2,725 research outputs found
Statistics of the Number of Zero Crossings : from Random Polynomials to Diffusion Equation
We consider a class of real random polynomials, indexed by an integer d, of
large degree n and focus on the number of real roots of such random
polynomials. The probability that such polynomials have no real root in the
interval [0,1] decays as a power law n^{-\theta(d)} where \theta(d)>0 is the
exponent associated to the decay of the persistence probability for the
diffusion equation with random initial conditions in space dimension d. For n
even, the probability that such polynomials have no root on the full real axis
decays as n^{-2(\theta(d) + \theta(2))}. For d=1, this connection allows for a
physical realization of real random polynomials. We further show that the
probability that such polynomials have exactly k real roots in [0,1] has an
unusual scaling form given by n^{-\tilde \phi(k/\log n)} where \tilde \phi(x)
is a universal large deviation function.Comment: 4 pages, 3 figures. Minor changes. Accepted version in Phys. Rev.
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Causal connectivity of evolved neural networks during behavior
To show how causal interactions in neural dynamics are modulated by behavior, it is valuable to analyze these interactions without perturbing or lesioning the neural mechanism. This paper proposes a method, based on a graph-theoretic extension of vector autoregressive modeling and 'Granger causality,' for characterizing causal interactions generated within intact neural mechanisms. This method, called 'causal connectivity analysis' is illustrated via model neural networks optimized for controlling target fixation in a simulated head-eye system, in which the structure of the environment can be experimentally varied. Causal connectivity analysis of this model yields novel insights into neural mechanisms underlying sensorimotor coordination. In contrast to networks supporting comparatively simple behavior, networks supporting rich adaptive behavior show a higher density of causal interactions, as well as a stronger causal flow from sensory inputs to motor outputs. They also show different arrangements of 'causal sources' and 'causal sinks': nodes that differentially affect, or are affected by, the remainder of the network. Finally, analysis of causal connectivity can predict the functional consequences of network lesions. These results suggest that causal connectivity analysis may have useful applications in the analysis of neural dynamics
Condensation of the roots of real random polynomials on the real axis
We introduce a family of real random polynomials of degree n whose
coefficients a_k are symmetric independent Gaussian variables with variance
= e^{-k^\alpha}, indexed by a real \alpha \geq 0. We compute exactly
the mean number of real roots for large n. As \alpha is varied, one finds
three different phases. First, for 0 \leq \alpha \sim
(\frac{2}{\pi}) \log{n}. For 1 < \alpha < 2, there is an intermediate phase
where grows algebraically with a continuously varying exponent,
\sim \frac{2}{\pi} \sqrt{\frac{\alpha-1}{\alpha}} n^{\alpha/2}. And finally for
\alpha > 2, one finds a third phase where \sim n. This family of real
random polynomials thus exhibits a condensation of their roots on the real line
in the sense that, for large n, a finite fraction of their roots /n are
real. This condensation occurs via a localization of the real roots around the
values \pm \exp{[\frac{\alpha}{2}(k+{1/2})^{\alpha-1} ]}, 1 \ll k \leq n.Comment: 13 pages, 2 figure
Sparse Exploratory Factor Analysis
Sparse principal component analysis is a very active research area in the last decade. It produces component loadings with many zero entries which facilitates their interpretation and helps avoid redundant variables. The classic factor analysis is another popular dimension reduction technique which shares similar interpretation problems and could greatly benefit from sparse solutions. Unfortunately, there are very few works considering sparse versions of the classic factor analysis. Our goal is to contribute further in this direction. We revisit the most popular procedures for exploratory factor analysis, maximum likelihood and least squares. Sparse factor loadings are obtained for them by, first, adopting a special reparameterization and, second, by introducing additional [Formula: see text]-norm penalties into the standard factor analysis problems. As a result, we propose sparse versions of the major factor analysis procedures. We illustrate the developed algorithms on well-known psychometric problems. Our sparse solutions are critically compared to ones obtained by other existing methods
Histopathological evaluation of placentas after diagnosis of maternal SARS-CoV-2 infection.
Background:The impact of maternal SARS-CoV-2 infection on placental histopathology is not well known. Objectives:To determine if significant placental histopathological changes occur after diagnosis of SARS-CoV-2 infection in pregnancy and whether these changes are correlated with the presence or absence of symptoms associated with infection. Study Design:Retrospective cohort study of women diagnosed with SARS-CoV-2 infection who delivered at a single center from April 9th to April 27th, 2020, and had placental specimens reviewed by pathology. Women with singleton gestations and laboratory-confirmed SARS-CoV-2 infection were eligible for inclusion. Historical controls selected from a cohort of women who delivered 6 months prior to the study period were matched in a 1:1 fashion by week of gestation at delivery. Histopathological characteristics were evaluated in each placenta and the incidence of these findings were compared between placentas after diagnosis of maternal SARS-CoV-2 infection and historical controls, as well as between placentas from patients with or without typical symptoms related to infection. Statistical analysis included use of Wilcoxon rank sum test and Fisher\u27s exact test for comparison of categorical and continuous variables. Statistical significance was defined as P value \u3c 0.05. Results:A total of 50 placentas after diagnosis of maternal SARS-CoV-2 infection and 50 historical controls were analyzed. Among placentas from patients diagnosed with SARS-CoV-2 infection, 3 (6%) were preterm (33 3/7, 34 6/7 and 36 6/7 weeks of gestation), 16 (32%) were from patients with typical symptoms related to infection and 34 (68%) were from patients without typical symptoms related to the infection. All patients had diagnosis of SARS-CoV-2 infection in the third trimester. Decidual vasculopathy was not visualized in any of the placentas from patients diagnosed with SARS-CoV-2 infection. There was no statistically significant difference in placental histopathological characteristics between the groups. SARS-CoV-2 testing for all neonates at 24 hours of life was negative. Conclusions:Based on our data, there are no significant placental histopathological changes that occur after diagnosis of SARS-CoV-2 infection in the third trimester of pregnancy compared to a gestational age-matched historical control group. Similar incidences of histopathological findings were also discovered when comparing placentas from patients with SARS-CoV-2 infection with or without the presence of symptoms typically related to infection
The lowest eigenvalue of Jacobi random matrix ensembles and Painlev\'e VI
We present two complementary methods, each applicable in a different range,
to evaluate the distribution of the lowest eigenvalue of random matrices in a
Jacobi ensemble. The first method solves an associated Painleve VI nonlinear
differential equation numerically, with suitable initial conditions that we
determine. The second method proceeds via constructing the power-series
expansion of the Painleve VI function. Our results are applied in a forthcoming
paper in which we model the distribution of the first zero above the central
point of elliptic curve L-function families of finite conductor and of
conjecturally orthogonal symmetry.Comment: 30 pages, 2 figure
CD-independent subsets in meet-distributive lattices
A subset of a finite lattice is CD-independent if the meet of any two
incomparable elements of equals 0. In 2009, Cz\'edli, Hartmann and Schmidt
proved that any two maximal CD-independent subsets of a finite distributive
lattice have the same number of elements. In this paper, we prove that if
is a finite meet-distributive lattice, then the size of every CD-independent
subset of is at most the number of atoms of plus the length of . If,
in addition, there is no three-element antichain of meet-irreducible elements,
then we give a recursive description of maximal CD-independent subsets.
Finally, to give an application of CD-independent subsets, we give a new
approach to count islands on a rectangular board.Comment: 14 pages, 4 figure
Stochastic Budget Optimization in Internet Advertising
Internet advertising is a sophisticated game in which the many advertisers
"play" to optimize their return on investment. There are many "targets" for the
advertisements, and each "target" has a collection of games with a potentially
different set of players involved. In this paper, we study the problem of how
advertisers allocate their budget across these "targets". In particular, we
focus on formulating their best response strategy as an optimization problem.
Advertisers have a set of keywords ("targets") and some stochastic information
about the future, namely a probability distribution over scenarios of cost vs
click combinations. This summarizes the potential states of the world assuming
that the strategies of other players are fixed. Then, the best response can be
abstracted as stochastic budget optimization problems to figure out how to
spread a given budget across these keywords to maximize the expected number of
clicks.
We present the first known non-trivial poly-logarithmic approximation for
these problems as well as the first known hardness results of getting better
than logarithmic approximation ratios in the various parameters involved. We
also identify several special cases of these problems of practical interest,
such as with fixed number of scenarios or with polynomial-sized parameters
related to cost, which are solvable either in polynomial time or with improved
approximation ratios. Stochastic budget optimization with scenarios has
sophisticated technical structure. Our approximation and hardness results come
from relating these problems to a special type of (0/1, bipartite) quadratic
programs inherent in them. Our research answers some open problems raised by
the authors in (Stochastic Models for Budget Optimization in Search-Based
Advertising, Algorithmica, 58 (4), 1022-1044, 2010).Comment: FINAL versio
A statistical mechanics approach to autopoietic immune networks
The aim of this work is to try to bridge over theoretical immunology and
disordered statistical mechanics. Our long term hope is to contribute to the
development of a quantitative theoretical immunology from which practical
applications may stem. In order to make theoretical immunology appealing to the
statistical physicist audience we are going to work out a research article
which, from one side, may hopefully act as a benchmark for future improvements
and developments, from the other side, it is written in a very pedagogical way
both from a theoretical physics viewpoint as well as from the theoretical
immunology one.
Furthermore, we have chosen to test our model describing a wide range of
features of the adaptive immune response in only a paper: this has been
necessary in order to emphasize the benefit available when using disordered
statistical mechanics as a tool for the investigation. However, as a
consequence, each section is not at all exhaustive and would deserve deep
investigation: for the sake of completeness, we restricted details in the
analysis of each feature with the aim of introducing a self-consistent model.Comment: 22 pages, 14 figur
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