Search results for "LYN"

A stronger Dunford-Pettis property

2008

Polynomial codimension growth and the Specht problem

2017

Abstract We construct a continuous family of algebras over a field of characteristic zero with slow codimension growth bounded by a polynomial of degree 4. This is achieved by building, for any real number α ∈ ( 0 , 1 ) a commutative nonassociative algebra A α whose codimension sequence c n ( A α ) , n = 1 , 2 , …  , is polynomially bounded and lim ⁡ log n ⁡ c n ( A α ) = 3 + α . As an application we are able to construct a new example of a variety with an infinite basis of identities.

Standard polynomials are characterized by their degree and exponent

2011

Abstract By the Giambruno–Zaicev theorem (Giambruno and Zaicev, 1999) [5] , the exponent exp ( A ) of a p.i. algebra A exists, and is always an integer. In Berele and Regev (2001) [2] it was shown that the exponent exp ( St n ) of the standard polynomial St n of degree n is not smaller than the exponent of any polynomial of degree n. Here it is proved that exp ( St n ) is strictly larger than the exponent of any other polynomial of degree n which is not a multiple of St n .

Pietsch's factorization theorem for dominated polynomials

2007

Abstract We prove that, like in the linear case, there is a canonical prototype of a p -dominated homogeneous polynomial through which every p -dominated polynomial between Banach spaces factors.

Nonlinear embeddings: Applications to analysis, fractals and polynomial root finding

2016

We introduce $\mathcal{B}_{\kappa}$-embeddings, nonlinear mathematical structures that connect, through smooth paths parameterized by $\kappa$, a finite or denumerable set of objects at $\kappa=0$ (e.g. numbers, functions, vectors, coefficients of a generating function...) to their ordinary sum at $\kappa \to \infty$. We show that $\mathcal{B}_{\kappa}$-embeddings can be used to design nonlinear irreversible processes through this connection. A number of examples of increasing complexity are worked out to illustrate the possibilities uncovered by this concept. These include not only smooth functions but also fractals on the real line and on the complex plane. As an application, we use $\mat…

The surjective hull of a polynomial ideal

2016

The aim of this paper is the study of surjective ideals of homogeneous polynomials between Banach spaces. To do so we define the surjective hull of a polynomial ideal and prove the main properties of this hull procedure. For a more comprehensive theory, new lifting properties of homogeneous polynomials are proved and applied to the description of the surjective hulls of the ideals of I-bounded polynomials and of composition polynomials ideals. Several applications are provided.

Polynomial identities on superalgebras: Classifying linear growth

2006

Abstract We classify, up to PI-equivalence, the superalgebras over a field of characteristic zero whose sequence of codimensions is linearly bounded. As a consequence we determine the linear functions describing the graded codimensions of a superalgebra.

Codimension and colength sequences of algebras and growth phenomena

2015

We consider non necessarily associative algebras over a field of characteristic zero and their polynomial identities. Here we describe some of the results obtained in recent years on the sequence of codimensions and the sequence of colengths of an algebra.

Deciding reachability for planar multi-polynomial systems

1996

In this paper we investigate the decidability of the reachability problem for planar non-linear hybrid systems. A planar hybrid system has the property that its state space corresponds to the standard Euclidean plane, which is partitioned into a finite number of (polyhedral) regions. To each of these regions is assigned some vector field which governs the dynamical behaviour of the system within this region. We prove the decidability of point to point and region to region reachability problems for planar hybrid systems for the case when trajectories within the regions can be described by polynomials of arbitrary degree.

On algebras of polynomial codimension growth

2016

Let A be an associative algebra over a field F of characteristic zero and let $$c_n(A), n=1, 2, \ldots $$ , be the sequence of codimensions of A. It is well-known that $$c_n(A), n=1, 2, \ldots $$ , cannot have intermediate growth, i.e., either is polynomially bounded or grows exponentially. Here we present some results on algebras whose sequence of codimensions is polynomially bounded.