Related papers: Classification of normal sequences
A sequence $(a_1, \ldots, a_n)$ of nonnegative integers is an {\em ascent sequence} if $a_0 =0$ and for all $i \geq 2$, $a_i$ is at most 1 plus the number of ascents in $(a_1, \ldots, a_{i-1})$. Ascent sequences were introduced by…
Aronson's sequence 1, 4, 11, 16, ... is defined by the English sentence ``t is the first, fourth, eleventh, sixteenth, ... letter of this sentence.'' This paper introduces some numerical analogues, such as: a(n) is taken to be the smallest…
The extent to which a sequence of finite length differs from a shifted version of itself is measured by its aperiodic autocorrelations. Of particular interest are sequences whose entries are 1 or -1, called binary sequences, and sequences…
A constant term sequence is a sequence of rational numbers whose $n$-th term is the constant term of $P^n(\boldsymbol{x}) Q(\boldsymbol{x})$, where $P(\boldsymbol{x})$ and $Q(\boldsymbol{x})$ are multivariate Laurent polynomials. While the…
Defined by Borel, a real number is normal to an integer base $b$, greater than or equal to $2$, if in its base-$b$ expansion every block of digits occurs with the same limiting frequency as every other block of the same length. We consider…
Generalising the work of Dey, we define the notion of ultra-synchronicity of sequences of real numbers. Let $B_{n,k},C_{n,k},P_{n,k},Q_{n,k}$ be the number of even permutations with $k$ descents, odd permutations with $k$ descents, even…
We describe a set $\mathcal{R}^{\infty}$ consisting of tuples of integer sequences and provide certain explicit maps on it. We show that this defines a semiregular crystal for $\mathfrak{sl}_{n+1}$ and $\mathfrak{sp}_{2n}$ respectively.…
We determine the Lie superalgebras that are graded by the root systems of the basic classical simple Lie superalgebras of type A$(m,n)$.
The run vector of a binary sequence reflects the run structure of the sequence, which is given by the set of all substrings of the run length encoding. The run vector and the aperiodic autocorrelations of a binary sequence are strongly…
Batch Normalization (BN) is a milestone technique in the development of deep learning, enabling various networks to train. However, normalizing along the batch dimension introduces problems --- BN's error increases rapidly when the batch…
A graph is called normal if its vertex set can be covered by cliques and also by stable sets, such that every such clique and stable set have non-empty intersection. This notion is due to Korner, who introduced the class of normal graphs as…
We define multilevel text normalization as sequence-to-sequence processing that transforms naturally noisy text into a sequence of normalized units of meaning (morphemes) in three steps: 1) writing normalization, 2) lemmatization, 3)…
Let $(x_n)_{n=1}^{\infty}$ be a sequence on the torus $\mathbb{T}$ (normalized to length 1). A sequence $(x_n)$ is said to have Poissonian pair correlation if, for all $s>0$, $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m…
Let n points be taken at random on a circle of unit circumference and clockwise ordered. Uniform spacings are defined as the clockwise arc-lengths between the successive points from this sample. We are interested in the asymptotic behavior…
Let $F_n$ be the $n$th Fibonacci number. Let $m, n$ be positive integers. Define a sequence $(G(k,n,m))_{k\geq 1}$ by $G(1,n,m) = F^m_n$, and $G(k+1,n,m) = F_{nG(k,n,m)}$ for all $k\geq 1$. We show that $F_n^{k+m-1}\mid G(k,n,m)$ for all…
A nut graph is a simple graph whose adjacency matrix is singular with $1$-dimensional kernel such that the corresponding eigenvector has no zero entries. In 2020, Fowler et al. characterised for each $d \in \{3,4,\ldots,11\}$ all values $n$…
A nondecreasing sequence of positive integers is $(\alpha,\beta)$-Conolly, or Conolly-like for short, if for every positive integer $m$ the number of times that $m$ occurs in the sequence is $\alpha + \beta r_m$, where $r_m$ is $1$ plus the…
From any given sequence of finite or infinite graphs, a nonstandard graph is constructed. The procedure is similar to an ultrapower construction of an internal set from a sequence of subsets of the real line, but now the individual entities…
Let $d, r \in \N$, $\|\cdot\|$ any norm on $\R^d$ and $B$ denote the unit ball with respect to this norm. We show that any sequence $v_1,v_2,...$ of vectors in $B$ can be partitioned into $r$ subsequences $V_1, ..., V_r$ in a balanced…
Binary sequences with minimal autocorrelations have applications in communication engineering, mathematics and computer science. In statistical physics they appear as groundstates of the Bernasconi model. Finding these sequences is a…