Related papers: The $\beta$-transformation with a hole

Given integers $k\geq 2$ and $a_1,\ldots,a_k\geq 1$, let $\boldsymbol{a}:=(a_1,\ldots,a_k)$ and $n:=a_1+\cdots+a_k$. An $\boldsymbol{a}$-multiset permutation is a string of length $n$ that contains exactly $a_i$ symbols $i$ for each…

Let $L$ be a number field. For a given prime $p$ we define integers $\alpha_{p}^{L}$ and $\beta_{p}^{L}$ with some interesting arithmetic properties. For instance, $\beta_{p}^{L}$ is equal to $1$ whenever $p$ does not ramify in $L$ and…

Given a positive integer $n$, we consider the group algebra of the symmetric group $S_{n}$. In this algebra, we define $n$ elements $t_{1},t_{2},\ldots,t_{n}$ by the formula \[…

Let $\beta>1$ be fixed. We consider the $(\frak{b, d})$ numeration system, where the base ${\frak b}=(b_k)_{k\geq 0}$ is a sequence of positive real numbers satisfying $\lim_{k\rightarrow \infty}b_{k+1}/b_k=\beta$, and the set of digits…

Using a recursive formula for the Mellin transform $T_{n,a}(s)$ of a spherical, principal series $GL(n,\mathbb{R})$ Whittaker function, we develop an explicit recurrence relation for this Mellin transform. This relation, for any $n\ge2$,…

Let $x \in [0,1)$ be an irrational number with continued fraction expansion $[a_1(x),a_2(x), \cdots,a_n(x),\cdots]$ and $q_n(x)$ be the denominator of its $n$-th convergent. We establish, for any $\alpha,\beta$ in $[0,+\infty]$, the…

Let X be a countably infinite set, and let f, g, and h be any three injective self-maps of X, each having at least one infinite cycle. (For instance, this holds if f, g, and h are not bijections.) We show that there are permutations a and b…

Let $f(x) = \alpha x + \beta \mod 1$ for fixed real parameters $\alpha$ and $\beta$. For any positive integer $n$, define the S\'os permutation $\pi$ to be the lexicographically first permutation such that $0 \leq f(\pi(0)) \leq f(\pi(1))…

An $n \times n$ matrix $H$ is Butson-Hadamard if its entries are $k^{\text{th}}$ roots of unity and it satisfies $HH^* = nI_n$. Write $BH(n, k)$ for the set of such matrices. Suppose that $k = p^{\alpha}q^{\beta}$ where $p$ and $q$ are…

The $\beta$-generalized quasi-geostrophic equation is studied in the range of $\alpha \in (0, 1), \beta \in (1/2, 1), 1/2 < \alpha + \beta < 3/2$. When $\alpha \in (1/2, 1), \beta \in (1/2, 1)$ such that $1 \leq \alpha + \beta < 3/2$, using…

We introduce a family of dynamical systems that generate negative $\beta$-expansions and study the support of the invariant measure which is absolutely continuous with respect to Lebesgue measure. We give a characterization of the set of…

We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers $b$ and $d$ there exists an integer $h(b,d)$ such that the following…

Pairs (A,B) of mutually annihilating operators AB=BA=0 on a finite dimensional vector space over an algebraically closed field were classified by Gelfand and Ponomarev [Russian Math. Surveys 23 (1968) 1-58] by method of linear relations.…

A new generalization of the modified Bessel function of the second kind $K_{z}(x)$ is studied. Elegant series and integral representations, a differential-difference equation and asymptotic expansions are obtained for it thereby…

For a fixed $\theta\neq 0$, we define the twisted divisor function $$ \tau(n, \theta):=\sum_{d\mid n}d^{i\theta}\ .$$ In this article we consider the error term $\Delta(x)$ in the following asymptotic formula $$ \sum_{n\leq x}^*|\tau(n,…

For a positive constant $\alpha$ a graph $G$ on $n$ vertices is called an $\alpha$-expander if every vertex set $U$ of size at most $n/2$ has an external neighborhood whose size is at least $\alpha\left|U\right|$. We study cycle lengths in…

This is an update on, and expansion of, our paper Open problems on $\beta\omega$ in the book Open Problems in Topology.

Expansions are provided for the moments of the number of collisions $X_n$ in the $\beta(2,b)$-coalescent restricted to the set $\{1,...,n\}$. We verify that $X_n/\mathbb{E}X_n$ converges almost surely to one and that $X_n$, properly…

For $\beta > 1$ a real algebraic integer ({\it the base}), the finite alphabets $\mathcal{A} \subset \mathbb{Z}$ which realize the identity $\mathbb{Q}(\beta) = {\rm Per}_{\mathcal{A}}(\beta)$, where ${\rm Per}_{\mathcal{A}}(\beta)$ is the…

For every positive integer $n$ and every $\delta \in [0,1]$, let $B(n, \delta)$ denote the probabilistic model in which a random set $A \subseteq \{1, \dots, n\}$ is constructed by choosing independently every element of $\{1, \dots, n\}$…