A Bose-Einstein Approach to the Random Partitioning of an Integer

Consider N equally-spaced points on a circle of circumference N. Choose at random n points out of $N$ on this circle and append clockwise an arc of integral length k to each such point. The resulting random set is made of a random number of connected components. Questions such as the evaluation of the probability of random covering and parking configurations, number and length of the gaps are addressed. They are the discrete versions of similar problems raised in the continuum. For each value of k, asymptotic results are presented when n,N both go to infinity according to two different regimes. This model may equivalently be viewed as a random partitioning problem of N items into n recipients. A grand-canonical balls in boxes approach is also supplied, giving some insight into the multiplicities of the box filling amounts or spacings. The latter model is a k-nearest neighbor random graph with N vertices and kn edges. We shall also briefly consider the covering problem in the context of a random graph model with N vertices and n (out-degree 1) edges whose endpoints are no more bound to be neighbors

Random walks pertaining to a class of deterministic weighted graphs

In this note, we try to analyze and clarify the intriguing interplay between some counting problems related to specific thermalized weighted graphs and random walks consistent with such graphs

Metastable wetting

Consider a droplet of liquid on top of a grooved substrate. The wetting or not of a groove implies the crossing of a potential barrier as the interface has to distort, to hit the bottom of the groove. We start with computing the free energies of the dry and wet states in the context of a simple thermodynamical model before switching to a random microscopic version pertaining to the Solid-on-Solid (SOS) model. For some range in parameter space (Young angle, pressure difference, aspect ratio), the dry and wet states both share the same free energy, which means coexistence. We compute these coexistence lines together with the metastable regions. In the SOS case, we describe the dynamic transition between coexisting states in wetting. We show that the expected time to switch from one state to the other grows exponentially with the free energy barrier between the stable states and the saddle state, proportional to the groove's width. This random time appears to have an exponential-like distribution

Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation

A detailed study is presented for a large class of uncoupled continuous-time random walks (CTRWs). The master equation is solved for the Mittag-Leffler survival probability. The properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed. Finally, some common objections found in the literature are thoroughly reviewed.Comment: Preprint version of an already published paper. 8 page

Nonextensivity and multifractality in low-dimensional dissipative systems

Power-law sensitivity to initial conditions at the edge of chaos provides a natural relation between the scaling properties of the dynamics attractor and its degree of nonextensivity as prescribed in the generalized statistics recently introduced by one of us (C.T.) and characterized by the entropic index $q$. We show that general scaling arguments imply that $1/(1-q) = 1/\alpha_{min}-1/\alpha_{max}$, where $\alpha_{min}$ and $\alpha_{max}$ are the extremes of the multifractal singularity spectrum $f(\alpha)$ of the attractor. This relation is numerically checked to hold in standard one-dimensional dissipative maps. The above result sheds light on a long-standing puzzle concerning the relation between the entropic index $q$ and the underlying microscopic dynamics.Comment: 12 pages, TeX, 4 ps figure

Exact Scale Invariance in Mixing of Binary Candidates in Voting Model

We introduce a voting model and discuss the scale invariance in the mixing of candidates. The Candidates are classified into two categories $\mu\in \{0,1\}$ and are called as `binary' candidates. There are in total $N=N_{0}+N_{1}$ candidates, and voters vote for them one by one. The probability that a candidate gets a vote is proportional to the number of votes. The initial number of votes (`seed') of a candidate $\mu$ is set to be $s_{\mu}$. After infinite counts of voting, the probability function of the share of votes of the candidate $\mu$ obeys gamma distributions with the shape exponent $s_{\mu}$ in the thermodynamic limit $Z_{0}=N_{1}s_{1}+N_{0}s_{0}\to \infty$. Between the cumulative functions $\{x_{\mu}\}$ of binary candidates, the power-law relation $1-x_{1} \sim (1-x_{0})^{\alpha}$ with the critical exponent $\alpha=s_{1}/s_{0}$ holds in the region $1-x_{0},1-x_{1}<<1$. In the double scaling limit $(s_{1},s_{0})\to (0,0)$ and $Z_{0} \to \infty$ with $s_{1}/s_{0}=\alpha$ fixed, the relation $1-x_{1}=(1-x_{0})^{\alpha}$ holds exactly over the entire range $0\le x_{0},x_{1} \le 1$. We study the data on horse races obtained from the Japan Racing Association for the period 1986 to 2006 and confirm scale invariance.Comment: 19 pages, 8 figures, 2 table

Fractional Levy motion through path integrals

Fractional Levy motion (fLm) is the natural generalization of fractional Brownian motion in the context of self-similar stochastic processes and stable probability distributions. In this paper we give an explicit derivation of the propagator of fLm by using path integral methods. The propagators of Brownian motion and fractional Brownian motion are recovered as particular cases. The fractional diffusion equation corresponding to fLm is also obtained.Comment: 9 pages, minor changes, published versio

The pregnane X receptor drives sexually dimorphic hepatic changes in lipid and xenobiotic metabolism in response to gut microbiota in mice.

The gut microbiota-intestine-liver relationship is emerging as an important factor in multiple hepatic pathologies, but the hepatic sensors and effectors of microbial signals are not well defined. By comparing publicly available liver transcriptomics data from conventional vs. germ-free mice, we identified pregnane X receptor (PXR, NR1I2) transcriptional activity as strongly affected by the absence of gut microbes. Microbiota depletion using antibiotics in Pxr &lt;sup&gt;+/+&lt;/sup&gt; vs Pxr &lt;sup&gt;-/-&lt;/sup&gt; C57BL/6J littermate mice followed by hepatic transcriptomics revealed that most microbiota-sensitive genes were PXR-dependent in the liver in males, but not in females. Pathway enrichment analysis suggested that microbiota-PXR interaction controlled fatty acid and xenobiotic metabolism. We confirmed that antibiotic treatment reduced liver triglyceride content and hampered xenobiotic metabolism in the liver from Pxr &lt;sup&gt;+/+&lt;/sup&gt; but not Pxr &lt;sup&gt;-/-&lt;/sup&gt; male mice. These findings identify PXR as a hepatic effector of microbiota-derived signals that regulate the host's sexually dimorphic lipid and xenobiotic metabolisms in the liver. Thus, our results reveal a potential new mechanism for unexpected drug-drug or food-drug interactions. Video abstract

Investigation of NRXN1 deletions: Clinical and molecular characterization

Deletions at 2p16.3 involving exons of NRXN1 are associated with susceptibility for autism and schizophrenia, and similar deletions have been identified in individuals with developmental delay and dysmorphic features. We have identified 34 probands with exonic NRXN1 deletions following referral for clinical microarray‐based comparative genomic hybridization. To more firmly establish the full phenotypic spectrum associated with exonic NRXN1 deletions, we report the clinical features of 27 individuals with NRXN1 deletions, who represent 23 of these 34 families. The frequency of exonic NRXN1 deletions among our postnatally diagnosed patients (0.11%) is significantly higher than the frequency among reported controls (0.02%; P  = 6.08 × 10 −7 ), supporting a role for these deletions in the development of abnormal phenotypes. Generally, most individuals with NRXN1 exonic deletions have developmental delay (particularly speech), abnormal behaviors, and mild dysmorphic features. In our cohort, autism spectrum disorders were diagnosed in 43% (10/23), and 16% (4/25) had epilepsy. The presence of NRXN1 deletions in normal parents and siblings suggests reduced penetrance and/or variable expressivity, which may be influenced by genetic, environmental, and/or stochastic factors. The pathogenicity of these deletions may also be affected by the location of the deletion within the gene. Counseling should appropriately represent this spectrum of possibilities when discussing recurrence risks or expectations for a child found to have a deletion in NRXN1 . © 2013 Wiley Periodicals, Inc.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/97220/1/35780_ftp.pd