the true self repelling motion we construct and study a continuous real valued random process which is of a new type it is self interacting self repelling but only in a local sense it only feels the self repellance due to its occu pation time measure density in the immediate neighbourhood of the point it is just visiting we focus on the most natural process with these properties that we call true self repelling motion this is the continuous counterpart to the integer valued true self avoiding walk which had been studied among others by the first author one of the striking properties of true self repelling motion is that although the coupleoxt occupation time measure of x at time tu is a continuous markov process x is not driven by a stochastic diaerential equation and is not a semi martingale it turns out for instance that it has a finite variation of order 3 2 which contrasts with the finite quadratic variation of semi martingales one of the key tools in the construction of x is a continuous system of coalescing brownian motions similar to those that have been constructed by arratia a1 a2 we derive various properties of x existence and properties of the occupation time densities ltoxu local variation etc and an identity that shows that the dynamics of x can be very loosely speaking described as follows dxt is equal to the gradient in space of ltoxu in a gener alized sense even though x7 ltoxu is not diaerentiable

sufficient second order optimality conditions for semilinear control problems with pointwise state constraints second order sufficient optimality conditions are established for the optimal control of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state in contrast to former publications on this subject the cone of critical directions is the smallest possible in the sense that the second order sufficient conditions are the closest to the associated necessary ones the theory is developed for elliptic distributed controls in domains up to dimension three moreover problems of elliptic boundary control and parabolic distributed control are discussed in spatial domains of dimension two and one respectively

cortical thickness analysis in autism with heat kernel smoothing we present a novel data smoothing and analysis framework for cortical thickness data defined on the brain cortical manifold gaussian kernel smoothing which weights neighboring observations according to their 3d euclidean distance has been widely used in 3d brain images to increase the signal to noise ratio when the observations lie on a convoluted brain surface however it is more natural to assign the weights based on the geodesic distance along the surface we therefore develop a framework for geodesic distance based kernel smoothing and statistical analysis on the cortical manifolds as an illustration we apply our methods in detecting the regions of abnormal cortical thickness in 16 high functioning autistic children via random field based multiple comparison correction that utilizes the new smoothing technique

second order analysis for optimal control problems improving results expected from abstract theory an abstract optimization problem of minimizing a functional on a convex subset of a banach space is considered we discuss natural assumptions on the functional that permit establishing sufficient second order optimality conditions with minimal gap with respect to the associated necessary ones though the two norm discrepancy is taken into account the obtained results exhibit the same formulation as the classical ones known from finite dimensional optimization we demonstrate that these assumptions are fulfilled in particular by important optimal control problems for partial differential equations we prove that in contrast to a widespread common belief the standard second order conditions formulated for these control problems imply strict local optimality of the controls not only in the sense of l infty but also of l 2

the λ coalescent speed of coming down from infinity consider a coalescent that comes down from infinity meani ng that it starts from a configuration containing infinitely many blocks at time 0 yet it has a finite number nt of blocks at any positive time t 0 we exhibit a deterministic function v 0 0 such that nt v t 1 almost surely and in l p for any p 1 as t 0 our approach relies on a novel martingale technique

difference formulas for the surface laplacian on a triangulated surface abstract different approximating expressions for the surface laplacian operator on a triangulated surface are derived they are evaluated on a triangulated spherical surface for which the analytical expression of the surface laplacian is known it is shown that in order to obtain accurate results due care has to be taken of irregularities present in the triangulation grid if this is done the approximation will equal the performance of an expression based on least squares which can be derived next the different approximations obtained are used as a regularization operator in the solution of an ill posed inverse problem in electrical volume conduction it is shown that in this application a crude approximation to the surface laplacian suffices

stochastic flows associated to coalescent processes we study a class of stochastic flows connected to the coalescent processes that have been studied recently by mohle pitman sagitov and schweinsberg in connection with asymptotic models for the genealogy of populations with a large fixed size we define a bridge to be a right continuous process b r r 0 1 with nondecreasing paths and exchangeable increments such that b 0 0 and b 1 1 we show that flows of bridges are in one to one correspondence with the so called exchangeable coalescents this yields an infinite dimensional version of the classical kingman representation for exchangeable partitions of we then propose a poissonian construction of a general class of flows of bridges and identify the associated coalescents we also discuss an important auxiliary measure valued process which is closely related to the genealogical structure coded by the coalescent and can be viewed as a generalized fleming viot process

universal tree structures in directed polymers and models of evolving populations by measuring or calculating coalescence times for several models of coalescence or evolution with and without selection we show that the ratios of these coalescence times become universal in the large size limit and we identify a few universality classes

hitting probability of a distant point for the voter model started with a single 1 the goal of this work is to find the asymptotics of the hitting probability of a distant point for the voter model on the integer lattice started from a single 1 at the origin in dimensions d 2 or 3 we obtain the precise asymptotic behavior of this probability we use the scaling limit of the voter model started from a single 1 at the origin in terms of super brownian motion under its excursion measure this invariance principle was stated by bramson cox and le gall as a consequence of a theorem of cox durrett and perkins less precise estimates are derived in dimension d 4

beta coalescents and continuous stable random trees coalescents with multiple collisions also known as a coalescents were introduced by pitman and sagitov in 1999 these processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block in the case that the measure a is the beta 2 a a distribution they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring here we use a recent result of birkner et al to prove that beta coalescents can be embedded in continuous stable random trees about which much is known due to the recent progress of duquesne and le gall our proof is based on a construction of the donnelly kurtz lookdown process using continuous random trees which is of independent interest this produces a number of results concerning the small time behavior of beta coalescents most notably we recover an almost sure limit theorem of the present authors for the number of blocks at small times and give the multifractal spectrum corresponding to the emergence of blocks with atypical size also we are able to find exact asymptotics for sampling formulae corresponding to the site frequency spectrum and the allele frequency spectrum associated with mutations in the context of population genetics

second order and stability analysis for optimal sparse control of the fitzhugh nagumo equation optimal sparse control problems are considered for the fitzhugh nagumo system including the so called schlogl model the nondifferentiable objective functional of tracking type includes a quadratic tikhonov regularization term and the l 1 norm of the control that accounts for the sparsity though the objective functional is not differentiable a theory of second order sufficient optimality conditions is established for tikhonov regularization parameter nu 0 and also for the case nu 0 in this context also local minima are discussed that are strong in the sense of the calculus of variations the second order conditions are used as the main assumption for proving the stability of locally optimal solutions with respect to nu to 0 and with respect to perturbations of the desired state functions the theory is confirmed by numerical examples that are resolved with high precision to confirm that the optimal solution obeys the system of necessary optimality conditions

what is not in the domain of the laplacian on sierpinski gasket type fractals abstract we consider the analog of the laplacian on the sierpinski gasket and related fractals constructed by kigami a function f is said to belong to the domain of if f is continuous and f is defined as a continuous function we show that if f is a nonconstant function in the domain of then f2 is not in the domain of we give two proofs of this fact the first is based on the analog of the pointwise identity f2 2f f f 2 where we show that f 2 does not exist as a continuous function in fact the correct interpretation of f2 is as a singular measure a result due to kusuoka we give a new proof of this fact the second is based on a dichotomy for the local behavior of a function in the domain of at a junction point x0 of the fractal in the typical case nonvanishing of the normal derivative we have upper and lower bounds for f x f x0 in terms of d x x0 for a certain value and in the nontypical case vanishing normal derivative we have an upper bound with an exponent greater than 2 this method allows us to show that general nonlinear functions do not operate on the domain of

a general framework for low level vision we introduce a new geometrical framework based on which natural flows for image scale space and enhancement are presented we consider intensity images as surfaces in the x i space the image is thereby a two dimensional 2 d surface in three dimensional 3 d space for gray level images and 2 d surfaces in five dimensions for color images the new formulation unifies many classical schemes and algorithms via a simple scaling of the intensity contrast and results in new and efficient schemes extensions to multidimensional signals become natural and lead to powerful denoising and scale space algorithms

optimal control of pdes in a complex space setting application to the schrodinger equation in this paper we discuss optimality conditions for abstract optimization problems over complex spaces we then apply these results to optimal control problems with a semigroup structure as an application we detail the case when the state equation is the schrodinger one with pointwise constraints on the bilinear control we derive first and second order optimality conditions and address in particular the case that the control enters the state equation and cost function linearly

optimal control problems with partially polyhedric constraints this paper discusses a class of state constrained optimal control problems for which it is possible to formulate second order necessary or sufficient conditions for local optimality or quadratic growth that do not involve all curvature terms for the constraints this kind of result is classical in the case of polyhedric control constraints our theory of optimization problems with partially polyhedric constraints allows to extend these results to the case when the control constraints are polyhedric in the presence of state constraints satisfying some specific hypotheses the analysis is based on the assumption that some strict semilinearized qualification condition is satisfied we apply the theory to some optimal control problems of elliptic equations with state and control constraints

the parisi formula in this chapter we prove the parisi formula which gives the limiting value of the free energy per site for the sherrington kirkpatrick model at each temperature starting with the famous result of guerra that the parisi formula provided a bound for this free energy we make full use of poisson dirichlet cascades we then use the same techniques to prove in a strong sense that in presence of an external field the overlaps are positive

geometric atlas modeling the cortex as an organized surface recent atlases of the cortical surface are based on a modelization of the cerebral cortex as a topological sphere this captures effectively its organization as a regular bidimensional sheet of layers parallel to the surface and with perpendicular cortical columns yet while in the vertical direction cortices are almost the same throughout phylia in the sense of its surface the cerebral cortex is one of the most variable and distinctive parts of the nervous system indeed gyri and sulci appear to have a crucial organizing role in an architectonic connectional and functional sense this organization is not explicitly captured by the surface model of the cortex we propose a geometric model of the cortical anatomy based on flat representations of principal sulci obtained from surface reconstructions of mri data and on neuroanatomical and theoretical considerations concerning the folding patterns of the cortex the cortex is modeled by a sphere where primary sulci are included as axes the arrangement of the axes is a simplification of the arrangement of principal sulci observed in flat stereographic representations of the whole cortical surface the position of secondary and tertiary sulci is then defined by a field of orientations parallel and orthogonal to the axes we consider the use of the geometric model as a synthetic reference cortex for addressing reconstructions of cortical surfaces we present a method which establishes a bijection between the geometric model and a cortical surface reconstruction by using the axes of the model as boundary conditions for a set of partial differential equations solved over both surfaces using the geometric model as atlas provides a natural parameterization of the cortical surface that unlike angular coordinates allows for a localization based on the surface distance to its main organizing landmarks and folding patterns

on an algorithm for optimal control using pontryagin s maximum principle y sakawa and y shindo recently proposed an algorithm to solve open loop optimal control problems using pontryagin s maximum principle it is established here that under some hypothesis the algorithm is well defined and globally converges in some weak sense when the stepsize of the algorithm tends to zero the displacements are equivalent to those of a gradient with projection method if the control enters in a linear way in the state equation and in a quadratic way in the criterion the algorithm can be interpreted as a gradient plus projection method in some new metric

on ruelle s probability cascades and an abstract cavity method we construct in this work a markov process which describes a clustering mechanism through which equivalence classes on are progressively lumped together this clustering process gives a new description of ruelle s continuous probability cascades it also enables to introduce an abstract cavity method which mimicks certain features of the cavity method developed by physicists in the context of the sherrington kirkpatrick model

second order analysis for strong solutions in the optimal control of parabolic equations in this paper we provide a second order analysis for strong solutions in the optimal control of parabolic equations we consider the case of box constraints on the control and final integral constraints on the state in contrast to sufficient conditions assuring quadratic growth in the weak sense i e when the cost increases at least quadratically for admissible controls uniformly near to the nominal one see e g 16 26 our main result provides a sufficient condition for quadratic growth of the cost for admissible controls whose associated states are uniformly near to the state of the nominal one as a consequence of our results for qualified problems with a strictly convex and quadratic hamiltonian we prove that both notions of quadratic growth coincide

hamilton jacobi skeleton on cortical surfaces in this paper we propose a new method to construct graphical representations of cortical folding patterns by computing skeletons on triangulated cortical surfaces in our approach a cortical surface is first partitioned into sulcal and gyral regions via the solution of a variational problem using graph cuts which can guarantee global optimality after that we extend the method of hamilton jacobi skeleton to subsets of triangulated surfaces together with a geometrically intuitive pruning process that can trade off between skeleton complexity and the completeness of representing folding patterns compared with previous work that uses skeletons of 3 d volumes to represent sulcal patterns the skeletons on cortical surfaces can be easily decomposed into branches and provide a simpler way to construct graphical representations of cortical morphometry in our experiments we demonstrate our method on two different cortical surface models its ability of capturing major sulcal patterns and its application to compute skeletons of gyral regions

second order sufficient optimality conditions for some state constrained control problems of semilinear elliptic equations this paper deals with a class of optimal control problems governed by elliptic equations with nonlinear boundary condition the case of boundary control is studied pointwise constraints on the control and certain equality and set constraints on the state are considered second order sufficient conditions for local optimality of controls are established

a primal sketch of the cortex mean curvature a morphogenesis based approach to study the variability of the folding patterns in this paper we propose a new representation of the cortical surface that may be used to study the cortex folding process and to recover some putative stable anatomical landmarks called sulcal roots usually buried in the depth of adult brains this representation is a primal sketch derived from a scale space computed for the mean curvature of the cortical surface this scale space stems from a diffusion equation geodesic to the cortical surface the primal sketch is made up of objects defined from mean curvature minima and saddle points the resulting sketch aims first at highlighting significant elementary cortical folds second at representing the fold merging process during brain growth the relevance of the framework is illustrated by the study of central sulcus sulcal roots from antenatal to adult age some results are proposed for ten different brains some preliminary results are also provided for superior temporal sulcus

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

asymptotic expansion for the solutions of control constrained semilinear elliptic problems with interior penalties in this work we consider the optimal control problem of a semilinear elliptic partial differential equation pde with a dirichlet boundary condition where the control variable is distributed over the domain and is constrained to be nonnegative the approach is to consider an associated family of penalized problems parametrized by varepsilon 0 whose solutions define a central path converging to the solution of the original problem our aim is to obtain an asymptotic expansion for the solutions of the penalized problems around the solution of the original problem this approach allows us to recover some known error bounds for the logarithmic barrier approximation and to obtain some new ones for a rather general class of barrier functions in this manner we generalize the results of f alvarez et al asymptotic expansions for interior penalty solutions of control constrained linear quadratic problems math program ser a to appear which were obtained in the ordinary differential equation ode framework

sufficient second order optimality conditions for an elliptic optimal control problem with pointwise control state constraints an optimal control problem for a semilinear elliptic equation is investigated where pointwise constraints are given on control and state the state constraints are of mixed bottleneck type where associated lagrange multipliers can be chosen as bounded and measurable functions based on this property a second order sufficient optimality condition is established that takes into account strongly active constraints

statistical sulcal shape comparisons application to the detection of genetic encoding of the central sulcus shape principal component analysis allows a quantitative description of shape variability with a restricted num ber of parameters or modes which can be used to quantify the difference between two shapes through the computation of a modal distance a statistical test can then be applied to this set of measurements in order to detect a statistically significant difference between two groups we have applied this methodol ogy to highlight evidence of genetic encoding of the shape of neuroanatomical structures to investigate genetic constraint we studied if shapes were more similar within 10 pairs of monozygotic twins than within interpairs and compared the results with those obtained from 10 pairs of dizygotic twins the statis tical analysis was performed using a mantel permuta tion test we show using simulations that this statis tical test applied on modal distances can detect a possible genetic encoding when applied to real data this study highlighted genetic constraints on the shape of the central sulcus we found from 10 pairs of monozygotic twins that the intrapair modal distance of the central sulcus was significantly smaller than the interpair modal distance for both the left central sul cus z 52 2 66 p 0 005 and the right central sulcus z 52 2 26 p 0 05 genetic constraints on the defi nition of the central sulcus shape were confirmed by applying the same experiment to 10 pairs of normal young individuals z 52 1 39 z 52 0 63 i e values not significant at the p 0 05 level and 10 pairs of dizy gotic twins z 5 0 47 z 5 0 03 i e values not signifi cant at the p 0 05 level 2000 academic press

the brownian web characterization and convergence the brownian web bw is the random network formally consist ing of the paths of coalescing one dimensional brownian motions start ing from every space time point in r r we extend the earlier work of arratia and of toth and werner by providing characterization and convergence results for the bw distribution including convergence of the system of all coalescing random walks to the bw under diffusive space time scaling we also provide characterization and convergence results for the double brownian web which combines the bw with its dual process of coalescing brownian motions moving backwards in time with forward and backward paths reflecting off each other for the bw deterministic space time points are almost surely of type 0 1 zero paths into the point from the past and exactly one path out of the point to the future we determine the hausdorff dimension for all types that actually occur dimension 2 for type 0 1 3 2 for 1 1 and 0 2 1 for 1 2 and 0 for 2 1 and 0 3

global divergence of spatial coalescents we study several fundamental properties of a class of stochastic processes called spatial coalescents in these models a number of particles perform independent random walks on some underlying graph g in addition particles on the same vertex merge randomly according to a given coalescing mechanism a remarkable property of mean field coalescent processes is that they may come down from infinity meaning that starting with an infinite number of particles only a finite number remains after any positive amount of time almost surely we show here however that in the spatial setting on any infinite and bounded degree graph the total number of particles will always remain infinite at all times almost surely moreover if g mathbb z d and the coalescing mechanism is kingman s coalescent then starting with n particles at the origin the total number of particles remaining is of order log n d at any fixed positive time where log is the inverse tower function at sufficiently large times the total number of particles is of order log n d 2 when d 2 we provide parallel results in the recurrent case d 2 the spatial beta coalescents behave similarly where log log n is replacing log n

the emerging discipline of computational functional anatomy computational functional anatomy cfa is the study of functional and physiological response variables in anatomical coordinates for this we focus on two things i the construction of bijections via diffeomorphisms between the coordinatized manifolds of human anatomy and ii the transfer group action and parallel transport of functional information into anatomical atlases via these bijections we review advances in the unification of the bijective comparison of anatomical submanifolds via point sets including points curves and surface triangulations as well as dense imagery we examine the transfer via these bijections of functional response variables into anatomical coordinates via group action on scalars and matrices in dti as well as parallel transport of metric information across multiple templates which preserves the inner product

coalescing brownian flows a new approach the coalescing brownian flow on r is a process which was introduced by arratia coalescing brownian motions on the line 1979 univ wisconsin madison and toth and werner probab theory related fields 111 1998 375 452 and which formally corresponds to starting coalescing brownian motions from every space time point we provide a new state space and topology for this process and obtain an invariance principle for coalescing random walks this result holds under a finite variance assumption and is thus optimal in previous works by fontes et al ann probab 32 2004 2857 2883 newman et al electron j probab 10 2005 21 60 the topology and state space required a moment of order 3 e for this convergence to hold the proof relies crucially on recent work of schramm and smirnov on scaling limits of critical percolation in the plane our approach is sufficiently simple that we can handle substantially more complicated coalescing flows with little extra work in particular similar results are obtained in the case of coalescing brownian motions on the sierpinski gasket this is the first such result where the limiting paths do not enjoy the noncrossing property

kinetic limit for a system of coagulating planar brownian particles we study a model of mass bearing coagulating planar brownian particles the coagulation occurs when two particles are within a distance of order e we assume that the initial number of particles n is of order loge under suitable assumptions of the initial distribution of particles and the microscopic coagulation propensities we show that the macroscopic particle densities satisfy a smoluchowski type equation

computational anatomy shape growth and atrophy comparison via diffeomorphisms computational anatomy ca is the mathematical study of anatomy i in i i alpha o g an orbit under groups of diffeomorphisms i e smooth invertible mappings g in g of anatomical exemplars i alpha in i the observable images are the output of medical imaging devices there are three components that ca examines i constructions of the anatomical submanifolds ii comparison of the anatomical manifolds via estimation of the underlying diffeomorphisms g in g defining the shape or geometry of the anatomical manifolds and iii generation of probability laws of anatomical variation p on the images i for inference and disease testing within anatomical models this paper reviews recent advances in these three areas applied to shape growth and atrophy

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

products of beta matrices and sticky flows a discrete model of brownian sticky flows on the unit circle is described it is constructed with products of beta matrices on the discrete torus sticky flows are defined by their moments which are consistent systems of transition kernels on the unit circle similarly the moments of the discrete model form a consistent system of transition matrices on the discrete torus a convergence of beta matrices to sticky kernels is shown at the level of the moments as the generators of the n point processes are defined in terms of dirichlet forms the proof is performed at the level of the dirichlet forms the evolution of a probability measure by the flow of beta matrices is described by a measure valued markov process a convergence result of its finite dimensional distributions is deduced

sticky flows on the circle and their noises this paper gives a construction of sticky flows on the circle sticky flows give examples of stochastic flows of kernels that interpolates between arratia s coalescing flow and the deterministic diffusion flow they are associated with systems of sticky independent brownian particles on the circle for some fixed parameter of stickyness it is proved that the noise generated by brownian sticky flows is black a new proof of the fact that the noise of arratia s coalescing flow is black is given

flows coalescence and noise we are interested in stationary fluid random evolutions with independent increments under some mild assumptions we show they are solutions of a stochastic differential equation sde there are situations where these evolutions are not described by flows of diffeomorphisms but by coalescing flows or by flows of probability kernels in an intermediate phase for which there exist a coalescing flow and a flow of kernels solution of the sde a classification is given all solutions of the sde can be obtained by filtering a coalescing motion with respect to a subnoise containing the gaussian part of its noise thus the coalescing motion cannot be described by a white noise

dominated variation and related concepts and tauberian theorems for laplace transforms abstract some abelian and tauberian theorems are proved under conditions of dominated variation and related concepts for example u is dominatedly varying if and only if its laplace stieltjes transform is dominatedly varying

the kinetic limit of a system of coagulating brownian particles we consider a random model of diffusion and coagulation a large number of small particles is randomly scattered in mathbb r d at an initial time each particle has some integer mass and moves as a brownian motion whose rate of diffusion is determined by that mass when any two particles are close they are liable to combine into a single particle that bears the mass of each of them the range of interaction between pairs of particles is chosen so that a typical particle is liable to interact with a unit order of other particles in a unit of time we determine the macroscopic evolution of the system in any dimension d 3 the density of particles evolves according to the smoluchowski system of partial differential equations indexed by the mass parameter in which the interaction term is a sum of products of densities central to the proof is the task of establishing the so called stosszahlansatz which asserts that at any given time the presence of particles of two given masses at any given point in macroscopic space is asymptotically independent as the initial number of particles is taken to be high nonetheless there is in a microscopic region about each particle a reduced probability of finding another particle determining this deficit precisely is necessary in computing the coefficients appearing in the interaction terms of the smoluchowski partial differential equation

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

the curvature primal sketch in this paper we introduce a novel representation of the significant changes in curvature along the bounding contour of planar shape we call the representation the curvature primal sketch because of the close analogy to the primal sketch representation advocated by marr for describing significant intensity changes we define a set of primitive parameterized curvature discontinuities and derive expressions for their convolutions with the first and second derivatives of a gaussian we describe an implemented algorithm that computes the curvature primal sketch by matching the multiscale convolutions of a shape and illustrate its performance on a set of tool shapes several applications of the representation are sketched

second order necessary and sufficient optimality conditions for optimization problems and applications to control theory this paper deals with a class of nonlinear optimization problems in a function space where the solution is restricted by pointwise upper and lower bounds and by finitely many equality and inequality constraints of functional type second order necessary and sufficient optimality conditions are established where the cone of critical directions is arbitrarily close to the form which is expected from the optimization in finite dimensional spaces the results are applied to some optimal control problems for ordinary and partial differential equations

plane answers to complex questions the theory of linear models providing a wide ranging introduction to the use of linear models in analyzing data this text presents a vector space and projections approach to the subject the topics covered include anova estimation hypothesis testing multiple comparison regression analysis and experimental design also covered are testing for lack of fit models with singular covariance matrices variance component estimation best linear prediction colinearity and variable selection

asymptotic density in a coalescing random walk model we consider a system of particles each of which performs a continuous time random walk on mathbb z d the particles interact only at times when a particle jumps to a site at which there are a number of other particles present if there are j particles present then the particle which just jumped is removed from the system with probability p j we show that if p j is increasing in j and if the dimension d is at least 6 and if we start with one particle at each site of mathbb z d then p t p there is at least one particle at the origin at time t sim c d t the constant c d is explicitly identified we think the result holds for every dimension d geq 3 and we briefly discuss which steps in our proof need to be sharpened to weaken our assumption d geq 6 the proof is based on a justification of a certain mean field approximation for dp t dt the method seems applicable to many more models of coalescing and annihilating particles

automated sulcal segmentation using watersheds on the cortical surface the human cortical surface is a highly complex folded structure sulci the spaces between the folds define location on the cortex and provide a parcellation into anatomically distinct areas a topic that has recently received increased attention is the segmentation of these sulci from magnetic resonance images with most work focusing on extracting either the sulcal spaces between the folds or curve representations of sulci unlike these methods we propose a technique that extracts actual regions of the cortical surface that surround sulci which we call sulcal regions the method is based on a watershed algorithm applied to a geodesic depth measure on the cortical surface a well known problem with the watershed algorithm is a tendency toward oversegmentation meaning that a single region is segmented as several pieces to address this problem we propose a postprocessing algorithm that merges appropriate segments from the watershed algorithm the sulcal regions are then manually labeled by simply selecting the appropriate regions with a mouse click and a preliminary study of sulcal depth is reported finally a scheme is presented for computing a complete parcellation of the cortical surface

larger is twistier spectral analysis of gyrification spangy applied to adult brain size polymorphism abstract the description of cortical folding pattern cfp is challenging because of geometric complexity and inter subject variability on a cortical surface mesh curvature estimation provides a good scalar proxy of cfp the oscillations of this function can be studied using a fourier like analysis to produce a power spectrum representative of the spatial frequency composition of cfp first we introduce an original method for the spectral analysis of gyrication spangy which performs a spectral decomposition of the mean curvature of the grey white interface mesh based on the laplace beltrami operator eigenfunctions spangy produces an ordered 7 bands power spectrum of curvature b0 b6 and provides an anatomically relevant segmentation of cfp based on local spectral composition a spatial frequency being associated with each eigenfunction the bandwidth design assumes frequency doubling between consecutive spectral bands next we observed that the last 3 spectral bands b4 5 and 6 accounted for 93 of the analyzed spectral power and were associated with fold related variations of curvature whereas the lower frequency bands were related to global brain shape the spectral segmentation of cfp revealed 1st 2nd and 3rd order elements associated with b4 b5 and b6 respectively these elements could be related to developmentally defined primary secondary and tertiary folds finally we used allometric scaling of frequency bands power and segmentation to analyze the relationship between the spectral composition of cfp and brain size in a large adult dataset total folding power followed a positive allometric scaling which did not divide up proportionally between the bands b4 contribution was constant b5 increased like total folding power and b6 much faster besides apparition of new elements of pattern with increasing size only concerned the 3rd order hence we demonstrate that large brains are twistier than smaller ones because of an increased number of high spatial frequency folds ramifications and kinks that accommodate the allometric increase of cortical surface

multimodal brain warping using the demons algorithm and adaptative intensity corrections this paper presents an original method for three dimensional elastic registrat ion of multimodal images we propose to make use of a scheme that iterates between correcting for intensity differences between images and performing standard monomodal registration the core of our contribution resides in providing a method that finds the transformation that maps the intensities of one image to those of another it makes the assumption that there are at most two functional dependences between the intensities of structures present in the images to register and relies on robust estimation techniques to evaluate these functions we provide results showing successful registratio n between several imaging modalities involving segmentations t1 magnetic resonance mr t2 mr proton density pd mr and computed tomography ct we also argue that our intensity modeling may be more appropriate than mutual information mi in the context of evaluating high dimensional deformations as it puts more constraints on the parameters to be estimated and thus permits a better search of the parameter space

on the scaling limits of planar percolation we prove tsirelson s conjecture that any scaling limit of the critical planar percolation is a black noise our theorems apply to a number of percolation models including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid we also suggest a natural construction for the scaling limit of planar percolation and more generally of any discrete planar model describing connectivity properties

smooth functional and structural maps on the neocortex via orthonormal bases of the laplace beltrami operator functional and structural maps such as a curvature cortical thickness and functional magnetic resonance imaging mri maps indexed over the local coordinates of the cortical manifold play an important role in neuropsychiatric studies due to the highly convoluted nature of the cerebral cortex and image quality these functions are generally uninterpretable without proper methods of association and smoothness onto the local coordinate system in this paper we generalized the spline smoothing problem wahba 1990 from a sphere to any arbitrary two dimensional 2 d manifold with boundaries we first seek a numerical solution to orthonormal basis functions of the laplace beltrami lb operator with neumann boundary conditions for a 2 d manifold m then solve the spline smoothing problem in a reproducing kernel hilbert space r k h s of real valued functions on manifold m with kernel constructed from the basis functions the explicit discrete lb representation is derived using the finite element method calculated directly on the manifold coordinates so that finding discrete lb orthonormal basis functions is equivalent to solving an algebraic eigenvalue problem and then smoothed functions in r k h s can be represented as a linear combination of the basis functions we demonstrate numerical solutions of spherical harmonics on a unit sphere and brain orthonormal basis functions on a planum temporale manifold then synthetic data is used to quantify the goodness of the smoothness compared with the ground truth and discuss how many basis functions should be incorporated in the smoothing we present applications of our approach to smoothing sulcal mean curvature cortical thickness and functional statistical maps on submanifolds of the neocortex

spatial and anatomical regularization of svm a general framework for neuroimaging data this paper presents a framework to introduce spatial and anatomical priors in svm for brain image analysis based on regularization operators a notion of proximity based on prior anatomical knowledge between the image points is defined by a graph e g brain connectivity graph or a metric e g fisher metric on statistical manifolds a regularization operator is then defined from the graph laplacian in the discrete case or from the laplace beltrami operator in the continuous case the regularization operator is then introduced into the svm which exponentially penalizes high frequency components with respect to the graph or to the metric and thus constrains the classification function to be smooth with respect to the prior it yields a new svm optimization problem whose kernel is a heat kernel on graphs or on manifolds we then present different types of priors and provide efficient computations of the gram matrix the proposed framework is finally applied to the classification of brain magnetic resonance mr images based on gray matter gm concentration maps and cortical thickness measures from 137 patients with alzheimer s disease ad and 162 elderly controls the results demonstrate that the proposed classifier generates less noisy and consequently more interpretable feature maps with high classification performances

characterization of local quadratic growth for strong minima in the optimal control of semi linear elliptic equations in this article we consider an optimal control problem of a semi linear elliptic equation with bound constraints on the control our aim is to characterize local quadratic growth for the cost function j in the sense of strong solutions this means that the function j growths quadratically over all feasible controls whose associated state is close enough to the nominal one in the uniform topology the study of strong solutions classical in the calculus of variations seems to be new in the context of pde optimization our analysis based on a decomposition result for the variation of the cost combines pontryagin s principle and second order conditions while these two ingredients are known we use them in such a way that we do not need to assume that the hessian of lagrangian of the problem is a legendre form or that it is uniformly positive on an extended set of critical directions

optical flow and advection on 2 riemannian manifolds a common framework dynamic pattern analysis and motion extraction can be efficiently addressed using optical flow techniques this paper presents a generalization of these questions to nonflat surfaces where optical flow is tackled through the problem of evolution processes on non euclidean domains the classical equations of optical flow in the euclidean case are transposed to the theoretical framework of differential geometry we adopt this formulation for the regularized optical flow problem prove its mathematical well posedness and combine it with the advection equation the optical flow and advection problems are dual a motion field may be retrieved from some scalar evolution using optical flow conversely a scalar field may be deduced from a velocity field using advection these principles are illustrated with qualitative and quantitative evaluations from numerical simulations bridging both approaches the proof of concept is further demonstrated with preliminary results from time resolved functional brain imaging data where organized propagations of cortical activation patterns are evidenced using our approach

a parameterization based numerical method for isotropic and anisotropic diffusion smoothing on non flat surfaces neuroimaging data such as 3d maps of cortical thickness or neural activation can often be analyzed more informatively with respect to the cortical surface rather than the entire volume of the brain any cortical surface based analysis should be carried out using computations in the intrinsic geometry of the surface rather than using the metric of the ambient 3d space we present parameterization based numerical methods for performing isotropic and anisotropic filtering on triangulated surface geometries in contrast to existing fem based methods for triangulated geometries our approach accounts for the metric of the surface in order to discretize and numerically compute the isotropic and anisotropic geometric operators we first parameterize the surface using a p harmonic mapping we then use this parameterization as our computational domain and account for the surface metric while carrying out isotropic and anisotropic filtering to validate our method we compare our numerical results to the analytical expression for isotropic diffusion on a spherical surface we apply these methods to smoothing of mean curvature maps on the cortical surface a step commonly required for analysis of gyrification or for registering surface based maps across subjects

meteor process on mathbb z d the meteor process is a model for mass redistribution on a graph the case of finite graphs was analyzed in billey et al on meteors earthworms and wimps ann appl probab 2014 this paper is devoted to the meteor process on mathbb z d the process is constructed and a stationary distribution is found convergence to this stationary distribution is proved for a large family of initial distributions the first two moments of the mass distribution at a vertex are computed for the stationary distribution for the one dimensional lattice mathbb z the net flow of mass between adjacent vertices is shown to have bounded variance as time goes to infinity an alternative representation of the process on mathbb z as a collection of non crossing paths is presented the distributions of a tracer particle in this system of non crossing paths are shown to be tight as time goes to infinity

exact solution for a random walk in a time dependent 1d random environment the point to point beta polymer we consider the beta polymer an exactly solvable model of directed polymer on the square lattice introduced by barraquand and corwin bc 2016 probab theory relat fields 1 16 we study the statistical properties of its point to point partition sum the problem is equivalent to a model of a random walk in a time dependent and in general biased 1d random environment in this formulation we study the sample to sample fluctuations of the transition probability distribution function pdf of the random walk using the bethe ansatz we obtain exact formulas for the integer moments and fredholm determinant formulas for the laplace transform of the directed polymer partition sum random walk transition probability the asymptotic analysis of these formulas at large time t is performed both i in a diffusive vicinity of the optimal direction in space time chosen by the random walk where the fluctuations of the pdf are found to be gamma distributed ii in the large deviations regime of the random walk where the fluctuations of the logarithm of the pdf are found to grow with time as t 1 3 and to be distributed according to the tracy widom gue distribution our exact results complement those of bc for the cumulative distribution function of the random walk in regime ii and in regime i they unveil a novel fluctuation behavior we also discuss the crossover regime between i and ii identified as our results are confronted to extensive numerical simulations of the model