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Register a new userOn polynomials that are not quite an identity on an associative algebra
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Let $f$ be a polynomial in the free algebra over a field $K$, and let $A$ be a $K$-algebra. We denote by $S_A(f)$, $A_A(f)$ and $I_A(f)$, respectively, the `verbal subspace, subalgebra, and ideal, in $A$, generated by the set of all $f$-values in $A$. We begin by studying the following problem: if $S_A(f)$ is finite-dimensional, is it true that $A_A(f)$ and $I_A(f)$ are also finite-dimensional? We then consider the dual to this problem for `marginal subspaces that are finite-codimensional in $A$. If $f$ is multilinear, the marginal subspace, $widehat{S}_A(f)$, of $f$ in $A$ is the set of all elements $z$ in $A$ such that $f$ evaluates to 0 whenever any of the indeterminates in $f$ is evaluated to $z$. We conclude by discussing the relationship between the finite-dimensionality of $S_A(f)$ and the finite-codimensionality of $widehat{S}_A(f)$.
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This paper continues the study of the lower central series quotients of an associative algebra A, regarded as a Lie algebra, which was started in math/0610410 by Feigin and Shoikhet. Namely, it provides a basis for the second quotient in the case when A is the free algebra in n generators (note that the Hilbert series of this quotient was determined earlier in math/0610410). Further, it uses this basis to determine the structure of the second quotient in the case when A is the free algebra modulo the relations saying that the generators have given nilpotency orders. Finally, it determines the structure of the third and fourth quotient in the case of 2 generators, confirming an answer conjectured in math/0610410. Finally, in the appendix, the results of math/0610410 are generalized to the case when A is an arbitrary associative algebra (under certain conditions on $A$).
Evolution algebras are non-associative algebras that describe non-Mendelian hereditary processes and have connections with many other areas. In this paper we obtain necessary and sufficient conditions for a given algebra $A$ to be an evolution algebra. We prove that the problem is equivalent to the so-called $SDC$ $problem$, that is, the $simultaneous$ $diagonalisation$ $via$ $congruence$ of a given set of matrices. More precisely we show that an $n$-dimensional algebra $A$ is an evolution algebra if, and only if, a certain set of $n$ symmetric $ntimes n$ matrices ${M_{1}, ldots, M_{n}}$ describing the product of $A$ are $SDC$. We apply this characterisation to show that while certain classical genetic algebras (representing Mendelian and auto-tetraploid inheritance) are not themselves evolution algebras, arbitrarily small perturbations of these are evolution algebras. This is intriguing as evolution algebras model asexual reproduction unlike the classical ones.
It has been suggested that all resistive-switching memory cells are memristors. The latter are hypothetical, ideal devices whose resistance, as originally formulated, depends only on the net charge that traverses them. Recently, an unambiguous test has been proposed [J. Phys. D: Appl. Phys. {bf 52}, 01LT01 (2019)] to determine whether a given physical system is indeed a memristor or not. Here, we experimentally apply such a test to both in-house fabricated Cu-SiO2 and commercially available electrochemical metallization cells. Our results unambiguously show that electrochemical metallization memory cells are not memristors. Since the particular resistance-switching memories employed in our study share similar features with many other memory cells, our findings refute the claim that all resistance-switching memories are memristors. They also cast doubts on the existence of ideal memristors as actual physical devices that can be fabricated experimentally. Our results then lead us to formulate two memristor impossibility conjectures regarding the impossibility of building a model of physical resistance-switching memories based on the memristor model.
In 1992, following earlier conjectures of Lichtman and Makar-Limanov, Klein conjectured that a noncommutative domain must contain a free, multiplicative, noncyclic subsemigroup. He verified the conjecture when the center is uncountable. In this note we consider the existence (or not) of free subsemigroups in associative $k$-algebras $R$, where $k$ is a field not algebraic over a finite subfield. We show that $R$ contains a free noncyclic subsemigroup in the following cases: (1) $R$ satisfies a polynomial identity and is noncommutative modulo its prime radical. (2) $R$ has at least one nonartinian primitive subquotient. (3) $k$ is uncountable and $R$ is noncommutative modulo its Jacobson radical. In particular, (1) and (2) verify Kleins conjecture for numerous well known classes of domains, over countable fields, not covered in the prior literature.
The main objective of this research note is to provide an identity for the H-function, which generalizes two identities involving H-function obtained earlier by Rathie and Rathie et al.
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