Bornological structures on many-valued sets

We introduce an approach to the concept of bornology in the framework of many-valued mathematical structures and develop the basics of the theory of many-valued bornological spaces and initiate the study of the category of many-valued bornological spaces and appropriately defined bounded "mappings" of such spaces. A scheme for constructing many-valued bornologies with prescribed properties is worked out. In particular, this scheme allows to extend an ordinary bornology of a metric space to a many-valued bornology on it

Extensions of tempered representations

Let $\pi, \pi'$ be irreducible tempered representations of an affine Hecke algebra H with positive parameters. We compute the higher extension groups $Ext_H^n (\pi,\pi')$ explicitly in terms of the representations of analytic R-groups corresponding to $\pi$ and $\pi'$. The result has immediate applications to the computation of the Euler-Poincar\'e pairing $EP(\pi,\pi')$, the alternating sum of the dimensions of the Ext-groups. The resulting formula for $EP(\pi,\pi')$ is equal to Arthur's formula for the elliptic pairing of tempered characters in the setting of reductive p-adic groups. Our proof applies equally well to affine Hecke algebras and to reductive groups over non-archimedean local fields of arbitrary characteristic. This sheds new light on the formula of Arthur and gives a new proof of Kazhdan's orthogonality conjecture for the Euler-Poincar\'e pairing of admissible characters.Comment: This paper grew out of "A formula of Arthur and affine Hecke algebras" (arXiv:1011.0679). In the second version some minor points were improve

Periodic behaviors

This paper studies behaviors that are defined on a torus, or equivalently, behaviors defined in spaces of periodic functions, and establishes their basic properties analogous to classical results of Malgrange, Palamodov, Oberst et al. for behaviors on R^n. These properties - in particular the Nullstellensatz describing the Willems closure - are closely related to integral and rational points on affine algebraic varieties.Comment: 13 page

Fuzzifying completeness and compactness in fuzzifying bornological linear spaces

The notions of completeness and compactness play important role in classical functional analysis. The main purpose of this paper is to generalize these notions to the setting of fuzzifying bornological linear spaces. At first, the concepts of fuzzifying Cauchy sequences and fuzzifying completeness are introduced and some interesting properties of them are studied. The relationships among fuzzifying completeness, separation axiom and fuzzifying bornological closed set are discussed. Then the notions of fuzzifying compactness and precompactness are presented, several properties of them are discussed. Particularly, it is demonstrated that a subset is fuzzifying bornological compact if and only if it is fuzzifying bornological precompact and bornological complete