3,199 research outputs found
1D periodic potentials with gaps vanishing at k=0
Appearance of energy bands and gaps in the dispersion relations of a periodic
potential is a standard feature of Quantum Mechanics. We investigate the class
of one-dimensional periodic potentials for which all gaps vanish at the center
of the Brillouin zone. We characterize them through a necessary and sufficient
condition. Potentials of the form we focus on arise in different fields of
Physics, from supersymmetric Quantum Mechanics, to Korteweg-de Vries equation
theory and classical diffusion problems. The O.D.E. counterpart to this problem
is the characterisation of periodic potentials for which coexistence occur of
linearly independent solutions of the corresponding Schroedinger equation
(Hill's equation). This result is placed in perspective of the previous related
results available in the literature.Comment: 29 pages, 4 figures, version accepted for publication in Memoirs on
Differential Equations and Mathematical Physic
Gene ranking and biomarker discovery under correlation
Biomarker discovery and gene ranking is a standard task in genomic high
throughput analysis. Typically, the ordering of markers is based on a
stabilized variant of the t-score, such as the moderated t or the SAM
statistic. However, these procedures ignore gene-gene correlations, which may
have a profound impact on the gene orderings and on the power of the subsequent
tests.
We propose a simple procedure that adjusts gene-wise t-statistics to take
account of correlations among genes. The resulting correlation-adjusted
t-scores ("cat" scores) are derived from a predictive perspective, i.e. as a
score for variable selection to discriminate group membership in two-class
linear discriminant analysis. In the absence of correlation the cat score
reduces to the standard t-score. Moreover, using the cat score it is
straightforward to evaluate groups of features (i.e. gene sets). For
computation of the cat score from small sample data we propose a shrinkage
procedure. In a comparative study comprising six different synthetic and
empirical correlation structures we show that the cat score improves estimation
of gene orderings and leads to higher power for fixed true discovery rate, and
vice versa. Finally, we also illustrate the cat score by analyzing metabolomic
data.
The shrinkage cat score is implemented in the R package "st" available from
URL http://cran.r-project.org/web/packages/st/Comment: 18 pages, 5 figures, 1 tabl
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