Economical adjunction of square roots to groups

How large must an overgroup of a given group be in order to contain a square root of any element of the initial group? We give an almost exact answer to this question (the obtained estimate is at most twice worse than the best possible) and state several related open questions.Comment: 5 pages. A Russian version of this paper is at http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm V2: minor correction

Arithmetical properties of finite groups and their representations

The hydrogasodynamics and technology a of melt treatment out of furnace by high-implused jets of neutral gas were studied. The conception of a jet synthesis, blowing regimes and devices to solve problems of melt treatment in ladle was developed. The appropriatenesses of melt degassing and homogenizig in ladle at the blowing by trans- and ultrasonic jets of neutral gas were obtained experimentaly. The approptiatenesses of gas-powder jets outflow were studied and a new class of blowing regimes and devices for powder material injection was developed. The principles of construction and optimization of regime parameters of blowing devices, regimes and devices for melt degassing by trans- and ultrasonic jets of neutral gas to inject the powder materials in a melt, technological decisions, concerning the correction of the chemical composition and temperature of a melt in ladle were developed. The principles of a temperature melting regime optimization were developed. The results obtained were applied in the West-Siberian Metallurgical works. The increase of the metal quality allowed to decrease the reject by 20-30%. The yield of steel was increased by 0,1-0,3%, the steel temperature on tap was decreased by 10-25% the specific expense of cast iron was decreased by 2-4 kg/t, the ferroalloy expense was reduced by 3-8%. The results may be use in steelmakingAvailable from VNTIC / VNTIC - Scientific & Technical Information Centre of RussiaSIGLERURussian Federatio

Global methods for solving systems of nonlinear algebraic equations

AbstractSystems of nonlinear algebraic equations (SNAE) are ubiquitous in the many applications requiring numerical simulation, and more robust and efficient methods for solving SNAE are continuously being sought. In this paper, we present an overview of existing algorithmic approaches for solving SNAE such as reduction to a Groebner basis, the multidimensional resultant method, and the spectral method. A major deficiency in all of these methods is the lack of a theoretical foundation that will allow a priori information about the number of solutions. In the present work, we recognize that SNAE are the principal object of an algebraic geometry and seek to derive qualitative criteria about the solution in an algebraic form. Desirable qualitative criteria include solvability and uniqueness. We show here that the problem of solving SNAE is equivalent to the problem of solving matrices of rank 1 in a given subspace of matrices. Recognizing such equivalencies is an important step to future success in developing improved methods for the solution of SNAE

Inapproximability Results for Equations over Finite Groups

An equation over a finite group G is an expression of form w_1 w_2... w_k = 1_G, where each w_i is a variable, an inverted variable, or a constant from G; such an equation is satisfiable if there is a setting of the variables to values in G so that the equality is realized. We study the problem of simultaneously satisfying a family of equations over a finite group G and show that it is NP-hard to approximate the number of simultaneously satisfiable equations to within |G|-ε for any ε > 0. This generalizes results of Håstad (2001, J. ACM, 48 (4)), who established similar bounds under the added condition that the group G is Abelian