Related papers: Naturalized bracket row and Motzkin triangle
It is known that the core of mathematics is natural numbers. And everything related to the natural number is interesting to mathematicians. In this paper, we draw parallels between natural numbers and elements of a non-numeric lexicographic…
In this paper, we perform an arithmetization of well-formed parenthesis strings with zeros (Motzkin words) and of corresponding Motzkin paths. The transformations used are reminiscent of G\"odel numbering for mathematical objects of some…
We continue to consider the ordered lexicographic sequence, which is constructed according to the formal characteristics of a series of natural numbers. For analysis, we selected balanced parentheses with zeros, Motzkin words. As you know,…
We establish a total lexicographical order on the set of Motzkin words. Elements are ordered similarly to Natural Numbers in accordance with known rules (axioms). As a result, we were able to obtain arithmetic and logical operations on the…
Bracketed words are basic structures both in mathematics (such as Rota-Baxter algebras) and mathematical physics (such as rooted trees) where the locations of the substructures are important. In this paper we give the classification of the…
The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has…
Natural numbers can be divided in two non-overlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as…
We review the linearization of Poisson brackets and related problems, in the formal, analytic and smooth categories.
This paper is about producing a new kind of the pairs which we call it MS-pairs. To produce these pairs, we use an algorithm for dividing a natural number $x$ by two for two arbitrary numbers and consider their related graphs. We present…
We present the pattern underlying some of the properties of natural numbers, using the framework of complex networks. The network used is a divisibility network in which each node has a fixed identity as one of the natural numbers and the…
A natural partial order on the set of prime numbers was derived by the author from the internal symmetries of the primary finite fields, independently of Ford a.a., who investigated Pratt trees for primality tests. It leads to a…
The Motzkin numbers can be derived as coefficients of hybrid polynomials. Such an identification allows the derivation of new identities for this family of numbers and offers a tool to investigate previously unnoticed links with the theory…
In this note the notion of Poisson brackets in Kontsevich's "Lie World" is developed. These brackets can be thought of as "universally" defined classical Poisson structures, namely formal expressions only involving the structure maps of a…
The article deals with a lexicographic order in various sequences. Consider the axiomatic of lexicographic series, based on the properties of the natural numbers. Elements of the set are ordered first the code length; further in each sign…
Naturally-occurring bracketings, such as answer fragments to natural language questions and hyperlinks on webpages, can reflect human syntactic intuition regarding phrasal boundaries. Their availability and approximate correspondence to…
This paper is devoted to a systematic study of a class of binary trees encoding the structure of rational numbers both from arithmetic and dynamical point of view. The paper is divided into two parts. The first one is a critical review of…
We consider the numbers arising in the problem of normal ordering of expressions in canonical boson creation and annihilation operators. We treat a general form of a boson string which is shown to be associated with generalizations of…
We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that…
The covariant canonical formalism is a covariant extension of the traditional canonical formalism of fields. In contrast to the traditional canonical theory, it has a remarkable feature that canonical equations of gauge theories or gravity…
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…