Abstract Algebra Research Papers - Academia.edu (original) (raw)

2025, Journal of Geometry and Physics

In this work we introduce the category of multiplicative sections of an LA-groupoid. We prove that these categories carry natural strict Lie 2-algebra structures, which are Morita invariant. As applications, we study the algebraic structure underlying multiplicative vector fields on a Lie groupoid and in particular vector fields on differentiable stacks. We also introduce the notion of geometric vector field on the quotient stack of a Lie groupoid, showing that the space of such vector fields is a Lie algebra. We describe the Lie algebra of geometric vector fields in several cases, including classifying stacks, quotient stacks of regular Lie groupoids and in particular orbifolds, and foliation groupoids.

2025, Differential Geometry and its Applications

We show in this paper that the correspondence between 2-term representations up to homotopy and VB-algebroids, established in , holds also at the level of morphisms. This correspondence is hence an equivalence of categories. As an application, we study foliations and distributions on a Lie algebroid, that are compatible both with the linear structure and the Lie algebroid structure. In particular, we show how infinitesimal ideal systems in a Lie algebroid A are related with subrepresentations of the adjoint representation of A.

2025

Let R be a Noetherian, integral domain which is also an algebra over ( is the field of rational numbers). Let σ be an automorphism of R and δ a σ-derivation of R. We also study the property of minimal prime ideals of R and prove the following in this direction: Let R be a Noetherian, integral domain which is also an algebra over . Let σ be an automorphism of R and δ a σ-derivation of R such that R is a (σ, δ)-ring. If P ∈ M in.Spec(R) is such that σ(P) = P, then δ(P) ⊆ P. Further if δ(P(R)) ⊆ P(R), then P[x; σ, δ] is a completely prime ideal of R[x; σ, δ].

2025

For a ring R, an endomorphism σ of R and δ a σ -derivation of R, we introduce a weak (σ , δ )-rigid ring, which generalizes the notion of (σ , δ )-rigid rings and investigate its properties. Moreover, we state and prove a necessary and sufficient condition for a weak (σ , δ )-rigid ring to be a (σ , δ )-rigid ring. We prove that a (σ , δ )-ring is a weak (σ , δ )-rigid ring and conversely that the prime radical of a weak(σ , δ )-rigid ring is a (σ , δ )-ring. We also find a relation between minimal prime ideals and completely prime ideals of a ring R, where R is a (σ , δ )-ring and R is a 2-primal weak (σ , δ )-rigid ring.

2025, arXiv (Cornell University)

This paper investigates the application of the theoretical algebraic notion of a separable ring extension, in the realm of cyclic convolutional codes or, more generally, ideal codes. We work under very mild conditions, that cover all previously known as well as new non trivial examples. It is proved that ideal codes are direct summands as left ideals of the underlying non-commutative algebra, in analogy with cyclic block codes. This implies, in particular, that they are generated by an idempotent element. Hence, by using a suitable separability element, we design an efficient algorithm for computing one of such idempotents.

2025, arXiv (Cornell University)

2025, Journal of Algebra

It is known that the recently discovered representations of the Artin groups of type An, the braid groups, can be constructed via BMW algebras. We introduce similar algebras of type Dn and En which also lead to the newly found faithful representations of the Artin groups of the corresponding types. We establish finite dimensionality of these algebras. Moreover, they have ideals I 1 and I 2 with I 2 ⊂ I 1 such that the quotient with respect to I 1 is the Hecke algebra and I 1 /I 2 is a module for the corresponding Artin group generalizing the Lawrence-Krammer representation. Finally we give conjectures on the structure, the dimension and parabolic subalgebras of the BMW algebra, as well as on a generalization of deformations to Brauer algebras for simply laced spherical type other than An.

2025, Linear Algebra and its Applications

We study evolution algebras of arbitrary dimension. We analyze in deep the notions of evolution subalgebras, ideals and non-degeneracy and describe the ideals generated by one element and characterize the simple evolution algebras. We also prove the existence and unicity of a direct sum decomposition into irreducible components for every non-degenerate evolution algebra. When the algebra is degenerate, the uniqueness cannot be assured. The graph associated to an evolution algebra (relative to a natural basis) will play a fundamental role to describe the structure of the algebra. Concretely, a non-degenerate evolution algebra is irreducible if and only if the graph is connected. Moreover, when the evolution algebra is finite-dimensional, we give a process (called the fragmentation process) to decompose the algebra into irreducible components.

2025

1. Determina las raíces de las siguientes ecuaciones de segundo grado y completa el trinomio cuadrado perfecto, donde x,y, z y w son variables y a y b constantes. Aplicando la fórmula general. Demostración de donde nace la formula general 2. Resuelva las ecuaciones siguientes aplicando la fórmula general. Factorización 3. Resuelve los siguientes ejercicios Solución de una ecuación de segundo grado incompleta Mixtas 4. Resuelve los siguientes ejercicios x1=0 x2=

2025, Rafi Kamindra

Dokumen ini membahas konsep-konsep lanjutan dalam transformasi linear, dengan fokus utama pada komposisi dan isomorfisma. Bagian awal mendefinisikan komposisi dari dua transformasi linear, (TS)(u)=T(S(u)), dan menunjukkan bahwa komposisi dari transformasi linear juga bersifat linear, yang pada representasi matriksnya setara dengan perkalian matriks. Selanjutnya, dokumen ini menguraikan konsep isomorfisma, yaitu transformasi linear yang bersifat bijektif. Konsep ini dibangun dari dua sifat fundamental: injektif (satu-satu), yang ekuivalen dengan memiliki kernel trivial (ker(T)={0}), dan surjektif (pada), yang ekuivalen dengan rank transformasi sama dengan dimensi kodomain. Teorema-teorema kunci disajikan, termasuk ekuivalensi antara injektivitas dan surjektivitas ketika domain dan kodomain memiliki dimensi yang sama. Dokumen ini dilengkapi dengan "responsi" atau kumpulan soal latihan yang mendalam, mencakup pengujian sifat injektif dan surjektif, penentuan rank dan nulitas, pencarian rumus transformasi, serta penentuan matriks representasi terhadap berbagai basis, termasuk dengan perubahan basis.

2025, arXiv (Cornell University)

Hidden sector glueball dark matter is well motivated by string theory, compactifications of which often have extra gauge groups uncoupled to the visible sector. We study the dynamics of glueballs in theories with a period of late time primordial matter domination followed by a low final reheating temperature due to a gravitationally coupled modulus. Compared to scenarios with a high reheating temperature, the required relic abundance is possible with higher hidden sector confinement scales, and less extreme differences in the entropy densities of the hidden and visible sectors. Both of these can occur in string derived models, and relatively light moduli are helpful for obtaining viable phenomenology. We also study the effects of hidden sector gluinos. In some parts of parameter space these can be the dominant dark matter component, while in others their abundance is much smaller than that of glueballs. Finally, we show that heavy glueballs produced from energy in the hidden sector prior to matter domination can have the correct relic abundance if they are sufficiently long lived.

2025

A graph G(V, E) is said to admit a prime distance labeling if there exists an oneto-one function h ∶ V (G) → Z such that |h(u) − h(v)| is a prime number for every pair of adjacent vertices u and v in G . In this paper, we investigate if the prime distance labeling of the non-commuting graph of non-abelian groups such as symmetric group (Sn, o), dihedral group D2n, n ∈ N exists or not.

2025, Bézout's identity

This here is a Proof of "Bézout's identity" in a diffrent way as i am using "Proof by induction"

2025, HAL (Le Centre pour la Communication Scientifique Directe)

2025

In this paper, we will be giving some exposition on categorical products and of overpacked lattices, or, "overloaded lattices". We will also be detailing some immediate results of the constructions.

2025, United International Journal for Research and Technology

This study was conducted to determine the effectiveness of HYPOWHEEL Model on selected topics in Physics. The Solomon Four Groups design was used to assign the participants into four groups through stratified random sampling. A 30-item pre-test and post-test were administered to obtain the needed data. Only the experimental groups were treated with Hypowheel model in the instruction and all the four groups took the post-test. The data revealed that the performance level in the pretest of the control and experimental was the same at low mastery. However, after the treatment of Hypowheel model to experimental group and no treatment to control group the performance level (PL) in the post-test of both groups was improved. Further, the experimental group got higher PL and performed better than control group. Specifically, the performance level in the post-test along the topic energy transformation, work and heat, and power of experimental groups was described as moving towards mastery while the control groups generally described as at average level. In addition, there was a significant difference between the pre-test and post-test result. There was also a significant difference of post-test result between each group. Using Scheffe's test, the difference lies between experimental group 1 and control group 2 with computed F of 22.83(Cohen's d=1.79) and, experimental group 2 and control group 2 with computed F of 25.25(Cohen's d=1.84). It recommended that the teacher and students may develop suitable instructional aids for physics instruction to enhance student's level of competency. Hence, they may use Hypowheel model to help achieve better performance on the said topics. Findings also suggest the need to conduct trainings on innovation, recycling, investigative project, and further studies about Hypowheel model.

2025, ELEMENTOS DE ÁLGEBRA LINEAL Y GEOMETRÍA I

Elementos de Álgebra Lineal y Geometría I y Elementos de Álgebra Lineal y Geometría II de los profesores D. Antonio Pardo Fraile, catedrático de Instituto de Bachillerato (cuando los catedráticos eran unos Sres. Catedráticos y cuando el... more

2025

We continue the development of the infinitesimal deformation theory of pasting diagrams of k-linear categories begun in . In the standard result that all obstructions are cocycles was established only for the elementary, composition-free parts of pasting diagrams. In the present work we give a proof for pasting diagrams in general. As tools we use the method developed by Shrestha [6] of representing formulas for obstructions, along with the corresponding cocycle and cobounding conditions by suitably labeled polygons, giving a rigorous exposition of the previously heuristic method, and deformations of pasting diagrams in which some cells are required to be deformed trivially.

2025, Variedades Algebraicas

Este manual es una introducción a la teoría de variedades algebraicas afines y proyectivas. Estudiamos las kkk-álgebras de tipo finito, la descomposición primaria y la teoría de la dimensión en variedades algebraicas afines y proyectivas. Se introducirán nuevas conceptos: el espectro primo de un anillo, la localización y el lenguaje categorial;
y las herramientas necesarias para los cálculos: la resultante de dos polinomios y las bases de Gröbner.

2025, Journal of Sciences, Islamic Republic of Iran

A ring R is said to be right McCoy‎, ‎if for every f(x),g(x) in the polynomial ring R[x], with f(x)g(x)=0 there exists a nonzero element cϵR with f(x)c=0‎. In this note‎, ‎we show that von Neumann regular McCoy rings are abelian‎. ‎This... more

2025, https://efficientnewlanguage.org/en/

This paper introduces a novel theoretical framework termed Mathematical Relativity, which reinterprets prime number distribution through a dynamic lens based on Modulo 6 structure. By treating number theory as a relativistic system with structural constraints and temporal symmetry, the research proposes a multi-layered spectrum of prime behavior linked with modular flows. This framework aims to bridge the gap between classical prime theory and dynamic cognition, offering a new way to model both numerical regularities and mathematical perception in a unified formalism.

2025

We derive expressions for the partition function p(n), with n in the form 7k+a, as (k+1)-dimensional determinants.

2025, Proceedings of the American Mathematical Society

We discuss the Kummer subspaces of tensor products of cyclic algebras, focusing mainly on the case of cyclic algebras of degree 3. We present a family of maximal spaces in the general case, classify all the monomial spaces in the case of tensor products of cyclic algebras of degree 3 using graph theory, and provide an upper bound for the dimension in the generic tensor product of cyclic algebras of degree 3.

2025, Communications in Algebra

We prove that the maximal dimension of a Kummer space in the generic tensor product of n cyclic algebras of degree 4 is 4n + 1.

2025

We study the subfields of quaternion algebras that are quadratic extensions of their center in characteristic 222. We provide examples of the following: Two nonisomorphic quaternion algebras that share all their quadratic subfields, two quaternion algebras that share all the inseparable but not all the separable quadratic subfields, and two algebras that share all the separable but not all the inseparable quadratic subfields. We also discuss quaternion algebras over global fields and fields of Laurent series over a perfect field of characteristic 222 and show that the quaternion algebras over these fields are determined by their separable quadratic subfields.

2025, Linear Algebra and its Applications

In this paper we study the generalized Clifford algebra defined by Pappacena of a monic (with respect to the first variable) homogeneous polynomial Φ(Z, X 1 , . . . , of degree d in n + 1 variables over some field F . We completely determine its structure in the following cases: n = 2 and d = 3 and either char (F ) = 3, 2 for some r, t, e ∈ F . Except for a few exceptions, this algebra is an Azumaya algebra of rank nine whose center is the coordinate ring of an affine elliptic curve. We also discuss representations of arbitrary generalized Clifford algebras assuming the base field F is algebraically closed of characteristic zero.

2025, Archiv der Mathematik

We study the behavior of square-central elements and Artin-Schreier elements in division algebras of exponent 2 and degree a power of 2. We provide chain lemmas for such elements in division algebras over 2-fields F of cohomological 2dimension cd 2 (F) ≤ 2, and deduce a common slot lemma for tensor products of quaternion algebras over such fields. We also extend to characteristic 2 a theorem proven by Merkurjev for characteristic not 2 on the decomposition of any central simple algebra of exponent 2 and degree a power of 2 over a field F with cd 2 (F) ≤ 2 as a tensor product of quaternion algebras.

2025

Misalkan R adalah suatu ring. Bilangan bulat positif terkecil m jika ada, sedemikian sehingga ma = 0 untuk setiap elemen a dalam ring tersebut dikatakan karakteristik dari ring tersebut. Jika bilangan bulat ini tidak ada, dikatakan... more

2025

Suatu subgrup H dari grup G adalah normal jika setiap koset kiri dari H dalam G sama dengan koset kanan dari H dalam G, yakni H subgrup normal dari G ↔ gH = Hg ∀g ∈ G.

2025

Definisi Jika G grup dengan operasi ⋆ dan H adalah suatu grup dengan operasi # maka suatu pemetaan ∅ : G  H adalah homomorfisme jika dan hanya jika ∅ (a ⋆ b) = ∅(a) # ∅(b), ∀a, b ϵ G.

2025

Operasi biner adalah operasi yang didefinisikan dengan baik, melibatkan satu atau lebih elemen dari himpunan yang tidak kosong dan menghasilkan elemen dalam himpunan yang sama.

2025

We shall prove a version of Gauß’s Lemma that recursively constructs

2025, Faruk Alpay

Alpay Algebra is introduced as a self-contained axiomatic framework with the ambition of serving as a universal foundation for mathematics. Developed in the spirit of Bourbaki's structural paradigm and Mac Lane's emphasis on form and function, Alpay Algebra posits a single abstract system from which diverse mathematical domains emerge. We present the precise axioms defining Alpay Algebra and develop its core constructs-including a recursive transformation operator ϕ, its transfinite iteration ϕ ∞ , an iterative state hierarchy χ λ , a limit object Ξ ∞ , and an evaluation functional ψ λ. From these primitives, we rigorously rebuild key fields: we derive fixed-point theorems and internal stability results, realize category theory within the algebra by interpreting compositional morphisms as iterative state transitions, recast homological algebra through cycles and invariants of the recursion, and outline an internal logic akin to topos theory grounded in stable truth values emerging from ψ λ. All definitions, theorems, and proofs are given entirely within the Alpay Algebra system without appeal to external frameworks. The development showcases how Alpay Algebra can subsume algebraic geometry, category theory, homological algebra, logic (including topos theory), and general structural mathematics under one unifying language. We conclude by highlighting new conjectures and problems that naturally arise from this universal algebraic perspective, underscoring the foundational depth and future potential of Alpay Algebra.

2025, Czechoslovak Mathematical Journal

2025, Academia.edu — Publicación Independiente

Se presenta una propuesta teórica basada en el Sistema López Chow Binomixoidal de
Anulación Cruzada (LCBAC) como método alternativo de análisis de la Hipótesis de Riemann.
Se define un modelo lógico binario no clásico donde los valores de la función zeta de Riemann
se categorizan bajo un criterio binario especial y se establece una operación de anulación cruzada
que permite explorar patrones de homogeneidad en los ceros no triviales de dicha función.

2025, Journal of Applied Mathematics

Davvaz (2008) introduced the concept of set-valued homomorphism andT-rough sets in a group. In this paper, we consider the set-valued homomorphismTonΓ-semihypergroupHto interpret the lower and upper approximations. We study the roughness of(m,n)bi-Γ-hyperideals and(m,n)quasi-Γ-hyperideals in terms of set-valued homomorphisms, which are extended notions of(m,n)bi-Γ-hyperideals and(m,n)quasi-Γ-hyperideals ofΓ-semihypergroups.

2025, Teoría Cuadrática de Smith: Una demostración a la Hipótesis de Riemann

Este trabajo presenta la Teoría Cuadrática de Smith, un marco riguroso y novedoso que ofrece una demostración de la Hipótesis de Riemann. La teoría construye un operador compacto y autoadjunto Tₐ en el espacio L²(R⁺), cuyo determinante... more

2025, Teoría Cuadrática de Smith: Una demostración a la Hipótesis de Riemann

2025

O presente parecer fixa o entendimento de que os contratos de cessão onerosa de bens público, mesmo com serviços auxiliares, são contratos de receita. Os contratos de receita são aqueles em que a Administração Pública busca arrecadar... more

2025

This paper contains all the data that is necessary for some computations done in the book 'quadratic and cyclotomic ring of integers'.

2025

This document lists all class groups of quadratic rings of integers with squarefree integer D in the range -10,000 to 10,000. It lists the generators of the class group and the classes of the class group. For positive integers D, the... more

2025, A presentation

A presentation

2025, Psychology and Education: A Multidisciplinary Journal

This study aimed to explore the factors affecting the learning behaviors among selected students in secondary school in General Luna, Quezon. Aspects studied were the demographic profile of the respondents. It investigated the factors affecting the learning behaviors among selected students. To achieve the researcher's goal, the researcher used a questionnaire to determine the respondents' demographic profile and administer the questionnaire for the factors affecting he learning behaviors. This involved 80 students from the selected secondary high school in General Luna, Quezon. The descriptive design on the main source of data and information. The result showed that most of the respondents are the age of 16-18years old and above, male, and female. According to the result of Kruskal Wallis all the null hypothesis are accepted with means that there is no significant difference on the factors affecting the learning behaviors when the respondents are grouped according to profile. Based on the results of the study, the following recommendations are suggested. The school administrators may be able to provide different intervention strategies to students to be more aware of learning behaviors. The parents they may help the students to learn the behavior in communication regularly at their home. The teachers they may motivate the students to be aware in their learning behaviors with positive learning environment. The students they may continue to learn about learning behaviors.

2025

For a hypergroup (H, •) we consider γ * , as the smallest equivalence relation on H such that the quotion (H/γ * , ⊗) is an abelian group. We study some more properties of γ *. Initially, it is investigated which subhypergroup the congruence relation modulo is strongly regular on, and its quotient results in an abelian group? This is directly related to the fundamental relation γ * , since such subhypergroups must contain S γ. Then, we examine the functor γ * from a categorical perspective and investigate properties such as continuity and cocontinuity concerning it using the decomposition γ = δ * β. For this purpose, we define the reduced words on strongly regular hypergroups. This has a direct application in studying how the functor γ * affects on the stalks of the sheaves of hypergroups.

2025

In this short paper, we present a simple but elegant identity linking the arithmetic mean and the geometric mean of a finite set of positive real numbers. This identity highlights a symmetric relationship between sums and products,... more

2025

This paper introduces a novel algebraic method for solving general polynomials of degree n, extending traditional approaches that are limited to degrees four or lower. The method leverages harmonic relationships, rational symmetric functions, and structured transformations to decompose the polynomial into solvable components. By formulating recurrence relations and leveraging auxiliary roots and coefficients, this approach provides exact, closed-form solutions under specific algebraic conditions. The technique is constructive and aligns with the fundamental theorem of algebra while opening a pathway to symbolic resolution for higher-degree polynomials, which are traditionally considered unsolvable in radicals. The paper also explores potential applications in computational algebra, integer factorization, and cryptography.

2025, Journal of Algebraic Hyperstructures and Logical Algebras

We introduce a new regular relation δ on a given group G and show that δ is a congruence relation on G, concerning module the commutator subgroup of G. Then we show that the effect of this relation on the fundamental relation β is equal to the fundamental relation γ. We conclude that, if ρ is an arbitrary strongly regular relation on the hypergroup H, then the effect of δ on ρ, results in a strongly regular relation on H such that its quotient is an abelian group.

2025

We have traced the mathematical lineage from simple trigonometric parameterizations, through classical roulettes, to a fully normalized, higher-dimensional geometric framework with dynamical feedback. This framework scaffolds spectral... more