Set Of Natural Numbers Research Articles

Likeℕ’s – a point of view on natural numbers

Abstract We define and study some simple structures which we call likens and which are conceptually near to both sets of natural numbers, i.e. ℕ with addition and ℕ* = ℕ \ {0} with multiplication. It appears that there are many different likens, which makes it possible to look on usual natural numbers from a more general point of view. In particular, we show that ℕ and ℕ* are related to some functionals on the space of likens. A similar idea is known for a long time as the Beurling generalized numbers. Our approach may be considered as a little more natural and more general, since it admits the finitely generated likens.

Radio Geometric Mean Labeling of Some Star Like Graphs

A radio Geometric Mean Labeling of a connected graph \(G\) is a one to one map \(f\) from the vertex set \(V(G)\) to the set of natural numbers \(N\) such that for two distinct vertices \(u\) and \(v\) of \(G\), \(d\left(u,v\right)+\left\lceil \sqrt{f(u)f(v)} \right\rceil \ge 1+\mathrm{diam}(G)\). The radio geometric mean number of \(f,\, r_{gmn} (f)\) is the maximum number assigned to any vertex of \(G\). The radio geometric mean number of \(G\), \(r_{gmn} (G)\) is the minimum value of \(r_{gmn} (f)\) taken over all radio geometric mean labeling \(f\) of \(G\). In this paper, we find the radio geometric mean number of some star like graphs.

Spiking Neural P Systems With Polarizations.

Spiking neural P (SN P) systems are a class of parallel computation models inspired by neurons, where the firing condition of a neuron is described by a regular expression associated with spiking rules. However, it is NP-complete to decide whether the number of spikes is in the length set of the language associated with the regular expression. In this paper, in order to avoid using regular expressions, two major and rather natural modifications in their form and functioning are proposed: the spiking rules no longer check the number of spikes in a neuron, but, in exchange, a polarization is associated with neurons and rules, one of the three electrical charges -, 0,+. Surprisingly enough, the computing devices obtained are still computationally complete, which are able to compute all Turing computable sets of natural numbers. On this basis, the number of neurons in a universal SN P system with polarizations is estimated. Several research directions are mentioned at the end of this paper.

Combinatorics of ideals --- selectivity versus density

This note is devoted to combinatorial properties of ideals on the set of natural numbers. By a result of Mathias, two such properties, selectivity and density, in the case of definable ideals, exclude each other. The purpose of this note is to measure the ``distance'' between them with the help of ultrafilter topologies of Louveau.

On integer sequences in product sets

Let B be a finite set of natural numbers or complex numbers. Product set corresponding to B is defined by B.B:={ab:a,b∈B}. In this paper we give an upper bound for longest length of consecutive terms of a polynomial sequence present in a product set accurate up to a positive constant. We give a sharp bound on the maximum number of Fibonacci numbers present in a product set when B is a set of natural numbers and a bound which is accurate up to a positive constant when B is a set of complex numbers.

Minimal complexity of equidistributed infinite permutations

An infinite permutation is a linear ordering of the set of natural numbers. An infinite permutation can be defined by a sequence of real numbers where only the order of elements is taken into account; such sequence of reals is called a representative of a permutation. In this paper we consider infinite permutations which possess an equidistributed representative on [0,1] (i.e., such that the prefix frequency of elements from each interval exists and is equal to the length of this interval), and we call such permutations equidistributed. Similarly to infinite words, a complexity p(n) of an infinite permutation is defined as a function counting the number of its subpermutations of length n. We show that, unlike for permutations in general, the minimal complexity of an equidistributed permutation α is pα(n)=n, establishing an analog of Morse and Hedlund theorem. The class of equidistributed permutations of minimal complexity coincides with the class of so-called Sturmian permutations, directly related to Sturmian words.

Metamathematical Properties of a Constructive Multi-typed Theory

This paper describes an axiomatic theory BT, which is a suitable formal theory for developing constructive mathematics, due to its expressive language with countable number of set types and its constructive properties such as the existence and disjunction properties, and consistency with the formal Church thesis. BT has a predicative comprehension axiom and usual combinatorial operations. BT has intuitionistic logic and is consistent with classical logic. BT is mutually interpretable with a so called theory of arithmetical truth PATr and with a second-order arithmetic SA that contains infinitely many sorts of sets of natural numbers. We compare BT with some standard second-order arithmetics and investigate the proof-theoretical strengths of fragments of BT, PATr and SA.

Topological graphs based on a new topology on Zn and its applications

Up to now there is no homotopy for Marcus-Wyse (for short M-) topological spaces. In relation to the development of a homotopy for the category of Marcus-Wyse (for short M-) topological spaces on Z2, we need to generalize the M-topology on Z2 to higher dimensional spaces X ? Zn, n ? 3 [18]. Hence the present paper establishes a new topology on Zn; n 2 N, where N is the set of natural numbers. It is called the generalized Marcus-Wyse (for short H-) topology and is denoted by (Zn, n). Besides, we prove that (Z3, 3) induces only 6- or 18-adjacency relations. Namely, (Z3, 3) does not support a 26-adjacency, which is quite different from the Khalimsky topology for 3D digital spaces. After developing an H-adjacency induced by the connectedness of (Zn; n), the present paper establishes topological graphs based on the H-topology, which is called an HA-space, so that we can establish a category of HA-spaces. By using the H-adjacency, we propose an H-topological graph homomorphism (for short HA-map) and an HA-isomorphism. Besides, we prove that an HA-map (resp. an HA-isomorphism) is broader than an H-continuous map (resp. an Hhomeomorphism) and is an H-connectedness preserving map. Finally, after investigating some properties of an HA-isomorphism, we propose both an HA-retract and an extension problem of an HA-map for studying HA-spaces.

The Unique Natural Number Set and Distributed Prime Numbers

The Natural number set has a new number set behind three main subset: odd, even and prime which are unique number set. It comes from a mathematical relationship between the two main types even and odd number set. It consists of prime and T-semi-prime numbers set. However, finding ways to understand how prime number was distributed was ambiguity and it is impossible, this new natural set show how the prime number was distributed and highlight focusing on T-semi-prime number as a new sub natural set.

Об аналоге теоремы Слупецкого для полиномиальных функций

Functional systems are among the main objects in discrete mathematics and mathematical cybernetics. The meaningful connection of functional systems with real cybernetic models of control systems, on the one hand, determines a series of essential requirements that are imposed on functional systems, and on the other hand, generates a class of important problems that have both theoretical and application significance. The theory of functional systems covers a wide range of problem areas, the main of them being the problems of completeness, relative completeness, expressibility, synthesis, analysis, and some others. The article considers the functional systems of polynomial functions falling in the following categories: (i) with natural coefficients in which the values of both the arguments and the functions themselves belong to the set of natural numbers; (ii) with integer coefficients in which the values of both the arguments and the functions themselves belong to the set of integer numbers; and (iii) with rational coefficients in which the values of both the arguments and the functions themselves belong to the set of rational numbers. The superposition operations (permutation of variables, renaming of variables without identifying them, identifying the variables, introducing a fictitious variable, removing a fictitious variable, and substituting one function into another) were taken as a set of operations in all these functional systems, and the problem of relative completeness is investigated for these systems. An algorithmic approach of this problem (an analog of Slupecki's theorem) is presented for the above-mentioned functional systems. In particular, it has been proven that the posed problem has a positive answer only for a functional system with natural coefficients, whereas the answer for the other two systems is negative.

Generalized Hyperarithmetical Computability Over Structures

We consider the class of approximation spaces generated by admissible sets, in particular by hereditarily finite superstructures over structures. Generalized computability on approximation spaces is conceived of as effective definability in dynamic logic. By analogy with the notion of a structure Σ-definable in an admissible set, we introduce the notion of a structure effectively definable on an approximation space. In much the same way as the Σ-reducibility relation, we can naturally define a reducibility relation on structures generating appropriate semilattices of degrees of structures (of arbitrary cardinality), as well as a jump operation. It is stated that there is a natural embedding of the semilattice of hyperdegrees of sets of natural numbers in the semilattices mentioned, which preserves the hyperjump operation. A syntactic description of structures having hyperdegree is given.

Nonconventional polynomial CLT

We obtain a functional central limit theorem (CLT) for sums of the form , where is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polynomial growth and certain regularity properties, is a certain centralizing constant and are arbitrary polynomials taking on positive integer values on positive integers, i.e. polynomials satisfying , where is the set of natural numbers. For polynomial ’s this CLT generalizes [The Annals of Probability, 2014, Vol. 42, No.2, 649–688] which allows only linear ’s to have the same polynomial degree. We also prove that exists and provide necessary and sufficient conditions for its positivity, which is equivalent to the statement that the weak limit of is not zero almost surely. Finally, we study independence properties of the increments of the limiting process. Our proofs require studying asymptotic densities of special subsets of , which is done in a separate section. As in [The Annals of Probability, 2014, Vol. 42, No.2, 649-688], our results hold true when , where T is a mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well as in the case when , where is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.

Natural Vector Spaces (inward power and Minkowski norm of a Natural Vector, Natural Boolean Hypercubes) and a Fermat’s Last Theorem conjecture

In order to use the structure and operations of Molecular Similarity semispaces, Natural Vector Semispaces are considered in this study as vector spaces defined over the set of natural numbers, with zero added if necessary. The complete sum and inward power of a vector, defined as basic tools in Quantum Molecular Similarity, are now applied to a Natural Vector to describe Minkowski norms in these vector spaces. The structure and behavior of the Minkowski norm of Natural Vector inward powers and the Boolean Hypercube vertex translation into natural numbers are further used to conjecture a plausible general set up of Fermat’s Last Theorem.

Genericity for Mathias forcing over general Turing ideals

In Mathias forcing, conditions are pairs (D, S) of sets of natural numbers, in which D is finite, S is infinite, and max D < min S. The Turing degrees and computational characteristics of generics for this forcing in the special (but important) case where the infinite sets S are computable were thoroughly explored by Cholak, Dzhafarov, Hirst and Slaman [2]. In this paper, we undertake a similar investigation for the case where the sets S are members of general countable Turing ideals, and give conditions under which generics for Mathias forcing over one ideal compute generics for Mathias forcing over another. It turns out that if I does not contain only the computable sets, then non-trivial information can be encoded into the generics for Mathias forcing over I. We give a classification of this information in terms of computability-theoretic properties of the ideal, using coding techniques that also yield new results about introreducibility. In particular, we extend a result of Slaman and Groszek and show that there is an infinite Δ 3 0 set with no introreducible subset of the same degree.

Generic muchnik reducibility and presentations of fields

We prove that if I is a countable ideal in the Turing degrees, then the field RI of real numbers in I is computable from exactly the degrees that list the functions (i.e., elements of ωω) in I. This implies, for example, that the degree spectrum of the field of computable real numbers consists exactly of the high degrees. We also prove that if I is a countable Scott ideal, then it is strictly easier to list the sets (i.e., elements of 2ω) in I than it is to list the functions in I. This allows us to answer a question of Knight, Montalban, and Schweber. They introduced generic Muchnik reducibility to extend the idea of Muchnik reducibility between countable structures to arbitrary structures. They asked if R is generically Muchnik reducible to the structure that consists of all sets of natural numbers. Our result for Scott ideals shows that this is not the case.

An application of Benford s law to fundamental accounting figures reported by Borsa Istanbul (BIST) companies

The financial reporting of public firms is argued for being subject to manipulation and fraud. Since adherence to this mysterious law is accepted as a sign for the data’s reliability, Benford’s Law has long been used by auditors as a tool to test the integrity of a dataset and to detect fraud. The expected distribution of digits in any set of natural numbers has initially been put forward by Benford (1938). Benford’s law states that probability distribution of digits’ occurrence is not uniform. Smaller digits are found to exist more frequently than the greater ones. This study tested the compliance of fundamental figures reported by Borsa Istanbul (BIST) companies with Benford’s Law, by means of a data between 2005 and 2015 covering 148 companies. According to the different testing approaches utilized, which imparted rather similar results, reported figures of current assets and net sales seemed to be almost in perfect conformity with Benford’s Law. However, the study detected several deviation points in the data of total assets and net profit figures from Benford’s Law. From the results of this study, we cannot conclude that they are extensively manipulative or they are in full conformity with the Benford’s Law. Nevertheless, this study suggests the possible point of interest for further researches. In application of Benford’s Law non-conformity should be evaluated with discretion. The deviations are only a signal to analyze the data further, and should not be seen as a solid proof of fraud or manipulation.

Multi-sorted version of second order arithmetic

This paper describes axiomatic theories SA and SAR, which are versions of second order arithmetic with countably many sorts for sets of natural numbers. The theories are intended to be applied in reverse mathematics because their multi-sorted language allows to express some mathematical statements in more natural form than in the standard second order arithmetic. We study metamathematical properties of the theories SA, SAR and their fragments. We show that SA is mutually interpretable with the theory of arithmetical truth PATr obtained from the Peano arithmetic by adding infinitely many truth predicates. Corresponding fragments of SA and PATr are also mutually interpretable. We compare the proof-theoretical strengths of the fragments; in particular, we show that each fragment SAs with sorts &lt;=s is weaker than next fragment SAs+1.

Asymptotic law of distribution of primes of special form

Let N0 be the set of natural numbers whose binary expansions have an even number of 1’s, and let N1 = N\N0. In this paper, we obtain asymptotic formulas for the number of primes p not exceeding X and such that p ∈ Ni, p + 1 ∈ Nj, where i and j take values 0 and 1 independently of each other.

Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set

Abstract In this paper, we study the maximal subsemigroups of several semigroups of order-preserving transformations on the natural numbers and the integers, respectively. We determine all maximal subsemigroups of the monoid of all order-preserving injections on the set of natural numbers as well as on the set of integers. Further, we give all maximal subsemigroups of the monoid of all bijections on the integers. For the monoid of all order-preserving transformations on the natural numbers, we classify also all its maximal subsemigroups, containing a particular set of transformations.

Torsion Subgroups of CM Elliptic Curves over Odd Degree Number Fields:

Let |$\mathscr{G}_{\operatorname{CM}}(d)$| denote the collection of groups (up to isomorphism) that appear as the torsion subgroup of a complex multiplication (CM) elliptic curve over a degree |$d$| number field. We completely determine |$\mathscr{G}_{\operatorname{CM}}(d)$| for odd integers |$d$| and deduce a number of statistical theorems about the behavior of torsion subgroups of CM elliptic curves. Here are three examples: (1) For each odd |$d$|⁠, the set of natural numbers |$d'$ with $\mathscr{G}_{\operatorname{CM}}(d')=\mathscr{G}_{\operatorname{CM}}(d)$| possesses a well-defined, positive asymptotic density. (2) Let |$T_{\operatorname{CM}}(d) = \max_{G \in \mathscr{G}_{\operatorname{CM}}(d)} \#G$|⁠; under the Generalized Riemann Hypothesis, (12eγπ)2/3≤lim supd→∞d oddTCM(d)(dlog⁡log⁡d)2/3≤(24eγπ)2/3. (3) For each |$\epsilon > 0$|⁠, we have |$\#\mathscr{G}_{\operatorname{CM}}(d) \ll_{\epsilon} d^{\epsilon}$| for all odd |$d$|⁠; on the other hand, for each |$A> 0$|⁠, we have |$\#\mathscr{G}_{\operatorname{CM}}(d) > (\log{d})^A$| for infinitely many odd |$d$|⁠.