Related papers: Low growth equational complexity

We study the complexity of multiplication in noncommutative group algebras which is closely related to the complexity of matrix multiplication. We characterize such semisimple group algebras of the minimal bilinear complexity and show…

Let $S$ be a finite semigroup and let $A$ be a finite dimensional $S$-graded algebra. We investigate the exponential rate of growth of the sequence of graded codimensions $c_n^S(A)$ of $A$, i.e $\lim\limits_{n \rightarrow \infty}…

We show that an arithmetic function which satisfies some weak multiplicativity properties and in addition has a non-decreasing or $\log$-uniformly continuous normal order is close to a function of the form $n\mapsto n^c$. As an application…

We investigate the class of finite dimensional not necessary associative algebras that have slowly growing length, that is, for any algebra in this class its length is less than or equal to its dimension. We show that this class is…

Given a finitely generated residually finite group $G$, the residual finiteness growth $\text{RF}_G: \mathbb{N} \to \mathbb{N}$ bounds the size of a finite group $Q$ needed to detect an element of norm at most $r$. More specifically, if…

Let G denote a closed, connected, self adjoint, noncompact subgroup of GL(n,R), and let d_{R} denote the canonical right invariant Riemannian metric on G. For v in R^{n} let G_{v} = {g in G : g(v) = v}. We obtain algebraically defined upper…

This paper proves that in a non-elementary relatively hyperbolic group, the logarithm growth rate of any non-elementary subgroup has a linear lower bound by the logarithm of the size of the corresponding generating set. As a consequence,…

We consider the question of which semigroups can occur as the semigroup $S_R(\nu)$ of positive values of a rank 1 valuation dominating a Noetherian local ring $R$. We give a number of bounds of polynomial type on the growth of…

By now, we have a product theorem in every finite simple group $G$ of Lie type, with the strength of the bound depending only in the rank of $G$. Such theorems have numerous consequences: bounds on the diameters of Cayley graphs, spectral…

We investigate how the behavior of the function d_A(n) that gives the size of a least size generating set for A^n, influences the structure of a finite solvable algebra A.

We prove lower bounds on complexity measures, such as the approximate degree of a Boolean function and the approximate rank of a Boolean matrix, using quantum arguments. We prove these lower bounds using a quantum query algorithm for the…

We study growth and complexity of \'etale groupoids in relation to growth of their convolution algebras. As an application, we construct simple finitely generated algebras of arbitrary Gelfand-Kirillov dimension $\ge 2$ and simple finitely…

The solvable Farb growth of a group quantifies how well-approximated the group is by its finite solvable quotients. In this note we present a new characterization of polycyclic groups which are virtually nilpotent. That is, we show that a…

We study product set growth in groups with acylindrical actions on quasi-trees and, more generally, hyperbolic spaces. As a consequence, we show that for every surface $S$ of finite type, there exist $\alpha,\beta>0$ such that for any…

In this note we present some uniqueness and comparison results for a class of problem of the form \begin{equation} \label{EE0} \begin{array}{c} - L u = H(x,u,\nabla u)+ h(x), \quad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega), \end{array}…

The equational probabilistic spectrum of a finite algebra is the set of probabilities with which equations are satisfied in the algebra. We study algebras with minimal spectrum, that is, spectra consisting only of the values $1$ and…

We study the representation growth of alternating and symmetric groups in positive characteristic and restricted representation growth for the finite groups of Lie type. We show that the the number of representations of dimension at most n…

For a class V of algebras, denote by Conc(V) the class of all semilattices isomorphic to the semilattice Conc(A) of all compact congruences of A, for some A in V. For classes V1 and V2 of algebras, we denote by crit(V1,V2) the smallest…

We give an upper bound of $n((n-1)!-(n-3)!)$ for the possible largest size of a subsemigroup of the full transformational semigroup over $n$ elements consisting only of nonpermutational transformations. As an application we gain the same…

The residual finiteness growth $\text{RF}_G: \mathbb{N} \to \mathbb{N}$ of a finitely generated group $G$ is a function that gives the smallest value of the index $[G:N]$ with $N$ a normal subgroup not containing a non-trivial element $g$,…