12,802 research outputs found
The Hopf structure of some dual operator algebras
We study the Hopf structure of a class of dual operator algebras
corresponding to certain semigroups. This class of algebras arises in dilation
theory, and includes the noncommutative analytic Toeplitz algebra and the
multiplier algebra of the Drury-Arveson space, which correspond to the free
semigroup and the free commutative semigroup respectively. The preduals of the
algebras in this class naturally form Hopf (convolution) algebras. The original
algebras and their preduals form (non-self-adjoint) dual Hopf algebras in the
sense of Effros and Ruan. We study these algebras from this perspective, and
obtain a number of results about their structure.Comment: 30 page
Improving English education in Thailand by modeling classroom behavior
English classroom behavior across Thailand is highly influenced by long-held traditions, culture and values. By comparing Thai secondary education classroom behavior with that of Chinese classes working at the same level, we have constructed a model of Thai students’ English classroom behavior. Comparison is made from surveys conducted by schools with students demonstrating similarly wide range of performance and ability in English. We also suggest effective ways to improve English teaching in Thailand for both Thai and foreign teachers of English by incorporating this model with the theories of J. Kunin, so as to help
avoid culture conflict, motivate students’ interest and make better use of existing standards and practices
Chalker scaling, level repulsion, and conformal invariance in critically delocalized quantum matter: Disordered topological superconductors and artificial graphene
We numerically investigate critically delocalized wavefunctions in models of
2D Dirac fermions, subject to vector potential disorder. These describe the
surface states of 3D topological superconductors, and can also be realized
through long-range correlated bond randomness in artificial materials like
molecular graphene. A frozen regime can occur for strong disorder in these
systems, wherein a single wavefunction presents a few localized peaks separated
by macroscopic distances. Despite this rarefied spatial structure, we find
robust correlations between eigenstates at different energies, at both weak and
strong disorder. The associated level statistics are always approximately
Wigner-Dyson. The system shows generalized Chalker (quantum critical) scaling,
even when individual states are quasilocalized in space. We confirm analytical
predictions for the density of states and multifractal spectra. For a single
Dirac valley, we establish that finite energy states show universal
multifractal spectra consistent with the integer quantum Hall plateau
transition. A single Dirac fermion at finite energy can therefore behave as a
Quantum Hall critical metal. For the case of two valleys and non-abelian
disorder, we verify predictions of conformal field theory. Our results for the
non-abelian case imply that both delocalization and conformal invariance are
topologically-protected for multivalley topological superconductor surface
states.Comment: 17 pages, 15 figures, published versio
Escaping the crunch: gravitational effects in classical transitions
During eternal inflation, a landscape of vacua can be populated by the
nucleation of bubbles. These bubbles inevitably collide, and collisions
sometimes displace the field into a new minimum in a process known as a
classical transition. In this paper, we examine some new features of classical
transitions that arise when gravitational effects are included. Using the
junction condition formalism, we study the conditions for energy conservation
in detail, and solve explicitly for the types of allowed classical transition
geometries. We show that the repulsive nature of domain walls, and the de
Sitter expansion associated with a positive energy minimum, can allow for
classical transitions to vacua of higher energy than that of the colliding
bubbles. Transitions can be made out of negative or zero energy (terminal)
vacua to a de Sitter phase, re-starting eternal inflation, and populating new
vacua. However, the classical transition cannot produce vacua with energy
higher than the original parent vacuum, which agrees with previous results on
the construction of pockets of false vacuum. We briefly comment on the possible
implications of these results for various measure proposals in eternal
inflation.Comment: 21 pages, 10 figure
Hypergeometric Functions of Matrix Arguments and Linear Statistics of Multi-Spiked Hermitian Matrix Models
This paper derives central limit theorems (CLTs) for general linear spectral
statistics (LSS) of three important multi-spiked Hermitian random matrix
ensembles. The first is the most common spiked scenario, proposed by Johnstone,
which is a central Wishart ensemble with fixed-rank perturbation of the
identity matrix, the second is a non-central Wishart ensemble with fixed-rank
noncentrality parameter, and the third is a similarly defined non-central
ensemble. These CLT results generalize our recent work to account for multiple
spikes, which is the most common scenario met in practice. The generalization
is non-trivial, as it now requires dealing with hypergeometric functions of
matrix arguments. To facilitate our analysis, for a broad class of such
functions, we first generalize a recent result of Onatski to present new
contour integral representations, which are particularly suitable for computing
large-dimensional properties of spiked matrix ensembles. Armed with such
representations, our CLT formulas are derived for each of the three spiked
models of interest by employing the Coulomb fluid method from random matrix
theory along with saddlepoint techniques. We find that for each matrix model,
and for general LSS, the individual spikes contribute additively to yield a
correction term to the asymptotic mean of the linear statistic, which we
specify explicitly, whilst having no effect on the leading order terms of the
mean or variance
A Robust Statistics Approach to Minimum Variance Portfolio Optimization
We study the design of portfolios under a minimum risk criterion. The
performance of the optimized portfolio relies on the accuracy of the estimated
covariance matrix of the portfolio asset returns. For large portfolios, the
number of available market returns is often of similar order to the number of
assets, so that the sample covariance matrix performs poorly as a covariance
estimator. Additionally, financial market data often contain outliers which, if
not correctly handled, may further corrupt the covariance estimation. We
address these shortcomings by studying the performance of a hybrid covariance
matrix estimator based on Tyler's robust M-estimator and on Ledoit-Wolf's
shrinkage estimator while assuming samples with heavy-tailed distribution.
Employing recent results from random matrix theory, we develop a consistent
estimator of (a scaled version of) the realized portfolio risk, which is
minimized by optimizing online the shrinkage intensity. Our portfolio
optimization method is shown via simulations to outperform existing methods
both for synthetic and real market data
Reducing Revenue to Welfare Maximization: Approximation Algorithms and other Generalizations
It was recently shown in [http://arxiv.org/abs/1207.5518] that revenue
optimization can be computationally efficiently reduced to welfare optimization
in all multi-dimensional Bayesian auction problems with arbitrary (possibly
combinatorial) feasibility constraints and independent additive bidders with
arbitrary (possibly combinatorial) demand constraints. This reduction provides
a poly-time solution to the optimal mechanism design problem in all auction
settings where welfare optimization can be solved efficiently, but it is
fragile to approximation and cannot provide solutions to settings where welfare
maximization can only be tractably approximated. In this paper, we extend the
reduction to accommodate approximation algorithms, providing an approximation
preserving reduction from (truthful) revenue maximization to (not necessarily
truthful) welfare maximization. The mechanisms output by our reduction choose
allocations via black-box calls to welfare approximation on randomly selected
inputs, thereby generalizing also our earlier structural results on optimal
multi-dimensional mechanisms to approximately optimal mechanisms. Unlike
[http://arxiv.org/abs/1207.5518], our results here are obtained through novel
uses of the Ellipsoid algorithm and other optimization techniques over {\em
non-convex regions}
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