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Concept# Finitary relation

Summary

In mathematics, a finitary relation over sets X1, ..., Xn is a subset of the Cartesian product X1 × ⋯ × Xn; that is, it is a set of n-tuples (x1, ..., xn) consisting of elements xi in Xi. Typically, the relation describes a possible connection between the elements of an n-tuple. For example, the relation "x is divisible by y and z" consists of the set of 3-tuples such that when substituted to x, y and z, respectively, make the sentence true.
The non-negative integer n giving the number of "places" in the relation is called the arity, adicity or degree of the relation. A relation with n "places" is variously called an n-ary relation, an n-adic relation or a relation of degree n. Relations with a finite number of places are called finitary relations (or simply relations if the context is clear). It is also possible to generalize the concept to infinitary relations with infinite sequences.
An n-ary rel

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The aim of this thesis is to build a system able to automatically and robustly track human motion in 3–D starting from monocular input. To this end two approaches are introduced, which tackle two different types of motion: The first is useful to analyze activities for which a characteristic pose, or key-pose, can be detected, as for example in the walking case. On the other hand the second can be used for cases in which such pose is not defined but there is a clear relation between some easily measurable image quantities and the body configuration, as for example in the skating case where the trajectory followed by a subject is highly correlated to how the subject articulates. In the first proposed technique we combine detection and tracking techniques to achieve robust 3D motion recovery of people seen from arbitrary viewpoints by a single and potentially moving camera. We rely on detecting key postures, which can be done reliably, using a motion model to infer 3D poses between consecutive detections, and finally refining them over the whole sequence using a generative model. We demonstrate our approach in the cases of golf motions filmed using a static camera and walking motions acquired using a potentially moving one. We will show that this approach, although monocular, is both metrically accurate because it integrates information over many frames and robust because it can recover from a few misdetections. The second approach is based on the fact that the articulated body models used to represent human motion typically have many degrees of freedom, usually expressed as joint angles that are highly correlated. The true range of motion can therefore be represented by latent variables that span a low-dimensional space. This has often been used to make motion tracking easier. However, learning the latent space in a problem independent way makes it non trivial to initialize the tracking process by picking appropriate initial values for the latent variables, and thus for the pose. In this thesis, it will be shown that by directly using observable quantities as latent variables, this issue can be eliminated.

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The nature of the pseudogap phase is a central problem in the effort to understand the high-transition-temperature (high-Tc) copper oxide superconductors. A fundamental question is what symmetries are broken when the pseudogap phase sets in, which occurs when the temperature decreases below a value T*. There is evidence from measurements of both polarized neutron diffraction and the polar Kerr effect that time-reversal symmetry is broken, but at temperatures that differ significantly from one another. Broken rotational symmetry was detected from both resistivity measurements and inelastic neutron scattering at low doping, and from scanning tunnelling spectroscopy at low temperature, but showed no clear relation to T*. Here we report the observation of a large in-plane anisotropy of the Nernst effect in YBa 2Cu3 Oy that sets in precisely at T* throughout the doping phase diagram. We show that the CuO chains of the orthorhombic lattice are not responsible for this anisotropy, which is therefore an intrinsic property of the CuO2 planes. We conclude that the pseudogap phase is an electronic state that strongly breaks four-fold rotational symmetry. This narrows the range of possible states considerably, pointing to stripe or nematic order. © 2010 Macmillan Publishers Limited. All rights reserved.

2010We perform first-principles molecular dynamics of liquid oxygen in which the magnetic structure evolves according to a generalized density-functional scheme allowing for noncollinear spin configurations. We investigate both structural correlations between the orientations of the molecular axes and magnetic correlations between the orientations of the molecular magnetic moments, demonstrating a clear relation between the local molecular configuration and the relative magnetic arrangement. The nuclear structure factor obtained from the simulation is found to agree well with the experimental one. The calculated magnetic structure factor shows antiferromagnetic correlations between molecules in the first shell, in accord with spin-polarized neutron scattering measurements. We observe the formation of dynamically coupled molecules, known as O-4 units, in which the molecular moments are aligned in an antiferromagnetic fashion. An analysis based on the life time of such units, revealed that in most cases the O-4 units occur as transient configurations during collisions. However, we also observed a small fraction of O-4 units surviving for relatively long periods. To account for electronic excitations which are missed in our density-functional scheme, we complement our description with a mean field model for the thermal fluctuations of the magnetic structure. The combined scheme is found to improve the description of the magnetic neutron structure factor and allows us to study the dependence of the magnetic susceptibility on temperature.

2004