arXiv subjects

A. Valenti

Publications and source records attributed to A. Valenti.

10 recordsLinked to original sources

arXiv2025

Star-Varieties of proper central exponent greater than two

Let $F$ be a field of characteristic zero and let $ \mathcal V^* $ be a variety of associative $F$-algebras with involution *. Associated to $ \mathcal V^* $ are three sequences: the sequence of $*$-codimensions $ c^{*}_n(\mathcal V^*) $, the sequence of central $*$-codimensions $ c^{*,z}_n(\mathcal V^*) $ and the sequence of proper central $*$-codimensions $ c^{*,\delta}_n(\mathcal V^*) $. These sequences provide information on the growth of, respectively, the *-polynomial identities, the central *-polynomial and the proper central *-polynomial of any generating algebra with involution $A$ of $ \mathcal V^*.$ In \cite{MR2022} it was proved that $exp^{*,\delta}(\mathcal V^*)=\lim_{n\to\infty}\sqrt[n]{c_n^{*,\delta}(\mathcal V^*)}$ exists and is an integer called the proper central $*$-exponent. The aim of this paper is to study the varieties of associative algebras with involution of proper central $*$-exponent greater than two. To this end we construct a finite list of algebras with involution and we prove that if $exp^{*,\delta}(\mathcal V^*) >2$, then at least one of these algebras belongs to $\mathcal V^*$.

math.RA↗

arXiv2025

Varieties of group-graded algebras of proper central exponent greater than two

Let $F$ be a field of characteristic zero and let $ \mathcal V $ be a variety of associative $F$-algebras graded by a finite abelian group $G$. To a variety $ \mathcal V $ is associated a numerical sequence called the sequence of proper central $G$-codimensions, $c^{G,\delta}_n(\mathcal V), \, n \ge 1.$ Here $c^{G,\delta}_n(\mathcal V)$ is the dimension of the space of multilinear proper central $G$-polynomials in $n$ fixed variables of any algebra $A$ generating the variety $\mathcal V.$ Such sequence gives information on the growth of the proper central $G$-polynomials of $A$ and in \cite{LMR} it was proved that $exp^{G,\delta}(\mathcal V)=\lim_{n\to\infty}\sqrt[n]{c_n^{G,\delta}(\mathcal V)}$ exists and is an integer called the proper central $G$-exponent. The aim of this paper is to characterize the varieties of associative $G$-graded algebras of proper central $G$-exponent greater than two. To this end we construct a finite list of $G$-graded algebras and we prove that $exp^{G,\delta}(\mathcal V) >2$ if and only if at least one of the algebras belongs to $\mathcal V$. Matching this result with the characterization of the varieties of almost polynomial growth given in \cite{GLP}, we obtain a characterization of the varieties of proper central $G$-exponent equal to two.

math.RA↗

arXiv2024

A characterization of varieties of algebras of proper central exponent equal to two

Let $F$ be a field of characteristic zero and let $ \mathcal V$ be a variety of associative $F$-algebras. In \cite{regev2016} Regev introduced a numerical sequence measuring the growth of the proper central polynomials of a generating algebra of $ \mathcal V$. Such sequence $c_n^\delta(\mathcal V), \, n \ge 1,$ is called the sequence of proper central polynomials of $ \mathcal V$ and in \cite{GZ2018}, \cite{GZ2019} the authors computed its exponential growth. This is an invariant of the variety. They also showed that $c_n^\delta(\mathcal V)$ either grows exponentially or is polynomially bounded. The purpose of this paper is to characterize the varieties of associative algebras whose exponential growth of $c_n^\delta(\mathcal V)$ is greater than two. As a consequence, we find a characterization of the varieties whose corresponding exponential growth is equal to two.

math.RA↗

arXiv2021

*-Graded Capelli Polynomials and their Asymptotic

Let $F\langle Y \cup Z, \ast \rangle$ be the free $\ast$-superalgebra over a field $F$ of characteristic zero and let $ Γ^\ast_{M^{\pm}, L^{\pm}} $ be the $T^\ast_{\mathbb{Z}_2}$-ideal generated by the set of the $\ast$-graded Capelli polynomials $Cap^{(\mathbb{Z}_2, \ast)}_{M^+} [Y^+,X]$, $Cap^{(\mathbb{Z}_2, \ast)}_{M^-} [Y^-,X]$, $Cap^{(\mathbb{Z}_2, \ast)}_{L^+} [Z^+,X]$, $Cap^{(\mathbb{Z}_2, \ast)}_{L^-} [Z^-,X]$ alternating on $M^+$ symmetric variables of homogeneous degree zero, on $M^-$ skew variables of homogeneous degree zero, on $L^+$ symmetric variables of homogeneous degree one and on $L^-$ skew variables of homogeneous degree one, respectively. We study the asymptotic behavior of the sequence of $\ast$-graded codimensions of $Γ^\ast_{M^{\pm}, L^{\pm}}.$ In particular we prove that the $\ast$-graded codimensions of the finite dimensional simple $\ast$-superalgebras are asymptotically equal to the $\ast$-graded codimensions of $Γ^\ast_{M^{\pm}, L^{\pm}}$, for some fixed natural numbers $M^+, M^-, L^+$ and $L^-$.

math.RA↗

arXiv2021

An NLP approach to quantify dynamic salience of predefined topics in a text corpus

The proliferation of news media available online simultaneously presents a valuable resource and significant challenge to analysts aiming to profile and understand social and cultural trends in a geographic location of interest. While an abundance of news reports documenting significant events, trends, and responses provides a more democratized picture of the social characteristics of a location, making sense of an entire corpus to extract significant trends is a steep challenge for any one analyst or team. Here, we present an approach using natural language processing techniques that seeks to quantify how a set of pre-defined topics of interest change over time across a large corpus of text. We found that, given a predefined topic, we can identify and rank sets of terms, or n-grams, that map to those topics and have usage patterns that deviate from a normal baseline. Emergence, disappearance, or significant variations in n-gram usage present a ground-up picture of a topic's dynamic salience within a corpus of interest.

cs.CL↗

arXiv2019

Asymptotics for Capelli Polynomials with Involution

Let $F\langle X, \ast \rangle$ be the free associative algebra with involution $\ast$ over a field $F$ of characteristic zero. We study the asymptotic behavior of the sequence of $\ast$-codimensions of the T-$\ast$-ideal $Γ_{M+1,L+1}^\ast$ of $F\langle X, \ast \rangle$ generated by the $\ast$-Capelli polynomials $Cap^\ast_{M+1} [Y,X]$ and $Cap^\ast_{L+1} [Z,X]$ alternanting on $M+1$ symmetric variables and $L+1$ skew variables, respectively. It is well known that, if $F$ is an algebraic closed field of characteristic zero, every finite dimensional $\ast$-simple algebra is isomorphic to one of the following algebras: \begin{itemize} \item [$\cdot$]$(M_{k}(F),t)$ the algebra of $k \times k$ matrices with the transpose involution; \item [$\cdot$]$(M_{2m}(F),s)$ the algebra of $2m \times 2m$ matrices with the symplectic involution; \item [$\cdot$]$(M_{h}(F)\oplus M_{h}(F)^{op}, exc)$ the direct sum of the algebra of $h \times h$ matrices and the opposite algebra with the exchange involution. \end{itemize} We prove that the $\ast$-codimensions of a finite dimensional $\ast$-simple algebra are asymptotically equal to the $\ast$-codimensions of $Γ_{M+1,L+1}^\ast$, for some fixed natural numbers $M$ and $L$. In particular: $$ c^{\ast}_n(Γ^{\ast}_{\frac{k(k+1)}{2} +1,\frac{k(k-1)}{2} +1})\simeq c^{\ast}_n((M_k(F),t)); $$ $$ c^{\ast}_n(Γ^{\ast}_{m(2m-1)+1,m(2m+1)+1})\simeq c^{\ast}_n((M_{2m}(F),s)); $$ and $$ c^{\ast}_n(Γ^{\ast}_{h^2+1,h^2+1})\simeq c^{\ast}_n((M_{h}(F)\oplus M_{h}(F)^{op},exc)). $$

math.RA↗

arXiv2017

Structure and Properties of DNA Molecules Over The Full Range of Biologically Relevant Supercoiling States

Topology affects physical and biological properties of DNA and impacts fundamental cellular processes, such as gene expression, genome replication, chromosome structure and segregation. In all organisms DNA topology is carefully modulated and the supercoiling degree of defined genome regions may change according to physiological and environmental conditions. Elucidation of structural properties of DNA molecules with different topology may thus help to better understand genome functions. Whereas a number of structural studies have been published on highly negatively supercoiled DNA molecules, only preliminary observations of highly positively supercoiled are available, and a description of DNA structural properties over the full range of supercoiling degree is lacking. Atomic Force Microscopy (AFM) is a powerful tool to study DNA structure at single molecule level. We here report a comprehensive analysis by AFM of DNA plasmid molecules with defined supercoiling degree, covering the full spectrum of biologically relevant topologies, under different observation conditions. Our data, supported by statistical and biochemical analyses, revealed striking differences in the behavior of positive and negative plasmid molecules.

q-bio.BM↗

arXiv2017

An uncountable family of almost nilpotent varieties of polynomial growth

A non-nilpotent variety of algebras is almost nilpotent if any proper subvariety is nilpotent. Let the base field be of characteristic zero. It has been shown that for associative or Lie algebras only one such variety exists. Here we present infinite families of such varieties. More precisely we shall prove the existence of 1) a countable family of almost nilpotent varieties of at most linear growth and 2) an uncountable family of almost nilpotent varieties of at most quadratic growth.

math.RA↗

arXiv2011

Self-adjointness of a generalized Camassa-Holm equation

It is well known that the Camassa-Holm equation possesses numerous remarkable properties characteristic for KdV type equations. In this paper we show that it shares one more property with the KdV equation. Namely, Ibragimov has shown that the KdV and the modified KdV equations are self-adjoint. Starting from the generalization of the Camassa-Holm equation, we prove that the Camassa-Holm equation is self-adjoint. This property is important, e.g. for constructing conservation laws associated with symmetries of the equation in question. Accordingly, we construct conservation laws for the generalized Camassa-Holm equation using its symmetries.

math-ph↗

arXiv2005

Equivalence transformations and differential invariants of a generalized nonlinear Schrödinger equation

By using the Lie's invariance infinitesimal criterion we obtain the continuous equivalence transformations of a class of nonlinear Schrödinger equations with variable coefficients. Starting from the equivalence generators we construct the differential invariants of order one. We apply these latter ones to find the most general subclass of variable coefficient nonlinear Schrödinger equations which can be mapped, by means of an equivalence transformation, to the well known cubic Schrödinger equation. We also provide the explicit form of the transformation.

nlin.SI↗