connectedness at infinity of complete kahler manifolds one of the main purposes of this paper is to prove that on a complete k ahler manifold of dimension m if the holomorphic bisectional curvature is bounded from below by 1 and the minimum spectrum lambda 1 m ge m 2 then it must either be connected at infinity or isometric to bbb r times n with a specialized metric with n being compact generalizations to complete k ahler manifolds satisfying a weighted poincar e inequality are also being considered

simplifying mixture models through function approximation the finite mixture model is widely used in various statistical learning problems however the model obtained may contain a large number of components making it inefficient in practical applications in this paper we propose to simplify the mixture model by minimizing an upper bound of the approximation error between the original and the simplified model under the use of the l 2 distance measure this is achieved by first grouping similar components together and then performing local fitting through function approximation the simplified model obtained can then be used as a replacement of the original model to speed up various algorithms involving mixture models during training e g bayesian filtering belief propagation and testing e g kernel density estimation support vector machine svm testing encouraging results are observed in the experiments on density estimation clustering based image segmentation and simplification of svm decision functions

results on a weighted poincare inequality of complete manifolds we study manifolds satisfying a weighted poincare inequality which was first introduced by li and wang we generalized their result by relaxing the ricci curvature bound condition only being satisfied outside a compact set and established a finitely many ends result we also proved a vanishing result for an l 2 harmonic 1 form provided that the weight function p is of sub quadratic growth of the distance function which generalized the li wang result on manifolds with a positive spectrum

weighted p harmonic functions and rigidity of smooth metric measure spaces abstract let m n g e f d v be a smooth metric measure space of dimensional n suppose that v is a positive weighted p eigenfunction associated to the eigenvalue 1 p on m namely e f d i v e f v p 2 v 1 p v p 1 in the distribution sense we first give a local gradient estimate for v provided the m dimensional bakry emery curvature r i c f m bounded from below consequently we show that when r i c f m 0 then v is constant if v is of sublinear growth at the same time we prove a harnack inequality for weighted p harmonic functions moreover we show global sharp gradient estimates for weighted p eigenfunctions then we use these estimates to study geometric structures at infinity when the first eigenvalue 1 p is maximal our achievements generalize several results proved earlier by li wang munteanu wang 11 12 17 18

the p eigenvalue estimates and lq p harmonic forms on submanifolds of hadamard manifolds abstract in this paper we first shows a lower bound estimate for the first eigenvalue of the p laplacian on a complete submanifold in a simply connected manifold of negative sectional curvature in terms of l n norm of the mean curvature in the second part we prove some vanishing theorems of l q p harmonic l forms on complete noncompact submanifolds in hadamard manifolds by assuming the index of certain schrodinger operators are zero or by assuming integral or pointwise pinching conditions on the mean curvature and the traceless second fundamental form

dynamic green s functions in anisotropic piezoelectric thermoelastic and poroelastic solids a procedure is described to generate fundamental solutions or green s functions for time harmonic point forces and sources the linearity of the field equations permits the green s function to be represented as an integral over the surface of a unit sphere where the integrand is the solution of a one dimensional impulse response problem the method is demonstrated for the theories of piezoelectricity thermoelasticity and poroelasticity time domain analogues are discussed and compared with known expressions for anisotropic elasticity

on a multivariate extension of the gamma and beta distributions the concept of liouville distribution of the first and second kind provides respective generalizations to the multivariate gamma distribution and the dirichlet distribution it is shown that these new classes of multivariate distributions defined by functional forms can be used to generate some well known statistical distributions

ends of metric measure spaces and sobolev inequalities generalizing work of li and wang we prove sharp volume growth decay rates for ends of metric measure spaces supporting a p p sobolev inequality a sharp result for q p sobolev inequalities is also proved

insights into multiple single lower bound approximation for extended variational inference in non gaussian structured data modeling for most of the non gaussian statistical models the data being modeled represent strongly structured properties such as scalar data with bounded support e g beta distribution vector data with unit length e g dirichlet distribution and vector data with positive elements e g generalized inverted dirichlet distribution in practical implementations of non gaussian statistical models it is infeasible to find an analytically tractable solution to estimating the posterior distributions of the parameters variational inference vi is a widely used framework in bayesian estimation recently an improved framework namely the extended vi evi has been introduced and applied successfully to a number of non gaussian statistical models evi derives analytically tractable solutions by introducing lower bound approximations to the variational objective function in this paper we compare two approximation strategies namely the multiple lower bounds mlbs approximation and the single lower bound slb approximation which can be applied to carry out the evi for implementation two different conditions the weak and the strong conditions are discussed convergence of the evi depends on the selection of the lower bound regardless of the choice of weak or strong condition we also discuss the convergence properties to clarify the differences between mlb and slb extensive comparisons are made based on some evi based non gaussian statistical models theoretical analysis is conducted to demonstrate the differences between the weak and strong conditions experimental results based on real data show advantages of the slb approximation over the mlb approximation

semi analytical elasticity solutions for bi directional functionally graded beams abstract elasticity solutions are presented for bending and thermal deformations of functionally graded beams with various end conditions using the state space based differential quadrature method the beams are assumed to be macroscopically isotropic with young s modulus varying exponentially along the thickness and longitudinal directions while poisson s ratio remaining constant the state space method is adopted to obtain analytically the thickness variation of the elastic field and when coupled with differential quadrature the longitudinal discretization can be analyzed in an approximate manner this approach is then validated by comparing the numerical results with the exact solutions for a special functionally graded beam and with finite element solutions the influences of material gradient indices on the response of bi directional functionally graded beams are finally investigated

complete manifolds with positive spectrum ii in this paper we continued our investigation of complete manifolds whose spectrum of the laplacian has an optimal positive lower bound in particular we proved a splitting type theorem for n dimensional manifolds that have a finite volume end this can be viewed as a study of the equality case of a theorem of cheng

steady and transient green s functions for anisotropic conduction in an exponentially graded solid abstract the problem of the determination of green s function in conduction for a rectilinearly anisotropic solid with an exponential grading along a certain direction is studied domains of an unbounded space and a half space either three dimensional or two dimensional are considered along the boundary of the domain homogeneous boundary conditions of the first and second kinds are imposed we find interestingly that under this specific type of grading the green s functions permit an algebraic decomposition which will in turn greatly simplify the formulation the method of fourier transform is employed for the green s function for a half space or a half plane although the derivation process is quite tedious we show analytically that the inverse transform can be found exactly and their resulting expressions are surprisingly neat and compact in addition both steady state and transient state field solutions are considered by taking laplace transform with respect to the time variable we show that the mathematical frameworks for the steady state and transient state green s functions are entirely analogous thereby the transient state green s function is readily obtained by taking laplace inverse transform back to the time domain these derived fundamental solutions will serve as benchmark results for modeling some inhomogeneous materials in the absence of grading term we have verified analytically that our solutions agree exactly with previously known green s functions for homogeneous media

incorporating mean template into finite mixture model for image segmentation the well known finite mixture model fmm has been regarded as a useful tool for image segmentation application however the pixels in fmm are considered independent of each other and the spatial relationship between neighboring pixels is not taken into account these limitations make the fmm more sensitive to noise in this brief we propose a simple and effective method to make the traditional fmm more robust to noise with the help of a mean template fmm can be considered a linear combination of prior and conditional probability from the expression of its mathematical formula we calculate these probabilities with two mean templates a weighted arithmetic mean template and a weighted geometric mean template thus in our model the prior probability or conditional probability of an image pixel is influenced by the probabilities of pixels in its immediate neighborhood to incorporate the local spatial and intensity information for eliminating the noise finally our algorithm is general enough and can be extended to any other fmm based models to achieve super performance experimental results demonstrate the improved robustness and effectiveness of our approach

real time vector quantization and clustering based on ordinary differential equations this brief presents a dynamical system approach to vector quantization or clustering based on ordinary differential equations with the potential for real time implementation two examples of different pattern clusters demonstrate that the model can successfully quantize different types of input patterns furthermore we analyze and study the stability of our dynamical system by discovering the equilibrium points for certain input patterns and analyzing their stability we have shown the quantizing behavior of the system with respect to its vigilance parameter the proposed system is applied to two real world problems providing comparable results to the best reported findings this validates the effectiveness of our proposed approach

a dirichlet process mixture of generalized dirichlet distributions for proportional data modeling in this paper we propose a clustering algorithm based on both dirichlet processes and generalized dirichlet distribution which has been shown to be very flexible for proportional data modeling our approach can be viewed as an extension of the finite generalized dirichlet mixture model to the infinite case the extension is based on nonparametric bayesian analysis this clustering algorithm does not require the specification of the number of mixture components to be given in advance and estimates it in a principled manner our approach is bayesian and relies on the estimation of the posterior distribution of clusterings using gibbs sampler through some applications involving real data classification and image databases categorization using visual words we show that clustering via infinite mixture models offers a more powerful and robust performance than classic finite mixtures

indentation of solids with gradients in elastic properties part i point force abstract analytical and computational results are presented on the evolution of stresses and displacements due to indentation from a normal point force on an elastic substrate whose young s modulus e varies as a function of depth z beneath the indented surface with the exception of variation of properties with depth the material is assumed to be linearly elastic and locally isotropic closed form solutions are derived for several fixed values of the poisson ratio v and variations in e which follow two prescribed functions of z 1 a simple power law e e 0 z k where 0 k e e 0 e z where e 0 is young s modulus at the surface 0 denotes a soft surface e g modulus variations measured as a function of depth beneath the earth s surface for solids and rocks the analytical solutions are checked with detailed finite element simulations it is shown that for the power law case there exists a critical poisson ratio v cr above which drastic changes occur in the stress distribution under the point load finite element results reveal however that the response is relatively less sensitive to the variation of v on either side of v cr than to the variation of e with depth applications of the present results are discussed wherever appropriate to surface treatments of engineering structures thick coatings thin film multilayers for microelectronic devices and settling of foundations in the context of soil mechanics and rock mechanics

shape matching and object recognition using shape contexts we present a novel approach to measuring similarity between shapes and exploit it for object recognition in our framework the measurement of similarity is preceded by 1 solving for correspondences between points on the two shapes 2 using the correspondences to estimate an aligning transform in order to solve the correspondence problem we attach a descriptor the shape context to each point the shape context at a reference point captures the distribution of the remaining points relative to it thus offering a globally discriminative characterization corresponding points on two similar shapes will have similar shape contexts enabling us to solve for correspondences as an optimal assignment problem given the point correspondences we estimate the transformation that best aligns the two shapes regularized thin plate splines provide a flexible class of transformation maps for this purpose the dissimilarity between the two shapes is computed as a sum of matching errors between corresponding points together with a term measuring the magnitude of the aligning transform we treat recognition in a nearest neighbor classification framework as the problem of finding the stored prototype shape that is maximally similar to that in the image results are presented for silhouettes trademarks handwritten digits and the coil data set

online learning of a dirichlet process mixture of beta liouville distributions via variational inference a large class of problems can be formulated in terms of the clustering process mixture models are an increasingly important tool in statistical pattern recognition and for analyzing and clustering complex data two challenging aspects that should be addressed when considering mixture models are how to choose between a set of plausible models and how to estimate the model s parameters in this paper we address both problems simultaneously within a unified online nonparametric bayesian framework that we develop to learn a dirichlet process mixture of beta liouville distributions i e an infinite beta liouville mixture model the proposed infinite model is used for the online modeling and clustering of proportional data for which the beta liouville mixture has been shown to be effective we propose a principled approach for approximating the intractable model s posterior distribution by a tractable one which we develop such that all the involved mixture s parameters can be estimated simultaneously and effectively in a closed form this is done through variational inference that enjoys important advantages such as handling of unobserved attributes and preventing under or overfitting we explain that in detail the effectiveness of the proposed work is evaluated on three challenging real applications namely facial expression recognition behavior modeling and recognition and dynamic textures clustering

improving on deterministic approximate bayesian inferences for mixture distributions this paper presents the branching approach which can improve deterministic implementation methods such as variational bayesian inference and expectation propagation method for bayesian mixture distributions this proposed approach utilizes a set of artificial conditions defined by using the latent variables of the mixture distribution this condition set is updated iteratively by branching based on a condition selected from the previous condition set the target approximate bayesian inference is obtained by merging the approximate conditional inferences under each condition in the condition set the proposed approach is compared with several standard implementation methods by using a numerical example and a real world example

on linking n dimensional anisotropic and isotropic green s functions for infinite space half space bimaterial and multilayer for conduction abstract we establish exact mathematical links between the n dimensional anisotropic and isotropic green s functions for diffusion phenomena for an infinite space a half space a bimaterial and a multilayered space the purpose of this work is not to attempt to present a solution procedure but to focus on the general conditions and situations in which the anisotropic physical problems can be directly linked with the green s functions of a similar configuration with isotropic constituents we show that for green s functions of an infinite and a half space and for all two dimensional configurations the exact correspondences between the anisotropic and isotropic ones can always be established without any regard to the constituent conductivities or any other information and thus knowing the isotropic green s functions will readily provide explicit expressions for anisotropic green s functions upon back transformation for three and higher dimensional bimaterials and layered spaces the correspondence can also be found but the constituent conductivities need to satisfy further algebraic constraints when these constraints are fully satisfied then the anisotropic green s functions can also be obtained from those of the isotropic ones or at least in principle

p harmonic l forms on riemannian manifolds with a weighted poincare inequality abstract given a riemannian manifold with a weighted poincare inequality in this paper we will show some vanishing type theorems for p harmonic l forms on such a manifold we also prove a vanishing result on submanifolds in euclidean space with flat normal bundle our results can be considered as generalizations of the work of lam li wang lin and vieira see lam 2008 li and wang 2001 lin 2015 vieira 2016 moreover we also prove a vanishing and splitting type theorem for p harmonic functions on manifolds with spin 9 holonomy provided a p p sobolev type inequality which can be considered as a general poincare inequality holds true

singular boundary method for heat conduction problems with certain spatially varying conductivity the singular boundary method sbm is a recent boundary type meshless collocation method in which the solution of a given problem is expanded as a linear combination of the fundamental solutions in terms of the source points the method circumvents the fictitious boundary long perplexing the method of fundamental solution mfs by the introduction of the concept of origin intensity factors oifs this paper documents the first attempt to extend the method to heat conduction problems in nonhomogeneous materials we derive the fundamental solutions of heat conduction problems with the thermal conductivity of the quadratic exponential and trigonometric material variations in three directions furthermore we firstly theoretically derive the value of the oif for the natural logarithm function which is later extended to the oifs for a group of fundamental solutions the feasibility accuracy and stability of the presented sbm formulation are confirmed for both two 2d and three dimensional 3d examples

count data modeling and classification using finite mixtures of distributions in this paper we consider the problem of constructing accurate and flexible statistical representations for count data which we often confront in many areas such as data mining computer vision and information retrieval in particular we analyze and compare several generative approaches widely used for count data clustering namely multinomial multinomial dirichlet and multinomial generalized dirichlet mixture models moreover we propose a clustering approach via a mixture model based on a composition of the liouville family of distributions from which we select the beta liouville distribution and the multinomial the novel proposed model which we call multinomial beta liouville mixture is optimized by deterministic annealing expectation maximization and minimum description length and strives to achieve a high accuracy of count data clustering and model selection an important feature of the multinomial beta liouville mixture is that it has fewer parameters than the recently proposed multinomial generalized dirichlet mixture the performance evaluation is conducted through a set of extensive empirical experiments which concern text and image texture modeling and classification and shape modeling and highlights the merits of the proposed models and approaches

on green s function for a three dimensional exponentially graded elastic solid the problem of a point force acting in an unbounded three dimensional isotropic elastic solid is considered kelvin solved this problem for homogeneous materials here the material is inhomogeneous it is functionally graded specifically the solid is exponentially graded which means that the lame moduli vary exponentially in a given fixed direction the solution for the green9s function is obtained by fourier transforms and consists of a singular part given by the kelvin solution plus a non singular remainder this grading term is not obtained in simple closed form but as the sum of single integrals over finite intervals of modified bessel functions and double integrals over finite regions of elementary functions knowledge of this new fundamental solution for graded materials permits the development of boundary integral methods for these technologically important inhomogeneous solids

discrete data clustering using finite mixture models finite mixture models have been applied for different computer vision image processing and pattern recognition tasks the majority of the work done concerning finite mixture models has focused on mixtures for continuous data however many applications involve and generate discrete data for which discrete mixtures are better suited in this paper we investigate the problem of discrete data modeling using finite mixture models we propose a novel well motivated mixture that we call the multinomial generalized dirichlet mixture the novel model is compared with other discrete mixtures we designed experiments involving spatial color image databases modeling and summarization and text classification to show the robustness flexibility and merits of our approach

refined kato inequalities and conformal weights in riemannian geometry abstract we establish refinements of the classical kato inequality for sections of a vector bundle which lie in the kernel of a natural injectively elliptic first order linear differential operator our main result is a general expression which gives the value of the constants appearing in the refined inequalities these constants are shown to be optimal and are computed explicitly in most practical cases

bayesian estimation methods for 2 2 contingency tables using mixtures of dirichlet distributions abstract in the estimation of cell probabilities from a 2 2 table a prior distribution is developed that can reflect prior beliefs about the cross classification structure in the table the posterior means using this prior shrink the classical estimates towards the association structure specified a priori by the user closed form approximations to the posterior means and credible regions are developed in the special case where the two variables of the table are believed independent