In this handout we shall discuss a result that was conjectured by Emil Artin in 1935 and proved shortly thereafter by Claude Chevalley. A refinement was given by Artin’s graduate student Ewald Warning, who, as the story goes, was the one whom Artin had intended to prove the theorem before Chevalley came visiting Göttingen and got Artin to say a little too much about the mathematics his student ...

In the present paper we prove the main structure theorem for Chevalley groups G = G(Φ, R) of types Φ = E6,E7 over a commutative ring R. More precisely, we describe subgroups in G normalized by the elementary subgroup E(Φ, R). This result is not new, since structure theorems are known for all Chevalley groups [25, 27, 28, 30], [38]–[40], and [58, 61] (see [42, 65, 34, 56] for further references)...

The Jordan decomposition states that a function f : R → R is of bounded variation if and only if it can be written as the difference of two monotone increasing functions. In this paper we generalize this property to real valued BV functions of many variables, extending naturally the concept of monotone function. Our result is an extension of a result obtained by Alberti, Bianchini and Crippa. A...

In this paper, the spectral expression of the frequency-limited H2-norm, firstly presented in [18], is extended to the case of LTI dynamical systems which are not diagonalizable. This extension is achieved by considering partial fraction decomposition of the transfer function associated to a system which is in a Jordan form. Besides, examples are presented to illustrate the behaviour of the fre...

In this paper we establish that the causal order determined by an Ol’shanski semigroup on the corresponding homogeneous space is globally hyperbolic. Using this fact, we present sufficient conditions for a special class of Lie semigroups to admit a canonical “triple decomposition,” namely those for which the Lie algebra is of Cayley type. This theory applies in particular to semigroups which ar...

We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on that of [Th]. We give new results about the stability condition, and propose a Jordan-Hölder-type decomposition of (special) Lagrangians. The main results are the uniqueness of special Lagrangians in hamiltonian deformation classes of Lagrangians, under mild conditions, and a proof of...

In this paper we present a neural network architecture that discovers a recursive decomposition of its input space. Based on a generalization of the modular architecture of Jacobs, Jordan, Nowlan, and Hinton (1991), the architecture uses competition among networks to recursively split the input space into nested regions and to learn separate associative mappings within each region. The learning...