spatiospectral concentration on a sphere we pose and solve the analogue of slepian s time frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere or alternatively of strictly spacelimited functions that are optimally concentrated in the spherical harmonic domain such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications as well as in medical imaging computer science cosmology and numerical analysis the spherical slepian functions can be found by solving either an algebraic eigenvalue problem in the spectral domain or a fredholm integral equation in the spatial domain the associated eigenvalues are a measure of the spatiospectral concentration when the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a sturm liouville operator this enables the eigenfunctions to be computed extremely accurately and efficiently even when their area bandwidth product or shannon number is large in the asymptotic limit of a small spatial region and a large spherical harmonic bandwidth the spherical concentration problem reduces to its planar equivalent which exhibits self similarity when the shannon number is kept invariant

super resolution on the sphere using convex optimization this paper considers the problem of recovering an ensemble of diracs on a sphere from its low resolution measurements the diracs can be located at any location on the sphere not necessarily on a grid we show that under a separation condition one can recover the ensemble with high precision by a three stage algorithm which consists of solving a semi definite program root finding and least square fitting the algorithm s computation time depends solely on the number of measurements and not on the required solution accuracy we also show that in the special case of non negative ensembles a sparsity condition is sufficient for recovery furthermore in the discrete setting we estimate the recovery error in the presence of noise as a function of the noise level and the super resolution factor

brownian motion on the hyperbolic plane and selberg trace formula abstract we will show that the relation of the heat kernels for the schrodinger operators with uniform magnetic fields on the hyperbolic plane h 2 the maass laplacians and for the schrodinger operators with morse potentials on r is given by means of a one dimensional fourier transform in the framework of stochastic analysis where the brownian motion on h 2 plays an important role by using this relation we will give the explicit forms of the green functions as a typical related problem we will discuss the selberg trace formula the close relation of the trace formula with the corresponding classical mechanics will also be discussed

homotopy techniques for tensor decomposition and perfect identifiability let t be a general complex tensor of format n1 nd when the fraction i ni 1 i ni 1 is an integer and a natural inequality called balancedness is satisfied it is expected that t has finitely many minimal decomposition as a sum of decomposable tensors we show how homotopy techniques allow us to find all the decompositions of t starting from a given one in particular this gives a guess true with high probability about the total number of these decompositions this guess matches exactly with all cases previously known and predicts several unknown cases some surprising experiments yielded two new cases of generic identifiability formats 3 4 5 and 2 2 2 3 which have a unique decomposition as the sum of 6 and 4 decomposable tensors respectively we conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds i e the only identifiable cases building on the computational experiments we also use basic tools from algebraic geometry to prove these two new cases are indeed generically identifiable

analyzing head related transfer function measurements using surface spherical harmonics a continuous functional representation of a large set of head related transfer function measurements hrtfs is developed the hrtfs are represented as a weighted sum of surface spherical harmonics sshs up to degree 17 a gaussian quadrature method is used to pick out a set of experimentally efficient measurement directions anechoic impulse responses are measured for these directions between a source loudspeaker and the entrance to the ear canal of a head and torso simulator hats three separate ssh analyses are carried out the first forms a ssh representation from the time responses with the variable onset delay caused by interaural differences intact by applying the analysis to each time sample in turn the second ssh model is formed in exactly the same way except using impulse responses in which the variable onset delays have been equalized the final ssh analysis is carried out in the frequency domain by applying the technique on a frequency bin by frequency bin basis to the magnitude and unwrapped phase responses of the hrtfs the accuracy and interpolation performance of each of the computed ssh models is investigated and the usefulness of the ssh technique in analyzing directional hearing and particularly in spatializing sounds is discussed

slepian spatial spectral concentration problem on the sphere analytical formulation for limited colatitude longitude spatial region in this paper we develop an analytical formulation for the slepian spatial spectral concentration problem on the sphere for a limited colatitude longitude spatial region on the sphere defined as the cartesian product of a range of positive colatitudes and longitudes the solution of the slepian problem is a set of functions that are optimally concentrated and orthogonal within a spatial or spectral region these properties make them useful for applications where measurements are taken within a spatially limited region of the sphere and or a signal is only to be analyzed within a region of the sphere to support localized spectral spatial analysis and estimation and sparse representation of localized data in these applications we exploit the expansion of spherical harmonics in the complex exponential basis to develop an analytical formulation for the slepian concentration problem for a limited colatitude longitude spatial region we also extend the analytical formulation for spatial regions that are comprised of a union of rotated limited colatitude longitude subregions by exploiting various symmetries of the proposed formulation we design a computationally efficient algorithm for the implementation of the proposed analytical formulation such a reduction in computation time is demonstrated through numerical experiments we present illustrations of our results with the help of numerical examples and show that the representation of a spatially concentrated signal is indeed sparse in the slepian basis

analyse quasi sure et l estimation du noyau de la chaleur pour temps petit the ito functionals can be redefined out of a slim set by the natural way quasi sure analysis is used to deal with the heat kernel asymptotic problems

local equations for equivariant evolutionary models abstract phylogenetic varieties related to equivariant substitution models have been studied largely in the last years one of the main objectives has been finding a set of generators of the ideal of these varieties but this has not yet been achieved in some cases for example for the general markov model this involves the open salmon conjecture see 2 and it is not clear how to use all generators in practice motivated by applications in biology we tackle the problem from another point of view the elements of the ideal that could be useful for applications in phylogenetics only need to describe the variety around certain points of no evolution see 13 we produce a collection of explicit equations that describe the variety on a zariski open neighborhood of these points see theorem 5 4 namely for any tree t on any number of leaves and any degrees at the interior nodes and for any equivariant model on any set of states we compute the codimension of the corresponding phylogenetic variety we prove that this variety is smooth at general points of no evolution and if a mild technical condition is satisfied d claw tree hypothesis we provide an algorithm to produce a complete intersection that describes the variety around these points

analysis of wiener functionals malliavin calculus and its applications to heat kernels on donne un expose facilement accessible du calcul de malliavin du calcul differentiel de dimension infinie sur l espace de wiener

stokes formula on the wiener space and n dimensional nourdin peccati analysis extensions of the nourdin peccati analysis to rn valued random variables are obtained by taking conditional expectation on the wiener space several proof techniques are explored from infinitesimal geometry to quasi sure analysis including a connection to stein s lemma to classical analysis on wiener space partial differential equations for the density of an rn valued centered random variable z z1 zn are obtained of particular importance is the function defined by the conditional expectation given z of the auxiliary random matrix dl 1zi dzj i j 1 2 n where d and l are respectively the derivative operator and the generator of the ornstein uhlenbeck semigroup on wiener space

closed form expansion conditional expectation and option valuation enlightened by the theory of watanabe watanabe s 1987 analysis of wiener functionals malliavin calculus and its applications to heat kernels ann probab 15 1 39 for analyzing generalized random variables and its further development in yoshida yoshida n 1992a asymptotic expansions for statistics related to small diffusions j japan statist soc 22 139 159 takahashi takahashi a 1995 essays on the valuation problems of contingent claims ph d thesis haas school of business university of california berkeley takahashi a 1999 an asymptotic expansion approach to pricing contingent claims asia pacific financial markets 6 115 151 as well as kunitomo and takahashi kunitomo n takahashi a 2001 the asymptotic expansion approach to the valuation of interest rate contingent claims math finance 11 1 117 151 kunitomo n takahashi a 2003 on validity of the asymptotic expansion approach in contingent claim analysis ann appl probab 13 3 914 952 etc we focus on a wide range of mult

phylogenetic complexity of the kimura 3 parameter model abstract in algebraic statistics the kimura 3 parameter model is one of the most interesting and classical phylogenetic models we prove that the ideals associated to this model are generated in degree four confirming a conjecture by sturmfels and sullivant

small time fluctuations for sub riemannian diffusion loops we study the small time fluctuations for diffusion processes which are conditioned by their initial and final positions under the assumptions that the diffusivity has a sub riemannian structure and that the drift vector field lies in the span of the sub riemannian structure in the case where the endpoints agree and the generator of the diffusion process is non elliptic at that point the deterministic malliavin covariance matrix is always degenerate we identify after a suitable rescaling another limiting malliavin covariance matrix which is non degenerate and we show that with the same scaling the diffusion malliavin covariance matrices are uniformly non degenerate we further show that the suitably rescaled fluctuations of the diffusion loop converge to a limiting diffusion loop which is equal in law to the loop we obtain by taking the limiting process of the unconditioned rescaled diffusion processes and condition it to return to its starting point the generator of the unconditioned limiting rescaled diffusion process can be described in terms of the original generator

chaotic field theory a sketch abstract spatio temporally chaotic dynamics of a classical field can be described by means of an infinite hierarchy of its unstable spatio temporally periodic solutions the periodic orbit theory yields the global averages characterizing the chaotic dynamics as well as the starting semiclassical approximation to the quantum theory new methods for computing corrections to the semiclassical approximation are developed in particular a nonlinear field transformation yields the perturbative corrections in a form more compact than the feynman diagram expansions

estimation de varadhan pour des diffusions a deux parametres sous des conditions de hormander generalisees la loi d une diffusion a parametre planaire possede une densite lisse en dehors des axes dans le cas elliptique nous etablissons une estimation de cette densite pur des petites valeurs du parametre celle ci constitue l analogue de l estimation de varadhan objet qui est bien connu dans le cas des diffusions ordinaires

a new delta expansion for multivariate diffusions via the ito taylor expansion abstract in this paper we develop a new delta expansion approach to deriving analytical approximation to the transition densities of multivariate diffusions using the ito taylor expansion of the conditional expectation of the dirac delta function our approach yields an explicit recursive formulas for the expansion coefficients and is universally applicable for a wide spectrum of models particularly the time inhomogeneous non affine irreducible multivariate diffusions we show that this new approach can be viewed as an extension of ait sahalia 2002 and lee et al 2014 to the case of multivariate models the derived expansions are proved to converge to the true probability density as the observational time interval shrinks the obtained approximations can thereby be used to carry out the maximum likelihood estimation for the diffusions with discretely observed data extensive numerical experiments demonstrate the accuracy and effectiveness of our approach

on the exponential decay of oscillatory integrals on an abstract wiener space abstract let x h be a real abstract wiener space the saddle point method is carried out on x and an exact formula for deformation of contour integration that is a formula transferring a fourier type integral to a fourier laplace one will be concluded as an application an exponentially fast decay of stochastic oscillatory integrals on x is studied

on tensors of border rank l in cm n l notice this is the author s version of a work that was accepted for publication in linear algebra and its applications changes resulting from the publishing process such as peer review editing corrections structural formatting and other quality control mechanisms may not be reflected in this document changes may have been made to this work since it was submitted for publication a definitive version was subsequently published in linear algebra and its applications june 23 2011 doi 10 1016 j laa 2011 05 013 the original publication is available at www elsevier com

a remark on the asymptotic expansion of density function of wiener functionals abstract we consider the asymptotic expansion of density function of wiener functionals as time tends to zero as in s kusuoka d w stroock precise asymptotics of certain wiener functionals j funct anal 99 1991 1 74 and give an explicit formula for the first coefficient

asymptotic expansion of bayes estimators for small diffusions using the malliavin calculus we derived asymptotic expansion of the distributions of the bayes estimators for small diffusions the second order efficiency of the bayes estimator is proved

a stochastic approach to the poincare hopf theorem in this paper the poincare hopf theorem hopf 8 is proved by a probabilistic method the proof of this theorem is reduced to an estimate of a supertrace of the fundamental solution we use the malliavian calculus to estimate the fundametal solution our proof can be applied to the case that the zero point set is the union of the submanifolds under some conditions

the discrete prolate spheroidal filter as a digital signal processing tool the discrete prolate spheroidal dps filter is one of the class of nonrecursive finite impulse response fir filters the dps filter first introduced by tufts and francis 1 is superior to other filters in this class in that it has maximum energy concentration in the frequency passband and minimum ringing in the time domain slepian 2 gives a complete discussion of dps function properties we give a mathematical development of the dps filter properties provide information required to easily and accurately construct even many coefficient filters and compare the properties of this filter with those of the more commonly used filters of the same class we note that use of the dps filter allows for particularly useful statements of data time frequency resolution cell values and that overall it forms an especially useful though little known tool for digital signal processing

perturbative analysis of anharmonic chains of oscillators out of equilibrium we compute the first order correction to the correlation functions of the stationary state of a stochastically forced harmonic chain out of equilibrium when a small on site anharmonic potential is added this is achieved by deriving a suitable formula for the covariance matrix of the invariant state we find that the first order correction of the heat current does not depend on the size of the system second the temperature profile is linear when the harmonic part of the on site potential is zero the sign of the gradient of the profile however is opposite to the sign of the temperature difference of the two heat baths

ranks of tensors secant varieties of segre varieties and fat points abstract a classical unsolved problem of projective geometry is that of finding the dimensions of all the higher secant varieties of the segre embeddings of an arbitrary product of projective spaces an important subsidiary problem is that of finding the smallest integer t for which the secant variety of projective t spaces fills the ambient projective space in this paper we give a new approach to these problems the crux of our method is the translation of a well known lemma of terracini into a question concerning the hilbert function of fat points in a multiprojective space our approach gives much new information on the classical problem even in the case of three factors a case also studied in the area of algebraic complexity theory

asymptotic behavior of divergences and cameron martin theorem on loop spaces we first prove the lp convergence p 1 and a fernique type exponential integrability of divergence functionals for all cameron martin vector fields with respect to the pinned wiener measure on loop spaces over a compact riemannian manifold we then prove that the driver flow is a smooth transform on path spaces in the sense of the malliavin calculus and has an quasi continuous modification which can be quasi surely well defined on path spaces this leads us to construct the driver flow on loop spaces through the corresponding flow on path spaces combining these two results with the cruzeiro lemma j funct anal 54 1983 206 227 we give an alternative proof of the quasi invariance of the pinned wiener measure under driver s flow on loop spaces which was established earlier by driver trans amer math soc 342 1994 375 394 and enchev and stroock adv math 119 1996 127 154 by doob s h processes approach together with the short time estimates of the gradient and the hessian of the logarithmic heat kernel on compact riemannian manifolds we also establish the lp convergence p 1 and a fernique type exponential integrability theorem for the stochastic anti development of pinned brownian motions on compact riemannian manifold with an explicit exponential exponent our results generalize and sharpen some earlier results due to gross j funct anal 102 1991 268 313 and hsu math ann 309 1997 331 339 our method does not need any heat kernel estimate and is based on quasi sure analysis and sobolev estimates on path spaces

quadratic variational framework for signal design on the 2 sphere this paper introduces a quadratic variational framework for solving a broad class of signal design problems on the 2 sphere the functional to be extremized combines energy concentration measures using a weighting function in the spatial domain multiplicative weights in the spectral domain and a total energy constraint this leads to two formulations of the signal design problem on the 2 sphere one a fredholm integral equation in the spatial domain and the other an infinite matrix equation in the spectral domain the framework is illustrated by deriving the key equations for the two classical spatio spectral concentration problems on the 2 sphere and for an isotropic filter design that maximizes the filtered energy in addition using the proposed framework we formulate a joint 3 d beamforming application which achieves optimal directivity and spatial resolution simultaneously

functional ito calculus and stochastic integral representation of martingales we develop a non anticipative calculus for functionals of a continuous semimartingale using an extension of the ito formula to path dependent functionals which possess certain directional derivatives the construction is based on a pathwise derivative introduced by b dupire for functionals on the space of right continuous functions with left limits we show that this functional derivative admits a suitable extension to the space of square integrable martingales this extension denes a weak derivative which is shown to be the inverse of the ito integral and which may be viewed as a non anticipative lifting of the malliavin derivative these results lead to a constructive martingale representation formula for ito processes by contrast with the clark haussmann ocone formula this representation only involves nonanticipative quantities which may be computed pathwise

decroissance non exponentielle du noyau de la chaleur in this paper we give estimates on the heat kernel on the diagonal in terms of exp logt 1 when1 vanishes moreover we compute by means of the stochastic calculus of variations the decay of the flatness of some heat kernels on the other hand we show that the computations in both cases are very similar

a proof of the set theoretic version of the salmon conjecture abstract we show that the irreducible variety of 4 4 4 complex valued tensors of border rank at most 4 is the zero set of polynomial equations of degree 5 the strassen commutative conditions of degree 6 the landsberg manivel polynomials and of degree 9 the symmetrization conditions

efficient computation of slepian functions for arbitrary regions on the sphere in this paper we develop a new method for the fast and memory efficient computation of slepian functions on the sphere slepian functions which arise as the solution of the slepian concentration problem on the sphere have desirable properties for applications where measurements are only available within a spatially limited region on the sphere and or a function is required to be analyzed over the spatially limited region slepian functions are currently not easily computed for large band limits for an arbitrary spatial region due to high computational and large memory storage requirements for the special case of a polar cap the symmetry of the region enables the decomposition of the slepian concentration problem into smaller subproblems and consequently the efficient computation of slepian functions for large band limits by exploiting the efficient computation of slepian functions for the polar cap region on the sphere we develop a formulation supported by a fast algorithm for the approximate computation of slepian functions for an arbitrary spatial region to enable the analysis of modern datasets that support large band limits for the proposed algorithm we carry out accuracy analysis of the approximation computational complexity analysis and review of memory storage requirements we illustrate through numerical experiments that the proposed method enables faster computation and has smaller storage requirements while allowing for sufficiently accurate computation of the slepian functions

differential and weighted slepian concentration problems on the sphere in this work we present a generalized formulation of the slepian concentration problem on the sphere for finding band limited functions with an optimal concentration in the spatial domain by introducing weighting functions in the formulation of classical slepian concentration problem and assigning different values to these weighting functions we present two variants of the concentration problem namely the differential and the weighted slepian concentration problem in the differential slepian concentration problem we consider two regions on the sphere and find band limited functions such that the energy is maximized in one region at the expense of the energy in the other region we propose non negative weighting using a spatial window function to formulate and solve the weighted slepian concentration problem each problem can be solved by formulating it in the harmonic domain as an eigenvalue problem the solution of which yields eigenfunctions that serve as alternative basis functions for the representation of band limited signals and are referred to as slepian functions we also present and analyse the properties of the slepian functions to support the applications in acoustics and cosmology we also provide a demonstration for the use of the proposed slepian functions for the robust signal modeling and the estimation of the energy spectrum of red and white stochastic processes on the sphere

bounded rank tensors are defined in bounded degree matrices of rank at most k are defined by the vanishing of polynomials of degree k 1 in their entries namely their k 1 k 1 subdeterminants regardless of the size of the matrix we prove a qualitative analogue of this statement for tensors of arbitrary dimension where matrices correspond to two dimensional tensors more specifically we prove that for each k there exists an upper bound d d k such that tensors of border rank at most k are defined by the vanishing of polynomials of degree at most d regardless of the dimension of the tensor and regardless of its size in each dimension our proof involves passing to an infinite dimensional limit of tensor powers of a vector space whose elements we dub infinite dimensional tensors and exploiting the symmetries of this limit in crucial ways

localized analysis of signals on the sphere over polygon regions to support localized signal analysis on the sphere for applications in geophysics cosmology acoustics and beyond we develop a framework for the analytic evaluation of the integral of spherical signals and for the analytic solution of the slepian spatial spectral concentration problem over simple spherical polygons we propose a polygon right angle triangulation method for the division of a simple spherical polygon into spherical right angle triangles which allows us to decompose the problem of integrating signals or solving the spatial spectral concentration problem over the polygon region into sub problems that require the evaluation of the integral of complex exponential functions over spherical right angle triangles of arbitrary orientation and position we derive closed form expressions for the evaluation of such integrals by appropriately choosing the rotations and using wigner d functions the proposed framework enables us to solve the slepian spatial spectral concentration problem for simple spherical polygons resulting in bandlimited spatially optimally concentrated basis functions for signal representation and reconstruction localized analysis and signal modeling on the sphere we also present convergence criterion for the infinite series expansions involved in the evaluation of the integral of complex exponential functions and establish the validity of the proposed developments by evaluating the integral and computing the slepian basis functions over the geographical region of australia and the volcanic plateau of tharsis using the earth and mars topography maps respectively