443 research outputs found

    Remarks on Two Nonstandard Versions of Periodicity in Words

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    In this paper, we study some periodicity concepts on words. First, we extend the notion of full tilings which was recently introduced by Karhumäki, Lifshits, and Rytter to partial tilings. Second, we investigate the notion of quasiperiods and show in particular that the set of quasiperiodic words is a context-sensitive language that is not context-free, answering a conjecture by Dömösi, Horváth and Ito

    On the K-theoretic classification of topological phases of matter

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    We present a rigorous and fully consistent KK-theoretic framework for studying gapped topological phases of free fermions such as topological insulators. It utilises and profits from powerful techniques in operator KK-theory. From the point of view of symmetries, especially those of time reversal, charge conjugation, and magnetic translations, operator KK-theory is more general and natural than the commutative topological theory. Our approach is model-independent, and only the symmetry data of the dynamics, which may include information about disorder, is required. This data is completely encoded in a suitable CC^*-superalgebra. From a representation-theoretic point of view, symmetry-compatible gapped phases are classified by the super-representation group of this symmetry algebra. Contrary to existing literature, we do not use KK-theory to classify phases in an absolute sense, but only relative to some arbitrary reference. KK-theory groups are better thought of as groups of obstructions between homotopy classes of gapped phases. Besides rectifying various inconsistencies in the existing literature on KK-theory classification schemes, our treatment has conceptual simplicity in its treatment of all symmetries equally. The Periodic Table of Kitaev is exhibited as a special case within our framework, and we prove that the phenomena of periodicity and dimension shifts are robust against disorder and magnetic fields.Comment: 41 pages, revised version with a new abstract, introductory sections and critique of the literatur

    Periodic boxcar deconvolution and diophantine approximation

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    We consider the nonparametric estimation of a periodic function that is observed in additive Gaussian white noise after convolution with a ``boxcar,'' the indicator function of an interval. This is an idealized model for the problem of recovery of noisy signals and images observed with ``motion blur.'' If the length of the boxcar is rational, then certain frequencies are irretreviably lost in the periodic model. We consider the rate of convergence of estimators when the length of the boxcar is irrational, using classical results on approximation of irrationals by continued fractions. A basic question of interest is whether the minimax rate of convergence is slower than for nonperiodic problems with 1/f-like convolution filters. The answer turns out to depend on the type and smoothness of functions being estimated in a manner not seen with ``homogeneous'' filters.Comment: Published at http://dx.doi.org/10.1214/009053604000000391 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Charged black holes in compactified spacetimes

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    We construct and investigate a compactified version of the four-dimensional Reissner-Nordstrom-NUT solution, generalizing the compactified Schwarzschild black hole that has been previously studied by several workers. Our approach to compactification is based on dimensional reduction with respect to the stationary Killing vector, resulting in three-dimensional gravity coupled to a nonlinear sigma model. Using that the original non-compactified solution corresponds to a target space geodesic, the problem can be linearized much in the same way as in the case of no electric nor NUT charge. An interesting feature of the solution family is that for nonzero electric charge but vanishing NUT charge, the solution has a curvature singularity on a torus that surrounds the event horizon, but this singularity is removed when the NUT charge is switched on. We also treat the Schwarzschild case in a more complete way than has been done previously. In particular, the asymptotic solution (the Levi-Civita solution with the height coordinate made periodic) has to our knowledge only been calculated up to a determination of the mass parameter. The periodic Levi-Civita solution contains three essential parameters, however, and the remaining two are explicitly calculated here.Comment: 20 pages, 3 figures. v2: Typo corrected, reference adde

    Resolutions of mesh algebras: periodicity and Calabi-Yau dimensions

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    A triangulated category is said to be Calabi-Yau of dimension d if the dth power of its suspension is a Serre functor. We determine which stable categories of self-injective algebras A of finite representation type are Calabi-Yau and compute their Calabi-Yau dimensions. We achieve this by studying the minimal projective resolution of the stable Auslander algebra of A over its enveloping algebra, and use covering theory to reduce to (generalized) preprojective algebras of Dynkin graphs. We also describe how this problem can be approached by realizing the stable categories in question as orbit categories of the bounded derived categories of hereditary algebras.Comment: Final version. To appear in Math.
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