Sequential and asynchronous processes driven by stochastic or quantum grammars and their application to genomics: a survey

We present the formalism of sequential and asynchronous processes defined in terms of random or quantum grammars and argue that these processes have relevance in genomics. To make the article accessible to the non-mathematicians, we keep the mathematical exposition as elementary as possible, focusing on some general ideas behind the formalism and stating the implications of the known mathematical results. We close with a set of open challenging problems.Comment: Presented at the European Congress on Mathematical and Theoretical Biology, Dresden 18--22 July 200

On the physical relevance of random walks: an example of random walks on a randomly oriented lattice

Random walks on general graphs play an important role in the understanding of the general theory of stochastic processes. Beyond their fundamental interest in probability theory, they arise also as simple models of physical systems. A brief survey of the physical relevance of the notion of random walk on both undirected and directed graphs is given followed by the exposition of some recent results on random walks on randomly oriented lattices. It is worth noticing that general undirected graphs are associated with (not necessarily Abelian) groups while directed graphs are associated with (not necessarily Abelian) $C^*$-algebras. Since quantum mechanics is naturally formulated in terms of $C^*$-algebras, the study of random walks on directed lattices has been motivated lately by the development of the new field of quantum information and communication

Random environment on coloured trees

In this paper, we study a regular rooted coloured tree with random labels assigned to its edges, where the distribution of the label assigned to an edge depends on the colours of its endpoints. We obtain some new results relevant to this model and also show how our model generalizes many other probabilistic models, including random walk in random environment on trees, recursive distributional equations and multi-type branching random walk on $\mathbb{R}$.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ101 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Bindweeds or random walks in random environments on multiplexed trees and their asympotics

We report on the asymptotic behaviour of a new model of random walk, we term the bindweed model, evolving in a random environment on an infinite multiplexed tree. The term \textit{multiplexed} means that the model can be viewed as a nearest neighbours random walk on a tree whose vertices carry an internal degree of freedom from the finite set $\{1,...,d\}$, for some integer $d$. The consequence of the internal degree of freedom is an enhancement of the tree graph structure induced by the replacement of ordinary edges by multi-edges, indexed by the set $\{1,...,d\}\times\{1,...,d\}$. This indexing conveys the information on the internal degree of freedom of the vertices contiguous to each edge. The term \textit{random environment} means that the jumping rates for the random walk are a family of edge-indexed random variables, independent of the natural filtration generated by the random variables entering in the definition of the random walk; their joint distribution depends on the index of each component of the multi-edges. We study the large time asymptotic behaviour of this random walk and classify it with respect to positive recurrence or transience in terms of a specific parameter of the probability distribution of the jump rates. This classifying parameter is shown to coincide with the critical value of a matrix-valued multiplicative cascade on the ordinary tree (\textit{i.e.} the one without internal degrees of freedom attached to the vertices) having the same vertex set as the state space of the random walk. Only results are presented here since the detailed proofs will appear elsewhere

Dynamical systems with heavy-tailed random parameters

Motivated by the study of the time evolution of random dynamical systems arising in a vast variety of domains --- ranging from physics to ecology ---, we establish conditions for the occurrence of a non-trivial asymptotic behaviour for these systems in the absence of an ellipticity condition. More precisely, we classify these systems according to their type and --- in the recurrent case --- provide with sharp conditions quantifying the nature of recurrence by establishing which moments of passage times exist and which do not exist. The problem is tackled by mapping the random dynamical systems into Markov chains on $\mathbb{R}$ with heavy-tailed innovation and then using powerful methods stemming from Lyapunov functions to map the resulting Markov chains into positive semi-martingales.Comment: 24 page

Equilibrium statistical mechanics of frustrated spin glasses; a survey of mathematical results

After a rapid introduction to the physical motivations and a succinct presentation of heuristic results, this survey summarises the main mathematical results known on the Edwards-Anderson and the Sherrington-Kirkpatrick models of spin glasses. Although not complete proofs but rather sketches of the relevant steps and important ideas are given, only results for which complete proofs are known --- and for which the author has been able to reproduce all the intermediate logical steps --- are presented in the sections entitled `mathematical results'. This paper is intended to both physicists, interested to know which articles among the multitude of papers published on the subject go beyond the heuristic arguments to obtain rigorous irrefutable results, but also to the mathematicians, interested in finding out how rich is the physical intuitive way of thinking and in being inspired by the heuristic results in view of a mathematical rigorisation. An extended, but not exhaustive, bibliography is included.Comment: 29 pages, postscrip

THERMODYNAMICS OF A BROWNIAN BRIDGE POLYMER MODEL IN A RANDOM ENVIRONMENT

We consider a directed random walk making either 0 or $+1$ moves and a Brownian bridge, independent of the walk, conditioned to arrive at point $b$ on time $T$. The Hamiltonian is defined as the sum of the square of increments of the bridge between the moments of jump of the random walk and interpreted as an energy function over the bridge connfiguration; the random walk acts as the random environment. This model provides a continuum version of a model with some relevance to protein conformation. The thermodynamic limit of the specific free energy is shown to exist and to be self-averaging, i.e. it is equal to a trivial --- explicitly computed --- random variable. An estimate of the asymptotic behaviour of the ground state energy is also obtained.Comment: 20 pages, uuencoded postscrip

MRCQuant- an accurate LC-MS relative isotopic quantification algorithm on TOF instruments

<p>Abstract</p> <p>Background</p> <p>Relative isotope abundance quantification, which can be used for peptide identification and differential peptide quantification, plays an important role in liquid chromatography-mass spectrometry (LC-MS)-based proteomics. However, several major issues exist in the relative isotopic quantification of peptides on time-of-flight (TOF) instruments: LC peak boundary detection, thermal noise suppression, interference removal and mass drift correction. We propose to use the Maximum Ratio Combining (MRC) method to extract MS signal templates for interference detection/removal and LC peak boundary detection. In our method, MRCQuant, MS templates are extracted directly from experimental values, and the mass drift in each LC-MS run is automatically captured and compensated. We compared the quantification accuracy of MRCQuant to that of another representative LC-MS quantification algorithm (msInspect) using datasets downloaded from a public data repository.</p> <p>Results</p> <p>MRCQuant showed significant improvement in the number of accurately quantified peptides.</p> <p>Conclusions</p> <p>MRCQuant effectively addresses major issues in the relative quantification of LC-MS-based proteomics data, and it provides improved performance in the quantification of low abundance peptides.</p

Cerebellum in timing control: Evidence from contingent negative variation after cerebellar tDCS

Background and aims Timing control is defined as the ability to quantify time. The temporal estimation of supra-seconds range is generally seen as a conscious cognitive process, while the sub-seconds range is a more automatic cognitive process. It is accepted that cerebellum contributes to temporal processing, but its function is still debated. The aim of this research was to better explore the role of cerebellum in timing control. We transitorily inhibited cerebellar activity and studied the effects on CNV components in healthy subjects. Methods Sixteen healthy subjects underwent a S1-S2 duration discrimination motor task, prior and after cathodal and sham cerebellar tDCS, in two separate sessions. In S1-S2 task they had to judge whether the duration of a probe interval trial was shorter (Short-ISI-trial:800 ms), longer (long-ISI-trail:1600 ms), or equal to the Target interval of 1200 ms. For each interval trial for both tDCS sessions, we measured: total and W2-CNV areas, the RTs of correct responses and the absolute number of errors prior and after tDCS. Results After cathodal tDCS a significant reduction in total-CNV and W2-CNV amplitudes selectively emerged for Short (p &lt; 0.001; p = 0.003 respectively) and Target-ISI-trial (total-CNV: p &lt; 0.001; W2-CNV:p = 0.003); similarly, a significant higher number of errors emerged for Short (p = 0.004) and Target-ISI-trial (p = 0.07) alone. No differences were detected for Longer-ISI-trials and after sham stimulation. Conclusions These data indicate that cerebellar inhibition selectively altered the ability to make time estimations for second and sub-second intervals. We speculate that cerebellum regulates the attentional mechanisms of automatic timing control by making predictions of interval timing