Related papers: Normal numbers and normality measure

Given a real number $0.a_1a_2 a_3\dots$ that is normal to base $b$, we examine increasing sequences $n_i$ so that the number $0.a_{n_1}a_{n_2}a_{n_3}\dots$ are normal to base $b$. Classically it is known that if the $n_i$ form an arithmetic…

Let $\alpha=0.a_1a_2a_3\ldots$ be an irrational number in base $b>1$, where $0\leq a_i<b$. The number $\alpha \in (0,1)$ is a \textit{normal number} if every block $(a_{n+1}a_{n+2}\ldots a_{n+k})$ of $k$ digits occurs with probability…

We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ \Delta^2 u=|u|^{p-1}u\ \{in} \ \R^n,$$ where $ p>1$ and $n\ge1$. We give a complete classification of stable and finite Morse index…

This work considers the problem of binary classification: given training data $x_1, \dots, x_n$ from a certain population, together with associated labels $y_1,\dots, y_n \in \left\{0,1 \right\}$, determine the best label for an element $x$…

Normality, in the colloquial sense, has historically been considered an aspirational trait, synonymous with ideality. The arithmetic average and, by extension, statistics including linear regression coefficients, have often been used to…

A new test of normality based on a standardised empirical process is introduced in this article. The first step is to introduce a Cram\'er-von Mises type statistic with weights equal to the inverse of the standard normal density function…

Ordinal classification problems, where labels exhibit a natural order, are prevalent in high-stakes fields such as medicine and finance. Accurate uncertainty quantification, including the decomposition into aleatoric (inherent variability)…

The dimension of a partially-ordered set $P$ is the smallest integer $d$ such that one can embed $P$ into a product of $d$ linear orders. We prove that the dimension of the divisibility order on the interval $\{1, \dotsc, n\}$ is bounded…

In this paper we study the property of normality of a number in base 2. A simple rule that associates a vector to a number is presented and the property of normality is stated for the vector associated to the number. The problem of testing…

We propose a new skewness test statistic for normality based on the Pearson measure of skewness. We obtain asymptotic first four moments of the null distribution for this statistic by using a computer algebra system and its normalizing…

Approximations to the modified signed likelihood ratio statistic are asymptotically standard normal with error of order $n^{-1}$, where $n$ is the sample size. Proofs of this fact generally require that the sufficient statistic of the model…

We prove a lower bound of $\tilde{\Omega}(n^{1/3})$ for the query complexity of any two-sided and adaptive algorithm that tests whether an unknown Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is monotone or far from monotone. This…

We prove that the Thue--Morse sequence $\mathbf t$ along subsequences indexed by $\lfloor n^c\rfloor$ is normal, where $1<c<3/2$. That is, for $c$ in this range and for each $\omega\in\{0,1\}^L$, where $L\geq 1$, the set of occurrences of…

Let $G$ be a graph. In a famous paper Collatz and Sinogowitz had proposed to measure its deviation from regularity by the difference of the (adjacency) spectral radius and the average degree: $\epsilon(G)=\rho(G)-\frac{2m}{n}$. We obtain…

Let $K_n$ denote the set of all nonsingular $n\times n$ lower triangular $(0,1)$-matrices. Hong and Loewy (2004) introduced the number sequence $$ c_n=\min\{\lambda\mid\lambda~\text{is an eigenvalue of}~XX^{\rm T},~X\in K_n\},\quad…

For $d\geq 2$ and any norm on $\mathbb R^d$, we prove that there exists a set of $n$ points that spans at least $(\tfrac d2-o(1))n\log_2n$ unit distances under this norm for every $n$. This matches the upper bound recently proved by Alon,…

A lower bound on the number of uncorrectable errors of weight half the minimum distance is derived for binary linear codes satisfying some condition. The condition is satisfied by some primitive BCH codes, extended primitive BCH codes,…

We show that for large enough $n$, the number of non-isomorphic pseudoline arrangements of order $n$ is greater than $2^{c\cdot n^2}$ for some constant $c > 0.2604$, improving the previous best bound of $c>0.2083$ by Dumitrescu and Mandal…

We consider sets of reals $X$ endowed with the Sorgenfrey lower limit topology denoted $X[\leq]$. Przymusi\'nski proved that if $X$ is a $Q$-set then $(X[\leq])^2$ is normal. While the converse is not in general true we consider examples of…

We give a construction of an absolutely normal real number $x$ such that for every integer $b $ greater than or equal to $2$, the discrepancy of the first $N$ terms of the sequence $(b^n x \mod 1)_{n\geq 0}$ is of asymptotic order…