On Lyapunov scaling factors of real symmetric matrices

A necessary and sufficient condition for the identity matrix to be the unique Lyapunov scaling factor of a given real symmetric matrix A is given. This uniqueness is shown to be equivalent to the uniqueness of the identity matrix as a scaling D for which the kernels of A and AD are identical.     

On the nonsingular symmetric factors of a real matrix

For a given real square matrix A this paper describes the following matrices: (¹) all nonsingular real symmetric (r.s.) matrices S such that A = S^?1T for some symmetric matrix T.All the signatures (defined as the absolute value of the difference of the number of positive eigenvalues and the number of negative eigenvalues) possible for feasible S in (¹) can be derived from the real Jordan normal form of A. In particular, for any A there is always a nonsingular r.s. matrix S with signature S ? 1 such that A = S^?1T.     

On real eigenvalues of real tridiagonal matrices

Lower bounds for the number of different real eigenvalues as well as for the number of real simple eigenvalues of a class of real irreducible tridiagonal matrices are given. Some numerical implications are discussed.     

Analogs of the cartan criteria forn-Lie algebras

It is shown that Cartan's criteria for finite-dimensional Lie algebras to be semisimple and solvable are fully adaptable to n-Lie algebras, provided that ideals of an n-Lie algebra are understood to be solvable in the sense of Kuz'min. Specifically, we present a characterization of the Kuz'min radical in terms of a trace form associated with some representation ρ, which is analogous to the characterization which we have in the case of Lie algebras. One more analog of the Cartan theorem is proved for n-Lie algebras which are solvable in the sense of Filippov. Translated fromAlgebra i Logika, Vol. 34, No. 3, pp. 274-287, May-June, 1995.     

On orientable real algebraicM-surfaces

Here we study relations between homology classes determined by real points of a real algebraicM-surface. We prove new congruences involving the Euler characteristics of the connected components of the set of these real points. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 510–526, October, 1997. Translated by S. S. Anisov     

On formally real modules

The aim of this paper is to develop the study of modules in real algebra. By introducing two kinds of reality for modules over commutative rings, we establish some results on semireal modules and real modules. Our results involve real radicals of submodules and orderings on modules.     

Some results on the cartan matrix of a frobenius algebra

Let F be a field and A a Frobenius algebra over F. The Jacobson radical of A is denoted by J = J(A) and the kth socle of A by S _k (A). Let [Abar]=A/J ^k or A/S _k (A). This article gives new interesting relations between the Cartan matrix of A and that of [Abar]. From these results we prove that the Cartan matrix of A is diagonal if A/Soc(A) is a symmetric algebra. Let G be a finite group. If A is a block of F|G] with the above condition, then the Cartan matrix of A is (n), where n is the order of the defect group of A and the least integer such that Jⁿ (A)=0.     

Generalized cartan type k lie algebras in characteristic 0

The Lie algebra of Cartan type K which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra F[x₀, x₁,…, x_n,x_n?1,…,x_?n], where F is a field of characteristic 0, was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials F[x₀,x₁,…, x_n,x_?1,…,x_?n,X₀ ^?1x₁ ^-1,…,x_n ^?1,…,x_?1 ^?1…,x_?n ^?1]A further generalization of these algebras is the main topic of this paper. We show when these algebras are simple, determine all possible     

On harmonic typically real mappings

In this paper, we prove that a univalent orientation-preserving harmonic mapping defined on the unit disk U with the normalization f(0)=0, , is a typically real mapping, if f(U) is a starlike domain with respect to the origin or f(U) is convex in one direction.     

On open real polynomial maps

We study open polynomial maps from ⁿ to ^p. For n = p we give a complete characterization, and for p = 2, n ≥ 3 we obtain some partial information.