Smooth Banach spaces and approximations

If E and F are real Banach spaces let C^p,q(E, F) O ≤ q ≤ p ≤ ∞, denote those maps from E to F which have p continuous Frechet derivatives of which the first q derivatives are bounded. A Banach space E is defined to be C^p,q smooth if C^p,q(E,R) contains a nonzero function with bounded support. This generalizes the standard C^p smoothness classification.

If an L^p space, p ≥ 1, is C^q smooth then it is also C^q,q smooth so that in particular L^p for p an even integer is C^∞,∞ smooth and L^p for p an odd integer is C^p-1,p-1 smooth. In general, however, a C^p smooth B-space need not be C^p,p smooth. C_o is shown to be a non-C^2,2 smooth B-space although it is known to be C^∞ smooth. It is proved that if E is C^p,1 smooth then C_o(E) is C^p,1 smooth and if E has an equivalent C^p norm then c_o(E) has an equivalent C^p norm.

Various consequences of C^p,q smoothness are studied. If f ϵ C^p,q(E,F), if F is C^p,q smooth and if E is non-C^p,q smooth, then the image under f of the boundary of any bounded open subset U of E is dense in the image of U. If E is separable then E is C^p,q smooth if and only if E admits C^p,q partitions of unity; E is C^p,psmooth, p ˂∞, if and only if every closed subset of E is the zero set of some C^P function.

f ϵ C^q(E,F), 0 ≤ q ≤ p ≤ ∞, is said to be C_p,q approximable on a subset U of E if for any ϵ ˃ 0 there exists a g ϵ C^p(E,F) satisfying

sup/xϵU, O≤k≤q ‖ D^k f(x) - D^k g(x) ‖ ≤ ϵ.

It is shown that if E is separable and C^p,q smooth and if f ϵ C^q(E,F) is C_p,q approximable on some neighborhood of every point of E, then F is C_p,q approximable on all of E.

In general it is unknown whether an arbitrary function in C¹(l², R) is C_2,1 approximable and an example of a function in C¹(l², R) which may not be C_2,1 approximable is given. A weak form of C_∞,q, q≥1, to functions in C^q(l², R) is proved: Let {U_α} be a locally finite cover of l² and let {T_α} be a corresponding collection of Hilbert-Schmidt operators on l². Then for any f ϵ C^q(l²,F) such that for all α

sup ‖ D^k(f(x)-g(x))[T_αh]‖ ≤ 1.

xϵU_α,‖h‖≤1, 0≤k≤q

Coincidence point theorems in quasi-metric spaces without assuming the mixed monotone property and consequences in G-metric spaces

In this paper, we present some coincidence point theorems in the setting of quasi-metric spaces that can be applied to operators which not necessarily have the mixed monotone property. As a consequence, we particularize our results to the field of metric spaces, partially ordered metric spaces and G-metric spaces, obtaining some very recent results. Finally, we show how to use our main theorems to obtain coupled, tripled, quadrupled and multidimensional coincidence point results.

Common Fixed Points of Generalized Rational Type Cocyclic Mappings in Multiplicative Metric Spaces

The aim of this paper is to present fixed point result of mappings satisfying a generalized rational contractive condition in the setup of multiplicative metric spaces. As an application, we obtain a common fixed point of a pair of weakly compatible mappings. Some common fixed point results of pair of rational contractive types mappings involved in cocyclic representation of a nonempty subset of a multiplicative metric space are also obtained. Some examples are presented to support the results proved herein. Our results generalize and extend various results in the existing literature.

Approximate Best Proximity Points of Cyclic Self-maps in Metric Spaces and Cyclic Asymptotic Regularity

3rd International Conference on Mathematical Modeling in Physical Sciences (IC-MSQUARE) Madrid, AUG 28-31, 2014 / editado por Vagenas, EC; Vlachos, DS; Bastos, C; Hofer, T; Kominis, Y; Kosmas, O; LeLay, G; DePadova, P; Rode, B; Suraud, E; Varga, K

On contractive cyclic fuzzy maps in metric spaces and some related results on fuzzy best proximity points and fuzzy fixed points

This paper investigates some properties of cyclic fuzzy maps in metric spaces. The convergence of distances as well as that of sequences being generated as iterates defined by a class of contractive cyclic fuzzy mapping to fuzzy best proximity points of (non-necessarily intersecting adjacent subsets) of the cyclic disposal is studied. An extension is given for the case when the images of the points of a class of contractive cyclic fuzzy mappings restricted to a particular subset of the cyclic disposal are allowed to lie either in the same subset or in its next adjacent one.

Convergence Properties and Fixed Points of Two General Iterative Schemes with Composed Maps in Banach Spaces with Applications to Guaranteed Global Stability

This paper investigates the boundedness and convergence properties of two general iterative processes which involve sequences of self-mappings on either complete metric or Banach spaces. The sequences of self-mappings considered in the first iterative scheme are constructed by linear combinations of a set of self-mappings, each of them being a weighted version of a certain primary self-mapping on the same space. The sequences of self-mappings of the second iterative scheme are powers of an iteration-dependent scaled version of the primary self-mapping. Some applications are also given to the important problem of global stability of a class of extended nonlinear polytopic-type parameterizations of certain dynamic systems.

An approach version of fuzzy metric spaces including an ad hoc fixed point theorem

In this paper, inspired by two very different, successful metric theories such us the real view-point of Lowen's approach spaces and the probabilistic field of Kramosil and Michalek's fuzzymetric spaces, we present a family of spaces, called fuzzy approach spaces, that are appropriate to handle, at the same time, both measure conceptions. To do that, we study the underlying metric interrelationships between the above mentioned theories, obtaining six postulates that allow us to consider such kind of spaces in a unique category. As a result, the natural way in which metric spaces can be embedded in both classes leads to a commutative categorical scheme. Each postulate is interpreted in the context of the study of the evolution of fuzzy systems. First properties of fuzzy approach spaces are introduced, including a topology. Finally, we describe a fixed point theorem in the setting of fuzzy approach spaces that can be particularized to the previous existing measure spaces.

Efficient sampling of atomic configurational spaces.

We describe a method to explore the configurational phase space of chemical systems. It is based on the nested sampling algorithm recently proposed by Skilling (AIP Conf. Proc. 2004, 395; J. Bayesian Anal. 2006, 1, 833) and allows us to explore the entire potential energy surface (PES) efficiently in an unbiased way. The algorithm has two parameters which directly control the trade-off between the resolution with which the space is explored and the computational cost. We demonstrate the use of nested sampling on Lennard-Jones (LJ) clusters. Nested sampling provides a straightforward approximation for the partition function; thus, evaluating expectation values of arbitrary smooth operators at arbitrary temperatures becomes a simple postprocessing step. Access to absolute free energies allows us to determine the temperature-density phase diagram for LJ cluster stability. Even for relatively small clusters, the efficiency gain over parallel tempering in calculating the heat capacity is an order of magnitude or more. Furthermore, by analyzing the topology of the resulting samples, we are able to visualize the PES in a new and illuminating way. We identify a discretely valued order parameter with basins and suprabasins of the PES, allowing a straightforward and unambiguous definition of macroscopic states of an atomistic system and the evaluation of the associated free energies.

Projective Limit Random Probabilities on Polish Spaces

A pivotal problem in Bayesian nonparametrics is the construction of prior distributions on the space M(V) of probability measures on a given domain V. In principle, such distributions on the infinite-dimensional space M(V) can be constructed from their finite-dimensional marginals---the most prominent example being the construction of the Dirichlet process from finite-dimensional Dirichlet distributions. This approach is both intuitive and applicable to the construction of arbitrary distributions on M(V), but also hamstrung by a number of technical difficulties. We show how these difficulties can be resolved if the domain V is a Polish topological space, and give a representation theorem directly applicable to the construction of any probability distribution on M(V) whose first moment measure is well-defined. The proof draws on a projective limit theorem of Bochner, and on properties of set functions on Polish spaces to establish countable additivity of the resulting random probabilities.

Reinforcement learning with reference tracking control in continuous state spaces

The contribution described in this paper is an algorithm for learning nonlinear, reference tracking, control policies given no prior knowledge of the dynamical system and limited interaction with the system through the learning process. Concepts from the field of reinforcement learning, Bayesian statistics and classical control have been brought together in the formulation of this algorithm which can be viewed as a form of indirect self tuning regulator. On the task of reference tracking using a simulated inverted pendulum it was shown to yield generally improved performance on the best controller derived from the standard linear quadratic method using only 30 s of total interaction with the system. Finally, the algorithm was shown to work on the simulated double pendulum proving its ability to solve nontrivial control tasks. © 2011 IEEE.

Thermal stratification produced by plumes and jets in enclosed spaces

The airflow and thermal stratification produced by a localised heat source located at floor level in a closed room is of considerable practical interest and is commonly referred to as a 'filling box'. In rooms with low aspect ratios H/R ≲ 1 (room height H to characteristic horizontal dimension R) the thermal plume spreads laterally on reaching the ceiling and a descending horizontal 'front' forms separating a stably stratified, warm upper region from cooler air below. The stratification is well predicted for H/R ≲ 1 by the original filling box model of Baines and Turner (J. Fluid. Mech. 37 (1968) 51). This model represents a somewhat idealised situation of a plume rising from a point source of buoyancy alone-in particular the momentum flux at the source is zero. In practical situations, real sources of heating and cooling in a ventilation system often include initial fluxes of both buoyancy and momentum, e.g. where a heating system vents warm air into a space. This paper describes laboratory experiments to determine the dependence of the 'front' formation and stratification on the source momentum and buoyancy fluxes of a single source, and on the location and relative strengths of two sources from which momentum and buoyancy fluxes were supplied separately. For a single source with a non-zero input of momentum, the rate of descent of the front is more rapid than for the case of zero source momentum flux and increases with increasing momentum input. Increasing the source momentum flux effectively increases the height of the enclosure, and leads to enhanced overturning motions and finally to complete mixing for highly momentum-driven flows. Stratified flows may be maintained by reducing the aspect ratio of the enclosure. At these low aspect ratios different long-time behaviour is observed depending on the nature of the heat input. A constant heat flux always produces a stratified interior at large times. On the other hand, a constant temperature supply ultimately produces a well-mixed space at the supply temperature. For separate sources of momentum and buoyancy, the developing stratification is shown to be strongly dependent on the separation of the sources and their relative strengths. Even at small separation distances the stratification initially exhibits horizontal inhomogeneity with localised regions of warm fluid (from the buoyancy source) and cool fluid. This inhomogeneity is less pronounced as the strength of one source is increased relative to the other. Regardless of the strengths of the sources, a constant buoyancy flux source dominates after sufficiently large times, although the strength of the momentum source determines whether the enclosure is initially well mixed (strong momentum source) or stably stratified (weak momentum source). © 2001 Elsevier Science Ltd. All rights reserved.