statistics on diffeomorphisms via tangent space representations in this paper we present a linear setting for statistical analysis of shape and an optimization approach based on a recent derivation of a conservation of momentum law for the geodesics of diffeomorphic flow once a template is fixed the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point along the geodesic is completely determined by the momentum at the origin through geodesic shooting equations the space of initial momentum provides a linear representation of the nonlinear diffeomorphic shape space in which linear statistical analysis can be applied specializing to the landmark matching problem of computational anatomy we derive an algorithm for solving the variational problem with respect to the initial momentum and demonstrate principal component analysis pca in this setting with three dimensional face and hippocampus databases

dirac type operators for spin manifolds associated to congruence subgroups of generalized modular groups fundamental solutions of dirac type operators are introduced for a class of conformally at spin manifolds this class consists of manifolds obtained by factoring out the upper half space of r n by congruence subgroups of generalized modular groups basic properties of these fundamental solutions are presented together with associated eisenstein and poincare type series

vahlen groups defined over commutative rings elements of a vahlen group are 2 times 2 matrices with entries in a clifford algebra satisfying some conditions they play a central role in the theory of higher dimensional harmonic automorphic forms traditionally they have come in both ordinary and paravector type and have been defined over clifford algebras over the real or complex numbers we extend the definition of both types to be over a commutative ring with an arbitrary quadratic form we show that they are indeed groups and identify in each case the group as the pin group spin group or another subgroup of the clifford group under some mild conditions for both types we show the equivalence of our definition with a suitably generalised version of the two standard definitions

higher dimensional analogues of fuchsian subgroups of 2 the problem of classification of 2 2 indefinite hermitian matrices over orders in clifford algebras is considered the unit groups of these matrices are analogous to maximal arithmetic fuchsian subgroups of psl 2 o where o is an order in a quadratic number field

time sequence diffeomorphic metric mapping and parallel transport track time dependent shape changes abstract serial mri human brain scans have facilitated the detection of brain development and of the earliest signs of neuropsychiatric and neurodegenerative diseases monitoring disease progression and resolving drug effects in clinical trials for preventing or slowing the rate of brain degeneration to track anatomical shape changes in serial images we introduce new point based time sequence large deformation diffeomorphic metric mapping ts lddmm to infer the time flow of within subject geometric shape changes that carry known observations through a period its euler lagrange equation is generalized for anatomies whose shapes are characterized by point sets such as landmarks curves and surfaces the time dependent momentum obtained from the ts lddmm encodes within subject shape changes for the purpose of across subject shape comparison we then propose a diffeomorphic analysis framework to translate within subject deformation in a global template without incorporating across subject anatomical variations via parallel transport technique the analysis involves the retraction of the within subject time dependent momentum along the ts lddmm trajectory from each time to the baseline the translation of the momentum in a global template and the reconstruction of the ts lddmm trajectory starting from the global template

diffeomorphisms groups and pattern matching in image analysis in a previous paper it was proposed to see the deformations of a common pattern as the action of an infinite dimensional group we show in this paper that this approac h can be applied numerically for pattern matching in image analysis of digital images using lie group ideas we construct a distance between deformations defined through a metric given the cost of infinitesimal deformations then we propose a numerical scheme to solve a variational problem involving this distance and leading to a sub optimal gradient pattern matching its links with fluid models are established

parametric regression on the grassmannian we address the problem of fitting parametric curves on the grassmann manifold for the purpose of intrinsic parametric regression we start from the energy minimization formulation of linear least squares in euclidean space and generalize this concept to general nonflat riemannian manifolds following an optimal control point of view we then specialize this idea to the grassmann manifold and demonstrate that it yields a simple extensible and easy to implement solution to the parametric regression problem in fact it allows us to extend the basic geodesic model to 1 a time warped variant and 2 cubic splines we demonstrate the utility of the proposed solution on different vision problems such as shape regression as a function of age traffic speed estimation and crowd counting from surveillance video clips most notably these problems can be conveniently solved within the same framework without any specifically tailored steps along the processing pipeline

a riemannian framework for tensor computing tensors are nowadays a common source of geometric information in this paper we propose to endow the tensor space with an affine invariant riemannian metric we demonstrate that it leads to strong theoretical properties the cone of positive definite symmetric matrices is replaced by a regular and complete manifold without boundaries null eigenvalues are at the infinity the geodesic between two tensors and the mean of a set of tensors are uniquely defined etc we have previously shown that the riemannian metric provides a powerful framework for generalizing statistics to manifolds in this paper we show that it is also possible to generalize to tensor fields many important geometric data processing algorithms such as interpolation filtering diffusion and restoration of missing data for instance most interpolation and gaussian filtering schemes can be tackled efficiently through a weighted mean computation linear and anisotropic diffusion schemes can be adapted to our riemannian framework through partial differential evolution equations provided that the metric of the tensor space is taken into account for that purpose we provide intrinsic numerical schemes to compute the gradient and laplace beltrami operators finally to enforce the fidelity to the data either sparsely distributed tensors or complete tensors fields we propose least squares criteria based on our invariant riemannian distance which are particularly simple and efficient to solve

weyl groups of some hyperbolic kac moody algebras abstract we use the theory of clifford algebras and vahlen groups to study weyl groups of hyperbolic kac moody algebras t n obtained by a process of double extension from a cartan matrix of finite type t n whose corresponding generalized cartan matrices are symmetric

landmark matching via large deformation diffeomorphisms this paper describes the generation of large deformation diffeomorphisms spl phi spl omega 0 1 sup 3 spl rlhar2 spl omega for landmark matching generated as solutions to the transport equation d spl phi x t dt spl nu spl phi x t t t spl isin 0 1 and spl phi x 0 x with the image map defined as spl phi spl middot 1 and therefore controlled via the velocity field spl nu spl middot t t spl isin 0 1 imagery are assumed characterized via sets of landmarks x sub n y sub n n 1 2 n the optimal diffeomorphic match is constructed to minimize a running smoothness cost spl par l spl nu spl par sup 2 associated with a linear differential operator l on the velocity field generating the diffeomorphism while simultaneously minimizing the matching end point condition of the landmarks both inexact and exact landmark matching is studied here given noisy landmarks x sub n matched to y sub n measured with error covariances spl sigma sub n then the matching problem is solved generating the optimal diffeomorphism spl phi spl circ x 1 spl int sub 0 sup 1 spl nu spl circ spl phi spl circ x t t dt x where spl nu spl circ spl middot argmin sub spl nu spl middot spl int sub 1 sup 1 spl int sub spl omega spl par l spl nu x t spl par sup 2 dxdt spl sigma sub n 1 sup n y sub n spl phi x sub n 1 sup t spl sigma sub n sup 1 y sub n spl phi x sub n 1 conditions for the existence of solutions in the space of diffeomorphisms are established with a gradient algorithm provided for generating the optimal flow solving the minimum problem results on matching two dimensional 2 d and three dimensional 3 d imagery are presented in the macaque monkey

on the two gap locus for the elliptic calogero moser model we give an analytical description of the locus of the two gap elliptic potentials associated with the corresponding flow of the calogero moser system we start with the description of treibich verdier two gap elliptic potentials the explicit formulae for the covers wavefunctions and lame polynomials are derived together with a new lax representation for the particle dynamics on the locus we then consider more general potentials within the weierstrass reduction theory of theta functions to lower genera the reduction conditions in the moduli space of the genus 2 algebraic curves are given this is a subvariety of the humbert surface which can be singled out by the condition of the vanishing of some theta constants

spinor valued l2 spaces on sm 1 and representations of spin 1 m abstract we show that the l 2 spaces of spinor valued functions on s m 1 provide models for the unitary principal series representations of spin 1 m which are characterized by the feature that they contain the fundamental spinor representation s of spin m the novelty of our approach is that we use tools characteristic of hypercomplex analysis the group spin 1 m is realized as a group of vahlen matrices and the harish chandra modules of such representations are expressed in terms of spherical monogenics which generalize the classical spherical harmonics by switching to the framework of hypercomplex analysis on the sphere we are able to extend part of the well known results for the scalar spherical principal series representations of so 1 m to the spinor setting in the process we introduce a clifford algebra valued riesz transform on the sphere

kloosterman sums for clifford algebras and a lower bound for the positive eigenvalues of the laplacian for congruence subgroups acting on hyperbolic spaces

diffeomorphic multi frame non rigid registration of cell nuclei in 2d and 3d live cell images to gain a better understanding of cellular and molecular processes it is important to quantitatively analyze the motion of subcellular particles in live cell microscopy image sequences since generally the subcellular particles move and cell nuclei move as well as deform it is important to decouple the movement of particles from that of the cell nuclei using non rigid registration methods we have developed a diffeomorphic multi frame approach for non rigid registration of cell nuclei in 2d and 3d live cell fluorescence microscopy images our non rigid registration approach is based on local optic flow estimation exploits information from multiple consecutive image frames and determines diffeomorphic transformations in the log domain which allows efficient computation of the inverse transformations to register single images of an image sequence to a reference image we use a temporally weighted mean image which is constructed based on inverse transformations and multiple consecutive frames using multiple consecutive frames improves the registration accuracy compared to pairwise registration and using a temporally weighted mean image significantly reduces the computation time compared with previous work in addition we use a flow boundary preserving method for regularization of computed deformation vector fields which prevents from over smoothing compared to standard gaussian filtering our approach has been successfully applied to 2d and 3d synthetic as well as real live cell microscopy image sequences and an experimental comparison with non rigid pairwise multi frame and temporal groupwise registration has been carried out

elliptic genera of toric varieties and applications to mirror symmetry the paper contains a proof that elliptic genus of a calabi yau manifold is a jacobi form finds in which dimensions the elliptic genus is determined by the hodge numbers and shows that elliptic genera of a calabi yau hypersurface in a toric variety and its mirror coincide up to sign the proof of the mirror property is based on the extension of elliptic genus to calabi yau hypersurfaces in toric varieties with gorenstein singularities

on the full asymptotics of analytic torsion abstract the purpose of this article is to study the asymptotic expansion of ray singer analytic torsion associated with powers p of a given positive line bundle over a compact n dimensional complex manifold as p here we prove that the asymptotic expansion contains only the terms of the form p n i log p p n i for i n for the first two leading terms it was proved by bismut vasserot we calculate the coefficients of the terms p n 1 log p p n 1 in the kahler case and thus answer the question posed in the recent work of klevtsov ma marinescu wiegmann about quantum hall effect our second result concerns the general asymptotic expansion of the analytic torsion for a compact complex orbifold

registration atlas estimation and variability analysis of white matter fiber bundles modeled as currents this paper proposes a generic framework for the registration the template estimation and the variability analysis of white matter fiber bundles extracted from diffusion images this framework is based on the metric on currents for the comparison of fiber bundles this metric measures anatomical differences between fiber bundles seen as global homologous structures across subjects it avoids the need to establish correspondences between points or between individual fibers of different bundles it can measure differences both in terms of the geometry of the bundles like its boundaries and in terms of the density of fibers within the bundle it is robust to fiber interruptions and reconnections in addition a recently introduced sparse approximation algorithm allows us to give an interpretable representation of the fiber bundles and their variations in the framework of currents first we used this metric to drive the registration between two sets of homologous fiber bundles of two different subjects a dense deformation of the underlying white matter is estimated which is constrained by the bundles seen as global anatomical landmarks by contrast the alignment obtained from image registration is driven only by the local gradient of the image second we propose a generative statistical model for the analysis of a collection of homologous bundles this model consistently estimates prototype fiber bundles called template which capture the anatomical invariants in the population a set of deformations which align the geometry of the template to that of each subject and a set of residual perturbations the statistical analysis of both the deformations and the residuals describe the anatomical variability in terms of geometry stretching torque etc and texture fiber density etc third this statistical modeling allows us to simulate new synthetic bundles according to the estimated variability this gives a way to interpret the anatomical features that the model detects consistently across the subjects this may be used to better understand the bias introduced by the fiber extraction methods and eventually to give anatomical characterization of the normal or pathological variability of fiber bundles

siegel automorphic form corrections of some lorentzian kac moody lie algebras we find automorphic form corrections which are generalized lorentzian kac moody superalgebras without odd real simple roots for two elliptic lorentzian kac moody algebras of rank 3 with a lattice weyl vector and calculate multiplicities of their simple and arbitrary roots these kac moody algebras are defined by hyperbolic symmetrized generalized cartan matrices inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of rank 3 both these algebras have elliptic type i e their weyl groups have fundamental polyhedra of finite volume in corresponding hyperbolic spaces and have a lattice weyl vector the correcting automorphic forms are siegel modular forms the form corresponding to g 1 is the classical siegel cusp form of weight 5 which is the product of ten even theta constants in particular we find an infinite product formula for this modular form

lra local rigid averaging of stretchable non rigid shapes we present a novel algorithm for generating the mean structure of non rigid stretchable shapes following an alignment process which supports local affine deformations we translate the search of the mean shape into a diagonalization problem where the structure is hidden within the kernel of a matrix this is the first step required in many practical applications where one needs to model bendable and stretchable shapes from multiple observations

analytic torsion for calabi yau threefolds after bershadsky cecotti ooguri vafa we introduce an invariant of calabi yau threefolds which we call the bcov invariant and which we obtain using analytic torsion we give an explicit formula for the bcov invariant as a function on the compactified moduli space when it is isomorphic to a projective line as a corollary we prove the formula for the bcov invariant of quintic mirror threefolds conjectured by bershadsky cecotti ooguri vafa

dirac and laplace operators on some non orientable conformally flat orbifolds abstract in this paper we present an explicit construction of the fundamental solution to the dirac and laplace operator on some non orientable conformally flat manifolds and special orbifolds we first treat a class of projective cylinders and tori where we can study monogenic sections with values in different pin bundles then we discuss the mobius strip the klein bottle and higher dimensional generalizations of them we present integral representation formulas together with some elementary tools of harmonic analysis on these manifolds resp orbifolds

k3 surfaces with involution equivariant analytic torsion and automorphic forms on the moduli space iii the case r m 18 we prove the automorphic property of the invariant of k3 surfaces with involution which we obtained using equivariant analytic torsion in the case where the dimension of the moduli space is less than or equal to 2

some conformally flat spin manifolds dirac operators and automorphic forms abstract in this paper we study clifford and harmonic analysis on some examples of conformal flat manifolds that have a spinor structure or more generally at least a pin structure the examples treated here are manifolds that can be parametrized by u where u is a subdomain of either s n or r n and is a kleinian group acting discontinuously on u the examples studied here include rp n and the hopf manifolds s 1 s n 1 also some hyperbolic manifolds will be treated special kinds of clifford analytic automorphic forms associated to the different choices of are used to construct explicit cauchy kernels cauchy integral formulas green s kernels and formulas together with hardy spaces and plemelj projection operators for l p spaces of hypersurfaces lying in these manifolds

trace formula for a dielectric microdisk with a point scatterer two dimensional dielectric microcavities are of widespread use in microoptics applications recently a trace formula has been established for dielectric cavities which relates their resonance spectrum to the periodic rays inside the cavity in this paper we extend this trace formula to a dielectric disk with a small scatterer this system has been introduced for microlaser applications because it has long lived resonances with strongly directional far field we show that its resonance spectrum contains signatures not only of periodic rays but also of diffractive rays that occur in keller s geometrical theory of diffraction we compare our results with those for a closed cavity with dirichlet boundary conditions

some modular varieties of low dimension ii the paper extends results obtained by frieder hermann and eberhard freitag about a six dimensional modular variety related to the orthogonal group of signature 2 6 the ring of modular forms of this variety turns out to be a weighted polynomial ring in 7 variables

a new class of hypercomplex analytic cusp forms in this paper we deal with a new class of clifford algebra valued automorphic forms on arithmetic subgroups of the ahlfors vahlen group the forms that we consider are in the kernel of the operator d delta k 2 for some even k is an element of z they will be called k holomorphic cliffordian automorphic forms k holomorphic cliffordian functions are well equipped with many function theoretical tools furthermore the real component functions also have the property that they are solutions to the homogeneous and inhomogeneous weinstein equations this function class includes the set of k hypermonogenic functions as a special subset while we have not been able so far to propose a construction for non vanishing k hypermonogenic cusp forms for k not equal 0 we are able to do so within this larger set of functions after having explained their general relation to hyperbolic harmonic automorphic forms we turn to the construction of poincare series these provide us with non trivial examples of cusp forms within this function class then we establish a decomposition theorem of the spaces of k holomorphic cliffordian automorphic forms in terms of a direct orthogonal sum of the spaces of k hypermonogenic eisenstein series and of k holomorphic cliffordian cusp forms

trace formula for dielectric cavities general properties the construction of the trace formula for open dielectric cavities is examined in detail using the krein formula it is shown that the sum over cavity resonances can be written as a sum over classical periodic orbits for the motion inside the cavity the contribution of each periodic orbit is the product of the two factors the first is the same as in the standard trace formula and the second is connected with the product of reflection coefficients for all points of reflection with the cavity boundary two asymptotic terms of the smooth resonance counting function related with the area and the perimeter of a convex cavity are derived the coefficient of the perimeter term differs from the one for closed cavities due to unusual high energy asymptotics of the s matrix for the scattering on the cavity corrections to the leading semi classical formula are briefly discussed obtained formulas agree well with numerical calculations for circular dielectric cavities

the graded ring of quaternionic modular forms of degree 2 in the 1960 s igusa i described the graded ring of siegel modular forms of degree 2 in the sequel a few more concrete examples of graded rings of modular forms with respect to paramodular groups or hermitian modular groups were given e g by freitag f aoki and ibukiyama ai or dern and krieg dk

a pattern theoretic characterization of biological growth mathematical and statistical modeling of biological growth is an important problem in medical diagnostics here we seek tools to analyze changes in anatomical parts using images collected over time we introduce a structured model called growth by random iterated diffeomorphisms grid that treats a cumulative growth deformation as a composition of several elementary deformations each elementary deformation applies to a small region by capturing deformation local to that region and is characterized by a seed and a radial deformation pattern around that seed these grid variables seed locations and radial deformation patterns are estimated from observed images in two steps 1 estimate a cumulative deformation over an observation interval 2 estimate grid variables using maximum likelihood criterion from this estimated cumulative deformation we demonstrate this framework using an mri image data of a rat s brain growth for future statistical analysis we propose a time varying poisson process for the seed placements and a random drawing from a predetermined catalog of deformations for the radial deformation patterns

six line configurations and string dualities we study the family of k3 surfaces of picard rank sixteen associated with the double cover of the projective plane branched along the union of six lines and the family of its van geemen sarti partners i e k3 surfaces with special nikulin involutions such that quotienting by the involution and blowing up recovers the former we prove that the family of van geemen sarti partners is a four parameter family of k3 surfaces with h oplus e 7 1 oplus e 7 1 lattice polarization we describe explicit weierstrass models on both families using even modular forms on the bounded symmetric domain of type iv we also show that our construction provides a geometric interpretation called geometric two isogeny for the f theory heterotic string duality in eight dimensions

toward a comprehensive framework for the spatiotemporal statistical analysis of longitudinal shape data this paper proposes an original approach for the statistical analysis of longitudinal shape data the proposed method allows the characterization of typical growth patterns and subject specific shape changes in repeated time series observations of several subjects this can be seen as the extension of usual longitudinal statistics of scalar measurements to high dimensional shape or image data the method is based on the estimation of continuous subject specific growth trajectories and the comparison of such temporal shape changes across subjects differences between growth trajectories are decomposed into morphological deformations which account for shape changes independent of the time and time warps which account for different rates of shape changes over time given a longitudinal shape data set we estimate a mean growth scenario representative of the population and the variations of this scenario both in terms of shape changes and in terms of change in growth speed then intrinsic statistics are derived in the space of spatiotemporal deformations which characterize the typical variations in shape and in growth speed within the studied population they can be used to detect systematic developmental delays across subjects in the context of neuroscience we apply this method to analyze the differences in the growth of the hippocampus in children diagnosed with autism developmental delays and in controls result suggest that group differences may be better characterized by a different speed of maturation rather than shape differences at a given age in the context of anthropology we assess the differences in the typical growth of the endocranium between chimpanzees and bonobos we take advantage of this study to show the robustness of the method with respect to change of parameters and perturbation of the age estimates

the fake monster superalgebra abstract we show that the physical states of a 10 dimensional superstring moving on a torus form a generalized kac moody superalgebra this gives the first explicit realizations of these algebras for a special torus the denominator function of this algebra is an automorphic form so that we can determine the simple roots we call this algebra the fake monster superalgebra

point perturbations of circle billiards the spectral statistics of the circular billiard with a point scatterer is investigated in the semiclassical limit the spectrum is demonstrated to be composed of two uncorrelated level sequences the first corresponds to states for which the scatterer is located in the classically forbidden region and its energy levels are not affected by the scatterer in the semiclassical limit while the second sequence contains the levels which are affected by the point scatterer the nearest neighbour spacing distribution which results from the superposition of these sequences is calculated analytically within some approximation and good agreement with the distribution that was computed numerically is found

well posedness in sobolev spaces of the full water wave problem in 3 d we consider the motion of the interface separating an inviscid incompressible irrotational fluid from a region of zero density in three dimensional space we assume that the fluid region is below the vacuum the fluid is under the influence of gravity and the surface tension is zero assume that the density of mass of the fluid is one the gravitational field is 0 0 1 the free interface is t at time t 0 and the fluid occupies the region t the motion of the fluid is described by vt v v 0 0 1 p on t t 0 euler s equation 1 1 div v 0 on t t 0 incompressible 1 2 curl v 0 on t t 0 irrotational 1 3 where v v1 v2 v3 is the fluid velocity p is the fluid pressure since we neglect the surface tension the pressure is zero on the interface so on the interface p 0 on t 1 4 1 v is tangent to the free surface t t 1 5 we want to find solutions of system 1 1 1 5 taking prescribed initial data such that for every fixed t 0 t approaches the xy plane at infinity and v x y z t 0 vt x y z t 0 as x y z since the fluid is assumed irrotational incompressible we can reduce the study of the entire motion to the motion of the free surface the above model is a 3 d water wave model it is generally known that when surface tension is neglected the motion of the interface between an inviscid fluid and vacuum under the influence of gravity can be subject to taylor instability 8 22 in a previous work 24 we studied the 2 dimensional water wave model we showed that for a 2 d water wave the sign condition relating to taylor instability always holds for nonself intersecting interface that is the motion of the interface is not subject to taylor instability we showed further that the 2 d full nonlinear water wave problem is uniquely solvable in sobolev spaces locally in time for any initially nonself intersecting interface earlier works on the well posedness in sobolev spaces of the 2 d water wave problem include nalimov 18 yosihara 25 and walter craig 6 where the main results concern the well posedness of the

bifurcations of periodic orbits and uniform approximations we derive uniform approximations for contributions to gutzwiller s periodic orbit sum for the spectral density which are valid close to bifurcations of periodic orbits in systems with mixed phase space there orbits lie close together and give collective contributions while the individual contributions of gutzwiller s type would diverge at the bifurcation new results for the tangent the period doubling and the period tripling bifurcation are given they are obtained by going beyond the local approximation and including higher order terms in the normal form of the action the uniform approximations obtained are tested on the kicked top and are found to be in excellent agreement with exact quantum results

on the geometry of a siegel modular threefold introduction 317 1 compactifications 318 2 humbert surfaces 324 3 the chern numbers of s n 331 4 symmetric quartic threefolds 334 5 the modular threefold of level 2 335 6 kummer surfaces 338 7 prym surfaces 340 8 humbert surfaces again 345

vahlen s group of clifford matrices and spin groups on modifie legerement la construction de vahlen pour definir les groupes sv n k q pour un corps k de caracteristique 2 et une forme quadratique non degeneree generale q pour k r et q q on demontre l equivalence de la definition avec celle de vahlen on etablit un isomorphisme des groupes sv n k q spin n 3 k q

application of a trace formula to the spectra of flat three dimensional dielectric resonators the length spectra of flat three dimensional dielectric resonators of circular shape were determined from a microwave experiment they were compared to a semiclassical trace formula obtained within a two dimensional model based on the effective index of refraction approximation and a good agreement was found it was necessary to take into account the dispersion of the effective index of refraction for the two dimensional approximation furthermore small deviations between the experimental length spectrum and the trace formula prediction were attributed to the systematic error of the effective index of refraction approximation in summary the methods developed in this article enable the application of the trace formula for two dimensional dielectric resonators also to realistic flat three dimensional dielectric microcavities and lasers allowing for the interpretation of their spectra in terms of classical periodic orbits