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We study the stability of one-dimensional linear hyperbolic systems with non-symmetric relaxation. Introducing a new frequency-dependent Kalman stability condition, we prove an abstract decay result underpinning a form of inhomogeneous hypocoercivity. In contrast with the homogeneous setting, the decay rates depend on how the Kalman condition is fulfilled and, in most cases, a loss of derivative occurs: one must assume an additional regularity assumption on the initial data to ensure the decay. Under structural assumptions, we refine our abstract result by providing an algorithm, of wide applicability, for the construction of Lyapunov functionals. This allows us to systematically establish decay estimates for a given system and uncover algebraic cancellations (beyond the reach of the Kalman-based approach) reducing the loss of derivatives in high frequencies. To demonstrate the applicability of our method, we derive new stability results for the Sugimoto model, which describes the propagation of nonlinear acoustic waves, and for a beam model of Timoshenko type with memory.

We present a general approach to establish the Central Limit Theorem with error bounds for sequential dynamical systems. The main tool we develop is the application to this setting of a projective metric on complex cones, following the ideas introduced by Rugh and Dubois. To demonstrate the power of the proposed setting, we apply it to both sequential expanding maps, where similar results are known, and to sequential dispersing billiards, for which no such results are currently known., Comment: 61 pages

In this paper, we introduce semi-autonomous neural ordinary differential equations (SA-NODEs), a variation of the vanilla NODEs, employing fewer parameters. We investigate the universal approximation properties of SA-NODEs for dynamical systems from both a theoretical and a numerical perspective. Within the assumption of a finite-time horizon, under general hypotheses we establish an asymptotic approximation result, demonstrating that the error vanishes as the number of parameters goes to infinity. Under additional regularity assumptions, we further specify this convergence rate in relation to the number of parameters, utilizing quantitative approximation results in the Barron space. Based on the previous result, we prove an approximation rate for transport equations by their neural counterparts. Our numerical experiments validate the effectiveness of SA-NODEs in capturing the dynamics of various ODE systems and transport equations. Additionally, we compare SA-NODEs with vanilla NODEs, highlighting the superior performance and reduced complexity of our approach., Comment: 22 pages, 16 pages

Probabilistic Graphical Bayesian models of causation have continued to impact on strategic analyses designed to help evaluate the efficacy of different interventions on systems. However, the standard causal algebras upon which these inferences are based typically assume that the intervened population does not react intelligently to frustrate an intervention. In an adversarial setting this is rarely an appropriate assumption. In this paper, we extend an established Bayesian methodology called Adversarial Risk Analysis to apply it to settings that can legitimately be designated as causal in this graphical sense. To embed this technology we first need to generalize the concept of a causal graph. We then proceed to demonstrate how the predicable intelligent reactions of adversaries to circumvent an intervention when they hear about it can be systematically modelled within such graphical frameworks, importing these recent developments from Bayesian game theory. The new methodologies and supporting protocols are illustrated through applications associated with an adversary attempting to infiltrate a friendly state.

We study piecewise injective, but not necessarily globally injective, contracting maps on a compact subset of \(\bR^d\). We prove that generically the attractor and the set of discontinuities of such a map are disjoint, and hence the attractor consists of periodic orbits. In addition, we prove that piecewise injective contractions are generically topologically stable.

We develop a bifurcation theory for infinite dimensional systems satisfying abstract hypotheses that are tailored for applications to mean field coupled chaotic maps. Our abstract theory can be applied to many cases, from globally coupled expanding maps to globally coupled Axiom A diffeomorphisms. To illustrate the range of applicability, we analyze an explicit example consisting of globally coupled Anosov diffeomorphisms. For such an example, we classify all the invariant measures as the coupling strength varies; we show which invariant measures are physical, and we prove that the existence of multiple invariant physical measures is a purely infinite dimensional phenomenon, i.e., the model exhibits phase transitions in the sense of statistical mechanics.

The aim of the present paper is to investigate the behavior of the spectrum of the Neumann Laplacian in domains with little holes excised from the interior. More precisely, we consider the eigenvalues of the Laplacian with homogeneous Neumann boundary conditions on a bounded, Lipschitz domain. Then, we singularly perturb the domain by removing Lipschitz sets which are "small" in a suitable sense and satisfy a uniform extension property. In this context, we provide an asymptotic expansion for all the eigenvalues of the perturbed problem which are converging to simple eigenvalues of the limit one. The eigenvalue variation turns out to depend on a geometric quantity resembling the notion of (boundary) torsional rigidity: understanding this fact is one of the main contributions of the present paper. In the particular case of a hole shrinking to a point, through a fine blow-up analysis, we identify the exact vanishing order of such a quantity and we establish some connections between the location of the hole and the sign of the eigenvalue variation., Comment: For the sake of clarity, this revised version makes more explicit a hypothesis - the connection of the complement in the case of scaling holes - that is implicit in the published version

We derive the heat equation for the thermal energy under diffusive space-time scaling for a purely deterministic microscopic dynamics satisfying Newton equations perturbed by an external chaotic force acting like a magnetic field.

Coffee extraction involves many complex physical and transport processes extremely difficult to model. Among the many factors that will affect the final quality of coffee, the microstructure of the coffee matrix is one of the most critical ones. In this article, we use X-ray micro-computed (microCT) technique to capture the microscopic details of coffee matrices at particle-level and perform fluid dynamics simulation based on the smoothed particle hydrodynamics method (SPH) with the 3D reconstructured data. Information like flow permeability and tortuosity of the matrices can be therefore obtained from our simulation. We found that inertial effects can be quite significant at the normal pressure gradient conditions typical for espresso brewing, and can provide an explanation for the inconsistency of permeability measurements seen in the literature. Several types of coffee powder are further examined, revealing their distinct microscopic details and resulting flow features. By comparing the microCT images of pre- and post-extraction coffee matrices, it is found that a decreasing porosity profile (from the bottom-outlet to the top-inlet) always develops after extraction. This counterintuitive phenomenon can be explained using a pressure-dependent erosion model proposed in our prior work. Our results show that microCT scan can provide useful microscopic details for coffee extraction modelling and establish the basis for a data-driven numerical framework to explore the link between coffee powder microstructure and extraction dynamics. The latter is the prerequisite to study the time evolution of both volatile and non-volatile organic compounds and then the flavour profile of coffee brews.

The exponential decay rate of the semigroup $S(t)=e^{t\mathbb{A}}$ generated by the abstract damped wave equation $$\ddot u + 2f(A) \dot u +A u=0 $$ is here addressed, where $A$ is a strictly positive operator. The continuous function $f$, defined on the spectrum of $A$, is subject to the constraints $$\inf f(s)>0\qquad\text{and}\qquad \sup f(s)/s <\infty$$ which are known to be necessary and sufficient for exponential stability to occur. We prove that the operator norm of the semigroup fulfills the estimate $$\|S(t)\|\leq Ce^{\sigma_*t}$$ being $\sigma_*<0$ the supremum of the real part of the spectrum of $\mathbb{A}$. This estimate always holds except in the resonant cases, where the negative exponential $e^{\sigma_*t}$ turns out to be penalized by a factor $(1+t)$. The decay rate is the best possible allowed by the theory.

Chain event graphs are a family of probabilistic graphical models that generalise Bayesian networks and have been successfully applied to a wide range of domains. Unlike Bayesian networks, these models can encode context-specific conditional independencies as well as asymmetric developments within the evolution of a process. More recently, new model classes belonging to the chain event graph family have been developed for modelling time-to-event data to study the temporal dynamics of a process. However, existing model selection algorithms for chain event graphs and its variants rely on all parameters having conjugate priors. This is unrealistic for many real-world applications. In this paper, we propose a mixture modelling approach to model selection in chain event graphs that does not rely on conjugacy. Moreover, we also show that this methodology is more amenable to being robustly scaled than the existing model selection algorithms used for this family. We demonstrate our techniques on simulated datasets.

The objective of disease mapping is to model data aggregated at the areal level. In some contexts, however, (e.g. residential histories, general practitioner catchment areas) when data is arising from a variety of sources, not necessarily at the same spatial scale, it is possible to specify spatial random effects, or covariate effects, at the areal level, by using a multiple membership principle (MM) (Petrof et al. 2020, Gramatica et al. 2021). A weighted average of conditional autoregressive (CAR) spatial random effects embeds spatial information for a spatially-misaligned outcome and estimate relative risk for both frameworks (areas and memberships). In this paper we investigate the theoretical underpinnings of these application of the multiple membership principle to the CAR prior, in particular with regard to parameterisation, properness and identifiability. We carry out simulations involving different numbers of memberships as compared to number of areas and assess impact of this on estimating parameters of interest. Both analytical and simulation study results show under which conditions parameters of interest are identifiable, so that we can offer actionable recommendations to practitioners. Finally, we present the results of an application of the multiple membership model to diabetes prevalence data in South London, together with strategic implications for public health considerations

We study infinite systems of globally coupled Anosov diffeomorphisms with weak coupling strength. Using transfer operators acting on anisotropic Banach spaces, we prove that the coupled system admits a unique physical invariant state, $h_\varepsilon$. Moreover, we prove exponential convergence to equilibrium for a suitable class of distributions and show that the map $\varepsilon\mapsto h_\varepsilon$ is Lipschitz continuous., Comment: To appear in Communications in Mathematical Physics

This errata fixes a mistake in the part of Giulietti, P.; Liverani, C.; Pollicott, M. Anosov flows and dynamical zeta functions. Ann. of Math. (2) {\bf 178} (2013), no. 2, 687--773, which proves a spectral gap for contact Anosov flows with respect to the measure of maximal entropy (Section 7). However, the first part of the paper, in which it is proved that the Ruelle zeta function is meromorphic, is unaffected.

The Dirichlet Process Mixture Model (DPMM) is a Bayesian non-parametric approach widely used for density estimation and clustering. In this manuscript, we study the choice of prior for the variance or precision matrix when Gaussian kernels are adopted. Typically, in the relevant literature, the assessment of mixture models is done by considering observations in a space of only a handful of dimensions. Instead, we are concerned with more realistic problems of higher dimensionality, in a space of up to 20 dimensions. We observe that the choice of prior is increasingly important as the dimensionality of the problem increases. After identifying certain undesirable properties of standard priors in problems of higher dimensionality, we review and implement possible alternative priors. The most promising priors are identified, as well as other factors that affect the convergence of MCMC samplers. Our results show that the choice of prior is critical for deriving reliable posterior inferences. This manuscript offers a thorough overview and comparative investigation into possible priors, with detailed guidelines for their implementation. Although our work focuses on the use of the DPMM in clustering, it is also applicable to density estimation., Comment: 35 pages

We study the energy transfer in the linear system $$ \begin{cases} \ddot u+u+\dot u=b\dot v\\ \ddot v+v-\epsilon \dot v=-b\dot u \end{cases} $$ made by two coupled differential equations, the first one dissipative and the second one antidissipative. We see how the competition between the damping and the antidamping mechanisms affect the whole system, depending on the coupling parameter $b$.

We address the energy transfer in the differential system $$ \begin{cases} u_{ttt}+\alpha u_{tt} - \beta \Delta u_t - \gamma \Delta u = -\eta \Delta \theta \\ \theta_t - \kappa \Delta \theta =\eta \Delta u_{tt}+ \alpha\eta \Delta u_t \end{cases} $$ made by a Moore-Gibson-Thompson equation in the supercritical regime, hence antidissipative, coupled with the classical heat equation. The asymptotic properties of the related solution semigroup depend on the strength of the coupling, ruling the competition between the Fourier damping and the MGT antidamping. Exponential stability will be shown always to occur, provided that the coupling constant is sufficiently large with respect to the other structural parameters. A fact of general interest will be also discussed, namely, the impossibility of attaining the optimal exponential decay rate of a given dissipative system via energy estimates.

We consider the MGT equation with memory $$\partial_{ttt} u + \alpha \partial_{tt} u - \beta \Delta \partial_{t} u - \gamma\Delta u + \int_{0}^{t}g(s) \Delta u(t-s) ds = 0.$$ We prove an existence and uniqueness result removing the convexity assumption on the convolution kernel $g$, usually adopted in the literature. In the subcritical case $\alpha\beta>\gamma$, we establish the exponential decay of the energy, without leaning on the classical differential inequality involving $g$ and its derivative $g'$, namely, $$g'+\delta g\leq 0,\quad\delta>0,$$ but only asking that $g$ vanishes exponentially fast.

We construct Birkhoff cones for dispersing billiards, which are contracted by the action of the transfer operator. This construction permits the study of statistical properties not only of regular dispersing billiards but also of sequential billiards (the billiard changes at each collision in a prescribed manner), open billiards (the dynamics exits some region or dies when hitting some obstacle) and many other examples. In particular, we include applications to chaotic scattering and the random Lorentz gas.

We study the spectrum of transfer operators associated to various dynamical systems. Our aim is to obtain precise information on the discrete spectrum. To this end we propose a unitary approach. We consider various settings where new information can be obtained following different branches along the proposed path. These settings include affine expanding Markov maps, uniformly expanding Markov maps, non-uniformly expanding or simply monotone maps, hyperbolic diffeomorphisms. We believe this approach could be greatly generalized., Comment: Final accepted version

We study a class of two dimensional partially hyperbolic systems, not necessarily skew products, trying to establish the germ of a general theory. To illustrate the scope of the theory, we apply our results to the case of fast-slow partially hyperbolic systems., Comment: Substantially revised version in which the arguments for fast-slow systems are clearly separated

Diabetes prevalence is on the rise in the UK, and for public health strategy, estimation of relative disease risk and subsequent mapping is important. We consider an application to London data on diabetes prevalence and mortality. In order to improve the estimation of relative risks we analyse jointly prevalence and mortality data to ensure borrowing strength over the two outcomes. The available data involves two spatial frameworks, areas (middle level super output areas, MSOAs), and general practices (GPs) recruiting patients from several areas. This raises a spatial misalignment issue that we deal with by employing the multiple membership principle. Specifically we translate area spatial effects to explain GP practice prevalence according to proportions of GP populations resident in different areas. A sparse implementation in Stan of both the MCAR and GMCAR allows the comparison of these bivariate priors as well as exploring the different implications for the mapping patterns for both outcomes. The necessary causal precedence of diabetes prevalence over mortality allows a specific conditionality assumption in the GMCAR, not always present in the context of disease mapping.

The Regression Discontinuity Design (RDD) is a quasi-experimental design that estimates the causal effect of a treatment when its assignment is defined by a threshold value for a continuous assignment variable. The RDD assumes that subjects with measurements within a bandwidth around the threshold belong to a common population, so that the threshold can be seen as a randomising device assigning treatment to those falling just above the threshold and withholding it from those who fall just below. Bandwidth selection represents a compelling decision for the RDD analysis as the results may be highly sensitive to its choice. A number of methods to select the optimal bandwidth, mainly originating from the econometric literature, have been proposed. However, their use in practice is limited. We propose a methodology that, tackling the problem from an applied point of view, consider units' exchangeability, i.e., their similarity with respect to measured covariates, as the main criteria to select subjects for the analysis, irrespectively of their distance from the threshold. We carry out clustering on the sample using a Dirichlet process mixture model to identify balanced and homogeneous clusters. Our proposal exploits the posterior similarity matrix, which contains the pairwise probabilities that two observations are allocated to the same cluster in the MCMC sample. Thus we include in the RDD analysis only those clusters for which we have stronger evidence of exchangeability. We illustrate the validity of our methodology with both a simulated experiment and a motivating example on the effect of statins to lower cholesterol level, using UK primary care data.

We establish sharp bounds on the mixing rates of a class of two dimensional non-uniformly hyperbolic symplectic maps. This provides a primer on how to investigate such questions in a concrete example and, at the same time, it solves a controversy between previous rigorous results and numerical experiments.

Given any smooth Anosov map we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that in the case of expanding maps it reduces exactly to the usual space of functions of bounded variation which has proven particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations which provides new information that could prove useful in treating hyperbolic systems with singularities.

By drawing on large-scale online data we construct and analyze the time-varying worldwide network of professional relationships among start-ups. The nodes of this network represent companies, while the links model the flow of employees and the associated transfer of know-how across companies. We use network centrality measures to assess, at an early stage, the likelihood of the long-term positive performance of a start-up, showing that the start-up network has predictive power and provides valuable recommendations doubling the current state of the art performance of venture funds. Our network-based approach not only offers an effective alternative to the labour-intensive screening processes of venture capital firms, but can also enable entrepreneurs and policy-makers to conduct a more objective assessment of the long-term potentials of innovation ecosystems and to target interventions accordingly.

This is a lightning introduction to some modern techniques used in the study of the statistical properties of hyperbolic dynamical systems. The emphasis is not in presenting a comprehensive theory but rather in fleshing out the main ideas in the simplest and fastest possible manner so that the reader can quickly get the intuition necessary to easily read the more technical (and more complete) accounts of the theory., Comment: I corrected several misprints

Objective: Alterations of Young's modulus (YM) and Poisson's ratio (PR) in biological tissues are often early indicators of the onset of pathological conditions. Knowledge of these parameters has been proven to be of great clinical significance for the diagnosis, prognosis and treatment of cancers. Currently, however, there are no non-invasive modalities that can be used to image and quantify these parameters in vivo without assuming incompressibility of the tissue, an assumption that is rarely justified in human tissues. Methods: In this paper, we develop a new method to simultaneously reconstruct YM and PR of a tumor and of its surrounding tissues, irrespective of the boundary conditions and the shape of the tumor based on ellipsoidal approximation. This new non-invasive method allows the generation of high spatial resolution YM and PR maps from axial and lateral strain data obtained via ultrasound elastography. The method was validated using finite element (FE) simulations and controlled experiments performed on phantoms with known mechanical properties. The clinical feasibility of the developed method was also demonstrated in an orthotopic mouse model of breast cancer. Results: Our results from simulations and controlled experiments demonstrate that the proposed reconstruction technique is accurate and robust. Conclusion: Availability of the proposed technique could address the clinical need of a non-invasive modality capable of imaging, quantifying and monitoring the mechanical properties of tumors with high spatial resolution and in real time. Significance: This technique can have a significant impact on the clinical translation of elasticity imaging methods., Comment: 38 pages, 34 figures, Submitted to IEEE Transactions on Biomedical Engineering

Motivated by the random Lorentz gas, we study deterministic walks in random environment and show that (in simple, yet relevant, cases) they can be reduced to a class of random walks in random environment where the jump probability depends (weakly) on the past. In addition, we prove few basic results (hopefully the germ of a general theory) on the latter, purely probabilistic, model., Comment: Accepted for publication on Annals of Probability

We discuss sufficient conditions that guarantee the existence of asymptotic expansions for the Central Limit Theorem for weakly dependent random variables including observations arising from sufficiently chaotic dynamical systems like piece-wise expanding maps, and strongly ergodic Markov chains. We primarily use spectral techniques to obtain the results., Comment: 45 pages

I will discuss, from a dynamical systems point of view, some recent attempts to rigorously derive the macroscopic laws of transport (e.g. the heat equation) from deterministic microscopic dynamics., Comment: Contribution to the ICM 2018

We consider a simple class of fast-slow partially hyperbolic dynamical systems and show that the (properly rescaled) behaviour of the slow variable is very close to a Friedlin--Wentzell type random system for times that are rather long, but much shorter than the metastability scale. Also, we show the possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon that turns out to be related to the lack of absolutely continuity of the central foliation., Comment: To appear in Journal of Statistical Physics

In this work we present a statistical approach to distinguish and interpret the complex relationship between several predictors and a response variable at the small area level, in the presence of i) high correlation between the predictors and ii) spatial correlation for the response. Covariates which are highly correlated create collinearity problems when used in a standard multiple regression model. Many methods have been proposed in the literature to address this issue. A very common approach is to create an index which aggregates all the highly correlated variables of interest. For example, it is well known that there is a relationship between social deprivation measured through the Multiple Deprivation Index (IMD) and air pollution; this index is then used as a confounder in assessing the effect of air pollution on health outcomes (e.g. respiratory hospital admissions or mortality). However it would be more informative to look specifically at each domain of the IMD and at its relationship with air pollution to better understand its role as a confounder in the epidemiological analyses. In this paper we illustrate how the complex relationships between the domains of IMD and air pollution can be deconstructed and analysed using profile regression, a Bayesian non-parametric model for clustering responses and covariates simultaneously. Moreover, we include an intrinsic spatial conditional autoregressive (ICAR) term to account for the spatial correlation of the response variable.

Ginzburg-Landau energy models arise as autonomous sto-chastic dynamics for the energies in coupled systems after a weak coupling limit (cf. [3, 6]). We prove here that, under certain conditions, the energy fluctuations of these stochastic dynamics are driven by the heat equation, under a diffusive space time scaling., Comment: Preliminary Draft

We prove exponential decay of correlations for the billiard flow associated with a two-dimensional finite horizon Lorentz Gas (i.e., the Sinai billiard flow with finite horizon). Along the way, we describe the spectrum of the generator of the corresponding semi-group L_t of transfer operators, i.e., the resonances of the Sinai billiard flow, on a suitable Banach space of anisotropic distributions. (Revised after referees' comments.), Comment: v2: Revised version after referees' comments. (The homogeneity layers do not appear anymore in the def. of the norm.) The text published in Invent. Math. is v2, but has more pages (due to smaller margins in v2). v3: Contains an additional 2-page note correcting flaws found by M. Jezequel (L 3.2) and D. Thomine (L 4.6 and L 7.5)

We investigate the relation between the distributions appearing in the study of ergodic averages of parabolic flows (e.g. in the work of Flaminio-Forni) and the ones appearing in the study of the statistical properties of hyperbolic dynamical systems (i.e. the eigendistributions of the transfer operator). In order to avoid, as much as possible, technical issues that would cloud the basic idea, we limit ourselves to a simple flow on the torus. Our main result is that, roughly, the growth of ergodic averages of a parabolic flows is controlled by the eigenvalues of a suitable transfer operator associated to the renormalising dynamics. The conceptual connection that we illustrate is expected to hold in considerable generality., Comment: An algebraic mistake in section 5 has been corrected and the statements have been corrected accordingly

We study the spectrum of one dimensional integral operators in bounded real intervals of length $2L$, for value of $L$ large. The integral operators are obtained by linearizing a non local evolution equation for a non conserved order parameter describing the phases of a fluid. We prove a Perron-Frobenius theorem showing that there is an isolated, simple minimal eigenvalue strictly positive for $L$ finite, going to zero exponentially fast in $L$. We lower bound, uniformly on $L$, the spectral gap by applying a generalization of the Cheeger's inequality. These results are usefulfor deriving spectral properties for non local Cahn-Hilliard type of equations in problems of interface dynamics., Comment: An serious error has been corrected and an author has been added

We prove several limit theorems for a simple class of partially hyperbolic fast-slow systems. We start with some well know results on averaging, then we give a substantial refinement of known large (and moderate) deviation results and conclude with a completely new result (a local limit theorem) on the distribution of the process determined by the fluctuations around the average. The method of proof is based on a mixture of standard pairs and Transfer Operators that we expect to be applicable in a much wider generality.

We consider a class of $\mathcal C^{4}$ partially hyperbolic systems on $\mathbb T^2$ described by maps $F_\varepsilon(x,\theta)=(f(x,\theta),\theta+\varepsilon\omega(x,\theta))$ where $f(\cdot,\theta)$ are expanding maps of the circle. For sufficiently small $\varepsilon$ and $\omega$ generic in an open set, we precisely classify the SRB measures for $F_\varepsilon$ and their statistical properties, including exponential decay of correlation for H\"older observables with explicit and nearly optimal bounds on the decay rate.

We present, in the simplest possible form, the so called martingale problem strategy to establish limit theorems. The presentation is specially adapted to problems arising in partially hyperbolic dynamical systems. We will discuss a simple partially hyperbolic example with fast-slow variables and use the martingale method to prove an averaging theorem and study fluctuations from the average. The emphasis is on ideas rather than on results. Also, no effort whatsoever is done to review the vast literature of the field., Comment: 26 pages

We study the Green-Kubo (GK) formula $\kappa (\varepsilon, \xi)$ for the heat conductivity of an infinite chain of $d$-dimensional finite systems (cells) coupled by a smooth nearest neighbour potential $\varepsilon V$. The uncoupled systems evolve according to Hamiltonian dynamics perturbed stochastically by an energy conserving noise of strength $\xi$. Noting that $\kappa (\varepsilon, \xi)$ exists and is finite whenever $\xi> 0$, we are interested in what happens when the strength of the noise $\xi \to 0$. For this, we start in this work by formally expanding $\kappa (\varepsilon, \xi)$ in a power series in $\varepsilon$, $\kappa (\varepsilon, \xi) = \varepsilon^2 \sum_{n\ge 2} \varepsilon^{n-2} \kappa_n (\xi)$ and investigating the (formal) equations satisfied by $\kappa_n (\xi$. We show in particular that $\kappa_2 (\xi)$ is well defined when no pinning potential is present, and coincides formally with the heat conductivity obtained in the weak coupling (van Hove) limit, where time is rescaled as $\varepsilon^{-2}t$, for the cases where the latter has been established \cite{LO, DL}. For one-dimensional systems, we investigate $\kappa_2 (\xi)$ as $\xi\to 0$ in three cases: the disordered harmonic chain, the rotor chain and a chain of strongly anharmonic oscillators. Moreover, we formally identify $\kappa_2 (\xi)$ with the conductivity obtained by having the chain between two reservoirs at temperature $T$ and $T+\delta T$, in the limit $\delta T\to 0$, $N \to \infty$, $\varepsilon \to 0$., Comment: New version with many improvements

In this work we obtain mixing (and in some cases sharp mixing rates) for a reasonable large class of invertible systems preserving an infinite measure. The examples considered here are the invertible analogue of both Markov and non Markov unit interval maps. Moreover, we obtain results on the decay of correlation in the finite case of invertible non Markov maps, which, to our knowledge, were not previously addressed. The present method consists of a combination of the framework of operator renewal theory, as introduced in the context of dynamical systems by Sarig [39], with the framework of function spaces of distributions developed in the recent years along the lines of Blank, Keller and Liverani [9]., Comment: extensively revised

In the limit of large particle numbers and low densities systems of cold atoms can be effectively described as macroscopic single particle systems in a mean-field approximation. In the case of a Bose-Hubbard system, modelling bosons on a discrete lattice with on-site interactions, this yields a discrete nonlinear Schr\"odinger equation of Gross-Pitaevskii type. It has been recently shown that the correspondence between the Gross-Pitaevskii equation and the Bose-Hubbard system breaks down for complex extensions. In particular, for a Bose-Hubbard dimer with complex on-site energy the mean-field approximation yields a generalised complex nonlinear Schr\"odinger equation. Conversely, a Gross-Pitaevskii equation with complex on-site energies arises as the mean-field approximation of many-particle Lindblad dynamics rather than a complex extension of the Bose-Hubbard system. Here we address the question of how the mean-field description is modified in the presence of a complex-valued particle interaction term for a Bose-Hubbard dimer. We derive the mean-field equations of motion leading to nonlinear dissipative Bloch dynamics, related to a nontrivial complex generalisation of the nonlinear Schr\"odinger equation. The resulting dynamics are analysed in detail. It is shown that depending on the parameter values there can be up to six stationary states, and for small values of the interaction strength there are limit cycles. Furthermore, we show how a Gross-Pitaevskii equation with a complex interaction term can be derived as the mean-field approximation of a Bose-Hubbard dimer with an additional Lindblad term modelling two-particle losses., Comment: 18 pages three figures; partly rewritten, additional material and references added

We consider the question of Markov chain Monte Carlo sampling from a general stick-breaking Dirichlet process mixture model, with concentration parameter alpha. This paper introduces a Gibbs sampling algorithm that combines the slice sampling approach of Walker (2007) and the retrospective sampling approach of Papaspiliopoulos and Roberts (2008). Our general algorithm is implemented as efficient open source C++ software, available as an R package, and is based on a blocking strategy similar to that suggested by Papaspiliopoulos (2008) and implemented by Yau et al (2011). We discuss the difficulties of achieving good mixing in MCMC samplers of this nature and investigate sensitivity to initialisation. We additionally consider the challenges when an additional layer of hierarchy is added such that joint inference is to be made on alpha. We introduce a new label switching move and compute the marginal model posterior to help to surmount these difficulties. Our work is illustrated using a profile regression (Molitor et al, 2010) application, where we demonstrate good mixing behaviour for both synthetic and real examples.

PReMiuM is a recently developed R package for Bayesian clustering using a Dirichlet process mixture model. This model is an alternative to regression models, non-parametrically linking a response vector to covariate data through cluster membership. The package allows Bernoulli, Binomial, Poisson, Normal and categorical response, as well as Normal and discrete covariates. Additionally, predictions may be made for the response, and missing values for the covariates are handled. Several samplers and label switching moves are implemented along with diagnostic tools to assess convergence. A number of R functions for post-processing of the output are also provided. In addition to fitting mixtures, it may additionally be of interest to determine which covariates actively drive the mixture components. This is implemented in the package as variable selection.

We provide abstract conditions which imply the existence of a robustly invariant neighbourhood of a global section of a fibre bundle flow. We then apply such a result to the bundle flow generated by an Anosov flow when the fibre is the space of jets (which are described by local manifolds). As a consequence we obtain sets of manifolds (e.g. approximations of stable manifolds) that are left invariant, {\bf for all} negative times, by the flow and its small perturbations. Finally, we show that the latter result can be used to easily fix a mistake recently uncovered in the paper {\em Smooth Anosov flows: correlation spectra and stability}, \cite{BuL}, by the present authors., Comment: This is a completely revised version of "Smooth Invariant Cone Fields: Errata Corrige to "Smooth Anosov Flows: Correlation Spectra and Stability"". The change of title reflects the emphasis on new results eventhough the paper still contains an errata to "Smooth Anosov Flows: Correlation Spectra and Stability"

I introduce Banach spaces on which it is possible to precisely characterize the spectrum of the transfer operator associated to a piecewise expanding map with H\"older weight., Comment: Added details and extended results

We study the Ruelle and Selberg zeta functions for $\Cs^r$ Anosov flows, $r > 2$, on a compact smooth manifold. We prove several results, the most remarkable being: (a) for $\Cs^\infty$ flows the zeta function is meromorphic on the entire complex plane; (b) for contact flows satisfying a bunching condition (e.g. geodesic flows on manifolds of negative curvature better than $\frac 19$-pinched) the zeta function has a pole at the topological entropy and is analytic in a strip to its left; (c) under the same hypotheses as in (b) we obtain sharp results on the number of periodic orbits. Our arguments are based on the study of the spectral properties of a transfer operator acting on suitable Banach spaces of anisotropic currents.

I provide a proof of the existence of absolutely continuous invariant measures (and study their statistical properties) for multidimensional piecewise expanding systems with not necessarily bounded derivative or distortion. The proof uses basic properties of multidimensional BV functions.

We prove exponential decay of correlations for a realistic model of piecewise hyperbolic flows preserving a contact form, in dimension three. This is the first time exponential decay of correlations is proved for continuous-time dynamics with singularities on a manifold. Our proof combines the second author's version of Dolgopyat's estimates for contact flows and the first author's work with Gou\"ezel on piecewise hyperbolic discrete-time dynamics. (Presentation revised.), Comment: 76 pages, 4 figures included in LaTeX file