322 research outputs found
The large-spin asymptotics of the ferromagnetic XXZ chain
We present new results and give a concise review of recent previous results
on the asymptotics for large spin of the low-lying spectrum of the
ferromagnetic XXZ Heisenberg chain with kink boundary conditions. Our main
interest is to gain detailed information on the interface ground states of this
model and the low-lying excitations above them. The new and most detailed
results are obtained using a rigorous version of bosonization, which can be
interpreted as a quantum central limit theorem.Comment: 30 pages, submitted to the proceedings of the workshop "Low-energy
states in quantum many-body systems", 29 January 2003, Cergy-Pontois
Central limit theorems for the large-spin asymptotics of quantum spins
We use a generalized form of Dyson's spin wave formalism to prove several
central limit theorems for the large-spin asymptotics of quantum spins in a
coherent state.Comment: 28 pages, uses package amsref
Restricted maximum-likelihood method for learning latent variance components in gene expression data with known and unknown confounders
Random effect models are popular statistical models for detecting and
correcting spurious sample correlations due to hidden confounders in
genome-wide gene expression data. In applications where some confounding
factors are known, estimating simultaneously the contribution of known and
latent variance components in random effect models is a challenge that has so
far relied on numerical gradient-based optimizers to maximize the likelihood
function. This is unsatisfactory because the resulting solution is poorly
characterized and the efficiency of the method may be suboptimal. Here we prove
analytically that maximum-likelihood latent variables can always be chosen
orthogonal to the known confounding factors, in other words, that
maximum-likelihood latent variables explain sample covariances not already
explained by known factors. Based on this result we propose a restricted
maximum-likelihood method which estimates the latent variables by maximizing
the likelihood on the restricted subspace orthogonal to the known confounding
factors, and show that this reduces to probabilistic PCA on that subspace. The
method then estimates the variance-covariance parameters by maximizing the
remaining terms in the likelihood function given the latent variables, using a
newly derived analytic solution for this problem. Compared to gradient-based
optimizers, our method attains greater or equal likelihood values, can be
computed using standard matrix operations, results in latent factors that don't
overlap with any known factors, and has a runtime reduced by several orders of
magnitude. Hence the restricted maximum-likelihood method facilitates the
application of random effect modelling strategies for learning latent variance
components to much larger gene expression datasets than possible with current
methods.Comment: 15 pages, 4 figures, 3 supplementary figures, 19 pages supplementary
methods; minor revision with expanded Discussion sectio
Mathematical Structure of Magnons in Quantum Ferromagnets
We provide the mathematical structure and a simple, transparent and rigorous
derivation of the magnons as elementary quasi-particle excitations at low
temperatures and in the infinite spin limit for a large class of Heisenberg
ferromagnets. The magnon canonical variables are obtained as fluctuation
operators in the infinite spin limit. Their quantum character is governed by
the size of the magnetization
Fluctuation Operators and Spontaneous Symmetry Breaking
We develop an alternative approach to this field, which was to a large extent
developed by Verbeure et al. It is meant to complement their approach, which is
largely based on a non-commutative central limit theorem and coordinate space
estimates. In contrast to that we deal directly with the limits of -point
truncated correlation functions and show that they typically vanish for provided that the respective scaling exponents of the fluctuation
observables are appropriately chosen. This direct approach is greatly
simplified by the introduction of a smooth version of spatial averaging, which
has a much nicer scaling behavior and the systematic developement of Fourier
space and energy-momentum spectral methods. We both analyze the regime of
normal fluctuations, the various regimes of poor clustering and the case of
spontaneous symmetry breaking or Goldstone phenomenon.Comment: 30 pages, Latex, a more detailed discussion in section 7 as to
possible scaling behavior of l-point function
Analysis of a Gibbs sampler method for model based clustering of gene expression data
Over the last decade, a large variety of clustering algorithms have been
developed to detect coregulatory relationships among genes from microarray gene
expression data. Model based clustering approaches have emerged as
statistically well grounded methods, but the properties of these algorithms
when applied to large-scale data sets are not always well understood. An
in-depth analysis can reveal important insights about the performance of the
algorithm, the expected quality of the output clusters, and the possibilities
for extracting more relevant information out of a particular data set. We have
extended an existing algorithm for model based clustering of genes to
simultaneously cluster genes and conditions, and used three large compendia of
gene expression data for S. cerevisiae to analyze its properties. The algorithm
uses a Bayesian approach and a Gibbs sampling procedure to iteratively update
the cluster assignment of each gene and condition. For large-scale data sets,
the posterior distribution is strongly peaked on a limited number of
equiprobable clusterings. A GO annotation analysis shows that these local
maxima are all biologically equally significant, and that simultaneously
clustering genes and conditions performs better than only clustering genes and
assuming independent conditions. A collection of distinct equivalent
clusterings can be summarized as a weighted graph on the set of genes, from
which we extract fuzzy, overlapping clusters using a graph spectral method. The
cores of these fuzzy clusters contain tight sets of strongly coexpressed genes,
while the overlaps exhibit relations between genes showing only partial
coexpression.Comment: 8 pages, 7 figure
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