Related papers: Growth behaviors in the range $e^{r^\alpha}$

We consider boundary Harnack inequalities for regional fractional Laplacian which are generators of censored stable-like processes on G taking \kappa(x,y)/|x-y|^{n+\alpha}dxdy, x,y\in G as the jumping measure. When G is a C^{1,\beta-1} open…

Young's convolution inequality provides an upper bound for the convolution of functions in terms of $L^p$ norms. It is known that for certain groups, including Heisenberg groups, the optimal constant in this inequality is equal to that for…

We investigate properties of ($\alpha,\beta$)-harmonic functions. First, we discuss the coefficient estimates for ($\alpha,\beta$)-harmonic functions. In particular, we obtain Heinz's inequality for ($\alpha,\beta$)-harmonic functions,…

We establish the central limit theorem for the number of groups at the equilibrium of a coagulation-fragmentation process given by a parameter function with polynomial rate of growth. The result obtained is compared with the one for random…

Let $K$ be a compact metric space. A real-valued function on $K$ is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. In this paper, we study two well known ordinal indices of…

For every 3/4 <= beta < 1 we construct a finitely generated group so that the expected distance of the simple random walk from its starting point is within a constant factor of n^beta. In fact, the speed can be set precisely to equal any…

Given a bounded open subset $\Omega$ and closed subsets $A,B$ of $\mathbb{R}^k$, we discuss when an estimate $u(x)\le g(dist(x,A\cup B))$, $x\in\Omega\setminus(A\cup B)$, for a function $u$ subharmonic on $\Omega\setminus B$, implies that…

We study the growth rate of some power-free languages. For any integer $k$ and real $\beta>1$, we let $\alpha(k,\beta)$ be the growth rate of the number of $\beta$-free words of a given length over the alphabet $\{1,2,\ldots, k\}$. Shur…

Let $\overline{p}(n)$ denote the overpartition function. In this paper, we obtain an inequality for the sequence $\Delta^{2}\log \ \sqrt[n-1]{\overline{p}(n-1)/(n-1)^{\alpha}}$ which states that \begin{equation*} \log…

In this paper we consider the following Dirichlet problem for the $p$-Laplacian in the positive parameters $\lambda$ and $\beta$: [{{array} [c]{rcll}% -\Delta_{p}u & = & \lambda h(x,u)+\beta f(x,u,\nabla u) & \text{in}\Omega u & = & 0 &…

In this short article, we study the extremal behavior $F_\Gamma(n)$ of divisibility functions $D_\Gamma$ introduced by the first author for finitely generated groups $\Gamma$. We show finitely generated subgroups of $\GL(m,K)$ for an…

Let $G$ be a finite group and $\alpha(G)=\frac{|C(G)|}{|G|}$\,, where $C(G)$ denotes the set of cyclic subgroups of $G$. In this short note, we prove that $\alpha(G)\leq\alpha(Z(G))$ and we describe the groups $G$ for which the equality…

The \emph{normal rank} of a group is the minimal number of elements whose normal closure coincides with the group. We study the relation between the normal rank of a group and its first $\ell^2$-Betti number and conjecture that inequality…

Two currently debated problems in galaxy evolution, the fundamentally local or global nature of the main sequence of star formation and the evolution of the mass-size relation of star forming galaxies (SFGs), are shown to be intimately…

Let A be an infinite set of generators for a group G, and let L_A(r) denote the number of elements of G whose word length with respect to A is exactly r. The purpose of this note is to determine all growth functions L_A(r) associated to…

We show that the growth inequality rate $$\limsup \frac{1}{n} \log (\# Fix (f^n))\geq \log d$$ holds for branched coverings of degree $d$ of the sphere $S^2$ having a completely invariant simply connected region $R$ with locally connected…

Every automaton group naturally acts on the space $X^\omega$ of infinite sequences over some alphabet $X$. For every $w\in X^\omega$ we consider the Schreier graph $\Gamma_w$ of the action of the group on the orbit of $w$. We prove that for…

The maximal normal subgroup growth type of a finitely generated group is $n^{\log n}$. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let $\Gamma$ be a group and $\Delta$…

We present examples of geometrically finite manifolds with pinched negative curvature, whose geodesic flow has infinite non-ergodic Bowen-Margulis measure and whose Poincar\'e series converges at the critical exponent $\delta_\Gamma$. We…

In [B] Bowen defined the growth rate of an endomorphism of a finitely generated group and related it to the entropy of a map $f:M \mapsto M$ on a compact manifold. In this note we study the purely group theoretic aspects of the growth rate…