Related papers: A note on depth-$b$ normal elements

We define $(\alpha_n)$ -regular sets in uniformly perfect metric spaces. This definition is quasisymmetrically invariant and the construction resembles generalized dyadic cubes in metric spaces. For these sets we then determine the…

In this paper, we consider rational functions $f$ with some minor restrictions over the finite field $\mathbb{F}_{q^n},$ where $q=p^k$ for some prime $p$ and positive integer $k$. We establish a sufficient condition for the existence of a…

Let $\mathbb{F}_q$ denote the finite field of $q$ elements and $\mathbb{F}_{q^n}$ the degree $n$ extension of $\mathbb{F}_q$. A normal basis of $\mathbb{F}_{q^n}$ over $\mathbb{F} _q$ is a basis of the form…

W. M. Schmidt, A. D. Pollington, and F. Schweiger have studied when normality with respect to one expansion is equivalent to normality with respect to another expansion. Following in their footsteps, we show that when $Q$ is an eventually…

It is well known that for any finite Galois extension field $K/F$, with Galois group $G = \mathrm{Gal}(K/F)$, there exists an element $\alpha \in K$ whose orbit $G\cdot\alpha$ forms an $F$-basis of $K$. Such an element $\alpha$ is called…

In this paper, the geometric meaning of (alpha,beta)-norms is made clear. On this basis, we introduce a new class of Finsler metrics called general (alpha,beta)-metrics, which are defined by a Riemannian metric and an 1-form. These metrics…

We show that the distance from an element of a C*-algebra to the set of regular elements is the infimum of the $\delta>0$ for which the $\delta$-cut-down of the element admits a polar decomposition within the algebra. This parallels results…

We introduce the notion of the depth of a finite group $G$, defined as the minimal length of an unrefinable chain of subgroups from $G$ to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups.…

We introduce the notion of the \textit{principal element} of a Frobenius Lie algebra $\f$. The principal element corresponds to a choice of $F\in \f^*$ such that $F[-,-]$ non-degenerate. In many natural instances, the principal element is…

The rank of an $A$-hypergeometric $D$-module $M_A(\beta)$, associated with a full rank $(d\times n)$-matrix $A$ and a vector of parameters $\beta\in \mathbb{C}^d$, is known to be the normalized volume of $A$, denoted $\mathrm{vol}(A)$, when…

In this article, we study the algebraic and dynamical structure of certain normal subgroups of the quasi-isometry group of Euclidean space $QI(\mathbb{R}^n)$. For \[ H = \Big\{ [f] \in QI(\mathbb{R}^n) : \lim_{\|x\|\to\infty}…

Bennett's notion of depth is usually considered to describe the usefulness and internal organization of the information encoded into an object such as an infinite binary sequence. We consider a natural way to relativize the notion of depth…

Let $q, n, m \in \mathbb{N}$ such that $q$ is a prime power, $m \geq 3$ and $a \in \mathbb{F}$. We establish a sufficient condition for the existence of a primitive normal pair ($\alpha$, $f(\alpha)$) in $\mathbb{F}_{q^m}$ over…

Fix $t\in [1,\infty]$. Let $S$ be an atomic commutative semigroup and, for all $x\in S$, let $\mathscr{L}_t(S):=\{\|f\|_t:f\in Z(x)\}$ be the "$t$-length set" of $x$ (using the standard $l_p$-space definition of $\|\cdot\|_t$). The…

We prove that if D is a domain in C, alpha>1 and c>0, then the family F of functions meromorphic in D such that |f'(z)|/(1+|f(z)|^alpha)>c for every z in D is normalin D. For alpha=1, the same assumptions imply quasi-normality but not…

We will examine a particular mathematical derivation in a paper by P. Falkensteiner and H. Grosse (F&G) [1]. In [1] a quantity "delta(A)" is defined. This quantity is generated when the normal ordered generalized charge operator undergoes a…

For the elements of a Coxeter group, we present a statistic called depth, defined in terms of factorizations of the elements into products of reflections. Depth is bounded above by length and below by the average of length and reflection…

The functional calculus for normal elements in $C^*$-algebras is an important tool of analysis. We consider polynomials $p(a,a^*)$ for elements $a$ with small self-commutator norm $\|[a,a^*]\| \le \delta$ and show that many properties of…

The depth statistic was defined by Petersen and Tenner for an element of an arbitrary Coxeter group in terms of factorizations of the element into a product of reflections. It can also be defined as the minimal cost, given certain…

Here we show that, given a finite homological system $({\cal P},\leq,\{\Delta_u\}_{u\in {\cal P}})$ for a finite-dimensional algebra $\Lambda$ over an algebraically closed field, the category ${\cal F}(\Delta)$ of $\Delta$-filtered modules…