In case of sequence of matrices we define the characteristic exponent ( ) ( ) N n n ∈ Δ = Δ ( ) Δ λ as a exponent of ( ) ( ) N n n ∈ Δ . It is well known [2] that the set of all Lyapunov exponents of system (1) contains at most s elements, say ) ( ... ) ( ) ( 2 1 A A A r λ λ λ < < < and the set { } ) ( ),..., ( 1 A A r λ λ is called the spectrum of (1). The greatest and the smallest exponent of...

After many unsuccessful attempts to obtain a proof in the late 30s-early 40s the following weaker version of The Burnside Problem was studied: Is it true that there are only finitely many 7??-generated finite groups of exponent nl In other words the question is whether there exists a universal finite m-generated group of exponent n having all other finite m-generated groups of exponent n as hom...

We study equations of the form (= x) which are single axioms for groups of exponent 4, where is a term in product only. Every such must have at least 9 variable occurrences, and there are exactly three such of this size, up to variable renaming and mirroring. These terms were found by an exhaustive search through all terms of this form. Automated techniques were used in two ways: to eliminate m...

A sufficient condition for the existence of HNN-extensions in the class of groups of odd exponent n ≫ 1 is given in the following form. Let Q be a group of odd exponent n > 2 and G be an HNN-extension of Q. If A ∈ G then let F(A) denote the maximal subgroup of Q which is normalized by A. By τA denote the automorphism of F(A) which is induced by conjugation by A. Suppose that for every A ∈ G, wh...

This paper designs and analyzes a quantum algorithm to solve a system of m quadratic equations in n variables over a finite field Fq. In the case m = n and q = 2, under standard assumptions, the algorithm takes time 2 on a mesh-connected computer of area 2, where t ≈ 0.45743 and a ≈ 0.01467. The area-time product has asymptotic exponent t+ a ≈ 0.47210. For comparison, the area-time product of G...