the scaling limit of the interface of the continuous space symbiotic branching model the continuous space symbiotic branching model describes the evolution of two interacting populations that can reproduce locally only in the simultaneous presence of each other if started with complementary heaviside initial conditions the interface where both populations coexist remains compact together with a diffusive scaling property this suggests the presence of an interesting scaling limit indeed in the present paper we show weak convergence of the diffusively rescaled populations as measure valued processes in the skorokhod respectively the meyer zheng topology for suitable parameter ranges the limit can be characterized as the unique solution to a martingale problem and satisfies a separation of types property this provides an important step toward an understanding of the scaling limit for the interface as a corollary we obtain an estimate on the moments of the width of an approximate interface

an algebraic approach to automatic construction of structural models we present algebraic approach to the inductive learning of structural models and automatic construction of shape prototypes for character recognition on the basis of the algebraic description of curve structure proposed by nishida and mori 1991 1992 a class in the structural models is a set of shapes that can be transformed continuously to each other we consider an algebraic representation of continuous transformation of components of the shape and give specific properties satisfied by each component in the class the generalization rules in the inductive learning are specified from the viewpoints of continuous transformation of components and relational structure among the components the learning procedure generalizes a pair of classes into one class incrementally and hierarchically in terms of the generalization rules we show experimental results on handwritten numerals

clustering of linearly interacting diffusions and universality of their long time limit distribution let k d d 1 be a compact convex set and a countable abelian group we study a stochastic process x in k equipped with the product topology where each coordinate solves a sde of the form dx i t j a j i x j t x i t dt x i t db i t here a is the kernel of a continuous time random walk on and is a continuous root of a diffusion matrix w on k if x t converges in distribution to a limit x and the symmetrized random walk with kernel a s i a i a i is recurrent then each component x i is concentrated on x k x 0 and the coordinates agree i e the system clusters both these statements fail if a s is transient under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of k we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components this distribution turns out to be universal in all recurrent kernels a s on abelian groups

interacting fisher wright diffusions in a catalytic medium we study the longtime behaviour of interacting systems in a randomly fluctuating space time medium and focus on models from population genetics there are two prototypes of spatial models in population genetics spatial branching processes and interacting fisher wright diffusions quite a bit is known on spatial branching processes where the local branching rate is proportional to a random environment catalytic medium

uniqueness for isotropic diffusions with a linear drift this paper proves that valued solutions to the sde are unique in distribution when d d is convex and open d c 0 is positive and locally lipschitz on d and zero on d and x d g x r is convex for r sufficiently small the proof for 0 is based on the transformation xt ectxt which removes the drift and a random time change although the set up is rather specialized the result gives uniqueness for some sde s that cannot be treated by any of the conventional techniques

segmentation through variable order surface fitting the solution of the segmentation problem requires a mechanism for partitioning the image array into low level entities based on a model of the underlying image structure a piecewise smooth surface model for image data that possesses surface coherence properties is used to develop an algorithm that simultaneously segments a large class of images into regions of arbitrary shape and approximates image data with bivariate functions so that it is possible to compute a complete noiseless image reconstruction based on the extracted functions and regions surface curvature sign labeling provides an initial coarse image segmentation which is refined by an iterative region growing method based on variable order surface fitting experimental results show the algorithm s performance on six range images and three intensity images

phase transitions for the long time behavior of interacting diffusions let xi t i zd t 0 be the system of interacting diffusions on 0 defined by the following collection of coupled stochastic differential equations dxi t j zd a i j xj t xi t dt bxi t dwi t i z t 0 here a is an irreducible random walk transition kernel on z z b 0 is a diffusion parameter and wi t i zd t 0 is a collection of independent standard brownian motions on r the initial condition is chosen such that xi 0 i zd is a shift invariant and shift ergodic random field on 0 with mean 0 the evolution preserves the mean we show that the long time behaviour of this system is the result of a delicate interplay between a and b in contrast to systems where the diffusion function is subquadratic in particular let i j 12 a i j a j i i j z denote the symmetrised transition kernel we show that a if is recurrent then for any b 0 the system locally dies out b if is transient then there exist b b2 0 such that b1 the system converges to an equilibrium with mean if 0 b b3 has a finite 2 nd moment if and only if 0 b2 the equilibrium is shown to be associated and mixing for all 0 b2 we further conjecture that the system locally dies out at b b for the case where a is symmetric and transient we further show that c there exists a sequence b2 b3 b4 0 such that c1 has a finite m th moment if and only if 0 bm c3 b2 m 1 bm 2 c4 limm m 1 bm c supm 2 m 1 bm mathematisches institut universitat erlangen nurnberg bismarckstrasse 1 1 2 d 91504 erlangen germany greven mi uni erlangen de mathematical institute leiden university p o box 9512 2300 ra leiden the netherlands denholla math leidenuniv nl eurandom p o box 513 5600 mb eindhoven the netherlands

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

algebraic description of curve structure the authors propose a compact and concise method of describing curves in terms of the quasi topological features and the structure of each singular point the quasi topological features are the convexity loop and connectivity the quasi topological structure is analyzed in a hierarchical way and algebraic structure is presented explicitly for each representation level the lower level representations are integrated into the higher level one in a systematic way when a curve has singular points branch points the curve is decomposed into components where each is a simple arc or a simple closed curve by decomposing each singular point the description scheme is applied to character recognition

computing exact aspect graphs of curved objects algebraic surfaces this article presents an algorithm for computing the exact aspect graph of an opaque solid bounded by a smooth algebraic surface orthographic projection is assumed the algorithm is based on a catalog of visual events available from singularity theory it uses curve tracing cell decomposition homotopy continuation and ray tracing to construct the regions of the view sphere delineated by visual event curves the algorithm has been fully implemented and examples are presented

a trotter type approach to infinite rate mutually catalytic branching dawson and perkins ann probab 26 1988 1088 1138 constructed a stochastic model of an interacting two type population indexed by a countable site space which locally undergoes a mutually catalytic branching mechanism in klenke and mytnik preprint 2008 arxiv 0901 0623 it is shown that as the branching rate approaches infinity the process converges to a process that is called the infinite rate mutually catalytic branching process imub it is most conveniently characterized as the solution of a certain martingale problem while in the latter reference a noise equation approach is used in order to construct a solution to this martingale problem the aim of this paper is to provide a trotter type construction the construction presented here will be used in a forthcoming paper klenke and mytnik preprint 2009 to investigate the long time behavior of imub coexistence versus segregation of types this paper is partly based on the ph d thesis of the second author 2008 where the trotter approach was first introduced

curve description based on directional features and quasi convexity concavity abstract qualitative and global features are appropriate for describing the shape of a complex and deformed object rather than quantitative and local features in particular quantized directional features and quasi convexity concavity are powerful and flexible for describing the shape of handwritten characters in this paper we present a method for structural analysis and description of simple open arcs or closed curves based on 2 m directional features m 1 2 3 and quasi convexity concavity we show some examples of shape description of handwritten characters and experimental results for selecting the optimal quantization of direction in handwritten character recognition

recurrence and ergodicity of interacting particle systems an important technical result proposition 2 3 and its proof present in the original submission was erroneously omitted when this paper was published in 2000 the missing text which should have appeared on page 243 directly before section 3 is included in this erratum together with a short account of its context simultaneously with the publication of this erratum the electronic version of the paper in vol 116 no 2 2000 will be completed by insertion of the missing text readers of probability theory and related fields who have access to the electronic version of this erratum will also have access via a url to the full intact paper

principal component analysis with missing data and its application to polyhedral object modeling observation based object modeling often requires integration of shape descriptions from different views to overcome the problems of errors and their accumulation we have developed a weighted least squares wls approach which simultaneously recovers object shape and transformation among different views without recovering interframe motion we show that object modeling from a range image sequence is a problem of principal component analysis with missing data pcamd which can be generalized as a wls minimization problem an efficient algorithm is devised after we have segmented planar surface regions in each view and tracked them over the image sequence we construct a normal measurement matrix of surface normals and a distance measurement matrix of normal distances to the origin for all visible regions over the whole sequence of views respectively these two matrices which have many missing elements due to noise occlusion and mismatching enable us to formulate multiple view merging as a combination of two wls problems a two step algorithm is presented after surface equations are extracted spatial connectivity among the surfaces is established to enable the polyhedral object model to be constructed experiments using synthetic data and real range images show that our approach is robust against noise and mismatching and generates accurate polyhedral object models

a cyclically catalytic super brownian motion in generalization of the mutually catalytic super brownian motion in r of dawson and perkins and mytnik a function valued cyclically catalytic model x is constructed as a strong markov solution to a martingale problem starting with a finite population x 0 each pair of neighboring types will globally segregate in the long term limit noncoexistence of neighboring types also finer extinction survival properties depending on x 0 are studied in the spirit of mueller and perkins in fact x 0 can be chosen in such a way that all types survive for all finite times on the other hand sufficient conditions on x 0 are stated for the following situation given a type k and a positive time t the kth subpopulation x k dies by time t with a large probability provided that its initial value x k 0 was sufficiently small

the scale space aspect graph currently the aspect graph is computed from the theoretical standpoint of perfect resolution in object shape the viewpoint and the projected image this means that the aspect graph may include details that an observer could never see in practice introducing the notion of scale into the aspect graph framework provides a mechanism for selecting a level of detail that is large enough to merit explicit representation this effectively allows control over the number of nodes retained in the aspect graph this paper introduces the concept of the scale space aspect graph defines three different interpretations of the scale dimension and presents a detailed example for a simple class of objects with scale defined in terms of the spatial extent of features in the image

on recognizing and positioning curved 3 d objects from image contours an approach for explicitly relating the shape of image contours to models of curved three dimensional objects is presented this relationship is used for object recognition and positioning object models consist of collections of parametric surface patches and their intersection curves this includes nearly all representations used in computer aided geometric design and computer vision the image contours considered are the projections of surface discontinuities and occluding contours elimination theory provides a method for constructing the implicit equation of these contours for an object observed under orthographic or perspective projection this equation is parameterized by the object s position and orientation with respect to the observer determining these parameters is reduced to a fitting problem between the theoretical contour and the observed data points the proposed approach readily extends to parameterized models it has been implemented for a simple world composed of various surfaces of revolution and tested on several real images

a structural approach to representation of curved objects abstract representation of complex curved objects is still a challenging and difficult problem in this paper we present a scheme for multi level structural qualitative representation of object surfaces the analysis begins with local face based descriptions of object surfaces surface primitive graphs and global high level structural and qualitative features are derived from these local descriptions the final descriptions global feature graphs are based on the quasi convexity concavity structures incorporating quantized directional features of closed boundaries of cross sections along several directions the global feature graphs have the following properties 1 they provide a characterization of global structural qualitative properties of the object 2 the graphs are composed of a few components nodes and edges with rich features and give compact descriptions of the objects 3 the global feature graph representations are robust and qualitative in that they are not affected by geometrical quantitative properties of the shape such as curvatures length and position 4 the representation has clear and rigorous properties furthermore we present some theoretical basis for object recognition using the global features graphs

determining modes for continuous data assimilation in 2d turbulence we study the number of determining modes necessary for continuous data assimilation in the two dimensional incompressible navier stokes equations our focus is on how the spatial structure of the body forcing affects the rate of continuous data assimilation and the number of determining modes we treat this problem analytically by proving a convergence result depending on the h 1 norm of f and computationally by considering a family of forcing functions with identical grashof numbers that are supported on different annuli in fourier space the rate of continuous data assimilation and the number of determining modes is shown to depend strongly on the length scales present in the forcing

infinite rate mutually catalytic branching in infinitely many colonies the longtime behavior consider the infinite rate mutually catalytic branching process imub constructed in infinite rate mutually catalytic branching in infinitely many colonies construction characterization and convergence 2008 preprint and ann probab 38 2010 479 497 for finite initial conditions we show that only one type survives in the long run if the interaction kernel is recurrent on the other hand under a slightly stronger condition than transience we show that both types can coexist

an example of absence of turbulence for any reynolds number ii we study a viscous incompressible fluid moving in a two dimensional flat torus 0 l 0 2 l 2 we show a set of external forces for which the stationary state is attractive for any reynolds numberr moreover the size of this set and the basin of attraction are independent ofr

a class of numerical algorithms for large time integration the nonlinear galerkin methods the nonlinear galerkin methods are numerical schemes well adapted to the long term integration of evolution partial differential equations they consist of looking for approximate solutions lying in nonlinear manifolds that are close in some sense to the attractor the aim of this paper is to extend these schemes to a large class of manifolds convergence results are derived for the schemes introduced

determination of the solutions of the navier stokes equations by a set of nodal values we consider the navier stokes equations of a viscous incompresible fluid and we want to see to what extent these solutions can be determined by a discrete set of nodal values of these solutions the results presented here are exact results and not approximate ones we show in several cases that the solutions are entirely determined by their values on a discrete set provided this set contains enough points and these points are sufficiently densely distributed in a sense described in the article two typical results are the following ones two stationary solutions coincide if they coincide on a set sufficiently dense but finite similarly if the large time behavior of the solutions to the navier stokes equations is known on an appropriate discrete set then the large time behavior of the solution itself is totally determined

on the rate of convergence of the nonlinear galerkin methods in this paper we provide estimates to the rate of convergence of the nonlinear galerkin approximation method in particular and by means of an illustrative example we show that the nonlinear galerkin method converges faster than the usual galerkin method

long time behavior and coexistence in a mutually catalytic branching model we study a system of two interacting populations which undergo random migration and mutually catalytic branching the branching rate of one population at a site is proportional to the mass of the other population at the site the system is modelled by an infinite system of stochastic differential equations allowing symmetric markov migration if the set of sites is discrete z d or by a stochastic partial differential equation with brownian migration if the set of sites is the real line a duality technique of leonid mytnik which gives uniqueness in law is used to examine the long time behavior of the solutions for example with uniform initial conditions the process converges to an equilibrium distribution as t and there is coexistence of types in the equilibrium iff the random migration is transient

a spherical representation for recognition of free form surfaces introduces a new surface representation for recognizing curved objects the authors approach begins by representing an object by a discrete mesh of points built from range data or from a geometric model of the object the mesh is computed from the data by deforming a standard shaped mesh for example an ellipsoid until it fits the surface of the object the authors define local regularity constraints that the mesh must satisfy the authors then define a canonical mapping between the mesh describing the object and a standard spherical mesh a surface curvature index that is pose invariant is stored at every node of the mesh the authors use this object representation for recognition by comparing the spherical model of a reference object with the model extracted from a new observed scene the authors show how the similarity between reference model and observed data can be evaluated and they show how the pose of the reference object in the observed scene can be easily computed using this representation the authors present results on real range images which show that this approach to modelling and recognizing 3d objects has three main advantages 1 it is applicable to complex curved surfaces that cannot be handled by conventional techniques 2 it reduces the recognition problem to the computation of similarity between spherical distributions in particular the recognition algorithm does not require any combinatorial search and 3 even though it is based on a spherical mapping the approach can handle occlusions and partial views

on a criterion for locating stable stationary solutions to the navier stokes equations on donne une estimation superieure uniforme explicite a la borne de constantin foias ternam et on donne une forme applicable explicite aux conditions suffisantes on considere seulement le cas a 2 dimensions

infinite rate mutually catalytic branching in infinitely many colonies construction characterization and convergence we construct a mutually catalytic branching process on a countable site space with infinite branching rate the finite rate mutually catalytic model in which the rate of branching of one population at a site is proportional to the mass of the other population at that site was introduced by dawson and perkins ann probab 26 3 1088 1138 1998 we show that our model is the limit for a class of models and in particular for the dawson perkins model as the rate of branching goes to infinity our process is characterized as the unique solution to a martingale problem we also give a characterization of the process as a weak solution of an infinite system of stochastic integral equations driven by a poisson noise

resolvent estimates for fleming viot operators and uniqueness of solutions to related martingale problems we define a two parameter scale of banach spaces contained in the contininuous functions on the space of probability measures on a compact metric space x and show that the resolvent of the fleming viot process is a bounded operator in the scale we use this result to prove uniqueness of solutions to the martingale problem for fleming viot operators whose mutation operator a a and sampling rate x y may depend on the measure variable m1 x provided they are sufficiently close to constant in a sense made precise by the norms defining the scale we prove an existence result with no such restriction on the coefficients

computing exact aspect graphs of curved objects solid of revolution this paper introduces a new approach to computing the exact orthographic aspect graph of curved objects curves corresponding to various visual events partition the view sphere into regions where the image structure is stable a catalog of these events for piecewise smooth objects is available from singularity theory for a solid of revolution whose generator is an algebraic curve each visual event is characterized by a system of polynomial equations whose roots can be computed by continuation methods within each region the stable image structure is characterized by a variation of cylindrical algebraic decomposition and ray tracing this approach has been implemented and several examples are presented

a fleming viot particle representation of the dirichlet laplacian we consider a model with a large number n of particles which move according to independent brownian motions a particle which leaves a domain d is killed at the same time a different particle splits into two particles for large n the particle distribution density converges to the normalized heat equation solution in d with dirichlet boundary conditions the stationary distributions converge as n to the first eigenfunction of the laplacian in d with the same boundary conditions

perception of 3 d surfaces from 2 d contours inference of 3 d shape from 2 d contours in a single image is an important problem in machine vision the authors survey classes of techniques proposed in the past and provide a critical analysis they show that two kinds of symmetries in figures which are known as parallel and skew symmetries give significant information about surface shape for a variety of objects they derive the constraints imposed by these symmetries and show how to use them to infer 3 d shape they also discuss the zero gaussian curvature zgc surfaces in depth and show results on the recovery of surface orientation for various zgc surfaces

asymptotic analysis of the navier stokes equations abstract new bounds are established on the number of modes which determine the solutions of the navier stokes equations in two dimensions the best bound available at present is nearly proportional to the generalized grashof number defined in the paper and less than logarithmically dependent on the spatial structure or the shape of the force driving the flow to the extent than for the case of 2 dimensional rayleigh benard convection the generalized grashof number may be identified with the usual grashof number the resulting bound on the number of modes is found to differ only slightly from a bound obtained earlier on heuristic grounds

on the dimension of the attractors in two dimensional turbulence abstract using a new version of the sobolev lieb thirring inequality we derive an upper bound for the dimension of the universal attractor for two dimensional space periodic navier stokes equations this estimate is optimal up to a logarithmic correction the relevance of this estimate to turbulence and related results are also briefly discussed

a space time property of a class of measure valued branching diffusions si d on montre que la diffusion ramifiee d dimensionnelle d indice etudiee par dawson et d autres distribue sa masse sur un support aleatoire de facon uniforme relativement a la mesure de hausdorff