4,860 research outputs found
A planar calculus for infinite index subfactors
We develop an analog of Jones' planar calculus for II_1-factor bimodules with
arbitrary left and right von Neumann dimension. We generalize to bimodules
Burns' results on rotations and extremality for infinite index subfactors.
These results are obtained without Jones' basic construction and the resulting
Jones projections.Comment: 56 pages, many figure
A Parameterized Study of Maximum Generalized Pattern Matching Problems
The generalized function matching (GFM) problem has been intensively studied starting with Ehrenfreucht and Rozenberg (Inf Process Lett 9(2):86–88, 1979). Given a pattern p and a text t, the goal is to find a mapping from the letters of p to non-empty substrings of t, such that applying the mapping to p results in t. Very recently, the problem has been investigated within the framework of parameterized complexity (Fernau et al. in FSTTCS, 2013). In this paper we study the parameterized complexity of the optimization variant of GFM (called Max-GFM), which has been introduced in Amir and Amihood (J Discrete Algorithms 5(3):514–523, 2007). Here, one is allowed to replace some of the pattern letters with some special symbols “?”, termed wildcards or don’t cares, which can be mapped to an arbitrary substring of the text. The goal is to minimize the number of wildcards used. We give a complete classification of the parameterized complexity of Max-GFM and its variants under a wide range of parameterizations, such as, the number of occurrences of a letter in the text, the size of the text alphabet, the number of occurrences of a letter in the pattern, the size of the pattern alphabet, the maximum length of a string matched to any pattern letter, the number of wildcards and the maximum size of a string that a wildcard can be mapped to
Semi-regular masas of transfinite length
In 1965 Tauer produced a countably infinite family of semi-regular masas in
the hyperfinite factor, no pair of which are conjugate by an
automorphism. This was achieved by iterating the process of passing to the
algebra generated by the normalisers and, for each , finding
masas for which this procedure terminates at the -th stage. Such masas are
said to have length . In this paper we consider a transfinite version of
this idea, giving rise to a notion of ordinal valued length. We show that all
countable ordinals arise as lengths of semi-regular masas in the hyperfinite
factor. Furthermore, building on work of Jones and Popa, we
obtain all possible combinations of regular inclusions of irreducible
subfactors in the normalising tower.Comment: 14 page
Subfactors of index less than 5, part 1: the principal graph odometer
In this series of papers we show that there are exactly ten subfactors, other
than subfactors, of index between 4 and 5. Previously this
classification was known up to index . In the first paper we give
an analogue of Haagerup's initial classification of subfactors of index less
than , showing that any subfactor of index less than 5 must appear
in one of a large list of families. These families will be considered
separately in the three subsequent papers in this series.Comment: 36 pages (updated to reflect that the classification is now complete
Spectral measures of small index principal graphs
The principal graph of a subfactor with finite Jones index is one of the
important algebraic invariants of the subfactor. If is the adjacency
matrix of we consider the equation . When has square
norm the spectral measure of can be averaged by using the map
, and we get a probability measure on the unit circle
which does not depend on . We find explicit formulae for this measure
for the principal graphs of subfactors with index , the
(extended) Coxeter-Dynkin graphs of type , and . The moment
generating function of is closely related to Jones' -series.Comment: 23 page
Group measure space decomposition of II_1 factors and W*-superrigidity
We prove a "unique crossed product decomposition" result for group measure
space II_1 factors arising from arbitrary free ergodic probability measure
preserving (p.m.p.) actions of groups \Gamma in a fairly large family G, which
contains all free products of a Kazhdan group and a non-trivial group, as well
as certain amalgamated free products over an amenable subgroup. We deduce that
if T_n denotes the group of upper triangular matrices in PSL(n,Z), then any
free, mixing p.m.p. action of the amalgamated free product of PSL(n,Z) with
itself over T_n, is W*-superrigid, i.e. any isomorphism between L^\infty(X)
\rtimes \Gamma and an arbitrary group measure space factor L^\infty(Y) \rtimes
\Lambda, comes from a conjugacy of the actions. We also prove that for many
groups \Gamma in the family G, the Bernoulli actions of \Gamma are
W*-superrigid.Comment: Final version. Some extra details have been added to improve the
expositio
Constraints on non-thermal Dark Matter from Planck lensing extraction
Distortions of CMB temperature and polarization anisotropy maps caused by
gravitational lensing, observable with high angular resolution and sensitivity,
can be used to constrain the sterile neutrino mass, offering several advantages
against the analysis based on the combination of CMB, LSS and Ly\alpha forest
power spectra. As the gravitational lensing effect depends on the matter
distribution, no assumption on light-to-mass bias is required. In addition,
unlike the galaxy clustering and Ly\alpha forest power spectra, the projected
gravitational potential power spectrum probes a larger range of angular scales,
the non-linear corrections being required only at very small scales. Taking
into account the changes in the time-temperature relation of the primordial
plasma and the modification of the neutrino thermal potential, we compute the
projected gravitational potential power spectrum and its correlation with the
temperature in the presence of DM sterile neutrino. We show that the
cosmological parameters are generally not biased when DM sterile neutrino is
included. From this analysis we found a lower limit on DM sterile neutrino mass
m_s >2.08 keV at 95% CL, consistent with the lower mass limit obtained from the
combined analysis of CMB, SDSS 3D power spectrum and SDSS Ly\alpha forest power
spectrum ( keV). We conclude that although the information that
can be obtained from lensing extraction is rather limited due to the high level
of the lensing noise of Planck experiment, weak lensing of CMB offers a
valuable alternative to constrain the dark matter sterile neutrino mass.Comment: 15 pages, 6 figure
Unbiased bases (Hadamards) for 6-level systems: Four ways from Fourier
In quantum mechanics some properties are maximally incompatible, such as the
position and momentum of a particle or the vertical and horizontal projections
of a 2-level spin. Given any definite state of one property the other property
is completely random, or unbiased. For N-level systems, the 6-level ones are
the smallest for which a tomographically efficient set of N+1 mutually unbiased
bases (MUBs) has not been found. To facilitate the search, we numerically
extend the classification of unbiased bases, or Hadamards, by incrementally
adjusting relative phases in a standard basis. We consider the non-unitarity
caused by small adjustments with a second order Taylor expansion, and choose
incremental steps within the 4-dimensional nullspace of the curvature. In this
way we prescribe a numerical integration of a 4-parameter set of Hadamards of
order 6.Comment: 5 pages, 2 figure
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