Related papers: Estimates in Shirshov height theorem

We first study i.i.d. products of finitely many invertible $2 \times 2$ matrices with positive entries, and prove that the top Lyapunov exponent admits an explicit, rapidly convergent Neumann-series-type representation involving an infinite…

If $x$ is a non-empty string then the repetition $xx$ is called a tandem repeat. Similarly, a tandem in a two dimensional array $X$ is a configuration consisting of a same primitive block $W$ that touch each other with one side or corner.…

Let $ L $ be an $ n $-dimensional nilpotent Lie algebra of nilpotency class $ c $ with the derived subalgebra of dimension $ m $. Recently, Rai proved that the dimension of Schur multiplier of $ L $ is bounded by $…

We show that inductive limits of virtually nilpotent groups have strongly quasidiagonal C*-algebras, extending results of the first author on solvable virtually nilpotent groups. We use this result to show that the decomposition rank of the…

We show that for some absolute (explicit) constant $C$, the following holds for every finitely generated group $G$, and all $d >0$: If there is some $ R_0 > \exp(\exp(Cd^C))$ for which the number of elements in a ball of radius $R_0$ in a…

Kahn and Kim (J. Comput. Sci., 1995) have shown that for a finite poset $P$, the entropy of the incomparability graph of $P$ (normalized by multiplying by the order of $P$) and the base-$2$ logarithm of the number of linear extensions of…

Revising Nekhoroshev's geometry of resonances, we provide a fully constructive and quantitative proof of Nekhoroshev's theorem for steep Hamiltonian systems proving, in particular, that the exponential stability exponent can be taken to be…

For integers $n\ge s\ge 2$ let $e(n,s)$ denote the maximum of $|\mathcal F|,$ where $\mathcal F$ is a family of subsets of an $n$-element set and $\mathcal F$ contains no $s$ pairwise disjoint members. Half a century ago, solving a…

The paper concerns the Gelfand-Kirillov dimension and the generating series of nonsymmetric operads. An analogue of Bergman's gap theorem is proved, namely, no finitely generated locally finite nonsymmetric operad has Gelfand-Kirillov…

Consider the interval of integers $I_{m,n} = \{m, m+1, m+2,\ldots, m+n-1 \}$. For fixed integers $h,k,m$, and $c$, let $\Phi_{h,k,m}^{(c)}(n)$ denote the number of solutions of the equation $(a_1+\cdots + a_h)- (a_{h+1} + \cdots +…

The height $H(n)$ of $n$, introduced by Pillai in 1929, is the smallest positive integer $i$ such that the $i$th iterate of Euler's totient function at $n$ is $1$. H. N. Shapiro (1943) studied the structure of the set of all numbers at a…

We characterize lower growth estimates for subsolutions in halfspaces of fully nonlinear partial differential equations on the form $$ F(x,u,Du,D^2u) = 0 $$ in terms of solutions to ordinary differential equations built solely upon a growth…

Since the work of Jennings (1955), it is well-known that any finitely generated torsion-free nilpotent group can be embedded into unitriangular integer matrices $UT_N(Z)$ for some $N$. In 2006, Nickel proposed an algorithm to calculate such…

In this paper, we consider the relatively free algebra of rank $n$, $F_n(\mathfrak{N}_p)$, in the variety of Lie nilpotent associative algebras of index $p$, denoted by $\mathfrak{N}_p$, over a field of characteristic zero. We describe an…

It follows from the Golod--Shafarevich theorem that if R is an associative algebra given by n generators and $d<\frac{n^2}{4}\cos^{-2}(\frac{\pi}{k+1})$ quadratic relations, then R is not k-step nilpotent. We show that the above estimate is…

Let $\mathbf{P}$ denote the set of prime numbers and, for an appropriate function $h$, define a set $\mathbf{P}_{h}=\{p\in\mathbf{P}: \exists_{n\in\mathbb{N}}\ p=\lfloor h(n)\rfloor\}$. The aim of this paper is to show that every subset of…

Let M be any closed, locally symmetric n-manifold (n>1) of nonpositive curvature. Assume that M has no locally Euclidean factors and no factors locally isometric to SL(3,R). Then for any closed Riemannian manifold N and any continuous map…

We show that the deterministic decision tree complexity of a (partial) function or relation $f$ lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation $f \circ g$ as long as the gadget $g$…

For a finite group $G$ and a non-negative integer $d$, denote by $\Omega_d(G)$ the number of functions $G^d\rightarrow G$ that are induced by substitution into a word with variables among $X_1,\ldots,X_d$. In this note, we show that…

We study some properties of the growth rate of $\mathcal{L}(\mathcal{A},\mathcal{F})$, that is, the language of words over the alphabet $\mathcal{A}$ avoiding the set of forbidden factors $\mathcal{F}$. We first provide a sufficient…