Utility functions on locally connected spaces

We investigate the role of local connectedness in utility theory and prove that any continuous total preorder on a locally connected separable space is continuously representable. This is a new simple criterion for the representability of continuous preferences, and is not a consequence of the standard theorems in utility theory that use conditions such as connectedness and separability, second countability, or path-connectedness. Finally we give applications to problems involving the existence of value functions in population ethics and to the problem of proving the existence of continuous utility functions in general equilibrium models with land as one of the commodities. (C) 2003 Elsevier B.V. All rights reserved.

Locally convex Riesz spaces and Archimedean quotient spaces

A Riesz space with a Hausdorff, locally convex topology determined by Riesz seminorms is called a locally convex Riesz space. A sequence {x_n} in a locally convex Riesz space L is said to converge locally to x ϵ L if for some topologically bounded set B and every real r ˃ 0 there exists N (r) and n ≥ N (r) implies x – x_n ϵ r^b. Local Cauchy sequences are defined analogously, and L is said to be locally complete if every local Cauchy sequence converges locally. Then L is locally complete if and only if every monotone local Cauchy sequence has a least upper bound. This is a somewhat more general form of the completeness criterion for Riesz – normed Riesz spaces given by Luxemburg and Zaanen. Locally complete, bound, locally convex Riesz spaces are barrelled. If the space is metrizable, local completeness and topological completeness are equivalent.

Two measures of the non-archimedean character of a non-archimedean Riesz space L are the smallest ideal A_o (L) such that quotient space is Archimedean and the ideal I (L) = { x ϵ L: for some 0 ≤ v ϵ L, n |x| ≤ v for n = 1, 2, …}. In general A_o (L) ᴝ I (L). If L is itself a quotient space, a necessary and sufficient condition that A_o (L) = I (L) is given. There is an example where A_o (L) ≠ I (L).

A necessary and sufficient condition that a Riesz space L have every quotient space Archimedean is that for every 0 ≤ u, v ϵ L there exist u₁ = sup (inf (n v, u): n = 1, 2, …), and real numbers m₁ and m₂ such that m₁ u₁ ≥ v₁ and m₂ v₁ ≥ u₁. If, in addition, L is Dedekind σ – complete, then L may be represented as the space of all functions which vanish off finite subsets of some non-empty set.

Structural Saliency: The Detection of Globally Salient Structures Using a Locally Connected Network

Certain salient structures in images attract our immediate attention without requiring a systematic scan. We present a method for computing saliency by a simple iterative scheme, using a uniform network of locally connected processing elements. The network uses an optimization approach to produce a "saliency map," a representation of the image emphasizing salient locations. The main properties of the network are: (i) the computations are simple and local, (ii) globally salient structures emerge with a small number of iterations, and (iii) as a by-product of the computations, contours are smoothed and gaps are filled in.

On Hamilton cycles in locally connected graphs with vertex degree constraints

It is shown that every connected, locally connected graph with the maximum vertex degree Δ(G)=5 and the minimum vertex degree δ(G)3 is fully cycle extendable. For Δ(G)4, all connected, locally connected graphs, including infinite ones, are explicitly described. The Hamilton Cycle problem for locally connected graphs with Δ(G)7 is shown to be NP-complete

On stratifiable locally convex spaces

In the present paper we prove several results on the stratifiability of locally convex spaces. In particular, we show that a free locally convex sum of an arbitrary set of stratifiable LCS is a stratifiable LCS, and that all locally convex F'-spaces whose bounded subsets are metrizable are stratifiable. Moreover, we prove that a strict inductive limit of metrizable LCS is stratifiable and establish the stratifiability of many important general and specific spaces used in functional analysis. We also construct some examples that clarify the relationship between the stratifiability and other properties.

Proximinality in Banach spaces

In this paper, we study approximatively τ-compact and τ-strongly Chebyshev sets, where τ is the norm or the weak topology. We show that the metric projection onto τ-strongly Chebyshev sets are norm-τ continuous. We characterize approximatively τ-compact and τ-strongly Chebyshev hyperplanes and use them to characterize factor reflexive proximinal subspaces in τ-almost locally uniformly rotund spaces. We also prove some stability results on approximatively τ-compact and τ-strongly Chebyshev subspaces.

CLOSED SPACES IN COSMOLOGY

This paper deals with two aspects of relativistic cosmologies with closed spatial sections. These spacetimes are based on the theory of general relativity, and admit a foliation into space sections S(t), which are spacelike hypersurfaces satisfying the postulate of the closure of space: each S(t) is a three-dimensional closed Riemannian manifold. The topics discussed are: (i) a comparison, previously obtained, between Thurston geometries and Bianchi-Kantowski-Sachs metrics for such three-manifolds is here clarified and developed; and (ii) the implications of global inhomogeneity for locally homogeneous three-spaces of constant curvature are analyzed from an observational viewpoint.

Extracting Salient Curves from Images: An Analysis of the Saliency Network

The Saliency Network proposed by Shashua and Ullman is a well-known approach to the problem of extracting salient curves from images while performing gap completion. This paper analyzes the Saliency Network. The Saliency Network is attractive for several reasons. First, the network generally prefers long and smooth curves over short or wiggly ones. While computing saliencies, the network also fills in gaps with smooth completions and tolerates noise. Finally, the network is locally connected, and its size is proportional to the size of the image. Nevertheless, our analysis reveals certain weaknesses with the method. In particular, we show cases in which the most salient element does not lie on the perceptually most salient curve. Furthermore, in some cases the saliency measure changes its preferences when curves are scaled uniformly. Also, we show that for certain fragmented curves the measure prefers large gaps over a few small gaps of the same total size. In addition, we analyze the time complexity required by the method. We show that the number of steps required for convergence in serial implementations is quadratic in the size of the network, and in parallel implementations is linear in the size of the network. We discuss problems due to coarse sampling of the range of possible orientations. We show that with proper sampling the complexity of the network becomes cubic in the size of the network. Finally, we consider the possibility of using the Saliency Network for grouping. We show that the Saliency Network recovers the most salient curve efficiently, but it has problems with identifying any salient curve other than the most salient one.

The Coupled Depth/Slope Approach to Surface Reconstruction

Reconstructing a surface from sparse sensory data is a well known problem in computer vision. Early vision modules typically supply sparse depth, orientation and discontinuity information. The surface reconstruction module incorporates these sparse and possibly conflicting measurements of a surface into a consistent, dense depth map. The coupled depth/slope model developed here provides a novel computational solution to the surface reconstruction problem. This method explicitly computes dense slope representation as well as dense depth representations. This marked change from previous surface reconstruction algorithms allows a natural integration of orientation constraints into the surface description, a feature not easily incorporated into earlier algorithms. In addition, the coupled depth/ slope model generalizes to allow for varying amounts of smoothness at different locations on the surface. This computational model helps conceptualize the problem and leads to two possible implementations- analog and digital. The model can be implemented as an electrical or biological analog network since the only computations required at each locally connected node are averages, additions and subtractions. A parallel digital algorithm can be derived by using finite difference approximations. The resulting system of coupled equations can be solved iteratively on a mesh-pf-processors computer, such as the Connection Machine. Furthermore, concurrent multi-grid methods are designed to speed the convergence of this digital algorithm.

The structure of compact disjointness preserving operators on continuous functions

Let T be a compact disjointness preserving linear operator from C0(X) into C0(Y), where X and Y are locally compact Hausdorff spaces. We show that T can be represented as a norm convergent countable sum of disjoint rank one operators. More precisely, T = Snd ?hn for a (possibly finite) sequence {xn }n of distinct points in X and a norm null sequence {hn }n of mutually disjoint functions in C0(Y). Moreover, we develop a graph theoretic method to describe the spectrum of such an operator

Design of associative memories using cellular neural networks

Cellular neural networks (CNNs) have locally connected neurons. This characteristic makes CNNs adequate for hardware implementation and, consequently, for their employment on a variety of applications as real-time image processing and construction of efficient associative memories. Adjustments of CNN parameters is a complex problem involved in the configuration of CNN for associative memories. This paper reviews methods of associative memory design based on CNNs, and provides comparative performance analysis of these approaches.

Propriedades de tranporte, caos e dissipação num sistema dinâmico não linear

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Class Notes for Math 871: General Topology, Instructor Jamie Radcliffe

Topics include: Topological space and continuous functions (bases, the product topology, the box topology, the subspace topology, the quotient topology, the metric topology), connectedness (path connected, locally connected), compactness, completeness, countability, filters, and the fundamental group.

On d- and D-separability

In this article we study relationships between d-separability and D-separability and present conditions under which these concepts are equivalent. We also study their relationship with D+-separability and define a generalization of discrete generability. (C) 2012 Elsevier B.V. All rights reserved.