A pair correlation problem, and counting lattice points with the zeta function

The pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form $$(a_n \alpha )_{n \ge 1}$$ ( a n α ) n ≥ 1 has been pioneered by Rudnick, Sarnak and Zaharescu. Here $$\alpha $$ α is a real parameter, and $$(a_n)_{n \ge 1}$$ ( a n ) n ≥ 1 is an integer sequence, often of arithmetic origin. Recently, a general framework was developed which gives criteria for Poissonian pair correlation of such sequences for almost every real number $$\alpha $$ α , in terms of the additive energy of the integer sequence $$(a_n)_{n \ge 1}$$ ( a n ) n ≥ 1 . In the present paper we develop a similar framework for the case when $$(a_n)_{n \ge 1}$$ ( a n ) n ≥ 1 is a sequence of reals rather than integers, thereby pursuing a line of research which was recently initiated by Rudnick and Technau. As an application of our method, we prove that for every real number $$\theta >1$$ θ > 1 , the sequence $$(n^\theta \alpha )_{n \ge 1}$$ ( n θ α ) n ≥ 1 has Poissonian pair correlation for almost all $$\alpha \in {\mathbb {R}}$$ α ∈ R .

Catalan words avoiding pairs of length three patterns

Catalan words are particular growth-restricted words counted by the eponymous integer sequence. In this article we consider Catalan words avoiding a pair of patterns of length 3, pursuing the recent initiating work of the first and last authors and of S. Kirgizov where (among other things) the enumeration of Catalan words avoiding a patterns of length 3 is completed. More precisely, we explore systematically the structural properties of the sets of words under consideration and give enumerating results by means of recursive decomposition, constructive bijections or bivariate generating functions with respect to the length and descent number. Some of the obtained enumerating sequences are known, and thus the corresponding results establish new combinatorial interpretations for them.

Bootstrapping of integer concepts: the stronger deviant-interpretation challenge (and how to solve it)

Beck (Cognition 158:110–121, 2017) presents an outline of the procedure of bootstrapping of integer concepts, with the purpose of explicating the account of Carey (The Origin of Concepts, 2009). According to that theory, integer concepts are acquired through a process of inductive and analogous reasoning based on the object tracking system (OTS), which allows individuating objects in a parallel fashion. Discussing the bootstrapping theory, Beck dismisses what he calls the "deviant-interpretation challenge"—the possibility that the bootstrapped integer sequence does not follow a linear progression after some point—as being general to any account of inductive learning. While the account of Carey and Beck focuses on the OTS, in this paper I want to reconsider the importance of another empirically well-established cognitive core system for treating numerosities, namely the approximate number system (ANS). Since the ANS-based account offers a potential alternative for integer concept acquisition, I show that it provides a good reason to revisit the deviant-interpretation challenge. Finally, I will present a hybrid OTS-ANS model as the foundation of integer concept acquisition and the framework of enculturation as a solution to the challenge.

Nonassociativity of the Norton Algebras of Some Distance Regular Graphs

A Norton algebra is an eigenspace of a distance regular graph endowed with a commutative nonassociative product called the Norton product, which is defined as the projection of the entrywise product onto this eigenspace. The Norton algebras are useful in finite group theory as they have interesting automorphism groups. We provide a precise quantitative measurement for the nonassociativity of the Norton product on the eigenspace of the second largest eigenvalue of the Johnson graphs, Grassman graphs, Hamming graphs, and dual polar graphs, based on the formulas for this product established in previous work of Levstein, Maldonado and Penazzi. Our result shows that this product is as nonassociative as possible except for two cases, one being the trivial vanishing case while the other having connections with the integer sequence A000975 on OEIS and the so-called double minus operation studied recently by Huang, Mickey, and Xu.

Grossberg–Karshon Twisted Cubes and Hesitant Jumping Walk Avoidance

Let $G$ be a complex simply-laced semisimple algebraic group of rank $r$ and $B$ a Borel subgroup. Let $\mathbf i \in [r]^n$ be a word and let $\boldsymbol{\ell} = (\ell_1,\dots,\ell_n)$ be a sequence of non-negative integers. Grossberg and Karshon introduced a virtual lattice polytope associated to $\mathbf i$ and $\boldsymbol{\ell}$ called a twisted cube, whose lattice points encode the character of a $B$-representation. More precisely, lattice points in the twisted cube, counted with sign according to a certain density function, yields the character of the generalized Demazure module determined by $\mathbf i$ and $\boldsymbol{\ell}$. In recent work, the author and Harada described precisely when the Grossberg–Karshon twisted cube is untwisted, i.e., the twisted cube is a closed convex polytope, in the situation when the integer sequence $\boldsymbol{\ell}$ comes from a weight $\lambda$ of $G$. However, not every integer sequence $\boldsymbol{\ell}$ comes from a weight of $G$. In the present paper, we interpret the untwistedness of Grossberg–Karshon twisted cubes associated with any word $\mathbf i$ and any integer sequence $\boldsymbol{\ell}$ using the combinatorics of $\mathbf i$ and $\boldsymbol{\ell}$. Indeed, we prove that the Grossberg–Karshon twisted cube is untwisted precisely when $\mathbf i$ is hesitant-jumping-$\boldsymbol{\ell}$-walk-avoiding.

BETA CANTOR SERIES EXPANSION AND ADMISSIBLE SEQUENCES

We introduce a numeration system, called the <em>beta Cantor series expansion</em>, that generalizes the classical positive and negative beta expansions by allowing non-integer bases in the Q-Cantor series expansion. In particular, we show that for a fix $\gamma \in \mathbb{R}$ and a sequence $B=\{\beta_i\}$ of real number bases, every element of the interval $x \in [\gamma,\gamma+1)$ has a <em>beta Cantor series expansion</em> with respect to B where the digits are integers in some alphabet $\mathcal{A}(B)$. We give a criterion in determining whether an integer sequence is admissible when $B$ satisfies some condition. We provide a description of the reference strings, namely the expansion of $\gamma$ and $\gamma+1$, used in the admissibility criterion.

2-Adic valuations of coefficients of certain integer powers of formal power series

Let [Formula: see text] be an integer sequence and [Formula: see text] be its ordinary generating function. In this paper, we study the behavior of 2-adic valuations of the sequence [Formula: see text], where [Formula: see text] is fixed and [Formula: see text] More precisely, we propose a method, which under suitable assumptions on the sequence [Formula: see text] allows us to prove boundedness of the sequence [Formula: see text] for certain values of [Formula: see text]. In particular, if [Formula: see text] is the classical Rudin–Shapiro sequence, then we prove that [Formula: see text] for given [Formula: see text] and all [Formula: see text]. A similar result is proved for a relative of the Rudin–Shapiro sequence recently introduced by Lafrance, Rampersad and Yee.

Universal Computing Machine for Number Genetics in the Proportiones Perfectus Theory

The proportiones perfectus law states that σ_x^y=(x+√(x^2+4y))/2 is a proportione perfectus if 1≤y≤x such that for an arbitrary positive integer , there exists an integer sequence defined simultaneously by the quasigeometric relation h_(n+1)=round(σ_x^y h_n ),n≥1 and the arithmetic relation h_(n+2)=〖xh〗_(n+1)+〖yh〗_n,n≥1. When x=y=1 σ_x^y is the golden mean. When x=2,y=1,σ_x^y is the silver mean. In previous works we introduced the theory of number genetics – a framework of logic within which the golden section is studied. In this work we apply the concept to all proportiones perfectus. Let be defined as above. Furthermore, let h1 satisfy {█(round(h_1/(σ_x^y ))=w@round(wσ_x^y)≠h_1 ). ┤Now let H'nbe Hn with h1=1 Again let be defined as A robust universal computing machine is herein developed for the purpose of establishing the relationship h_i=g_(n+i-1)±h_i^',i,n≥1, which relationship is key to the logic protocol. It is clear therefore that this system of logic presents as the building block of every (including itself) within a particular σ_x^y regime. Clearly number genetics takes central place in the proportiones perfectus theory.

Role of Graphic Integer Sequence in the Determination of Graph Integrity

Networks have an important role in our daily lives. The effectiveness of the network decreases with the breaking down of some vertices or links. Therefore, a less vulnerable communication network is required for greater stability. Vulnerability is the measure of resistance of the network after failure of communication links. In this article, a graph has been taken for modeling a network and integrity as a measure of vulnerability. The approach is to estimate the integrity or upper bound of integrity of at least one connected graph or network constructed from the given graphic integer sequence. Experiments have been done with random graphs, complex networks and also a comparison between two parameters, namely the vertex connectivity and graph integrity as a measure of the network vulnerability have been carried out by removing vertices randomly from various complex networks. A comparison with the existing method shows that the algorithm proposed in this article provides a much better integrity measurement.