Compact Metrics Research Articles

Compact probabilistic metrics on bounded closed intervals of distribution functions

A Menger space (a special type of probabilistic metric spaces) is said to be compact if its strong uniformity is compact. We construct, in a natural way, Menger T-metrics on the set of distribution functions, one for each copula T (a special type of continuous t-norms). We show that, on each bounded closed interval of distribution functions, the strong uniformities of all our spaces are induced by the modified Lévy metric. Hence, they are compact. We establish an alternative description and a number of good properties of our Menger spaces, which may render them well-behaved examples for workers in the field.

Compact probabilistic metrics on bounded closed intervals of distribution functions

A Menger space (a special type of probabilistic metric spaces) is said to be compact if its strong uniformity is compact. We construct, in a natural way, Menger T -metrics on the set of distribution functions, one for each copula T (a special type of continuous t-norms). We show that, on each bounded closed interval of distribution functions, the strong uniformities of all our spaces are induced by the modified Lévy metric. Hence, they are compact. We establish an alternative description and a number of good properties of our Menger spaces, which may render them well-behaved examples for workers in the field.

Poincaré–Einstein metrics and the Schouten tensor

We examine the space of conformally compact metrics g on the interior of a compact manifold with boundary which have the property that the k th elementary symmetric function of the Schouten tensor A g is constant. When k = 1 this is equivalent to the familiar Yamabe problem, and the corresponding metrics are complete with constant negative scalar curvature. We show for every k that the deformation theory for this problem is unobstructed, so in particular the set of conformal classes containing a solution of any one of these equations is open in the space of all conformal classes. We then observe that the common intersection of these solution spaces coincides with the space of conformally compact Einstein metrics, and hence this space is a finite intersection of closed analytic submanifolds.

ANY COMPACT GROUP IS A GAUGE GROUP

The assignment of the local observables in the vacuum sector, fulfilling the standard axioms of local quantum theory, is known to determine uniquely a compact group G of gauge transformations of the first kind together with a central involutive element k of G, and a complete normal algebra of fields carrying the localizable charges, on which k defines the Bose/Fermi grading. We show here that any such pair {G, k}, where G is compact metrizable, does actually appear. The corresponding model can be chosen to fulfill also the split property. This is not a dynamical phenomenon: a given {G, k} arises as the gauge group of a model where the local algebras of observables are a suitable subnet of local algebras of a possibly infinite product of free field theories.

3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients

Using 3-Sasakian reduction techniques we obtain infinite familiesof new 3-Sasakian manifolds M (p1,p2, p3) andM (p1,p2, p3, p4) in dimension 11 and 15 respectively. The metric cone on (p1,p2, p3) is a generalization ofthe Kronheimer hyperkahler metric on the regular maximalnilpotent orbit of sl (3, C)whereas the cone on M (p1,p2, p3, p4)generalizes the hyperkahler metric onthe 16-dimensional orbit of so(6, C).These are the first examples of 3-Sasakian metrics which are neither homogeneous nor toric. In addition we consider some further U(1)-reductions of M(p1,p2, p3).These yield examples of nontoric 3-Sasakian orbifold metrics in dimensions 7. As a result we obtain explicit families O(Θ) of compact self-dual positivescalar curvature Einstein metrics with orbifoldsingularities and with only one Killing vector field.

Scattering Theory for Conformally Compact Metrics with Variable Curvature at Infinity

We develop the scattering theory of a general conformally compact metric by treating the Laplacian as a degenerate elliptic operator (with non-constant indicial roots) on a compact manifold with boundary. Variability of the roots implies that the resolvent admits only a partial meromorphic continuation, and the bulk of the paper is devoted to studying the structure of the resolvent, Poisson, and scattering kernels for frequencies outside the region of meromorphy. For low frequencies the scattering matrix is shown to be a pseudodifferential operator with frequency dependent domain. In particular, generalized eigenfunctions exhibit L2 decay in directions where the asymptotic curvature is sufficiently negative. We explicitly construct the resolvent kernel for generic frequency in this part of the continuous spectrum.

On some dimensional properties of [formula omitted]-manifolds

It is shown, under the assumption of Jensen's principle ◊, that if for a complex L with [L]≥[S 4] there exists a metrizable compactum whose extension dimension is [L], then there exists a differentiable, countably compact, perfectly normal and hereditarily separable 4-manifold whose extension dimension is also [L].

On conforming axiomatics

We construct conforming axiomatics of the covering dimension dim and the D-dimension (Henderson, 1968) on the class of all metrizable compacta.

Compactifications and universal spaces in extension theory

We show that for each countable simplicial complex P the following conditions are equivalent: (1) $P \in AE(X)$ iff $P \in AE(\beta X)$ for any space X; (2) There exists a P-invertible map of a metrizable compactum X with $P \in AE(X)$ onto the Hilbert cube.

Closed maps on spaces with point-countable bases

We prove a version of Lašnev's theorem for spaces with point-countable bases. Theorem. Let X have a point-countable base, f: X→ Y be a closed map. Then there is an ω 1-relatively-discrete subspace Z of Y such that | Z|≤ d( Y) and f −1( y) is compact for every y∈ Y\\ Z. Then we study the subspaces of closed images of regular spaces with point-countable bases and show that every such subspace has countable π-character and a point-countable π-base. The latter result is extended to a wider class of spaces which is invariant under closed maps and products with metrizable compacta. The proofs use a structure which controls the convergence properties of the space and those of its closed image.

Fuzzy Optimal Control in L2 Space

This article presents a framework for studying the existence of optimal control based on fuzzy rules. The framework consists of two propositions: (1) A set of fuzzy membership functions which are selected out of L2 [a,b] is convex and compact metrizable for the weak topology. (2) The defuzzification function is continuous on the set above. Then, the existence of fuzzy optimal control is proved.

On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties

We construct a family of Kählerian quasi-Einstein metrics with an isometry group U( n) acting linearly on the holomorphic coordinates. Suitable restrictions on the parameters give rise to complete non-compact as well as compact metrics whose geometrical structure is studied in detail. The two-loop renormalizability properties are discussed according to whether one considers the bosonic σ-models or their (2,2) supersymmetric extension.

Compact extremal versus compact Einstein metrics

Using appropriate non-holomorphic coordinates we give an explicit form for Calabi's extremal metrics. Several particular cases of these metrics appear to have been studied later on in the literature. In the four-dimensional case we show, by giving the explicit change in coordinates, that one compact bolt - bolt extremal metric is conformal to Page's compact Einstein metric.

Zero-dimensional covers of finite dimensional dynamical systems

AbstractIf (X, f) is a compact metric, finite-dimensional dynamical system with a zero-dimensional set of periodic points, then there is a zero-dimensional compact metric dynamical system (C, g) and a finite-to-one (in fact, at most (n + l)n-to-one) surjection h: C → X such that h o g = f o h. An example shows that the requirement on the set of periodic points is necessary.

The Power Substitution for Rings of Complex and Real Functions on Compact Metric Spaces

The weak power substitution property for rings of matrices over the ring of functions on a compact metric space X is given in terms of cohomological dimension. A compactum with the ring of complex functions $C(X)$ having the following property is constructed: the units of $C(X)$ are not dense in $C(X)$ and they are dense among squares.

Reduction of Exponential Rank in Direct Limits of C*-Algebras

AbstractWe prove the following result. Let A be a direct limit of direct sums of C*-algebras of the form C(X) ⊗ Mn, with the spaces X being compact metric. Suppose that there is a finite upper bound on the dimensions of the spaces involved, and suppose that A is simple. Then the C* exponential rank of A is at most 1 + ε, that is, every element of the identity component of the unitary group of A is a limit of exponentials. This is true regardless of whether the real rank of A is 0 or 1.

Shape properties of nonmetrizable spaces

We prove a spectral theorem in the SHAPE category. Using it we extend some results concerning shape properties of metrizable compacta to the nonmetrizable case.

Uncountably many pairwise disjoint copies of one metrizable compactum in another

We prove that if K is a compact metrizable space and if X is separable and completely metrizable and contains uncountably many pairwise disjoint homeomorphs of K then X contains a copy of ω 2 × K. We also present applications of this result.

Differentiability Properties of the Pressure of a Continuous Transformation on a Compact Metric Space

We study the topological pressure of a continuous transformation T with finite topological entropy on a compact metric space X. We give a family of examples which have a unique tangent functional at the constant function zero and the unique tangent functional is not ergodic. We obtain several necessary and sufficient conditions for the pressure of T to be Fréchet differentiable at a given f ɛ C(X).

Controlled semi-Markov models under long-run average rewards

Let the state space S be a Borel subset of a complete separable metric space, the action space A compact metric. Existence of stationary optimal policies is proved and a dynamic programming equation derived for general semi-Markov models under the long-run average reward criterion, focusing on it as a limiting case of optimization under discounting as the discount factor goes to one. This extends many earlier results. An example of Reed (1974) on harvesting a natural resource provides an application not covered by earlier results.