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Nonparametric estimation of the tree structure of a nested Archimedean copulaApr 04 2013Dec 17 2013One of the features inherent in nested Archimedean copulas, also called hierarchical Archimedean copulas, is their rooted tree structure. A nonparametric, rank-based method to estimate this structure is presented. The idea is to represent the target structure ... More

Inference for heavy tailed stationary time series based on sliding blocksJun 06 2017Feb 26 2018The block maxima method in extreme value theory consists of fitting an extreme value distribution to a sample of block maxima extracted from a time series. Traditionally, the maxima are taken over disjoint blocks of observations. Alternatively, the blocks ... More

Regularly varying time series in Banach spacesJan 19 2010When a spatial process is recorded over time and the observation at a given time instant is viewed as a point in a function space, the result is a time series taking values in a Banach space. To study the spatio-temporal extremal dynamics of such a time ... More

Polar decomposition of regularly varying time series in star-shaped metric spacesApr 01 2016Feb 02 2017There exist two ways of defining regular variation of a time series in a star-shaped metric space: either by the distributions of finite stretches of the series or by viewing the whole series as a single random element in a sequence space. The two definitions ... More

An M-estimator of spatial tail dependenceMar 08 2014Jan 09 2015Tail dependence models for distributions attracted to a max-stable law are fitted using observations above a high threshold. To cope with spatial, high-dimensional data, a rank-based M-estimator is proposed relying on bivariate margins only. A data-driven ... More

One- versus multi-component regular variation and extremes of Markov treesFeb 06 2019A Markov tree is a random vector indexed by the nodes of a tree whose distribution is determined by the distributions of pairs of neighbouring variables and a list of conditional independence relations. Upon an assumption on the tails of the Markov kernels ... More

Max-stable models for multivariate extremesApr 02 2012Multivariate extreme-value analysis is concerned with the extremes in a multivariate random sample, that is, points of which at least some components have exceptionally large values. Mathematical theory suggests the use of max-stable models for univariate ... More

Nonparametric Inference for Max-Stable DependenceAug 17 2012Discussion of "Statistical Modeling of Spatial Extremes" by A. C. Davison, S. A. Padoan and M. Ribatet [arXiv:1208.3378].

On the asymptotic distribution of the mean absolute deviation about the meanJun 16 2014The mean absolute deviation about the mean is an alternative to the standard deviation for measuring dispersion in a sample or in a population. For stationary, ergodic time series with a finite first moment, an asymptotic expansion for the sample mean ... More

Hybrid Copula EstimatorsMay 08 2014Nov 28 2014An extension of the empirical copula is considered by combining an estimator of a multivariate cumulative distribution function with estimators of the marginal cumulative distribution functions for marginal estimators that are not necessarily equal to ... More

Asymptotics of empirical copula processes under non-restrictive smoothness assumptionsDec 09 2010Jul 05 2012Weak convergence of the empirical copula process is shown to hold under the assumption that the first-order partial derivatives of the copula exist and are continuous on certain subsets of the unit hypercube. The assumption is non-restrictive in the sense ... More

Multivariate regular variation of heavy-tailed Markov chainsJan 15 2007The upper extremes of a Markov chain with regulary varying stationary marginal distribution are known to exhibit under general conditions a multiplicative random walk structure called the tail chain. More generally, if the Markov chain is allowed to switch ... More

Measuring Association between Random VectorsJul 21 2011This paper suggests five measures of association between two random vectors X = (X_1, ..., X_p) and Y = (Y_1, ..., Y_q). They are copula based and therefore invariant with respect to the marginal distributions of the components X_i and Y_j. The measures ... More

Nonparametric estimation of multivariate extreme-value copulasJul 12 2011Nov 29 2011Extreme-value copulas arise in the asymptotic theory for componentwise maxima of independent random samples. An extreme-value copula is determined by its Pickands dependence function, which is a function on the unit simplex subject to certain shape constraints ... More

On the weak convergence of the empirical conditional copula under a simplifying assumptionNov 20 2015When the copula of the conditional distribution of two random variables given a covariate does not depend on the value of the covariate, two conflicting intuitions arise about the best possible rate of convergence attainable by nonparametric estimators ... More

Marginal standardization of upper semicontinuous processes. With application to max-stable processesMar 14 2016In the field of spatial extremes, stochastic processes with upper semicontinuous (usc) trajectories have been proposed as random shape functions for max-stable models. In the literature dealing with usc processes, max-stability is defined via a sequences ... More

Weak convergence of the weighted empirical beta copula processMay 19 2017Jan 11 2018The empirical copula has proved to be useful in the construction and understanding of many statistical procedures related to dependence within random vectors. The empirical beta copula is a smoothed version of the empirical copula that enjoys better finite-sample ... More

Tails of correlation mixtures of elliptical copulasDec 17 2009Correlation mixtures of elliptical copulas arise when the correlation parameter is driven itself by a latent random process. For such copulas, both penultimate and asymptotic tail dependence are much larger than for ordinary elliptical copulas with the ... More

On the covariance of the asymptotic empirical copula processMar 19 2009Mar 16 2010Conditions are given under which the empirical copula process associated with a random sample from a bivariate continuous distribution has a smaller asymptotic covariance function than the standard empirical process based on observations from the copula. ... More

On the maximum likelihood estimator for the Generalized Extreme-Value distributionJan 21 2016Mar 15 2017The vanilla method in univariate extreme-value theory consists of fitting the three-parameter Generalized Extreme-Value (GEV) distribution to a sample of block maxima. Despite claims to the contrary, the asymptotic normality of the maximum likelihood ... More

Tails of multivariate Archimedean copulasJan 12 2009A complete and user-friendly directory of tails of Archimedean copulas is presented which can be used in the selection and construction of appropriate models with desired properties. The results are synthesized in the form of a decision tree: Given the ... More

Extreme value copula estimation based on block maxima of a multivariate stationary time seriesNov 13 2013May 08 2014The core of the classical block maxima method consists of fitting an extreme value distribution to a sample of maxima over blocks extracted from an underlying series. In asymptotic theory, it is usually postulated that the block maxima are an independent ... More

Word-level Symbolic Trajectory EvaluationMay 29 2015Symbolic trajectory evaluation (STE) is a model checking technique that has been successfully used to verify industrial designs. Existing implementations of STE, however, reason at the level of bits, allowing signals to take values in {0, 1, X}. This ... More

Markov tail chainsApr 29 2013Feb 03 2014The extremes of a univariate Markov chain with regulary varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper, we extend this fact to ... More

Extreme-Value CopulasNov 05 2009Dec 07 2009Being the limits of copulas of componentwise maxima in independent random samples, extreme-value copulas can be considered to provide appropriate models for the dependence structure between rare events. Extreme-value copulas not only arise naturally in ... More

Nonparametric estimation of an extreme-value copula in arbitrary dimensionsOct 05 2009Inference on an extreme-value copula usually proceeds via its Pickands dependence function, which is a convex function on the unit simplex satisfying certain inequality constraints. In the setting of an iid random sample from a multivariate distribution ... More

On the maximum likelihood estimator for the Generalized Extreme-Value distributionJan 21 2016Oct 13 2016The vanilla method in univariate extreme-value theory consists of fitting the three-parameter Generalized Extreme-Value (GEV) distribution to a sample of block maxima. Despite claims to the contrary, the asymptotic normality of the maximum likelihood ... More

Maximum likelihood estimation for the Fréchet distribution based on block maxima extracted from a time seriesNov 24 2015Sep 16 2016The block maxima method in extreme-value analysis proceeds by fitting an extreme-value distribution to a sample of block maxima extracted from an observed stretch of a time series. The method is usually validated under two simplifying assumptions: the ... More

Monte Carlo integration with a growing number of control variatesJan 05 2018Mar 23 2018The use of control variates is a well-known variance reduction technique in Monte Carlo integration. If the optimal linear combination of control variates is estimated by ordinary least squares and if the number of control variates is allowed to grow ... More

On the weak convergence of the empirical conditional copula under a simplifying assumptionNov 20 2015May 16 2017When the copula of the conditional distribution of two random variables given a covariate does not depend on the value of the covariate, two conflicting intuitions arise about the best possible rate of convergence attainable by nonparametric estimators ... More

Rank-based inference for bivariate extreme-value copulasJul 27 2007Aug 26 2009Consider a continuous random pair $(X,Y)$ whose dependence is characterized by an extreme-value copula with Pickands dependence function $A$. When the marginal distributions of $X$ and $Y$ are known, several consistent estimators of $A$ are available. ... More

Marginal standardization of upper semicontinuous processes. with application to max-stable processesMar 14 2016Dec 22 2016Extreme-value theory for random vectors and stochastic processes with continuous trajectories is usually formulated for random objects all of whose univariate marginal distributions are identical. In the spirit of Sklar's theorem from copula theory, such ... More

Regularly varying multivariate time seriesJul 26 2007Jul 26 2007A multivariate, stationary time series is said to be jointly regularly varying if all its finite-dimensional distributions are multivariate regularly varying. This property is shown to be equivalent to weak convergence of the conditional distribution ... More

Generalised regular variation of arbitrary orderJan 11 2009Let $f$ be a measurable, real function defined in a neighbourhood of infinity. The function $f$ is said to be of generalised regular variation if there exist functions $h \not\equiv 0$ and $g > 0$ such that $f(xt) - f(t) = h(x) g(t) + o(g(t))$ as $t \to ... More

Nonparametric estimation of pair-copula constructions with the empirical pair-copulaJan 24 2012A pair-copula construction is a decomposition of a multivariate copula into a structured system, called regular vine, of bivariate copulae or pair-copulae. The standard practice is to model these pair-copulae parametrically, which comes at the cost of ... More

Tails of random sums of a heavy-tailed number of light-tailed termsMar 01 2007Oct 09 2007The tail of the distribution of a sum of a random number of independent and identically distributed nonnegative random variables depends on the tails of the number of terms and of the terms themselves. This situation is of interest in the collective risk ... More

Stability and tail limits of transport-based quantile contoursNov 29 2018We extend Robert McCann's treatment of the existence and uniqueness of an optimal transport map between two probability measures on a Euclidean space to a class of possibly infinite measures, finite outside neighbourhoods of the origin. For convergent ... More

Tails of optimal transport plans for regularly varying probability measuresNov 29 2018May 02 2019For the basic case of $L_2$ optimal transport between two probability measures on a Euclidean space, the regularity of the coupling measure and the transport map in the tail regions of these measures is studied. For this purpose, Robert McCann's classical ... More

On the smallest poles of Igusa's p-adic zeta functionsSep 02 2005Let K be a p-adic field. We explore Igusa's p-adic zeta function, which is associated to a K-analytic function on an open and compact subset of K^n. First we deduce a formula for an important coefficient in the Laurent series of this meromorphic function ... More

A functional limit theorem for dependent sequences with infinite variance stable limitsJan 08 2010Oct 11 2012Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is ... More

Multivariate peaks over thresholds modelsMar 21 2016May 03 2017Multivariate peaks over thresholds modeling based on generalized Pareto distributions has up to now only been used in few and mostly 2-dimensional situations. This paper contributes theoretical understanding, physically based models, inference tools, ... More

Radial-angular decomposition of regularly varying time series in star-shaped metric spacesApr 01 2016There exist two ways of defining regular variation of a time series in a star-shaped metric space: either by the distributions of finite stretches of the series or by viewing the whole series as a single random element in a sequence space. The two definitions ... More

Statistics for Tail Processes of Markov ChainsMay 29 2014Dec 10 2014At high levels, the asymptotic distribution of a stationary, regularly varying Markov chain is conveniently given by its tail process. The latter takes the form of a geometric random walk, the increment distribution depending on the sign of the process ... More

Risk Concentration and Diversification: Second-Order PropertiesOct 13 2009Dec 19 2009The quantification of diversification benefits due to risk aggregation plays a prominent role in the (regulatory) capital management of large firms within the financial industry. However, the complexity of today's risk landscape makes a quantifiable reduction ... More

Bayesian inference for bivariate ranksFeb 09 2018A recommender system based on ranks is proposed, where an expert's ranking of a set of objects and a user's ranking of a subset of those objects are combined to make a prediction of the user's ranking of all objects. The rankings are assumed to be induced ... More

Multivariate peaks over thresholds modelsMar 21 2016Multivariate peaks over thresholds modeling based on generalized Pareto distributions has up to now only been used in few and mostly low-dimensional situations. This paper contributes to the theoretical understanding, physically based models, inference ... More

Second-order refined peaks-over-threshold modelling for heavy-tailed distributionsJan 12 2009Modelling excesses over a high threshold using the Pareto or generalized Pareto distribution (PD/GPD) is the most popular approach in extreme value statistics. This method typically requires high thresholds in order for the (G)PD to fit well and in such ... More

Identifying groups of variables with the potential of being large simultaneouslyFeb 27 2018Identifying groups of variables that may be large simultaneously amounts to finding out which joint tail dependence coefficients of a multivariate distribution are positive. The asymptotic distribution of a vector of nonparametric, rank-based estimators ... More

Large-sample tests of extreme-value dependence for multivariate copulasMay 11 2011Starting from the characterization of extreme-value copulas based on max-stability, large-sample tests of extreme-value dependence for multivariate copulas are studied. The two key ingredients of the proposed tests are the empirical copula of the data ... More

Multivariate peaks over thresholds modelsMar 21 2016Oct 20 2016Multivariate peaks over thresholds modeling based on generalized Pareto distributions has up to now only been used in few and mostly 2-dimensional situations. This paper contributes theoretical understanding, physically based models, inference tools, ... More

When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographsMay 28 2013Aug 14 2014In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions equipped with the ... More

Max-factor individual risk models with application to credit portfoliosDec 10 2014Individual risk models need to capture possible correlations as failing to do so typically results in an underestimation of extreme quantiles of the aggregate loss. Such dependence modelling is particularly important for managing credit risk, for instance, ... More

Multivariate generalized Pareto distributions: parametrizations, representations, and propertiesMay 22 2017Multivariate generalized Pareto distributions arise as the limit distributions of exceedances over multivariate thresholds of random vectors in the domain of attraction of a max-stable distribution. These distributions can be parametrized and represented ... More

Nonparametric estimation of extremal dependenceNov 03 2014There is an increasing interest to understand the dependence structure of a random vector not only in the center of its distribution but also in the tails. Extreme-value theory tackles the problem of modelling the joint tail of a multivariate distribution ... More

Maximum empirical likelihood estimation of the spectral measure of an extreme-value distributionDec 18 2008Sep 01 2009Consider a random sample from a bivariate distribution function $F$ in the max-domain of attraction of an extreme-value distribution function $G$. This $G$ is characterized by two extreme-value indices and a spectral measure, the latter determining the ... More

The Empirical Beta CopulaJul 15 2016Given a sample from a multivariate distribution $F$, the uniform random variates generated independently and rearranged in the order specified by the componentwise ranks of the original sample look like a sample from the copula of $F$. This idea can be ... More

The Empirical Beta CopulaJul 15 2016Nov 18 2016Given a sample from a multivariate distribution $F$, the uniform random variates generated independently and rearranged in the order specified by the componentwise ranks of the original sample look like a sample from the copula of $F$. This idea can be ... More

An estimator of the stable tail dependence function based on the empirical beta copulaSep 12 2017The replacement of indicator functions by integrated beta kernels in the definition of the empirical stable tail dependence function is shown to produce a smoothed version of the latter estimator with the same asymptotic distribution but superior finite-sample ... More

Nonparametric Bayesian Inference on Bivariate ExtremesNov 17 2009May 11 2012The tail of a bivariate distribution function in the domain of attraction of a bivariate extreme-value distribution may be approximated by the one of its extreme-value attractor. The extreme-value attractor has margins that belong to a three-parameter ... More

On the longest gap between power-rate arrivalsMar 28 2017Sep 21 2017Let $L_t$ be the longest gap before time $t$ in an inhomogeneous Poisson process with rate function $\lambda_t$ proportional to $t^{\alpha-1}$ for some $\alpha\in(0,1)$. It is shown that $\lambda_tL_t-b_t$ has a limiting Gumbel distribution for suitable ... More

An M-estimator for tail dependence in arbitrary dimensionsDec 05 2011Oct 04 2012Consider a random sample in the max-domain of attraction of a multivariate extreme value distribution such that the dependence structure of the attractor belongs to a parametric model. A new estimator for the unknown parameter is defined as the value ... More

Semiparametric Gaussian copula models: Geometry and efficient rank-based estimationJun 27 2013Oct 01 2014We propose, for multivariate Gaussian copula models with unknown margins and structured correlation matrices, a rank-based, semiparametrically efficient estimator for the Euclidean copula parameter. This estimator is defined as a one-step update of a ... More

A Sliding Blocks Estimator for the Extremal IndexDec 22 2008In extreme value statistics for stationary sequences, blocks estimators are usually constructed by using disjoint blocks because exceedances over high thresholds of different blocks can be assumed asymptotically independent. In this paper we focus on ... More

A method of moments estimator of tail dependenceOct 10 2007Nov 14 2008In the world of multivariate extremes, estimation of the dependence structure still presents a challenge and an interesting problem. A procedure for the bivariate case is presented that opens the road to a similar way of handling the problem in a truly ... More

Peaks over thresholds modelling with multivariate generalized Pareto distributionsDec 06 2016The multivariate generalized Pareto distribution arises as the limit of a suitably normalized vector conditioned upon at least one component of that vector being extreme. Statistical modelling using multivariate generalized Pareto distributions constitutes ... More

A continuous updating weighted least squares estimator of tail dependence in high dimensionsJan 19 2016Likelihood-based procedures are a common way to estimate tail dependence parameters. They are not applicable, however, in non-differentiable models such as those arising from recent max-linear structural equation models. Moreover, they can be hard to ... More

Peaks over thresholds modelling with multivariate generalized Pareto distributionsDec 06 2016Feb 06 2018When assessing the impact of extreme events, it is often not just a single component, but the combined behaviour of several components which is important. Statistical modelling using multivariate generalized Pareto (GP) distributions constitutes the multivariate ... More

Bayesian model averaging over tree-based dependence structures for multivariate extremesMay 30 2017Jul 22 2018Describing the complex dependence structure of extreme phenomena is particularly challenging. To tackle this issue we develop a novel statistical algorithm that describes extremal dependence taking advantage of the inherent hierarchical dependence structure ... More

A Euclidean likelihood estimator for bivariate tail dependenceApr 16 2012The spectral measure plays a key role in the statistical modeling of multivariate extremes. Estimation of the spectral measure is a complex issue, given the need to obey a certain moment condition. We propose a Euclidean likelihood-based estimator for ... More

Detecting changes in cross-sectional dependence in multivariate time seriesJun 12 2012May 21 2014Classical and more recent tests for detecting distributional changes in multivariate time series often lack power against alternatives that involve changes in the cross-sectional dependence structure. To be able to detect such changes better, a test is ... More

Connected, Disconnected and Strange Quark Contributions to HVPSep 06 2016Nov 03 2016We calculate all neutral vector two-point functions in Chiral Perturbation Theory (ChPT) to two-loop order and use these to estimate the ratio of disconnected to connected contributions as well as contributions involving the strange quark. We extend the ... More

Chiral Perturbation Theory at Finite Volume and/or with Twisted Boundary ConditionsNov 18 2016In this talk we discuss a number of ChPT calculations relevant for lattice QCD. These include the finite volume corrections at two-loop order for masses and decay constants. The second part is about hadronic vacuum polarization where we present the two-loop ... More

Vector two-point functions in finite volume using partially quenched chiral perturbation theoryOct 12 2017Dec 14 2017We calculate vector-vector correlation functions at two loops using partially quenched chiral perturbation theory including finite volume effects and twisted boundary conditions. We present expressions for the flavor neutral cases and the flavor charged ... More

Masses, decay constants and electromagnetic form-factors with twisted boundary conditionsSep 24 2015Jan 07 2016We discuss some of the effects of twisted boundary conditions in finite volume using continuum SU(3) Chiral Perturbation Theory. We point out how broken cubic symmetry affects the definitions of quantities such as form-factors. Using the $\pi^+$ as an ... More

Connected, Disconnected and Strange Quark Contributions to HVPSep 06 2016We calculate all neutral vector two-point functions in Chiral Perturbation Theory (ChPT) to two-loop order and use these to estimate the ratio of disconnected to connected contributions as well as contributions involving the strange quark. We extend the ... More

Operations and single particle interferometryDec 15 2003Jan 16 2006Interferometry of single particles with internal degrees of freedom is investigated. We discuss the interference patterns obtained when an internal state evolution device is inserted into one or both the paths of the interferometer. The interference pattern ... More

Radiative and semileptonic decays in Chiral Perturbation TheoryJul 03 2007I give a short overview of what has been done in radiative and semileptonic Kaon decays in Chiral Perturbation Theory. This includes for semileptonic decays the work which has been done to order $p^6$ including preliminary results of isospin breaking ... More

Eta and Eta' physicsOct 22 2007This talk describes the reasons why $\eta$ and $\eta^\prime$ decays are an interesting topic of study for both theory and experiment. The main part discusses the results of the recent calculation of $\eta\to3\pi$ at two-loop order in ChPT. Some puzzling ... More

QCD and Weak Interactions of Light QuarksApr 05 2002This review contains an overview of strong interaction effects in weak decays starting with a historical introduction. It contains a short overview of semileptonic decays and their relevance for measuring CKM matrix elements. The main part is devoted ... More

I : Chiral Perturbation for Kaons II: The $ΔI=1/2$-rule in the Chiral LimitJul 27 1999I : Chiral Perturbation Theory is introduced and its applications to semileptonic and nonleptonic kaon decays are discussed. II: The method of large $N_c$ is used to calculate $K\to\pi\pi$ nonleptonic matrix elements, in particular the matching procedure ... More

Chiral Perturbation TheoryFeb 27 1995A short overview of the current state of Chiral Perturbation Theory is given. This includes a description of the basic assumptions, the usefulness of the external field method is emphasized using a simple lowest order example. Then at next-to-leading ... More

An improved description of charged Higgs boson productionOct 11 2004Many extensions of the Standard Model predict the existence of charged Higgs bosons. In order to be able to find those particles, an accurate description of their production is needed. In Monte Carlo simulations of charged Higgs boson production at hadron ... More

Aspects of nonrelativistic quantum gravityOct 22 2009A nonrelativistic approach to quantum gravity is studied. At least for weak gravitational fields it should be a valid approximation. Such an approach can be used to point out problems and prospects inherent in a more exact theory of quantum gravity, yet ... More

Newtonian Quantum GravityDec 04 2006A Newtonian approach to quantum gravity is studied. At least for weak gravitational fields it should be a valid approximation. Such an approach could be used to point out problems and prospects inherent in a more exact theory of quantum gravity, yet to ... More

A hierarchy of cosmic compact objects - without black holesMar 14 2006We make the case for the existence of a, hitherto unknown and unobserved, hierarchy of ever more compact cosmic objects in the universe. This hypothesis is based on i) the assumption of "elementary" particle sub-constituents on several levels below the ... More

A simple solution to color confinementNov 03 2000We show that color confinement is a direct result of the nonabelian, i.e. nonlinear, nature of the color interaction in quantum chromodynamics. This makes it in general impossible to describe the color field as a collection of elementary quanta (gluons). ... More

Reply to comment on ``A simple explanation of the non-appearance of physical gluons and quarks"Feb 13 2003This is the reply to a comment by Andreas Aste [hep-th/0302103] on a previous article of mine in Can.J.Phys. The counter-arguments used by Aste utilize a mathematical limit without physical meaning. We still contend that in QCD, the particles ``gluons'' ... More

Turan's problem 10 revisitedSep 11 2006Jun 28 2007In this paper we prove that inf_{|z_k| => 1} max_{v=1,...,n^2} |sum_{k=1}^n z_k^v| = sqrt n+O(n^{0.2625+epsilon}). This improves on the bound O(sqrt (n log n)) of Erdos and Renyi. In the special case of $n+1$ being a prime we have previously proved the ... More

On a problem of Ramachandra and approximation of functions by Dirichlet polynomials with bounded coefficientsJul 19 2012We prove effective results on when a function can be approximated by a Dirichlet polynomial with bounded coefficients. Assuming that \Phi(n) is an increasing function we prove that the set of polynomials {\sum_{n=2}^N a_n n^{it-1}: N \geq 2, |a_n| \leq ... More

The number of unbounded components in the Poisson Boolean model in hyperbolic spaceOct 05 2006Nov 05 2007We consider the Poisson Boolean continuum percolation model in n-dimensional hyperbolic space. In 2 dimensions we show that there are intensities for the underlying Poisson process for which there are infinitely unbounded components in the covered and ... More

On compatibility of the $\ell$-adic realisations of an abelian motiveJun 28 2017Sep 08 2017In this article we introduce the notion of a quasi-compatible system of Galois representations. The quasi-compatibility condition is a slight relaxation of the classical compatibility condition in the sense of Serre. The main theorem that we prove is ... More

Physics-inspired derivations of some algorithms for computing the permanentJun 21 2017We provide physics-inspired derivations of a number of algorithms for computing the permanent of a matrix. In particular we formulate the computation of the permanent as a Grassmann integral that may be viewed as an interacting many-fermion problem. Applying ... More

Group Compactifications and Moduli SpacesJun 03 2017We give a summary of joint work with Michael Thaddeus that realizes toroidal compactifcations of split reductive groups as moduli spaces of framed bundles on chains of rational curves. We include an extension of this work that covers Artin stacks with ... More

Attracting Currents and Equilibrium Measures for Quasi-attractors of $\mathbb P^k$Sep 02 2016Let $f$ be a holomorphic endomorphism of $\mathbb P^k$ of degree $d.$ For each quasi-attractor of $f$ we construct a finite set of currents with attractive behaviors. To every such an attracting current is associated an equilibrium measure which allows ... More

On questions of Cassels and Drungilas-DubickasJun 08 2016We answer a question of Drungilas-Dubickas in the affirmative under the assumption of standard conjectures on smooth numbers in polynomial sequences. This gives evidence against the "Dubickas Conjecture", which Ka\v{c}inskait\.e and Laurin\v{c}ikas proved ... More

The Case for Meta-Cognitive Machine Learning: On Model Entropy and Concept Formation in Deep LearningNov 04 2017Machine learning is usually defined in behaviourist terms, where external validation is the primary mechanism of learning. In this paper, I argue for a more holistic interpretation in which finding more probable, efficient and abstract representations ... More

A short note on extended probability theorySep 29 2015We propose two distinct interpretations of extended probabilities which are realistic for the physical world.

A note on the Brush Number of Jaco Graphs, $J_n(1), n \in \Bbb NDec 18 2014The concept of the brush number $b_r(G)$ was introduced for a simple connected undirected graph $G$. This note extends the concept to a special family of directed graphs and declares that the brush number $b_r(J_n(1))$ of a finite Jaco graph, $J_n(1), ... More

The homotopy groups of the spectrum TmfDec 15 2012We use the structure of the homotopy groups of the connective spectrum tmf of topological modular forms and the elliptic and Adams-Novikov spectral sequences to compute the homotopy groups of the non-connective version, Tmf, of that spectrum. This is ... More

On the Entropy of Random Fibonacci WordsJan 20 2010The random Fibonacci chain is a generalisation of the classical Fibonacci substitution and is defined as the rule mapping $0\mapsto 1$ and $1 \mapsto 01$ with probability $p$ and $1 \mapsto 10$ with probability $1-p$ for $0<p<1$ and where the random rule ... More