29,936 research outputs found
Anderson localization through Polyakov loops: lattice evidence and Random matrix model
We investigate low-lying fermion modes in SU(2) gauge theory at temperatures
above the phase transition. Both staggered and overlap spectra reveal
transitions from chaotic (random matrix) to integrable (Poissonian) behavior
accompanied by an increasing localization of the eigenmodes. We show that the
latter are trapped by local Polyakov loop fluctuations. Islands of such "wrong"
Polyakov loops can therefore be viewed as defects leading to Anderson
localization in gauge theories. We find strong similarities in the spatial
profile of these localized staggered and overlap eigenmodes. We discuss
possible interpretations of this finding and present a sparse random matrix
model that reproduces these features.Comment: 11 pages, 23 plots in 11 figures; some comments and references added,
some axis labels corrected; journal versio
The distribution of word matches between Markovian sequences with periodic boundary conditions
Word match counts have traditionally been proposed as an alignment-free measure of similarity for biological sequences. The D2 statistic, which simply counts the number of exact word matches between two sequences, is a useful test bed for developing rigorous mathematical results, which can then be extended to more biologically useful measures. The distributional properties of the D2 statistic under the null hypothesis of identically and independently distributed letters have been studied extensively, but no comprehensive study of the D2 distribution for biologically more realistic higher-order Markovian sequences exists. Here we derive exact formulas for the mean and variance of the D2 statistic for Markovian sequences of any order, and demonstrate through Monte Carlo simulations that the entire distribution is accurately characterized by a PĂłlya-Aeppli distribution for sequence lengths of biological interest. The approach is novel in that Markovian dependency is defined for sequences with periodic boundary conditions, and this enables exact analytic formulas for the mean and variance to be derived. We also carry out a preliminary comparison between the approximate D2 distribution computed with the theoretical mean and variance under a Markovian hypothesis and an empirical D2 distribution from the human genome
Two-level system with noise: Blue's function approach
By using the random matrix approach and generalized Blue's function
representation we solve analytically the model of an effective two-level system
coupled to a noise reservoir. We show that calculated spectral properties of
the system are in agreement with the numerically simulated results. We outline
possible applications of the model in the field of condensed phase reactions.Comment: 17 pages LaTeX, 5 EPS figures include
Decay properties of spectral projectors with applications to electronic structure
Motivated by applications in quantum chemistry and solid state physics, we
apply general results from approximation theory and matrix analysis to the
study of the decay properties of spectral projectors associated with large and
sparse Hermitian matrices. Our theory leads to a rigorous proof of the
exponential off-diagonal decay ("nearsightedness") for the density matrix of
gapped systems at zero electronic temperature in both orthogonal and
non-orthogonal representations, thus providing a firm theoretical basis for the
possibility of linear scaling methods in electronic structure calculations for
non-metallic systems. We further discuss the case of density matrices for
metallic systems at positive electronic temperature. A few other possible
applications are also discussed.Comment: 63 pages, 13 figure
Validation of Soft Classification Models using Partial Class Memberships: An Extended Concept of Sensitivity & Co. applied to the Grading of Astrocytoma Tissues
We use partial class memberships in soft classification to model uncertain
labelling and mixtures of classes. Partial class memberships are not restricted
to predictions, but may also occur in reference labels (ground truth, gold
standard diagnosis) for training and validation data.
Classifier performance is usually expressed as fractions of the confusion
matrix, such as sensitivity, specificity, negative and positive predictive
values. We extend this concept to soft classification and discuss the bias and
variance properties of the extended performance measures. Ambiguity in
reference labels translates to differences between best-case, expected and
worst-case performance. We show a second set of measures comparing expected and
ideal performance which is closely related to regression performance, namely
the root mean squared error RMSE and the mean absolute error MAE.
All calculations apply to classical crisp classification as well as to soft
classification (partial class memberships and/or one-class classifiers). The
proposed performance measures allow to test classifiers with actual borderline
cases. In addition, hardening of e.g. posterior probabilities into class labels
is not necessary, avoiding the corresponding information loss and increase in
variance.
We implement the proposed performance measures in the R package
"softclassval", which is available from CRAN and at
http://softclassval.r-forge.r-project.org.
Our reasoning as well as the importance of partial memberships for
chemometric classification is illustrated by a real-word application:
astrocytoma brain tumor tissue grading (80 patients, 37000 spectra) for finding
surgical excision borders. As borderline cases are the actual target of the
analytical technique, samples which are diagnosed to be borderline cases must
be included in the validation.Comment: The manuscript is accepted for publication in Chemometrics and
Intelligent Laboratory Systems. Supplementary figures and tables are at the
end of the pd
Google matrix analysis of directed networks
In past ten years, modern societies developed enormous communication and
social networks. Their classification and information retrieval processing
become a formidable task for the society. Due to the rapid growth of World Wide
Web, social and communication networks, new mathematical methods have been
invented to characterize the properties of these networks on a more detailed
and precise level. Various search engines are essentially using such methods.
It is highly important to develop new tools to classify and rank enormous
amount of network information in a way adapted to internal network structures
and characteristics. This review describes the Google matrix analysis of
directed complex networks demonstrating its efficiency on various examples
including World Wide Web, Wikipedia, software architecture, world trade, social
and citation networks, brain neural networks, DNA sequences and Ulam networks.
The analytical and numerical matrix methods used in this analysis originate
from the fields of Markov chains, quantum chaos and Random Matrix theory.Comment: 56 pages, 58 figures. Missed link added in network example of Fig3
Index Theorem and Overlap Formalism with Naive and Minimally Doubled Fermions
We present a theoretical foundation for the Index theorem in naive and
minimally doubled lattice fermions by studying the spectral flow of a Hermitean
version of Dirac operators. We utilize the point splitting method to implement
flavored mass terms, which play an important role in constructing proper
Hermitean operators. We show the spectral flow correctly detects the index of
the would-be zero modes which is determined by gauge field topology. Using the
flavored mass terms, we present new types of overlap fermions from the naive
fermion kernels, with a number of flavors that depends on the choice of the
mass terms. We succeed to obtain a single-flavor naive overlap fermion which
maintains hypercubic symmetry.Comment: 27 pages, 17 figures; references added, version accepted in JHE
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