97,841 research outputs found

    Mereological foundations of point-free geometry via multi-valued logic

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    We suggest possible approaches to point-free geometry based on multi-valued logic. The idea is to assume as primitives the notion of a region together with suitable vague predicates whose meaning is geometrical in nature, e.g. ‘close’, ‘small’, ‘contained’. Accordingly, some first-order multi-valued theories are proposed. We show that, given a multi-valued model of one of these theories, by a suitable definition of point and distance we can construct a metrical space in a natural way. Taking into account that interesting metrical approaches to geometry exist, this looks to be promising for a point-free foundation of the notion of space. We hope also that this way to face point-free geometry provides a tool to illustrate the passage from a naïve and ‘qualitative’ approach to geometry to the ‘quantitative’ approach of advanced science

    Spatial Aggregation: Theory and Applications

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    Visual thinking plays an important role in scientific reasoning. Based on the research in automating diverse reasoning tasks about dynamical systems, nonlinear controllers, kinematic mechanisms, and fluid motion, we have identified a style of visual thinking, imagistic reasoning. Imagistic reasoning organizes computations around image-like, analogue representations so that perceptual and symbolic operations can be brought to bear to infer structure and behavior. Programs incorporating imagistic reasoning have been shown to perform at an expert level in domains that defy current analytic or numerical methods. We have developed a computational paradigm, spatial aggregation, to unify the description of a class of imagistic problem solvers. A program written in this paradigm has the following properties. It takes a continuous field and optional objective functions as input, and produces high-level descriptions of structure, behavior, or control actions. It computes a multi-layer of intermediate representations, called spatial aggregates, by forming equivalence classes and adjacency relations. It employs a small set of generic operators such as aggregation, classification, and localization to perform bidirectional mapping between the information-rich field and successively more abstract spatial aggregates. It uses a data structure, the neighborhood graph, as a common interface to modularize computations. To illustrate our theory, we describe the computational structure of three implemented problem solvers -- KAM, MAPS, and HIPAIR --- in terms of the spatial aggregation generic operators by mixing and matching a library of commonly used routines.Comment: See http://www.jair.org/ for any accompanying file

    Poincaré on the Foundation of Geometry in the Understanding

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    This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study of groups of operations. In place of the established view I offer a revised view, according to which Poincaré held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincaré. Poincaré’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them

    Holographic fermions at strong translational symmetry breaking: a Bianchi-VII case study

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    It is presently unknown how strong lattice potentials influence the fermion spectral function of the holographic strange metals predicted by the AdS/CFT correspondence. This embodies a crucial test for the application of holography to condensed matter experiments. We show that for one particular momentum direction this spectrum can be computed for arbitrary strength of the effective translational symmetry breaking potential of the so-called Bianchi-VII geometry employing ordinary differential equations. Deep in the strange metal regime we find rather small changes to the single-fermion response computed by the emergent quantum critical IR, even when the potential becomes relevant in the infra-red. However, in the regime where holographic quasi-particles occur, defining a Fermi surface in the continuum, they acquire a finite lifetime at any finite potential strength. At the transition from irrelevancy to relevancy of the Bianchi potential in the deep infra-red the quasi-particle remnants disappear completely and the fermion spectrum exhibits a purely relaxational behaviour.Comment: 30 pages, 10 figure

    Geometry-induced localization of thermal fluctuations in ultrathin superconducting structures

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    Thermal fluctuations of the order parameter in an ultrathin triangular shaped superconducting structure are studied near TcT_{c}, in zero applied field. We find that the order parameter is prone to much larger fluctuations in the corners of the structure as compared to its interior. This geometry-induced localization of thermal fluctuations is attributed to the fact that condensate confinement in the corners is characterised by a lower effective dimensionality, which favors stronger fluctuations.Comment: 9 pages, 5 figure

    Ab Initio Calculations on the H_(2)+D_(2)=2HD Four‐Center Exchange Reaction. I. Elements of the Reaction Surface

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    We present the results of ab initio calculations on some interesting regions of the reaction surface for the four‐center exchange reaction H_(2)+D_(2)=2HD. These calculations, which use a minimum basis set of Slater orbitals, indicate that for all geometries appropriate to the transition state of the reaction, a barrier height of at least 148 kcal/mole is present. This is far greater than the energy required to produce free radicals and more than three times the experimental energy of activation, 42 kcal/mole. Considering the sources and magnitudes for errors due to correlation and basis set restrictions, we estimate the barrier height for this exchange reaction to be 132 ± 20 kcal/mole exclusive of zero‐point energies. In this paper we discuss the surface as determined by configuration interaction techniques. We find that the most favorable geometries for the exchange reactions are the square, rhombus, and kite configurations. However, all of these states are unstable with respect to H_(2) + 2H. In addition we find no evidence of collision complexes for any of the likely transition state geometries. In the following paper we will examine the G1 wavefunctions for this system in order to obtain an understanding of the factors responsible for the shape of the surface

    Fermi level alignment in single molecule junctions and its dependence on interface structure

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    The alignment of the Fermi level of a metal electrode within the gap of the highest occupied and lowest unoccupied orbital of a molecule is a key quantity in molecular electronics. Depending on the type of molecule and the interface structure of the junction, it can vary the electron transparency of a gold/molecule/gold junction by at least one order of magnitude. In this article we will discuss how Fermi level alignment is related to surface structure and bonding configuration on the basis of density functional theory calculations for bipyridine and biphenyl dithiolate between gold leads. We will also relate our findings to quantum-chemical concepts such as electronegativity.Comment: 5 pages, 2 figures, presented at the ICN+T 2006 conferenc
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