives. A difference scheme is constructed by elimination of all the partial derivatives. The elimination can be achieved by selecting a proper elimination ranking and by computing a Gröbner basis of the linear difference ideal generated by the polynomials in the discrete system. For these purposes we use the difference form of Janet-like Gröbner bases and their implementation in Maple. As illustration of the described methods and algorithms, we construct a number of difference schemes for Burgers and Falkowich-Karman equations and discuss their numerical properties.

math.RA↗

arXiv2006

On Computing Janet Bases for Degree Compatible Orderings

We consider three modifications of our involutive algorithm for computing Janet bases. These modifications are related to degree compatible monomial orders and specify selection strategies for non-multiplicative prolongations. By using the standard data base of polynomial benchmarks for \Gr bases software we compare the modifications and confront them with Magma that implements Faugère's $F_4$ algorithm.

math.AC↗

arXiv2005

A Software Package to Construct Polynomial Sets over Z_2 for Determining the Output of Quantum Computations

A C# package is presented that allows a user for an input quantum circuit to generate a set of multivariate polynomials over the finite field Z_2 whose total number of solutions in Z_2 determines the output of the quantum computation defined by the circuit. The generated polynomial system can further be converted to the canonical Groebner basis form which provides a universal algorithmic tool for counting the number of common roots of the polynomials.

quant-ph↗