existence of solutions for a p x kirchhoff type equation this paper is concerned with the existence and multiplicity of solutions to a class of p x kirchhoff type problem with dirichlet boundary data by means of a direct variational approach and the theory of the variable exponent sobolev spaces we establish conditions ensuring the existence and multiplicity of solutions for the problem

on nonlocal p x laplacian equations abstract this paper deals with the nonlocal anisotropic p x laplacian dirichlet problems with non variational form i 1 n a i u d i d i u p i x 2 d i u b u f x u x in u 0 and with variational form i 1 n a i d i u p i x p i x d x d i d i u p i x 2 d i u b f x u d x f x u x in u 0 where f x t 0 t f x s d s and a i is allowed to be singular at zero using s mapping theory and the variational method some results on existence and multiplicity for the problems are obtained

a multibody factorization method for independently moving objects the structure from motion problem has been extensively studied in the field of computer vision yet the bulk of the existing work assumes that the scene contains only a single moving object the more realistic case where an unknown number of objects move in the scene has received little attention especially for its theoretical treatment in this paper we present a new method for separating and recovering the motion and shape of multiple independently moving objects in a sequence of images the method does not require prior knowledge of the number of objects nor is dependent on any grouping of features into an object at the image level for this purpose we introduce a mathematical construct of object shapes called the shape interaction matrix which is invariant to both the object motions and the selection of coordinate systems this invariant structure is computable solely from the observed trajectories of image features without grouping them into individual objects once the matrix is computed it allows for segmenting features into objects by the process of transforming it into a canonical form as well as recovering the shape and motion of each object the theory works under a broad set of projection models scaled orthography paraperspective and affine but they must be linear so it excludes projective cameras

two view multibody structure from motion we present an algebraic geometric approach to 3 d motion estimation and segmentation of multiple rigid body motions from noise free point correspondences in two perspective views our approach exploits the algebraic and geometric properties of the so called multibody epipolar constraint and its associated multibody fundamental matrix which are natural generalizations of the epipolar constraint and of the fundamental matrix to multiple motions we derive a rank constraint on a polynomial embedding of the correspondences from which one can estimate the number of independent motions as well as linearly solve for the multibody fundamental matrix we then show how to compute the epipolar lines from the first order derivatives of the multibody epipolar constraint and the epipoles by solving a plane clustering problem using generalized pca gpca given the epipoles and epipolar lines the estimation of individual fundamental matrices becomes a linear problem the clustering of the feature points is then automatically obtained from either the epipoles and epipolar lines or from the individual fundamental matrices although our approach is mostly designed for noise free correspondences we also test its performance on synthetic and real data with moderate levels of noise

infinitely many solutions for a p x laplacian equation in rn abstract this paper deals with a p x laplacian equation in r n by means of a direct variational approach and the theory of the variable exponent sobolev spaces we establish the existence of infinitely many distinct homoclinic radially symmetric solutions whose w 1 p x r n norms tend to zero to infinity respectively under weaker hypotheses about nonlinearity at zero at infinity respectively

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

multiple solutions to some degenerate quasilinear equation with variable exponents via perturbation method abstract in this paper we prove the existence of signed and sign changing solutions to some degenerate quasilinear and elliptic equation containing a gradient term and involving a variable exponent our main idea is to make use of the perturbation method together with invariant sets of descending flow the fact that the exponent is not constant plays an important role

spectral clustering on multiple manifolds spectral clustering sc is a large family of grouping methods that partition data using eigenvectors of an affinity matrix derived from the data though sc methods have been successfully applied to a large number of challenging clustering scenarios it is noteworthy that they will fail when there are significant intersections among different clusters in this paper based on the analysis that sc methods are able to work well when the affinity values of the points belonging to different clusters are relatively low we propose a new method called spectral multi manifold clustering smmc which is able to handle intersections in our model the data are assumed to lie on or close to multiple smooth low dimensional manifolds where some data manifolds are separated but some are intersecting then local geometric information of the sampled data is incorporated to construct a suitable affinity matrix finally spectral method is applied to this affinity matrix to group the data extensive experiments on synthetic as well as real datasets demonstrate the promising performance of smmc

existence and multiplicity of solutions for p x laplacian equations in rn abstract this paper investigates the existence and multiplicity of solutions for p x laplacian equations in r n

parameterized principal component analysis abstract when modeling multivariate data one might have an extra parameter of contextual information that could be used to treat some observations as more similar to others for example images of faces can vary by age and one would expect the face of a 40 year old to be more similar to the face of a 30 year old than to a baby face we introduce a novel manifold approximation method parameterized principal component analysis ppca that models data with linear subspaces that change continuously according to the extra parameter of contextual information e g age instead of ad hoc atlases special care has been taken in the loss function and the optimization method to encourage smoothly changing subspaces across the parameter values the approach ensures that each observation s projection will share information with observations that have similar parameter values but not with observations that have large parameter differences we tested ppca on artificial data based on known smooth functions of an added parameter as well as on two real datasets with different types of parameters we compared ppca to pca sparse pca and to independent principal component analysis ipca an atlas based method that groups observations by their parameter values and projects each group using pca with no sharing of information between groups ppca recovers the known functions with less error and projects the datasets test set observations with consistently less reconstruction error than ipca does in some cases where the manifold is truly nonlinear pca outperforms all the other manifold approximation methods compared

asymptotic stability for anisotropic kirchhoff systems we study the question of asymptotic stability as time tends to infinity of solutions of dissipative anisotropic kirchhoff systems involving the p x laplacian operator governed by time dependent nonlinear damping forces and strongly nonlinear power like variable potential energies this problem had been considered earlier for potential energies which arise from restoring forces whereas here we allow also the effect of amplifying forces global asymptotic stability can then no longer be expected and should be replaced by local stability the results are further extended to the more delicate problem involving higher order damping terms

relations among some low rank subspace recovery models recovering intrinsic low dimensional subspaces from data distributed on them is a key preprocessing step to many applications in recent years a lot of work has modeled subspace recovery as low rank minimization problems we find that some representative models such as robust principal component analysis r pca robust low rank representation r lrr and robust latent low rank representation r latlrr are actually deeply connected more specifically we discover that once a solution to one of the models is obtained we can obtain the solutions to other models in closed form formulations since r pca is the simplest our discovery makes it the center of low rank subspace recovery models our work has two important implications first r pca has a solid theoretical foundation under certain conditions we could find globally optimal solutions to these low rank models at an overwhelming probability although these models are nonconvex second we can obtain significantly faster algorithms for these models by solving r pca first the computation cost can be further cut by applying low complexity randomized algorithms for example our novel filtering algorithm to r pca although for the moment the formal proof of our filtering algorithm is not yet available experiments verify the advantages of our algorithm over other state of the art methods based on the alternating direction method

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

positive and negative solutions of a class of nonlocal problems abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper still by using the invariant sets of descent flow we obtain positive and negative solutions of a class of nonlocal quasilinear elliptic boundary value problems as follows a b u 2 u f x u in u 0 on

a general framework for subspace detection in unordered multidimensional data the analysis of large volumes of unordered multidimensional data is a problem confronted by scientists and data analysts every day often it involves searching for data alignments that emerge as well defined structures or geometric patterns in datasets for example straight lines circles and ellipses represent meaningful structures in data collected from electron backscatter diffraction particle accelerators and clonogenic assays also customers with similar behavior describe linear correlations in e commerce databases we describe a general approach for detecting data alignments in large unordered noisy multidimensional datasets in contrast to classical techniques such as the hough transforms which are designed for detecting a specific type of alignment on a given type of input our approach is independent of the geometric properties of the alignments to be detected as well as independent of the type of input data thus it allows concurrent detection of multiple kinds of data alignments in datasets containing multiple types of data given its general nature optimizations developed for our technique immediately benefit all its applications regardless the type of input data

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

a new approach to dimensionality reduction theory and algorithms this paper applies whitney s embedding theorem to the data reduction problem and introduces a new approach motivated in part by the constructive proof of the theorem the notion of a good projection is introduced which involves picking projections of the high dimensional system that are optimized such that they are easy to invert the basic theory of the approach is outlined and algorithms for finding the projections are presented and applied to several test cases a method for constructing the inverse projection is detailed and its properties including a new measure of complexity are discussed finally well known methods of data reduction are compared with our approach within the context of whitney s theorem

a geometric analysis of subspace clustering with outliers this paper considers the problem of clustering a collection of unlabeled data points assumed to lie near a union of lower dimensional planes as is common in computer vision or unsupervised learning applications we do not know in advance how many subspaces there are nor do we have any information about their dimensions we develop a novel geometric analysis of an algorithm named sparse subspace clustering ssc 11 which signicantly broadens the range of problems where it is provably eective for instance we show that ssc can recover multiple subspaces each of dimension comparable to the ambient dimension we also prove that ssc can correctly cluster data points even when the subspaces of interest intersect further we develop an extension of ssc that succeeds when the data set is corrupted with possibly overwhelmingly many outliers underlying our analysis are clear geometric insights which may bear on other sparse recovery problems a numerical study complements our theoretical analysis and demonstrates the eectiveness of these methods

multiframe motion segmentation with missing data using powerfactorization and gpca we consider the problem of segmenting multiple rigid body motions from point correspondences in multiple affine views we cast this problem as a subspace clustering problem in which point trajectories associated with each motion live in a linear subspace of dimension two three or four our algorithm involves projecting all point trajectories onto a 5 dimensional subspace using the svd the powerfactorization method or ransac and fitting multiple linear subspaces representing different rigid body motions to the points in 5 using gpca unlike previous work our approach does not restrict the motion subspaces to be four dimensional and independent instead it deals gracefully with all the spectrum of possible affine motions from two dimensional and partially dependent to four dimensional and fully independent our algorithm can handle the case of missing data meaning that point tracks do not have to be visible in all images by using the powerfactorization method to project the data in addition our method can handle outlying trajectories by using ransac to perform the projection we compare our approach to other methods on a database of 167 motion sequences with full motions independent motions degenerate motions partially dependent motions missing data outliers etc on motion sequences with complete data our method achieves a misclassification error of less that 5 for two motions and 29 for three motions

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

segmentation of multivariate mixed data via lossy data coding and compression in this paper based on ideas from lossy data coding and compression we present a simple but effective technique for segmenting multivariate mixed data that are drawn from a mixture of gaussian distributions which are allowed to be almost degenerate the goal is to find the optimal segmentation that minimizes the overall coding length of the segmented data subject to a given distortion by analyzing the coding length rate of mixed data we formally establish some strong connections of data segmentation to many fundamental concepts in lossy data compression and rate distortion theory we show that a deterministic segmentation is approximately the asymptotically optimal solution for compressing mixed data we propose a very simple and effective algorithm that depends on a single parameter the allowable distortion at any given distortion the algorithm automatically determines the corresponding number and dimension of the groups and does not involve any parameter estimation simulation results reveal intriguing phase transition like behaviors of the number of segments when changing the level of distortion or the amount of outliers finally we demonstrate how this technique can be readily applied to segment real imagery and bioinformatic data

remarks on an elliptic equation of kirchhoff type abstract we present a short survey on the nonlocal elliptic boundary value problem m u 2 d x u f x u in u 0 on where is a smooth bounded domain of r n m is a positive function and f has subcritical growth

transformation invariant subspace clustering subspace clustering has achieved great success in many computer vision applications however most subspace clustering algorithms require well aligned data samples which is often not straightforward to achieve this paper proposes a transformation invariant subspace clustering framework by jointly aligning data samples and learning subspace representation by alignment the transformed data samples become highly correlated and a better affinity matrix can be obtained the joint problem can be reduced to a sequence of least squares regression problems which can be efficiently solved we verify the effectiveness of the proposed method with extensive experiments on unaligned real data demonstrating its higher clustering accuracy than the state of the art subspace clustering and transformation invariant clustering algorithms highlightsa transformation invariant subspace clustering algorithm is proposed alignment and subspace clustering are mutually dependent state of the art clustering results on the unconstrained face dataset

multiscale hybrid linear models for lossy image representation in this paper we introduce a simple and efficient representation for natural images we view an image in either the spatial domain or the wavelet domain as a collection of vectors in a high dimensional space we then fit a piece wise linear model i e a union of affine subspaces to the vectors at each downsampling scale we call this a multiscale hybrid linear model for the image the model can be effectively estimated via a new algebraic method known as generalized principal component analysis gpca the hybrid and hierarchical structure of this model allows us to effectively extract and exploit multimodal correlations among the imagery data at different scales it conceptually and computationally remedies limitations of many existing image representation methods that are based on either a fixed linear transformation e g dct wavelets or an adaptive uni modal linear transformation e g pca or a multimodal model that uses only cluster means e g vq we will justify both quantitatively and experimentally why and how such a simple multiscale hybrid model is able to reduce simultaneously the model complexity and computational cost despite a small overhead of the model our careful and extensive experimental results show that this new model gives more compact representations for a wide variety of natural images under a wide range of signal to noise ratios than many existing methods including wavelets we also briefly address how the same hybrid linear modeling paradigm can be extended to be potentially useful for other applications such as image segmentation

existence of three solutions for some equation of kirchhoff type involving variable exponents abstract in this paper we are concerned with some nonlocal equation involving variable exponents the equation studied could be regarded as an eigenvalue problem a result of existence of three nonnegative solutions is established using variational technique

mixtures of probabilistic principal component analyzers principal component analysis pca is one of the most popular techniques for processing compressing and visualizing data although its effectiveness is limited by its global linearity while nonlinear variants of pca have been proposed an alternative paradigm is to capture data complexity by a combination of local linear pca projections however conventional pca does not correspond to a probability density and so there is no unique way to combine pca models therefore previous attempts to formulate mixture models for pca have been ad hoc to some extent in this article pca is formulated within a maximum likelihood framework based on a specific form of gaussian latent variable model this leads to a well defined mixture model for probabilistic principal component analyzers whose parameters can be determined using an expectationmaximization algorithm we discuss the advantages of this model in the context of clustering density modeling and local dimensionality reduction and we demonstrate its application to image compression and handwritten digit recognition

generalized principal component analysis gpca this paper presents an algebro geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points we represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point when the number of subspaces is known we show that these polynomials can be estimated linearly from data hence subspace segmentation is reduced to classifying one point per subspace we select these points optimally from the data set by minimizing certain distance function thus dealing automatically with moderate noise in the data a basis for the complement of each subspace is then recovered by applying standard pca to the collection of derivatives normal vectors extensions of gpca that deal with data in a high dimensional space and with an unknown number of subspaces are also presented our experiments on low dimensional data show that gpca outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as k subspaces and expectation maximization we also present applications of gpca to computer vision problems such as face clustering temporal video segmentation and 3d motion segmentation from point correspondences in multiple affine views

robust kernel low rank representation recently low rank representation lrr has shown promising performance in many real world applications such as face clustering however lrr may not achieve satisfactory results when dealing with the data from nonlinear subspaces since it is originally designed to handle the data from linear subspaces in the input space meanwhile the kernel based methods deal with the nonlinear data by mapping it from the original input space to a new feature space through a kernel induced mapping to effectively cope with the nonlinear data we first propose the kernelized version of lrr in the clean data case we also present a closed form solution for the resultant optimization problem moreover to handle corrupted data we propose the robust kernel lrr rklrr approach and develop an efficient optimization algorithm to solve it based on the alternating direction method in particular we show that both the subproblems in our optimization algorithm can be efficiently and exactly solved and it is guaranteed to obtain a globally optimal solution besides our proposed algorithm can also solve the original lrr problem which is a special case of our rklrr when using the linear kernel in addition based on our new optimization technique the kernelization of some variants of lrr can be similarly achieved comprehensive experiments on synthetic data sets and real world data sets clearly demonstrate the efficiency of our algorithm as well as the effectiveness of rklrr and the kernelization of two variants of lrr

multiplicity result for some nonlocal anisotropic equation via nonsmooth critical point theory approach abstract in this paper we study quasilinear problems involving variable exponent growth conditions and nonlocal terms on the whole space r n a multiplicity result is established all the coefficients involved in the terms of the equation depend both on the variable x and the unknown function u our main argument is nonsmooth critical point theory

infinitely many positive solutions for a p x kirchhoff type equation this paper is concerned with the existence of infinitely many positive solutions to a class of p x kirchhoff type problem by means of a direct variational approach and the theory of the variable exponent sobolev spaces we establish the existence of infinitely many distinct positive solutions whose w1 p x norms and l norms tend to zero under suitable hypotheses about nonlinearity

linear time subspace clustering via bipartite graph modeling we present a linear time subspace clustering approach that combines sparse representations and bipartite graph modeling the signals are modeled as drawn from a union of low dimensional subspaces and each signal is represented by a sparse combination of basis elements termed atoms which form the columns of a dictionary matrix the sparse representation coefficients are arranged in a sparse affinity matrix which defines a bipartite graph of two disjoint sets 1 atoms and 2 signals subspace clustering is obtained by applying low complexity spectral bipartite graph clustering that exploits the small number of atoms for complexity reduction the complexity of the proposed approach is linear in the number of signals thus it can rapidly cluster very large data collections performance evaluation of face clustering and temporal video segmentation demonstrates comparable clustering accuracies to state of the art at a significantly lower computational load

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

infinitely many non negative solutions for a p x kirchhoff type problem with dirichlet boundary condition abstract in this paper we consider the dirichlet problem involving the p x kirchhoff type a b 1 p x u p x d x div u p x 2 u f x u in u 0 on we prove the existence of infinitely many non negative solutions of the problem by applying a general variational principle due to b ricceri and the theory of the variable exponent sobolev spaces

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

on nonlocal p x laplacian dirichlet problems abstract this paper deals with the nonlocal p x laplacian dirichlet problems with non variational form a u p x u x b u f x u x in u 0 and with variational form a u p x p x d x p x u x b f x u d x f x u x in u 0 where f x t 0 t f x s d s and a is allowed to be singular at zero using s mapping theory and the variational method some results on existence and multiplicity for the problems are obtained under weaker hypotheses our results are also new even for the case when p x p is a constant

pca meets rg a system with many degrees of freedom can be characterized by a covariance matrix principal components analysis focuses on the eigenvalues of this matrix hoping to find a lower dimensional description but when the spectrum is nearly continuous any distinction between components that we keep and those that we ignore becomes arbitrary it then is natural to ask what happens as we vary this arbitrary cutoff we argue that this problem is analogous to the momentum shell renormalization group following this analogy we can define relevant and irrelevant operators where the role of dimensionality is played by properties of the eigenvalue density these results also suggest an approach to the analysis of real data as an example we study neural activity in the vertebrate retina as it responds to naturalistic movies and find evidence of behavior controlled by a nontrivial fixed point applied to financial data our analysis separates modes dominated by sampling noise from a smaller but still macroscopic number of modes described by a non gaussian distribution

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

multiplicity of solutions for quasilinear elliptic equations in rn in the present paper we study some quasilinear elliptic problem for which we prove the existence of infinitely many weak solutions on rn all the coefficient involved the unknown function so the using of a nonsmooth critical point theory approach

density of smooth functions in wk p x ω kovacik rakosnik investigated the spaces l p x omega of functions which are integrable with variable power p x and the corresponding counterparts of the sobolev spaces w k p x omega we continue that investigation and describe a class of functions p x for which the set of smooth functions on omega is dense in w k p x omega as a corollary we obtain in terms of the distance function a condition on elements of w k p x omega sufficient to ensure that they belong to w 0 k p x omega