Search results for "Codimension"

Separation properties of continuous maps in codimension 1 and geometrical applications

1992

Abstract Nuno Ballesteros, J.J. and M.C. Romero Fuster, Separation properties of continuous maps in codimension 1 and geometrical applications, Topology and its Applications 46 (1992) 107-111. We show that the image of a proper closed continuous map, f , from an n -manifold X to an ( n + 1)-manifold Y , such that H 1 (Y; Z 2 ) =0 , separates Y into at least two connected components provided the self-intersections set of f is not dense in any connected component of Y . We also obtain some geometrical applications.

Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3

1987

AbstractA cusp type germ of vector fields is a C∞ germ at 0∈ℝ2, whose 2-jet is C∞ conjugate toWe define a submanifold of codimension 5 in the space of germs consisting of germs of cusp type whose 4-jet is C0 equivalent toOur main result can be stated as follows: any local 3-parameter family in (0, 0) ∈ ℝ2 × ℝ3 cutting transversally in (0, 0) is fibre-C0 equivalent to

Uniform rectifiability implies Varopoulos extensions

2020

We construct extensions of Varopolous type for functions $f \in \text{BMO}(E)$, for any uniformly rectifiable set $E$ of codimension one. More precisely, let $\Omega \subset \mathbb{R}^{n+1}$ be an open set satisfying the corkscrew condition, with an $n$-dimensional uniformly rectifiable boundary $\partial \Omega$, and let $\sigma := \mathcal{H}^n\lfloor_{\partial \Omega}$ denote the surface measure on $\partial \Omega$. We show that if $f \in \text{BMO}(\partial \Omega,d\sigma)$ with compact support on $\partial \Omega$, then there exists a smooth function $V$ in $\Omega$ such that $|\nabla V(Y)| \, dY$ is a Carleson measure with Carleson norm controlled by the BMO norm of $f$, and such th…

Codimension growth of two-dimensional non-associative algebras

2007

Let F be a field of characteristic zero and let A be a two-dimensional non-associative algebra over F. We prove that the sequence c n (A), n =1,2,..., of codimensions of A is either bounded by n + 1 or grows exponentially as 2 n . We also construct a family of two-dimensional algebras indexed by rational numbers with distinct T-ideals of polynomial identities and whose codimension sequence is n + 1, n > 2.

Time minimal synthesis with target of codimension one under generic conditions

1995

Graded polynomial identities and codimensions: Computing the exponential growth

2010

Abstract Let G be a finite abelian group and A a G-graded algebra over a field of characteristic zero. This paper is devoted to a quantitative study of the graded polynomial identities satisfied by A. We study the asymptotic behavior of c n G ( A ) , n = 1 , 2 , … , the sequence of graded codimensions of A and we prove that if A satisfies an ordinary polynomial identity, lim n → ∞ c n G ( A ) n exists and is an integer. We give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple G × Z 2 -graded algebra related to A.

Graded algebras with polynomial growth of their codimensions

2015

Abstract Let A be an algebra over a field of characteristic 0 and assume A is graded by a finite group G . We study combinatorial and asymptotic properties of the G -graded polynomial identities of A provided A is of polynomial growth of the sequence of its graded codimensions. Roughly speaking this means that the ideal of graded identities is “very large”. We relate the polynomial growth of the codimensions to the module structure of the multilinear elements in the relatively free G -graded algebra in the variety generated by A . We describe the irreducible modules that can appear in the decomposition, we show that their multiplicities are eventually constant depending on the shape obtaine…

Polynomial growth and identities of superalgebras and star-algebras

2009

Abstract We study associative algebras with 1 endowed with an automorphism or antiautomorphism φ of order 2, i.e., superalgebras and algebras with involution. For any fixed k ≥ 1 , we construct associative φ -algebras whose φ -codimension sequence is given asymptotically by a polynomial of degree k whose leading coefficient is the largest or smallest possible.

PI-algebras with slow codimension growth

2005

Let $c_n(A),\ n=1,2,\ldots,$ be the sequence of codimensions of an algebra $A$ over a field $F$ of characteristic zero. We classify the algebras $A$ (up to PI-equivalence) in case this sequence is bounded by a linear function. We also show that this property is closely related to the following: if $l_n(A), \ n=1,2,\ldots, $ denotes the sequence of colengths of $A$, counting the number of $S_n$-irreducibles appearing in the $n$-th cocharacter of $A$, then $\lim_{n\to \infty} l_n(A)$ exists and is bounded by $2$.

Minimal varieties of algebras of exponential growth

2003

Abstract The exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer. Through this scale we can now classify all minimal varieties of given exponent and of finite basic rank. As a consequence, we describe the corresponding T-ideals of the free algebra and we compute the asymptotics of the related codimension sequences, verifying in this setting some known conjectures. We also show that the number of these minimal varieties is finite for any given exponent. We finally point out some relations between the exponent of a variety and the Gelfand–Kirillov dimension of the corresponding relatively free algebras of finite rank.