order based dependent dirichlet processes in this article we propose a new framework for bayesian nonparametric modeling with continuous covariates in particular we allow the nonparametric distribution to depend on covariates through ordering the random variables building the weights in the stick breaking representation we focus mostly on the class of random distributions that induces a dirichlet process at each covariate value we derive the correlation between distributions at different covariate values and use a point process to implement a practically useful type of ordering two main constructions with analytically known correlation structures are proposed practical and efficient computational methods are introduced we apply our framework through mixtures of these processes to regression modeling the modeling of stochastic volatility in time series data and spatial geostatistical modeling

global existence for energy critical waves in 3 d domains we prove that the defocusing quintic wave equation with dirichlet boundary conditions is globally well posed on for any smooth compact domain the main ingredient in the proof is an spectral projector estimate obtained recently by smith and sogge combined with a precise study of the boundary value problem

on the strongly damped wave equation we prove the existence of the universal attractor for the strongly damped semilinear wave equation in the presence of a quite general nonlinearity of critical growth when the nonlinearity is subcritical we prove the existence of an exponential attractor of optimal regularity having a basin of attraction coinciding with the whole phase space as a byproduct the universal attractor is regular and of finite fractal dimension moreover we carry out a detailed analysis of the asymptotic behavior of the solutions in dependence of the damping coefficient

non gaussian spatiotemporal modelling through scale mixing we construct non gaussian processes that vary continuously in space and time with nonseparable covariance functions starting from a general and flexible way of constructing valid nonseparable covariance functions through mixing over separable covariance functions the resulting models are generalized by allowing for outliers as well as regions with larger variances we induce this through scale mixing with separate positive valued processes smooth mixing processes are applied to the underlying correlated processes in space and in time thus leading to regions in space and time of increased spread an uncorrelated mixing process on the nugget effect accommodates outliers posterior and predictive bayesian inference with these models is implemented through a markov chain monte carlo sampler an application to temperature data in the basque country illustrates the potential of this model in the identification of outliers and regions with inflated variance and shows that this improves the predictive performance

algebraic structures for 1d quasiperiodic systems the structure of certain quasiperiodic systems is analysed and interpreted by algebraic means all systems considered have a free monoid and semigroup structure with n generators as treated in the field of combinatorics on words an extension from free monoids to free groups and their automorphisms as treated by combinatorial group theory applies to a class of iterative systems the group structures yield algebraic relations and conserved quantities examples encompass the fibonacci system and its generalizations motivated by the applications to the physics of these systems homomorphisms from the free groups into linear operators of su 2 and su 1 1 are studied the systems are also viewed in a geometry due to fricke and klein the homomorphism to the group su 1 1 is the basis for the determination of 1d electronic spectra the class types their multiplication and the commutator properties are given the physics of these systems corresponds to the continuous schrodinger equation the quasiperiodic structure gives rise to exact relations between continuous and discrete systems

global existence for energy critical waves in 3 d domains neumann boundary conditions we prove that the defocusing quintic wave equation with neumann boundary conditions is globally well posed on h 1 l 2 for any smooth compact domain r 3 the proof relies on one hand on l p estimates for the spectral projector and on the other hand on a precise analysis of the boundary value problem which turns out to be much more delicate than in the case of dirichlet boundary conditions

localization problem in one dimension mapping and escape a one dimensional schr odinger equation in a discontinuous quasiperiodic potential is reduced to a recursion relation for transfer matrices and then to one for traces of these matrices when the potential is periodic the bandwidth goes to zero as an algebraic function of the period with a critical index which depends upon the potential strength this critical index is also evaluated as the solution to an escape rate problem for the recursion relations

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

fricke klein geometry for the group sl 2 c the homomorphisms from the free group f2 with two generators to the complex unimodular group sl 2 c are classified the action induced by the automorphism group phi 2 of f2 on pairs of elements from sl 2 c is expressed in a geometry inspired by fricke and klein

generators and subgroups for aut f3 the automorphisms of the free group f3 with three generators form the group aut f3 new involutive generators and relations for aut f3 are given which explain the relations given by nielsen finite coxeter groups and normal subgroups are derived which admit generalizations to n or 3

gaussian predictive process models for large spatial data sets with scientific data available at geocoded locations investigators are increasingly turning to spatial process models for carrying out statistical inference over the last decade hierarchical models implemented through markov chain monte carlo methods have become especially popular for spatial modelling given their flexibility and power to fit models that would be infeasible with classical methods as well as their avoidance of possibly inappropriate asymptotics however fitting hierarchical spatial models often involves expensive matrix decompositions whose computational complexity increases in cubic order with the number of spatial locations rendering such models infeasible for large spatial data sets this computational burden is exacerbated in multivariate settings with several spatially dependent response variables it is also aggravated when data are collected at frequent time points and spatiotemporal process models are used with regard to this challenge our contribution is to work with what we call predictive process models for spatial and spatiotemporal data every spatial or spatiotemporal process induces a predictive process model in fact arbitrarily many of them the latter models project process realizations of the former to a lower dimensional subspace thereby reducing the computational burden hence we achieve the flexibility to accommodate non stationary non gaussian possibly multivariate possibly spatiotemporal processes in the context of large data sets we discuss attractive theoretical properties of these predictive processes we also provide a computational template encompassing these diverse settings finally we illustrate the approach with simulated and real data sets

approximate bayesian inference with the weighted likelihood bootstrap we introduce the weighted likelihood bootstrap wlb as a way to simulate approximately from a posterior distribution this method is often easy to implement requiring only an algorithm for calculating the maximum likelihood estimator such as iteratively reweighted least squares in the generic weighting scheme the wlb is first order correct under quite general conditions inaccuracies can be removed by using the wlb as a source of samples in the sampling importance resampling sir algorithm which also allows incorporation of particular prior information the sir adjusted wlb can be a competitive alternative to other integration methods in certain models asymptotic expansions elucidate the second order properties of the wlb which is a generalization of rubin s bayesian bootstrap the calculation of approximate bayes factors for model comparison is also considered we note that given a sample simulated from the posterior distribution the required marginal likelihood may be simulation consistently estimated by the harmonic mean of the associated likelihood values a modification of this estimator that avoids instability is also noted these methods provide simple ways of calculating approximate bayes factors and posterior model probabilities for a very wide class of models

poisson gamma random field models for spatial statistics summary doubly stochastic bayesian hierarchical models are introduced to account for uncertainty and spatial variation in the underlying intensity measure for point process models inhomogeneous gamma process random fields and more generally markov random fields with infinitely divisible distributions are used to construct positively autocorrelated intensity measures for spatial poisson point processes these in turn are used to model the number and location of individual events a data augmentation scheme and markov chain monte carlo numerical methods are employed to generate samples from bayesian posterior and predictive distributions the methods are developed in both continuous and discrete settings and are applied to a problem in forest ecology

maximal functions associated to filtrations abstract let t be a bounded linear or sublinear operator from lp y to lq x a maximal operator t f x supj t f yj x is associated to any sequence of subsets yj of y under the hypotheses that q p and the sets yj are nested we prove that t is also bounded classical theorems of menshov and zygmund are obtained as corollaries multilinear generalizations of this theorem are also established these results are motivated by applications to the spectral analysis of schrodinger operators

well posedness and attractors for a super cubic weakly damped wave equation with h 1 source term abstract this work is devoted to dirichlet problem for the sup cubic semi linear wave equation with h 1 forcing in a bounded domain of r 3 the global well posedness of the weak solution with some translational regularity is obtained moreover the existence of a global attractor as well as exponential attractor in the natural phase space are also considered the results are crucially based on the strichartz estimates for the linear wave equation in bounded domain of r 3

involutive generators and actions for the group phi 2 the automorphisms of the free group f2 form the group phi 2 three involutions are shown to generate phi 2 the homomorphism h2 phi 2 to gl 2 z induced by the abelianization h1 f2 to z2 is generated by three affine reflections r3 to r3 the actions of phi 2 induced by the homomorphisms f2 to su 2 su 1 1 sl 2 c are generated by algebraic involutions and geometric reflections

space time covariance functions this work considers a number of properties of space time covariance functions and how these relate to the spatial temporal interactions of the process first it examines how the smoothness away from the origin of a space time covariance function affects for example temporal correlations of spatial differences models that are not smoother away from the origin than they are at the origin such as separable models have a kind of discontinuity to certain correlations that one might wish to avoid in some circumstances smoothness away from the origin of a covariance function is shown to follow from the corresponding spectral density having derivatives with finite moments these results are used to obtain a parametric class of spectral densities whose corresponding space time covariance functions are infinitely differentiable away from the origin and that allows for essentially arbitrary and possibly different degrees of smoothness for the process in space and time second this work considers models that

an interpretation of 2d quasiperiodic patterns in terms of automorphisms of free groups the penrose and triangle patterns are interpreted as strings in the context of free grammars the set of production rules of such grammars is induced by certain automorphisms of free groups

dispersion for the wave equation inside strictly convex domains i the friedlander model case we consider a model case for a strictly convex domain r d of dimension d 2 with smooth boundary 6 and we describe dispersion for the wave equation with dirichlet boundary conditions more specically we obtain the optimal xed time decay rate for the smoothed out green function a t 1 4 loss occurs with respect to the boundary less case due to repeated occurrences of swallowtail type singularities in the wave front set

the double torus as a 2d cosmos groups geometry and closed geodesics the double torus provides a relativistic model for a closed 2d cosmos with topology of genus 2 and constant negative curvature its unfolding into an octagon extends to an octagonal tessellation of its universal covering the hyperbolic space h2 the tessellation is analysed with tools from hyperbolic crystallography actions on h2 of groups subgroups are identified for su 1 1 for a hyperbolic coxeter group acting also on su 1 1 and for the homotopy group 2 whose extension is normal in the coxeter group closed geodesics arise from links on h2 between octagon centres the direction and length of the shortest closed geodesics is computed dedicated to marcos moshinsky on the occasion of his 80th birthday

asymptotic regularity for some dissipative equations abstract this paper is devoted to proving some asymptotic regularity for both reaction diffusion equation with a polynomially growing nonlinearity of arbitrary order and strongly damped wave equation with critical nonlinearity which excel the sharp regularity allowed by the corresponding stationary equations equilibrium points based on this regularity the existence of the finite dimensional global and exponential attractors can be obtained easily