Related papers: Self-Matching Properties of Beatty Sequences

The Fibonacci numbers satisfy the famous recurrence $F_n = F_{n - 1} + F_{n - 2}$. The theory of C-finite sequences ensures that the Fibonacci numbers whose indices are divisible by $m$, namely $F_{mn}$, satisfy a similar recurrence for…

This paper is devoted to the study of a discrepancy-type characteristic -- the fixed volume discrepancy -- of the Fibonacci point set in the unit square. It was observed recently that this new characteristic allows us to obtain optimal rate…

The list of properties of Fibonacci numbers F(n) (with multifaceted relevance in physics) is complemented by an empirical observation that in combination with the "next" family of the "delayed Fibonacci" numbers G(n) called, for…

We introduce a class of Jacobi operators with discrete spectra which is characterized by a simple convergence condition. With any operator J from this class we associate a characteristic function as an analytic function on a suitable…

We analyze statistical properties of complex eigenvalues of random matrices $\hat{A}$ close to unitary. Such matrices appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with…

We study Jacobi matrices on star-like graphs, which are graphs that are given by the pasting of a finite number of half-lines to a compact graph. Specifically, we extend subordinacy theory to this type of graphs, that is, we find a…

In this article, a sublinear expectation induced by $G$-expectation is introduced, which is called $G$-evaluation for convenience. As an application, we prove that any $\xi\in L^\beta_G(\Omega_T)$ with some $\beta>1$ the decomposition…

We study limiting distribution of pair counting statistics of the form $ \sum_{1\leq i\neq j\leq N} f(L_N\*(\theta_i-\theta_j))$ for the circular $\beta$-ensemble (C$\beta$E) of random matrices for sufficiently smooth test function $f$ and…

In this paper we study the set of digit frequencies that are realised by elements of the set of $\beta$-expansions. The main result of this paper demonstrates that as $\beta$ approaches $1,$ the set of digit frequencies that occur amongst…

Let $1/2\leq\beta<1$, $p$ be a generic prime number and $f_\beta$ be a random multiplicative function supported on the squarefree integers such that $(f_\beta(p))_{p}$ is an i.i.d. sequence of random variables with distribution…

We introduce the notion of Differential Sequences of ordinary differential equations. This is motivated by related studies based on evolution partial differential equations. We discuss the Riccati Sequence in terms of symmetry analysis,…

We complement the recent paper of Zheng and Wu [Uniform recurrence properties for beta-transformation, Nonlinearity 33 (2020), 4590--4612], where the authors study, from the metrical point of view, the uniform recurrence properties of the…

Let $\mathbb{M}$ be the group of multiplicative characters of a finite field $\mathbb{F}$, and let $\mathbb{J}(\alpha, \beta)$ be the Jacobi sum, for $\alpha, \beta \in \mathbb{M}$. We observe that the function $\mathbb{J} \colon \mathbb{M}…

A time series delta(n), the fluctuation of the nth unfolded eigenvalue was recently characterized for the classical Gaussian ensembles of NxN random matrices (GOE, GUE, GSE). It is investigated here for the beta-Hermite ensemble as a…

Embeddings in AI convert symbolic structures into fixed-dimensional vectors, effectively fusing multiple signals. However, the nature of this fusion in real-world data is often unclear. To address this, we introduce two methods: (1)…

Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: $$c_{\delta,\beta}^{(n)} \prod_{1\leq…

We define zeta functions for the adjoint action of GL(n) on its Lie algebra and study their analytic properties. For n<4 we are able to fully analyse these functions, and recover the Shintani zeta function for the prehomogeneous vector…

The loop equations for the $\beta$-ensembles are conventionally solved in terms of a $1/N$ expansion. We observe that it is also possible to fix $N$ and expand in inverse powers of $\beta$. At leading order, for the one-point function…

We study one class of continuous functions $f$ defined on segment $[0,1]$ by equality $$ f(x)=\delta_{\alpha_1(x)1}+\sum^{\infty}_{k=2}\left[\delta_{\alpha_k(x)k}\prod^{k-1}_{j=1}g_{\alpha_j…

We study spectral and transport properties of one-dimensional tight-binding $\mathcal{PT}$-symmetric chains with alternating couplings. Based on the transfer matrix method, we have analytically developed the expressions for the transmission…