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[6] viXra:2312.0128 [pdf] replaced on 2024-01-06 10:00:52
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
[5] viXra:2312.0117 [pdf] submitted on 2023-12-21 03:09:50
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
[4] viXra:2312.0085 [pdf] replaced on 2023-12-20 20:58:31
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
[3] viXra:2312.0051 [pdf] submitted on 2023-12-09 11:35:02
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
[2] viXra:2312.0030 [pdf] submitted on 2023-12-05 01:39:32
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
[1] viXra:2312.0025 [pdf] submitted on 2023-12-05 17:06:25
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
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