Positive Integer Research Articles

Ramanujan Sums of All Natural Numbers with Grandi Series

We are all know that the Ramanujan’s theory of calculating the sum of all the natural number. When we came to known that sum of all-natural number is ,that time everyone thinks about two things, one is how it possible that sum of positive number is negative & other is how it is very close to zero. Another research paper gives value of all natural number is then I started the study of this sum, then I find it is as zero. Quit interestingly but this value has more important than Ramanujan’s sum because I did not consider the Grandi series for calculating this sum. And then it is very easy to calculate the sum of negative integer & sum of all odd natural number & sum of all even natural number.

Value distribution of a pair of meromorphic functions

The well-known Picard theorem shows only what happens on the Picard exceptional value of a meromorphic function f. In this paper, we mainly consider what happens on the common or different Picard exceptional values of two or three meromorphic functions with certain types. For instance, we will provide some new results and discussions for the generalized Picard exceptional values or small functions of a pair of meromorphic functions fng(k) and gmf(k), these two functions are the crossed variants of the complex differential polynomials in Hayman's conjecture, where n, m are positive integers and k≥0. We give more details on the case that fg′ and gf′ when f and g are exponential polynomials in particular.

Single-neuron representation of nonsymbolic and symbolic number zero in the human medial temporal lobe

The number zero holds a special status among numbers, indispensable for developing a comprehensive number theory.1,2,3,4 Despite its importance in mathematics, the neuronal foundation of zero in the human brain is unknown. We conducted single-neuron recordings in neurosurgical patients5,6,7 while they made judgments involving nonsymbolic number representations (dot numerosity), including the empty set, and symbolic numbers (Arabic numerals), including numeral zero. Neurons showed responsiveness to either the empty set or numeral zero, but not both. Neuronal activity to zero in both nonsymbolic and symbolic formats exhibited a numerical distance effect, indicating that zero representations are integrated together with countable numerosities and positive integers at the low end of the number line.8,9 A boundary in neuronal coding existed between the nonsymbolic empty set and small numerosities, correlating with the relative difficulty in discriminating numerosity zero behaviorally. Conversely, no such boundary was found for symbolic zero activity, suggesting that symbolic representations integrate zero with other numerals along the number line, reconciling its outlier role. The status of zero as a special nonsymbolic numerical quantity is reflected in the activity of neurons in the human brain, which seems to serve as a scaffold for more advanced representations of zero as a symbolic number.

An Improved Driving Training-Based Optimization Algorithm for the Minimum Gap Graph Connected Partition Problem

In this paper, we consider the minimum gap partition problem in an undirected connected graph with nonnegative vertex weights. This problem involves in dividing the given graph into a specified number of connected subgraphs such that the total difference between the largest and the smallest vertex weights in each subgraph is minimized. Namely, let G = (V, E, W) be a given undirected connected graph with vertex set V , edge set E, and nonnegative vertex weight function W : V → R+, and let k ≥ 2 be a given positive integer. The minimum gap partition problem consists of constructing a partition P = {V1, V2, . . . , Vk} of non-empty and pairwise disjoint subsets of V such that each vertex set Vi, i = 1, 2, . . . , k, induces a connected subgraph G[Vi] of G. The objective of the problem is to find such a partition of V that minimizes the total difference 1≤i≤k maxv∈Viw(v) − minv∈Viw(v). This problem, as many other variants of graph partition problems, is known to be NP-hard problem. We designed an efficient metaheuristic algorithm for finding approximate solution of large-scale instances of this problem. The quality of the proposed algorithm is assessed comparing with the previous algorithms proposed for the problem.

On the number of prime factors with a given multiplicity over h-free and h-full numbers

Let k and n be natural numbers. Let ωk(n) denote the number of distinct prime factors of n with multiplicity k as studied by Elma and the third author [5]. We obtain asymptotic estimates for the first and the second moments of ωk(n) when restricted to the set of h-free and h-full numbers. We prove that ω1(n) has normal order log⁡log⁡n over h-free numbers, ωh(n) has normal order log⁡log⁡n over h-full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions ωk(n) with 1<k<h do not have normal order over h-free numbers and ωk(n) with k>h do not have normal order over h-full numbers.

Reliability Analysis of (n, k)-Bubble-Sort Networks Based on Extra Conditional Fault

Given a graph G=(V(G),V(E)), a non-negative integer g and a set of faulty vertices F⊆V(G), the g-extra connectivity of G, denoted by κg(G), is the smallest cardinality of F, whose value of deletion, if exists, will disconnect G and give each remaining component at least g+1 vertices. The g-extra diagnosability of the graph G, denoted by tg(G), is the maximum cardinality of the set F of fault vertices that the graph can guarantee to identify under the condition that each fault-free component has more than g vertices. In this paper, we determine that g-extra connectivity of (n,k)-bubble-sort network Bn,k is κg(Bn,k)=n+g(k−2)−1 for 4≤k≤n−1 and 0≤g≤n−k. Afterwards, we show that g-extra diagnosability of Bn,k under the PMC model (4≤k≤n−1 and 0≤g≤n−k) and MM* model (4≤k≤n−1 and 0≤g≤min{n−k−1,k−2}) is tg(Bn,k)=n+g(k−1)−1, respectively.

Addendum on ‘Natural number dynamics: reconstructing physics from generalized atomicity’ (2021 Eur. J. Phys. 42 025402)

The derivation of the Schrödinger equation is revisited with particular emphasis on the role of the Riccati equation and the relationship between the Hamiltonian and natural number operators.

L(3,2,1)-labeling of certain planar graphs

Given a graph G=(V,E) of maximum degree Δ, denoting by d(x,y) the distance in G between nodes x,y∈V, an L(3,2,1)-labeling of G is an assignment l from V to the set of non-negative integers such that |l(x)−l(y)|≥3 if x and y are adjacent, |l(x)−l(y)|≥2 if d(x,y)=2, and |l(x)−l(y)|≥1 if d(x,y)=3, for all x and y in V. The L(3,2,1)-number λ(G) is the smallest positive integer such that G admits an L(3,2,1)-labeling with labels from {0,1,…,λ(G)}.In this paper, the L(3,2,1)-number of certain planar graphs is determined, proving that it is linear in Δ, although the general upper bound for the L(3,2,1)-number of planar graphs is quadratic in Δ.

Examples and counterexamples in Ehrhart theory

This article provides a comprehensive exposition about inequalities that the coefficients of Ehrhart polynomials and h^{*} -polynomials satisfy under various assumptions. We pay particular attention to the properties of Ehrhart positivity as well as unimodality, log-concavity, and real-rootedness for h^{*} -polynomials.We survey inequalities that arise when the polytope has different normality properties. We include statements previously unknown in the Ehrhart theory setting as well as some original contributions to this topic. We address numerous variations of the conjecture asserting that IDP polytopes have a unimodal h^{*} -polynomial and construct concrete examples showing that these variations of the conjecture are false. Emphasis is put on polytopes arising within algebraic combinatorics.Furthermore, we describe and construct polytopes having pathological properties on their Ehrhart coefficients and roots, and we indicate for the first time a connection between the notions of Ehrhart positivity and h^{*} -real-rootedness. We investigate the log-concavity of the sequence of evaluations of an Ehrhart polynomial at the non-negative integers. We conjecture that IDP polytopes have a log-concave Ehrhart series. Many additional problems and challenges are proposed.

Multistability of state-dependent switched neural networks with Mexican-hat-type activation functions

This paper investigates the issue of multistability in state-dependent switched neural networks (SSNNs) with Mexican-hat-type activation functions (AFs). It establishes the coexistence and stability of multiple equilibrium points (EPs). Initially, the state space is partitioned based on the geometric characteristics of the Mexican-hat-type AF, enabling to determine the positions of the EPs. Secondly, the coexistence of 9h17h25h33h4 EPs for n-neurons SSNNs under specific sufficient conditions is proved with the Brouwer’s fixed-point theorem. Next, by using diagonally dominant matrix theory and Gershgorin circle theorem, it is proven that there are 5h14h23h32h4 asymptotically stable EPs under some conditions, where h1,h2,h3 and h4 are nonnegative integers satisfying 0≤h1+h2+h3+h4≤n. Therefore, we can obtain that SSNNs can have larger storage capacity by selecting the appropriate parameters hi,i=1,2,3,4. Finally, the correctness of the results in this paper is verified through two numerical examples.

Generalized boson and fermion operators with a maximal total occupation property

Abstract We propose a new generalization of the standard (anti-)commutation relations for creation and annihilation operators of bosons and fermions. These relations preserve the usual symmetry properties of bosons and fermions. Only the standard (anti-)commutator relation involving one creation and one annihilation operator is deformed by introducing fractional coefficients, containing a positive integer parameter p. The Fock space is determined by the classical definition. The new relations are chosen in such a way that the total occupation number in the system has the maximum value p. From the actions of creation and annihilation operators in the Fock space, a group theoretical framework is determined, and from here the correspondence with known particle statistics is established.

Odd-Even Congruence Labeling of Some Graphs

Introduction: Graph labeling is one of the fastest growing areas in the field of graph theory. A graph labeling is basically an assignment of integers to the nodes or edges or both, subject to certain conditions. Different types of graph labeling techniques have been developed by several authors. Objectives: The applications of labeling of graphs has diverse fields in the study of data base management, secret sharing schemes, physical cosmology, debug circuit design, X-ray, crystallography, astronomy, radar, broadcasting network, secret, coding theory and much more. The graph labeling techniques serve as useful mathematical models and solve various graph related problems.Methods: The strength of this research paper is based on Odd-even congruence labeling of different types of graphs. Assignment of natural numbers as labels for the edges and vertices of a graph. It is based on modular arithmetic property known as congruence graph labelling of a graph.For congruence graph labeling it entails the assignment of odd integers to vertices and even integers to edges with property of congruence graph labeling. Results: This labeling method has been identified on friendship graph, shell graph, generalized butterfly graph, fan graph, P_2+mK_1 graph

Finite Groups Containing No Blocks of Defect Zero

The irreducible characters of finite groups are always contained in blocks of defects which are nonnegative integers. Even though blocks always exist in finite groups, it is not the case that blocks of defect zero would always exist as well. Blocks of defect zero contain only one irreducible ordinary character each of defect zero and the defect group of blocks of defect zero is always the trivial subgroup of a finite group. Some finite groups do not have characters of defect zero and hence no blocks of defect zero either. The object in this paper is to study finite groups containing no blocks of defect zero. Finite abelian groups and p-groups will serve as special cases in this regard, with all blocks of finite abelian groups being of full/highest defect. We shall also determine an upper bound for the number of blocks in finite groups which contain no blocks of defect zero.

On Closed Forms of Some Trigonometric Series

We have derived alternative closed-form formulas for the trigonometric series over sine or cosine functions when the immediate replacement of the parameter appearing in the denominator with a positive integer gives rise to a singularity. By applying the Choi–Srivastava theorem, we reduce these trigonometric series to expressions over Hurwitz’s zeta function derivative.

Graded Weakly Strongly Quasi-Primary Ideals over Commutative Graded Rings

In this article, we introduce and examine the concept of graded weakly strongly quasi primary ideals. A proper graded ideal P of R is said to be a graded weakly strongly quasi primary (shortly, Gwsq-primary) ideal if whenever 0≠xy∈P, for some homogeneous elements x,y∈R, then x2∈P or yn∈P, for some positive integer n. Many examples and properties of Gwsq-primary ideals are given. Among several results, we compare Gwsq-primary ideals and other classical graded ideals such as graded strongly quasi primary ideals, graded weakly primary ideals and graded weakly 2-prime ideals etc.

Periodicity of bipartite walk on biregular graphs with conditional spectra

In this paper we study a class of discrete quantum walks, known as bipartite walks. These include the well-known Grover’s walks. A discrete quantum walk is given by the powers of a unitary matrix U indexed by arcs or edges of the underlying graph. The walk is periodic if U k = I for some positive integer k. Kubota has given a characterization of periodicity of Grover’s walk when the walk is defined on a regular bipartite graph with at most five eigenvalues. We extend Kubota’s results—if a biregular graph G has eigenvalues whose squares are algebraic integers with degree at most two, we characterize periodicity of the bipartite walk over G in terms of its spectrum. We apply periodicity results of bipartite walks to get a characterization of periodicity of Grover’s walk on regular graphs.

Constacyclic codes over F q S T and their applications

Let q = p r , where p is an odd prime and r is a positive integer. Consider a ring F q S T , where S = F q + u F q + v F q + uv F q with u 2 = 1 , v 2 = 1 , uv = vu and T = F q + u F q + v F q + w F q + uv F q + uw F q + vw F q + uvw F q with u 2 = 1 , v 2 = 1 , w 2 = 1 , uv = vu , uw = wu , vw = wv . In this paper, we examine the algebraic structure of constacyclic codes over the ring F q S T of block length ( α , β , γ ) . Further, a construction of quantum error-correcting codes (QECCs) from constacyclic codes over F q S T is also given. Moreover, we derive a number of new QECCs.

Multiplication cubes and multiplication automata

Abstract We extend previously known two-dimensional multiplication tiling systems that simulate multiplication by two natural numbers p and q in base $pq$ to higher dimensional multiplication tessellation systems. We develop the theory of these systems and link different multiplication tessellation systems with each other via macrotile operations that glue cubes in one tessellation system into larger cubes of another tessellation system. The macrotile operations yield topological conjugacies and factor maps between cellular automata performing multiplication by positive numbers in various bases.

The ϕS polar decomposition when S is skew-symmetric

Let F be a field with characteristic not equal to 2, and S∈M2n(F) be skew-symmetric and nonsingular. For X∈M2n(F), we show that X has a ϕS polar decomposition if and only if (a) ϕS(X)X has a ϕS-symmetric square root, (b) XϕS(X) is similar to ϕS(X)X, and (c) rank [XϕS(X)]tX is even for all nonnegative integers t.

Investigating Secant Varieties

A variety is a geometric object that locally looks like the set of points that are the solutions of some polynomial equations. Any given variety gives rise to a sequence of secant varieties indexed by the nonnegative integers. The kth secant variety is the union of all planes spanned by k+1 points on the original variety. Like vector spaces in linear algebra, varieties have an intrinsic dimension. A naive parameter count produces an upper bound on the dimension of the kth secant variety called its expected dimension. A secant variety is called defective if its actual dimension is less than its expected dimension. Despite being well understood in a few interesting cases, the classification of varieties with defective secant varieties in general remains far from complete. For a specific class of varieties called smooth toric varieties, our goal is to determine which ones admit defective secant varieties. Using the computer algebra system Macaulay2, we conducted experiments on this class of varieties and collected data on defectiveness, focusing on varieties of low dimension and Picard rank. Toric varieties have strong connections to combinatorics and convex polytopes; for any toric variety, there is a corresponding polytope that contains data about the variety. From the polytope, we are able to construct a particular kind of matrix that encodes information about the variety and its secants, which allowed us to prove defectivity for various Picard rank 2 toric varieties. Conversely, applying results from a paper by Laface–Massarenti–Rischter (2022) to the polytopes, we were able to prove non-defectivity of all secant varieties of certain toric surfaces with Picard rank 2. These statements together yield a classification of defective secant varieties for a subset of 2-dimensional smooth toric varieties with Picard rank 2.