63 research outputs found
-Martin boundary of killed random walks in the quadrant
We compute the -Martin boundary of two-dimensional small steps random
walks killed at the boundary of the quarter plane. We further provide explicit
expressions for the (generating functions of the) discrete -harmonic
functions. Our approach is uniform in , and shows that there are three
regimes for the Martin boundary.Comment: 18 pages, 2 figures, to appear in S\'eminaire de Probabilit\'e
Passage time from four to two blocks of opinions in the voter model and walks in the quarter plane
A random walk in spatially homogeneous in the interior, absorbed at
the axes, starting from an arbitrary point and with step
probabilities drawn on Figure 1 is considered. The trivariate generating
function of probabilities that the random walk hits a given point at a given time is made explicit. Probabilities of absorption
at a given time and at a given axis are found, and their precise asymptotic
is derived as the time . The equivalence of two typical ways of
conditioning this random walk to never reach the axes is established. The
results are also applied to the analysis of the voter model with two candidates
and initially, in the population , four connected blocks of same opinions.
Then, a citizen changes his mind at a rate proportional to the number of its
neighbors that disagree with him. Namely, the passage from four to two blocks
of opinions is studied.Comment: 11 pages, 1 figur
On the functions counting walks with small steps in the quarter plane
Models of spatially homogeneous walks in the quarter plane
with steps taken from a subset of the set of jumps to the eight
nearest neighbors are considered. The generating function of the numbers of such walks starting at the origin and
ending at after steps is studied. For all
non-singular models of walks, the functions and are continued as multi-valued functions on having
infinitely many meromorphic branches, of which the set of poles is identified.
The nature of these functions is derived from this result: namely, for all the
51 walks which admit a certain infinite group of birational transformations of
, the interval of variation of splits into
two dense subsets such that the functions and are shown to be holonomic for any from the one of them and
non-holonomic for any from the other. This entails the non-holonomy of
, and therefore proves a conjecture of
Bousquet-M\'elou and Mishna.Comment: 40 pages, 17 figure
Step-wise responses in mesoscopic glassy systems: a mean field approach
We study statistical properties of peculiar responses in glassy systems at
mesoscopic scales based on a class of mean-field spin-glass models which
exhibit 1 step replica symmetry breaking. Under variation of a generic external
field, a finite-sized sample of such a system exhibits a series of step wise
responses which can be regarded as a finger print of the sample. We study in
detail the statistical properties of the step structures based on a low
temperature expansion approach and a replica approach. The spacings between the
steps vanish in the thermodynamic limit so that arbitrary small but finite
variation of the field induce infinitely many level crossings in the
thermodynamic limit leading to a static chaos effect which yields a
self-averaging, smooth macroscopic response. We also note that there is a
strong analogy between the problem of step-wise responses in glassy systems at
mesoscopic scales and intermittency in turbulent flows due to shocks.Comment: 50 pages, 18 figures, revised versio
Martin boundary of a reflected random walk on a half-space
The complete representation of the Martin compactification for reflected
random walks on a half-space is obtained. It is shown that the
full Martin compactification is in general not homeomorphic to the ``radial''
compactification obtained by Ney and Spitzer for the homogeneous random walks
in : convergence of a sequence of points to a
point of on the Martin boundary does not imply convergence of the sequence
on the unit sphere . Our approach relies on the large
deviation properties of the scaled processes and uses Pascal's method combined
with the ratio limit theorem. The existence of non-radial limits is related to
non-linear optimal large deviation trajectories.Comment: 42 pages, preprint, CNRS UMR 808
Mass measurements of As, Se and Br nuclei and their implication on the proton-neutron interaction strength towards the N=Z line
Mass measurements of the nuclides 69As, 70,71Se, and 71Br, produced via fragmentation of a 124Xe primary beam at the Fragment Separator (FRS) at GSI, have been performed with the multiple-reflection time-of-flight mass spectrometer (MR-TOF-MS) of the FRS Ion Catcher with an unprecedented mass resolving power of almost 1000000. Such high resolving power is the only way to achieve accurate results and resolve overlapping peaks of short-lived exotic nuclei, whose total number of accumulated events is always limited. For the nuclide 69As, this is the first direct mass measurement. A mass uncertainty of 22 keV was achieved with only ten events. For the nuclide 70Se, a mass uncertainty of 2.6 keV was obtained, corresponding to a relative accuracy of δm/m=4.0×10−8, with less than 500 events. The masses of the nuclides 71Se and 71Br have been measured with an uncertainty of 23 and 16 keV, respectively. Our results for the nuclides 70,71Se and 71Br are in good agreement with the 2016 Atomic Mass Evaluation, and our result for the nuclide 69As resolves the discrepancy between the previous indirect measurements. We measured also the mass of the molecule 14N15N40Ar (A=69) with a relative accuracy of δm/m=1.7×10−8, the highest yet achieved with an MR-TOF-MS. Our results show that the measured restrengthening of the proton-neutron interaction (δVpn) for odd-odd nuclei along the N=Z line above Z=29 (recently extended to Z=37) is hardly evident at the N−Z=2 line, and not evident at the N−Z=4 line. Nevertheless, detailed structure of δVpn along the N−Z=2 and N−Z=4 lines, confirmed by our mass measurements, may provide a hint regarding the ongoing ≈500 keV discrepancy in the mass value of the nuclide 70Br, which prevents including it in the world average of Ft value for superallowed 0+→0+β decays. The reported work sets the stage for mass measurements with the FRS Ion Catcher of nuclei at and beyond the N=Z line in the same region of the nuclear chart, including the nuclide 70Br.peerReviewe
Pb and Cd on growth, leaf ultrastructure and essential oil yield mint (Mentha arvensis L.)
ATHENA: A knowledge-based hybrid backpropagation-grammatical evolution neural network algorithm for discovering epistasis among quantitative trait Loci
<p>Abstract</p> <p>Background</p> <p>Growing interest and burgeoning technology for discovering genetic mechanisms that influence disease processes have ushered in a flood of genetic association studies over the last decade, yet little heritability in highly studied complex traits has been explained by genetic variation. Non-additive gene-gene interactions, which are not often explored, are thought to be one source of this "missing" heritability.</p> <p>Methods</p> <p>Stochastic methods employing evolutionary algorithms have demonstrated promise in being able to detect and model gene-gene and gene-environment interactions that influence human traits. Here we demonstrate modifications to a neural network algorithm in ATHENA (the Analysis Tool for Heritable and Environmental Network Associations) resulting in clear performance improvements for discovering gene-gene interactions that influence human traits. We employed an alternative tree-based crossover, backpropagation for locally fitting neural network weights, and incorporation of domain knowledge obtainable from publicly accessible biological databases for initializing the search for gene-gene interactions. We tested these modifications <it>in silico </it>using simulated datasets.</p> <p>Results</p> <p>We show that the alternative tree-based crossover modification resulted in a modest increase in the sensitivity of the ATHENA algorithm for discovering gene-gene interactions. The performance increase was highly statistically significant when backpropagation was used to locally fit NN weights. We also demonstrate that using domain knowledge to initialize the search for gene-gene interactions results in a large performance increase, especially when the search space is larger than the search coverage.</p> <p>Conclusions</p> <p>We show that a hybrid optimization procedure, alternative crossover strategies, and incorporation of domain knowledge from publicly available biological databases can result in marked increases in sensitivity and performance of the ATHENA algorithm for detecting and modelling gene-gene interactions that influence a complex human trait.</p
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