58 research outputs found

    Constrained Rough Paths

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    A combinatorial approach to geometric rough paths and their controlled paths

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    We develop the structure theory for transformations of weakly geometric rough paths of bounded 1<p1 < p-variation and their controlled paths. Our approach differs from existing approaches as it does not rely on smooth approximations. We derive an explicit combinatorial expression for the rough path lift of a controlled path, and use it to obtain fundamental identities such as the associativity of the rough integral, the adjunction between pushforwards and pullbacks, and a change of variables formula for rough differential equations (RDEs). As applications we define rough paths, rough integration and RDEs on manifolds, extending the results of [CDL15] to the case of arbitrary pp

    Dimension-independent Harnack inequalities for subordinated semigroups

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    Dimension-independent Harnack inequalities are derived for a class of subordinate semigroups. In particular, for a diffusion satisfying the Bakry-Emery curvature condition, the subordinate semigroup with power α\alpha satisfies a dimension-free Harnack inequality provided α∈(1/2,1)\alpha \in(1/2, 1), and it satisfies the log-Harnack inequality for all α∈(0,1).\alpha \in (0,1). Some infinite-dimensional examples are also presented

    Time separation as a hidden variable to the Copenhagen school of quantum mechanics

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    The Bohr radius is a space-like separation between the proton and electron in the hydrogen atom. According to the Copenhagen school of quantum mechanics, the proton is sitting in the absolute Lorentz frame. If this hydrogen atom is observed from a different Lorentz frame, there is a time-like separation linearly mixed with the Bohr radius. Indeed, the time-separation is one of the essential variables in high-energy hadronic physics where the hadron is a bound state of the quarks, while thoroughly hidden in the present form of quantum mechanics. It will be concluded that this variable is hidden in Feynman's rest of the universe. It is noted first that Feynman's Lorentz-invariant differential equation for the bound-state quarks has a set of solutions which describe all essential features of hadronic physics. These solutions explicitly depend on the time separation between the quarks. This set also forms the mathematical basis for two-mode squeezed states in quantum optics, where both photons are observable, but one of them can be treated a variable hidden in the rest of the universe. The physics of this two-mode state can then be translated into the time-separation variable in the quark model. As in the case of the un-observed photon, the hidden time-separation variable manifests itself as an increase in entropy and uncertainty.Comment: LaTex 10 pages with 5 figure. Invited paper presented at the Conference on Advances in Quantum Theory (Vaxjo, Sweden, June 2010), to be published in one of the AIP Conference Proceedings serie

    Coherent states for compact Lie groups and their large-N limits

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    The first two parts of this article surveys results related to the heat-kernel coherent states for a compact Lie group K. I begin by reviewing the definition of the coherent states, their resolution of the identity, and the associated Segal-Bargmann transform. I then describe related results including connections to geometric quantization and (1+1)-dimensional Yang--Mills theory, the associated coherent states on spheres, and applications to quantum gravity. The third part of this article summarizes recent work of mine with Driver and Kemp on the large-N limit of the Segal--Bargmann transform for the unitary group U(N). A key result is the identification of the leading-order large-N behavior of the Laplacian on "trace polynomials."Comment: Submitted to the proceeding of the CIRM conference, "Coherent states and their applications: A contemporary panorama.

    Brownian bridges to submanifolds

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    We introduce and study Brownian bridges to submanifolds. Our method involves proving a general formula for the integral over a submanifold of the minimal heat kernel on a complete Riemannian manifold. We use the formula to derive lower bounds, an asymptotic relation and derivative estimates. We also see a connection to hypersurface local time. This work is motivated by the desire to extend the analysis of path and loop spaces to measures on paths which terminate on a submanifold

    A review of elliptical and disc galaxy structure, and modern scaling laws

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    A century ago, in 1911 and 1913, Plummer and then Reynolds introduced their models to describe the radial distribution of stars in `nebulae'. This article reviews the progress since then, providing both an historical perspective and a contemporary review of the stellar structure of bulges, discs and elliptical galaxies. The quantification of galaxy nuclei, such as central mass deficits and excess nuclear light, plus the structure of dark matter halos and cD galaxy envelopes, are discussed. Issues pertaining to spiral galaxies including dust, bulge-to-disc ratios, bulgeless galaxies, bars and the identification of pseudobulges are also reviewed. An array of modern scaling relations involving sizes, luminosities, surface brightnesses and stellar concentrations are presented, many of which are shown to be curved. These 'redshift zero' relations not only quantify the behavior and nature of galaxies in the Universe today, but are the modern benchmark for evolutionary studies of galaxies, whether based on observations, N-body-simulations or semi-analytical modelling. For example, it is shown that some of the recently discovered compact elliptical galaxies at 1.5 < z < 2.5 may be the bulges of modern disc galaxies.Comment: Condensed version (due to Contract) of an invited review article to appear in "Planets, Stars and Stellar Systems"(www.springer.com/astronomy/book/978-90-481-8818-5). 500+ references incl. many somewhat forgotten, pioneer papers. Original submission to Springer: 07-June-201

    Insomnia and its correlates in a representative sample of the Greek population

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    <p>Abstract</p> <p>Background</p> <p>Insomnia is a major public health concern affecting about 10% of the general population in its chronic form. Furthermore, epidemiological surveys demonstrate that poor sleep and sleep dissatisfaction are even more frequent problems (10-48%) in the community. This is the first report on the prevalence of insomnia in Greece, a southeastern European country which differs in several socio-cultural and climatic aspects from the rest of European Community members. Data obtained from a national household survey (n = 1005) were used to assess the relationship between insomnia symptoms and a variety of sociodemographic variables, life habits, and health-related factors.</p> <p>Methods</p> <p>A self-administered questionnaire with questions pertaining to general health and related issues was given to the participants. The Short Form-36 (Mental Health subscale), the Athens Insomnia Scale (AIS) as a measure of insomnia-related symptoms, and the International Physical Activity Questionnaire (IPAQ) were also used for the assessment.</p> <p>Results</p> <p>The prevalence of insomnia in the total sample was 25.3% (n = 254); insomnia was more frequent in women than men (30.7% vs. 19.5%) and increased with age. Multiple regression analysis revealed a significant association of insomnia with low socio-economical status and educational level, physical inactivity, existence of a chronic physical or mental disease and increased number of hospitalizations in the previous year.</p> <p>Conclusions</p> <p>The present study confirms most findings reported from other developed countries around the world regarding the high prevalence of insomnia problems in the general population and their association with several sociodemographic and health-related predisposing factors. These results further indicate the need for more active interventions on the part of physicians who should suspect and specifically ask about such symptoms.</p

    Epidemiology and etiology of Parkinson’s disease: a review of the evidence

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    A Note on Stochastic Calculus in Vector Bundles

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