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.

Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof

In this paper, we present a new unified approach and an elementary proof of a very general theorem on the existence of a semicontinuous or continuous utility function representing a preference relation. A simple and interesting new proof of the famous Debreu Gap Lemma is given. In addition, we prove a new Gap Lemma for the rational numbers and derive some consequences. We also prove a theorem which characterizes the existence of upper semicontinuous utility functions on a preordered topological space which need not be second countable. This is a generalization of the classical theorem of Rader which only gives sufficient conditions for the existence of an upper semicontinuous utility function for second countable topological spaces. (C) 2002 Elsevier Science B.V. All rights reserved.

Multiscale histograms: Summarizing topological relations in large spatial datasets

Summarizing topological relations is fundamental to many spatial applications including spatial query optimization. In this paper, we present several novel techniques to eectively construct cell density based spatial histograms for range (window) summarizations restricted to the four most important topological relations: contains, contained, overlap, and disjoint. We rst present a novel framework to construct a multiscale histogram composed of multiple Euler histograms with the guarantee of the exact summarization results for aligned windows in constant time. Then we present an approximate algorithm, with the approximate ratio 19/12, to minimize the storage spaces of such multiscale Euler histograms, although the problem is generally NP-hard. To conform to a limited storage space where only k Euler histograms are allowed, an effective algorithm is presented to construct multiscale histograms to achieve high accuracy. Finally, we present a new approximate algorithm to query an Euler histogram that cannot guarantee the exact answers; it runs in constant time. Our extensive experiments against both synthetic and real world datasets demonstrated that the approximate mul- tiscale histogram techniques may improve the accuracy of the existing techniques by several orders of magnitude while retaining the cost effciency, and the exact multiscale histogram technique requires only a storage space linearly proportional to the number of cells for the real datasets.

Hamiltonian mappings and circle packing phase spaces.

We introduce three area preserving maps with phase space structures which resemble circle packings. Each mapping is derived from a kicked Hamiltonian system with one of the three different phase space geometries (planar, hyperbolic or spherical) and exhibits an infinite number of coexisting stable periodic orbits which appear to ‘pack’ the phase space with circular resonances.

The relationship between hetero-oligomer formation and function of the topological specificity domain of the Escherichia coli MinE protein

MinE is an oligomeric protein that, in conjunction with other Min proteins, is required for the proper placement of the cell division site of Escherichia coli. We have examined the self-association properties of MinE by analytical ultracentrifugation and by studies of hetero-oligomer formation in non-denaturing polyacrylamide gets. The self-association properties of purified MinE predict that cytoplasmic MinE is likely to exist as a mixture of monomers and dimers. Consistent with this prediction, the C-terminal MinE(22-88) fragment forms hetero-oligomers with MinE(+) when the proteins are co-expressed. In contrast, the MinE(36-88) fragment does not form MinE(+)/MinE(36-88) hetero-oligomers, although MinE36-88 affects the topological specificity of septum placement as shown by its ability to induce minicell formation when co-expressed with MinE(+) in wild-type cells. Therefore, hetero-oligomer formation is not necessary for the induction of mini-celling by expression of MinE(36-88) in wild-type cells. The interference with normal septal placement is ascribed to competition between MinE(36-88),nd the corresponding domain in the complete MinE protein for a component required for the topological specificity of septal placement.

The dimerization and topological specificity functions of MinE reside in a structurally autonomous C-terminal domain

Correct placement of the division septum in Escherichia coli requires the co-ordinated action of three proteins, MinC, MinD and MinE. MinC and MinD interact to form a non-specific division inhibitor that blocks septation at all potential division sites. MinE is able to antagonize MinCD in a topologically sensitive manner, as it restricts MinCD activity to the unwanted division sites at the cell poles, Here, we show that the topological specificity function of MinE residues in a structurally autonomous, trypsin-resistant domain comprising residues 31-88, Nuclear magnetic resonance (NMR) and circular dichroic spectroscopy indicate that this domain includes both alpha and beta secondary structure, while analytical ultracentrifugation reveals that it also contains a region responsible for MinE homodimerization. While trypsin digestion indicates that the anti-MinCD domain of MinE (residues 1-22) does not form a tightly folded structural domain, NMR analysis of a peptide corresponding to MinE(1-22) indicates that this region forms a nascent helix in which the peptide rapidly interconverts between disordered (random coil) and alpha-helical conformations, This suggests that the N-terminal region of MinE may be poised to adopt an alpha-helical conformation when it interacts with the target of its anti-MinCD activity, presumably MinD.

Unitary R-matrices for topological quantum computing

The main problem with current approaches to quantum computing is the difficulty of establishing and maintaining entanglement. A Topological Quantum Computer (TQC) aims to overcome this by using different physical processes that are topological in nature and which are less susceptible to disturbance by the environment. In a (2+1)-dimensional system, pseudoparticles called anyons have statistics that fall somewhere between bosons and fermions. The exchange of two anyons, an effect called braiding from knot theory, can occur in two different ways. The quantum states corresponding to the two elementary braids constitute a two-state system allowing the definition of a computational basis. Quantum gates can be built up from patterns of braids and for quantum computing it is essential that the operator describing the braiding-the R-matrix-be described by a unitary operator. The physics of anyonic systems is governed by quantum groups, in particular the quasi-triangular Hopf algebras obtained from finite groups by the application of the Drinfeld quantum double construction. Their representation theory has been described in detail by Gould and Tsohantjis, and in this review article we relate the work of Gould to TQC schemes, particularly that of Kauffman.

Topological charge and angular momentum of light beams carrying optical vortices

We analyze the properties of light beams carrying phase singularities, or optical vortices. The transformations of topological charge during free-space propagation of a light wave, which is a combination of a Gaussian beam and a multiple charged optical vortex within a Gaussian envelope, are studied both in theory and experiment. We revise the existing knowledge about topological charge conservation, and demonstrate possible scenarios where additional vortices appear or annihilate during free propagation of such a combined beam. Coaxial interference of optical vortices is also analyzed, and the general rule for angular-momentum density distribution in a combined beam is established. We show that, in spite of any variation in the number of vortices in a combined beam, the total angular momentum is constant during the propagation. [S1050-2947(97)09910-1].