Eigenfunctions of the Laplacian and associated Ruelle operator

Let $\Gamma$ be a co-compact Fuchsian group of isometries on the Poincar\'e disk \DD and $\Delta$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $\Delta$, equivariant by $\Gamma$ with real eigenvalue $\lambda=-s(1-s)$, where $s={1/2}+ it$, admits an integral representation by a distribution \dd_{f,s} (the Helgason distribution) which is equivariant by $\Gamma$ and supported at infinity \partial\DD=\SS^1. The geodesic flow on the compact surface \DD/\Gamma is conjugate to a suspension over a natural extension of a piecewise analytic map T:\SS^1\to\SS^1, the so-called Bowen-Series transformation. Let $\ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-s\ln |T'|$. M. Pollicott showed that \dd_{f,s} is an eigenfunction of the dual operator $\ll_s^*$ for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction $\psi_{f,s}$ of $\ll_s$ for the eigenvalue 1, given by an integral formula \psi_{f,s} (\xi)=\int \frac{J(\xi,\eta)}{|\xi-\eta|^{2s}} \dd_{f,s} (d\eta), \noindent where $J(\xi,\eta)$ is a $\{0,1\}$-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface \DD/\Gamma

On the complexity of some birational transformations

Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines, and relies mainly on univariate polynomial algebra, the second approach is a singularity analysis, and the third method is more numerical, using integer arithmetics. Each method has its own domain of application, but they give corroborating results, and lead us to a conjecture on the complexity of a class of maps constructed from matrix inversions

Supersymmetric Intersecting Branes on the Waves

We construct a general family of supersymmetric solutions in time- and space-dependent wave backgrounds in general supergravity theories describing single and intersecting p-branes embedded into time-dependent dilaton-gravity plane waves of an arbitrary (isotropic) profile, with the brane world-volume aligned parallel to the propagation direction of the wave. We discuss how many degrees of freedom we have in the solutions. We also propose that these solutions can be used to describe higher-dimensional time-dependent "black holes", and discuss their property briefly.Comment: 12 pages, LaTe

Cross sections for geodesic flows and \alpha-continued fractions

We adjust Arnoux's coding, in terms of regular continued fractions, of the geodesic flow on the modular surface to give a cross section on which the return map is a double cover of the natural extension for the \alpha-continued fractions, for each $\alpha$ in (0,1]. The argument is sufficiently robust to apply to the Rosen continued fractions and their recently introduced \alpha-variants.Comment: 20 pages, 2 figure

Evaluation of elicitation methods to quantify Bayes linear models

The Bayes linear methodology allows decision makers to express their subjective beliefs and adjust these beliefs as observations are made. It is similar in spirit to probabilistic Bayesian approaches, but differs as it uses expectation as its primitive. While substantial work has been carried out in Bayes linear analysis, both in terms of theory development and application, there is little published material on the elicitation of structured expert judgement to quantify models. This paper investigates different methods that could be used by analysts when creating an elicitation process. The theoretical underpinnings of the elicitation methods developed are explored and an evaluation of their use is presented. This work was motivated by, and is a precursor to, an industrial application of Bayes linear modelling of the reliability of defence systems. An illustrative example demonstrates how the methods can be used in practice

Spatial Guilds in the Serengeti Food Web Revealed by a Bayesian Group Model

Food webs, networks of feeding relationships among organisms, provide fundamental insights into mechanisms that determine ecosystem stability and persistence. Despite long-standing interest in the compartmental structure of food webs, past network analyses of food webs have been constrained by a standard definition of compartments, or modules, that requires many links within compartments and few links between them. Empirical analyses have been further limited by low-resolution data for primary producers. In this paper, we present a Bayesian computational method for identifying group structure in food webs using a flexible definition of a group that can describe both functional roles and standard compartments. The Serengeti ecosystem provides an opportunity to examine structure in a newly compiled food web that includes species-level resolution among plants, allowing us to address whether groups in the food web correspond to tightly-connected compartments or functional groups, and whether network structure reflects spatial or trophic organization, or a combination of the two. We have compiled the major mammalian and plant components of the Serengeti food web from published literature, and we infer its group structure using our method. We find that network structure corresponds to spatially distinct plant groups coupled at higher trophic levels by groups of herbivores, which are in turn coupled by carnivore groups. Thus the group structure of the Serengeti web represents a mixture of trophic guild structure and spatial patterns, in contrast to the standard compartments typically identified in ecological networks. From data consisting only of nodes and links, the group structure that emerges supports recent ideas on spatial coupling and energy channels in ecosystems that have been proposed as important for persistence.Comment: 28 pages, 6 figures (+ 3 supporting), 2 tables (+ 4 supporting

Studies of the dose-effect relation

Dose-effect relations and, specifically, cell survival curves are surveyed with emphasis on the interplay of the random factors — biological variability, stochastic reaction of the cell, and the statistics of energy deposition —that co-determine their shape. The global parameters mean inactivation dose, , and coefficient of variance, V, represent this interplay better than conventional parameters. Mechanisms such as lesion interaction, misrepair, repair overload, or repair depletion have been invoked to explain sigmoid dose dependencies, but these notions are partly synonymous and are largely undistinguishable on the basis of observed dose dependencies. All dose dependencies reflect, to varying degree, the microdosimetric fluctuations of energy deposition, and these have certain implications, e.g. the linearity of the dose dependence at small doses, that apply regardless of unresolved molecular mechanisms of cellular radiation action

Early developmental pathways to childhood symptoms of attention-deficit hyperactivity disorder, anxiety and autism spectrum disorder

Background Children with autism spectrum disorder (ASD) often have co-occurring symptoms of attention-deficit/hyperactivity disorder (ADHD) and/or anxiety. It is unclear whether these disorders arise from shared or distinct developmental pathways. We explored this question by testing the specificity of early-life (infant and toddler) predictors of mid-childhood ADHD and anxiety symptoms compared to ASD symptoms. Methods Infants (n = 104) at high and low familial risk for ASD took part in research assessments at 7, 14, 24 and 38 months, and 7 years of age. Symptoms of ASD, ADHD and anxiety were measured by parent report at age 7. Activity levels and inhibitory control, also measured by parent report, in infancy and toddlerhood were used as early-life predictors of ADHD symptoms. Fearfulness and shyness measured in infancy and toddlerhood were used as early-life predictors of anxiety symptoms. Correlations and path analysis models tested associations between early-life predictors and mid-childhood ADHD and anxiety symptoms compared to mid-childhood ASD symptoms, and the influence of controlling for ASD symptoms on those associations. Results Increased activity levels and poor inhibitory control were correlated with ADHD symptoms and not ASD or anxiety; these associations were unchanged in path models controlling for risk-group and ASD symptoms. Increased fearfulness and shyness were correlated with anxiety symptoms, but also ASD symptoms. When controlling for risk-group in path analysis, the association between shyness and anxiety became nonsignificant, and when further controlling for ASD symptoms the association between fearfulness and anxiety became marginal. Conclusions The specificity of early-life predictors to ADHD symptoms suggests early developmental pathways to ADHD might be distinct from ASD. The overlap in early-life predictors of anxiety and ASD suggests that these disorders are difficult to differentiate early in life, which could reflect the presence of common developmental pathways or convergence in early behavioural manifestations of these disorders

On tree amplitudes with gluons coupled to gravitons

In this paper, we study the tree amplitudes with gluons coupled to gravitons. We first study the relations among the mixed amplitudes. With BCFW on-shell recursion relation, we will show the color-order reversed relation, $U(1)$-decoupling relation and KK relation hold for tree amplitudes with gluons coupled to gravitons. We then study the disk relation which expresses mixed amplitudes by pure gluon amplitudes. More specifically we will prove the disk relation for mixed amplitudes with gluons coupled to one graviton. Using the disk relation and the properties of pure gluon amplitudes, the color-order reversed relation, $U(1)$-decoupling relation and KK relation for mixed amplitudes can also be proved. Finally, we give some brief discussions on BCJ-like relation for mixed amplitudes.Comment: 33pages,no figur