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Concept

Exterior algebra

In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v , denoted by u \wedge v, is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors. The magnitude of u \wedge v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like t

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Sur quelques équations aux dérivées partielles en lien avec le lemme de Poincaré

David Valentin Strütt

In this thesis, we study two distinct problems. The first problem consists of studying the linear system of partial differential equations which consists of taking a k-form, and applying the exterior derivative 'd' to it and add the wedge product with a 1-form 'a'. The study of this differential operator is linked to the study of the multiplication by a two form, that is the system of linear equations where we take a k-form and apply the exterior wedge product by 'da', the exterior derivative of 'a'. We establish links between the partial differential equation and the linear system. The second problem is a generalization of the symmetric gradient and the curl equation. The equation of a symmetric gradient consists of taking a vector field, apply the gradient and then add the transpose of the gradient, whereas in the curl equation we subtract the transpose of the gradient. Both can be seen as an equation of the form A * grad u + (grad u)t * A, where A is a symmetric matrix for the case of the symmetric gradient and skew symmetric for the curl equation. We generalize to the case where A verifies no symmetry assumption and more significantly add a Dirichlet condition on the boundary.

EPFL2018

Jordan blocks of unipotent elements in some irreducible representations of classical groups in good characteristic

Mikko Tapani Korhonen

Let

G

be a classical group with natural module

V

over an algebraically closed field of good characteristic. For every unipotent element

u

G

, we describe the Jordan block sizes of

u

on the irreducible

G

-modules which occur as composition factors of

V \otimes V^*

\wedge ^2(V)

, and

S^2(V)

. Our description is given in terms of the Jordan block sizes of the tensor square, exterior square, and the symmetric square of

u

, for which recursive formulae are known.

2019

Tensor Products of Weil Modules

Alex Monnard

The aim of this work is to understand the decomposition, in irreducible modules, of tensor products of Weil modules for Sp2n(3). To do this, we begin by computing the Weil modules for Sp4(3), Sp6(3) and Sp8(3) in order to understand how tensor products decompose for these cases. This leads us to some results and hypotheses for the general case. The understanding of the decomposition of two-fold tensor product of Weil modules for Sp2n(3) has been treated by Kay Magaard and Pham Huu Tiep in “Irreducible tensor products of representations of finite quasi-simple groups of Lie type” (to appear in the Proceedings of the Virginia Symposium on Representations of Finite Groups). Following it, we try to understand as much as possible the decomposition of three-fold tensor products of Weil modules for Sp2n(3).

2010

CIVIL-450: Thermodynamics of comfort in buildings

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Study the basics of representation theory of groups and associative algebras.

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L'objectif de ce cours est d'apprendre à réaliser de manière rigoureuse et critique des analyses par éléments finis de problèmes concrets en mécanique des solides à l'aide d'un logiciel CAE moderne.

Differential form

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pione

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In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such

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In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. I