Algèbre géométrique (structure)

Une algèbre géométrique est, en mathématiques, une structure algébrique, similaire à une algèbre de Clifford réelle, mais dotée d'une interprétation géométrique mise au point par David Hestenes, reprenant les travaux de Hermann Grassmann et William Kingdon Clifford (le terme est aussi utilisé dans un sens plus général pour décrire l'étude et l'application de ces algèbres : l'algèbre géométrique est l'étude des algèbres géométriques). Le but avoué de ce physicien théoricien et pédagogue est de fonder un langage propre à unifier les manipulations symboliques en physique, dont les nombreuses branches pratiquent aujourd'hui, pour des raisons historiques, des formalismes différents (tenseurs, matrices, torseurs, analyse vectorielle, utilisation de nombres complexes, spineurs, quaternions, formes différentielles…). Le nom choisi par David Hestenes (geometric algebra) est celui que Clifford voulait donner à son algèbre. L'algèbre géométrique se veut utile dans les problèmes de physique qui

Geometric spects A of Similarity Problems

Peter Clemens Schlicht

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.

OXFORD UNIV PRESS2018

Reductive overgroups of distinguished unipotent elements in simple algebraic groups

Mikko Tapani Korhonen

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

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

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

G

that contain a distinguished unipotent element

u

G

. This leads us to consider connected reductive overgroups

X

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

EPFL2018

STEP-NC compliant process planning of additive manufacturing: remanufacturing

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.

Springer London Ltd2017

Reductive overgroups of distinguished unipotent elements in simple algebraic groups

Mikko Tapani Korhonen

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

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

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)

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

X

G

that contain a distinguished unipotent element

u

G

. This leads us to consider connected reductive overgroups

X

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

(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

EPFL2018