Cohomology with twisted coefficients of the classifying space of a fusion system

We study the cohomology with twisted coefficients of the geometric realization of a linking system associated to a saturated fusion system $\mathcal{F}$. More precisely, we extend a result due to Broto, Levi and Oliver to twisted coefficients. We generalize the notion of $\mathcal{F}$-stable elements to $\mathcal{F}^c$-stable elements in a setting of cohomology with twisted coefficients by an action of the fundamental group.% or, in other word, with locally constant coefficients. We then study the problem of inducing an idempotent from an $\mathcal{F}$-characteristic $(S,S)$-biset and we show that, if the coefficient module is nilpotent, then the cohomology of the geometric realization of a linking system can be computed by $\mathcal{F}^c$-stable elements. As a corollary, we show that for any coefficient module, the cohomology of the classifying space of a $p$-local finite group can be computed by these $\mathcal{F}^c$-stable elements.Comment: 18 pages. Published in Topology and its Application

On graded centres and block cohomology

We extend the group theoretic notions of transfer and stable elements to graded centers of triangulated categories. When applied to the center H∗Db(B)) of the derived bounded category of a block algebra B we show that the block cohomology H∗(B) is isomorphic to a quotient of a certain subalgebra of stable elements of H∗(Db(B)) by some nilpotent ideal, and that a quotient of H∗(Db(B)) by some nilpotent ideal is Noetherian over H∗(B)

Most Stable Elements First Approach

After the introduction of the hypothesis upon which our research is based, we describe the conceptual framework of the methodology we propose. The framework presents an overview of computer-based automation where the most important artefact is a non-formal document describing concepts and requirements. Follows a description of an approach based upon a classification of requirements, concepts, objectives and constraints. Based upon a stability criterion three categories are established: Hard Core, Protective Belt, Fluctuating Elements. The levels of stability impose an order on the implementation. The artefacts for the requirements of the three categories of stability are outlined. The approach named “Most Stable Element Firs

A Generalization of Mathieu Subspaces to Modules of Associative Algebras

We first propose a generalization of the notion of Mathieu subspaces of associative algebras $\mathcal A$, which was introduced recently in [Z4] and [Z6], to $\mathcal A$-modules $\mathcal M$. The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets $\sigma(N)$ and $\tau(N)$ of stable elements and quasi-stable elements, respectively, for all $R$-subspaces $N$ of $\mathcal A$-modules $\mathcal M$, where $R$ is the base ring of $\mathcal A$. We then prove some general properties of the sets $\sigma(N)$ and $\tau(N)$. Furthermore, examples from certain modules of the quasi-stable algebras [Z6], matrix algebras over fields and polynomial algebras are also studied.Comment: A new case has been added; some mistakes and misprints have been corrected. Latex, 31 page

Cohomology of linking systems with twisted coefficients by a pp-solvable action

In this paper we study the cohomology of the geometric realization of linking systems with twisted coefficients. More precisely, given a prime $p$ and a $p$-local finite group $(S,\mathcal{F},\mathcal{L})$, we compare the cohomology of $\mathcal{L}$ with twisted coefficients with the submodule of $\mathcal{F}^c$-stable elements in the cohomology of $S$. We start with the particular case of constrained fusion systems. Then, we study the case of $p$-solvable actions on the coefficients.Comment: 20 pages. a few typos corrected. Published in Homology Homotopy and Application

Stabilized finite element schemes with LBB-stable elements for incompressible flows

AbstractWe study stabilized FE approximations of SUPG type to the incompressible Navier–Stokes problem. Revisiting the analysis for the linearized model, we show that for conforming LBB-stable elements the design of the stabilization parameters for many practical flows differs from that commonly suggested in literature and initially designed for the case of equal-order approximation. Then we analyze a reduced SUPG scheme often used in practice for LBB-stable elements. To provide the reduced scheme with appropriate stability estimates we introduce a modified LBB condition which is proved for a family of FE approximations. The analysis is given for the linearized equations. Numerical experiments for some linear and nonlinear benchmark problems support the theoretical results

Dynamical Networks in Function Dynamics

As a first step toward realizing a dynamical system that evolves while spontaneously determining its own rule for time evolution, function dynamics (FD) is analyzed. FD consists of a functional equation with a self-referential term, given as a dynamical system of a 1-dimensional map. Through the time evolution of this system, a dynamical graph (a network) emerges. This graph has three interesting properties: i) vertices appear as stable elements, ii) the terminals of directed edges change in time, and iii) some vertices determine the dynamics of edges, and edges determine the stability of the vertices, complementarily. Two aspects of FD are studied, the generation of a graph (network) structure and the dynamics of this graph (network) in the system.Comment: 29 pages, 10 figure