443 research outputs found
Remarks on Two Nonstandard Versions of Periodicity in Words
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
We present a rigorous and fully consistent -theoretic framework for
studying gapped topological phases of free fermions such as topological
insulators. It utilises and profits from powerful techniques in operator
-theory. From the point of view of symmetries, especially those of time
reversal, charge conjugation, and magnetic translations, operator -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 -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 -theory to classify phases in an absolute sense,
but only relative to some arbitrary reference. -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
-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
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
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
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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