hamiltonian stationary cones and self similar solutions in higher dimension in lw we construct examples of two dimensional hamiltonian stationary self shrinkers and self expanders for lagrangian mean curvature flows which are asymptotic to the union of two schoen wolfson cones these self shrinkers and self expanders can be glued together to yield solutions of the brakke flow a weak formulation of the mean curvature flow moreover there is no mass loss along the brakke flow in this paper we generalize these results to higher dimension we construct new higher dimensional hamiltonian stationary cones of different topology as generalizations of the schoen wolfson cones hamiltonian stationary self shrinkers and self expanders that are asymptotic to these hamiltonian stationary cones are also constructed they can also be glued together to produce eternal solutions of the brakke flow without mass loss finally we show the same conclusion holds for those lagrangian self similar examples recently found by joyce tsui and the first author in jlt

reaction limited catalytic reaction in one dimension the authors analyse a mathematical model of reaction limited catalytic reaction on a one dimensional lattice which can be transformed into a system of random walkers with pair annihilation and spontaneous branchings time evolution of the correlation function and reaction rate are obtained theoretically and compared with numerical results

rescaled contact processes converge to super brownian motion in two or more dimensions we show that in dimensions two or more a sequence of long range contact processes suitably rescaled in space and time converges to a super brownian motion with drift as a consequence of this result we can improve the results of bramson durrett and swindle 1989 by replacing their order of magnitude estimates of how close the critical value is to 1 with sharp asymptotics

mean exit times for particles driven by weakly colored noise consider a particle whose velocity is an ornstein uhlenbeck process with correlation time varepsilon 2 frac dx dt frac sigma z varepsilon qquad frac dz dt frac z varepsilon 2 frac sqrt 2 xi t varepsilon where xi t is gaussian white noise this represents brownian motion with memory i e a non markovian brownian motion singular perturbation techniques reduce the mean exit time problem to a boundary layer problem which turns out to be the half range expansion problem for the fokker planck operator this is solved via complex variable theory it is found that the spatial density has a milne extrapolation length varepsilon sigma zeta tfrac 1 2 varepsilon sigma 1 46035 cdots where zeta is the riemann zeta function it is also found that the average time at which x t first escapes an interval a 0 b ven that it starts at x 0 0 and z 0 z 0 is e t text ex frac 1 2 sigma 2 left a varepsilo

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

rescaled lotka volterra models converge to super brownian motion we show that a sequence of stochastic spatial lotka volterra models suitably rescaled in space and time converges weakly to super brownian motion with drift the result includes both long range and nearest neighbor models the latter for dimensions three and above these theorems are special cases of a general convergence theorem for perturbations of the voter model

analytic representation of bayes labeling and bayes clustering operators for random labeled point processes clustering algorithms typically group points based on some similarity criterion but without reference to an underlying random process to make clustering algorithms rigorously predictive in fact there exists a probabilistic theory of clustering in the context of random labeled point sets in which clustering error is defined in terms of the process in the present paper given an underlying point process we develop a general analytic procedure for finding an optimal clustering operator the bayes clusterer that corresponds to the bayes classifier in classification theory we provide detailed solutions under gaussian models owing to computational complexity we also develop approximations of the bayes clusterer

spatial models for species area curves the relationship between species number and area is an old problem in biology we propose here an interacting particle system the multitype voter model with mutation as a mathematical model to study this problem we analyze the species area curves of this model as the mutation rate a tends to zero we obtain two basic types of behavior depending on the size of the spatial region under consideration if the region is a square with area r r 1 then for small the number of species is of order l r log 2 whereas if r 1 the number of species is bounded

stochastic lattice gas model for a predator prey system we propose a stochastic lattice gas model to describe the dynamics of two animal species population one being a predator and the other a prey this model comprehends the mechanisms of the lotka volterra model our analysis was performed by using a dynamical mean field approximation and computer simulations our results show that the system exhibits an oscillatory behavior of the population densities of prey and predators for the sets of parameters used in our computer simulations these oscillations occur at a local level mean field results predict synchronized collective oscillations

crabgrass measles and gypsy moths an introduction to modern probability on etudie cinq modeles qui permettent d expliquer des processus lies aux systemes de particules en interaction ainsi que d autres processus

the analytic solutions of some boundary layer problems in the theory of brownian motion the authors give analytic solutions of the generalised albedo and milne problems for the uhlenbeck ornstein process and discuss the relation between these and the original wang uhlenbeck 1945 problem of determining the distribution of first passage times

on the convergence in distribution of measurable multifunctions random sets normal integrands stochastic processes and stochastic infima the concept of the distribution function of a closed valued measurable multifunction is introduced and used to study the convergence in distribution of sequences of multifunctions and the epi convergence in distribution of normal integrands and stochastic processes in particular various compactness criteria are exhibited the connections with the classical convergence theory for stochastic processes are analyzed and for purposes of illustration we apply the theory to sketch out a modified derivation of donsker s theorem brownian motion as a limit of normalized random walks we also suggest the potential application of the theory to the study of the convergence of stochastic infima

the contact process in high dimensions on renforce le theoreme de convergence complete pour le processus de contact en grandes dimensions en elargissant l etendue du parametre d infection dans l hypothese du theoreme

a bayesian hierarchical nonoverlapping random disc growth model a methodology is proposed to efficiently model a random set via a multistage hierarchical bayesian model we define a nonoverlapping random disk model nordm which is similar in spirit to the well known poisson boolean model this model is formulated in a conditional setting that facilitates bayesian sampling of important parameters in the model this framework can accommodate any object not just those with disk shapes although the model can be easily extended to include any known compact convex set instead of the disc e g polygons or ellipses we further propose a growth model that is conceptually simple and allows straightforward estimation of parameters without the need for tedious calculations of hitting or inclusion probabilities the model is applied to severe storm cell development as obtained from weather radar

the density profile for the klein kramers equation near an absorbing wall we derive asymptotic series for the expansion coefficients of a function in terms of the pagani functions which occur in the boundary layer solutions of the klein kramers equation the results enable us to determine the density profile in the stationary solution of this equation near an absorbing wall from the numerically determined velocity distribution at the wall with an accuracy of about 2 we also obtain information about the analytic behavior of the density profile this profile increases near the wall with the square root of the distance to the wall finally the asymptotic analysis leads to an understanding of the slow convergence of variational approximations to the solution of the absorbing wall problem and of the exponents that occur when one studies the variational approximations to various quantities of interest as functions of the number of terms in the variational ansatz this is used to obtain a better variational estimate for the density at the wall

binary mosaics and the spatial pattern of heather an idealized model for the spatial pattern of heather in a planar region is proposed in this model the subregiorl occupied by heather is represented by the union of randomly located discs of randomly varying radii some previous applications of the model are noted techniques for parameter estimation and for testing goodness of fit are suggested these techniques are applied to data showing the incidence of heather in a lomx20m rectangle at jadraas sweden thereby demonstrating the extension to binary mosaics of methodology previously illustrated with spatial point processes

residual analysis for spatial point processes with discussion summary we define residuals for point process models fitted to spatial point pattern data and we propose diagnostic plots based on them the residuals apply to any point process model that has a conditional intensity the model may exhibit spatial heterogeneity interpoint interaction and dependence on spatial covariates some existing ad hoc methods for model checking quadrat counts scan statistic kernel smoothed intensity and berman s diagnostic are recovered as special cases diagnostic tools are developed systematically by using an analogy between our spatial residuals and the usual residuals for non spatial generalized linear models the conditional intensity plays the role of the mean response this makes it possible to adapt existing knowledge about model validation for generalized linear models to the spatial point process context giving recommendations for diagnostic plots a plot of smoothed residuals against spatial location or against a spatial covariate is effective in diagnosing spatial trend or co variate effects q q plots of the residuals are effective in diagnosing interpoint interaction

stochastic p d e s arising from the long range contact and long range voter processes a long range contact process and a long range voter process are scaled so that the distance between sites decreases and the number of neighbors of each site increases the approximate densities of occupied sites under suitable tine scaling converge to continuous space time densities which solve stochastic p d e s for the contact process the limiting equation is the kolmogorov petrovskii piscuinov equation driven by branching white noise for the voter process the limiting equation is the heat equation driven by fisher wright white noise

convergence rates for the optimal values of allocation processes the optimal allocation process is of the following type a sequence of functions production functions f1 f2 is given each mapping r n into 0 along with it a vector q 0 in r n vector of resources is provided for each integer k we consider the problem mbox maximize sum k i 1 f i x i quad mbox subject to sum k i 1 x i leq kq here r n is the nonnegative orthant of the n dimensional euclidean space inequalities between vectors are taken coordinatewise in this paper we consider the case where the functions fi are subject to randomness specifically they are generated as independent random drawings from a common distribution in particular fi fi x with being in a probability space and the solutions xi might depend on u as well as on k and q the detailed technical conditions are displayed in 2 we are interested in limit properties as k of the optimal values of the allocation process

universal first passage properties of discrete time random walks and levy flights on a line statistics of the global maximum and records abstract in these lecture notes i will discuss the universal first passage properties of a simple correlated discrete time sequence x 0 0 x 1 x 2 x n up to n steps where x i represents the position at step i of a random walker hopping on a continuous line by drawing independently at each time step a random jump length from an arbitrary symmetric and continuous distribution it includes e g the levy flights i will focus on the statistics of two extreme observables associated with the sequence i its global maximum and the time step at which the maximum occurs and ii the number of records in the sequence and their ages i will demonstrate how the universal statistics of these observables emerge as a consequence of pollaczek spitzer formula and the associated sparre andersen theorem

lattice green s functions for kagome diced and hyperkagome lattices the lattice green s functions lgfs and density of states dos for noninteracting models on three related lattices are presented the dos and lgfs at the origin for the kagome and diced lattices are rederived furthermore from the form obtained for the dos of the hyperkagome lattice it seems highly suggestive that a closed form expression can be obtained if a particular complicated yet highly symmetrical integral can be solved the results have been checked numerically the continuum of energy states in the hyperkagome lattice and the appearance of a zero in the kagome lattice may be explicitly seen from the formulas obtained

law of large numbers for random sets and allocation processes in this paper we establish a strong law of large numbers for unbounded random sets and then apply it to an optimization problem arising in allocation processes under uncertainty

asymptotic granulometric mixing theorem morphological estimation of sizing parameters and mixture proportions abstract if a random set binary image is composed of randomly sized disjoint translates arising as homothetics of a finite number of compact primitives and a granulometry is generated by a convex compact set then the granulometric moments of the random set can be expressed in terms of the granulometric moments of the generating primitives their areas and the radii of the homethetics moreover if there is only a single generating primitive then the granulometric moments of the random set are asymptotically normal relative to an increasing number of grains translates and methods exist to express their asymptotic moments the present paper extends the asymptotic theory to multiple primitive images the extension is significant in two regards first the single primitive asymptotic theory was based on the classical moment theory of cramer the new theory uses an approach based on analytic decomposition on the granulometric moments second not only does the new theory demonstrate asymptotic normality but it also permits method of moment estimation of the parameters of the distributions governing grain sizes and mixture proportions among grain types in the present setting granulometric method of moment estimation is based on asymptotic representation of expectations of granulometric moments in terms of grain sizing model parameters mixture proportions and geometric constants method of moment estimation for sizing parameters and mixture proportions is achieved by replacing granulometric moment expectations with moment estimations from image realizations and then solving the resulting system of equations for the model parameters the theory is applied to both normal and gamma sizing distributions

approximations of upper and lower probabilities by measurable selections a random set can be regarded as the result of the imprecise observation of a random variable following this interpretation we study to which extent the upper and lower probabilities induced by the random set keep all the information about the values of the probability distribution of the random variable we link this problem to the existence of selectors of a multi valued mapping and with the inner approximations of the upper probability and prove that under fairly general conditions although not in all cases the upper and lower probabilities are an adequate tool for modelling the available information in doing this we generalise a number of results from the literature finally we study the particular case of consonant random sets and we also derive a relationship between aumann and choquet integrals

the structure of two dimensional witten sander aggregates a relatively large number 102 103 of reasonably large about 105 particles or sites two dimensional witten sander aggregates are simulated using new algorithms this enables the reduction of statistical errors by an order of magnitude and provides an opportunity to determine some new quantities describing in more detail the structure of witten sander clusters a quantitate description of the distortion of large lattice based witten sander clusters into a diamond like shape is provided the orientational correlation in the bond directions for off lattice witten sander clusters is also measured this quantity has a power law dependence on distance with an exponent of about 0 37

random set theory and problems of modeling the three or four dimensional world in which we live is full of objects to be measured and summarized very often a parsimonious finite collection of measurements is enough for scientific investigation into an object s genesis and evolution there is a growing need however to describe and model objects through their form as well as their size the purpose of this article is to show the potentials and limitations of a probabilistic and statistical approach collections of objects the data are assimilated to a random set the model whose parameters provide description and or explanation

branching exclusion process on a strip we consider a model of stochastically interacting particles on an infinite strip of 2 in this model known as a branching exclusion process particles jump to each empty neighboring site with rate 4 and also can create a new particle with rate 1 4 at each one of these sites the initial configuration is assumed to have a rightmost particle and we study the process as seen from the rightmost vertical line occupied we prove that this process has exactly one invariant measure with the property thath the number of empty sites to the left of the rightmost particle has an exponential moment this refines a result presented by bramson eaet al who proved that ford 1 h is finite with probability 1

the kinetic boundary layer for the fokker planck equation with absorbing boundary the fokker planck equation for the distribution of position and velocity of a brownian particle is a particularly simple linear transport equation its normal solutions and an apparently complete set of stationary boundary layer solutions can be determined explicitly by a numerical algorithm we select linear combinations of them that approximately fulfill the boundary condition for a completely absorbing plane wall and that approach a linearly increasing position space density far from the wall various aspects of these approximate solutions are discussed in particular we find that the extrapolated asymptotic density reaches zero at a distance xm beyond the wall we find xm 1 46 in units of the velocity persistence length of the brownian particle this study was motivated by certain problems in the theory of diffusion controlled reactions and the results might be used to test approximate theories employed in that field

random set framework for multiple instance learning multiple instance learning mil is a technique used for learning a target concept in the presence of noise or in a condition of uncertainty while standard learning techniques present the learner with individual samples mil alternatively presents the learner with sets of samples although sets are the primary elements used for analysis in mil research in this area has focused on using standard analysis techniques in the following a random set framework for multiple instance learning rsf mil is proposed that can directly perform analysis on sets the proposed method uses random sets and fuzzy measures to model the mil problem thus providing a more natural mathematical framework a more general mil solution and a more versatile learning tool comparative experimental results using rsf mil are presented for benchmark data sets rsf mil is further compared to the state of the art in landmine detection using ground penetrating radar data

assessment of local influence summary statistical models usually involve some degree of approximation and therefore are nearly always wrong because of this inexactness an assessment of the influence of minor perturbations of the model is important we discuss a method for carrying out such an assessment the method is not restricted to linear regression models and it seems to provide a relatively simple unified approach for handling a variety of problems

non equilibrium behaviour of a many particle process density profile and local equilibria one considers a simple exclusion particle jump process on where the underlying one particle motion is a degenerate random walk that moves only to the right one starts with the configuration in which the left halfline is completely occupied and the right one free it is shown that the number of particles at time t between site u t and v t divided by t converges a s to int limits u nu f w dw where f might be called the density profile it is explicitely determined and shown to be an affine function secondly we prove that the distribution of the process looked at by an observer travelling at constant speed u converges weakly to the bernoulli measure with density f u as the time tends to infinity

stochastic spatial models in the models we will consider space is represented by a grid of sites that can be in one of a finite number of states and that change at rates that depend on the states of a finite number of sites our main aim here is to explain an idea of durrett and levin 1994 the behavior of these models can be predicted from the properties of the mean field ode i e the equations for the densities of the various types that result from pretending that all sites are always independent we will illustrate this picture through a discussion of eight families of examples from statistical mechanics genetics population biology epidemiology and ecology some of our findings are only conjectures based on simulation but in a number of cases we are able to prove results for systems with fast stirring by exploiting connections between the spatial model and an associated reaction diffusion equation

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

diffusive clustering in the two dimensional voter model on donne des resultats de simulation et une enquete concernant les resultats connus sur le modele du votant dans trois cas distincts trois ou plusieurs dimensions une dimension et deux dimensions

ergodic theorems for the multitype contact process this paper studies a process involving competition of two types of particles 1 and 2 for the empty space 0 each site of the latticez d is therefore in one of three possible states 0 1 or 2 particles of each type die with rate 1 while an empty site becomes occupied by a particle of typei with rate i proportion of neighbors of typei the set of neighbors of a sitex is of the form y x y j for a positive integerj and a norm assuming there are only 0 s and 1 s present at the beginning the process reduces to the contact process with the critical rate of survival of 1 s being c the basic problem we address is the existence of equilibria in which both types of particles coexist without loss of generality one can restrict to the case 2 1 c and in this case we show 1 if 2 1 and the initial state is translation invariant and contains infinitely many 2 s then the 1 s go away and the process approaches the invariant measure of the contact process with only 2 s and 0 s present 2 if 2 1 andd 2 then clustering occurs starting from a translation invariant initial measure with no mass on all 0 s the process converges weakly to a convex combination of the two invariant measures obtained with only one type of particles present and 3 if 2 1 andd 3 then there is a one parameter family of invariant measures including both types

an analysis of the fixation probability of a mutant on special classes of non directed graphs there is a growing interest in the study of evolutionary dynamics on populations with some non homogeneous structure in this paper we follow the model of lieberman et al lieberman et al 2005 na

generalized voter models we examine a one dimensional class of interacting particle systems which generalize some voter models this class includes a particular case in the class of models of catalytic surfaces introduced by swindle and grannan we show that this class has the clustering property of ordinary finite range voter models at least when one is concerned with translation invariant measures on the state space

on the stability of a population growth model with sexual reproduction on z d d geq 2 we continue our study on the stability properties of a population growth model with sexual reproduction on z d d 2 in the author s previous work it was proved that in the type iv process the two dimensional symmetric model on z 2 the vacant state o is stable under perturbation of the initial state the first kind of perturbation and it is unstable under perturbation of the birth rate the second kind of perturbation in this paper we prove that in the type iii process on z 2 the vacant state o is stable under the second kind of perturbation and in three or higher dimensional symmetric models the vacant state o is unstable under the first kind of perturbation these results combined with the results obtained earlier provide a fairly complete picture concerning the stability properties of these models

a phase transition for a stochastic pde related to the contact process we consider the one dimensional heat equation with a semilinear term and with a nonlinear white noise term r durrett conjectured that this equation arises as a weak limit of the contact process with longrange interactions we show that our equation possesses a phase transition to be more precise we assume that the initial function is nonnegative with bounded total mass if a certain parameter in the equation is small enough then the solution dies out to 0 in finite time with probability 1 if this parameter is large enough then the solution has a positive probability of never dying out to 0 this result answers a question of durett

self organized critical forest fire model a forest fire model is introduced which contains a lightning probability f this leads to a self organized critical state in the limit f ensuremath rightarrow 0 provided that the time scales of tree growth and burning down of forest clusters are separated we derive scaling laws and calculate all critical exponents the values of the critical exponents are confirmed by computer simulations for a two dimensional system we show that the forest density in the critical state assumes its minimum possible value i e that energy dissipation is maximum

self similar solutions and translating solitons for lagrangian mean curvature flow we construct many self similar and translating solitons for lagrangian mean curvature flow including self expanders and translating solitons with arbitrarily small oscillation on the lagrangian angle our translating solitons play the same role as cigar solitons in ricci flow and are important in studying the regularity of lagrangian mean curvature flow given two transverse lagrangian planes wn in cn with sum of characteristic angles less than a we show there exists a lagrangian self expander asymptotic to this pair of planes the maslov class of these self expanders is zero thus they can serve as local models for surgeries on lagrangian mean curvature flow families of self shrinkers and self expanders with different topologies are also constructed this paper generalizes the work of anciaux i joyce 12 lawlor 15 and lee and wang 17 18

on multidimensional branching random walks in random environment we study branching random walks in random i i d environment in z d d geq 1 for this model the population size cannot decrease and a natural definition of recurrence is introduced we prove a dichotomy for recurrence transience depending only on the support of the environmental law we give sufficient conditions for recurrence and for transience in the recurrent case we study the asymptotics of the tail of the distribution of the hitting times and prove a shape theorem for the set of lattice sites which are visited up to a large time

survival and coexistence in stochastic spatial lotka volterra models a spatially explicit stochastic lotka volterra model was introduced by neuhauser and pacala in neuhauser and pacala ann appl probab 9 1226 1259 1999 a low density limit theorem for this process was proved by the authors in cox and perkins ann probab 33 904 947 2005 showing that certain generalized rescaled lotka volterra models converge to super brownian motion with drift here we use this convergence result to extend what is known about the parameter regions for the lotka volterra process where i survival of one type holds and ii coexistence holds

escape of a uniform random walk from an interval we study the first passage properties of a random walk in the unit interval in which the length of a single step is uniformly distributed over the finite range a a for a of the order of one the exit probabilities to each edge of the interval and the exit time from the interval exhibit anomalous properties stemming from the change in the minimum number of steps to escape the interval as a function of the starting point as a decreases first passage properties approach those of continuum diffusion but non diffusive effects remain because of residual discreteness effects

flux to a trap the flux of particles to a single trap is investigated for two systems 1 particles in 3d space which jump a fixed step lengthl the rayleigh flight and are adsorbed by a spherical surface and 2 particles on a lattice jumping to nearest neighbor sites with a single adsorbing site initially the particles are uniformly distributed outside the traps when the jump length goes to zero both processes go over to regular diffusion and the first case yields the diffusive flux to a sphere as solved by smoluchowski for nonzero step length the flux for large times is given by a modified form of smoluchowski s result with the effective radius replaced byr cl wherec 0 29795219 andcl is the milne extrapolation length for this problem for the second problem a similar expression for the flux is found with the effective trap radius a function of the lattice sc bcc fcc being considered

coexistence results for catalysts in this paper we consider a modification of ziff gulari and barshad s 1986 model of oxidation of carbon monoxide on a catalyst surface in which the reactants are mobile on the catalyst surface we find regions in the parameter space in which poisoning occurs the catalyst surface becomes completely occupied by one type of atom and another in which there is a translation invariant stationary distribution in which the two atoms have positive density the last result is proved by exploiting a connection between the particle system with fast stirring and a limiting system of reaction diffusion equations

self shrinkers for the mean curvature flow in arbitrary codimension in this paper we generalize colding minicozzi s recent results about codimension 1 self shrinkers for the mean curvature flow to higher codimension in particular we prove that the sphere bf s n sqrt 2n is the only complete embedded connected f stable self shrinker in mathbf r n k with mathbf h ne 0 polynomial volume growth flat normal bundle and bounded geometry we also discuss some properties of symplectic self shrinkers proving that any complete symplectic self shrinker in mathbf r 4 with polynomial volume growth and bounded second fundamental form is a plane as a corollary we show that there is no finite time type i singularity for symplectic mean curvature flow which has been proved by chen li using different method we also study lagrangian self shrinkers and prove that for lagrangian mean curvature flow the blow up limit of the singularity may be not f stable

overview of theoretical models for reaction rates there is enormous recent interest in the development of models for rate processes because rates are an almost universal characterization in the physical and biological sciences in this paper we provide an introduction to several of the problems to be discussed in greater depth by other speakers at a symposium held at the national institutes of health on may 6 8 1985 this review will focus on 1 the smoluchowski model for reaction rates together with its extension by onsager 2 first passage time formalism for discrete and continuous master equations and fokker planck equations 3 the kramers model and its extensions 4 diffusion in the presence of trapping centers

numerical analysis of the rebellious voter model the rebellious voter model introduced by sturm and swart 2008 is a variation of the standard one dimensional voter model in which types that are locally in the minority have an advantage it is related both through duality and through the evolution of its interfaces to a system of branching annihilating random walks that is believed to belong to the parity conservation universality class this paper presents numerical data for the rebellious voter model and for a closely related one sided version of the model both models appear to exhibit a phase transition between noncoexistence and coexistence as the advantage for minority types is increased for the one sided model but not for the original two sided rebellious voter model it appears that the critical point is exactly a half and two important functions of the process are given by simple explicit formulas a fact for which we have no explanation

improved upper bounds for the contact process critical value the best known upper bound for the critical value c of the basic one dimensional contact process is 2 most techniques for finding bounds on critical values have the property that they can be modified in order to obtain improved bounds this seemed not to be the case for the approach which yielded c 2 for the basic contact process in this paper we propose a technique for generating better bounds in this context to illustrate its use we carry out the full program in one case with the conclusion that c 1 942

the central limit theorem for supercritical oriented percolation in two dimensions we consider the cardinality of supercritical oriented bond percolation in two dimensions we show that whenever the the origin is conditioned to percolate the process appropriately normalized converges asymptotically in distribution to the standard normal law this resolves a longstanding open problem pointed out to in several instances in the literature the result applies also to the continuous time analog of the process viz the basic one dimensional contact process we also derive general random indices central limit theorems for associated random variables as byproducts of our proof

kriging nonstationary data abstract spatial data modeled to have come from a random function with a nonstationary mean are considered the spatial prediction method known as kriging exploits second order spatial correlation structure to obtain minimum variance predictions of certain average values of the random function but to do so it must be assumed that either the mean function the drift is known up to a constant or the second order structure the variogram is known exactly knowledge of the drift allows the stationary variogram to be estimated and leads to ordinary kriging knowledge of the variogram allows the drift to be estimated and leads to universal kriging more usually neither is known this article shows how median polish of gridded spatial data provides a resistant and relatively bias free way of kriging in the presence of drift yet yields results as good as the mathematically optimal but operationally difficult universal kriging comparisons are performed on two data sets

a spatial statistical analysis of tumor growth abstract growth models for tumors commonly are developed for a one dimensional summary e g number of cells volume at the supracellular level however ignoring tumor shape leads to oversimplification of the growth mechanism for example an oncologist would view the discovery of a compact tumor of volume v differently from the discovery of two osculatory compact tumors each of volume v 2 this article presents a growth model that uses shape information at a previous time to describe the tumor at the present time an analysis is given of three successive two dimensional images of cell islands which are obtained from in vitro growth of human breast cancer cells

contact processes in several dimensions this paper deals primarily with the basic contact processes introduced and studied by harris 12t4 these are random evolutions on the state space s a l l subsets of z a zd the d dimensional integer lattice with extremely simple local dynamics namely if we think of the contact process as representing the spread of an infection t es denoting the set of infected sites at time t then x e it becomes healthy at exponential rate 1 while x z a t is infected at a rate proport ional to the number of sites neighboring x where infection is present the proportionali ty constant 2 is called the infection parameter contact processes are perhaps the simplest s valued markov processes which exhibit a critical phenomenon infection emanating from a single site dies out with probabili ty one for small positive 2 but has positive probability of surviving for all time when is large there is a critical value 2 d where the transition occurs principal objectives of analysis are the precise formulation of the critical phenomenon and detailed description of the ergodic behavior both below and above a c our recent survey articles 10 and i 6 provide introductions to contact processes and to the general field of interacting s valued systems respectively we will make constant use of notation and techniques from the surveys assuming that the reader is familiar with thern the article 10 describes in some detail the current state of knowledge concerning one dimensional contact processes the theory for d 1 is now fairly complete with two major exceptions 1 is not even determined to one decimal place 1 18 2 e there is an

rescaled voter models converge to super brownian motion we show that a sequence of voter models suitably rescaled in space and time converges weakly to super brownian motion the result includes both nearest neighbor and longer range voter models and complements a limit theorem of mueller and tribe in one dimension

rigorous results on mathematical models of catalytic surfaces two types of mathematical models of catalytic surfaces are considered conditions guaranteeing either convergence to traps with all sites occupied by a single reactant poisoning and or coexistence in equilibrium are established