Let (R,m) be a commutative Noetherian local ring. We establish some bounds for the sequence of Bass numbers and their dual for a finitely generated R-module. Introduction Throughout this paper, (R,m, k) is a non-trivial commutative Noetherian local ring with unique maximal ideal m and residue field k. Several authors have obtained results on the growth of the sequence of Betti numbers {βn(k)} (...

1. Fréchet space C∞[a, b] 2. Fréchet space of smooth functions E = C∞(Rn) 3. Fréchet space of Schwartz functions S on R 4. LF-space of test functions D on R 5. Generalized functions (distributions) D∗, S ∗, E∗ as linear functionals 6. Generalized functions D∗, S ∗, E∗ as completions of D 7. Fourier transforms of tempered distributions 8. Distributions supported at a point 9. Appendix: weak dual...

It is no exaggeration to say that the theory of separably injective spaces is quite different from that of injective spaces. In this chapter we will explain why. Indeed, we will enter now in the main topic of the monograph, namely, separably injective spaces and their “universal” version. After giving the main definitions and taking a look at the first natural examples one encounters, we presen...

Let K be a compact Lie group acting on a finite dimensional Hermitian vector space V via some unitary representation. Now K acts by automorphisms on the associated Heisenberg group HV = V × R and we say that (K,HV ) is a Gelfand pair when the algebra LK(HV ) of integrable K-invariant functions on HV commutes under convolution. In this situation an application of the Orbit Method yields a inject...

A realization of a graph G is pair (G,p) where p maps the vertices into Euclidean space Rd. The injective if and quasi-injective for each edge G, endpoints to different points in space. globally rigid any (G,q) Rd with same lengths congruent (G,p). In this paper we characterize graphs that have an (quasi-injective, respectively) non-globally R1, show problem recognizing these NP-complete both c...

The notion of approximations relative to a functor is introduced and several characterizations of relative (dual) Maschke functors are given by using them. As an application, the injective objects in the category of comodules over a coring are described.

It is no exaggeration to say that the theory of separably injective spaces is quite different from that of injective spaces. In this chapter we will explain why. Indeed, we will enter now in the main topic of the monograph, namely, separably injective spaces and their “universal” version. After giving the main definitions and taking a look at the first natural examples one encounters, we presen...

In the category Top0 of T0-spaces and continuous maps, embeddings are just those morphisms with respect to which the Sierpiński space is Kan-injective, and the Kan-injective hull of the Sierpiński space is the category of continuous lattices and maps preserving directed suprema and arbitrary infima. In the category Loc of locales and localic maps, we give an analogous characterization of flat e...

In this paper we consider some finite generalized polygons, defined over a field with characteristic 2, that admit an embedding in a projective or affine space over a field with characteristic unequal to 2. In particular, we classify the (lax) embeddings of the unique generalized quadrangle H(3, 4) of order (4, 2). We also classify all (lax) embeddings of both the split Cayley hexagon H(2) and ...

In this paper, we introduce and characterize the concept of fuzzy almost generalized $e$-continuous mappings. Several interesting properties of these mappings are also given. Examples and counter examples are also given to illustrate the concepts introduced in the paper. We also introduce the concept of fuzzy $f T_{frac{1}{2}}e$-space, fuzzy $ge$-space, fuzzy regular $ge$-space and  fuzzy gener...