Algebra

The G-Drazin Invertibility of Sum with Orthogonal Conditions

Authors: Guo Shile, Huanyin Chen
Comments: 15 Pages.

In this article, we present novel additive properties for the g-Drazin inverse within the context of Banach algebras. Subsequently, we provide representations for the g-Drazin inverse of block operator matrices defined over Banach spaces. These findings build upon and extend various well-known results, such as those by Bu, Feng, and Bai (J. Comput. Appl. Math., 218(2012), 10226--10237) and Dopazo and Martínez-Serrano (Linear Algebra Appl., 432(2010), 1896--1904).
Category: Algebra

Pseudo Drazin Invertibility of Anti-Triangular Matrix Over a Ring and Its Applications

Authors: Guo Shilu, Huanyin Chen
Comments: 21 Pages. In Chinese

In this paper, we investigate the p-Drazin inverse of the anti-triangular operator matrix $left(begin{array}{cc} a & 1 b & 0 end{array} ight)$. We identify two new class of necessary and sufficient conditions for an anti-triangular matrix to possess a p-Drazin inverse. Subsequently, we generalize several established results to a broader context. As practical applications, we demonstrate the p-Drazin invertibility of specific block operator matrices $left( begin{array}{cc} A & B C & D end{array}ight)in mathcal{B}(Xoplus X)$.
Category: Algebra

Sumthin: A Context Dependent Slang Used on the Streets to Depict Unknown Variables

Authors: Makafui Apedo
Comments: 11 Pages.

Variables are the fundamental blocks upon which Algebra is built. Algebra as we know it today, was created using basic language of the common people. Early Arabic texts in Algebra were written in plain language; It was later that mathematicians used letters, Hindu numerals, and notations to represent words. So, it was easier to read equations then than it is now. This also means that every human language, whether formal or informal can enable Algebra. In this paper, I demonstrated the word ‘sumthin’ (pronounced "sum"), a context-dependent slang that depicts unknown amount of money that change, as a synonym for a variable.
Category: Algebra

Algebraic Conception of the Five Common Notions of Euclid

Authors: Juan Elias Millas Vera
Comments: 1 Page.

Following Hilbert’s path to make algebraic the classical texts I express literal notions of Euclid in simple equations.
Category: Algebra

Goedelian Encryption & Goldbach's Conjecture

Authors: Paris S. Miles-Brenden
Comments: 16 Pages.

The qualifications to a mathematical context, in a theorem of elimination, remain the orderable indexed assumptive of a sequence in N, and a count, among hyperbolic and geometric spaces in general, to which this reproves to the theoretical limitations of the Turing Model, that the precalculability of machine states is in one exceptional qualifier, prohibitionally entitled of an Axiom of Choice, to Leonard Cohen's work.
Category: Algebra

Para Complexes Numbers

Authors: Ahcene Ait saadi
Comments: 9 Pages. (Note by viXra Admin: For the last time, please cite and list scientific references!)

In this article, I study the numbers that I have titled: para-complex numbers. A para- complex number consists of a real part and para- imaginary part. Algebraic form of a para- complex number:z=a+jb with j^2=1 and j different from 1 "J : is a pure para-imaginary."a, is the real part; b, is the imaginary part. In an orthonormal coordinate system, the x-axis represents the real numbers, while the y-axis represents the para-imaginary number. This work allows me to find a large number of mathematical formulas. This is just the beginning; I hope that researchers will improve it and derive more interesting mathematical formulas that will serve science.
Category: Algebra

Algebra zero

Authors: Ahcene Ait saadi
Comments: 6 Pages. (Note by viXra Admin: Author name should be between article title and the asbstract)

In ordinary arithmetic, the expression 0/0 does not make sense. It is an indeterminate form, for there is no such thing as a number that multiplied by 0, gives a number . So, the division Of 0/0 is indefinite (undetermined) in mathematics. I will show with three examples, that this theory is completely wrong. I will show from the following, three examples which are my discoveries, That ,this theory is completely wrong. 1) System of three equations.2) Resolution of the quadratic equation with a new formula.3) Pseudo differential calculus. Mathematical tool that I discoveredSee http://viXra.org/abs/2409.0003With this discovery, I give the definition of absolute zero.I hope that these discoveries, will open up new horizons for scientific research in general and for mathematics in particular.
Category: Algebra

Solving Cubic Equation Using Cardano’s Method

Authors: Miloš Čojanović
Comments: 5 Pages.

A cubic equation is solvable by radicals. This means that the solutions of the cubic equation can be obtained using four basic arithmetic operations which include addition, subtraction, multiplication and division, and taking the square and cube roots.
Category: Algebra

Group Inverses and Generalized Group Inverses of Anti-Triangular Matrices

Authors: Huanyin Chen, Dayong Liu
Comments: 24 Pages.

We provide representations for the group inverse and generalized group inverse of an anti-triangular operator matrix of the form$left(begin{array}{cc}a&bc&0end{array}ight)$ under the conditions $ca=c$ or $ab=b$. Specifically, we present the weak group inverse for these types of anti-triangular block operator matrices.
Category: Algebra

The Simplest Proof of Fermat's Last Theorem

Authors: Xiaochun Mei
Comments: 2 Pages. In Chinese

In this paper, the Fermat equation is written in the form of quadratic indeterminate equation, and its solutions are compared with the Pythagorean numbers of quadratic equation. It is proved in a very simple method that the Fermat equation has no integer solutions in case of p>2.
Category: Algebra

Generalized Group Inverse of Block Operator Matrices

Authors: Huanyin Chen
Comments: 18 Pages.

We derive the generalized group inverse of a triangular block matrix over a Banach algebra. We apply this formula in order to find the generalized group inverse of $2times 2$ block operators under some conditions. In particular, the weak group inverse of certain block operator matricesare given.
Category: Algebra

On the Arc Length of an Ellipse

Authors: Tai-Choon Yoon
Comments: 4 Pages.

Elliptic integral is an integral equation that appears in the process of calculating the length of an ellipse. It does not provide an exact solution, and the approximation equation for the solution is complicated. The arc length of an ellipse is given as l=aθE(k). And the entire arc length of an ellipse is l=2aπE(k).
Category: Algebra

Additive Property of Generalized Core-ep Inverse in Banach *-Algebra

Authors: Huanyin Chen
Comments: 18 Pages.

We present new necessary and sufficient conditionsunder which the sum of two generalized core-EP invertible elements in a Banach *-algebra has generalized core-EP inverse. As an application, the generalized core-EP invertibility for the matrices with generalized core-EP invertible entriesis investigated.
Category: Algebra

Generalized W-Core Inverse in Banach Algebras with Involution

Authors: Huanyin Chen, Marjan Sheibani
Comments: 24 Pages.

In this paper, we introduce the generalized $w$-core inverse in a Banach *-algebra. We characterize this new generalized inverse by using the generalized weighted core decomposition and present the representations by the weighted g-Drazin inverse. The generalized $w$-core orders are investigated as well. These extend the weighted core inverse and pseudo core inverse for complex matrices and linear bounded operators to more general setting.
Category: Algebra

A New Way of Looking at Complex Numbers

Authors: Aziz Arbai, Amina Bellekbir
Comments: 8 Pages.

A new way to teach and to solve problems in complex numbers, as how to use the Moivre Formula, Newton's Binomial and Euler formula for the linearization. We will also share new results like The "Magic Formula" of the second root of any complex number, and the true method to solve the equation of second degree in general case (with complex coefficients) with the explicit formula of the solution and an algorithm to program it.
Category: Algebra

Weighted Weak Group Inverse in a Ring with Proper Involution

Authors: Huanyin Chen
Comments: 24 Pages.

In this paper, we introduce weighted weak group inverse in a ring with proper involution.This is a natural generalization of weak group inverse for a complex matrix and weighted weak group inverse for a Hilbert operator.We characterize this weighted generalized by using a kind of decomposition involving weighted group inverses and nilpotents. The relations among weighted weak group inverse, weighted Drazin inverse and weighted core-EP inverse are thereby presented.
Category: Algebra

Weighted M-Generalized Group Inverse in *-Banach Algebras

Authors: Huanyin Chen, Yueming Xiang
Comments: 22 Pages.

Recently, Gao, Zuo and Wang introduced the $W$-weighted $m$-weak group inverse for complex matrices which generalizedthe (weighted) core-EP inverse and the WC inverse. The main purpose of this paper is to extend the concept of $W$-weighted $m$-weak group inverse for complex matrices to elements in a Banach *-algebra. This extension is called $w$-weighted $m$-generalized group inverse. We present various properties, presentations of such new weighted generalized inverse. Related (weighted) $m$-generalized core inverses are investigated as well. Many properties of the $W$-weighted $m$-weak group inverse are thereby extended to wider cases.
Category: Algebra

Mathematics Involvement in Machine Learning

Authors: Abhishek Kumar
Comments: 4 Pages. Keywords : Mathematics in Machine Learning, Statistics, Calculus, Linear Algebra, Probability

Machine learning (ML) is a prominent branch of artificial intelligence (AI) that has drastically transformed various fields by providing sophisticated tools for data analysis and prediction. This paper reviews the pivotal role of mathematics in the development and refinement of machine learning algorithms. The core objective is to illustrate how mathematical principles underpin the processes of training and optimizing ML models, ensuring their effectiveness in recognizing patterns and making autonomous decisions from data.
Category: Algebra

Ideals of the Algebra II: Prime Ideal

Authors: Shao-Dan Lee
Comments: 4 Pages.

We have constructed an ideal with respect to a subset of binary operations. In this paper, we construct a prime ideal with respect to a nonempty subset of binary operations in an algebra. Let P and Q be two prime ideals with respect to Φ and Ψ, respectively. Then we have that P ∪ Q is a prime ideal if some conditions hold.
Category: Algebra

Rational Formal Power Series

Authors: Jānis Buls, Aigars Valainis
Comments: 35 Pages.

We are following Bini G., Flamini F. and Hou X—D., Lopez—Permouth S. R., Parra—Avila B. R. Nevertheless, we are interested only inthe clarification of proofs.
Category: Algebra

On the Classification of the Operational Spaces

Authors: Mathis Antonetti
Comments: 11 Pages.

In this paper, we introduce operational spaces that are a special case of what is usually called quotient topological spaces with an equivalence relation of interest. Then we derive some results and conjectures on their classification and an application to define the convergence of a sequence towards a cycle in the general case. We also provide a perspective on the still unaddressed conjectures concerning the operational spaces and we argue that the commonly useless non-uniform almost periodicity is actually useful in this context to understand the classification of such spaces.
Category: Algebra

Fukaya Categories, Reidemeister Moves, and the Novikov Ring

Authors: Ryan J. Buchanan
Comments: 13 Pages.

We construct a Novikov-type ring and generalize Dirac's $delta$ function into a functor. By applying the generalized $delta$-functor, we are able to reconstruct the Fukaya category $mathscr{B}_0$.
Category: Algebra

Ideals of the Algebra

Authors: Shao-Dan Lee
Comments: 6 Pages.

We construct an algebra A such that A has a nonempty finite set Δ of associative and commutative binary operations. Then we may define an ideal with respect to a nonempty subset of Δ. If some hypotheses are satisfied, then we have that a union of the ideals is an ideal. An ideal M is maximal with respect to a subset of Δ if there is not an ideal J ≠ A such that J contains M. And an algebra is local with respect to a subset of Δ if it has a unique maximal ideal. Suppose that the algebra A is local with respect to Φ and Ψ, M and N are the maximal ideals, respectively, and J is an ideal with respect to Φ∪Ψ. Then we have that J ⊆ M∩ N if some conditions hold. Let A be a local algebra with respect to Φ, M the maximal ideal. For all Ψ with Φ ⊆ Ψ⊂ Δ, if M is an ideal with respect to Ψ, then A is local with respect to Ψ. A preimage of an ideal with respect to Φ under a homomorphism is an ideal with respect to Φ.
Category: Algebra

Geometric Entity Dualization in the Geometric Algebra PGA G(3,0,1)

Authors: Robert Benjamin Easter, Daranee Pimchangthong
Comments: 23 Pages.

In Geometric Algebra, the degenerate-metric algebra G(3,0,1) is known as the Projective Geometric Algebra (PGA) for 3D space (3DPGA). In PGA, there is a point-based geometric algebra (point-based PGA) and a plane-based geometric algebra (plane-based PGA). Both algebras have homogeneous geometric entities for points, lines, and planes. The two algebras of PGA are dual to each other through a new geometric entity dualization operation J_e, which is introduced in this paper as its main subject and contribution. The new dualization J_e is an anti-involution with inverse −J_e = D_e. Using J_e, the dual of a point-based PGA entity is its corresponding plane-based PGA entity representing the same geometry (point, line, or plane) with the same orientation. Using D_e = −J_e, the inverse dual (undual) of a plane-based PGA entity is its corresponding point-based PGA entity with the same orientation. The new dualization operation Je maintains the correct orientation of an entity. J_e is defined by a table of duals that are found empirically by observation to maintain correct entity orientation through the dualization. We define a Hodge star dualization operation to be purely an involution, or else purely an anti-involution, between all basis blades and their dual basis blades. As an anti-involution, J_e is also implemented by algebraic methods using Hodge star dualizations in non-degenerate algebras that correspond to PGA. In the prior literature, there are other definitions for the duals in PGA that may not maintain the correct entity orientation and are different than J_e.
Category: Algebra

Dual Quaternion Geometric Algebra in PGA G(3,0,1)

Authors: Robert Benjamin Easter, Daranee Pimchangthong
Comments: 28 Pages.

In Geometric Algebra, the algebra G(3,0,1) is known as PGA, the plane-based and point-based geometric algebras, or projective geometric algebra, of points, lines, and planes in 3D space. The even-grades subalgebra of PGA, which we call Dual Quaternion Geometric Algebra (DQGA), represents Dual Quaternion Algebra (DQA). In the plane-based algebra of PGA, there are entities for points, lines, and planes and many operations on them, including dualization to the point-based entities, reflections in planes, rotations, translations, projections, rejections, and intersections (meet products). In this paper, we derive a complete set of identities that relate all of the plane-based entities and operations in PGA to their corresponding entities and operations in DQGA. Therefore, this paper contributes into the literature on dual quaternions and PGA the complete details on how to use DQA or DQGA as a geometric algebra of points, lines, and planes with many useful operations. All DQGA entities and operations are defined or derived such that the orientations of the entities are maintained correctly through all of the operations. We also define three new part operators for taking the point, line, or plane part of a dual quaternion, which may improve the computational efficiency of intersection (meet) operations. Dual quaternions already have some applications in computer graphics and kinematics. This paper expands on the understanding of dual quaternions and introduces DQA as a versatile geometric algebra of points, lines, and planes with many new operations that do not appear in prior literature, expanding the possible applications of dual quaternions.
Category: Algebra

L[n]garithms

Authors: Juan Elias Millas Vera
Comments: 3 Pages.

I introduce in this paper a new perception of the concept of logarithm. Generalization of the log_a(b) and doing some combinations we can assume that the log concept is just a case of a more abstract idea.
Category: Algebra

Geometric Product of Two Oriented Points in Conformal Geometric Algebra

Authors: Eckhard Hitzer
Comments: 15 Pages. Submitted to Advances in Applied Clifford Algebras, January 2023.

We compute and explore the full geometric product of two oriented points in conformal geometric algebra Cl(4,1) of three-dimensional Euclidean space. We comment on the symmetry of the various components, and state for all expressions also a representation in terms of point pair center and radius vectors.
Category: Algebra

Linear Algebra and Group Theory

Authors: Teo Banica
Comments: 400 Pages.

This is an introduction to linear algebra and group theory. We first review the linear algebra basics, namely the determinant, the diagonalization procedure and more, and with the determinant being constructed as it should, as a signed volume. We discuss then the basic applications of linear algebra to questions in analysis. Then we get into the study of the closed groups of unitary matrices $Gsubset U_N$, with some basic algebraic theory, and with a number of probability computations, in the finite group case. In the general case, where $Gsubset U_N$ is compact, we explain how the Weingarten integration formula works, and we present some basic $Ntoinfty$ applications.
Category: Algebra

Space Time PGA in Geometric Algebra G(1,3,1)

Authors: Robert B. Easter, Daranee Pimchangthong
Comments: 10 Pages.

In Geometric Algebra, G(1,3,1) is a degenerate-metric geometric algebra being introduced in this paper as Space Time PGA [STPGA], based on 3D Homogeneous PGA G(3,0,1) [3DPGA] and 4D Conformal Spacetime CGA G(2,4,0) [CSTA]. In CSTA, there are flat (linear) geometric entities for hyperplane, plane, line, and point as inner product null space (IPNS) geometric entities and dual outer product null space (OPNS) geometric entities. The IPNS CSTA geometric entities are closely related, in form, to the STPGA plane-based geometric entities. Many other aspects of STPGA are borrowed and adapted from 3DPGA, including a new geometric entity dualization operation J_e that is an involution in STPGA. STPGA includes operations for spatial rotation, spacetime hyperbolic rotation (boost), and spacetime translation as versor operators. This short paper only introduces the basics of the STPGA algebra. Further details and applications may appear in a later extended paper or in other papers. This paper is intended as a quick and practical introduction to get started, including explicit forms for all entities and operations. Longer papers are cited for further details.
Category: Algebra

Solvable Quintic Equation X^5 45X + 108 = 0

Authors: Quang Nguyen Van
Comments: 2 Pages.

We have previously proposed a quintic equation that is outside the available arguments of the solvable quintic equation . In this article, we give another quintic equation in Bring - Jerrard form and its root.
Category: Algebra

Geometric Entity Dualization and Dual Quaternion Geometric Algebra in PGA G(3,0,1) with Double PGA G(6,0,2) for General Quadrics

Authors: Robert B. Easter, Daranee Pimchangthong
Comments: 76 Pages. Original Research Paper, Version v1, 15 Dec 2023.

In Geometric Algebra, G(3,0,1) is a degenerate-metric algebra known as PGA, originally called Projective Geometric Algebra in prior literature. It includes within it a point-based algebra, plane-based algebra, and a dual quaternion geometric algebra (DQGA). In the point-based algebra of PGA, there are outer product null space (OPNS) geometric entities based on a 1-blade point entity, and the join (outer product) of two or three points forms a 2-blade line or 3-blade plane. In the plane-based algebra of PGA, there are commutator product null space (CPNS) geometric entities based on a 1-blade plane entity, and the meet (outer product) of two or three planes forms a 2-blade line or 3-blade point. The point-based OPNS entities are dual to the plane-based CPNS entities through a new geometric entity dualization operation J_e that is defined by careful observation of the entity duals in same orientation and collected in a table of basis-blade duals. The paper contributes the new operation J_e and its implementations using three different nondegenerate algebras {G(4),G(3,1),G(1,3)} as forms of Hodge star dualizations, which in geometric algebra are various products of entities with nondegenerate unit pseudoscalars, taking a grade k entity to its dual grade 4-k entity copied back into G(3,0,1). The paper contributes a detailed development of DQGA. DQGA represents and emulates the dual quaternion algebra (DQA) as a geometric algebra that is entirely within the even-grades subalgebra of PGA G(3,0,1). DQGA has a close relation to the plane-based CPNS PGA entities through identities, which allows to derive dual quaternion representations of points, lines, planes, and many operations on them (reflection, rotation, translation, intersection, projection), all within the dual quaternion algebra. In DQGA, all dual quaternion operations are implemented by using the larger PGA algebra. The DQGA standard operations include complex conjugate, quaternion conjugate, dual conjugate, and part operators (scalar, vector, tensor, unit, real, imaginary), and some new operations are defined for taking more parts (point, plane, line) and taking the real component of the imaginary part by using the new operation J_e. All DQGA entities and operations are derived in detail. It is possible to easily convert any point-based OPNS PGA entity to and from its dual plane-based CPNS PGA entity, and then also convert any CPNS PGA entity to and from its DQGA entity form, all without changing orientation of the entities. Thus, each of the three algebras within PGA can be taken advantage of for what it does best, made possible by the operation J_e and identities relating CPNS PGA to DQGA. PGA G(3,0,1) is then doubled into a Double PGA (DPGA) G(6,0,2) including a Double DQGA (DDQGA), which feature two closely related forms of a general quadric entity that can be rotated, translated, and intersected with planes and lines. The paper then concludes with final remarks.
Category: Algebra

Minimal Polynomials and Multivector Inverses in Non-Degenerate Clifford Algebras

Authors: Dimiter Prodanov
Comments: 22 Pages.

Clifford algebras are an active area of mathematical research with numerous applications in mathematical physics and computer graphics among many others.The paper demonstrates an algorithm for the computation of inverses of such numbers in a non-degenerate Clifford algebra of an arbitrary dimension. This is achieved by the translation of the classical Faddeev-LeVerrier-Souriau (FVS) algorithm for characteristic polynomial computation in the language of the Clifford algebra. The FVS algorithm is implemented using the Clifford package in the open-source Computer Algebra System Maxima.Symbolic and numerical examples in different Clifford algebras are presented.
Category: Algebra

A 2-Pitch Structure

Authors: Shao-Dan Lee
Comments: 5 Pages.

We have constructed a pitch structure. In this paper, we define a binary relation on the set of steps, thus the set become a circle set. And we define the norm of a key transpose. To apply the norm, we define a scale function on the circle set. Hence we may construct the 2-pitch structure over the circle set.
Category: Algebra

Some Remarks on the Generalization of Atlases

Authors: Ryan J. Buchanan
Comments: 6 Pages.

We generalize atlases for flat stacks over smooth bundles by constructing local-global bijections between modules of differing order. We demonstrate an adjunction between a special mixed module and a holonomy groupoid.
Category: Algebra

Sufficient Conditions and Necessary Conditions for Extreme Value Problems with Constraints

Authors: Chengshen Xu
Comments: 8 Pages.

In this paper, we prove that the sufficient conditions for the extreme value problem with constraints requires that the projection of the gradient of the point in the final constraint surface is zero and that the second-order partial derivative matrix--Hessian matrix on the local linear subspace in the constraint surface is positive definite or negative definite. The necessary conditions require that the projection of the gradient of the point in the final constraint surface be zero and that the second-order partial derivative matrix--Hessian matrix on the local linear subspace in the constraint surface be semi-positive definite or semi-negative definite. Finally, we discuss the reconstruction of the micro-base vector on the local linear subspace in the constraint surface and the local coordinate system.
Category: Algebra

Quasi-diagonalization and Quasi-Jordanization of Real Matrices in Real Number Field

Authors: Chengshen Xu
Comments: 8 Pages.

A real matrix may not be similar to a diagonal matrix or a Jordan canonical matrix in the real number field. However, it is valuable to discuss the quasi-diagonalization and quasi-Jordanization of matrices in the field of real numbers. Because the characteristic polynomial of a real matrix is a real coefficient polynomial, the complex eigenvalues and eigenvector chains occur in complex conjugate pairs. So we can re-select the base vectors to quasi-diagonalize it or quasi-Jordanize it into blocks whose dimensions are no larger than 2. In this paper, we prove these conclusions and give the method of finding transition matrix from the Jordan canonical form matrix to the quasi-diagonalized matrix.
Category: Algebra

Ebu's Suggestion: Real and Imaginary Rectangles

Authors: Ebubekir Kaya
Comments: 6 Pages. In English and German

The suggestion real and imaginary rectangles is an extension of complex numbers. With this representation, polynomial functions can be visualized. And there is a special relationship between these functions.
Category: Algebra

Number Notation for Operations and Hyperoperations

Authors: Juan Elias Millas Vera
Comments: 4 Pages.

In this paper I will apply some of notation tools in hyperoperators to the understanding of a deep and developed theory in notation. Using [a] or [-a] for an a Natural we will see the richness of this algebraic writing form.
Category: Algebra

Theory on Quantum Complexes

Authors: Parker Emmerson
Comments: 10 Pages. (Note by viXra Admin: Please fill out author name as follows: First name and last name)

It is from the definitions δ = Bθ (α) −→ δ |∆(κ,κi)= Bθ (αi) and the chain of definitions∫ τ λu2032 ∩ (∀ρ) → τ ≤i−j 1 ≤ℓ l(δ), that can be expressed by Φ(τ iδℓ), such that ∀τ <ν (Λ − β), then for some p(w) → p−1(v), there is some vector u of positive variational uπ such that i−a = vn(τ ). This implies that the numberof binary connections from ia,ν , lowers the complexity of φ. ∀(τ <ν (Λ−β)) ∃ p(w) → p−1(v) ∃ u, where uπ > 0 and i−a = vn(τ ) so that φ is less complex. This process is facilitated by the idea that the functions Bθ (α) and Φ(τ iδℓ)can be used to express changes in the system and yield new solutions. By specifying certain values of τ , δ and ℓ, as well as using the relation τ ≤i−j 1 ≤ℓ l(δ), a set of parameters which are applicable in various contexts canbe constructed. This allows for an easier analysis of the system, which can subsequently be used to develop more efficient solutions. Thus, these definitions and functions can be used to construct useful parameters which can enhance the performance of the system.
Category: Algebra

Proof of Beal's Conjecture

Authors: Vedernikov Sergey Ivanovich
Comments: 11 Pages. In Russian

Применение некоторых моментов, не имеющих распространения в современной алгебре и теории чисел, по крайней мере в справочной литературе, для доказательства теоремы Ферма и гипотезы Била.

The purpose of this work is to prove Beal's conjecture. The proof is based on methods not currently used by elementary mathematics. The concept of them is absent in the reference sources. Firstly, this is an idea of the possibility of expressing an even number that has a factor of 8 by the difference of the squares oftwo odd numbers. In addition, a feature of the integer solution of the quadratic equation by Pythagorean triples is applied. As a result, Beale's conjecture was completely proved, taking into account the accompanying Fermat's Last Theorem, by simple methods. Keywords: difference of squares of two odd numbers, factorization, even numbers, odd numbers, Pythagorean triples.
Category: Algebra

Calculation of Nth Partial Sums ���� of Power Series and Its Relationship with the Calculation of Bernoulli Numbers

Authors: Carlos Oscar Rodríguez Leal
Comments: 2 Pages. In Spanish

In this work, the general formula of the n-th partial sums ��_n of sums of powers of the form 1^n+ 2^n + . . . + m^n is obtained by an algebraic method, and said formula is applied to the obtaining the Bernoulli numbers by a new simple method.

En este trabajo se obtiene la fórmula general de las n-ésimas sumas parciales ��^n de sumas de potencias de la forma 1^n + 2^n + . . . + m^n mediante un método algebraico y dicha fórmula se aplica a la obtención de los números de Bernoulli por un método alternativo recursivo sencillo.
Category: Algebra

Inner Product of Two Oriented Points in Conformal Geometric Algebra in Detail

Authors: Eckhard Hitzer
Comments: 19 Pages. accepted for D. DaSilva, D. Hildenbrand, E. Hitzer (eds.), Proceedings of ICACGA 2022, Springer Proceedings in Mathematics & Statistics, Springer, Heidelberg, 2023.

We study in full detail the inner product of oriented points in conformal geometric algebra and its geometric meaning. The notion of oriented point is introduced and the inner product of two general oriented points is computed, analyzed (including symmetry) and graphed in terms of point to point distance, and angles between the distance vector and the local orientation planes of the two points. Seven examples illustrate the results obtained. Finally, the results are extended from dimension three to arbitrary dimensions n.
Category: Algebra

A Solvable Sextic Equation

Authors: Tai-Choon Yoon
Comments: 6 Pages.

This paper presents a solvable sextic equation under the condition that several coefficients of such polynomials are restricted to become dependent on the preceding or following coefficients. We can solve a sextic equation by restricting one or two in total seven coefficients available, and by solving a bisextic equation and a quintic equation. And we can also find the arbitrary coupling coefficients that generate a new solvable sextic equation as well.
Category: Algebra

An Algebraic Structure of Music Theory

Authors: Shao-Dan Lee
Comments: 8 Pages.

We may define a binary relation. Then a nonempty finite set equipped with the binary relation is called a circle set. And we define a bijective mapping of the circle set, and the mapping is called a shift. We may construct a pitch structure over a circle set. And we may define a tonic and step of a pitch structure. Then the ordered pair of the tonic and step is called the key of the pitch structure. Then we define a key transpose along a shift. And a key transpose is said to be regular if it consists of stretches, shrinks and a shift. A key transpose is regular if and only if it satisfies some hypotheses.
Category: Algebra

Null Algebra Resolutions of Complex Exponentiation: Null Algebra Extension III

Authors: Robert S. Miller
Comments: 32 Pages. Contains detailed resolutions for i^i with explanations for where all resolutions of the expression to a real number exist.

This paper explores the Null Algebra and traditional Algebra resolutions for the complex number i^i. It explains the apparent differences between the Null Algebra resolutions, and those of traditional Algebra, which uses the substitution i=e^(i π/2). The methods shown herein explore the full set of subspace equations implied by the given equation, as well as why powers of i resulting from the trigonometric substitution must be considered in deriving those equations. It is shown that the values obtained from the various possible resolutions to i^i are found on some aspect of the given equation, or its expanded subspace equation sets. It shows i^i equals +1, -1, +4.81047738 and +0.20788.
Category: Algebra

Growth in Matrix Algebras and a Conjecture of Perez-Garcia, Verstraete, Wolf and Cirac

Authors: Yaroslav Shitov
Comments: 12 Pages.

Let S be a family of n x n matrices over a field such that, for some integer l, the products of the length l of the matrices in S span the full n x n matrix algebra. We show this for any positive integer l > n^2 + 2n − 5.
Category: Algebra

Skolems Solution for Integer-Linear-Recurrences, with Commensurable Arguments for Characteristic-Roots of the Same Modulus

Authors: Deepak Ponvel Chermakani
Comments: 8 Pages. (Correction made by viXra Admin)

For a homogeneous linear-recurrence f_n with integer coefficients and integer starting points, we derive a deterministic algorithm that finds the upper bound of the last non-periodic position n where f_n=0, for a large family of special cases. First, when theta is a given irrational constant, then we show that, the eventual lower bound ofminimum(absolute(cos(m PI theta)), over positive integers m less than n), for large positive integers n, is (2 theta / (sqrt(5) n)). Our deterministic algorithm is based on the key concept that this lower bound decreases at a lower rate than the nth power of the ratio of root moduli since the ratio is lesser than 1. Our deterministic algorithm is developed for the special cases where G(x), the characteristic polynomial of f_n, has either equal absolute values of arguments or commensurate arguments of those complex roots, whose moduli are equal. In an attempt to extend this algorithm as a general solution to Skolems problem, we obtain the lower bound of the distance between a zero and the next (2^(m+1))th zero, in the weighted sum of m continuous cosine functions, where the weights are given real-algebraic constants.
Category: Algebra

Mathematical Proof of Proof 1^i=1

Authors: Robert S. Miller
Comments: 5 Pages. (Correction made by viXra Admin - Please conform!)

This paper details the expression 1^i=1. It uses a known trigonometric substitutions in a mathematical proof showing the accuracy of the expression.
Category: Algebra

A Solvable Quintic Equation

Authors: Tai-Choon Yoon
Comments: 7 Pages.

This article presents a solvable quintic equation under the conditions that several coef-ficients of a quintic equation are restricted to become dependent on the other coefficients.We can solve a quintic equation by restricting two coefficients among total four coefficientsavailable. If a quintic equation has a quadratic factor (x^2 + b_1 x + b_0), then we get a twosimultaneous equations, which can be solved by using a sextic equation under restriction.
Category: Algebra

Higher Rank Substitutions for Tensor Decompositions. I. Direct Sum Conjectures

Authors: Yaroslav Shitov
Comments: 85 Pages.

The substitution method of tensor rank computation is a higher dimensional analogue of Gaussian elimination, and it builds on the fact that the removal of a rank one slice s and a subsequent addition of arbitrary scalar multiples of s to all other slices of the same direction decreases the minimum rank exactly by one. We explain how to embed an initial tensor T to a larger linear space and replace its higher rank slice g by a family f of rank one slices in the new space so that the substitutions performed with respect to g in every direction of T have the same effect on the minimum rank as the corresponding substitutions with respect to f. We present several applications, which include a resolution of the well known and widely studied direct sum conjecture for Waring ranks and a strong form of counterexamples to Strassen’s conjecture.
Category: Algebra

Attempt to Find Quantum Group

Authors: Alexey Kuzmin
Comments: 18 Pages. (Corrections made by viXra Admin to conform with scholarly norm)

I have tried to find a new compact matrix quantum group within actions on a braided noncommutative quadric associated to a solution of Yang- Baxter equation outside of the 8-vertex model. The result appeared to be isomorphic to the circle group.
Category: Algebra

A Category is a Partial Algebra

Authors: Shao-Dan Lee
Comments: 7 Pages.

A category consists of arrows and objects. We may define a language L B {dom, cod, ◦}. Then a category is a partial algebra of the language L. Hence a functor is a homomorphism of partial algebras. And a natural transformation of functors is a natural transformation of homomorphisms. And we may define a limit of a homomorphism like a limit of functor. Then a limit of a homomorphism forms a homomorphism of partial algebras.
Category: Algebra

Prime Number — Large Number Factorization

Authors: Seung-pyo Hong
Comments: 26 Pages.

The method of factoring large numbers is not known; however, using the method included in this document, it is possible to make a small number of judgments in a short time. The programming language is composed of Java.
Category: Algebra

Foundations of Differential Geometric Algebra

Authors: Michael Reed
Comments: 28 Pages.

Tools built on these foundations enable computations based on multi-linear algebra and spin groups using the geometric algebra known as Grassmann algebra or Clifford algebra. This foundation is built on a direct-sum parametric type system for tangent bundles, vector spaces, and also projective and differential geometry. Geometric algebra is a mathematical foundation for differential geometry, which can be used to simplify the Maxwell equations to a single wave equation due to the geometric product. Introduction of geometric algebra to engineering science disciplines will be easier with programmable foundations.In order to devise an expressive and performance oriented language for efficient discrete differential geometric algebra with the Grassmann elements, an efficient computer algebra representation was programmed. With this unifying mathematical foundation, it is possible to improve efficiency of multi-disciplinary research using geometric tensor calculus by relying on universal mathematical principles. Tools built on universal differential geometric algebra provide a natural geometric language for the Helmholtz decomposition and Hodge-DeRahm co/homology.
Category: Algebra

Null Algebra Extension II

Authors: Robert S. Miller
Comments: 48 Pages.

This extension to Null Algebra more deeply examines the the application, and consequences of division by zero and the solutions to the negative radical within the complex plane. The paper takes this application to its natural result of resolving the complex plane to a real hyper-plane formed in three directions form the union of real and subspace axis. A later version of this paper will explore the negative area these concepts mandate must exist.
Category: Algebra

Золото-комплексное сечение (Complex Golden Section)

Authors: Sergey Y. Kotkovsky
Comments: 9 Pages. In Russian

Для широко известной пары чисел золотого сечения и обратной к нему величины обнаружен их комплексный аналог. На основе изучения комплекснозначных бисимметричных матриц второго порядка показана неразрывная связь вещественного и комплексного золотых сечений.

For a well—known pair of numbers of the golden section and the inverse of it, their complex analogue has been found. Based on the study of complex-valued symmetric matrices of the second order, we show the deep inseparable connection between real and complex golden sections.
Category: Algebra

A Boolean Algebra over a Theory

Authors: Shao-Dan Lee
Comments: 9 Pages.

Suppose that L is a first-order language. Let Lu2020 denote the union of L and {t, f} where t(true), f(false) are the nullary operations. We may define a binary relation ‘≤’ such that the sentences set Φ of the language Lu2020 is a preordered set. And we may construct a boolean algebra Φ/∼, denoted Φ ̃ , by an equivalence relation ‘∼’. Then Φ ̃ is a partial ordered set. Let A be a structure of the language L. If Th(A) is a theory of A, then Thu2020(A) is an ultrafilter. If Ψ ⊂ Φ ̃ is a finitely generated filter, then Ψ is principal. We may define a kernel of a homomorphism of the boolean algebra Φ ̃ such that the kernel is a filter. And a filter is a kernel if it is satisfied by some assumptions.
Category: Algebra

On Tetration Theory

Authors: Juan Elias Millas Vera
Comments: 24 Pages.

In this paper I am going to explain and compare some of the different tetration notations and properties for the tetration concept. Then I am going to incorporate some tables of reference for numerical tetration.
Category: Algebra

NeutroAlgebra is a Generalization of Partial Algebra

Authors: Florentin Smarandache
Comments: 11 Pages.

In this paper we recall, improve, and extend several definitions, properties and applications of our previous 2019 research referred to NeutroAlgebras and AntiAlgebras (also called NeutroAlgebraic Structures and respectively AntiAlgebraic Structures). Let be an item (concept, attribute, idea, proposition, theory, etc.). Through the process of neutrosophication, we split the nonempty space we work on into three regions.
Category: Algebra

Generalizations and Alternatives of Classical Algebraic Structures to NeutroAlgebraic Structures and AntiAlgebraic Structures

Authors: Florentin Smarandache
Comments: 3 Pages.

In this paper we present the development from paradoxism to neutrosophy, which gave birth to neutrosophic set and logic and especially to NeutroAlgebraic Structures (or NeutroAlgebras) and AntiAlgebraic Structures (or AntiAlgebras) that are generalizations and alternatives of the classical algebraic structures.
Category: Algebra

Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures (revisited)

Authors: Florentin Smarandache
Comments: 25 Pages.

In all classical algebraic structures, the Laws of Compositions on a given set are well-defined. But this is a restrictive case, because there are many more situations in science and in any domain of knowledge when a law of composition defined on a set may be only partially-defined (or partially true) and partially-undefined (or partially false), that we call NeutroDefined, or totally undefined (totally false) that we call AntiDefined.
Category: Algebra

Neutro-BCK-Algebra

Authors: Mohammad Hamidi, Florentin Smarandache
Comments: 8 Pages.

This paper introduces the novel concept of Neutro-BCK-algebra. In Neutro-BCK-algebra, the outcome of any given two elements under an underlying operation (neutro-sophication procedure) has three cases, such as: appurtenance, non-appurtenance, or indeterminate. While for an axiom: equal, non-equal, or indeterminate. This study investigates the Neutro-BCK-algebra and shows that Neutro-BCK-algebra are different from BCKalgebra. The notation of Neutro-BCK-algebra generates a new concept of NeutroPoset and Neutro-Hassdiagram for NeutroPosets. Finally, we consider an instance of applications of the Neutro-BCK-algebra.
Category: Algebra

A New Trend to Extensions of CI-algebras

Authors: Florentin Smarandache, Akbar Rezaei, Hee Sik Kim
Comments: 8 Pages.

In this paper, as an extension of CI-algebras, we discuss the new notions of Neutro-CI-algebras and Anti-CI-algebras. First, some examples are given to show that these definitions are different. We prove that any proper CI-algebra is a Neutro-BE-algebra or Anti-BE-algebra. Also, we show that any NeutroSelf-distributive and AntiCommutative CIalgebras are not BE-algebras.
Category: Algebra

Neutrosophic Lattices

Authors: Vasantha Kandasamy, Florentin Smarandache
Comments: 6 Pages.

In this paper authors for the first time define a new notion called neutrosophic lattices. We define few properties related with them. Three types of neutrosophic lattices are defined and the special properties about these new class of lattices are discussed and developed. This paper is organised into three sections. First section introduces the concept of partially ordered neutrosophic set and neutrosophic lattices. Section two introduces different types of neutrosophic lattices and the final section studies neutrosophic Boolean algebras. Conclusions and results are provided in section three.
Category: Algebra

Neutrosophic Measure and Neutrosophic Integral

Authors: Florentin Smarandache
Comments: 5 Pages.

Since the world is full of indeterminacy, the neutrosophics found their place into contemporary research. We now introduce for the first time the notions of neutrosophic measure and neutrosophic integral. Neutrosophic Science means development and applications of neutrosophic logic/set/measure/integral/ probability etc. and their applications in any field. It is possible to define the neutrosophic measure and consequently the neutrosophic integral and neutrosophic probability in many ways, because there are various types of indeterminacies, depending on the problem we need to solve.
Category: Algebra

Universal NeutroAlgebra and Universal AntiAlgebra

Authors: Florentin Smarandache
Comments: 5 Pages.

This paper introduces the Universal NeutroAlgebra that studies the common properties of the NeutroAlgebra structures, and the Universal AntiAlgebra that studies the common properties of the AntiAlgebraic structures.
Category: Algebra

Introducción a la Super-Hiper-Álgebra Y la Super-HiperÁlgebra Neutrosófica
Introduction to Super-Hyper-Algebra and Neutrosophic Superhyper-Algebra

Authors: Florentin Smarandache
Comments: 6 Pages. In Spanish

In this article, the concepts of Nth Power Set of a Set, Super-Hyper-Oper-Operation, Super-Hyper-Axiom, SuperHyper-Algebra, and their corresponding Neutrosophic Super-Hyper-Oper-Operation, Neutrosophic Super-Hyper-Axiom and Neutrosophic Super-Hyper-Algebra are reviewed. In general, in any field of knowledge, really what are found are Super-HyperStructures (or more specifically Super-Hyper-Structures (m, n)).
Category: Algebra

The Superhyperfunction and the Neutrosophic Superhyperfunction (Revisited Again)

Authors: Florentin Smarandache
Comments: 7 Pages.

In this paper, one recalls the general definition of the SuperHyperAlgebra with its SuperHyperOperations and SuperHyperAxioms. Then one introduces for the first time the SuperHyperTopology and especially the SuperHyperFunction and Neutrosophic SuperHyperFunction. One gives a numerical example of a Neutro-SuperHyperGroup.
Category: Algebra

Introduction to SuperHyperAlgebra and Neutrosophic SuperHyperAlgebra

Authors: Florentin Smarandache
Comments: 8 Pages.

In this paper we recall our concepts of n th-Power Set of a Set, SuperHyperOperation, SuperHyperAxiom, SuperHyperAlgebra, and their corresponding Neutrosophic SuperHyperOperation, Neutrosophic SuperHyperAxiom and Neutrosophic SuperHyperAlgebra. In general, in any field of knowledge, one actually encounters SuperHyperStructures (or more accurately (m, n)- SuperHyperStructures).
Category: Algebra

Discovering and Programming the Cubic Formula

Authors: Timothy W. Jones
Comments: 11 Pages.

Solving a cubic polynomial using a formula is possible; a formula exists. In this article we connect various dots from a pre-calculus course and attempt to show how the formula could be discovered. Along the way we make a TI-84 CE menu driven program that allows for experiments, confirmations of speculations, and eventually a working program that solves all cubic polynomials.
Category: Algebra

What is the Factorization Of(x^n + Y^n) When N is an Even Positive Integer?

Authors: Mohamed E. Hassani
Comments: 9 Pages.

The main motivation behind this paper is the question ‘What is the factorization of (��^��+��^��) when n is an even positive integer ?’ which was and is frequently asked on the Internet by many high school and university students and, to my knowledge, even the specialized textbooks and research articles have not yet answered the question, and with time the question itself transformed into a problem that needs to be solved. In the present article, the question is positively answered and the problem is solved through the detailed study of the factorization that leads directly to an apparently new type of indefinite irrational integrals.
Category: Algebra

Hidden Premises in Galois Theory

Authors: Timothy W. Jones
Comments: 4 Pages.

This is a primer for Chapter 3 of Hadlock's book Field Theory and Its Classical Problems: Solution by Radicals. We take a rather naive perspective and consider the linear and quadratic cases afresh and evolve what is really met by solving a polynomial by radicals. There are what we consider to be several hidden premises that some students might be subconsciously puzzled about.
Category: Algebra

A Compact Solution of a Cubic Equation

Authors: Tai-Choon Yoon
Comments: 5 Pages.

In this article, a simple solution of cubic equation is presented by the use of a new substitution $y = (sqrt[3]{alpha} + s/sqrt[3]{alpha})$, which can replace a complicated solution presented by G. Cardano, and François Viète's Vieta substitution. This paper also shows that one of the existing solution of the trigonometric function is to be changed to $- cos(phi - frac{pi}{3})$ instead of $cos(phi - frac{4pi}{3})$ due to the range limit of the inverse trigonometric function.
Category: Algebra

A Solution of a Quartic Equation

Authors: Tai-Choon Yoon
Comments: 4 Pages.

This solution is equal to L. Ferrari's if we simply change the inner square root $sqrt{w}$ to $sqrt{alpha + 2y}$. This article shows the shortest way to have a resolvent cubic for a quartic equation as well as the solution of a quartic equation.
Category: Algebra

The Expression of Binomial Formula (a + B)^n When N is a Prime

Authors: Quang Nguyen Van
Comments: 3 Pages.

We give the expression of binomial formula (a + b)^n when n is a prime number.
Category: Algebra

Cauchy Functions Compared to the Gaussian for X-Ray Powder Diffraction Line Profile Fitting: An Exercise

Authors: Hans Hermann Otto
Comments: 9 Pages.

The best-known profile function is the Gaussian function, which can be used, for instance, to fit successfully optical absorption bands or neutron scattering patterns. However, peaks of X-ray powder pattern can hardly be fitted well with such a simple function. Whereas combined functions are widely in use for such purpose, we applied Cauchy functions to fit our well resolved Guinier powder diffraction data. The Cauchy function of second order is well suited and will be described in more detail as a didactic exercise in crystallography as well as mathematics. In addition, a profile function with an exotic non-integer exponent based on the golden mean is supplemented. This contribution will be continuously revised before its final publication to react of Fewster’s new diffraction theory that will change the matter in future. As an example beyond crystallography the author fitted German Covid-19 data to correlate virus infection peak maxima with causal events.
Category: Algebra

Generalized (σ,τ)-Derivations on Associative Rings Satisfying Certain Identities

Authors: Mehsin Jabel Atteya
Comments: 24 Pages.

The main purpose of this paper is to study a number of results concerning the generalized (σ, τ )-derivation D associated with the derivation d of semiprime ring and prime ring R such that D and d are zero power valued on R, where the mappings σ and τ act as automorphism mappings.Precisely, this article divided into two sections, in the first section, we emphasize on generalized (σ, τ )-derivation D associated with the derivation d of the semiprime ring and prime ring R while in the second section, we study the effect of the compositions of generalized (σ, τ )-derivations of semiprime ring and prime ring R such that D is period (n − 1) on R, for some positive integer n.
Category: Algebra

A Sheaf on a Lattice

Authors: Shao-Dan Lee
Comments: 14 Pages.

A sheaf is constructed on a topological space. But a topological space is a bounded distributive lattice. Hence we may construct a sheaf of lattices on a bounded dis- tributive lattice. Then we define a stalk of the sheaf at a chain in a bounded distributive lattice. And we define a morphism of the sheaves, that the morphism is induced by a homo- morphism of the bounded distributive lattices. Then the kernel and image of the morphism are the subsheaves. A sheaf is obtained by gluing sheaves together.
Category: Algebra

Octonions with Associative Property Using Geometric Algebra

Authors: Jesús Sánchez
Comments: 10 Pages.

In this paper, we will use geometric algebra to derive sets of octonions that have associative property (unlike the original ones that cannot keep it). The price to pay is that is not possible to keep all the elements of the diagonal with -1 but at least one of them has to be +1. Also, the anticommutative property is affected.
Category: Algebra

Some Facts about Relations and Operations of Algebras

Authors: Shao-Dan Lee
Comments: 5 Pages.

Let A be a σ-algebra. Suppose that Θ is a congruence of A. Then Θ is a subalgebra of A×A. If φ is an automorphism from A to A, then (φ,φ) is an automorphism of A×A. And it is obvious that (φ,φ)(Θ) is a congruence of A. Let B be a σ-algebra and ψ a homomorphism from A to B. Then B′ := ψ(A) is a subalgebra of B. And (ψ,ψ)(Θ) is a congruence of B′. If ψ is an epimorphism, then (ψ,ψ)(Θ) is a congruence of B. Suppose that A is a category of all σ-algebras. Let A,B ∈ A and ψ: A → B be a homomorphism. Then the pullback A ⊓B A is isomorphic to a congruence of A. An n-ary relation of an algebra A is a subset of An. If satisfies some conditions, then is a subalgebra of An. The set of languages is a lattice. If is the set of the compositions of the operations in a language σ, then is an algebra.
Category: Algebra

Larger Types of Infinities and Its Impact on Society

Authors: Murilo Leandro
Comments: 3 Pages.

In this paper, we consider an extended real number denoting "larger types of infinity" through elements of a polynomial in a way that justify concepts as infinity plus one, or multiplication of infinities. Then, we equip the set with sums and multiplication such that it forms a ring, and finally, define its order relations.
Category: Algebra

Special Affine Fourier Transform for Space-Time Algebra Signals in Detail

Authors: Eckhard Hitzer
Comments: 18 Pages. To be published in Adv. Appl. Clifford Algebras, 2022.

We generalize the space-time Fourier transform (SFT) [1] to a special affine Fourier transform (SASFT, also known as offset linear canonical transform) for 16-dimensional space-time multivector Cl(3,1)-valued signals over the domain of space-time (Minkowski space) R^{3,1}. We establish how it can be computed in terms of the SFT, and introduce its properties of multivector coefficient linearity, shift and modulation, inversion, Rayleigh (Parseval) energy theorem, partial derivative identities, a directional uncertainty principle and its specialization to coordinates. All important results are proven in full detail. [1] E. Hitzer, Quaternion Fourier Transform on Quaternion Fields and Generalizations. Adv. Appl. Clifford Algebras 17(3), pp. 497-517 (2007), DOI: https://doi.org/10.1007/s00006-007-0037-8.
Category: Algebra

Division by Zero

Authors: Tim Olsson
Comments: 1 Page.

This short paper presents an algebraic theory of extended complex numbers.
Category: Algebra

On Class Field Theory from a Group Theoretical Viewpoint

Authors: Lucian M Ionescu
Comments: 4 Pages.

The main goal of Class Field Theory, of characterizing abelian field extensions in terms of the arithmetic of the rationals, is achieved via the correspondence between Arithmetic Galois Theory and classical (algebraic) Galois Theory, as formulated in its traditional form by Artin. The analysis of field extensions, primarily of the way rational primes decompose in field extensions, is proposed, in terms of an invariant of the Galois group encoding its structure. Prospects of the non-abelian case are given in terms of Grothendieck's Anabelian Theory.
Category: Algebra

On Finite Groups and Galois Theory

Authors: Lucian M Ionescu
Comments: 5 Pages. Preliminary version

We comment on Artin's reformulation of Galois Theory incorporating MacLane's non-abelian extensions theory, and Eilenberg's Category Theory ideology.
Category: Algebra

Arithmetic Galois Theory (Part II)

Authors: Lucian M Ionescu
Comments: 20 Pages. Beamer (LaTex) presentation

A brief historic introduction to Galois Theory is followed by "Arithmetic Galois Theory", which applies the concepts of Galois objects to the category Z of cyclic groups.
Category: Algebra

A Decomposition Formula for Third Order Real Antysimmetric Matrices

Authors: Luca Pettinari
Comments: 8 Pages.

A decomposition formula for an antisymmetric matrix Aω ∈ A3(R) is provided, where its axial vector is expressed as ω = Mν, with M symmetric and ν ∈ R3. The proof is based mainly on vector projection through Frobenius inner product. In the end, a vectorial identity involving cross product is proved as a corollary of the decomposition formula.
Category: Algebra

On Weyl Zeros

Authors: Lucian M Ionescu
Comments: 4 Pages. With SAGE / CoCalc support for computations.

We investigate the zeros of the Betti portion of the Weil rational zeta function for elliptic curves, towards a direct understanding of the Weil conjectures. Examples are provided and various directions of investigations are considered.
Category: Algebra

On Weil Conjectures

Authors: Lucian M Ionescu
Comments: 21 Pages.

We review and comment on the Weil conjectures.
Category: Algebra

On Riemann Zeros and Weil Conjectures

Authors: Lucian M Ionescu
Comments: 25 Pages.

The article aims to motivate the study of the relations between the Riemann zeros, and the zeros of the Weil polynomial of a hyper-elliptic curve over finite fields, beyond the well-known formal analogy. The non-trivial distribution of the p-sectors of the Riemann spectrum recently studied by various authors, represent evidence of a yet unknown algebraic structure exhibited by the Riemann spectrum, supporting the above investigations. This preparatory article consists essentially in a review of the topics involved, and the ``maize'' of relationships to be clarified subsequently. Examples are provided and further directions of investigation are suggested. It is, if successful, a viable, possibly new approach to proving the Riemann Hypothesis, with hindsight from the proof in finite characteristic and function fields.
Category: Algebra

A Note on Rings in Which Each Element is a Sum of Two Idempotents

Authors: Santosh Kumar Pandey
Comments: 3 Pages. Liense:NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)

In this paper we consider a result on rings in which each element is a sum of two idempotents ( appeared in [1] ) and we improve the result by providing a counterexample.
Category: Algebra

A Note on Invo-Regular Rings

Authors: Santosh Kumar Pandey
Comments: 3 Pages. Liense:NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)

In this paper we provide an important and significant observation on a result related to invo-regular rings [1].
Category: Algebra

On Galois Theory with an Invitation to Category Theory

Authors: Lucian M Ionescu
Comments: 8 Pages.

Galois theory in the category of cyclic groups studies the automorphism groups of the cyclic group extensions and the corresponding Galois connection. The theory can be rephrased in dual terms of quotients, corresponding to extensions, when viewed as covering maps. The computation of Galois groups and stating the associated Galois connection are based on already existing work regarding the automorphism groups of finite p-adic groups. The initial goals for developing such a theory were: pedagogical, to introduce the basic language of Category Theory, while exposing the student to core ideas of Galois Theory, but also targeting applications to the Galois Theory of cyclotomic extensions. Some aspects of Abelian Class Field Theory and Anabelian Geometry are also mentioned.
Category: Algebra

On Factorization of Multivectors in Cl(2,1) by Exponentials and Idempotents

Authors: Eckhard Hitzer
Comments: 17 Pages. E. Hitzer, On Factorization of Multivectors in Cl(2,1), byTo be published in Mathematical Methods in the Applied Sciences, 2022.

In this paper we consider general multivector elements of Clifford algebras Cl(2,1), and look for possibilities to factorize multivectors into products of blades, idempotents and exponentials, where the exponents are frequently blades of grades zero (scalar) to n (pseudoscalar). We will succeed mostly, with a minor open case remaining.
Category: Algebra

Current Survey of Clifford Geometric Algebra Applications

Authors: Eckhard Hitzer, Carlile Lavor, Dietmar Hildenbrand
Comments: To be published in Mathematical Methods in the Applied Sciences, 37 pages, (2022).

We extensively survey applications of Clifford Geometric algebra in recent years (mainly 2019–2022). This includes engineering, electric engineering, optical fibers, geographic information systems, geometry, molecular geometry, protein structure, neural networks, artificial intelligence, encryption, physics, signal-, image- and video processing, and software.
Category: Algebra

Natural Ways of Mapping Subsets to Subsets

Authors: Pierre-Yves Gaillard
Comments: 2 Pages.

If X is a set, M its monoid of self-maps and P its power set, then P can be viewed as a left M-set L or as a right M-set R. We compute the monoids End L and End R.
Category: Algebra

A Note on Strongly Invo-Clean Rings

Authors: Santosh Kumar Pandey
Comments: 4 Pages. This paper offers substantial result in Algebra (Ring Theory).

In this note, some important observations have been reported on recent works related to strongly invo-clean rings [1-3].
Category: Algebra

Proof of Fermat's Last Theorem (Using 6 Methods)

Authors: Mantzakouras Nikos
Comments: 26 Pages.

The Pythagorean theorem is perhaps the best known theorem in the vast world of mathematics.A simple relation of square numbers, which encapsulates all the glory of mathematical science, isalso justifiably the most popular yet sublime theorem in mathematical science. The starting pointwas Diophantus’ 20 th problem (Book VI of Diophantus’ Arithmetica), which for Fermat is for n= 4 and consists in the question whether there are right triangles whose sides can be measuredas integers and whose surface can be square. This problem was solved negatively by Fermat inthe 17 th century, who used the wonderful method (ipse dixit Fermat) of infinite descent. Thedifficulty of solving Fermat’s equation was first circumvented by Willes and R. Taylor in late1994 ([1],[2],[3],[4]) and published in Taylor and Willes (1995) and Willes (1995). We presentthe proof of Fermat’s last theorem and other accompanying theorems in 4 different independentways. For each of the methods we consider, we use the Pythagorean theorem as a basic principleand also the fact that the proof of the first degree Pythagorean triad is absolutely elementary anduseful. The proof of Fermat’s last theorem marks the end of a mathematical era; however, theurgent need for a more educational proof seems to be necessary for undergraduates and students ingeneral. Euler’s method and Willes’ proof is still a method that does not exclude other equivalentmethods. The principle, of course, is the Pythagorean theorem and the Pythagorean triads, whichform the basis of all proofs and are also the main way of proving the Pythagorean theorem in anunderstandable way. Other forms of proofs we will do will show the dependence of the variableson each other. For a proof of Fermat’s theorem without the dependence of the variables cannotbe correct and will therefore give undefined and inconclusive results . It is, therefore, possible to prove Fermat's last theorem more simply and equivalently than the equation itself, without monomorphisms. "If one cannot explain something simply so that the last student can understand it, it is not called an intelligible proof and of course he has not understood it himself." R.Feynman Nobel Prize in Physics .1965.
Category: Algebra

The Dominating Number of Expansions

Authors: Theophilus Agama
Comments: 5 Pages.

In this paper, we study the notion of dominating number of expansions.
Category: Algebra

Proof of Beal's Conjecture

Authors: Nikos Mantzakouras
Comments: 31 Pages.

The difference between the Beal equation and the Fermat equation is the different exponents of the variables and the method of solving it. As we will show, for the proof of the Beal equation to be complete, Fermat's theorem will be must hold. There are only 10 known solutions and all of them appear with exponent 2. This very fact is proved here using a uniform method. Therefore, Beal's conjecture is true under the above conditions because it accepts that there is no solution if the condition that all exponent values are greater than 2 occurs, the truth of which is proved in Theorem 6, based on the results of Theorem 5. The primary purpose for solving the equation is to see what happens for solving the equation ax + by = cz i.e. for Pythagorean triples of degree 1. This is the generator of the theorems and programs that follow.
Category: Algebra

The Index of Expansions

Authors: Theophilus Agama
Comments: 6 Pages.

In this paper, we study the notion of an index of sub-expansions in an expansion. We prove the index inequality as an application.
Category: Algebra

Extending Lasenby’s Embedding of Octonions in Space-Time Algebra Cl(1,3), to All Three and Four Dimensional Clifford Geometric Algebras Cl(p,q), N = P + Q = 3,4

Authors: Eckhard Hitzer
Comments: 24 Pages. 6 figures, 7 tables.

We study the embedding of octonions in the Clifford geometric algebra for spacetime STA Cl(1,3), as suggested by Anthony Lasenby at AGACSE 2021. As far as possible, we extend the approach to similar octonion embeddings for all three- and four dimensional Clifford geometric algebras Cl(p,q), n = p + q = 3,4. Noticeably, the lack of a quaternionic subalgebra in Cl(2,1), seems to prevent the construction of an octonion embedding in this case, and necessitates a special approach in Cl(2,2). As examples, we present for Cl(3,0) the non-associativity of the octonionic product in terms of multivector grade parts with cyclic symmetry, show how octonion products and involutions can be combined to make the opposite transition from octonions to the Clifford geometric algebra Cl(3,0), and how octonionic multiplication can be represented with (complex) biquaternions or Pauli matrix algebra.
Category: Algebra

Exact Expansions

Authors: Theophilus Agama
Comments: 6 Pages.

In this paper, we continue the development of multivariate expansivity theory. We introduce and study the notion of an exact expansion and exploit some applications.
Category: Algebra

New Applications of Clifford’s Geometric Algebra

Authors: Stephane Breuils, Kanta Tachibana, Eckhard Hitzer
Comments: 43 Pages. Revised (expanded) version of journal article https://doi.org/10.1007/s00006-021-01196-7.

The new applications of Clifford's geometric algebra surveyed in this paper include kinematics and robotics, computer graphics and animation, neural networks and pattern recognition, signal and image processing, applications of versors and orthogonal transformations, spinors and matrices, applied geometric calculus, physics, geometric algebra software and implementations, applications to discrete mathematics and topology, geometry and geographic information systems, encryption, and the representation of higher order curves and surfaces.
Category: Algebra

R-Algebra of Dimension 1, 2, 4 or 8

Authors: Philippe Gibone
Comments: 12 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]

We will show that it is possible to find a few parameters that describes several "interesting" R-Algebras of dimension 4 and 8 (we will recall the very easy and well known 1 and 2 cases).
Category: Algebra

On Fermat’s Last Theorem and Gӧdel’s Incompleteness Theorem

Authors: Richard Wayte
Comments: 7 Pages.

Fermat’s Last Theorem is proved using elementary arithmetic. Connection of this proof to Gӧdel’s Incompleteness Theorem is mentioned.
Category: Algebra

On the Converse of Lagrange’s Theorem: An Elementary Proof

Authors: Lamarr Widmer
Comments: 1 Page.

The smallest counterexample to the converse of Lagrange’s Theorem is the alternating group of permutations of four elements. We offer an elementary proof that this group has no subgroup of order 6.
Category: Algebra

Genalgebra: A System of Generalized Algebraic Numbers and Operators

Authors: Huhnkie Lee
Comments: 23 Pages. Thank you-

An historical account of numbers and algebra is narrated. Properties of numbers and operators are reviewed. Expansion of extant number and operator systems is introduced. Philosophy of mathematics is discussed.
Category: Algebra

An Algebra of Infinity

Authors: Huhnkie Lee
Comments: 18 Pages. Thank you very much and a Happy New Year //:-D

Correction of Dirac Delta Function is presented. An algebra of infinity, i.e., Alpha Algebra, is introduced. An algebra of infinitesimality, a.k.a., Omega Algebra, is illustrated.
Category: Algebra

Higher Multiplicative Series

Authors: Vishal Pandey
Comments: 12 Pages.

In the Fibonacci series, we have two numbers by adding them we get a series consisting of even and odd numbers in this it goes up to infinity we can track any n the number by Binet’s formula. I have just thought of the multiplication of the first two terms and continued till where I can go, it means that the first two terms in the form (a, b) we will continue the multiplication as we do the addition in the Fibonacci series. As a result, we will get the big integers from the 7th term approximately which is obvious by multiplying to its previous one it will come to a very big integer which cannot be accountable by some range. If we do the multiplication the first two terms will be the same however from the third term it can be written as the power of those integers in which the powers will be following the Fibonacci series in this we can also find the nth term for the multiplicative series. Here the first two terms will be in the same order as they will be given to find the series by changing the order it will violate the rule of the restricted term. The meaning of the restricted here is that the order of (a, b) will be the same throughout the calculation of the whole series we cannot alter that if we do so then it will not be a more restricted term. So there are two concepts in the multiplicative series restricted and non-restricted series. If the (a, b) is there and the operation is going on then it can be said as the restricted series if it is given (a, b) and asked for the (b, a) series then it is said as non-restricted series. I have considered 4 possible criteria to check the pairing of the variables (a, b). We will get to know about the series and also the nth term value of that series for all possible solutions.
Category: Algebra

Four-Operator Algebra

Authors: Juan Elias Millas Vera
Comments: 5 Pages.

In this paper I am going to present a new algebra based in four different operators. As you can imagine the first three operators are the basic and existing: addition, product and power. The fourth is which I am going to develop based in the idea of Graham's number. Also I will to explain the inverse definition for the main three operators and introduce the concept of the inverse of the fourth. Even I will explain some characteristics of this four operator and its inverse.
Category: Algebra

On Factorization of Multivectors in Cl(3,0), Cl(1,2) and Cl(0,3), by Exponentials and Idempotents

Authors: Eckhard Hitzer
Comments: 25 Pages. 4 tables. accepted for publication by Complex Variables and Elliptic Equations, Oct. 2021, DOI: https://doi.org/10.1080/17476933.2021.2001462

In this paper we consider general multivector elements of Clifford algebras Cl(3,0), Cl(1,2) and Cl(0,3), and look for possibilities to factorize multivectors into products of blades, idempotents and exponentials, where the exponents are frequently blades of grades zero (scalar) to n (pseudoscalar).
Category: Algebra

Reconstructing Mythic Algebra

Authors: Michael Griffin
Comments: 11 Pages. contact email mdg46@juno.com

This is the tenth in a series of papers on an algebra, derived from mythology, that can model symbolic processes. Previous papers used literary semantics, mythology, semiotics, philosophy and mathematics. The main features of the algebra are set-based elements and making association a new operation. While such a system can be reductive, it need not be. It may reconstruct into a more useful tool. Various implications for its foundations are considered. 6 Keywords: algebra, association, mathematical modeling, numbers, reduction.
Category: Algebra

The Link Between Algebraic and Rational Homogeneous Linear Recurrences

Authors: Deepak Ponvel Chermakani
Comments: 3 Theorems and 7 pages with 1 example [Correction made by viXra Admin.]

We show that the nth term of an algebraic homogeneous linear-recurrence, can be expressed as a weighted sum of the nth terms of a finite number of rational homogeneous linear recurrences. The weights in this weighted sum belong to a finite set of algebraic constants, no two of which are rational multiples of each other.
Category: Algebra

On Fermat's Last Theorem (III)

Authors: Richard Wayte
Comments: 6 Pages.

Fermat’s Last Theorem is proved using elementary arithmetic.
Category: Algebra

Pinhole Cameras and Division by Zero Calculus

Authors: Saburou Saitoh
Comments: 4 Pages.

From the elementary example of pinhole cameras, the essential fact of the division by zero calculus may be looked and at the same time, some great impacts to rational mappings are referred with the basic interrelation with zero and infinity. Some strong discontinuity property at infinity may be looked as a very interesting property.
Category: Algebra

Introduction to Clifford’s Geometric Algebra

Authors: Stephane Breuils, Kanta Tachibana, Eckhard Hitzer
Comments: 10 Pages. To accompany "New Applications of Clifford’s Geometric Algebra" by the same authors.

A brief application-oriented introduction to W.K. Clifford's geometric algebras, including conformal geometric algebra (CGA).
Category: Algebra

A New Solvable Quintic Equation of the Bring Jerrard Form X^5 + ax + B = 0

Authors: Quang Nguyen Van
Comments: 2 Pages.

In the previous post, we gave one more irreducible equation of the shape x^5 + ax^2 + b = 0, which is solvable. In this paper, we give an irreducible equation of the shape x^5 + ax + b = 0, which is also solvable, contrary to some available arguments.
Category: Algebra

Бивекторная алгебра (Bivector Algebra)

Authors: S. Y. Kotkovsky
Comments: 17 Pages. In Russian

В настоящей работе изучается алгебра икватернионов ненулевой меры с их главной подалгеброй в виде комплекснозначных трехмерных векторов, которые в свою очередь подразделяются на моновекторы и бивекторы. Исследуются свойства комплексных векторов аналогичные параллельности и ортогональности обычных вещественных векторов. Найдены векторные структуры, цикличные относительно произведения, и доказана теорема о тождественности векторного цикла и ориентированного базиса. Как мы выяснили, базисы комплексного векторного пространства так же, как и в вещественном случае распадаются на две ориентации, непереводимые друг в друга непрерывными преобразованиями. Сравнение свойств бивекторов и нульвекторов при унитарных преобразованиях и их циклических структур позволяет говорить об однозначном соответствии этих алгебр заряженным частицам и свету. Тем самым даётся алгебраическое обоснование ключевого для физики векторного характера электромагнитного поля.

In this paper, we study the algebra of nonzero measure icaternions with their principal subalgebra in the form of complex-valued three-dimensional vectors, which in turn are subdivided into monovectors and bivectors. The properties of complex vectors similar to the parallelism and orthogonality of ordinary real vectors are investigated. Vector structures that are cyclic with respect to the product are found, and a theorem on the identity of a vector cycle and an oriented basis is proved. As we have found out, the bases of the complex vector space, as in the real case, split into two orientations, which cannot be translated into each other by continuous transformations. Comparison of the properties of bivectors and zero vectors under unitary transformations and their cyclic structures allows us to speak about the unambiguous correspondence of these algebras to charged particles and light. Thus, an algebraic substantiation of the key vector nature of the electromagnetic field for physics is given.
Category: Algebra

Heuristic Instruction as Pedagogical Praxis in the Teaching of Absolute Value Equations: A Case Study in High Schools in Cabinda, Angola

Authors: F. Maciala, A. Puindi
Comments: 15 Pages.

Mathematics is an area of knowledge whose learning is done in a phased manner. Each of these phases is filled with learning from mathematical entities or entities that in turn serve as a support for learning other new concepts, considered more complex in relation to those already seen. However, for the learning of the new concepts to occur without difficulties, it is necessary that the basic concepts are very well retained. And for the retention of these new concepts, this work presents a proposal in which one can work with modular equations, implementing heuristic instruction, supporting the literary G. Polya. The proposed approach is applied to the treatment of modular equations in the Second Cycle of Secondary Education.
Category: Algebra

Explicit Correspondence of Pauli Matrices, the Basis of Cl(3,0) to 7 Esoteric Principles and to the Bagua.

Authors: Alex Kritov
Comments: 4 Pages. Text is in Russian

The article shows the correspondence of the seven principles of the esoteric doctrines, the Chinese trigrams Gua Fu Xi with Pauli matrices, which are the basis of the corresponding Lie algebras and groups, and widely used in modern fundamental physics. This article is intended for a very narrow category of readers, namely, for those who are familiar with the mathematical apparatus of the theory of groups and algebras Clifford in their application to the physical world, with binary (bit) operations and, at the same time, with the esoteric teachings of Theosophy as presented by H.P. Blavatsky, Subba Row. For mathematicians, this work may be interesting in connection with the proposed method for enumerating Pauli matrices using bitwise operations. [in Russian]
Category: Algebra

How Hard is the Tensor Rank?

Authors: Yaroslav Shitov
Comments: 18 Pages. The results from an older version arXiv:1611.01559 are still here, almost everything is rewritten, new results added

We build a combinatorial technique to solve several long-standing problems on the complexity of tensor decompositions. These include the polynomial time equivalence between the problem of computing the tensor rank over an integral domain R and the solvability of a system of polynomial equations over R. In particular, the tensor rank is undecidable over Z, which answers a question posed by Gonzalez and Ja’Ja’ in 1980, and another special case R = Q answers a question of Blaser. Also, we determine the algorithmic complexity of the symmetric rank, which confirms the NP-hardness conjecture of Hillar and Lim. As a byproduct of our approach, we answer a question discussed by Buss, Frandsen, Shallit in 1999 and determine the algorithmic complexity of the minimal rank matrix completion, and we solve two problems of Grossmann and Woerdeman on the fractional minimal rank.
Category: Algebra

Analytic Expansions and an Application to Function Theory

Authors: Theophilus Agama
Comments: 9 Pages.

In this paper we introduce and study the notion of singularity, the kernel and analytic expansions. We provide an application to the existence of singularities of solutions to certain polynomial equations.
Category: Algebra

The Cohomology of an Endomorphism

Authors: Antoine Balan
Comments: 1 page, written in french

We define a cohomology for an endomorphism of complex, not only a differential.
Category: Algebra

Strange Enough i^xi

Authors: Miroslaw Kozlowski
Comments: Pages.

In this paper I consider the strange real function f(x)= ixi , i=Sqrt[-1] and x is real . It is hard to find its application. But it is an example the subtle connection of the real and imaginary universe
Category: Algebra

Solvable Form of the Polynomial Equation X^n + an-1x^(n-1) + ...+a1x + A0= 0 (n = 2k + 1)

Authors: Quang Nguyen Van
Comments: 3 Pages.

It is know, there is no solution in radicals to general polynomial equation of degree five or higher with arbitrary coefficient. In this article, we give a form of the polynomial equations with odd degree can be solved in radicals. From there, we come up some solvable equations with one or more zero coefficients, especially for the quintic and septic equations.
Category: Algebra

Cramer's Rule for Overdetermined Systems of Linear Equations

Authors: Yanko Popov
Comments: 10 Pages.

In this article, with the help of introduced cross product in n-dimensional Euclidean space, the Jacobi identity and Cramer's rule for over-determined system of linear equations are displayed.
Category: Algebra

A Simple Criteria of Prime Numbers

Authors: Masami Yamane
Comments: 2 Pages. We present a simple criteria of prime numbers.

In this short note, we will propose a simple criteria for prime numbers and our mehtod seems to be that is practical. Our idea will have some connection with the famous Goldbach conjecture.
Category: Algebra

The D'alembert-Gauss Theorem

Authors: Antoine Balan
Comments: 2 pages, written in french

We give a geometric and topological proof of the d'Alembert-Gauss theorem: a complex polynomial has a root.
Category: Algebra

A Theorem on the Number of Distinct Eigenvalues

Authors: Rachid Marsli
Comments: 17 Pages.

A theorem on the number of distinct eigenvalues of diagonalizable matrices is obtained. Some applications related to matrices with simple eigenvalues, triangular defective matrices, adjacency matrices and graphs are discussed. Other ideas and examples are provided.
Category: Algebra

The Theory of Cyclic Isomorphism

Authors: Franz Hermann
Comments: 22 Pages.

In this paper, we introduce the notion of cyclic isomorphism of subgroups of some finite abstract group. Examples of matrix representation of such a simple isomorphism are shown on the example of Clifford-Pauli matrices
Category: Algebra

Uncovering the True Value of Euler’s Famous Identity

Authors: Mueiz Gafer KamalEldeen
Comments: 2 Pages. [Corrections made by viXra Admin to conform with scholarly norm - Please conform]]

It is shown that the great claims made for the importance and beauty of Euler’s Identity are only misconceptions based on apparent and superficial view on the subject which don’t look at the core of the identity which comes into view by the analysis of the its elements.
Category: Algebra

A Universality Theorem for Nonnegative Matrix Factorizations

Authors: Yaroslav Shitov
Comments: 24 Pages. a full version of arXiv:1606.09068

Let A be a nonnegative matrix, that is, a matrix with nonnegative real entries. A nonnegative factorization of size k is a representation of A as a sum of k nonnegative rank-one matrices. The space of all such factorizations is a bounded semialgebraic set, and we prove that spaces arising in this way are universal. More presicely, we show that every bounded semialgebraic set U is rationally equivalent to the set of nonnegative size-k factorizations of some matrix A up to a permutation of matrices in the factorization. Our construction is effective, and we can compute a pair (A, k) in polynomial time from a given description of U as a system of polynomial inequalities with coefficients in Q. This result gives a complete description of the algorithmic complexity of several important problems, including the nonnegative matrix factorization, completely positive rank, nested polytope problem, and it also leads to a complete resolution of the problem of Cohen and Rothblum on nonnegative factorizations over different ordered fields.
Category: Algebra

Multivariate Expansivity Theory

Authors: Theophilus Agama
Comments: 16 Pages.

In this paper we launch an extension program for single variable expansivity theory. We study this notion under tuples of polynomials belonging to the ring $\mathbb{R}[x_1,x_2,\ldots,x_n]$.
Category: Algebra

A Group of Order 1575 with a Normal Sylow 3-subgroup is Abelian

Authors: Henry Wong
Comments: 2 Pages.

In this paper we lay out the proof of this result in group theory using an old-fashioned approach.
Category: Algebra

The Elementary of Fermat's Last Theorem (Thai Version)

Authors: Sattawat Suntisurat
Comments: 5 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]

Proof of Fermat's last theorem by using basic of algebra.
Category: Algebra

Isomorphisms Between Dual Spaces of a Vector Space

Authors: Eduardo Magalhães
Comments: 5 Pages.

In this small paper, it's deduced that for every finite-dimensional vector space V, the i-th and the j-th dual spaces of V are isomorphic. Tho other minor lemmas are also proven: 1) Every vector space V with dimension n over a field K is isomorphic to K^n, and 2) The i-th dual space of a finite-dimensional vector space V is isomorphic to the i+1-th dual space of V.
Category: Algebra

The 2N Conjecture on Spectrally Arbitrary Sign Patterns Is False

Authors: Yaroslav Shitov
Comments: 21 Pages.

A sign pattern is a matrix with entries in {+, −, 0}. An n × n sign pattern S is spectrally arbitrary if, for any monic polynomial f of degree n with real coefficients, one can replace the + and − signs in S with real numbers of the corresponding signs so that the resulting matrix has characteristic polynomial f. This paper refutes a long-standing conjecture with a construction of an n × n spectrally arbitrary sign pattern with less than 2n entries nonzero.
Category: Algebra

A New Solvable Quintic Equation of the Shape X^5 + Ax^2 + B = 0

Authors: Quang Nguyen Van
Comments: 2 Pages.

So far, there are in all five solvable quintics of the shape x^5 + ax^2 + b = 0. We have found one more. In this paper, we give that equation and its solution.
Category: Algebra

The 2-Morphisms

Authors: Antoine Balan
Comments: 1 Page. Written in French

The 2-morphisms are generalizations of morphisms between algebras.
Category: Algebra

Solvable Sextic Equation X^6 + Px^4 + Qx^3 + Rx^2 + PQx/3 +PR/3-2P^3/27=0

Authors: Quang Nguyen Van
Comments: 1 Page.

We give a new solvable sextic equation and its solution.
Category: Algebra

A Journey to the Pierce-Birkhoff Conjecture

Authors: Theophilus Agama
Comments: 7 Pages.

This paper initializes the study of the Pierce-Birkhoff conjecture. We start by introducing the notion of the area and volume induced by a multivariate expansion and develop some inequalities for our next studies. In particular we obtain the inequality \begin{align} \sum \limits_{\substack{i,j\in [1,n]\\a_{i_{\sigma(s)}}<a_{j_{\sigma(s)}}\\s\in [1,l]\\v\neq i,j\\v\in [1,n] }}\bigg | \bigg |\vec{a}_{i} \diamond \vec{a}_{j}\diamond \cdots \diamond \vec{a}_v\bigg |\bigg |\sum \limits_{k=1}^{n}\int \limits_{a_{i_{\sigma(l)}}}^{a_{j_{\sigma(l)}}}\int \limits_{a_{i_{\sigma(l-1)}}}^{a_{j_{\sigma(l-1)}}}\cdots \int \limits_{a_{i_{\sigma(1)}}}^{a_{j_{\sigma(1)}}}g_kdx_{\sigma(1)}dx_{\sigma(2)}\cdots dx_{\sigma(l)}\nonumber\\ \leq 2C\times \binom{n}{2}\times \sqrt{n}\times \nonumber \\ \times \int \limits_{a_{i_{\sigma(l)}}}^{a_{j_{\sigma(l)}}}\int \limits_{a_{i_{\sigma(l-1)}}}^{a_{j_{\sigma(l-1)}}}\cdots \int \limits_{a_{i_{\sigma(1)}}}^{a_{j_{\sigma(1)}}}\sqrt{\bigg(\sum \limits_{k=1}^{n}(\mathrm{max}(g_k))^2\bigg)}dx_{\sigma(1)}dx_{\sigma(2)}\cdots dx_{\sigma(l)}\nonumber \end{align}for some constant $C>0$, where $\sigma:\{1,2,\ldots,l\}\longrightarrow \{1,2,\ldots,l\}$ is a permutation for $g_k\in \mathbb{R}[x_1,x_2,\ldots,x_l]$ and $\vec{a}_{i} \diamond \vec{a}_{j}\diamond \cdots \diamond \vec{a}_k \diamond \vec{a}_{v}$ is the cross product of any of the $n-1$ fixed spots in $\mathbb{R}^{l}$ including the spots $\vec{a}_i,\vec{a}_j$.
Category: Algebra

The Waring Rank of the 3 x 3 Determinant

Authors: Yaroslav Shitov
Comments: 23 Pages.

Let f be a homogeneous polynomial of degree d with coefficients in C. The Waring rank of f is the smallest integer r such that f is a sum of r powers of linear forms. We show that the Waring rank of the polynomial x1 y2 z3 − x1 y3 z2 + x2 y3 z1 − x2 y1 z3 + x3 y1 z2 − x3 y2 z1 is at least 18, which matches the known upper bound.
Category: Algebra

Geometric Product of Two Oriented Points in Conformal Geometric Algebra

Authors: Eckhard Hitzer
Comments: 15 Pages. Accepted for publication by Adv. Appl. Clifford Algebras, Oct. 2024.

We compute and explore the full geometric product of two oriented points in conformal geometric algebra Cl(4,1) of three-dimensional Euclidean space. We comment on the symmetry of the various components, and state for all expressions also a representation in terms of point pair center and radius vectors.
Category: Algebra

Linear Algebra and Group Theory

Authors: Teo Banica
Comments: 400 Pages.

This is an introduction to linear algebra and group theory. We first review the linear algebra basics, namely the determinant, the diagonalization procedure and more, and with the determinant being constructed as it should, as a signed volume. We discuss then the basic applications of linear algebra to questions in analysis. Then we get into the study of the closed groups of unitary matrices $Gsubset U_N$, with some basic algebraic theory, and with a number of probability computations, in the finite group case. In the general case, where $Gsubset U_N$ is compact, we explain how the Weingarten integration formula works, and we present some basic $Ntoinfty$ applications.
Category: Algebra

Space Time PGA in Geometric Algebra G(1,3,1)

Authors: Robert Benjamin Easter, Daranee Pimchangthong
Comments: Pages.

In Geometric Algebra, G(1,3,1) is a degenerate-metric geometric algebra being introduced in this paper as Space Time PGA [STPGA], based on 3D Homogeneous PGA G(3,0,1) [3DPGA] and 4D Conformal Spacetime CGA G(2,4,0) [CSTA]. In CSTA, there are flat (linear) geometric entities for hyperplane, plane, line, and point as inner product null space (IPNS) geometric entities and dual outer product null space (OPNS) geometric entities. The IPNS CSTA geometric entities are closely related, in form, to the STPGA plane-based geometric entities. Many other aspects of STPGA are borrowed and adapted from 3DPGA, including a new geometric entity dualization operation J_e that is an involution in STPGA. STPGA includes operations for spatial rotation, spacetime hyperbolic rotation (boost), and spacetime translation as versor operators. This short paper only introduces the basics of the STPGA algebra. Further details and applications may appear in a later extended paper or in other papers. This paper is intended as a quick and practical introduction to get started, including explicit forms for all entities and operations. Longer papers are cited for further details.
Category: Algebra

Geometric Entity Dualization and Dual Quaternion Geometric Algebra in PGA G(3,0,1) with Double PGA G(6,0,2) for General Quadrics

Authors: Robert Benjamin Easter, Daranee Pimchangthong
Comments: 76 Pages.

In Geometric Algebra, G(3,0,1) is a degenerate-metric algebra known as PGA, originally called Projective Geometric Algebra in prior literature. It includes within it a point-based algebra, plane-based algebra, and a dual quaternion geometric algebra (DQGA). In the point-based algebra of PGA, there are outer product null space (OPNS) geometric entities based on a 1-blade point entity, and the join (outer product) of two or three points forms a 2-blade line or 3-blade plane. In the plane-based algebra of PGA, there are commutator product null space (CPNS) geometric entities based on a 1-blade plane entity, and the meet (outer product) of two or three planes forms a 2-blade line or 3-blade point. The point-based OPNS entities are dual to the plane-based CPNS entities through a new geometric entity dualization operation J_e that is defined by careful observation of the entity duals in same orientation and collected in a table of basis-blade duals. The paper contributes the new operation J_e and its implementations using three different nondegenerate algebras {G(4),G(3,1),G(1,3)} as forms of Hodge star dualizations, which in geometric algebra are various products of entities with nondegenerate unit pseudoscalars, taking a grade k entity to its dual grade 4-k entity copied back into G(3,0,1). The paper contributes a detailed development of DQGA. DQGA represents and emulates the dual quaternion algebra (DQA) as a geometric algebra that is entirely within the even-grades subalgebra of PGA G(3,0,1). DQGA has a close relation to the plane-based CPNS PGA entities through identities, which allows to derive dual quaternion representations of points, lines, planes, and many operations on them (reflection, rotation, translation, intersection, projection), all within the dual quaternion algebra. In DQGA, all dual quaternion operations are implemented by using the larger PGA algebra. The DQGA standard operations include complex conjugate, quaternion conjugate, dual conjugate, and part operators (scalar, vector, tensor, unit, real, imaginary), and some new operations are defined for taking more parts (point, plane, line) and taking the real component of the imaginary part by using the new operation J_e. All DQGA entities and operations are derived in detail. It is possible to easily convert any point-based OPNS PGA entity to and from its dual plane-based CPNS PGA entity, and then also convert any CPNS PGA entity to and from its DQGA entity form, all without changing orientation of the entities. Thus, each of the three algebras within PGA can be taken advantage of for what it does best, made possible by the operation J_e and identities relating CPNS PGA to DQGA. PGA G(3,0,1) is then doubled into a Double PGA (DPGA) G(6,0,2) including a Double DQGA (DDQGA), which feature two closely related forms of a general quadric entity that can be rotated, translated, and intersected with planes and lines. The paper then concludes with final remarks.
Category: Algebra

New atom model and new SU(5) model

Authors: Wan-Chung Hu
Comments: 23 Pages.

This manuscript provides a new determinative atom model. The magic number 2, 8, 8, 18, 18, 32, 32 can be well explained without using quantum mechanics. In addition, spin-orbit coupling can also be deducted without quantum mechanics. In the final part of the manuscript, modified su(5) model called Hu SU(5) model includes all the fundamental particles and explain mass origin and decay mode in a clear picture.
Category: Algebra

Skolems Solution for Integer-Linear-Recurrences, with Commensurable Arguments for Characteristic-Roots of the Same Modulus

Authors: Deepak Ponvel Chermakani
Comments: 9 Pages. Explained more on the proofs

For a homogeneous linear-recurrence f(n) with integer coefficients and integer starting points, we derive a deterministic algorithm that finds the upper bound of the last non-periodic position n where f(n)=0, for a large family of special cases. First, when theta is a given irrational constant, then we show that, an eventual lower bound of minimum(absolute(cos(m PI theta)), over positive integers m less than n), for large positive integers n, is (2 theta / (sqrt(5) n)). Our deterministic algorithm is based on the key concept that this lower bound decreases at a lower rate than the nth power of the ratio of root-moduli since the ratio is lesser than 1. Our deterministic algorithm is developed for the special cases where G(x), the characteristic polynomial of f(n), has either equal absolute values of arguments or commensurable arguments of those complex roots whose moduli are equal. In an attempt to extend this algorithm as a general solution to Skolems problem, we obtain the lower bound of the distance between a zero and the next (2^(m+1))th zero, in the weighted sum of m continuous cosine functions, where the weights are given real-algebraic constants.
Category: Algebra

Золото-комплексное сечение (Complex Golden Section)

Authors: Sergey Y. Kotkovsky
Comments: 12 Pages. In Russian. Corrections and additions made.

Для широко известной пары чисел золотого сечения и обратной к нему величины обнаружен их комплексный аналог. На основе изучения комплекснозначных бисимметричных матриц второго порядка показана неразрывная связь вещественного и комплексного золотых сечений.

For a well-known pair of numbers of the golden section and the inverse of it, their complex analogue has been found. Based on the study of complex-valued symmetric matrices of the second order, we show the deep inseparable connection between real and complex golden sections.
Category: Algebra

Discovering and Programming the Cubic Formula

Authors: Timothy W. Jones
Comments: 12 Pages. Some suggestions were given by a reader and a solution to the puzzle of the earlier version is added.

Solving a cubic polynomial using a formula is possible; a formula exists. In this article we connect various dots from a pre-calculus course and attempt to show how the formula could be discovered. Along the way we make a TI-84 CE menu driven program that allows for experiments, confirmations of speculations, and eventually a working program that solves all cubic polynomials.
Category: Algebra

Hidden Premises in Galois Theory

Authors: Timothy W. Jones
Comments: 6 Pages. I've added a section that clarifies earlier statements concerning algebraic versus transcendental numbers. Thanks for any patience.

This is a primer for Chapter 3 of Hadlock's book Field Theory and Its Classical Problems: Solution by Radicals. We take a rather naive perspective and consider the linear and quadratic cases afresh and evolve what is really met by solving a polynomial by radicals. There are what we consider to be several hidden premises that some students might be subconsciously puzzled about.
Category: Algebra

The Expression of Binomial Formula ( a + B)^n When N is a Prime

Authors: Quang Nguyen Van
Comments: 3 Pages.

We give the expression of binomial formula (a + b)^n when n is a prime number.
Category: Algebra

Division by Zero

Authors: Tim Olsson
Comments: 1 Page. Mistakes corrected; added that (−1)^∞ = 0.

An algebraic theory of extended complex numbers.
Category: Algebra

Division by Zero

Authors: Tim Olsson
Comments: 1 Page.

An algebraic theory of extended complex numbers.
Category: Algebra

A Decomposition Formula for Third Order Real Antysimmetric Matrices

Authors: Luca Pettinari
Comments: 8 Pages.

A decomposition formula for an antisymmetric matrix Aω ∈ A3(R) is provided, where its axial vector is expressed as ω = Mν, with M symmetric and ν ∈ R3. The proof is based mainly on vector projection through Frobenius inner product. In the end, a vectorial identity involving cross product is proved as a corollary of the decomposition formula.
Category: Algebra

Provability of the Received Fermat's Last Theorem

Authors: Richard Wayte
Comments: 7 Pages.

The received Theorem is transformed into a new symmetrical expression, for it to be compared with a parallel all-integer expression. Differences in the configurations of the expressions prove that components of the Theorem cannot all be integers.
Category: Algebra

How Hard is the Tensor Rank?

Authors: Yaroslav Shitov
Comments: 26 Pages. added results on matrix rigidity

We build a combinatorial technique to solve several long standing problems in linear algebra with a particular focus on algorithmic complexity of matrix completion and tensor decomposition problems. For all appropriate integral domains R, we show the polynomial time equivalence of the problem of the solvability of a system of polynomial equations over R to • the minimum rank matrix completion problem (in particular, we answer a question asked by Buss, Frandsen, Shallit in 1999), • the determination of matrix rigidity (we answer a question posed by Mahajan, Sarma in 2010 by showing the undecidability over Z, and we solve recent problems of Ramya corresponding to Q and R), • the computation of tensor rank (we answer a question asked by Gonzalez, Ja'Ja' in 1980 on the undecidability over Z, and, additionally, the special case with R = Q solves a problem posed by Blaser in 2014), • the computation of the symmetric rank of a symmetric tensor, whose algorithimic complexity remained open despite an extensive discussion in several foundational papers. In particular, we prove the NP-hardness conjecture proposed by Hillar, Lim in 2013. In addition, we solve two problems on fractional minimal ranks of incomplete matrices recently raised by Grossmann, Woerdeman, and we answer, in a strong form, a recent question of Babai, Kivva on the dependence of the solution to the matrix rigidity problem on the choice of the target field.
Category: Algebra

Multivariate Expansivity Theory

Authors: Theophilus Agama
Comments: 20 Pages. An application added

In this paper we launch an extension program for single variable expansivity theory. We study this notion under tuples of polynomials belonging to the ring $\mathbb{R}[x_1,x_2,\ldots,x_n]$. As an application we show that \begin{align}\mathrm{min}\{\mathrm{max}\{\mathrm{Ind}_{f_k}(x_{\sigma(i)})\}_{k=1}^{s}+1\}_{i=1}^{l}&<\frac{1}{l}\sum \limits_{i=1}^{l}\mathrm{max}\{\mathrm{Ind}_{f_k}(x_{\sigma(i)})\}_{k=1}^{s}+2+\mathcal{J}\nonumber \end{align}where $\mathcal{J}:=\mathcal{J}(l)\geq 0$ and $\mathrm{Ind}_{f_k}(x_j)$ is the largest power of $x_j$~($1\leq j\leq n$) in the polynomial $f_k\in \mathbb{R}[x_1,x_2,\ldots,x_n]$.
Category: Algebra

The Elementary Proof of Fermat's Last Theorem (Complete)

Authors: Sattawat Suntisurat
Comments: 5 Pages.

Proof of Fermat's Last Theorem by using basic of algebra.
Category: Algebra

A New Solvable Quintic Equation of the Shape X^5 + aX^2 + b = 0

Authors: Quang Nguyen Van
Comments: 2 Pages.

So far, there are in all five solvable quintics of the shape x^5 + ax^4 + b = 0. We have found one more. In this paper, we give that equation and its solutions.
Category: Algebra

The 2-Morphisms

Authors: Antoine Balan
Comments: 2 Pages. In French

The 2-morphisms are generalizations of morphisms between algebras. They are morphisms of algebras if we make a tensor product.
Category: Algebra

The 2-Morphisms

Authors: Antoine Balan
Comments: Pages.

The 2-morphisms are generalizations of morphisms between algebras.
Category: Algebra