singular solutions for some semilinear elliptic equations let b r x n x r with n 2 consider a function u which satisfies cd egin array 20 c u n c 2 b r ackslash 0 u eqslant 0 on b r ackslash 0 elta u u p 0 on b r ackslash 0 nd array 1

perturbed homoclinic solutions in reversible 1 1 resonance vector fields we consider a smooth reversible vector field in r 4 such that the origin is a fixed point the differential at the origin has two double pure imaginary eigenvalues iq for the critical value 0 of the parameter we show by a normal form analysis that the vector field can be approximated by an integrable field in r 4 for which we know all solutions specially interesting ones are the homoclinics to 0 and homoclinics to periodic solutions depending on the sign of a leading nonlinear coefficient we prove in particular the persistence of these homoclinics for the full vector field

pattern formation on the free surface of a ferrofluid spatial dynamics and homoclinic bifurcation abstract we establish the existence of spatially localised one dimensional free surfaces of a ferrofluid near onset of the rosensweig instability assuming a general nonlinear magnetisation law it is shown that the ferrohydrostatic equations can be derived from a variational principle that allows one to formulate them as an infinite dimensional spatial hamiltonian system in which the unbounded free surface direction plays the role of time a centre manifold reduction technique converts the problem for small solutions near onset to an equivalent hamiltonian system with finitely many degrees of freedom normal form theory yields the existence of homoclinic solutions to the reduced system which correspond to spatially localised solutions of the ferrohydrostatic equations

singular solutions of some nonlinear elliptic equations abstract we give a classification of isolated singularities of any solution of the equation u u u q 1 0 if 0 is the singular point we prove that there exist two types of singularities when 1 n n 2 and that we have essentially the following in 0 i either u x l q n x 2 q 1 with l q n 2 q 1 2q q 1 n 1 q 1 ii or u x c x where c is any constant and the fundamental solution of the laplacian in r n

solutions with time dependent singular sets for the heat equation with absorption abstract we consider the heat equation with a superlinear absorption term t u u u p in r n and study the existence of nonnegative solutions with an m dimensional time dependent singular set where n m 3 we prove that if 1 p n m n m 2 then there are two types of singular solutions moreover we show the uniqueness of the solutions and specify the exact behavior of the solutions near the singular set

resolution of a semilinear equation in l1 let n or be a bounded regular open set of n and let x s be a continuous nondecreasing function in s measurable in x such that x 0 0 almost everywhere we solve for f l 1 the problem p u u f in u 0 on in fact for this result instead of assuming that is nondecreasing in s we need only that x s s 0 we deduce an almost necessary and sufficient condition on in order that p has a solution roughly speaking this condition is f v g with g l 1 and v l 1

nonexistence theorems for singular solutions of quasilinear partial differential equations on etudie les etats fondamentaux singuliers de l equation de poisson semilineaire u f u 0 et de son analogue quasi lineaire div a du du f u 0 ou est le laplacien a n dimensions et a p et f u sont des fonctions donnees

isolated singularities for weighted quasilinear elliptic equations abstract we classify all the possible asymptotic behavior at the origin for positive solutions of quasilinear elliptic equations of the form div u p 2 u b x h u in 0 where 1 p n and is an open subset of r n with 0 our main result provides a sharp extension of a well known theorem of friedman and veron for h u u q and b x 1 and a recent result of the authors for p 2 and b x 1 we assume that the function h is regularly varying at with index q that is lim t h t h t q for every 0 and the weight function b x behaves near the origin as a function b 0 x varying regularly at zero with index greater than p this condition includes b x x and some of its perturbations for instance b x x log x m for any m r our approach makes use of the theory of regular variation and a new perturbation method for constructing sub and super solutions

exact filtering for partially observed continuous time models summary the forward backward algorithm is an exact filtering algorithm which can efficiently calculate likelihoods and which can be used to simulate from posterior distributions using a simple result which relates gamma random variables with different rates we show how the forward backward algorithm can be used to calculate the distribution of a sum of gamma random variables and to simulate from their joint distribution given their sum one application is to calculating the density of the time of a specific event in a markov process as this time is the sum of exponentially distributed interevent times this enables us to apply the forward backward algorithm to a range of new problems we demonstrate our method on three problems calculating likelihoods and simulating allele frequencies under a non neutral population genetic model analysing a stochastic epidemic model and simulating speciation times in phylogenetics

pattern selection in ferrofluids pattern formation by surface instabilities in a ferrofluid is studied as a function of an applied vertical magnetic field h the effects of sidewalls are neglected the problem is formulated as a bifurcation problem on an appropriately chosen doubly periodic lattice normal forms for both square and hexagonal lattices are given and the necessary coefficients computed from the partial differential equations on the square lattice solutions in the form of parallel ridges are never stable near onset for a relative magnetic permeability 1 4 stable squares are formed on the hexagonal lattice a finite amplitude instability produces hexagons analysis of a degenerate bifurcation predicts a hysteretic transition to ridges with increasing h

asymptotic behavior of solutions of nonlinear elliptic equations we study and obtain formulas for the asymptotic behavior as x of c2 solutions of the semilinear equation u f x u xe where is the complement of some ball in n and f is continuous and nonlinear in u if for large x f is nearly radially symmetric in x we give conditions under which each positive solution of is asymptotic as x to some radially symmetric function our results can also be useful when f is only bounded above or below by a function which is radially symmetric in x or when the solution oscillates in sign examples when f has power like growth or exponential growth in the variables x and u usefully illustrate our results

some models for discretized series of events a discretized series of events is a binary time series that indicates whether or not events of a point process in the line occur in successive intervals such data are common in environmental applications we describe a class of models for them based on an unobserved continuous time discrete state markov process which determines the rate of a doubly stochastic poisson process from which the binary time series is constructed by discretization we discuss likelihood inference for these processes and their second order properties and extend them to multiple series an application involves modeling the times of exposures to air pollution at a number of receptors in western europe

statistical inference for discretely observed markov jump processes summary likelihood inference for discretely observed markov jump processes with finite state space is investigated the existence and uniqueness of the maximum likelihood estimator of the intensity matrix are investigated this topic is closely related to the imbedding problem for markov chains it is demonstrated that the maximum likelihood estimator can be found either by the em algorithm or by a markov chain monte carlo procedure when the maximum likelihood estimator does not exist an estimator can be obtained by using a penalized likelihood function or by the markov chain monte carlo procedure with a suitable prior the methodology and its implementation are illustrated by examples and simulation studies

marginal likelihood from the gibbs output abstract in the context of bayes estimation via gibbs sampling with or without data augmentation a simple approach is developed for computing the marginal density of the sample data marginal likelihood given parameter draws from the posterior distribution consequently bayes factors for model comparisons can be routinely computed as a by product of the simulation hitherto this calculation has proved extremely challenging our approach exploits the fact that the marginal density can be expressed as the prior times the likelihood function over the posterior density this simple identity holds for any parameter value an estimate of the posterior density is shown to be available if all complete conditional densities used in the gibbs sampler have closed form expressions to improve accuracy the posterior density is estimated at a high density point and the numerical standard error of resulting estimate is derived the ideas are applied to probit regression and finite mixture models

bayesian inference for markov processes with diffusion and discrete components summary data arising in certain radio tracking experiments consist of both a continuous spatial component and a discrete component related to behaviour this leads naturally to stochastic models with a state space which is a product of continuous and discrete components we consider a class of such models in continuous time which can be thought of as diffusions in random environments they are related to switching diffusion or hidden markov models but observations are made on both components at discrete time points so that neither component is completely hidden we describe and illustrate an approach to fully bayesian inference for these general models the algorithm used is a hybrid markov chain monte carlo method the diffusion parameters the environment parameters and the sample path of the environment process itself are updated separately in sequence and the individual steps are a mixture of gibbs and random walk metropolishastings types some implementation and model checking issues are discussed and an example using data arising from a radio tracking experiment is described

exact singular behavior of positive solutions to nonlinear elliptic equations with a hardy potential abstract in this paper we study the singular behavior at x 0 of positive solutions to the equation u x 2 u x u p x 0 where r n n 3 is a bounded domain with 0 and p 1 2 are given constants for the case n 2 2 4 the singular behavior of all the positive solutions is completely classified in the recent paper 5 here we determine the exact singular behavior of all the positive solutions for the remaining case n 2 2 4 in sharp contrast to the case n 2 2 4 where several converging blow up rates of u x are possible as x 0 we show that when n 2 2 4 every positive solution u x blows up in the same fashion lim x 0 x 2 p 1 u x 2 p 1 2 p 1 2 n 1 p 1

classification of singular solutions of a nonlinear heat equation on considere toutes les solutions non negatives de l equation de la chaleur non lineaire avec absorption u t u m u p dans q r n 0 t pour n 1 m p 1 avec des conditions initiales u x 0 0 pour x 0

the dimension of the boundary of super brownian motion we show that the hausdorff dimension of the boundary of d dimensional super brownian motion is 0 if d 1 4 2 sqrt 2 if d 2 and 9 sqrt 17 2 if d 3

an exact gibbs sampler for the markov modulated poisson process a markov modulated poisson process mmpp is a poisson process whose intensity varies according to a markov process we present a novel technique for simulating from the exact distribution of a continuous time markov chain over an interval given the start and end states and the infinitesimal generator and use this to create a gibbs sampler which samples from the exact distribution of the hidden markov chain in an mmpp we apply the gibbs sampler to modelling the occurrence of a rare dna motif the chi site and to inferring regions of the genome with evidence of high or low intensities for occurrences of this site

asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity we consider the semilinear elliptic equationu h u in 0 w here is an open subset of r n n 2 containing the origin and h is locally lipschitz continuous on 0 positive in 0 w e give a complete classification of isolated singularities of positive solutions when h varies regularly at infinity of in dex q 1 c n that is limu h u h u q f or every 0 where cn denotes either n n 2 if n 3o r if n 2 our result extends a well known theorem of veron for the case h u u q

asymptotic behavior of the least energy solution of a problem with competing powers we consider the problem e2 u uq up 0 in u 0 in u 0 on here is a smooth bounded domain in rn 1 q p n 2n 2 if n 3 and e is a small positive parameter we study the asymptotic behavior of the least energy solution as e goes to zero in the case q nn 2 we show that the limiting behavior is dominated by the singular solution g gq 0 in p g 0 on the reduced energy is of nonlocal type