it is firstly proved that the multi-input-single-output (miso) fuzzy systems based on interval-valued $r$- and $s$-implications can approximate any continuous function defined on a compact set to arbitrary accuracy.  a formula to compute the lower upper bounds on the number  of interval-valued fuzzy sets needed to achieve a pre-specified approximation  accuracy for an arbitrary multivariate con...

1 Orbital integrals and the Harish-Chandra transform. This section is devoted to a rapid review of some of the basic analysis that is necessary in representation theory and the basic theory of automorphic forms. Even though the material below looks complicated it is just the tip of the iceberg. 1.1 Left invariant measures. Let X be a locally compact topological space with a countable basis for ...

We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the multiplicity one case and extends to higher multiplicity their dilation approach. We prove several results for operator-valued frames concerning duality, disjointeness, complementar...

"&('*),+/.0 1 )213&4 a.e. 576 Our main results assert that is path-connected if 8!" :9 while it has the same homotopy classes as ;=<> *? @ if 8!BA 9 . We also present some results and open problems about density of smooth maps in C D E? F . Sobre algunas cuestiones topológicas para espacios de Sobolev fracionarios con valores en GIH Resumen. El propósito de este artı́culo es describir las clase...

Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F ⊆ X and every point x ∈ X \ F , there exist f ∈ Cp(X,G) and g ∈ G \ {e} such that f(x) = g and f(F ) ⊆ {e}; (b) G-regular provided that t...

In this paper we review the definition of the invariants “bar codes” and “Jordan cells” of real and angle valued tame maps as proposed in [1] and [4] and prove the homotopy invariance of the sums ]Bc r + ]Bo r−1 and of the set of Jordan cells. Here Bc r resp. Bo r denote the sets of closed resp. open bar codes in dimension r. In addition we provide calculation of some familiar topological invar...

Using graph representations a new class of computable topological invariants associated with a tame real or angle valued map were recently introduced, providing a theory which can be viewed as an alternative to MorseNovicov theory for real or angle valued Morse maps. The invariants are ”barcodes” and ”Jordan cells”. From them one can derive all familiar topological invariants which can be deriv...

Let G be a finite connected graph, and let ∆ be its universal covering tree. The edges of ∆ are directed and each geometric edge corresponds to two directed edges δ and δ. Let ∆ be the set of vertices and ∆ the set of directed edges of ∆. The boundary ∂∆ is the set of equivalence classes of infinite semi-geodesics in ∆, where two semi-geodesics are equivalent if they agree except on finitely ma...

We study circle valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real valued maps, circle valued maps enjoy a different class of invariants called Jordan cell...

We study circle valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real valued maps, circle valued maps enjoy a different class of invariants called Jordan cell...