In this chapter we give a categorical definition of the integral cohomology ring of a stack. For quotient stacks [X/G] the categorical cohomology ring may be identified with the equivariant cohomology H∗ G(X). Identifying the stack cohomology ring with equivariant cohomology allows us to prove that the cohomology ring of a quotient Deligne-Mumford stack is rationally isomorphic to the cohomolog...

Hecke operators play an important role in the theory of automorphic forms, and automorphic forms are closely linked to various cohomology groups. This paper is mostly a survey of Hecke operators acting on certain types of cohomology groups. The class of cohomology on which Hecke operators are introduced includes the group cohomology of discrete subgroups of a semisimple Lie group, the de Rham c...

Bilinear systems offer a promising approach for nonlinear control because a broad class of nonlinear problems can be reformulated and approximated in bilinear form. System identification is a technique to obtain such a bilinear approximation for a nonlinear system from input-output data. Recent discrete-time bilinear model identification methods rely on Input-Output-to-State Representations (IO...

Bilinear systems offer a promising approach for nonlinear control because a broad class of nonlinear problems can be reformulated in bilinear form. In this paper system identification is shown to be a technique to obtain such a bilinear approximation of a nonlinear system. Recent discrete-time bilinear model identification methods rely on Input-Output-to-State Representations. These IOSRs are e...

The study of discriminants has been a central part of algebraic number theory (c.f. [30]), and has recently led to striking results in arithmetic geometry (e.g. [6], [33], [2]). In this article we summarize two different generalizations ([16], [11]) of discriminants to arithmetic schemes having a tame action by a finite group. We also discuss the results proved in [11, 15, 16] concerning the co...

The third real de Rham cohomology of compact homogeneous spaces is studied. Given $$M=G/K$$ with G semisimple, we first show that each bi-invariant symmetric bilinear form Q on $${\mathfrak {g}} $$ such $$Q|_{{\mathfrak {k}} \times {\mathfrak }=0$$ naturally defines a G-invariant closed 3-form $$H_Q$$ M, which plays the role so called Cartan $$Q([\cdot ,\cdot ],\cdot )$$ Lie group G. Indeed, ev...

Introduction:  The  advent  of  dual-modality  PET/CT  scanners  has  revolutionized  clinical  oncology  by  improving lesion localization and facilitating treatment planning for radiotherapy. In addition, the use of  CT images for CT-based attenuation correction (CTAC) decreases the overall scanning time and creates  a noise-free  attenuation  map  (6map).  CTAC  methods  include  scaling,  s...

Cryptographic protocols depend on the hardness of some computational problems for their security. Joux briefly summarized known relations between assumptions related bilinear map in a sense that if one problem can be solved easily, then another problem can be solved within a polynomial time [6]. In this paper, we investigate additional relations between them. Firstly, we show that the computati...