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Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees…
Embedding spaces contain interpretable dimensions indicating gender, formality in style, or even object properties. This has been observed multiple times. Such interpretable dimensions are becoming valuable tools in different areas of…
A double occurrence word $w$ over a finite alphabet $\Sigma$ is a word in which each alphabet letter appears exactly twice. Such words arise naturally in the study of topology, graph theory, and combinatorics. Recently, double occurrence…
A two-dimensional string is simply a two-dimensional array. We continue the study of the combinatorial properties of repetitions in such strings over the binary alphabet, namely the number of distinct tandems, distinct quartics, and runs.…
Every encoding has priori information if the encoding represents any semantic information of the unverse or object. Encoding means mapping from the unverse to the string or strings of digits. The semantic here is used in the model-theoretic…
In [2], while studying a relevant class of polyominoes that tile the plane by translation, i.e., double square polyominoes, the authors found that their boundary words, encoded by the Freeman chain coding on a four letters alphabet, have…
The binomial notation (w u) represents the number of occurrences of the word u as a (scattered) subword in w. We first introduce and study possible uses of a geometrical interpretation of (w ab) and (w ba) when a and b are distinct letters.…
By defining the dimension of natural numbers as the number of prime factors, all natural numbers smaller than 2^(n+1) (n is a natural number) can be classified by their dimensions, and the count of numbers of each dimension gives a…
In this article we give two different ways of representations of circular words. Representations with tuples are intended as a compact notation, while representations with trees give a way to easily process all conjugates of a word. The…
Neural codes, represented as collections of binary strings called codewords, are used to encode neural activity. A code is called convex if its codewords are represented as an arrangement of convex open sets in Euclidean space. Previous…
If $x$ is a non-empty string then the repetition $xx$ is called a tandem repeat. Similarly, a tandem in a two dimensional array $X$ is a configuration consisting of a same primitive block $W$ that touch each other with one side or corner.…
Tandem duplication is an evolutionary process whereby a segment of DNA is replicated and proximally inserted. The different configurations that can arise from this process give rise to some interesting combinatorial questions. Firstly, we…
We investigate the possibility of modelling the syntax and semantics of natural language by constraints, or rules, imposed by the multi-dimensional type theory Nabla. The only multiplicity we explicitly consider is two, namely one dimension…
In recent years several novel models were developed to process natural language, development of accurate language translation systems have helped us overcome geographical barriers and communicate ideas effectively. These models are…
The words of a language reflect the structure of the human mind, allowing us to transmit thoughts between individuals. However, language can represent only a subset of our rich and detailed cognitive architecture. Here, we ask what kinds of…
We solve the word problem for free double categories without equations between generators by translating it to the word problem for 2-categories. This yields a quadratic algorithm deciding the equality of diagrams in a free double category.…
We give tight bounds on the relation between the primal and dual of various combinatorial dimensions, such as the pseudo-dimension and fat-shattering dimension, for multi-valued function classes. These dimensional notions play an important…
In this article, we present a fresh perspective on language, combining ideas from various sources, but mixed in a new synthesis. As in the minimalist program, the question is whether we can formulate an elegant formalism, a universal…
Pre-trained word embeddings are widely used for transfer learning in natural language processing. The embeddings are continuous and distributed representations of the words that preserve their similarities in compact Euclidean spaces.…
A problem of reconstructing words from their subwords involves determining the minimum amount of information needed, such as multisets of scattered subwords of a specific length or the frequency of scattered subwords from a given set, in…