Topics in Geometric Algebra over Rings

Abstract

This paper generalizes former versions of the so-called fundamental theorem of projective geometry. It shows that generalized semilinear maps between vector spaces correspond to certain partial lineations between the associated projective spaces. In particular, it gives a characterization of all nondegenerate lineations between arguesian projective spaces.

The fundamental theorem of projective geometry

Structural Properties of Models of N1-Categorical Theories

Chapter 2 - Projective and Affine Geometry over Division Rings

Several different generalizations of finite factorization domains (i.e., integral domains where every nonzero nonunit has only finitely many divisors up to associates) have been defined for commutative rings with zero divisors. We study these notions in the context of commutative monoid rings with zero divisors, utilizing semigroup theory to simultaneously generalize and extend many past results about “finite factorization” properties in commutative polynomial rings. Along the way, we expand upon the general theory of factorization in commutative rings with zero divisors, providing new characterizations and results about several kinds of “finite factorization rings.”

Given a quadratic form on a vector space, the geometric algebra of the corresponding pseudo-euclidean space is defined in terms of a simple set of rules which characterizes the geometric product of vectors. We develop geometric algebra in such a way that it augments, but remains fully compatible with, the more traditional tools of matrix algebra. Indeed, matrix multiplication arises naturally from the geometric multiplication of vectors by introducing a spectral basis of mutually annihiliating idempotents in the geometric algebra. With the help of a few more algebraic identities, and given the proper geometric interpretation, the geometric algebra can be applied to the study of affine, projective, conformal and other geometries. The advantage of geometric algebra is that it provides a single algebraic framework with a comprehensive, but flexible, geometric interpretation. For example, the affine plane of rays is obtained from the euclidean plane of points by adding a single anti-commuting vector to the underlying vector space. The key to the study of noneuclidean geometries is the definition of the operations of meet and join, in terms of which incidence relationships are expressed. The horosphere provides a homogeneous model of euclidean space, and is obtained by adding a second anti-commuting vector to the underlying vector space of the affine plane. Linear orthogonal transformations on the higher dimensional vector space correspond to conformal or Mobius transformations on the horosphere. The horosphere was first constructed by F.A. Wachter (1792–1817), but has only recently attracted attention byo ering a host of new computational tools in projective and hyperbolic geometries when formulated in terms of geometric algebra.

Let R be a commutative ring with identity. A short exact sequence of R-modules is said to be neat (respectively, -pure) if the sequence (respectively, ) is exact for every simple R-module S. The characterization of the integral domains over which neatness and -purity coincide has been given by László Fuchs: they are the integral domains where every maximal ideal is projective (and so also finitely generated). We show that for a commutative ring R, every maximal ideal of R is finitely generated and projective if and only if R has projective socle and neatness and -purity coincide. For a commutative ring R, we prove that neatness and -purity coincide if and only if every maximal ideal of R is finitely generated and the unique maximal ideal PP of the local ring RP is a principal ideal for every maximal ideal P of R. This result is proved firstly over commutative local rings and then using localization over any commutative ring. The Auslander-Bridger transpose of simple modules is used in proving these equivalences because it is an essential tool for the passage between proper classes of short exact sequences of modules that are projectively generated and these that are flatly generated by a set of finitely presented modules. For a commutative ring R, we prove that neatness and -purity coincide if and only if every simple R-module S is finitely presented and an Auslander-Bridger transpose of S is projectively equivalent to S.

Over the last 20 years, our learning environment for linear algebra has changed dramatically mathematical tools take an important role in our classes. Sage is popular mathematical software which was released in 2005. This software has efficient features to adapt the internet environment and it can cover most of mathematical problems, for example, algebra, combinatorics, numerical mathematics, calculus and linear algebra. Nowadays there are more mobile/smartphones than the number of personal computers in the world. Furthermore, the most sophisticated smartphones have almost the same processing power as personal computer and it can be connected to the internet. For example, we can connect from mobile phone to any Sage server through the internet. We have developed over the years on Mobile mathematics with Smartphone for teaching linear algebra (Ko et al. 2009; Lee and Kim 2009; Lee et al. 2001). In this article, we introduce Sage and how we can use it in our linear algebra classes. We aim to show the mobile infrastructure of the Sage and the mobile-learning environment. We shall also introduce mobile contents for linear algebra using Sage. In fact, almost all the concepts of linear algebra can be easily covered.

Let R R be a semiprime K K -algebra with unity, d d a nonzero derivation of R R , and f ( x 1 , … , x t ) f({x_1}, \ldots ,{x_t}) a monic multilinear polynomial over K K such that d ( f ( a 1 , … , a t ) ) ≠ 0 d(f({a_1}, \ldots ,{a_t})) \ne 0 for some a 1 , … , a t ∈ R {a_1}, \ldots ,{a_t} \in R . It is shown that if for every r 1 , … , r t {r_1}, \ldots ,{r_t} in R R either d ( f ( r 1 , … , r t ) ) = 0 d(f({r_1}, \ldots ,{r_t})) = 0 or d ( f ( r 1 , … , r t ) ) d(f({r_1}, \ldots ,{r_t})) is invertible in R R , then R R is either a division ring D D or M 2 ( D ) {M_2}(D) , the ring of 2 × 2 2 \times 2 matrices over D D , unless f ( x 1 , … , x t ) f({x_1}, \ldots ,{x_t}) is a central polynomial for R R . Moreover, if R = M 2 ( D ) R = {M_2}(D) , where 2 R ≠ 0 2R \ne 0 and f ( x 1 , … , x t ) f({x_1}, \ldots ,{x_t}) is not a central polynomial for D D , then d d is an inner derivation of R R .

We consider twisted conjugacy classes and the R ∞-property for classical linear groups. In particular, it is stated that the general linear group GL n (K) and the special linear group SL n (K), where $$ n \geqslant 3 $$ , possess the R ∞-property if either K is an infinite integral domain with trivial automorphism group or K is an integral domain containing a subring of integers, whose automorphism group Aut(K) is finite. By an integral domain we mean a commutative ring with identity which has no zero divisors.

Torsion theories over commutative rings

Geometric Algebra (GA) is an algebra that encodes geometry much better than standard techniques, which are mainly based on linear algebra with various extensions. Compared to standard techniques, GA has clearer semantics and a richer, more consistent language. This expresses itself, among others, in a much greater genericity of functions over the algebra. Exploiting this genericity efficiently is a problem that can be solved through generative programming.This paper describes our Geometric Algebra Implementation Generator Gaigen 2. Gaigen 2 synthesizes highly efficient GA implementations from the specification of the algebra. Functions over such algebras can be defined in a high-level coordinate-free domain-specific language, and Gaigen 2 transforms these functions into low-level coordinate-based code. This code can be emitted in any target language through a custom back-end. Benchmarks of our implementation show that the combination of GA and Gaigen 2 can rival the performance of standard geometry techniques, despite the greater abstraction and genericity of GA.To obtain this high performance, Gaigen 2 must adapt the generated code to the program that links to it. This is done via a profiling feedback loop. While running, the generated code makes a connection to the code generator. The generated code sends information about functions that should be optimized. The code generator registers this information and sends back new type information. After the program terminates, the code is regenerated according to the recorded profile. This profiling feedback technique may also be useful to implement other types of algebras.

SummaryThis paper presents and discusses the usage of Geometric Algebra (GA) for the analysis of electrical alternating current (AC) circuits. The potential benefits of this novel approach are highlighted in the study of linear and nonlinear circuits with sinusoidal and non‐sinusoidal sources in the frequency domain, which are important issues in electrical engineering undergraduate courses. The analysis and understanding of how AC circuits operate in steady state are of a paramount importance for all the electrical engineers and practitioners around the world. Typically, lecturers of most undergraduate courses teach circuit theory using complex phasors, vector calculus, or linear algebra. However, these approaches have some important limitations in practice, which requires the development of strategies to improve teaching‐learning process related to AC circuit analysis. By formulating a new mathematical framework, the paper presents and discusses how GA can be of help in this approach. It is also described how the results obtained by using GA are validated using computer‐based simulations. It is highlighted how the proposed teaching methodology based on GA theory can be effective in helping students learn AC circuit analysis since it has several potential benefits derived from its simplicity, compactness, and use of easily identifiable geometrical concepts.

This paper presents the validation of a novel code for the dynamic analysis of flexible multibody systems, which leverages projective geometric algebra for the kinematic description of the system. The theoretical foundations of geometric algebra are introduced briefly. A closed-form solution for the interpolation of motion is derived within the proposed framework and a finite element formulation of geometrically exact beams is developed. The accuracy of this code is assessed by comparing its prediction with those of eight multibody dynamics codes developed by independent researchers. Results are presented for three benchmark problems: the flexible four-bar mechanism, the lateral buckling of a thin beam, and the stability of a rotating shaft. Excellent agreement is found between the predictions of the proposed code and the previously published results, confirming the accuracy of the PGA-based approach. These findings demonstrate the potential of geometric algebra as a robust and elegant basis for the analysis of complex multibody systems.