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Über dieses Buch

This book constitutes the proceedings of the 16th International Workshop on Computer Algebra in Scientific Computing, CASC 2014, held in Warsaw, Poland, in September 2014. The 33 full papers presented were carefully reviewed and selected for inclusion in this book.

The papers address issues such as Studies in polynomial algebra are represented by contributions devoted to factoring sparse bivariate polynomials using the priority queue, the construction of irreducible polynomials by using the Newton index, real polynomial root finding by means of matrix and polynomial iterations, application of the eigenvalue method with symmetry for solving polynomial systems arising in the vibration analysis of mechanical structures with symmetry properties, application of Gröbner systems for computing the (absolute) reduction number of polynomial ideals, the application of cylindrical algebraic decomposition for solving the quantifier elimination problems, certification of approximate roots of overdetermined and singular polynomial systems via the recovery of an exact rational univariate representation from approximate numerical data, new parallel algorithms for operations on univariate polynomials (multi-point evaluation, interpolation) based on subproduct tree techniques.



Computable Infinite Power Series in the Role of Coefficients of Linear Differential Systems

We consider linear ordinary differential systems over a differential field of characteristic 0. We prove that testing unimodularity and computing the dimension of the solution space of an arbitrary system can be done algorithmically if and only if the zero testing problem in the ground differential field is algorithmically decidable. Moreover, we consider full-rank systems whose coefficients are computable power series and we show that, despite the fact that such a system has a basis of formal exponential-logarithmic solutions involving only computable series, there is no algorithm to construct such a basis.

Sergei A. Abramov, Moulay A. Barkatou

Relation Algebra, RelView, and Plurality Voting

We demonstrate how relation algebra and a supporting tool can be combined to solve problems of voting systems. We model plurality voting within relation algebra and present relation-algebraic specifications for some computational tasks. They can be transformed immediately into the programming language of the BDD-based Computer Algebra system


, such that this tool can be used to solve the problems in question and to visualize the computed results. The approach is extremely formal, very flexible and especially appropriate for prototyping, experimentation, scientific research, and education.

Rudolf Berghammer

An Algorithm for Converting Nonlinear Differential Equations to Integral Equations with an Application to Parameter Estimation from Noisy Data

This paper provides a contribution to the parameter estimation methods for nonlinear dynamical systems. In such problems, a major issue is the presence of noise in measurements. In particular, most methods based on numerical estimates of derivations are very noise sensitive. An improvement consists in using integral equations, acting as noise filtering, rather than differential equations. Our contribution is a pair of algorithms for converting fractions of differential polynomials to integral equations. These algorithms rely on an improved version of a recent differential algebra algorithm. Their usefulness is illustrated by an application to the problem of estimating the parameters of a nonlinear dynamical system, from noisy data.

François Boulier, Anja Korporal, François Lemaire, Wilfrid Perruquetti, Adrien Poteaux, Rosane Ushirobira

Truth Table Invariant Cylindrical Algebraic Decomposition by Regular Chains

A new algorithm to compute cylindrical algebraic decompositions (CADs) is presented, building on two recent advances. Firstly, the output is truth table invariant (a TTICAD) meaning given formulae have constant truth value on each cell of the decomposition. Secondly, the computation uses regular chains theory to first build a cylindrical decomposition of complex space (CCD) incrementally by polynomial. Significant modification of the regular chains technology was used to achieve the more sophisticated invariance criteria. Experimental results on an implementation in the


Library for


verify that combining these advances gives an algorithm superior to its individual components and competitive with the state of the art.

Russell Bradford, Changbo Chen, James H. Davenport, Matthew England, Marc Moreno Maza, David Wilson

Computing the Topology of an Arrangement of Implicit and Parametric Curves Given by Values

Curve arrangement studying is a subject of great interest in Computational Geometry and CAGD. In our paper, a new method for computing the topology of an arrangement of algebraic plane curves, defined by implicit and parametric equations, is presented. The polynomials appearing in the equations are given in the Lagrange basis, with respect to a suitable set of nodes. Our method is of sweep-line class, and its novelty consists in applying algebra by values for solving systems of two bivariate polynomial equations. Moreover, at our best knowledge, previous works on arrangements of curves consider only implicitly defined curves.

Jorge Caravantes, Mario Fioravanti, Laureano Gonzalez–Vega, Ioana Necula

Finding a Deterministic Generic Position for an Algebraic Space Curve

Checking whether an algebraic space curve is in a generic position or not is an important step for computing the topology of real algebraic space curve. In this paper, we present an algorithm to find a deterministic generic position for an algebraic space curve.

Jin-San Cheng, Kai Jin

Optimal Estimations of Seiffert-Type Means By Some Special Gini Means

Let us consider the logarithmic mean


the identric mean


the trigonometric means




defined by H. J. Seiffert, the hyperbolic mean


defined by E. Neuman and J. Sándor, and the Gini mean


. The optimal estimations of these means by power means


and also some of the optimal estimations by Lehmer means


are known. We prove the rest of optimal estimations by Lehmer means and the optimal estimations by some other special Gini means


. In proving some of the results we used the computer algebra system


. We believe that some parts of the proofs couldn’t be done without the help of such a computer algebra system (at least by following our way of proving those results).

Iulia Costin, Gheorghe Toader

CAS Application to the Construction of High-Order Difference Schemes for Solving Poisson Equation

In the present work, a computer algebra system (CAS) is applied for constructing a new difference scheme of high-accuracy order for solving boundary-value problem for Poisson equation. The formulae of the difference scheme are constructed in the symbol form in CAS. CAS are used for translation of complex formulae to C++ language operators, calculation of arithmetic values of the constructed scheme coefficients and matrix elements of a system of linear algebraic equations for the discrete problem approximating the initial difference problem. Efficiency of the CAS application and of the schemes constructed with its help is shown.

Grigoriy M. Drozdov, Vasily P. Shapeev

On Symbolic Solutions of Algebraic Partial Differential Equations

In this paper we present a general procedure for solving first-order autonomous algebraic partial differential equations in two independent variables. The method uses proper rational parametrizations of algebraic surfaces and generalizes a similar procedure for first-order autonomous ordinary differential equations. We will demonstrate in examples that, depending on certain steps in the procedure, rational, radical or even non-algebraic solutions can be found. Solutions computed by the procedure will depend on two arbitrary independent constants.

Georg Grasegger, Alberto Lastra, J. Rafael Sendra, Franz Winkler

Eigenvalue Method with Symmetry and Vibration Analysis of Cyclic Structures

We present an application of the eigenvalue method with symmetry for solving polynomial systems arising in the vibration analysis of mechanical structures with symmetry properties. The search for solutions is conducted by the so called multiplication matrix method in which the symmetry of the system is taken into account by introducing a symmetry group and by working with the set of invariant polynomials under the action of this group. Using this method, we compute the periodic solutions of a simple dynamic system modeling a cyclic mechanical structure subjected to nonlinearities.

Aurelien Grolet, Philippe Malbos, Fabrice Thouverez

Symbolic-Numerical Solution of Boundary-Value Problems with Self-adjoint Second-Order Differential Equation Using the Finite Element Method with Interpolation Hermite Polynomials

We present a symbolic algorithm generating finite-element schemes with interpolating Hermite polynomials intended for solving the boundary-value problems with self-adjoint second-order differential equation and implemented in the Maple computer algebra system. Recurrence relations for the calculation in analytical form of the interpolating Hermite polynomials with nodes of arbitrary multiplicity are derived. The integrals of interpolating Hermite polynomials are used for constructing the stiffness and mass matrices and formulating a generalized algebraic eigenvalue problem. The algorithm is used to generate Fortran routines that allow solution of the generalized algebraic eigenvalue problem with matrices of large dimension. The efficiency of the programs generated in Maple and Fortran is demonstrated by the examples of exactly solvable quantum-mechanical problems with continuous and piecewise continuous potentials.

Alexander A. Gusev, Ochbadrakh Chuluunbaatar, Sergue I. Vinitsky, Vladimir L. Derbov, Andrzej Góźdź, Luong Le Hai, Vitaly A. Rostovtsev

Sporadic Examples of Directed Strongly Regular Graphs Obtained By Computer Algebra Experimentation

We report about the results of the application of modern computer algebra tools for construction of directed strongly regular graphs. The suggested techniques are based on the investigation of noncommutative association schemes and Cayley graphs over non-Abelian groups. We demonstrate examples of directed strongly regular graphs for 28 different parameter sets, for which the existence of a corresponding digraph has not been known before.

Štefan Gyürki, Mikhail Klin

On the Parallelization of Subproduct Tree Techniques Targeting Many-Core Architectures

We propose parallel algorithms for operations on univariate polynomials (multi-point evaluation, interpolation) based on subproduct tree techniques and targeting many-core GPUs. On those architectures, we demonstrate the importance of adaptive algorithms, in particular the combination of parallel plain arithmetic and parallel FFT-based arithmetic. Experimental results illustrate the benefits of our algorithms.

Sardar Anisul Haque, Farnam Mansouri, Marc Moreno Maza

Deterministically Computing Reduction Numbers of Polynomial Ideals

We discuss the problem of determining reduction numbers of a polynomial ideal




variables. We present two algorithms based on parametric computations. The first one determines the absolute reduction number of


and requires computations in a polynomial ring with (






parameters and




variables. The second one computes via a Gröbner system the set of all reduction numbers of the ideal


and thus in particular also its big reduction number. However, it requires computations in a ring with




parameters and



Amir Hashemi, Michael Schweinfurter, Werner M. Seiler

A Note on Global Newton Iteration Over Archimedean and Non-Archimedean Fields

In this paper, we study iterative methods on the coefficients of the

rational univariate representation (RUR)

of a given algebraic set, called a

global Newton iterations

. We compare two natural approaches to define locally quadratically convergent iterations: the first one involves Newton iteration applied to the approximate roots individually and then interpolation to find the RUR of these approximate roots; the second one considers the coefficients in the exact RUR as zeroes of a high dimensional map defined by polynomial reduction and applies Newton iteration on this map. We prove that over fields with a p-adic valuation these two approaches give the same iteration function. However, over fields equipped with the usual Archimedean absolute value they are not equivalent. In the latter case, we give explicitly the iteration function for both approaches. Finally, we analyze the parallel complexity of the different versions of the global Newton iteration, compare them, and demonstrate that they can be efficiently computed. The motivation for this study comes from the certification of approximate roots of overdetermined and singular polynomial systems via the recovery of an exact RUR from approximate numerical data.

Jonathan D. Hauenstein, Victor Y. Pan, Agnes Szanto

Invariant Manifolds in the Classic and Generalized Goryachev–Chaplygin Problem

With the aid of computer algebra methods, we have conducted qualitative analysis of the phase space for the classic and generalized Goryachev–Chaplygin problem. In particular, we have found a series of new invariant manifolds of various dimension which possess some extremal property. Motions on a one-dimensional invariant manifold have been investigated. It was shown that these motions are asymptotically stable on this manifold, and one of equilibrium points on the manifold is a limit point for these motions.

Valentin Irtegov, Tatyana Titorenko

Coherence and Large-Scale Pattern Formation in Coupled Logistic-Map Lattices via Computer Algebra Systems

Three quantitative measures of the spatiotemporal behavior of the coupled map lattices: reduced density matrix, reduced wave function, and an analog of particle number, have been introduced. Making extensive use of two computer algebra systems (Maxima and Mathematica) various properties of the above mentioned parameters have been thoroughly studied. Their behavior suggests that the logistic coupled-map lattices approach the states which resemble the condensed states of systems of Bose particles. In addition, pattern formation in two-dimensional coupled map lattices based on the logistic mapping has been investigated with respect to the non-linear parameter, the diffusion constant and initial as well as boundary conditions.

Maciej Janowicz, Arkadiusz Orłowski

On the Computation of the Determinant of a Generalized Vandermonde Matrix

“Vandermonde” matrix is a matrix whose (




)th entry is in the form of


. The matrix has a lot of applications in many fields such as signal processing and polynomial interpolations. This paper generalizes the matrix, and let its (




) entry be






) where





) is a polynomial of


. We present an efficient algorithm to compute the determinant of the generalized Vandermonde matrix. The algorithm is composed of two sub-algorithms: the one that depends on given polynomials





) and the one that does not. The latter algorithm (the one does not depend on





)) can be performed beforehand, and the former (the one that depends on





)) is mainly composed of the computation of determinants of numerical matrices. Determinants of the generalized Vandermonde matrices can be used, for example, to compute the optimal






norm of a system achievable by a static feedback controller (for details, see [18],[19]).

Takuya Kitamoto

Towards Conflict-Driven Learning for Virtual Substitution

We consider satisfiability modulo theory-solving for linear real arithmetic. Inspired by related work for the Fourier–Motzkin method, we combine virtual substitution with learning strategies. For the first time, we present virtual substitution—including our learning strategies—as a formal calculus. We prove soundness and completeness for that calculus. Some standard linear programming benchmarks computed with an experimental implementation of our calculus show that the integration of learning techniques into virtual substitution gives rise to considerable speedups. Our implementation is open-source and freely available.

Konstantin Korovin, Marek Kos̆ta, Thomas Sturm

Sharpness in Trajectory Estimation for Planar Four-points Piecewise-Quadratic Interpolation

This paper discusses the problem of fitting non-parametric planar data

$Q_m = \{q_i\}^m_{i=0}$

with four-points piecewise-quadratic interpolant to estimate an unknown convex curve


in Euclidean space



sampled more-or-less uniformly. The derivation of the interpolant involves non-trivial algebraic and symbolic computations. As it turns out, exclusive symbolic computations with

Wolfram Mathematica 9

are unable to explicitly construct the interpolant in question. The alternative solution involves human and computer interaction. The theoretical asymptotic analysis concerning this interpolation scheme as already demonstrated yields quartic orders of convergence for trajectory estimation. This paper verifies in affirmative the sharpness of the above asymptotics via numerical tests and independently via analytic proof based on symbolic computations. Finally, we prove the necessity of admitting more-or-less uniformity and strict convexity to attain at least quartic order of convergence for trajectory approximation. In case of violating strict convexity of


we propose a corrected interpolant


which preserves quartic order of convergence.

Ryszard Kozera, Lyle Noakes, Piotr Szmielew

Scheme for Numerical Investigation of Movable Singularities of the Complex Valued Solutions of Ordinary Differential Equations

We present structure of integration scheme suitable for ordinary differential equations in some bounded region of the complex plane. The program which bases on these ideas can help to obtain qualitative information about the structure of singularities of solutions in the complex plane. It was tested on two representative examples.

Radosław Antoni Kycia

Generalized Mass-Action Systems and Positive Solutions of Polynomial Equations with Real and Symbolic Exponents (Invited Talk)

Dynamical systems arising from chemical reaction networks with mass action kinetics are the subject of chemical reaction network theory (CRNT). In particular, this theory provides statements about uniqueness, existence, and stability of positive steady states for all rate constants and initial conditions. In terms of the corresponding polynomial equations, the results guarantee uniqueness and existence of positive solutions for all positive parameters.

We address a recent extension of CRNT, called generalized mass-action systems, where reaction rates are allowed to be power-laws in the concentrations. In particular, the (real) kinetic orders can differ from the (integer) stoichiometric coefficients. As with mass-action kinetics, complex balancing equilibria are determined by the graph Laplacian of the underlying network and can be characterized by binomial equations and parametrized by monomials. In algebraic terms, we focus on a constructive characterization of positive solutions of polynomial equations with real and symbolic exponents.

Uniqueness and existence for all rate constants and initial conditions additionally depend on sign vectors of the stoichiometric and kinetic-order subspaces. This leads to a generalization of Birch’s theorem, which is robust with respect to certain perturbations in the exponents. In this context, we discuss the occurrence of multiple complex balancing equilibria.

We illustrate our results by a running example and provide a MAPLE worksheet with implementations of all algorithmic methods.

Stefan Müller, Georg Regensburger

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 (




) and three arbitrary function in


. 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.

Dominik L. Michels, Dmitry A. Lyakhov, Vladimir P. Gerdt, Gerrit A. Sobottka, Andreas G. Weber

Real Polynomial Root-Finding by Means of Matrix and Polynomial Iterations

Frequently one seeks approximation to all


real roots of a polynomial of degree


with real coefficients, which also has nonreal roots. We split a polynomial into two factors, one of which has degree


and has


real roots. We approximate them at a low cost, and then decrease the arithmetic time of the known algorithms for this popular problem by roughly a factor of




, if


iterations prepare splitting.


is a small integer unless some nonreal roots lie close to the real axis, but even if there nonreal roots near the real axis, we substantially accelerate the known algorithms. We also propose a dual algorithm, operating with the associated structured matrices. At the price of minor increase of the arithmetic time, it facilitates numerical implementation. Our analysis and tests demonstrate the efficiency of our approach.

Victor Y. Pan

On Testing Uniqueness of Analytic Solutions of PDE with Boundary Conditions

We consider linear partial differential equations with polynomial coefficients and prove algorithmic undecidability of the following problem: to test whether a given equation of considered form has no more than one solution that is analytic on a domain and that satisfies some fixed boundary conditions. It is assumed that a polynomial which vanishes at each point of the domain boundary is known.

Serge V. Paramonov

Continuous Problems: Optimality, Complexity, Tractability (Invited Talk)

Information-based complexity



) is a branch of computational complexity that studies continuous problems for which available information is partial, noisy, and priced. We present basic ideas of IBC and give some important results on optimal algorithms, complexity, and tractability of such problems. The focus is on numerical integration of univariate and multivariate functions.

Leszek Plaskota

On Integrability of Evolutionary Equations in the Restricted Three-Body Problem with Variable Masses

The satellite version of the restricted three-body problem formulated on the basis of classical Gylden–Meshcherskii problem is considered. Motion of the point



of infinitesimal mass about the point



is described in the first approximation in terms of the osculating elements of the aperiodic quasi-conical motion, and an influence of the point



gravity on this motion is analyzed. Long-term evolution of the orbital elements is determined by the differential equations written in the Hill approximation and averaged over the mean anomalies of points






. Integrability of the evolutionary equations is analyzed, and the laws of mass variation have been found for which the evolutionary equations are integrable. All relevant symbolic calculations and visualizations are done with the computer algebra system Mathematica.

Alexander N. Prokopenya, Mukhtar Zh. Minglibayev, Baglan A. Beketauov

Factoring Sparse Bivariate Polynomials Using the Priority Queue

We revisit the polytope method for factoring sparse bivariate polynomials over finite fields, and address the bottleneck arising from solving the Hensel lifting equations using the sparse distributed polynomial representation. We revise the analysis when polynomials are represented as such, which reveals how performing the polynomial multiplications and ensuing additions in separate (serialised) phases causes the Hensel lifting phase to suffer from poor work, space, and I/O complexity, and hinges on the size of the intermediary output, as size is defined in the sparse distributed representation. We propose to overlap all polynomial arithmetic in one Hensel lifting step using a MAX priority queue. The overlapping approach adapts not only to the growth in the degree of the input polynomial but also to irregularities in the sparsity of intermediary output. It also results in evading expression swell and reducing the overall work and space complexity by an order of magnitude. When the priority queue is implemented as a cache-oblivious data structure, the overlapping approach achieves an order of magnitude improvement in I/O over the serialised approach, even when the latter is using cache efficient structures to assist in polynomial multiplications and additions. We present empirical results for the polytope method using a max-heap implementation of the global priority queue, which demonstrate extremely superior performance, and specifically against Magma, for sufficiently sparse input polynomials of very high degrees.

Fatima K. Abu Salem, Khalil El-Harake, Karl Gemayel

Solving Parametric Sparse Linear Systems by Local Blocking

In solving parametric sparse linear systems, we want 1) to know relations on parametric coefficients which change the system largely, 2) to express the parametric solution in a concise form suitable for theoretical and numerical analysis, and 3) to find simplified systems which show characteristic features of the system. The block triangularization is a standard technique in solving the sparse linear systems. In this paper, we attack the above problems by introducing a concept of local blocks. The conventional block corresponds to a strongly connected maximal subgraph of the associated directed graph for the coefficient matrix, and our local blocks correspond to strongly connected non-maximal subgraphs. By determining local blocks in a nested way and solving subsystems from low to higher ones, we replace sub-expressions by solver parameters systematically, obtaining the solution in a concise form. Furthermore, we show an idea to form simple systems which show characteristic features of the whole system.

Tateaki Sasaki, Daiju Inaba, Fujio Kako

Analytical Calculations in Maple to Implement the Method of Adiabatic Modes for Modelling Smoothly Irregular Integrated Optical Waveguide Structures

This paper presents analytical calculations in CAS Maple. The calculations are used for the method of adiabatic waveguide modes, applied in mathematical modeling of smoothly irregular integrated-optical waveguides. Such structures drew researchers’ attention at the end of the 20th century, when several models were proposed to describe the coherent laser radiation in such structures. But these models could not adequately characterize the phenomena of depolarization and hybridization of guided modes.

The proposed model of adiabatic waveguide modes, on the contrary, describes these experimentally observed phenomena, but on the other hand, we have more complicated analytical expressions. So we have developed a special program in Maple to computerize analytical calculations. The program is presented in this work.

Leonid A. Sevastyanov, Anton L. Sevastyanov, Anastasiya A. Tyutyunnik

CAS Application to the Construction of the Collocations and Least Residuals Method for the Solution of the Burgers and Korteweg–de Vries–Burgers Equations

In the present work, the computer algebra system (CAS) is applied for constructing a new version of the analytic-numerical method of collocations and least residuals (CLR) for solving the Burgers equation and the Korteweg–de Vries–Burgers equation. The CAS is employed at all stages from writing, deriving, and verifying the formulas of the method to their translation into arithmetic operators of the Fortran language. The verification of derived formulas of the method has been done on the test problem solutions. Comparisons of the results of numerical computations by the new CLR method with the exact solutions of test problems show a high accuracy of the developed method.

Vasily P. Shapeev, Evgenii V. Vorozhtsov

An Algorithm for Computing the Truncated Annihilating Ideals for an Algebraic Local Cohomology Class



be an algebraic local cohomology class, and


a natural number. The purpose of this paper is to present an algorithm for computing the right




generated by linear differential operators annihilating


and of order less than or equal to


. This algorithm is based on Matlis duality theorem, and is applicable to the case where


has parameters in its coefficients. Our main interest is where algebraic local cohomology classes


is a generator of the dual space of the Milnor algebra of a hypersurface isolated singularity.

Takafumi Shibuta, Shinichi Tajima

Applications of the Newton Index to the Construction of Irreducible Polynomials

We use properties of the Newton index associated to a polynomial with coefficients in a discrete valuation domain for generating classes of irreducible polynomials. We obtain factorization properties similar to the case of bivariate polynomials and we give new applications to the construction of families of irreducible polynomials over various discrete valuation domains. The examples are obtained using the package



Doru Ştefănescu

Symbolic-Numeric Algorithm for Solving the Problem of Quantum Tunneling of a Diatomic Molecule through Repulsive Barriers

Symbolic-numeric algorithm for solving the boundary-value problems that describe the model of quantum tunneling of a diatomic molecule through repulsive barriers is described. Two boundary-value problems (BVPs) in Cartesian and polar coordinates are formulated and reduced to 1D BVPs for different systems of coupled second-order differential equations (SCSODEs) that contain potential matrix elements with different asymptotic behavior. A symbolic algorithm implemented in CAS Maple to calculate the required asymptotic behavior of adiabatic basis, the potential matrix elements, and the fundamental solutions of the SCSODEs is elaborated. Comparative analysis of the potential matrix elements calculated in the Cartesian and polar coordinates is presented. Benchmark calculations of quantum tunneling of a diatomic molecule with the nuclei coupled by Morse potential through Gaussian barriers below dissociation threshold are carried out in Cartesian and polar coordinates using the finite element method, and the results are discussed.

Sergue Vinitsky, Alexander Gusev, Ochbadrakh Chuluunbaatar, Luong Le Hai, Andrzej Góźdź, Vladimir Derbov, Pavel Krassovitskiy

Enumeration of Schur Rings Over Small Groups

By optimizing the algorithms used in COCO and COCO-II, we enumerated all Schur rings over the groups of orders up to 63. A few statistical views of results with respect to Schur property, amount and type of generators and primitivity are presented.

Discussion of the details of the old algorithms and the improvements we implemented in order to achieve those results is included. We compare the results to similar computerized efforts (Hanaki and Miyamoto, Pech and Reichard, Heinze), as well as to theoretical classifications of Schur groups.

The computer based results may assist the theoretical efforts to classify all Schur groups, over abelian and non-abelian groups.

Matan Ziv-Av


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