On codimensions k immersions of m-manifolds for k=1 and k=m-2

Let us consider M a closed smooth connected m-manifold, N a smooth ( 2m-2)-manifold and f: M -> N a continuous map, with m equivalent to 1( 4). We prove that if f*: H(1)(M; Z(2)) -> H(1)(f(M); Z(2)) is injective, then f is homotopic to an immersion. Also we give conditions to a map between manifolds of codimension one to be homotopic to an immersion. This work complements some results of Biasi et al. (Manu. Math. 104, 97-110, 2001; Koschorke in The singularity method and immersions of m-manifolds into manifolds of dimensions 2m-2, 2m-3 and 2m-4. Lecture Notes in Mathematics, vol. 1350. Springer, Heidelberg, 1988; Li and Li in Math. Proc. Camb. Phil. Soc. 112, 281-285, 1992).

Hopf Bifurcations in a Watt Governor with a Spring

This paper pursues the study carried out in [ 10], focusing on the codimension one Hopf bifurcations in the hexagonal Watt governor system. Here are studied Hopf bifurcations of codimensions two, three and four and the pertinent Lyapunov stability coefficients and bifurcation diagrams. This allows to determine the number, types and positions of bifurcating small amplitude periodic orbits. As a consequence it is found an open region in the parameter space where two attracting periodic orbits coexist with an attracting equilibrium point.

Finite-dimensional non-associative algebras and codimension growth

Let A be a (non-necessarily associative) finite-dimensional algebra over a field of characteristic zero. A quantitative estimate of the polynomial identities satisfied by A is achieved through the study of the asymptotics of the sequence of codimensions of A. It is well known that for such an algebra this sequence is exponentially bounded. Here we capture the exponential rate of growth of the sequence of codimensions for several classes of algebras including simple algebras with a special non-degenerate form, finite-dimensional Jordan or alternative algebras and many more. In all cases such rate of growth is integer and is explicitly related to the dimension of a subalgebra of A. One of the main tools of independent interest is the construction in the free non-associative algebra of multialternating polynomials satisfying special properties. (C) 2010 Elsevier Inc. All rights reserved.

Axioms for the coincidence index of maps between manifolds of the same dimension

We study the coincidence theory of maps between two manifolds of the same dimension from an axiomatic viewpoint. First we look at coincidences of maps between manifolds where one of the maps is orientation true, and give a set of axioms such that characterizes the local index (which is an integer valued function). Then we consider coincidence theory for arbitrary pairs of maps between two manifolds. Similarly we provide a set of axioms which characterize the local index, which in this case is a function with values in Z circle plus Z(2). We also show in each setting that the group of values for the index (either Z or Z circle plus Z(2)) is determined by the axioms. Finally, for the general case of coincidence theory for arbitrary pairs of maps between two manifolds we provide a set of axioms which characterize the local Reidemeister trace which is an element of an abelian group which depends on the pair of functions. These results extend known results for coincidences between orientable differentiable manifolds. (C) 2012 Elsevier B.V. All rights reserved.

Asymptotic Behaviour of Colength of Varieties of Lie Algebras

This project was partially supported by RFBR, grants 99-01-00233, 98-01-01020 and 00-15-96128.

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

Partially supported by grant RFFI 98-01-01020.

The Variety of Leibniz Algebras Defined by the Identity x(y(zt)) ≡ 0

2000 Mathematics Subject Classification: Primary: 17A32; Secondary: 16R10, 16P99, 17B01, 17B30, 20C30

Involution Matrix Algebras – Identities and Growth

2000 Mathematics Subject Classification: 16R50, 16R10.