Thomas Decomposition of Algebraic and Differential Systems

In this paper we consider disjoint decomposition of algebraic and non-linear partial differential systems of equations and inequations into so-called simple subsystems. We exploit Thomas decomposition ideas and develop them into a new algorithm. For algebraic systems simplicity means triangularity, squarefreeness and non-vanishing initials. For differential systems the algorithm provides not only algebraic simplicity but also involutivity. The algorithm has been implemented in Maple

Computer classification of integrable coupled KdV-like systems

The foundations of the symmetry approach to the classification problem of integrable non-linear evolution systems are briefly described. Within the framework of the symmetry approach the ten-parametric family of the third order non-linear evolution coupled KdV-like systems is investigated. The necessary integrability conditions lead to an over-determined non-linear algebraic system. To solve that system an effective method based on its structure has been used. This allows us to obtain the complete list of integrable systems of a given type. All computation has been completed on the basis of computer algebra systems FORMAC and REDUCE

Computer algebra application for classification of integrable non-linear evolution equations

The application of computer algebra for classification of integrable non-linear evolution equations is discussed. Algorithms for testing conditions of formal integrability, to calculate the Lie-Bäcklund symmetries and conservation law densities are developed and implemented on the basis of the computer algebra system PL/FORMAC

An Algorithm to Construct Groebner Bases for Solving Integration by Parts Relations

This paper is a detailed description of an algorithm based on a generalized Buchberger algorithm for constructing Groebner-type bases associated with polynomials of shift operators. The algorithm is used for calculating Feynman integrals and has proven itself efficient in several complicated cases.Comment: LaTeX, 9 page

Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras

The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be published in Lecture Notes in Computer Scienc

Milestone Developments in Quantum Information and No-Go Theorems

In this article we present milestone developments in the theory and application 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. © 2020, Springer Nature Switzerland AG