Automorphisms of the Planar Tree Power Series Algebra and the Non-Associative Logarithm

2000 Mathematics Subject Classification: 17A50, 05C05.

A Characterization of Varieties of Associative Algebras of Exponent two

∗The first author was partially supported by MURST of Italy; the second author was par- tially supported by RFFI grant 99-01-00233.

A Computer Algebra Application to Determination of Lie Symmetries of Partial Differential Equations

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006

Exponents of Subvarieties of Upper Triangular Matrices over Arbitrary Fields are Integral

Partially supported by grant RFFI 98-01-01020.

The Automorphism Group of the Free Algebra of Rank Two

The theorem of Czerniakiewicz and Makar-Limanov, that all the automorphisms of a free algebra of rank two are tame is proved here by showing that the group of these automorphisms is the free product of two groups (amalgamating their intersection), the group of all affine automorphisms and the group of all triangular automorphisms. The method consists in finding a bipolar structure. As a consequence every finite subgroup of automorphisms (in characteristic zero) is shown to be conjugate to a group of linear automorphisms.

Operational Methods in the Environment of a Computer Algebra System

This article presents the principal results of the doctoral thesis “Direct Operational Methods in the Environment of a Computer Algebra System” by Margarita Spiridonova (Institute of mathematics and Informatics, BAS), successfully defended before the Specialised Academic Council for Informatics and Mathematical Modelling on 23 March, 2009.

Quasi-Monte Carlo Methods for some Linear Algebra Problems. Convergence and Complexity

We present quasi-Monte Carlo analogs of Monte Carlo methods for some linear algebra problems: solving systems of linear equations, computing extreme eigenvalues, and matrix inversion. Reformulating the problems as solving integral equations with a special kernels and domains permits us to analyze the quasi-Monte Carlo methods with bounds from numerical integration. Standard Monte Carlo methods for integration provide a convergence rate of O(N^(−1/2)) using N samples. Quasi-Monte Carlo methods use quasirandom sequences with the resulting convergence rate for numerical integration as good as O((logN)^k)N^(−1)). We have shown theoretically and through numerical tests that the use of quasirandom sequences improves both the magnitude of the error and the convergence rate of the considered Monte Carlo methods. We also analyze the complexity of considered quasi-Monte Carlo algorithms and compare them to the complexity of the analogous Monte Carlo and deterministic algorithms.

Applying Associative Classifier PGN for Digitised Cultural Heritage Resource Discovery

Resource discovery is one of the key services in digitised cultural heritage collections. It requires intelligent mining in heterogeneous digital content as well as capabilities in large scale performance; this explains the recent advances in classification methods. Associative classifiers are convenient data mining tools used in the field of cultural heritage, by applying their possibilities to taking into account the specific combinations of the attribute values. Usually, the associative classifiers prioritize the support over the confidence. The proposed classifier PGN questions this common approach and focuses on confidence first by retaining only 100% confidence rules. The classification tasks in the field of cultural heritage usually deal with data sets with many class labels. This variety is caused by the richness of accumulated culture during the centuries. Comparisons of classifier PGN with other classifiers, such as OneR, JRip and J48, show the competitiveness of PGN in recognizing multi-class datasets on collections of masterpieces from different West and East European Fine Art authors and movements.

On Generalized Models and Singular Products of Distributions in Colombeau Algebra G(R)

MSC 2010: 46F30, 46F10

Convexity around the Unit of a Banach Algebra

2000 Mathematics Subject Classification: Primary: 46B20. Secondary: 46H99, 47A12.