Let G be a reductive algebraic group associated to a self-adjoint homogeneous cone defined over Q, and let Γ ⊂ G be a appropriate neat arithmetic subgroup. We present two algorithms to compute the action of the Hecke operators on H(Γ;Z) for all i. This simultaneously generalizes the modular symbol algorithm of Ash-Rudolph [7] to a larger class of groups, and provides techniques to compute the H...

If a Quillen model category is defined via a suitable right adjoint over a sheafifiable homotopy model category (in the sense of part I of this paper), it is sheafifiable as well; that is, it gives rise to a functor from the category of topoi and geometric morphisms to Quillen model categories and Quillen adjunctions. This is chiefly useful in dealing with homotopy theories of algebraic structu...

Let G be a reductive algebraic group associated to a self-adjoint homogeneous cone defined over Q, and let Γ ⊂ G be a appropriate neat arithmetic subgroup. We present two algorithms to compute the action of the Hecke operators on H(Γ;Z) for all i. This simultaneously generalizes the modular symbol algorithm of Ash-Rudolph [7] to a larger class of groups, and provides techniques to compute the H...

We show that considering labelled transition systems as relational presheaves captures several recently studied examples in a general setting. This approach takes into account possible algebraic structure on labels. We show that left (2-)adjoints to change-of-base functors between categories of relational presheaves with relational morphisms always exist and, as an application, that weak closur...

Let G be a simple, simply connected and connected algebraic group over an algebraically closed field of characteristic p > 0, and let V be a rational G-module such that dim V ≤ p. According to a result of Jantzen, V is completely reducible, and H(G, V ) = 0. In this paper we show that H(G, V ) = 0 unless some composition factor of V is a non-trivial Frobenius twist of the adjoint representation...

We define and study a family of partitions of the wonderful compactification G of a semi-simple algebraic group G of adjoint type. The partitions are obtained from subgroups of G × G associated to triples (A1, A2, a), where A1 and A2 are subgraphs of the Dynkin graph Γ of G and a : A1 → A2 is an isomorphism. The partitions of G of Springer and Lusztig correspond respectively to the triples (∅, ...

Let H and Y be separable Hilbert spaces, U finite dimensional. Let A ∈ L(H), B ∈ L(U, H), C ∈ L(H,Y ), D ∈ L(U,Y ), and suppose that the open loop transfer function D(z) := D + zC(I − zA)B ∈ H∞(U, Y ). Let J ≥ 0 be a cost operator. We study a subset of self adjoint solutions P of the discrete time algebraic Riccati equation (DARE)

The aim of this note is to classify all weakly closed unipotent subgroups in the split Chevalley groups. In an application we show under some mild assumptions on the characteristic that 2 dimX + dim cg(X) < dim g for X a non-trivial unipotent subgroup of the connected simple algebraic group G. This shows the failure of the analogue of the so called “2F-condition” for finite groups for the adjoi...

This paper provides a description of an algebraic setting for the Lagrangian formalism over graded algebras and is intended as the necessary first step towards the noncommutative C-spectral sequence (variational bicomplex). A noncommutative version of integration procedure, the notion of adjoint operator, Green’s formula, the relation between integral and differential forms, conservation laws, ...

This paper provides a description of an algebraic setting for the Lagrangian formalism over graded algebras and is intended as the necessary rst step towards the noncommutative C-spectral sequence (variational bicomplex). A noncommutative version of integration procedure, the notion of adjoint operator, Green's formula, the connection between integral and di erential forms, conservation laws, E...