wave propagation on riemannian symmetric spaces abstract let x g k be a riemannian symmetric space of the noncompact type and let l x be the laplace beltrami operator on x we consider on x r a differential equation of the type l x u p t u where p t is a second order differential operator in t r using the radon transform on x we relate huygens principle for the solutions u to this equation to the corresponding question for the solutions v to the equation l a v p t r v on a maximal flat subspace a where r is a certain explicit constant in particular we conclude that huygens principle holds for solutions to the modified wave equation on x obtained by letting p t t 2 r when x is odd dimensional and g has one conjugacy class of cartan subgroups

weighted strichartz estimates for the schrodinger and wave equations on damek ricci spaces we prove strichartz estimates for radial solutions of the schrodinger and wave equations on damek ricci spaces and in particular on symmetric spaces of noncompact type and rank one using the perturbative theory with potentials the curvature of the noncompact manifold has an influence on the dispersive properties and indeed we obtain strichartz estimates with weights at spatial infinity which are stronger than the standard ones in the flat case

pixelwise adaptive blind optical flow assuming nonstationary statistics we address some of the major issues in optical flow within a new framework assuming nonstationary statistics for the motion field and for the errors problems addressed include the preservation of discontinuities model data errors outliers confidence measures and performance evaluation in solving these problems we assume that the statistics of the motion field and the errors are not only spatially varying but also unknown we thus derive a blind adaptive technique based on generalized cross validation for estimating an independent regularization parameter for each pixel our formulation is pixelwise and combines existing first and second order constraints with a new second order temporal constraint we derive a new confidence measure for an adaptive rejection of erroneous and outlying motion vectors and compare our results to other techniques in the literature a new performance measure is also derived for estimating the signal to noise ratio for real sequences when the ground truth is unknown

on the estimation of optical flow relations between different approaches and some new results abstract a common basis is suggested for the optical flow estimation approaches of nagel 1983 haralick and lee 1983 and tretiak and pastor 1984 based on a discussion of these approaches an exact solution for the system of partial differential equations proposed by horn and schunck 1981 is given at gray value corners and extrema the insight gained by this solution results in a modification of the oriented smoothness approach of nagel 1983 which thereby becomes considerably simpler in addition the optical flow estimation approach of hildreth 1983 1984 can be shown to represent a kind of special case of this modified oriented smoothness approach in a more direct manner than discussed in nagel 1984

formulae for the derivatives of heat semigroups abstract formulae for the derivatives of solutions of diffusion equations are derived which clearly exhibit and allow estimation of the equations smoothing properties these also give formulae for the logarithmic gradient of the corresponding heat kernels extending and giving a very elementary proof of bismut s well known formula corresponding formulae are derived for solutions of heat equations for differential forms and their exterior derivatives

green s functions biot savart operators and linking numbers on negatively curved symmetric spaces we construct radial fundamental solutions for the differential form laplacian on negatively curved symmetric spaces at least one of these green s functions also yields a biot savart operator i e a right inverse of the exterior differential on closed forms with image in the kernel of the codifferential any biot savart operator gives rise to a gauss linking integral

pinning class of the wiener measure by a functional related martingales and invariance properties for a given functional y on the path space we define the pinning class of the wiener measure as the class of probabilities which admit the same conditioning given y as the wiener measure using stochastic analysis and the theory of initial enlargement of filtration we study the transformations not necessarily adapted which preserve this class we prove in this non markov setting a stochastic newton equation and a stochastic noether theorem we conclude the paper with some non canonical representations of brownian motion closely related to our study

integrability and huygens principle on symmetric spaces the explicit formulas for fundamental solutions of the modified wave equations on certain symmetric spaces are found these symmetric spaces have the following characteristic property all multiplicities of their restricted roots are even as a corollary in the odd dimensional case one has that the huygens principle in hadamard s sense for these equations is fulfilled we consider also the heat and laplace equations on such a symmetric space and give explicitly the corresponding fundamental solutions heat kernel and green s function this continues our previous investigations 16 of the spherical functions on the same symmetric spaces based on the fact that the radial part of the laplace beltrami operator on such a space is related to the algebraically integrable case of the generalised calogero sutherland moser quantum system in the last section of this paper we apply the methods of heckman and opdam to extend our results to some other symmetric spaces in particular to complex and quaternian grassmannians

symmetries in the stochastic calculus of variations given a stochastic action integral we define a notion of invariance of this action under a family of one parameter space time transformations and a notion of prolonged transformations which extend the existing analogs in classical calculus of variations we prove that a family of prolonged transformations leaves the action integral invariant if and only if it leaves invariant the heat equation associated to it as well as the structure of the extremals we then prove a stochastic version of noether theorem to each family of transformations leaving the action invariant or symmetries we can associate a function which gives a martingale when taken along a process minimizing the action under endpoint constraints

conditioning and initial enlargement of filtration on a riemannian manifold we extend to riemannian manifolds the theory of conditioned stochastic differential equations we also provide some enlargement formulas for the brownian filtration in this nonflat setting

orbits of the geodesic flow and chains on the boundary of a grauert tube a c omega riemannian manifold x g determines an integrable complex structure on a tubular neighborhood t epsilon x of the zero section in t x and a cr structure on the boundary m epsilon there are two natural families of curves on m epsilon the orbits of the geodesic flow and a cr invariant family called chains it is natural to ask whether they are related we show that if orbits of the geodesic flow are chains on m epsilon for all epsilon sufficiently small then x g is einstein as a partial converse we show that if x g is harmonic then orbits of the geodesic flow are chains to prove this we study the fefferman metric associated with m epsilon

huygens principle for wave equations on symmetric spaces let x gk be a symmetric space lx the laplace betrami operator on x and 2 the sum of the positive restricted roots of x with multiplicity using the fourier transform on x we write down the solution of the modified wave equation 2u t2 lx 2 u on x it is proved using the paley wiener theorem for this fourier transform s helgason ann of math 98 1973 451 480 that the equation satisfies huygens principle if dim x is odd and all cartan subgroups of g are conjugate

the atiyah singer theorems a probabilistic approach ii the lefschetz fixed point formulas abstract in this paper the lefschetz fixed point formulas of m f atiyah and i m singer ann of math 87 1968 484 530 546 604 for classical elliptic complexes are proved by using probabilistic methods

isospectral pairs of metrics on balls spheres and other manifolds with different local geometries the first isospectral pairs of metrics are constructed on the most simple simply connected domains namely on balls and spheres this long standing problem concerning the existence of such pairs has been solved by a new method called anticommutator technique among the wide range of such pairs the most striking examples are provided on the spheres s4k 1 where k 3 one of these metrics is homogeneous since it is the metric on the geodesic sphere of a 2 point homogeneous space while the other is locally inhomogeneous these examples demonstrate the surprising fact that no information about the isometries is encoded in the spectrum of the laplacian acting on functions in other words the group of isometries even the local homogeneity property is lost to the nonaudible in the debate of audible versus nonaudible geometry in this article we settle a long standing and intensively studied problem of spectral geometry by constructing the first examples of nontrivial isospectral metrics on the simplest simply connected manifolds namely on balls and spheres this result is achieved by a new isospectral construction technique called anticommutator technique it provides a whole cornucopia of such examples not just on balls and spheres but on other types of manifolds too the most striking examples are constructed on the geodesic spheres of the hyperbolic quaternionic 2 point homogeneous spaces the induced metrics on these spheres are obviously homogeneous while the metrics constructed isospectrally to these are locally inhomogeneous this shows the weakness of the information about the local geometry encoded in the spectrum of laplacian acting on functions

universality and self similarity in the bifurcations of circle maps abstract the bifurcation structure in a two parameter family of circle maps is considered these maps have a topological degree that may be different from one a generalization of the rotation number is given and symmetries of the bifurcations in parameter space are described continuity arguments are used to establish the existence of periodic orbits by plotting the locus of parameter values associated with superstable cycles self similar bifurcations are found these bifurcations are a generalization of the familiar period doubling cascade in maps with one extrema to two parameter maps with two extrema finally a scheme for the global organization of bifurcation in these maps is proposed

submanifolds isoperimetric inequalities and optimal transportation abstract the aim of this paper is to prove isoperimetric inequalities on submanifolds of the euclidean space using mass transportation methods we obtain a sharp weighted isoperimetric inequality and a nonsharp classical inequality similar to the one obtained in michael and simon 1973 19 the proof relies on the description of a solution of the problem of monge when the initial measure is supported in a submanifold and the final one supported in a linear subspace of the same dimension

improved accuracy in gradient based optical flow estimation optical flow estimation by means of first derivatives can produce surprisingly accurate and dense optical flow fields in particular recent empirical evidence suggests that the method that is based on local optimization of first order constancy constraints is among the most accurate and reliable methods available nevertheless a systematic investigation of the effects of the various parameters for this algorithm is still lacking this paper reports such an investigation performance is assessed in terms of flow field accuracy density and resolution the investigation yields new information regarding pre filter differentiator least squares neighborhood and reliability test selection several changes to previously employed parameter settings result in significant overall performance improvements while they simultaneously reduce the computational cost of the estimator

geometric properties of rank one asymptotically harmonic manifolds in this article we consider asymptotically harmonic manifolds which are simply connected complete riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature h we prove the following equivalences for asymptotically harmonic manifolds x under the additional assumption that their curvature tensor together with its covariant derivative are uniformly bounded a x has rank one b x has anosov geodesic flow c x is gromov hyperbolic d x has purely exponential volume growth with volume entropy equals h this generalizes earlier results by g knieper for noncompact harmonic manifolds and by a zimmer for asymptotically harmonic manifolds admitting compact quotients

symmetric like riemannian manifolds and geodesic symmetries we treat several classes of riemannian manifolds whose shape operators of geodesic spheres or jacobi operators share some properties with the ones on symmetric spaces

support theorem for the fundamental solution to the schrodinger equation on certain compact symmetric spaces abstract in this paper we construct the fundamental solution to the schodinger equation on a compact symmetric space with even root multiplicities using shift operators of heckman and opdam next we prove that the support of the fundamental solution becomes a lower dimensional subset at a rational time whereas its support and its singular support coincide with the whole symmetric space at an irrational time moreover we also show that generalized gauss sums appear in the expression of the fundamental solution

contour tracking with a spatio temporal intensity moment standard edge detection operators such as the laplacian of gaussian and the gradient of gaussian can be used to track contours in image sequences when using edge operators a contour which is determined on a frame of the sequence is simply used as a starting contour to locate the nearest contour on the subsequent frame however the strategy used to look for the nearest edge points may not work when tracking contours of non isolated gray level discontinuities in these cases strategies derived from the optical flow equation which look for similar gray level distributions appear to be more appropriate since these can work with a lower frame rate than that needed for strategies based on pure edge detection operators however an optical flow strategy tends to propagate the localization errors through the sequence and an additional edge detection procedure is essential to compensate for such a drawback in this paper a spatio temporal intensity moment is proposed which integrates the two basic functions of edge detection and tracking

some observations on the questions is ventricular fibrillation chaos abstract ventricular fibrillation has been modeled as cardiac chaos occuring after a series of subharmonic bifurcations however previous experimental studies have suggested that fibrillatory oscillations have a relatively narrow band frequency spectrum inconsistent with a turbulent process similarly during the first minute of canine fibrillation we observed only a few localized frequency peaks from the epicardial and body surface electrocardiogram rather than a broadband type of spectrum as would be predicted for chaotic dynamics further narrowing of the frequency spectrum occured during the second minute of fibrillation the frequency spectrum of ventricular fibrillation contrasts with scaled broadband spectra observed in normal cardiac function we suggest that ventricular fibrillation may serve as a general model for transitions from broadband stability to certain types of pathological periodicities in other physiological perturbations

integrability in the theory of schrodinger operator and harmonic analysis the algebraic integrability for the schrodinger equation in n and the role of the quantum calogero sutherland problem and root systems in this context are discussed for the special values of the parameters in the potential the explicit formula for the eigenfunction of the corresponding sutherland operator is found as an application the explicit formula for the zonal spherical functions on the symmetric spacessu 2n spn type a ii in cartan notations is presented

point matching under large image deformations and illumination changes to solve the general point correspondence problem in which the underlying transformation between image patches is represented by a homography a solution based on extensive use of first order differential techniques is proposed we integrate in a single robust m estimation framework the traditional optical flow method and matching of local color distributions these distributions are computed with spatially oriented kernels in the 5d joint spatial color space the estimation process is initiated at the third level of a gaussian pyramid uses only local information and the illumination changes between the two images are also taken into account subpixel matching accuracy is achieved under large projective distortions significantly exceeding the performance of any of the two components alone as an application the correspondence algorithm is employed in oriented tracking of objects

bifurcation and chaos in a periodically stimulated cardiac oscillator abstract periodic stimulation of an aggregate of spontaneously beating cultured cardiac cells displays phase locking period doubling bifurcations and aperiodic chaotic dynamics at different values of the stimulation parameters this behavior is analyzed by considering an experimentally determined one dimensional poincare or first return map a simplified version of the experimentally determined poincare map is proposed and several features of the bifurcations of this map are described

the atiyah singer theorems a probabilistic approach i the index theorem abstract the atiyah singer index theorem for classical elliptic complexes is proved by using probabilistic methods the general idea is to use a probabilistic construction of the heat equation kernel which permits the direct derivation of the index formulas without using the theory of invariants of gilkey stochastic calculus on the exterior algebra is then used to find the classical local formula for the index theorem the lefschetz fixed point formulas of atiyah singer will be proved in part ii

a cornucopia of isospectral pairs of metrics on spheres with different local geometries this article concludes the comprehensive study started in sz5 where the first nontrivial isospectral pairs of metrics are constructed on balls and spheres these investigations incorporate four different cases since these balls and spheres are considered both on 2 step nilpotent lie groups and on their solvable extensions in sz5 the considerations are completely concluded in the ball case and in the nilpotent case the other cases were mostly outlined in this paper the isospectrality theorems are completely established on spheres also the important details required about the solvable extensions are concluded in this paper

transition to topological chaos for circle maps abstract many biperiodic flows can be modelled by maps of a circle to itself for such maps the transition from zero to positive topological entropy can be achieved in several ways we describe all the possible routes for smooth circle maps and discuss the relevance of our results to the transition to chaos for two frequency systems

on asymptotically harmonic manifolds of negative curvature we study asymptotically harmonic manifolds of negative curvature without any cocompactness or homogeneity assumption we show that asymptotic harmonicity provides a lot of information on the asymptotic geometry of these spaces in particular we determine the volume entropy the spectrum and the relative densities of visual and harmonic measures on the ideal boundary then we prove an asymptotic analogue of the classical mean value property of harmonic manifolds and we characterize asymptotically harmonic manifolds among cartan hadamard spaces of strictly negative curvature by the existence of an asymptotic equivalent tau u mathrm e er for the volume density of geodesic spheres with tau constant in case dr m is bounded finally we show the existence of a margulis function and explicitly compute it for all asymptotically harmonic manifolds

nonlinear dynamics chaos and complex cardiac arrhythmias periodic stimulation of a nonlinear cardiac oscillator in vitro gives rise to complex dynamics that is well described by one dimensional finite difference equations as stimulation parameters are varied a large number of different phase locked and chaotic rhythms is observed similar rhythms can be observed in the intact human heart when there is interaction between two pacemaker sites simplified models are analysed which show some correspondence to clinical observations the normal adult human heart at rest usually beats at a rate of between 50 and 100 times per minute in many circumstances some of which are lifethreatening but most of which are not the normal rhythmicity is altered resulting in abnormal rhythms called cardiac arrhythmias the point of this paper is to show that a branch of mathematics called nonlinear dynamics may be useful in the analysis of physiological processes believed to underlie normal heart rate regulation and some cardiac arrhythmias the idea that mathematical analysis can play a role in understanding cardiac arrhythmias is not novel indeed in the 1920s it was demonstrated that as parameters in mathematical models for the heart were varied several different rhythms that resembled clinically observed arrhythmias could be generated mobitz 1924 van de pol van der mark 1928 in nonlinear mathematics these changes in the qualitative features of the rhythms that are observed as parameters vary are called bifurcations thus the problem of understanding cardiac arrhythmias in the human heart is identified with understanding the bifurcations and complex dynamics in mathematical models of the human heart one type of dynamic behaviour that is the object of intensive analysis in mathematics is chaos loosely chaos is defined as aperiodic dynamics in deterministic systems in which there is sensitive dependence to the initial conditions this means that although in principle one could determine precisely the future evolution of the system starting from some initial condition for chaotic dynamics any difference in the initial condition no matter how small will eventually lead to marked differences in the future evolution of the system although the existence of chaos was known to poincare and others since the end of the last century in the

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

computation of component image velocity from local phase information we present a technique for the computation of 2d component velocity from image sequences initially the image sequence is represented by a family of spatiotemporal velocity tuned linear filters component velocity computed from spatiotemporal responses of identically tuned filters is expressed in terms of the local first order behavior of surfaces of constant phase justification for this definition is discussed from the perspectives of both 2d image translation and deviations from translation that are typical in perspective projections of 3d scenes the resulting technique is predominantly linear efficient and suitable for parallel processing moreover it is local in space time robust with respect to noise and permits multiple estimates within a single neighborhood promising quantiative results are reported from experiments with realistic image sequences including cases with sizeable perspective deformation

stochastic aspects of cardiac arrhythmias abnormal cardiac rhythms cardiac arrhythmias often display complex changes over time that can have a random or haphazard appearance mathematically these changes can on occasion be identified with bifurcations in difference or differential equation models of the arrhythmias one source for the variability of these rhythms is the fluctuating environment however in the neighborhood of bifurcation points the fluctuations induced by the stochastic opening and closing of individual ion channels in the cell membrane which results in membrane noise may lead to randomness in the observed dynamics to illustrate this we consider the effects of stochastic properties of ion channels on the resetting of pacemaker oscillations and on the generation of early afterdepolarizations the comparison of the statistical properties of long records showing arrhythmias with the predictions from theoretical models should help in the identification of different mechanisms underlying cardiac arrhythmias

jost maps ball homogeneous and harmonic manifolds abstract given a real number e 0 small enough an associated jost map j e between two riemannian manifolds is defined then we prove that connected riemannian manifolds for which the center of mass of each small geodesic ball is the center of the ball i e for which the identity is a j e map are ball homogeneous in the analytic case we characterize such manifolds in terms of the euclidean laplacian and we show that they have constant scalar curvature under some restriction on the ricci curvature we prove that riemannian analytic manifolds for which the center of mass of each small geodesic ball is the center of the ball are locally and weakly harmonic

heat equation derivative formulas for vector bundles we use martingale methods to give bismut type derivative formulas for differentials and co differentials of heat semigroups on forms and more generally for sections of vector bundles the formulas are mainly in terms of weitzenbock curvature terms in most cases derivatives of the curvature are not involved in particular our results improve b k driver s formula in 1997 j math pures appl 9 76 703 737 for logarithmic derivatives of the heat kernel measure on a riemannian manifold our formulas also include the formulas of k d elworthy and x m li 1998 c r acad sci paris ser i math 327 87 92