116,273 research outputs found

    Ways to restrict the differential path

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    People had developed some attack methods to attack hash function. These methods need to choose some differential pattern [Dau05]. We present a way to restrict the collisions that hold the differential pattern . At the same time, to build a hash function that meet the different needs, we propose a construction

    Between Laws and Models: Some Philosophical Morals of Lagrangian Mechanics

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    I extract some philosophical morals from some aspects of Lagrangian mechanics. (A companion paper will present similar morals from Hamiltonian mechanics and Hamilton-Jacobi theory.) One main moral concerns methodology: Lagrangian mechanics provides a level of description of phenomena which has been largely ignored by philosophers, since it falls between their accustomed levels--``laws of nature'' and ``models''. Another main moral concerns ontology: the ontology of Lagrangian mechanics is both more subtle and more problematic than philosophers often realize. The treatment of Lagrangian mechanics provides an introduction to the subject for philosophers, and is technically elementary. In particular, it is confined to systems with a finite number of degrees of freedom, and for the most part eschews modern geometry. But it includes a presentation of Routhian reduction and of Noether's ``first theorem''.Comment: 106 pages, no figure

    Supersymmetry, homology with twisted coefficients and n-dimensional knots

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    Let nn be any natural number. Let KK be any nn-dimensional knot in Sn+2S^{n+2}. We define a supersymmetric quantum system for KK with the following properties. We firstly construct a set of functional spaces (spaces of fermionic \{resp. bosonic\} states) and a set of operators (supersymmetric infinitesimal transformations) in an explicit way. Thus we obtain a set of the Witten indexes for KK. Our Witten indexes are topological invariants for nn-dimensional knots. Our Witten indexes are not zero in general. If KK is equivalent to the trivial knot, all of our Witten indexes are zero. Our Witten indexes restrict the Alexander polynomials of nn-knots. If one of our Witten indexes for an nn-knot KK is nonzero, then one of the Alexander polynomials of KK is nontrivial. Our Witten indexes are connected with homology with twisted coefficients. Roughly speaking, our Witten indexes have path integral representation by using a usual manner of supersymmetric theory.Comment: 10pages, no figure

    Symmetry Protected Topological phases and Generalized Cohomology

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    We discuss the classification of SPT phases in condensed matter systems. We review Kitaev's argument that SPT phases are classified by a generalized cohomology theory, valued in the spectrum of gapped physical systems. We propose a concrete description of that spectrum and of the corresponding cohomology theory. We compare our proposal to pre-existing constructions in the literature.Comment: 27 pages, 10 figures. v2: citation updat

    Symmetry Protected Topological phases and Generalized Cohomology

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    We discuss the classification of SPT phases in condensed matter systems. We review Kitaev's argument that SPT phases are classified by a generalized cohomology theory, valued in the spectrum of gapped physical systems. We propose a concrete description of that spectrum and of the corresponding cohomology theory. We compare our proposal to pre-existing constructions in the literature.Comment: 27 pages, 10 figures. v2: citation updat

    On Conformal Field Theory and Stochastic Loewner Evolution

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    We describe Stochastic Loewner Evolution on arbitrary Riemann surfaces with boundary using Conformal Field Theory methods. We propose in particular a CFT construction for a probability measure on (clouded) paths, and check it against known restriction properties. The probability measure can be thought of as a section of the determinant bundle over moduli spaces of Riemann surfaces. Loewner evolutions have a natural description in terms of random walk in the moduli space, and the stochastic diffusion equation translates to the Virasoro action of a certain weight-two operator on a uniformised version of the determinant bundle.Comment: 24 pages, 4 figures, LaTeX; v2: added section 4.1, references and minor clarifications, version to appear in NP

    Multiple scattering in random mechanical systems and diffusion approximation

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    This paper is concerned with stochastic processes that model multiple (or iterated) scattering in classical mechanical systems of billiard type, defined below. From a given (deterministic) system of billiard type, a random process with transition probabilities operator P is introduced by assuming that some of the dynamical variables are random with prescribed probability distributions. Of particular interest are systems with weak scattering, which are associated to parametric families of operators P_h, depending on a geometric or mechanical parameter h, that approaches the identity as h goes to 0. It is shown that (P_h -I)/h converges for small h to a second order elliptic differential operator L on compactly supported functions and that the Markov chain process associated to P_h converges to a diffusion with infinitesimal generator L. Both P_h and L are selfadjoint (densely) defined on the space L2(H,{\eta}) of square-integrable functions over the (lower) half-space H in R^m, where {\eta} is a stationary measure. This measure's density is either (post-collision) Maxwell-Boltzmann distribution or Knudsen cosine law, and the random processes with infinitesimal generator L respectively correspond to what we call MB diffusion and (generalized) Legendre diffusion. Concrete examples of simple mechanical systems are given and illustrated by numerically simulating the random processes.Comment: 34 pages, 13 figure
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