A version of the Stone-Weierstrass theorem in fuzzy analysis

Let C ( K , E 1 ) be the space of continuous functions defined between a compact Hausdorff space K and the space of fuzzy numbers E 1 endowed with the supremum metric. We provide a set of sufficient conditions on a subspace of C ( K , E 1 ) in order that it be dense. We also obtain a similar result for interpolating families of C ( K , E 1 ) . As a corollary of the above results we prove that certain fuzzy-number-valued neural networks can approximate any continuous fuzzy-number-valued function defined on a compact subspace of R

Completeness, metrizability and compactness in spaces of fuzzy-number-valued functions

Fuzzy-number-valued functions, that is, functions defined on a topological space taking values in the space of fuzzy numbers, play a central role in the development of Fuzzy Analysis. In this paper we study completeness, metrizability and compactness of spaces of continuous fuzzy-number-valued functions

Sequentially compact subsets and monotone functions: An application to fuzzy theory

Let (X,<,τO) be a first countable compact linearly ordered topological space. If (Y,D) is a uniform sequentially compact linearly ordered space with weight less than the splitting number s, then we characterize the sequentially compact subsets of the space M(X,Y) of all monotone functions from X into Y endowed with the topology of the uniform convergence induced by the uniformity D. In particular, our results are applied to identify the compact subsets of M([0,1],Y) for a wide class of linearly ordered topological spaces, including Y=R. This allows us to provide a characterization of the compact subsets of an extended version of the fuzzy number space (with the supremum metric) where the reals are replaced by certain linearly ordered topological spaces, which corrects some characterizations which appear in the literature. Since fuzzy analysis is based on the notion of fuzzy number just as much as classical analysis is based on the concept of real number, our results open new possibilities of research in this field

Bilinear isometries on spaces of vector-valued continuous functions

Let X, Y, Z be compact Hausdorff spaces and let E1, E2, E3 be Banach spaces. If T:C(X,E1)×C(Y,E2)→C(Z,E3) is a bilinear isometry which is stable on constants and E3 is strictly convex, then there exist a nonempty subset Z0 of Z, a surjective continuous mapping h:Z0→X×Y and a continuous function ω:Z0→Bil(E1×E2,E3) such that T(f,g)(z)=ω(z)(f(πX(h(z))),g(πY(h(z)))) for all z∈Z0 and every pair (f,g)∈C(X,E1)×C(Y,E2). This result generalizes the main theorems in Cambern (1978) [2] and Moreno and Rodríguez (2005) [7]

Fully covariant and conformal formulation of the Z4 system in a reference-metric approach: comparison with the BSSN formulation in spherical symmetry

We adopt a reference-metric approach to generalize a covariant and conformal version of the Z4 system of the Einstein equations. We refer to the resulting system as ``fully covariant and conformal", or fCCZ4 for short, since it is well suited for curvilinear as well as Cartesian coordinates. We implement this fCCZ4 formalism in spherical polar coordinates under the assumption of spherical symmetry using a partially-implicit Runge-Kutta (PIRK) method and show that our code can evolve both vacuum and non-vacuum spacetimes without encountering instabilities. Our method does not require regularization of the equations to handle coordinate singularities, nor does it depend on constraint-preserving outer boundary conditions, nor does it need any modifications of the equations for evolutions of black holes. We perform several tests and compare the performance of the fCCZ4 system, for different choices of certain free parameters, with that of BSSN. Confirming earlier results we find that, for an optimal choice of these parameters, and for neutron-star spacetimes, the violations of the Hamiltonian constraint can be between 1 and 3 orders of magnitude smaller in the fCCZ4 system than in the BSSN formulation. For black-hole spacetimes, on the other hand, any advantages of fCCZ4 over BSSN are less evident.Comment: 13 pages, 10 figure

Multilinear isometries on function algebras

Let be function algebras (or more generally, dense subspaces of uniformly closed function algebras) on locally compact Hausdorff spaces , respectively, and let Z be a locally compact Hausdorff space. A -linear map is called a multilinear (or k-linear) isometry if (Formula presented.) Based on a new version of the additive Bishop’s Lemma, we provide a weighted composition characterization of such maps. These results generalize the well-known Holsztyński’s theorem and the bilinear version of this theorem provided in Moreno and Rodríguez [Studia Math. 2005;166:83–91] by a different approach.Research of J.J. Font and M. Sanchis was partially supported by the Spanish Ministry of Science and Education [grant number MTM2011-23118], and by Bancaixa [Projecte P11B2011-30]