strings mathematical theory and statistical examples strings in the sense of the present paper are sequences of multiarrays with two types of indices tensorial and structural and they are characterized by a transformation law that generalizes those for tensors affine connections and derivatives of scalars the original definition barndorff nielsen proc r soc lond a 406 127 137 1986 is here extended and a systematic study of the mathematics of strings is undertaken in particular a convolutive multiplication of strings is introduced and is used in the discussion of intertwining a type of operation that produces tensors from strings and strings from tensors and connection strings a special kind of string it appears that tensors and connection strings have a role respectively as coordinates and coordinate frames in the calculus of strings a definition of covariant differentiation of strings is proposed and is related to convolutive multiplication and to intertwining the general theory is illustrated by various examples from the context of statistical inference finally a brief comparison is made between strings and the somewhat related concept of extensors

entropy dissipation and long range interactions we study boltzmann s collision operator for long range interactions i e without grad s angular cut off assumption we establish a functional inequality showing that the entropy dissipation controls smoothness of the distribution function in a precise sense our estimate is optimal and gives a unified treatment of both the linear and the nonlinear cases we also give simple and self contained proofs of several useful results that were scattered in previous works as an application we obtain several helpful estimates for the cauchy problem and for the landau approximation in plasma physics

mathematics of granular materials this is a short and somewhat informal review on the most mathematical parts of the kinetic theory of granular media intended for physicists and for mathematicians outside the field

a bayesian approach to geometric subspace estimation this paper presents a geometric approach to estimating subspaces as elements of the complex grassmann manifold with each subspace represented by its unique complex projection matrix variation between the subspaces is modeled by rotating their projection matrices via the action of unitary matrices elements of the unitary group u n subspace estimation or tracking then corresponds to inferences on u n taking a bayesian approach a posterior density is derived on u n and certain expectations under this posterior are empirically generated for the choice of the hilbert schmidt norm on u n to define estimation errors an optimal mmse estimator is derived it is shown that this estimator achieves a lower bound on the expected squared errors associated with all possible estimators the estimator and the bound are computed using metropolis adjusted langevin s diffusion algorithm for sampling from the posterior for use in subspace tracking a prior model on subspace rotation that utilizes newtonian dynamics is suggested

the boltzmann grad limit of a hard sphere system analysis of the correlation error we present a quantitative analysis of the boltzmann grad low density limit of a hard sphere system we introduce and study a set of functions the correlation errors measuring the deviations in time from the statistical independence of particles propagation of chaos in the context of the bbgky hierarchy a correlation error of order k measures the event where k particles are connected by a chain of interactions preventing the factorization we show that provided k varepsilon alpha such an error flows to zero with the average density varepsilon for short times as varepsilon gamma k for some positive alpha gamma in 0 1 this provides an information on the size of chaos namely j different particles behave as dictated by the boltzmann equation even when j diverges as a negative power of varepsilon the result requires a rearrangement of lanford perturbative series into a cumulant type expansion and an analysis of many recollision events

generalized selection combining for cognitive relay networks over nakagami m fading we consider transmit antenna selection with receive generalized selection combining tas gsc for cognitive decode and forward df relaying in nakagami m fading channels in an effort to assess the performance the probability density function and the cumulative distribution function of the end to end snr are derived using the moment generating function from which new exact closed form expressions for the outage probability and the symbol error rate are derived we then derive a new closed form expression for the ergodic capacity more important by deriving the asymptotic expressions for the outage probability and the symbol error rate as well as the high snr approximations of the ergodic capacity we establish new design insights under the two distinct constraint scenarios 1 proportional interference power constraint and 2 fixed interference power constraint several pivotal conclusions are reached for the first scenario the full diversity order of the outage probability and the symbol error rate is achieved and the high snr slope of the ergodic capacity is 1 2 for the second scenario the diversity order of the outage probability and the symbol error rate is zero with error floors and the high snr slope of the ergodic capacity is zero with capacity ceiling

characterizing the statistical properties of mutual information in mimo channels we consider gaussian multiple input multiple output mimo frequency selective spatially correlated fading channels assuming that the channel is unknown at the transmitter and perfectly known at the receiver for gaussian codebooks using results from multivariate statistics we derive an analytical expression for a tight lower bound on the ergodic capacity of such channels at any signal to noise ratio snr we show that our bound is tighter than previously reported analytical lower bounds and we proceed to analytically quantify the impact of spatial fading correlation on ergodic capacity based on a closed form approximation of the variance of mutual information in correlated flat fading mimo channels we provide insights into the multiplexing diversity tradeoff for gaussian code books furthermore for a given total number of antennas we consider the problem of finding the optimal ergodic capacity maximizing number of transmit and receive antennas and we reveal the snr dependent nature of the maximization strategy finally we present numerical results and comparisons between our capacity bounds and previously reported bounds

a maxwellian lower bound for solutions to the boltzmann equation we prove that the solution of the spatially homogeneous boltzmann equation is bounded pointwise from below by a maxwellian i e a function of the formc1 exp c2v2 this holds for any initial data with bounded mass energy and entropy and for any positive timet t0 the constantsc1 andc2 depend on the mass energy and entropy of the initial data and ont0 0 only

rate of convergence to equilibrium for the spatially homogeneous boltzmann equation with hard potentials for the spatially homogeneous boltzmann equation with hard potentials and grad s cutoff e g hard spheres we give quantitative estimates of exponential convergence to equilibrium and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation on which we provide a lower bound our approach is based on establishing spectral gap like estimates valid near the equilibrium and then connecting the latter to the quantitative nonlinear theory this leads us to an explicit study of the linearized boltzmann collision operator in functional spaces larger than the usual linearization setting

intrinsic analysis of statistical estimation the parametric statistical models with suitable regularity conditions have a natural riemannian manifold structure given by the information metric since the parameters are merely labels for the probability measures an inferential statement should be formulated through intrinsic objects invariant under reparametrizations in this context the estimators will be random objects valued on the manifold corresponding to the statistical model in spite of these considerations classical measures of an estimator s performance like the bias and the mean square error are clearly dependent on the statistical model parametrizations in this paper the authors work with extended notions of mean value and moments of random objects which take values on a hausdorff and connected manifold equipped with an affine connection in particular the riemannian manifold case is considered this extension is applied to the bias and the mean square error study in statistical point estimation theory under this approach an intrinsic version of the cramer rao lower bound is obtained a lower bound which depends on the intrinsic bias and the curvature of the statistical model for the mean square of the rao distance the invariant measure analogous to the mean square error further the behavior of the mean square of the rao distance of an estimator when conditioning with respect to a sufficient statistic is considered obtaining intrinsic versions of the rao blackwell and lehmann scheffe theorems asymptotic properties complete the study

large sample theory of intrinsic and extrinsic sample means on manifolds ii this article develops nonparametric inference procedures for estimation and testing problems for means on manifolds a central limit theorem for frechet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on riemannian manifolds central limit theorems are also obtained for extrinsic sample means w r t an arbitrary embedding of a differentiable manifold in a euclidean space bootstrap methods particularly suitable for these problems are presented applications are given to distributions on the sphere s d directional spaces real projective space rp n 1 axial spaces complex projective space cp k 2 planar shape spaces w r t veronese whitney embeddings and a three dimensional shape space 4 3

the mahalanobis distance and elliptic distributions summary the mahalanobis distance is shown to be an appropriate measure of distance between two elliptic distributions having different locations but a common shape this extends a result long familiar in multivariate analysis to a class of nonnormal distributions it can also be used to show that the sample version of the mahalanobis distance is appropriate under both estimative and predictive approaches to estimation for the family of multivariate normal distributions differing only in location

covariance subspace and intrinsic crame spl acute r rao bounds crame spl acute r rao bounds on estimation accuracy are established for estimation problems on arbitrary manifolds in which no set of intrinsic coordinates exists the frequently encountered examples of estimating either an unknown subspace or a covariance matrix are examined in detail the set of subspaces called the grassmann manifold and the set of covariance positive definite hermitian matrices have no fixed coordinate system associated with them and do not possess a vector space structure both of which are required for deriving classical crame spl acute r rao bounds intrinsic versions of the crame spl acute r rao bound on manifolds utilizing an arbitrary affine connection with arbitrary geodesics are derived for both biased and unbiased estimators in the example of covariance matrix estimation closed form expressions for both the intrinsic and flat bounds are derived and compared with the root mean square error rmse of the sample covariance matrix scm estimator for varying sample support k the accuracy bound on unbiased covariance matrix estimators is shown to be about 10 log 10 n k sup 1 2 db where n is the matrix order remarkably it is shown that from an intrinsic perspective the scm is a biased and inefficient estimator and that the bias term reveals the dependency of estimation accuracy on sample support observed in theory and practice the rmse of the standard method of estimating subspaces using the singular value decomposition svd is compared with the intrinsic subspace crame spl acute r rao bound derived in closed form by varying both the signal to noise ratio snr of the unknown p dimensional subspace and the sample support in the simplest case the crame spl acute r rao bound on subspace estimation accuracy is shown to be about p n p sup 1 2 1 2 snr sup 1 2 rad for p dimensional subspaces it is seen that the svd based method yields accuracies very close to the crame spl acute r rao bound establishing that the principal invariant subspace of a random sample provides an excellent estimator of an unknown subspace the analysis approach developed is directly applicable to many other estimation problems on manifolds encountered in signal processing and elsewhere such as estimating rotation matrices in computer vision and estimating subspace basis vectors in blind source separation

zf detectors over correlated k fading mimo channels this paper provides a systematic characterization of zero forcing zf detectors over multiple input multiple output mimo channels that experience both small and large scale fading in particular we consider the generic k distribution rayleigh gamma distribution to model the composite fading fluctuations and also assume the general case of semi correlated small scale fading in the following novel exact analytical expressions for the achievable sum rate are derived followed by asymptotic expressions in the high and low signal to noise ratio snr regimes in these limiting cases two common and insightful affine expansions are studied followed by new closed form upper and lower bounds on the sum rate that remain tight for all snrs in the second part of the paper we present exact tractable expressions along with first order expansions for the symbol error rate ser and outage probability we also quantify the performance of zf detectors in terms of diversity order and array or coding gain the implications of the model parameters on the zf detector performance are investigated via monte carlo simulations which also validate the theoretical analysis

cercignani s conjecture is sometimes true and always almost true we establish several new functional inequalities comparing boltzmann s entropy production functional with the relative h functional first we prove a longstanding conjecture by cercignani under the nonphysical assumption that the boltzmann collision kernel is superquadratic at infinity the proof rests on the method introduced in 39 combined with a novel use of the blachman stam inequality if the superquadraticity assumption is not satisfied then it is known that cercignani s conjecture is not true however we establish a slightly weakened version of it for all physically relevant collision kernels thus extending previous results from 39 finally we consider the entropy entropy production version of kac s spectral gap problem and obtain estimates about the dependence of the constants with respect to the dimension the first two results are sharp in some sense and the third one is likely to be too they contain all previously known entropy estimates as particular cases this gives a first coherent picture of the study of entropy production according to a program started by carlen and carvalho 12 ten years ago these entropy inequalities are one step in our study of the trend to equilibrium for the boltzmann equation in both its spatially homogeneous and spatially inhomogeneous versions

l estimates for the space homogeneous boltzmann equation this paper studies the boundedness of solutionsf of the initial value problem for the space homogeneous boltzmann equation for inverse kth power forces whenk 5 and under angular cutoff the main result is that if the initial value isf0 0 with 1 2 0 el1 and 1 sf0e l for somes 2 then 1 s ftel fort 0 and ess tsup 1 s f t 5

a weak formulation of the boltzmann equation based on the fourier transform in this article we present an alternative formulation of the spatially homogeneous boltzmann equation rewriting the weak form of the equation with shifted test functions and using fourier techniques it turns out that the transformed problem contains only a three fold integral

analysis of spectral methods for the homogeneous boltzmann equation the development of accurate and fast algorithms for the boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis recently several works were devoted to the derivation of spectrally accurate schemes for the boltzmann equation but very few of them were concerned with the stability analysis of the method in particular there was no result of stability except when the method is modified in order to enforce the posivity preservation which destroys the spectral accuracy in this paper we propose a new method to study the stability of homogeneous boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution this method takes advantage of the spreading property of the collision together with estimates on regularity and entropy production as an application we prove stability and convergence of spectral methods for the boltzmann equation when the discretization parameter is large enough with explicit bound

uniqueness of the self similar profile for a kinetic annihilation model abstract we prove the uniqueness of the self similar profile solution for a modified boltzmann equation describing probabilistic ballistic annihilation such a model describes a system of hard spheres such that whenever two particles meet they either annihilate with probability 0 1 or they undergo an elastic collision with probability 1 the existence of a self similar profile for smaller than an explicit threshold value 1 has been obtained in our previous contribution 6 we complement here our analysis of such a model by showing that for some explicit the self similar profile is unique for 0

sample mean versus sample frechet mean for combining complex wishart matrices a statistical study the space of covariance matrices is a non euclidean space the matrices form a manifold which if equipped with a riemannian metric becomes a riemannian manifold and recently this idea has been used for comparison and clustering of complex valued spectral matrices which at a given frequency are typically modeled as complex wishart distributed random matrices identically distributed sample complex wishart matrices can be combined via a standard sample mean to derive a more stable overall estimator however using the riemannian geometry their so called sample frechet mean can also be found we derive the expected value of the determinant of the sample frechet mean and the expected value of the sample frechet mean itself the population frechet mean is shown to be a scaled version of the true covariance matrix the risk under convex loss functions for the standard sample mean is never larger than for the frechet mean in simulations the sample mean also performs better for the estimation of an important functional derived from the estimated covariance matrix namely partial coherence

fast numerical method for the boltzmann equation on non uniform grids we introduce a new fast numerical method for computing discontinuous solutions to the boltzmann equation and illustrate it by numerical examples a combination of adaptive grids for approximation of the distribution function and an approximate fast fourier transform on non uniform grids for computing smooth terms in the boltzmann collision integral is used

boltzmann model for viscoelastic particles asymptotic behavior pointwise lower bounds and regularity we investigate the long time behavior of a system of viscoelastic particles modeled with the homogeneous boltzmann equation we prove the existence of a universal maxwellian intermediate asymptotic state with explicit rate of convergence towards it exponential lower pointwise bounds and propagation of regularity are also studied these results can be seen as a generalization of several classical results holding for the pseudo maxwellian and constant normal restitution models

information geometry of q gaussian densities and behaviors of solutions to related diffusion equations this paper presents new geometric aspects of the behaviors of solutions to the porous medium equation pme and its associated equation first we discuss thermostatistical structure with information geometry on a manifold of generalized exponential densities a dualistic relation between the two existing formalisms is elucidated next by equipping the manifold of q gaussian densities with such a structure we derive several physically and geometrically interesting properties of the solutions the manifold is proved invariant and attracting for the evolving solutions which play crucial roles in our analysis we demonstrate that the moment conserving projection of a solution coincides with a geodesic curve on the manifold further the evolutional velocities of the second moments and the convergence rate to the manifold are evaluated in terms of the bregman divergence finally we show that the self similar solution is geometrically special in the sense that it is simultaneously geodesic with respect to the mutually dual two affine connections

characterizing volume and surface deformations in an atlas framework theory applications and implementation given deformations for mapping images or surfaces into an atlas configuration methods are described for characterizing the mean deformation and deviations from this mean jacobian matrices are used to characterize the deformations locally and the method can be applied to any image warping method for which jacobian matrices can be computed the method makes use of the fact that each matrix descriptor of the local deformation required to match an image to the atlas corresponds to a point on a semi riemannian manifold by assuring that the mean matrix lies within this manifold fundamental geometric properties common to all of the images can be preserved local deviations from the mean can be characterized in a euclidean space tangent to the semi riemannian manifold at the mean and can be accumulated globally across multiple sampling locations within the atlas to generate a global multivariate characterization of how each image deviates from the mean

on the trend to global equilibrium for spatially inhomogeneous kinetic systems the boltzmann equation as part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations started in 21 we derive estimates on the rate of convergence to equilibrium for solutions of the boltzmann equation like o t our results hold conditionally to some strong but natural estimates of smoothness decay at large velocities and strict positivity which at the moment have only been established in certain particular cases among the most important steps in our proof are 1 quantitative variants of boltzmann s h theorem as proven in 52 60 based on symmetry features hypercontractivity and information theoretical tools 2 a new quantitative version of the instability of the hydrodynamic description for non small knudsen number 3 some functional inequalities with geometrical content in particular the korn type inequality which we established in 22 and 4 the study of a system of coupled differential inequalities of second order by a treatment inspired from 21 we also briefly point out the particular role of conformal velocity fields when they are allowed by the geometry of the problem

stability convergence to self similarity and elastic limit for the boltzmann equation for inelastic hard spheres we consider the spatially homogeneous boltzmann equation for inelastic hard spheres in the framework of so called constant normal restitution coefficients alpha in 0 1 in the physical regime of a small inelasticity that is alpha in alpha 1 for some constructive alpha in 0 1 we prove uniqueness of the self similar profile for given values of the restitution coefficient alpha in alpha 1 the mass and the momentum therefore we deduce the uniqueness of the self similar solution up to a time translation

on measure solutions of the boltzmann equation part ii rate of convergence to equilibrium abstract the paper considers the convergence to equilibrium for measure solutions of the spatially homogeneous boltzmann equation for hard potentials with angular cutoff we prove the exponential sharp rate of strong convergence to equilibrium for conservative measure solutions having finite mass and energy the proof is based on the regularizing property of the iterated collision operators exponential moment production estimates and some previous results on the exponential rate of strong convergence to equilibrium for square integrable initial data we also obtain a lower bound of the convergence rate and deduce that no eternal solutions exist apart from the trivial stationary solutions given by the maxwellian equilibrium the constants in these convergence rates depend only on the collision kernel and conserved quantities mass momentum and energy we finally use these convergence rates in order to deduce global in time strong stability of measure solutions

cooling process for inelastic boltzmann equations for hard spheres part i the cauchy problem we develop the cauchy theory of the spatially homogeneous inelastic boltzmann equation for hard spheres for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity as well as the sticky particles model we prove local in time non concentration estimates in orlicz spaces from which we deduce weak stability and existence theorem strong stability together with uniqueness and instantaneous appearance of exponential moments are proved under additional smoothness assumption on the initial datum for a restricted class of collision rates concerning the long time behaviour we give conditions for the cooling process to occur or not in finite time

optimal local well posedness theory for the kinetic wave equation abstract we prove local existence and uniqueness results for the space homogeneous 4 wave kinetic equation in wave turbulence theory we consider collision operators defined by radial but general dispersion relations satisfying suitable bounds and we prove two local well posedness theorems in nearly critical weighted spaces

on the global existence of mild solutions to the boltzmann equation for small data in ld we develop a new theory of existence of global solutions to the boltzmann equation for small initial data these new mild solutions are analogous to the mild solutions for the navier stokes equations the existence comes as a result of the study of the competing phenomena of dispersion due to the transport operator and of singularity formation due to the nonlinear boltzmann collision operator it is the joint use of the so called dispersive estimates with new convolution inequalities on the gain term of the collision operator that allows to obtain uniform bounds on the solutions and thus demonstrate the existence of solutions

regularization properties of the 2d homogeneous boltzmann equation without cutoff we consider the 2 dimensional spatially homogeneous boltzmann equation for hard potentials we assume that the initial condition is a probability measure that has some exponential moments and is not a dirac mass we prove some regularization properties for a class of very hard potentials the solution instantaneously belongs to hr for some r in 1 2 depending on the parameters of the equation our proof relies on the use of a well suited malliavin calculus for jump processes

on the mimo channel capacity for the nakagami m channel this paper presents the multiple input multiple output mimo channel capacity over the nakagami m fading channel the joint eigenvalue density function of w hh where h is the channel matrix is derived in closed form for h 2 times 2 and any integer values of m as well as for h 2 times 3 with to 2 and to 3 the marginal eigenvalue distribution of w is also derived in a closed form expression all the results are validated by numerical monte carlo simulations and are in excellent agreement

on the capacity of generalized k fading mimo channels this correspondence explores the ergodic capacity of multiple input multiple output mimo systems operating in generalized k fading conditions using some recent results on majorization theory we derive an analytical capacity bound which is applicable for arbitrary values of the signal to noise ratio snr and number of antenna elements in addition we deduce simple bound approximations in the high snr regime and demonstrate that the effects of small and large scale fading are decoupled a similar statistical analysis is carried out for mimo channels under k fading which represents a special case of generalized k fading that can be tackled via the wishart matrix theory the implications of the model parameters on the bound performance are also investigated via monte carlo simulations

a model for perturbed production or measurement processes involving compound normal distributions to solve an easy to understand real world problem we invite the reader to travel through many useful concepts related to the important models provided by the compound normal distributions this class of distributions is a subclass of the multivariate elliptical distributions that allows a clearcut interpretation of a data generating mechanism useful in certain situations a multivariate elliptical distribution is characterized by the existence of a particular function f in the special case of a compound normal distribution the function f appears to be simply the one sided laplace stieltjes transform lst of the mixing distribution this fact allows in particular to easily express the covariance matrix and the kurtosis parameter of the compound normal in terms of derivatives evaluated at the origin of the lst of the mixing distribution an example of application of the results is the asymptotic theory for canonical correlation analysis the asymptotic distributions of the sample canonical correlation coefficients and of statistics used for testing hypotheses about the population coefficients have very simple forms in the case of compound normal distributions they depend on the lst of the mixing distribution through the kurtosis parameter here we focus our attention on the inference on the simple correlation coefficient r of the components of bivariate compound normal distributions we illustrate with a simple example in industrial engineering how inference on r is affected by the choice of the associated mixing distribution

monte carlo extrinsic estimators of manifold valued parameters monte carlo mc methods have become an important tool for inferences in non gaussian and non euclidean settings we study their use in those signal image processing scenarios where the parameter spaces are certain riemannian manifolds finite dimensional lie groups and their quotient sets we investigate the estimation of means and variances of the manifold valued parameters using two popular sampling methods independent and importance sampling using euclidean embeddings we specify a notion of extrinsic means employ monte carlo methods to estimate these means and utilize large sample asymptotics to approximate the estimator covariances experimental results are presented for target pose estimation orthogonal groups and signal subspace estimation grassmann manifolds asymptotic covariances are utilized to construct confidence regions to compare estimators and to determine the sample size for mc sampling

propagation of gevrey regularity for solutions of the boltzmann equation for maxwellian molecules we prove that gevrey regularity is propagated by the boltzmann equation with maxwellian molecules with or without angular cut off the proof relies on the wild expansion of the solution to the equation and on the characterization of gevrey regularity by the fourier transform

fast algorithms for computing the boltzmann collision operator the development of accurate and fast numerical schemes for the five fold boltzmann collision integral represents a challenging problem in scientific computing for a particular class of interactions including the so called hard spheres model in dimension three we are able to derive spectral methods that can be evaluated through fast algorithms these algorithms are based on a suitable representation and approximation of the collision operator explicit expressions for the errors in the schemes are given and spectral accuracy is proved parallelization properties and adaptivity of the algorithms are also discussed

on the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds we prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails in particular boltzmann type equations with smoothed soft potentials we compensate the lack of uniform in time estimates by the use of precise logarithmic sobolev type inequalities and the assumption that the initial datum decays rapidly at large velocities our method not only gives explicit results on the times of convergence but is also able to cover situations in which compactness arguments apparently do not apply even mere convergence to equilibrium was an open problem for soft potentials

combined tensor fitting and tv regularization in diffusion tensor imaging based on a riemannian manifold approach in this paper we consider combined tv denoising and diffusion tensor fitting in dti using the affine invariant riemannian metric on the space of diffusion tensors instead of first fitting the diffusion tensors and then denoising them we define a suitable tv type energy functional which incorporates the measured dwis using an inverse problem setup and which measures the nearness of neighboring tensors in the manifold to approach this functional we propose generalized forward backward splitting algorithms which combine an explicit and several implicit steps performed on a decomposition of the functional we validate the performance of the derived algorithms on synthetic and real dti data in particular we work on real 3d data to our knowledge the present paper describes the first approach to tv regularization in a combined manifold and inverse problem setup

a particle filtering scheme for processing time series corrupted by outliers the literature in engineering and statistics is abounding in techniques for detecting and properly processing anomalous observations in the data most of these techniques have been developed in the framework of static models and it is only in recent years that we have seen attempts that address the presence of outliers in nonlinear time series for a target tracking problem described by a nonlinear state space model we propose the online detection of outliers by including an outlier detection step within the standard particle filtering algorithm the outlier detection step is implemented by a test involving a statistic of the predictive distribution of the observations such as a concentration measure or an extreme upper quantile we also provide asymptotic results about the convergence of the particle approximations of the predictive distribution and its statistics and assess the performance of the resulting algorithms by computer simulations of target tracking problems with signal power observations

geometric variability of the scoliotic spine using statistics on articulated shape models this paper introduces a method to analyze the variability of the spine shape and of the spine shape deformations using articulated shape models the spine shape was expressed as a vector of relative poses between local coordinate systems of neighboring vertebrae spine shape deformations were then modeled by a vector of rigid transformations that transforms one spine shape into another because rigid transformations do not naturally belong to a vector space conventional mean and covariance could not be applied the frechet mean and a generalized covariance were used instead the spine shapes of a group of 295 scoliotic patients were quantitatively analyzed as well as the spine shape deformations associated with the cotrel dubousset corrective surgery 33 patients the boston brace 39 patients and the scoliosis progression without treatment 26 patients the variability of intervertebral poses was found to be inhomogeneous lumbar vertebrae were more variable than the thoracic ones and anisotropic with maximal rotational variability around the coronal axis and maximal translational variability along the axial direction finally brace and surgery were found to have a significant effect on the frechet mean and on the generalized covariance in specific spine regions where treatments modified the spine shape

estimates for the boltzmann collision operator via radial symmetry and fourier transform abstract we extend the l p theory of the boltzmann collision operator by using classical techniques based in the carleman representation and fourier analysis allied to new ideas that exploit the radial symmetry of this operator we are then able to greatly simplify existent technical proofs in this theory extend the range and obtain explicit sharp constants in some convolution like inequalities for the gain part of the boltzmann collision operator

probability convexity and harmonic maps with small image i uniqueness and fine existence this paper uses probability theory to provide an approach to the dirichlet problem for harmonic maps the probabilistic tools used are those of manifold valued brownian motion and martingales probabilistic proofs are given of uniqueness continuous dependence on data and existence of generalized solutions finely harmonic maps for target domains which have convex geometry this class of domain includes all regular geodesic balls links are made with new results in other fields the martingale dirichlet problem riemannian centres of mass

a direct method for the regularity of the gain term in the boltzmann equation this paper presents an essentially simple and direct method to prove the regularizing properties of the collision gain term in the first equation of the boltzmann hierarchy some corollaries of the main results of this paper are given to improve the previous results on the regularity of the gain term in the boltzmann equation an extension of a new fundamental result concerning separated variable kernels is also given to some general kernels

convolution inequalities for the boltzmann collision operator we study integrability properties of a general version of the boltzmann collision operator for hard and soft potentials in n dimensions a reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution where the weight is given by an operator invariant under rotations using a symmetrization technique in l p we prove a young s inequality for hard potentials which is sharp for maxwell molecules in the l 2 case further we find a new hardy littlewood sobolev type of inequality for boltzmann collision integrals with soft potentials the same method extends to radially symmetric non increasing potentials that lie in some l s weak or l s the method we use resembles a brascamp lieb and luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting consequently it is closely connected to the classical analysis of young and hardy littlewood sobolev inequalities in all cases the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section in the spirit of grad cut off as an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions

regularity of solutions to the spatially homogeneous boltzmann equation for non maxwellian molecules without angular cutoff abstract in this paper we use the pseudo differential calculus to analyze the smoothing property of weak solutions to the spatially homogeneous boltzmann equation precisely we show that for the non maxwellian molecules with debye yukawa potential if the positive weak solution is lipschitz continuous in the velocity variable then it lies in the sobolev space h loc r 3 and hence it is automatically smooth

on the boltzmann equation for diffusively excited granular media we study the boltzmann equation for a space homogeneous gas of inelastic hard spheres with a diffusive term representing a random background forcing under the assumption that the initial datum is a nonnegative l 2 n function with bounded mass and kinetic energy second moment we prove the existence of a solution to this model which instantaneously becomes smooth and rapidly decaying under a weak additional assumption of bounded third moment the solution is shown to be unique we also establish the existence but not uniqueness of a stationary solution in addition we show that the high velocity tails of both the stationary and time dependent particle distribution functions are overpopulated with respect to the maxwellian distribution as conjectured by previous authors and we prove pointwise lower estimates for the solutions

some applications of the method of moments for the homogeneous boltzmann and kac equations using the method of moments we prove that any polynomial moment of the solution of the homogeneous boltzmann equation with hard potentials or hard spheres is bounded provided that a moment of order strictly higher than 2 exists initially we also give partial results of convergence towards the maxwellian equilibrium in the case of soft potentials finally exponential as well as maxwellian estimates are introduced for the kac equation

consistency of restricted maximum likelihood estimators of principal components in this paper we consider two closely related problems estimation of eigenvalues and eigenfunctions of the covariance kernel of functional data based on possibly irregular measurements and the problem of estimating the eigenvalues and eigenvectors of the covariance matrix for high dimensional gaussian vectors in 23 a restricted maximum likelihood reml approach has been developed to deal with the flrst problem in this paper we establish consistency and derive rate of convergence of the reml estimator for the functional data case under appropriate smoothness conditions moreover we prove that when the number of measurements per sample curve is bounded under squared error loss the rate of convergence of the reml estimators of eigenfunctions is near optimal in the case of gaussian vectors asymptotic consistency and an e cient score representation of the estimators are obtained under the assumption that the efiective dimension grows at a rate slower than the sample size these results are derived through an explicit utilization of the intrinsic geometry of the parameter space which is non euclidean moreover the results derived in this paper suggest an asymptotic equivalence between the inference on functional data with dense measurements and that of the high dimensional gaussian vectors

moshfit algorithms for occlusion tolerant mean shape and rigid motion from 3d movement data this work addresses the use of 3d point data to measure rigid motions in the presence of occlusion and without reference to a prior model of relative point locations this is a problem where cluster based measurement techniques are used e g for measuring limb movements and no static calibration trial is available the same problem arises when performing the task known as roving capture in which a mobile 3d movement analysis system is moved through a volume with static markers in unknown locations and the ego motion of the system is required in order to understand biomechanical activity in the environment to provide a solution for both of these applications the new concept of a visibility graph is introduced and is combined with a generalised procrustes method adapted from ones used by the biological shape statistics and computer graphics communities recent results on shape space manifolds are applied to show sufficient conditions for convergence to unique solution algorithm source code is available and referenced here processing speed and rate of convergence are demonstrated using simulated data positional and angular accuracy are shown to be equivalent to approaches which require full calibration to within a small fraction of input resolution typical processing times for sub micron convergence are found to be fractions of a second so the method is suitable for workflows where there may be time pressure such as in sports science and clinical analysis

entropy dissipation and moment production for the boltzmann equation leth f m flog f m dv be the relative entropy off and the maxwellian with the same mass momentum and energy and denote the corresponding entropy dissipation term in the boltzmann equation byd f q f f logf dv an example is presented which shows that d f h f m can be arbitrarily small this example is a sequence of isotropic functions and the estimates are very explicitly given by a simple formula ford which holds for such functions the paper also gives a simplified proof of the so called povzner inequality which is a geometric inequality for the magnitudes of the velocities before and after an elastic collision that inequality is then used to prove that f v v s dt c t wheref is the solution of the spatially homogeneous boltzmann equation herec t is an explicitly given function dependings and the mass energy and entropy of the initial data

a distance between elliptical distributions based in an embedding into the siegel group this paper describes two different embeddings of the manifolds corresponding to many elliptical probability distributions with the informative geometry into the manifold of positive definite matrices with the siegel metric generalizing a result published previously elsewhere these new general embeddings are applicable to a wide class of elliptical probability distributions in which the normal t student and cauchy are specific examples a lower bound for the rao distance is obtained which is itself a distance and through these embeddings a number of statistical tests of hypothesis are derived

the effect of macrodiversity on the performance of maximal ratio combining in flat rayleigh fading the performance of maximal ratio combining mrc in rayleigh channels with co channel interference cci is well known for receive arrays which are co located recent work in network mimo edge excited cells and base station collaboration is increasing interest in macrodiversity systems hence in this paper we consider the effect of macrodiversity on mrc performance in rayleigh fading channels with cci we consider the uncoded symbol error rate ser as our performance measure of interest and investigate how different macrodiversity power profiles affect ser performance this is the first analytical work in this area we derive approximate and exact symbol error rate results for m qam bpsk modulations and use the analysis to provide a simple power metric numerical results verified by simulations are used in conjunction with the analysis to gain insight into the effects of the link powers on performance

regularity theory for the spatially homogeneous boltzmann equation with cut off we develop the regularity theory of the spatially homogeneous boltzmann equation with cut off and hard potentials for instance hard spheres by i revisiting the lp theory to obtain constructive bounds ii establishing propagation of smoothness and singularities iii obtaining estimates on the decay of the singularities of the initial datum our proofs are based on a detailed study of the regularity of the gain operator an application to the long time behavior is presented

well posedness of spatially homogeneous boltzmann equation with full range interaction in the present work we give the coercivity estimates for the boltzmann collision operator q without angular cut off to clarify in which case the functional langle q g f f rangle will become the truly sub elliptic based on this observation and commutator estimates in alexandre et al arch rat mech anal 198 39 123 2010 the upper bound estimates for the collision operator in chen and he arch rat mech anal 201 2 501 548 2011 and the stability results in desvillettes and mouhot arch rat mech anal 193 2 227 253 2009 in the function space l 1 q cap h n we establish global well posedness or local well posedness for the spatially homogeneous boltzmann equation with full range interaction covering most of physical collision kernels

ergodic capacity upper bound for dual mimo ricean systems simplified derivation and asymptotic tightness an analytical upper bound on the ergodic capacity of multiple input multiple output mimo systems is deduced with the aid of a simplified approach that relies on a fundamental power normalization given their high practical usability we are particularly interested in dual configurations where both ends deploy two antenna elements contrary to the majority of related studies where only the common case of rayleigh fading is considered our analysis is extended to account for ricean fading where a deterministic line of sight los component exists in the communication link and both ends are affected by spatial correlation in the following it is shown that the proposed bound is applicable for any arbitrary signal to noise ratio snr and rank of the mean channel matrix in fact we consider both conventional and optimized mimo configurations with equal los eigenvalues moreover the tightness of the bound is explored where it is demonstrated that as the snr tends to zero the bound becomes asymptotically tight at high snrs the offset between empirical capacity and the bound is analytically computed which implies that an explicit asymptotic capacity expression can ultimately be obtained

stability and uniqueness for the spatially homogeneous boltzmann equation with long range interactions in this paper we prove some a priori stability estimates in weighted sobolev spaces for the spatially homogeneous boltzmann equation without angular cutoff covering all physical collision kernels these estimates are conditional on some regularity estimates on the solutions and therefore reduce the stability and uniqueness issue to one of proving suitable regularity bounds on the solutions we then prove such regularity bounds for a class of interactions including the so called non cutoff and non mollified hard potentials and moderately soft potentials in particular we obtain the first result of global existence and uniqueness for these long range interactions

global l p properties for the spatially homogeneous boltzmann equation this paper studies the lp behavior for 1 p of solutions of the nonlinear spatially homogeneous boltzmann equation for a class of collision kernels including inverse kth power forces with k 5 and angular cut off the following topics are treated differentiability in lp together with global boundedness in time for lp moments that exist initially translation continuity in lp uniformly in time and strong convergence to equilibrium