a new approach to computing the euler characteristic in this paper we describe a new approach to computing the euler characteristic of a three dimensional digital image our approach is based on computing the change in numbers of black components tunnels and cavities in 3 x 3 x 3 neighborhood of an object black point due to its deletion the existing algorithms to computing the euler characteristic of a 3d digital image are based on counting the numbers of all k dimensional elements 0 k 3 in a polyhedral representation of the image our approach can be modified for 6 26 18 6 6 18 and other connectivity relations of grid points a parallel implementation of the algorithm is described using the concept of sub fields

asymptotically one dimensional diffusions on the sierpinski gasket and the abc gaskets diffusion processes on the sierpinski gasket and theabc gaskets are constructed as limits of random walks in terms of the associated renormalization group the present method uses the inverse trajectories which converge to unstable fixed points corresponding to the random walks on one dimensional chains in particular non degenerate fixed points are unnecessary for the construction a limit theorem related to the discrete time multi type non stationary branching processes is applied

on maximum bifurcation delay in real planar singularly perturbed vector fields abstract in singularly perturbed vector fields where the unperturbed vector field has a curve of singularities a critical curve orbits tend to be attracted towards or repelled away from this curve depending on the sign of the divergence of the vector field at the curve when at some point this sign bifurcates from negative to positive orbits will typically be repelled away immediately after passing the bifurcation point turning point atypical behaviour is nevertheless observed as well when orbits follow the critical curve for some distance after the turning point before they repel away from it a delay in the bifurcation is present interesting are systems that have a maximum bifurcation delay i e there is a point on the critical curve beyond which orbits cannot stay close to the critical curve this behaviour is known to appear in some systems in dimension 3 see e benoit ed dynamic bifurcations in lecture notes in mathematics vol 1493 springer verlag berlin 1991 and it is commonly believed that it is not an issue in real planar systems beside making the observation that it does appear in non analytic planar systems it is shown that whenever bifurcation delay appears it has no non trivial maximum for analytic planar vector fields the proof is based on the notion of family blow up at the turning point on formal power series in terms of blow up variables the study of their gevrey properties and analytic continuation of their borel transform these results complement existing results concerning the equivalence of local and global canard solutions in a fruchard r schafke overstability and resonance ann inst fourier grenoble 53 1 2003 227 264

estimates of transition densities for brownian motion on nested fractals we obtain upper and lower bounds for the transition densities of brownian motion on nested fractals compared with the estimate on the sierpinski gasket the results require the introduction of a new exponent dj related to the shortest path metric and chemical exponent on nested fractals further holder order of the resolvent densities sample paths and local times are obtained the results are obtained using the theory of multi type branching processes

tauberian properties for monomial summability with applications to pfaffian systems abstract in this paper we will show that monomial summability processes with respect to different monomials are not compatible except in the trivial case of a convergent series we will apply this fact to the study of solutions of pfaffian systems with normal crossings focusing in the implications of the complete integrability condition on these systems

summability of canard heteroclinic saddle connections abstract for a given real analytic slow fast system x e f x y e y g x y e that admits a slow fast saddle and that satisfies some mild assumptions the borel summability properties of the saddle separatrix tangent in the direction of the critical curve are investigated 1 summability is shown it is also shown that slow fast saddle connections of canard type have summability properties in contrast to the typical lack of borel summability for canard solutions of general equations

canard solutions at non generic turning points this paper deals with singular perturbation problems for vector fields on 2 dimensional manifolds canard solutions are solutions that starting near an attracting normally hyperbolic branch of the singular curve cross a turning point and follow for a while a normally repelling branch of the singular curve following the geometric ideas developed by dumortier and roussarie in 1996 for the study of canard solutions near a generic turning point we study canard solutions near non generic turning points characterization of manifolds of canard solutions is given in terms of boundary conditions their regularity properties are studied and the relation is described with the more traditional asymptotic approach it reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve we also study their contact and its relation to the control curve

detection of 3 d simple points for topology preserving transformations with application to thinning the problems of 3 d digital topology preservation under binary transformations and 3 d object thinning are considered in this correspondence first the authors establish the conditions under which transformation of an object voxel to a non object voxel or its inverse does not affect the image topology an efficient algorithm to detect a simple point has been proposed on the basis of those conditions in this connection some other interesting properties of 3 d digital geometry are also discussed using these properties and the simple point detection algorithm the authors have proposed an algorithm to generate a surface skeleton so that the topology of the original image is preserved the shape of the image is maintained as much as possible and the results are less affected by noise

coupling and harnack inequalities for sierpinski carpets uniform harnack inequalities for harmonic functions on the preand graphical sierpinski carpets are proved using a probabilistic coupling argument various results follow from this including the construction of brownian motion on sierpinski carpets embedded in rd d 3 estimates on the fundamental solution of the heat equation and sobolev and poincare inequalities the sierpinski carpets scs we will study are generalizations of the cantor set let f0 0 1 be the unit cube in r d 2 centered at z0 1 2 1 2 let k a be integers with 1 a k and a k even divide f0 into k equal subcubes remove a central block of a subcubes and let f1 be what remains thus f1 f0 k a 2k k a 2k now repeat this operation on each of the k a remaining subcubes to obtain f2 iterating we obtain a decreasing sequence of closed sets fn then f n 0 fn is a sierpinski carpet and has hausdorff dimension df df f log k a log k when d 2 k 3 and a 1 we get the usual sierpinski carpet let fn kfn 0 and define the pre sierpinski carpet by f n 1 fn see 10 the graphical sierpinski carpet is the graph g v e with vertex set v z0 z f and edge set e x y v x y 1 thus int f is a domain in r with a large scale structure which mimics the small scale structure of f we are interested in the behavior of solutions of the laplace and heat equations on f f and g one reason for this is applications to transport phenomena in disordered media see 6 another is the new type of behavior of the heat kernel on these spaces let w be brownian motion on f with normal reflection on f and let q t x y be the transition density of w so that q solves the heat equation on f with neumann boundary conditions on f 1991 mathematics subject classification primary 60b99 secondary 60j35

time analysis and entry exit relation near planar turning points abstract the paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions we follow a geometric approach based on the use of c k normal forms centre manifolds and family blow up as we did in trans amer math soc to appear in trans amer math soc to appear we considered the existence of manifolds of canard solutions for given appropriate boundary conditions these manifolds need not be smooth at the turning point in this paper we essentially study the transition time along such manifolds as well as the divergence integral providing a structure theorem for these integrals as a consequence we get a nice structure theorem for the transition equation governing the canard solutions it permits to compare different control manifolds and to obtain a precise description of the entry exit relation of different canard solutions attention is also given to the special case in which the canard manifolds are smooth i e when formal canard solutions exist

topology preservation in 3d digital space abstract in a 3d two tone digital space the conditions under which sequential transformation of an object point to a non object point or its inverse does not affect the image topology are investigated the concept of border connectivity is introduced and an efficient algorithm for checking a simple point is proposed in this connection some other interesting properties of 3d digital geometry are also discussed this investigation may be used in thinning and topology preserving morphological transformations in 3d digital space

brownian motion on the sierpinski gasket we construct a brownian motion taking values in the sierpinski gasket a fractal subset of 2 and study its properties this is a diffusion process characterized by local isotropy and homogeneity properties we show for example that the process has a continuous symmetric transition density p t x y with respect to an appropriate hausdorff measure and obtain estimates on p t x y

borel summability of divergent solutions for singular first order partial differential equations with variable coefficients part i abstract under a certain restriction singular first order linear partial differential equations of nilpotent type with two variables are divided into two classes in the previous paper part i we dealt with the one class and comprehended that there was a close affinity between the borel summability of divergent solutions and global analytic continuation properties for coefficients in this part ii we give a similar consideration on the other class more precise global estimates than those given in part i for coefficients will be required to prove the borel summability of divergent solutions

digital topology and geometry in medical imaging a survey digital topology and geometry refers to the use of topologic and geometric properties and features for images defined in digital grids such methods have been widely used in many medical imaging applications including image segmentation visualization manipulation interpolation registration surface tracking object representation correction quantitative morphometry etc digital topology and geometry play important roles in medical imaging research by enriching the scope of target outcomes and by adding strong theoretical foundations with enhanced stability fidelity and efficiency this paper presents a comprehensive yet compact survey on results principles and insights of methods related to digital topology and geometry with strong emphasis on understanding their roles in various medical imaging applications specifically this paper reviews methods related to distance analysis and path propagation connectivity surface tracking image segmentation boundary and centerline detection topology preservation and local topological properties skeletonization and object representation correction and quantitative morphometry a common thread among the topics reviewed in this paper is that their theory and algorithms use the principle of digital path connectivity path propagation and neighborhood analysis

recent progress in the theory of formal solutions for ode and pde in this article we describe the theory of multisummability of formal power series and discuss its application to power series solutions of ordinary and partial differential equations in particular we give some new application to a singular perturbation of a regular singular point

digital fundamental group and euler characteristic of a connected sum of digital closed surfaces unlike the connected sum in classical topology its digital version is shown to have some intrinsic feature in this paper we study both the digital fundamental group and the euler characteristic of a connected sum of digital closed k i surfaces i 0 1

an extension of borel laplace methods and monomial summability abstract in this paper we will show that monomial summability can be characterized using borel laplace like integral transformations depending of a parameter 0 s 1 we will apply this result to prove 1 summability in a monomial of formal solutions of a family of partial differential equations

asymptotic behaviour of solutions near a turning point the example of the brusselator equation abstract the brusselator equation is an example of a singularly perturbed differential equation with an additional parameter it has two turning points at x 0 and x 1 we study some properties of so called canard solutions that remain bounded in a full neighbourhood of 0 and in the largest possible domain the main goal is the complete asymptotic expansion of the difference between two values of the additional parameter corresponding to such solutions for this purpose we need a study of behaviour of the solutions near a turning point here we prove that for a large class of equations if 0 is a turning point of order p any solution y not exponentially large has in some sector centred at 0 an asymptotic behaviour when e 0 of the form y n x e e n where e p 1 e for x e x with x large enough but independent of e in the brusselator case we moreover compute a stokes constant for a particular nonlinear differential equation

harmonic calculus on p c f self similar sets the main object of this paper is the laplace operator on a class of fractals first we establish the concept of the renormalization of difference operators on post critically finite p c f for short self similar sets which are large enough to include finitely ramified self similar sets and extend the results for sierpinski gasket given in 10 to this class under each invariant operator for renormalization the laplace operator green function dirichlet form and neumann derivatives are explicitly constructed as the natural limits of those on finite pre self similar sets which approximate the p c f self similar sets also harmonic functions are shown to be finite dimensional and they are characterized by the solution of an infinite system of finite difference equations

the gevrey asymptotics in the case of singular perturbations abstract in this paper we present results in the gevrey asymptotics which correspond to some existing results concerning asymptotic solutions in the poincare asymptotics of singularly perturbed ordinary differential equations the main idea is based on a characterization of the gevrey flat functions and a characterizaton of the gevrey asymptotic expansions as it had been already exhibited by y sibuya 1990 lecture notes in pure and applied math vol 124 pp 393 401 dekker new york and p f hsieh and y sibuya 1999 basic theory of ordinary differential equations chap xi sect xi 2 and chap xii sect xii 5 springer verlag berlin

gevrey properties of the asymptotic critical wave speed in a family of scalar reaction diffusion equations abstract we consider front propagation in a family of scalar reaction diffusion equations in the asymptotic limit where the polynomial degree of the potential function tends to infinity we investigate the gevrey properties of the corresponding critical propagation speed proving that the formal series expansion for that speed is gevrey 1 with respect to the inverse of the degree moreover we discuss the question of optimal truncation finally we present a reliable numerical algorithm for evaluating the coefficients in the expansion with arbitrary precision and to any desired order and we illustrate that algorithm by calculating explicitly the first ten coefficients our analysis builds on results obtained previously in f dumortier n popovic t j kaper the asymptotic critical wave speed in a family of scalar reaction diffusion equations j math anal appl 326 2 2007 1007 1023 and makes use of the blow up technique in combination with geometric singular perturbation theory and complex analysis while the numerical evaluation of the coefficients in the expansion for the critical speed is based on rigorous interval arithmetic

gevrey normal forms of vector fields with one zero eigenvalue abstract we study normal forms of isolated singularities of vector fields in r n or c n when all eigenvalues of the linear part of the vector field are nonzero one can eliminate all so called nonresonant terms from the equation provided some spectral condition like siegel is satisfied in this paper we discuss the case where there is one zero eigenvalue in that case siegel s condition is not satisfied and show that the formal normalizing transformations are either convergent or divergent of at most gevrey type in some cases we show the summability of the normalizing transformations which leads to the existence of analytic normal forms in complex sectors around the singularity

monomial summability and doubly singular differential equations abstract in this work we consider systems of differential equations that are doubly singular i e that are both singularly perturbed and exhibit an irregular singular point if the irregular singular point is at the origin they have the form e x r 1 d y d x f x e y f 0 0 0 0 with f analytic in some neighborhood of 0 0 0 if the jacobian d f d y 0 0 0 is invertible we show that the unique bivariate formal solution is monomially summable i e summable with respect to the monomial t e x r in a new sense that will be defined as a preparation poincare asymptotics and gevrey asymptotics in a monomial are studied

gevrey properties of real planar singularly perturbed systems abstract by applying geometric techniques to real analytic singularly perturbed vector fields on the plane we develop a way to give a bound on the gevrey type of the taylor development of canard manifolds at degenerate planar turning points by blowing up the phase space at the turning point we find asymptotic estimates even when such expansions w r t traditional phase space variables do not exist the asymptotic estimates are then used to give a sufficient and necessary condition on the existence of local canard solutions

dirichlet forms on fractals poincare constant and resistance we study dirichlet forms associated with random walks on fractal like finite grahs we consider related poincare constants and resistance and study their asymptotic behaviour we construct a markov semi group on fractals as a subsequence of random walks and study its properties finally we construct self similar diffusion processes on fractals which have a certain recurrence property and plenty of symmetries

a new shape preserving parallel thinning algorithm for 3d digital images this paper is concerned with a new parallel thinning algorithm for three dimensional digital images that preserves the topology and maintains their shape we introduce an approach of selecting shape points and outer layer used for erosion during each iteration the approach produces good skeleton for different types of corners the concept of using two image versions in thinning is introduced and its necessity in parallel thinning is justified the robustness of the algorithm under pseudo random noise as well as rotation with respect to shape properties is studied and the results are found to be satisfactory

multisummability of formal solutions of singular perturbation problems abstract we investigate the summability of the unique formal power series solution of a singular perturbation problem for certain linear systems of ordinary meromorphic differential equations using a new type of summability of power series in several here two variables