Sum Of Ideals Research Articles

Cauchy’s Residue Theorem and Its Application

One of the key theorems in complex analysis, Cauchy’s Residue Theorem, which refers to the integral value of an analytic function along any simple closed contour surrounding an isolated singularity in a certain ring domain divided by 2πi, will greatly simplify the process of computing integrals on contour surrounding singularities. In the field of basic mathematics, the Cauchy’ Residue Theorem plays a key role in the integral calculation of analytic functions, nonuniform complex functions and the argument principle, which establishes a connection between a curve’s winding number and the quantity of zeros and poles inside the curve. Furthermore, Residue Theorem can combine with various subjects, including electromagnetism. This paper gives an overview of the Cauchy’s Residue Theorem and its application. Firstly, it talks about the definition and proof of the Cauchy’ Residue Theorem, along with the definition of residue. Then two examples of using Residue Theorem to integrate functions will be given. Finally, this paper involves application of Residue Theorem in trigonometric sum identities and a remark on the method.

Convolution identities for divisor sums and modular forms

We consider certain convolution sums that are the subject of a conjecture by Chester, Green, Pufu, Wang, and Wen in string theory. We prove a generalized form of their conjecture, explicitly evaluating absolutely convergent sums [Formula: see text]where [Formula: see text] is a Laurent polynomial with logarithms. Contrary to original expectations, such convolution sums, suitably extended to [Formula: see text], do not vanish, but instead, they carry number theoretic meaning in the form of Fourier coefficients of holomorphic cusp forms.

Lattice sums of I-Bessel functions, theta functions, linear codes and heat equations

We extend a certain type of identities on sums of I-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of theta functions such as the Dedekind eta function can be given by I-Bessel lattice sum identities with characters. We consider analogues of theta functions of lattices coming from linear codes and show that sums of I-Bessel functions defined by linear codes can be expressed by complete weight enumerators. We also prove that I-Bessel lattice sums appear as solutions of heat equations on general lattices. As a further application, we obtain an explicit solution of the heat equation on Zn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{Z}}^n$$\\end{document} whose initial condition is given by a linear code.

Strong metric dimension of the prime ideal sum graphs of commutative rings

Strong metric dimension of the prime ideal sum graphs of commutative rings

On rings whose prime ideal sum graphs are line graphs

Let [Formula: see text] be a commutative ring with unity. The prime ideal sum graph of the ring [Formula: see text] is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text], [Formula: see text] are adjacent if and only if [Formula: see text] is a prime ideal of [Formula: see text]. In this paper, we characterize all commutative Artinian rings whose prime ideal sum graphs are line graphs. Finally, we give a description of all commutative Artinian rings whose prime ideal sum graph is the complement of a line graph.

The Riemann zeta function and exact exponential sum identities of divisor functions

We prove an explicit integral formula for computing the product of two shifted Riemann zeta functions everywhere in the complex plane. We show that this formula implies the existence of infinite families of exact exponential sum identities involving the divisor functions, and we provide examples of these identities. We conjecturally propose a method to compute divisor functions by matrix inversion, without employing arithmetic techniques.

On the Leonardo quaternions sequence

In the present work, a new sequence of quaternions related to the Leonardo numbers -- named the Leonardo quaternions sequence -- is defined and studied. Binet's formula and certain sum and binomial-sum identities, some of which derived from the mentioned formula, are established. Tagiuri-Vajda's identity and, as consequences, Catalan's identity, d'Ocagne's identity and Cassini's identity are presented. Furthermore, applying Catalan's identity, and the connection between composition algebras and vector cross product algebras, Gelin-Cesàro's identity is also stated and proved. Finally, the generating function, the exponential generating function and the Poisson generating function are deduced. In addition to the results on Leonardo quaternions, known results on Leonardo numbers and on Fibonacci quaternions are extended.

Computing degree based topological indices of algebraic hypergraphs

Topological indices are numerical parameters that indicate the topology of graphs or hypergraphs. A hypergraph H=(V(H),E(H)) consists of a vertex set V(H) and an edge set E(H), where each edge e∈E(H) is a subset of V(H) with at least two elements. In this paper, our main aim is to introduce a general hypergraph structure for the prime ideal sum (PIS)- graph of a commutative ring. The prime ideal sum hypergraph of a ring R is a hypergraph whose vertices are all non-trivial ideals of R and a subset of vertices Ei with at least two elements is a hyperedge whenever I+J is a prime ideal of R for each non-trivial ideal I, J in Ei and Ei is maximal with respect to this property. Moreover, we also compute some degree-based topological indices (first and second Zagreb indices, forgotten topological index, harmonic index, Randić index, Sombor index) for these hypergraphs. In particular, we describe some degree-based topological indices for the newly defined algebraic hypergraph based on prime ideal sum for Zn where n=pα,pq,p2q,p2q2,pqr,p3q, p2qr,pqrs for the distinct primes p,q,r and s.

Complex and rational hypergeometric functions on root systems

We consider some new limits for the elliptic hypergeometric integrals on root systems. After the degeneration of elliptic beta integrals of type I and type II for root systems An and Cn to the hyperbolic hypergeometric integrals, we apply the limit ω1→−ω2 for their quasiperiods (corresponding to b→i in the two-dimensional conformal field theory) and obtain complex beta integrals in the Mellin–Barnes representation admitting exact evaluation. Considering type I elliptic hypergeometric integrals of a higher order obeying nontrivial symmetry transformations, we derive their descendants to the level of complex hypergeometric functions and prove the Derkachov–Manashov conjectures for functions emerging in the theory of non-compact spin chains. We describe also symmetry transformations for a type II complex hypergeometric function on the Cn-root system related to the recently derived generalized complex Selberg integral. For some hyperbolic beta integrals we consider a special limit ω1→ω2 (or b→1) and obtain new hypergeometric identities for sums of integrals of rational functions.

Bounds For Spectral Radius and Energy of $PIS$ Graphs

Once the spectral radius and energy of a graph structure have been defined, many properties have been studied. The spectral radius and energy of a graph are related to the eigenvalues of the adjacency matrix of the graph. In this paper, we define an adjacency matrix for a prime ideal sum ($PIS$) graph and then extend the concepts of spectral radius and energy to $PIS$ graphs. Some bound theorems on the energy and spectral radius of $PIS$ graph structures are given. A SageMath code for plotting these graphs is also provided.

Characterization of rings with planar, toroidal or projective planar prime ideal sum graphs

Let R be a commutative ring with unity. The prime ideal sum graph PIS ( R ) of the ring R is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent if and only if I + J is a prime ideal of R. In this paper, we study some interplay between algebraic properties of rings and graph-theoretic properties of their prime ideal sum graphs. In this connection, we classify non-local commutative Artinian rings R such that PIS ( R ) is of crosscap at most two. We prove that there does not exist a non-local commutative Artinian ring whose prime ideal sum graph is projective planar. Further, we classify non-local commutative Artinian rings of genus one prime ideal sum graphs.

Forgotten Topological and Wiener Indices of Prime Ideal Sum Graph of Zn.

Chemical graph theory is a sub-branch of mathematical chemistry, assuming each atom of a molecule is a vertex and each bond between atoms as an edge. Owing to this theory, it is possible to avoid the difficulties of chemical analysis because many of the chemical properties of molecules can be determined and analyzed via topological indices. Due to these parameters, it is possible to determine the physicochemical properties, biological activities, environmental behaviours and spectral properties of molecules. Nowadays, studies on the zero divisor graph of Z_n via topological indices is a trending field in spectral graph theory. For a commutative ring R with identity, the prime ideal sum graph of R is a graph whose vertices are nonzero proper ideals of R and two distinctvertices I and J are adjacent if and only if I+J is a prime ideal of R. In this study the forgotten topological index and Wiener index of the prime ideal sum graph of Z_n are calculated for n=p^α,pq,p^2 q,p^2 q^2,pqr,p^3 q,p^2 qr,pqrs where p,q,r and s are distinct primes and a Sage math code is developed for designing graph and computing the indices. In the light of this study, it is possible to handle the other topological descriptors for computing and developing new algorithms for next studies and to study some spectrum and graph energies of certain finite rings with respect to PIS-graph easily.

Some New Identities Related to Dedekind Sums Modulo a Prime

The main purpose of this article is to use some identities of the classical Gauss sums, the properties of character sums, and Dedekind sums (modulo an odd prime) to study the computational problem of one-kind mean values related to Dedekind sums and give some interesting identities for them.

Some Finite Summation Identities Comprising Binomial Coefficients for Integrals of the Bernstein Polynomials and Their Applications

Certain finite sums, including the Catalan numbers, factorial functions, binomial coefficients, and their computational formulas are of indispensable importance both in probability and statistics applications and in other branches of science. The primary aim of this article is to give the integral representation of the finite sum containing the products of the Bernstein polynomials, given in our article, by applying the Beta function and the Euler gamma functions. Other aims of this paper are to bring to light novel finite sum formulae containing binomial coefficients by analyzing and unifying this integral representation. Finally, some relations among these sums, binomial coefficients, and the Catalan numbers are given. We also give the Wolfram language codes. By applying these codes to the finite sums, we give some numerical values.

Lattice Ordered G􀀀Semirings

Objectives: The main objective of this paper is to derive some of the results of lattice ordered semirings, distributive lattice, lattice ideals and morphisms. Methods: To establish the results, we use some conditions like commutativity, simple, multiplicative idempotent, additively idempotent, and finally, use the concept of lattice ideal in semirings. Findings: First we give some examples of lattice ordered semirings and then study some results regarding lattices, distributive lattices, commutative lattice ordered semirings and finally lattice ideals and morphisms. The unique feature of this study is that the concept of gamma is new for the study of lattices. Novelty: We consider a condition (c.f. Theorem 4.1.5) for an additively idempotent semiring due to which it becomes a distributive lattice ordered semiring. Again, in general, the sum of ideals of a semiring need not be ideal. Indeed, and are ideals of is a set of non-negative integers. Clearly, (say) is not a ideal, because , but . However, this condition does not hold in the case of a lattice ordered semiring. AMS Mathematics subject classification (2020): 16Y60. Keywords: Lattices, additive idempotent, multiplicative Γ-idempotent, k-ideal, lattice ideal, Γ-morphism

A New Family of Appell-Type Changhee Polynomials with Geometric Applications

Recently, Appell-type polynomials have been investigated and applied in several ways. In this paper, we consider a new extension of Appell-type Changhee polynomials. We introduce two-variable generalized Appell-type λ-Changhee polynomials (2VGATλCHP). The generating function, series representations, and summation identities related to these polynomials are explored. Further, certain symmetry identities involving two-variable generalized Appell-type λ-Changhee polynomials are established. Finally, Mathematica was used to examine the zero distributions of two-variable truncated-exponential Appell-type Changhee polynomials.

A sharp bound for the resurgence of sums of ideals

We prove a sharp upper bound for the resurgence of sums of ideals involving disjoint sets of variables, strengthening work of Bisui–Hà–Jayanthan–Thomas [Collect. Math. 72 (2021), pp. 605–614]. Complete solutions are delivered for two conjectures proposed by these authors. For given real numbers a a and b b , we consider the set R e s ( a , b ) Res(a,b) of possible values of the resurgence of I + J I+J where I I and J J are ideals in disjoint sets of variables having resurgence a a and b b , respectively. Some questions and partial results about R e s ( a , b ) Res(a,b) are discussed.

Metric dimension in a prime ideal sum graph of a commutative ring

The prime ideal sum graph of a commutative unital ring [Formula: see text], denoted by [Formula: see text], is an undirect and simple graph whose vertices are nontrivial ideals of [Formula: see text] and there exists an edge between two distinct vertices if and only if their sum is a prime ideal of [Formula: see text]. In this paper, the metric dimension of [Formula: see text] is discussed and some formulae for this parameter in [Formula: see text] are given.

Computing Topological Descriptors of Prime Ideal Sum Graphs of Commutative Rings

Let n≥1 be a fixed integer. The main objective of this paper is to compute some topological indices and coindices that are related to the graph complement of the prime ideal sum (PIS) graph of Zn, where n=pα,p2q,p2q2,pqr,p3q,p2qr, and pqrs for the different prime integers p,q,r, and s. Moreover, we construct M-polynomials and CoM-polynomials using the PIS-graph structure of Zn to avoid the difficulty of computing the descriptors via formulas directly. Furthermore, we present a geometric comparison for representations of each surface obtained by M-polynomials and CoM-polynomials. Finally, we discuss the applicability of algebraic graphs to chemical graph theory.

Using a Level-based RPG Educational Game to Enhance Students’ Knowledge of Mathematics

This study utilized RPG Maker MZ to develop a simulated mathematical educational game that enables learners to immerse in the digital game while also acquiring knowledge about all units in Mathematics 3A, such as radian measure, trigonometric function graphs, trigonometric angle sum and difference identities, superposition of sine and cosine functions, exponential function, logarithms and its laws, logarithmic function, and plane vector operations, inner products, and applications. The said mathematical educational game aims to increase students’ learning interests and assist them in achieving better learning efficiency. Ninety-three second-graders from a vocational senior high school participated in the study and were given questionnaires to fill out afterward. This study discovered that mathematical educational games enable students to not only go through a variety of situations and experiences but also improve their learning efficiency and their ability to think about the problems around them through virtual life situation-based games. This type of interaction can assist students in better understanding and mastering the flexible application of mathematical concepts and knowledge in a variety of life situations.