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).

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 $left( begin{array}{cc} A & B C & D end{array}ight)in mathcal{B}(Xoplus X)$.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 $mathscr{B}_0$.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.<p>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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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. <p> 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.

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.

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.

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.

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.

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.

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.

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 $f_k\in \mathbb{R}[x_1,x_2,\ldots,x_n]$.

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.

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.

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 $\vec{a}_i,\vec{a}_j$.

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.

Let f be a homogeneous polynomial of degree d with coefficients in a field F satisfying char F = 0 or char F > d. The Waring rank of f is the smallest integer r such that f is a linear combination of r powers of F-linear forms. We show that the Waring rank of the polynomial x1 y2 z3 + x1 y3 z2 + x2 y1 z3 + x2 y3 z1 + x3 y1 z2 + x3 y2 z1 is at least 16, which matches the known upper bound.

Lattice is a partially ordered set with two operations defined on it that satisfy certain conditions. Lattice theory itself is a branch of abstract algebra. In this paper we present solutions to three classical problems in lattice theory. The solutions are by no means novel.

In this paper some properties and the chain rule for the hessian tensor for combined vector functions are derived. We will derive expressions for H(T + L) , H(aT) , and H(T ◦ L) (chain rule for hessian tensors) and show some specific examples of the chain rule in certain types of composite maps.

In this paper we consider general multivector elements of Clifford algebras Cl(p,q), p+q <3, and study multivector factorization into products of exponentials and idempotents, where the exponents are blades of grades zero (scalar) to n (pseudoscalar).

We introduce the notion of an expansion in a specified and mixed directions. This is a piece of an extension program of \textbf{single~variable~expansivity~theory} developed by the author.

In this paper we introduce a particular class of matrices. We study the concept of a matrix to be balanced. We study some properties of this concept in the context of matrix operations. We examine the behaviour of various matrix statistics in this setting. The crux will be to understanding the determinants and the eigenvalues of balanced matrices. It turns out that there does exist a direct communication among the leading entry, the trace, determinants and, hence, the eigenvalues of these matrices of order $2times 2$. These matrices have an interesting property that enables us to predict their quadratic forms, even without knowing their entries but given their spectrum.

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.

In this note we show that under certain conditions the inequality holds \begin{align}\sum \limits_{\lambda_i\in \mathrm{Spec}(ab^{T})}\mathrm{min}\{\log |t-\lambda_i|\}_{[||a||,||b||]}&\leq \# \mathrm{Spec}(ab^T)\log\bigg(\frac{||b||+||a||}{2}\bigg)\nonumber \\&+\frac{1}{||b||-||a||}\sum \limits_{\lambda_i\in \mathrm{Spec}(ab^T)}\log \bigg(1-\frac{2\lambda_i}{||b||+||a||}\bigg).\nonumber \end{align}Also under the same condition, the inequality also holds\begin{align}\int \limits_{||a||}^{||b||}\log|\mathrm{det}(ab^{T}-tI)|dt&\leq \# \mathrm{Spec}(ab^T)(||b||-||a||)\log\bigg(\frac{||b||+||a||}{2}\bigg)\nonumber \\&+\sum \limits_{\lambda_i\in \mathrm{Spec}(ab^T)}\log \bigg(1-\frac{2\lambda_i}{||b||+||a||}\bigg).\nonumber \end{align}

This article takes an oblique sidestep from two previous papers, wherein an approach to reformulation of game theory in terms of information theory, topology, as well as a few other notions was indicated. In this document a description is provided as to how one might determine an approach for an agent to choose a policy concerning which actions to take in a game that constrains behaviour of subsidiary agents. It is then demonstrated how these results in algebraic information theory, together with previous investigations in geometric and topological information theory, can be unified into a single cohesive framework.

In this paper we consider general multivector elements of Clifford algebras $Cl(p,q)$, $n=p+q \leq 3$, and study multivector equivalents of polar decompositions and factorization into products of exponentials, where the exponents are frequently blades of grades zero (scalar) to $n$ (pseudoscalar).

The Two Couriers Problem is an algebra problem, originally stated in 1746 by the French mathematician Clairaut. For over a century, the Two Couriers Problem has been re-used in various forms as a mathemat- ical problem, in textbooks and journals, by different mathematicians and authors. The Two Couriers Problem involves cases where division by zero arises in practice, where each has a real-world, actual result for the solution. Thus the Two Couriers Problem is a centuries old algebra problem with actual applied results that involve division by zero. It is an excellent mathematical problem to evaluate different methods for dividing by zero. Division by zero has many different mathematical approaches. Conventional mathematics handles division by zero as an indeterminate or undefined result. Transmathematics defines division by zero as either nul- lity or explicitly positive or negative infinity. Two other approaches are by Saitoh, who defines division by zero simply as zero, and Barukčić who defines division by zero as either unity or explicitly positive or implicitly negativity infinity. The question is, which approach is best to solve the mathematical problem of division by zero? The paramount goal of this paper is to use the Two Couriers Problem as an objective test to examine and evaluate mathematical approaches to division by zero – and find which one is best.

We introduce a unimodular Determinant=1 8x8 rotation matrix to produce four 4 dimensional copies of H4 600-cells from the 240 vertices of the Split Real Even E8 Lie group. Unimodularity in the rotation matrix provides for the preservation of the 8 dimensional volume after rotation, which is useful in the application of the matrix in various fields, from theoretical particle physics to 3D visualization algorithm optimization.

There are natural lead-ins to abstract algebra that occur in elementary algebra. We explore function composition using linear functions and permutations on letters in misspellings of words. Groups and the central idea of abstract algebra, proving 5th degree and greater polynomials are unsolvable, are put into focus for college students.

The main purpose of this paper is to study and investigate certain results concerning the (σ,τ)-generalized derivation D associated with the (σ,τ)-derivation d of semiprime and prime rings R, where σ and τ act as two automorphism mappings of R. We focus on the composition of (σ,τ)-generalized derivations of the Leibniz’s formula, where we introduce the general formula to compute the composition of the (σ,τ)-generalized derivation D of R.

To each monoid M we attach an inclusion A --> B of Q-algebras, and ask: Is B flat over A? If our monoid M is a group, A is von Neumann regular, and the answer is trivially Yes in this case.

Assuming known algebraic expressions for multivector inverses in any Clifford algebra over an even dimensional vector space R^{p',q'), n' = p' +q' = 2m, we derive a closed algebraic expression for the multivector inverse over vector spaces one dimension higher, namely over R^{p,q}, n = p+q = p'+q'+1 = 2m+1. Explicit examples are provided for dimensions n' = 2,4,6, and the resulting inverses for n = n' +1 = 3,5,7. The general result for n = 7 appears to be the first ever reported closed algebraic expression for a multivector inverse in Clifford algebras Cl(p,q), n = p + q = 7, only involving a single addition of multivector products in forming the determinant.

Rationality problems of algebraic k-tori are closely related to rationality problems of the invariant field, also known as Noether's Problem. We describe how a function field of algebraic k-tori can be identified as an invariant field under a group action and that a k-tori is rational if and only if its function field is rational over k. We also introduce character group of k-tori and numerical approach to determine rationality of k-tori.

In this work I axiomatize the result of $x \cdot 0$ ($x\in\real_{\ne 0}$) as a number $\inull(x)$ that has a null real part (denoted as $\Re(\inull(x))=0$) but that is not real. This implies that $y+\Re(\inull(x)) = y$ but $y+\inull(x) = y + x\cdot 0 \not=y$, $y\in\real_{\ne 0}$. From this I define the set of null imaginary numbers $\nullset=\{\inull(x)=x\cdot 0|\forall x\in\real_{\ne 0}\}$ and present its elementary algebra taking the axiom of uniqueness as base (i.e., if $x\ne y\Leftrightarrow \inull(x)\ne \inull(y)$). Under the condition of existence of $\nullset$ I show that division by zero can be defined without causing inconsistencies in elementary algebra.

The Triple Conformal Geometric Algebra (TCGA) for the Euclidean R^2-plane extends CGA as the product of three orthogonal CGAs, and thereby the representation of geometric entities to general cubic plane curves and certain cyclidic (or roulette) quartic, quintic, and sextic plane curves. The plane curve entities are 3-vectors that linearize the representation of non-linear curves, and the entities are inner product null spaces (IPNS) with respect to all points on the represented curves. Each IPNS entity also has a dual geometric outer product null space (OPNS) form. Orthogonal or conformal (angle-preserving) operations (as versors) are valid on all TCGA entities for inversions in circles, reflections in lines, and, by compositions thereof, isotropic dilations from a given center point, translations, and rotations around arbitrary points in the plane. A further dimensional extension of TCGA, also provides a method for anisotropic dilations. Intersections of any TCGA entity with a point, point pair, line or circle are possible. TCGA defines commutator-based differential operators in the coordinate directions that can be combined to yield a general n-directional derivative.

The purpose of the paper is to present new results (exponential, real structure, Cartan algebra,...) but, as the definitions are sill varying with the authors, the paper covers all the domain, and can be read as a comprehensive presentation of Clifford algebras.

Search engines are huge power factors on the Web, guiding people to information and services. Google is the most successful search engine in recent years,his research results are very complete and precise. When Google was an early research project at Stanford, several articles have been written describing the underlying algorithms. The dominant algorithm has been called PageRank and is still the key to providing accurate rankings for search results. A key feature of web search engines is sorting results associated with a query in order of importance or relevance. We present a model allowing to define a quantification of this concept (Pagerank) a priori fuzzy and elements of formalization for the numerical resolution of the problem. We begin with a natural first approach unsatisfactory in some cases. A refinement of the algorithm is introduced to improve the results.

We present notes on the Marcus-de Oliveira conjecture. The conjecture concerns the region in the complex plane covered by the determinants of the sums of two normal matrices with prescribed eigenvalues. Call this region ∆. This paper focuses on boundary matrices of ∆. We prove 2 theorems regarding these boundary matrices.

This document is the first in what is intended to be a collection of solutions of high-school-level problems via Geometric Algebra (GA). GA is very much "overpowered" for such problems, but students at that level who plan to go into more-advanced math and science courses will benefit from seeing how to "translate" basic problems into GA terms, and to then solve them using GA identities and common techniques.

This paper introduces the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), based in the $mathcal{G}_{8, 2}$ Clifford geometric algebra. DCGA is an extension of CGA and has entities representing points and general (quartic) Darboux cyclide surfaces in Euclidean 3D space, including circular tori and all quadrics, and all surfaces formed by their inversions in spheres. Dupin cyclides are quartic surfaces formed by inversions in spheres of torus, cylinder, and cone surfaces. Parabolic cyclides are cubic surfaces formed by inversions in spheres that are centered on points of other surfaces. All DCGA entities can be transformed by versors, and reflected in spheres and planes. Keywords: Conformal geometric algebra, Darboux Dupin cyclide, Quadric surface Math. Subj. Class.: 15A66, 53A30, 14J26, 53A05, 51N20, 51K05

This note very briefly describes or sketches the general ideas of some applications of the G(p,q) Geometric Algebra (GA) of a complex vector space C^(p,q) of signature (p,q), which is also known as the Clifford algebra Cl(p,q). Complex number scalars are only used for the anisotropic dilation (directed scaling) operation and to represent infinite distances, but otherwise only real number scalars are used. The anisotropic dilation operation is implemented in Minkowski spacetime as hyperbolic rotation (boost) by an imaginary rapidity (+/-)f = atanh(sqrt(1-d^2)) for dilation factor d>1, using +f in the Minkowski spacetime of signature (1,n) and -f in the signature (n,1). The G(k(p+q+2),k(q+p+2)) Mother Algebra of CGA (k-MACGA) is a generalization of G(p+1,q+1) Conformal Geometric Algebra (CGA) having k orthogonal G(p+1,q+1):p>q Euclidean CGA (ECGA) subalgebras and k orthogonal G(q+1,p+1) anti-Euclidean CGA (ACGA) subalgebras with opposite signature. Any k-MACGA has an even 2k total count of orthogonal subalgebras and cannot have an odd 2k+1 total count of orthogonal subalgebras. The more generalized G(l(p+1)+m(q+1),l (q+1)+m(p+1)):p>q k-CGA algebra, for even or odd k=l+m, has any l orthogonal G(p+1,q+1) ECGA subalgebras and any m orthogonal G(q+1,p+1) ACGA subalgebras with opposite signature. Any 2k-CGA with even 2k orthogonal subalgebras can be represented as a k-MACGA with different signature, requiring some sign changes. All of the orthogonal CGA subalgebras are corresponding by representing the same vectors, geometric entities, and transformation versors in each CGA subalgebra, which may differ only by some sign changes. A k-MACGA or a 2k-CGA has even-grade 2k-vector geometric inner product null space (GIPNS) entities representing general even-degree 2k polynomial implicit hypersurface functions F for even-degree 2k hypersurfaces, usually in a p-dimensional space or (p+1)-spacetime. Only a k-CGA with odd k has odd-grade k-vector GIPNS entities representing general odd-degree k polynomial implicit hypersurface functions F for odd-degree k hypersurfaces, usually in a p-dimensional space or (p+1)-spacetime. In any k-CGA, there are k-blade GIPNS entities representing the usual G(p+1,q+1) CGA GIPNS 1-blade entities, but which are representing an implicit hypersurface function F^k with multiplicity k and the k-CGA null point entity is a k-point entity. In the conformal Minkowski spacetime algebras G(p+1,2) and G(2,p+1), the null 1-blade point embedding is a GOPNS null 1-blade point entity but is a GIPNS null 1-blade hypercone entity.

The G_{8,2} Geometric Algebra, also called the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), has entities that represent conic sections. DCGA also has entities that represent planar sections of Darboux cyclides, which are called cyclidic sections in this paper. This paper presents these entities and many operations on them. Operations include projection, rejection, and intersection with respect to spheres and planes. Other operations include rotation, translation, and dilation. Possible applications are introduced that include orthographic and perspective projections of conic sections onto view planes, which may be of interest in computer graphics or other computational geometry subjects.

Paravectors just like integers have a ring structure. By introducing an integrated product we get geometric properties which make paravectors similar to vectors. The concepts of parallelism, perpendicularity and the angle are conceptually similar to vector counterparts, known from the Euclidean geometry. Paravectors meet the idea of parallelogram law, Pythagorean theorem and many other properties well-known to everyone from school.

An AG-groupoid is an algebraic structure that lies in between a groupoid and a commutative semigroup. It has many characteristics similar to that of a commutative semigroup. If we consider x^2y^2= y^2x^2, which holds for all x, y in a commutative semigroup, on the other hand one can easily see that it holds in an AG-groupoid with left identity e and in AG**-groupoids. This simply gives that how an AG-groupoid has closed connections with commutative algebras. We extend now for the first time the AG-Groupoid to the Neutrosophic AG-Groupoid. A neutrosophic AG-groupoid is a neutrosophic algebraic structure that lies between a neutrosophic groupoid and a neutrosophic commutative semigroup.

In this book, we introduce the concept of (/in, /in /or q_k)-fuzzy ideals and (/in_gamma, in_gamma /or q_delta)-fuzzy ideals in a non-associative algebraic structure called Abel Grassmann’s groupoid, discuss several important features of a regular AG-groupoid, investigate some characterizations of regular and intra-regular AG-groupoids and so on.

Each finite group is a subgroup of some symmetric group, known as the Cayley theorem. We find the symmetric group of smallest order which hosts the finite groups in that sense for most groups of order less than 37. For each of these small groups this is made concrete by providing a permutation group with a minimum number of moved elements in terms of a list of generators of the permutation group in reduced cycle notation.

A substitution map applied to the simplest algebraic identities is shown to yield second- and third-order equations that share an interesting property at the minimum 137.036.

Exactly 125 years ago G. Peano introduced the modern concept of vectors in his 1888 book "Geometric Calculus - According to the Ausdehnungslehre (Theory of Extension) of H. Grassmann". Unknown to Peano, the young British mathematician W. K. Clifford (1846-1879) in his 1878 work "Applications of Grassmann's Extensive Algebra" had already 10 years earlier perfected Grassmann's algebra to the modern concept of geometric algebras, including the measurement of lengths (areas and volumes) and angles (between arbitrary subspaces). This leads currently to new ideal methods for vector field computations in geometric algebra, of which several recent exemplary results will be introduced.

In this paper, we defined the Smarandache seminormal subgroupoids. We have proved some results for finding the Smarandache seminormal subgroupoids in Z(n) when n is even and n is odd.

This paper about indefinite summation describes a natural approach to discrete calculus. Two natural operators for discrete difference and summation are defined. They preserve symmetry and show a duality in contrast to the classical operators. Several summation and differentiation algorithms will be presented.

Quarks are described mathematically by (3 x 3) matrices. To include these quarkonian mathematical structures into Geometric Algebra it is helpful to restate Geometric Algebra in the mathematical language of (3 x 3) matrices. It will be shown in this paper how (3 x 3) permutation matrices can be interpreted as unit vectors. <b> Special emphasis will be given to the definition of some wedge products which fit better to this algebra of (3 x 3) matrices than the usual Geometric Algebra wedge product. </b> And as S3 permutation symmetry is flavour symmetry a unified flavour picture of Geometric Algebra will emerge.

It is briefly shown that, due to the growth conditions in their definition, the Colombeau algebras cannot handle arbitrary Lie groups, and in particular, cannot allow the formulation, let alone, solution of Hilbert's Fifth Problem.

It is briefly shown that, due to the growth conditions in their definition, the Colombeau algebras cannot handle arbitrary analytic nonlinear PDEs, and in particular, cannot allow the formulation, let alone, give the proof of the global Cauchy-Kovalevskaia theorem.

We introduce the notion of Smarandache GT-algebras, and the notion of Smarandache GT-Filters of the Smarandache GT-algebra related to the Tarski algebra, and related some properties are investigated.

We consider the intuitionistic fuzzi?cation of the concept of several Γ-ideals in Γ-LA-semigroup S, and investigate some related properties of such Γ-ideals. We also prove in this paper the set of all intuitionistic fuzzy left(right) Γ-ideal of S is become LA-semigroup. We prove In Γ-LA band intuitionistic fuzzy right and left Γ-ideals are coincide..

Using a personal computer and freely available software, the author factored some members of the Smarandache consecutive sequence and the reverse Smarandache sequence. Nearly complete factorizations are given up to Sm(80) and RSm(80). Both sequences were excessively searched for prime members, with only one prime found up to Sm(840) and RSm(750): RSm(82) = 828180 ... 10987654321.