Set Of Natural Numbers Research Articles

K-Zumkeller graphs through mycielski transformation

Let G = (V, E) be a simple graph. A 1-1 function f : V → ℕ , where ℕ is the set of natural numbers, is said to induce a k-Zumkeller graph G if the induced edge function f * : E → ℕ defined by f* (xy) = f (x) f (y) satisfies the following conditions: (i) f* (xy) is a Zumkeller number for every xy ∈ E. (ii) The total number of distinct Zumkeller numbers on the edges of G is k. A Mycielski transformation of a graph is a larger graph having more vertices and edges. In this article, the Mycielski transformation of a graphs such as path, cycle and star graphs have been computed and their k-Zumkeller graphs have been investigated by reducing the number of distinct Zumkeller numbers. AMS Subject Classification: 05C78

Generalized Dynamic Inequalities of Copson Type on Time Scales

This paper introduces novel generalizations of dynamic inequalities of Copson type within the framework of time scales delta calculus. The proposed generalizations leverage mathematical tools such as Hölder’s inequality, Minkowski’s inequality, the chain rule on time scales, and the properties of power rules on time scales. As special cases of our results, particularly when the time scale T equals the real line (T=R), we derive some classical continuous analogs of previous inequalities. Furthermore, when T corresponds to the set of natural numbers including zero (T=N0), the obtained results, to the best of the authors’ knowledge, represent innovative contributions to the field.

АЛГОРИТМИ ПОБУДОВИ МІНІМАЛЬНОЇ ПОРОДЖУВАЛЬНОЇ МНОЖИНИ РОЗВ’ЯЗКІВ СИСТЕМ ЛІНІЙНИХ РІВНЯНЬ

We consider optimizing transformations of the algorithm for construction of minimal generating sets of solutions of systems of linear homogeneous equations (SLHE) over the set of natural numbers. The features of such SLHEs are described, optimization transforms are substantiated, and examples of algorithm operation before and after optimization transforms are given. The application of the algorithm is illustrated by examples of the analysis of the properties of Petri nets and the construction of a set of basic solutions in the fields of complex, real, and rational numbers and over finite fields. Keywords: systems of linear equations, algorithms, solutions, optimization, complexity.

СТИСНЕННЯ ПРИРОДНОМОВНИХ ТЕКСТІВ РЕВЕРСНИМИ МУЛЬТИРОЗДІЛЬНИКОВИМИ КОДАМИ

We study a class of binary reverse multi-delimiter (RMD) data compression codes in application to natural language text compression. The RMD-codewords start with delimiters, i.e., prefixes of the form that cannot occur in other places of the codeword. The position of the delimiter in an RMD codeword differs from its position in “direct” multi-delimiter (MD) codes, where delimiters are codeword suffixes. RMD and MD codes possess many useful properties, such as unique decodability, completeness, universality, synchronizability, asymptotic densities, and finite automaton acceptability. For RMD-codes, we construct a monotonic mapping from the set of natural numbers to the set of codewords. For original MD-codes, hitherto, this was an open question. The discovered mapping and the byte quantification of a decoding automaton allow us to develop very fast byte-aligned algorithms for decoding and direct Boyer–Moore style search in compressed files. Compared with the known byte (SCDC) and Fibonacci codes, RMD codes demonstrate the best compression ratio on natural language texts (more than four times closer to the entropy bound than that of SCDC). Computer experiments demonstrate that RMD codes can be decoded almost as fast as SCDC and times faster than Fibonacci codes. In natural language text compression, we also practiced the RMD-encoding as a preprocessing tool, which improves the performance of the known modern powerful archivers.

From Diophantian Equations to Matrix Equations (III) - Other Diophantian Quadratic Equations and Diophantian Equations of Higher Degree

In this paper, we propose to continue the steps started in the first two papers with the same generic title and symbolically denoted by (I) and (II), namely, the presentation of ways of achieving a systemic vision on a certain mathematical notional content, a vision that to motivate and mobilize the activity of those who teach in the classroom, thus facilitating both the teaching and the assimilation of notions, concepts, scientific theories approached by the educational disciplines that present phenomena and processes from nature. Thus, we will continue in the same systemic approach, solving some Diophantine equations of higher degree, more precisely some generalizations of the Pythagorean equation and some quadratic Diophantine equations, in the set of natural numbers, then of the whole numbers, in order to "submerge" a such an equation in a ring of matrices and try to find as many matrix solutions as possible. In this way we will solve 12 large classes of Diophantine quadratic or higher order equations. For attentive readers interested in these matters, at the end of the paper we will propose 6 open problems. The solution of each of these open problems represents, in fact, a vast research activity and that can open the way to solving such more complicated Diophantine and / or matrix equations.

Uniqueness and homogeneity of non-separable Urysohn universal ultrametric spaces

We first introduce the new notion of petaloid ultrametric spaces, which is intended to be a standard class of non-separable Urysohn universal ultrametric spaces. Next we prove that all petaloid spaces are isometric to each other and homogeneous for all finite subspaces (and compact subspaces). Moreover, we show that the following spaces are petaloid, and hence they are isometric to each other and homogeneous: (1) The space of all continuous functions, whose images contain the zero, from the Cantor set into the space of non-negative real numbers equipped with the nearly discrete topology, (2) the space of all continuous ultrametrics on a zero-dimensional infinite compact metrizable space, (3) the non-Archimedean Gromov–Hausdorff space, and (4) the space of all maps from the set of non-negative real numbers into the set of natural numbers whose supports are finite or decreasing sequences convergent to the zero.

Integrability of Sequences

Abstract This paper deals with density on the set of natural numbers and its connections to the distribution of sequences. Under the assumption of independence, some formulas are derived.

Disjoint hypercyclicity, Sidon sets and weakly mixing operators

AbstractWe prove that a finite set of natural numbers J satisfies that $J\cup \{0\}$ is not Sidon if and only if for any operator T, the disjoint hypercyclicity of $\{T^j:j\in J\}$ implies that T is weakly mixing. As an application we show the existence of a non-weakly mixing operator T such that $T\oplus T^2\oplus\cdots \oplus T^n$ is hypercyclic for every n.

Linearity of the Position of Prime Numbers. The Order of Prime and Composite Numbers

In this paper it was shown that all prime numbers lie on 96 half-lines. At the same time, it was shown that if a given number does not lie on any of the above half-lines, then it is a composite number. A corresponding linear mathematical relationship was also derived and it was shown that all prime numbers must satisfy it. If a given number does not satisfy the above dependence, then it is a composite number. One linear equation and 8 numbers are enough to carry out all of the above-mentioned analyses. In addition, the astonishing cyclical nature of the occurrence of composite numbers in the set of natural numbers has been demonstrated.

Linearity of the Position of Prime Numbers. The Order of Prime and Composite Numbers

In this paper it was shown that all prime numbers lie on 96 half-lines. At the same time, it was shown that if a given number does not lie on any of the above half-lines, then it is a composite number. A corresponding linear mathematical relationship was also derived and it was shown that all prime numbers must satisfy it. If a given number does not satisfy the above dependence, then it is a composite number. One linear equation and 8 numbers are enough to carry out all of the above-mentioned analyses. In addition, the astonishing cyclical nature of the occurrence of composite numbers in the set of natural numbers has been demonstrated.

Modification of algorithm for constructing minimal supported set of solutions of SLE over the set of natural numbers

Modification of algorithm for constructing minimal supported set of solutions of SLE over the set of natural numbers

Generic algebraic properties in spaces of enumerated groups

We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well-known examples from combinatorial group theory, combined with the Baire category theorem, we obtain a plethora of results demonstrating that several phenomena in group theory are generic. In effect, we provide a new topological framework for the analysis of various well known problems in group theory. We also provide a connection between genericity in these spaces, the word problem for finitely generated groups and model-theoretic forcing. Using these connections, we investigate a natural question raised by Osin: when does a certain space of enumerated groups contain a comeager isomorphism class? We obtain a sufficient condition that allows us to answer Osin’s question in the negative for the space of all enumerated groups and the space of left orderable enumerated groups. We document several open questions in connection with these considerations.

Tentang Rumus Induksi Matematika

This paper presents various forms of induction formulas and at the same time proves the equivalence of these formulas by proving that some of these formulas are equivalent to the Well-Ordering Principle which applies to natural number set.

Prime Numbers Among Fractional, Integer and Natural Numbers

Series of primes are considered in three levels of sets of numbers (N)(Z)(Q), where – series of prime numbers in the set of natural numbers (PN), – a series of integer primes in the set of integers (WPN), – a series of rational primes in the set of rational (fractional) numbers (FPN). In series and the abscissa is taken as a complete series of integers , and for a series of prime numbers the positive right side of the x-axis is taken as a series of natural numbers . It is proved that the FPN set contains four types in the quadrants of the Cartesian coordinate system, the WPN set includes two mirror-opposite lines (the left one in quadrants II and IV and the right one in quadrants I and III). Subtraction from the next term WPN and PN of the previous component of the series gives new series of increments. Converting from decimal to binary gives for the series WPN and PN, as well as their increments, the rational root 1/2 of the Riemann zeta function on the second vertical of the binary expansion. Such a transformation makes it possible to obtain the block structure WPN and PN of partitioning the original series into blocks, and then the groups starting from the number 2. Using the example of block No. 8168 for the series PN, we proved the decomposition of groups into separate quanta, starting from the number 2, by asymmetric wavelets in the form solitary waves (solitons) with variable amplitude and oscillation period with a maximum relative error of 0.1%.

Some Applications of the Maynard–Tao Theorem

Let and be the set of natural numbers and prime numbers, respectively. For a positive integer n, define the following two representation functions and An old result of Erdős showed that is unbounded. As an elementary application of the Maynard–Tao theorem, we prove that is also unbounded, which confirms a conjecture of Chen in 2010.

A Novel Radio Geometric Mean Algorithm for a Graph

Radio antennas switch signals in the form of radio waves using different frequency bands of the electromagnetic spectrum. To avoid interruption, each radio station is assigned a unique number. The channel assignment problem refers to this application. A radio geometric mean labeling of a connected graph G is an injective function h from the vertex set, V(G) to the set of natural numbers N such that for any two distinct vertices x and y of G, h(x)·h(y)≥diam+1−d(x,y). The radio geometric mean number of h, rgmn(h), is the maximum number assigned to any vertex of G. The radio geometric mean number of G, rgmn(G) is the minimum value of rgmn(h), taken over all radio geometric mean labeling h of G. In this paper, we present two theorems for calculating the exact radio geometric mean number of paths and cycles. We also present a novel algorithm for determining the upper bound for the radio geometric mean number of a given graph. We verify that the upper bounds obtained from this algorithm coincide with the exact value of the radio geometric mean number for paths, cycles, stars, and bi-stars.

How to approximate fuzzy sets: mind-changes and the Ershov Hierarchy

Computability theorists have introduced multiple hierarchies to measure the complexity of sets of natural numbers. The Kleene Hierarchy classifies sets according to the first-order complexity of their defining formulas. The Ershov Hierarchy classifies limit computable sets with respect to the number of mistakes that are needed to approximate them. Biacino and Gerla extended the Kleene Hierarchy to the realm of fuzzy sets, whose membership functions range in a complete lattice. In this paper, we combine the Ershov Hierarchy and fuzzy set theory, by introducing and investigating the Fuzzy Ershov Hierarchy.

Disproof of a conjecture on the rainbow triangles in arc-colored digraphs

Let D be a digraph. The arc number a(D) of D is defined as the number of arcs of D. An arc-coloring of D is a mapping C: A(D)→N, where N is the set of natural numbers. The color number c(D) of D is defined as the number of colors assigned to the arcs of D. A rainbow triangle in D is a directed triangle in which every pair of arcs has distinct colors. Recently, Li et al. (2022) proved that if a(D)+c(D)=n(n−1)+⌊n24⌋+2 for an arc-colored digraph D of order n=3,4 without containing rainbow triangles, then D≅Kn↔. Thus, they conjectured that if a(D)+c(D)=n(n−1)+⌊n24⌋+1 for an arc-colored digraph D of order n≥5 without containing rainbow triangles, then D≅Kn↔. In this note, we disprove the conjecture by constructing a family of counterexamples. In fact, we show that there is not a function f(n) such that D≅Kn↔ if a(D)+c(D)=f(n) for an arc-colored digraph D of order n≥5 without containing rainbow triangles.

Collatz Conjecture Redefinition on Prime Numbers

The definition of Collatz Operator, the mathematical avatar of the Collatz Algorithm, permits the transformation of the Collatz conjecture, which is delineated over the whole natural number set, into an equivalent inference restricted to the odd prime number set only. Based on this redefinition, one can describe an empirical-heuristic proof of the Collatz conjecture.

On the set of exceptions in the product of sets of natural numbers with asymptotic density 1

On the set of exceptions in the product of sets of natural numbers with asymptotic density 1