Ergodic transition in a simple model of the continuous double auction

We study a phenomenological model for the continuous double auction, whose aggregate order process is equivalent to two independent M/M/1 queues. The continuous double auction defines a continuous-time random walk for trade prices. The conditions for ergodicity of the auction are derived and, as a consequence, three possible regimes in the behavior of prices and logarithmic returns are observed. In the ergodic regime, prices are unstable and one can observe a heteroskedastic behavior in the logarithmic returns. On the contrary, non-ergodicity triggers stability of prices, even if two different regimes can be seen

Biased estimators on Quotient spaces

International audienceUsual statistics are defined, studied and implemented on Euclidean spaces. But what about statistics on other mathematical spaces, like manifolds with additional properties: Lie groups, Quotient spaces, Stratified spaces etc. How can we describe the interaction between statistics and geometry? The structure of Quotient space in particular is widely used to model data, for example every time one deals with shape data. These can be shapes of constellations in Astronomy, shapes of human organs in Computational Anatomy, shapes of skulls in Palaeontology, etc. Given this broad field of applications, statistics on shapes -and more generally on observations belonging to quotient spaces- have been studied since the 1980's. However, most theories model the variability in the shapes but do not take into account the noise on the observations themselves. In this paper, we show that statistics on quotient spaces are biased and even inconsistent when one takes into account the noise. In particular, some algorithms of template estimation in Computational Anatomy are biased and inconsistent. Our development thus gives a first theoretical geometric explanation of an experimentally observed phenomenon. A biased estimator is not necessarily a problem. In statistics, it is a general rule of thumb that a bias can be neglected for example when it represents less than 0.25 of the variance of the estimator. We can also think about neglecting the bias when it is low compared to the signal we estimate. In view of the applications, we thus characterize geometrically the situations when the bias can be neglected with respect to the situations when it must be corrected

Minimal Obstructions for Partial Representations of Interval Graphs

Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals. We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solving this problem. In this paper, we characterize the minimal obstructions which make partial representations non-extendible. This generalizes Lekkerkerker and Boland's characterization of the minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to a linear-time certifying algorithm for partial representation extension

Dual random fragmentation and coagulation and an application to the genealogy of Yule processes

The purpose of this work is to describe a duality between a fragmentation associated to certain Dirichlet distributions and a natural random coagulation. The dual fragmentation and coalescent chains arising in this setting appear in the description of the genealogy of Yule processes.Comment: 14 page

Signal processing with Levy information

Levy processes, which have stationary independent increments, are ideal for modelling the various types of noise that can arise in communication channels. If a Levy process admits exponential moments, then there exists a parametric family of measure changes called Esscher transformations. If the parameter is replaced with an independent random variable, the true value of which represents a "message", then under the transformed measure the original Levy process takes on the character of an "information process". In this paper we develop a theory of such Levy information processes. The underlying Levy process, which we call the fiducial process, represents the "noise type". Each such noise type is capable of carrying a message of a certain specification. A number of examples are worked out in detail, including information processes of the Brownian, Poisson, gamma, variance gamma, negative binomial, inverse Gaussian, and normal inverse Gaussian type. Although in general there is no additive decomposition of information into signal and noise, one is led nevertheless for each noise type to a well-defined scheme for signal detection and enhancement relevant to a variety of practical situations.Comment: 27 pages. Version to appear in: Proc. R. Soc. London

Initial conditions, Discreteness and non-linear structure formation in cosmology

In this lecture we address three different but related aspects of the initial continuous fluctuation field in standard cosmological models. Firstly we discuss the properties of the so-called Harrison-Zeldovich like spectra. This power spectrum is a fundamental feature of all current standard cosmological models. In a simple classification of all stationary stochastic processes into three categories, we highlight with the name ``super-homogeneous'' the properties of the class to which models like this, with $P(0)=0$, belong. In statistical physics language they are well described as glass-like. Secondly, the initial continuous density field with such small amplitude correlated Gaussian fluctuations must be discretised in order to set up the initial particle distribution used in gravitational N-body simulations. We discuss the main issues related to the effects of discretisation, particularly concerning the effect of particle induced fluctuations on the statistical properties of the initial conditions and on the dynamical evolution of gravitational clustering.Comment: 28 pages, 1 figure, to appear in Proceedings of 9th Course on Astrofundamental Physics, International School D. Chalonge, Kluwer, eds N.G. Sanchez and Y.M. Pariiski, uses crckapb.st pages, 3 figure, ro appear in Proceedings of 9th Course on Astrofundamental Physics, International School D. Chalonge, Kluwer, Eds. N.G. Sanchez and Y.M. Pariiski, uses crckapb.st

Exact, time-independent estimation of clone size distributions in normal and mutated cells

Biological tools such as genetic lineage tracing, three dimensional confocal microscopy and next generation DNA sequencing are providing new ways to quantify the distribution of clones of normal and mutated cells. Population-wide clone size distributions in vivo are complicated by multiple cell types, and overlapping birth and death processes. This has led to the increased need for mathematically informed models to understand their biological significance. Standard approaches usually require knowledge of clonal age. We show that modelling on clone size independent of time is an alternative method that offers certain analytical advantages; it can help parameterize these models, and obtain distributions for counts of mutated or proliferating cells, for example. When applied to a general birth-death process common in epithelial progenitors this takes the form of a gamblers ruin problem, the solution of which relates to counting Motzkin lattice paths. Applying this approach to mutational processes, an alternative, exact, formulation of the classic Luria Delbruck problem emerges. This approach can be extended beyond neutral models of mutant clonal evolution, and also describe some distributions relating to sub-clones within a tumour. The approaches above are generally applicable to any Markovian branching process where the dynamics of different "coloured" daughter branches are of interest

Theoretical size distribution of fossil taxa: analysis of a null model

BACKGROUND: This article deals with the theoretical size distribution (of number of sub-taxa) of a fossil taxon arising from a simple null model of macroevolution. MODEL: New species arise through speciations occurring independently and at random at a fixed probability rate, while extinctions either occur independently and at random (background extinctions) or cataclysmically. In addition new genera are assumed to arise through speciations of a very radical nature, again assumed to occur independently and at random at a fixed probability rate. CONCLUSION: The size distributions of the pioneering genus (following a cataclysm) and of derived genera are determined. Also the distribution of the number of genera is considered along with a comparison of the probability of a monospecific genus with that of a monogeneric family

Societal Learning in Epidemics: Intervention Effectiveness during the 2003 SARS Outbreak in Singapore

BACKGROUND: Rapid response to outbreaks of emerging infectious diseases is impeded by uncertain diagnoses and delayed communication. Understanding the effect of inefficient response is a potentially important contribution of epidemic theory. To develop this understanding we studied societal learning during emerging outbreaks wherein patient removal accelerates as information is gathered and disseminated. METHODS AND FINDINGS: We developed an extension of a standard outbreak model, the simple stochastic epidemic, which accounts for societal learning. We obtained expressions for the expected outbreak size and the distribution of epidemic duration. We found that rapid learning noticeably affects the final outbreak size even when learning exhibits diminishing returns (relaxation). As an example, we estimated the learning rate for the 2003 outbreak of severe acute respiratory syndrome (SARS) in Singapore. Evidence for relaxation during the first eight weeks of the outbreak was inconclusive. We estimated that if societal learning had occurred at half the actual rate, the expected final size of the outbreak would have reached nearly 800 cases, more than three times the observed number of infections. By contrast, the expected outbreak size for societal learning twice as effective was 116 cases. CONCLUSION: These results show that the rate of societal learning can greatly affect the final size of disease outbreaks, justifying investment in early warning systems and attentiveness to disease outbreak by both government authorities and the public. We submit that the burden of emerging infections, including the risk of a global pandemic, could be efficiently reduced by improving procedures for rapid detection of outbreaks, alerting public health officials, and aggressively educating the public at the start of an outbreak

The broad-spectrum antimicrobial potential of [Mn(CO)4(S2CNMe(CH2CO2H))], a water-soluble CO-releasing molecule (CORM-401): intracellular accumulation, transcriptomic and statistical analyses, and membrane Polarization

Aims: Carbon monoxide (CO)-releasing molecules (CORMs) are candidates for animal and antimicrobial therapeutics. We aimed to probe the antimicrobial potential of a novel manganese CORM. Results: [Mn(CO)4S2CNMe(CH2CO2H)], CORM-401, inhibits growth of Escherichia coli and several antibiotic-resistant clinical pathogens. CORM-401 releases CO that binds oxidases in vivo, but is an ineffective respiratory inhibitor. Extensive CORM accumulation (assayed as intracellular manganese) accompanies antimicrobial activity. CORM-401 stimulates respiration, polarizes the cytoplasmic membrane in an uncoupler-like manner, and elicits loss of intracellular potassium and zinc. Transcriptomics and mathematical modeling of transcription factor activities reveal a multifaceted response characterized by elevated expression of genes encoding potassium uptake, efflux pumps, and envelope stress responses. Regulators implicated in stress responses (CpxR), respiration (Arc, Fnr), methionine biosynthesis (MetJ), and iron homeostasis (Fur) are significantly disturbed. Although CORM-401 reduces bacterial growth in combination with cefotaxime and trimethoprim, fractional inhibition studies reveal no interaction. Innovation: We present the most detailed microbiological analysis yet of a CORM that is not a ruthenium carbonyl. We demonstrate CO-independent striking effects on the bacterial membrane and global transcriptomic responses. Conclusions: CORM-401, contrary to our expectations of a CO delivery vehicle, does not inhibit respiration. It accumulates in the cytoplasm, acts like an uncoupler in disrupting cytoplasmic ion balance, and triggers multiple effects, including osmotic stress and futile respiration