Concept# Geometric algebra

Summary

In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions.
The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). Clifford defined the Clifford algebra and its product as a unification of the Grassmann algebra and Hamilton's quate

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related people (4)

Related concepts (81)

Cross product

In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented

Quaternion

In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in thre

Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body

Related publications (38)

Loading

Loading

Loading

Related courses (24)

MATH-341: Linear models

Regression modelling is a fundamental tool of statistics, because it describes how the law of a random variable of interest may depend on other variables. This course aims to familiarize students with linear models and some of their extensions, which lie at the basis of more general regression model

MATH-111(en): Linear algebra (english)

The purpose of the course is to introduce the basic notions of linear algebra and its applications.

MATH-408: Regression methods

General graduate course on regression methods

Related units (3)

Related lectures (29)

This article presents a geometric approach to some similarity problems involving metric arguments in the non-positively curved space of positive invertible operators of an operator algebra and the canonical isometric action by invertible elements on the cone given by g . a = gag*. Through this approach, we extend and put into a geometric framework results by Pisier and partially answer a question by Andruchow et al. about minimality properties of canonical unitarizers.

Ian Anthony Stroud, Jumyung Um

Additive manufacturing is becoming one of the key methods for reproducing repair sections in remanufacturing processes. The major advantage of using additive processes is to minimize production time and waste. However, the surface quality and shape accuracy are usually insufficient for the final product because the approximated representation format causes the accumulation of the error during the geometric operations of the process planning. This limitation is a barrier to utilize additive processes as finishing processes, such as general metal cutting. There is need to improve the final quality of parts obtained with additive manufacturing. In this paper, STEP-based numerical control (STEP-NC)-based process planning is applied to the additive manufacturing. ISO 14649 (STEP-NC) describes part programs with geometric data directly and also contains the information necessary for the intelligent process planning. This paper proposes the STEP-NC-based representation method of additive manufacturing and the series of geometric reasoning to automate the derivation of the repair section. The proposed representation has the benefits to provide a high accuracy for the final surface and to describe multiple materials. Topological data maintain low error during the series of process planning through the CAD-CAM-CNC chain. The proposed platform supports consideration of the process tolerance and comparison of the selected plan with alternative processes. In order to show the practical advantages, an analysis of the remanufacturing process is carried out. The case study of remanufacturing a pocket part is presented in order to validate the proposed process plan. The result of the case study shows the improvement in terms of automatic process planning and surface quality accuracy.

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geq 0$. In this thesis, we investigate closed connected reductive subgroups $X < G$ that contain a given distinguished unipotent element $u$ of $G$. Our main result is the classification of all such $X$ that are maximal among the closed connected subgroups of $G$.
When $G$ is simple of exceptional type, the result is easily read from the tables computed by Lawther (J. Algebra, 2009). Our focus is then on the case where $G$ is simple of classical type, say $G = \operatorname{SL}(V)$, $G = \operatorname{Sp}(V)$, or $G = \operatorname{SO}(V)$. We begin by considering the maximal closed connected subgroups $X$ of $G$ which belong to one of the families of the so-called \emph{geometric subgroups}. Here the only difficult case is the one where $X$ is the stabilizer of a tensor decomposition of $V$. For $p = 2$ and $X = \operatorname{Sp}(V_1) \otimes \operatorname{Sp}(V_2)$, we solve the problem with explicit calculations; for the other tensor product subgroups we apply a result of Barry (Comm. Algebra, 2015).
After the geometric subgroups, the maximal closed connected subgroups that remain are the $X < G$ such that $X$ is simple and $V$ is an irreducible and tensor indecomposable $X$-module. The bulk of this thesis is concerned with this case. We determine all triples $(X, u, \varphi)$ where $X$ is a simple algebraic group, $u \in X$ is a unipotent element, and $\varphi: X \rightarrow G$ is a rational irreducible representation such that $\varphi(u)$ is a distinguished unipotent element of $G$. When $p = 0$, this was done in previous work by Liebeck, Seitz and Testerman (Pac. J. Math, 2015).
In the final chapter of the thesis, we consider the more general problem of finding all connected reductive subgroups $X$ of $G$ that contain a distinguished unipotent element $u$ of $G$. This leads us to consider connected reductive overgroups $X$ of $u$ which are contained in some proper parabolic subgroup of $G$. Testerman and Zalesski (Proc. Am. Math. Soc, 2013) have shown that when $u$ is a regular unipotent element of $G$, no such $X$ exists. We give several examples which show that their result does not generalize to distinguished unipotent elements. As an extension of the Testerman-Zalesski result, we show that except for two known examples which occur in the case where $(G, p) = (C_2, 2)$, a connected reductive overgroup of a distinguished unipotent element of order $p$ cannot be contained in a proper parabolic subgroup of $G$.