random walks on graphs electric networks and fractals this paper is devoted to the problem of the nearest neighbour random walk on infinite graphs we investigate the rw x n started from a fixed vertex x0 x v of the graph g v e and the expected value of the first exit time t n from the n ball b n in g it will be shown that if g is sufficiently regular then d underset raise0 3em hbox smash scriptscriptstyle geqslant d omega d r underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and d r d 2 d omega where d r is the rw dimension d is the fractal dimension and d is the exponent of the growth of the resistance of b n though the method of the paper was developed for the case when x0 x is a fixed vertex of the graph we hope that the result can be generalized

sub gaussian estimates of heat kernels on infinite graphs we prove that a two sided sub gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the green kernel admits a uniform polynomial decay

learning robust and discriminative low rank representations for face recognition with occlusion for robust face recognition tasks we particularly focus on the ubiquitous scenarios where both training and testing images are corrupted due to occlusions previous low rank based methods stacked each error image into a vector and then used l1 or l2 norm to measure the error matrix however in the stacking step the structure information of the error image can be lost depart from the previous methods in this paper we propose a novel method by exploiting the low rankness of both the data representation and each occlusion induced error image simultaneously by which the global structure of data together with the error images can be well captured in order to learn more discriminative low rank representations we formulate our objective such that the learned representations are optimal for classification with the available supervised information and close to an ideal code regularization term with strong structure information preserving and discrimination capabilities the learned robust and discriminative low rank representation rdlrr works very well on face recognition problems especially with face images corrupted by continuous occlusions together with a simple linear classifier the proposed approach is shown to outperform several other state of the art face recognition methods on databases with a variety of face variations we focus on face recognition scenarios where both training and testing image are corrupted duo to occlusions we propose to learn robust and discriminative representation based on low rank matrix recovery model we solve the proposed model by using alm and the complexity analysis and convergence analysis are also provided experimental results demonstrate the effectiveness of our method

divergence form operators on fractal like domains we consider elliptic operators l in divergence form on certain domains in rd with fractal volume growth the domains we look at are pre sierpinski carpets which are derived from higher dimensional sierpinski carpets we prove a harnack inequality for non negative l harmonic functions on these domains and establish upper and lower bounds for the corresponding heat equation

maximum correntropy criterion for robust face recognition in this paper we present a sparse correntropy framework for computing robust sparse representations of face images for recognition compared with the state of the art l1norm based sparse representation classifier src which assumes that noise also has a sparse representation our sparse algorithm is developed based on the maximum correntropy criterion which is much more insensitive to outliers in order to develop a more tractable and practical approach we in particular impose nonnegativity constraint on the variables in the maximum correntropy criterion and develop a half quadratic optimization technique to approximately maximize the objective function in an alternating way so that the complex optimization problem is reduced to learning a sparse representation through a weighted linear least squares problem with nonnegativity constraint at each iteration our extensive experiments demonstrate that the proposed method is more robust and efficient in dealing with the occlusion and corruption problems in face recognition as compared to the related state of the art methods in particular it shows that the proposed method can improve both recognition accuracy and receiver operator characteristic roc curves while the computational cost is much lower than the src algorithms

robust regression for face recognition in this paper we address the problem of robust face recognition by formulating the pattern recognition task as a problem of robust estimation using a fundamental concept that in general patterns from a single object class lie on a linear subspace barsi and jacobs 2003 1 we develop a linear model representing a probe image as a linear combination of class specific galleries in the presence of noise the well conditioned inverse problem is solved using the robust huber estimation and the decision is ruled in favor of the class with the minimum reconstruction error the proposed robust linear regression classification rlrc algorithm is extensively evaluated for two important cases of robustness i e illumination variations and random pixel corruption illumination invariant face recognition is demonstrated on three standard databases under exemplary evaluation protocols reported in the literature comprehensive comparative analysis with the state of art illumination tolerant approaches indicates a comparable performance index for the proposed rlrc algorithm the efficiency of the proposed approach in the presence of severe random noise is validated under several exemplary noise models such as dead pixel problem salt and pepper noise speckle noise and additive white gaussian noise awgn the rlrc algorithm is found to be favorable compared with the benchmark generative approaches

estimates of transition densities for brownian motion on nested fractals we obtain upper and lower bounds for the transition densities of brownian motion on nested fractals compared with the estimate on the sierpinski gasket the results require the introduction of a new exponent dj related to the shortest path metric and chemical exponent on nested fractals further holder order of the resolvent densities sample paths and local times are obtained the results are obtained using the theory of multi type branching processes

robust nuclear norm based matrix regression with applications to robust face recognition face recognition fr via regression analysis based classification has been widely studied in the past several years most existing regression analysis methods characterize the pixelwise representation error via l 1 norm or l 2 norm which overlook the 2d structure of the error image recently the nuclear norm based matrix regression model is proposed to characterize low rank structure of the error image however the nuclear norm cannot accurately describe the low rank structural noise when the incoherence assumptions on the singular values does not hold since it overpenalizes several much larger singular values to address this problem this paper presents the robust nuclear norm to characterize the structural error image and then extends it to deal with the mixed noise the majorization minimization mm method is applied to derive a iterative scheme for minimization of the robust nuclear norm optimization problem then an efficiently alternating direction method of multipliers admm method is used to solve the proposed models we use weighted nuclear norm as classification criterion to obtain the final recognition results experiments on several public face databases demonstrate the effectiveness of our models in handling with variations of structural noise occlusion illumination and so on and mixed noise

obtaining upper bounds of heat kernels from lower bounds we show that a near diagonal lower bound of the heat kernel of a dirichlet form on a metric measure space with a regular measure implies an on diagonal upper bound if in addition the dirichlet form is local and regular then we obtain a full off diagonal upper bound of the heat kernel provided the dirichlet heat kernel on any ball satisfies a near diagonal lower estimate this reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel c 2007 wiley periodicals inc

linear regression for face recognition in this paper we present a novel approach of face identification by formulating the pattern recognition problem in terms of linear regression using a fundamental concept that patterns from a single object class lie on a linear subspace we develop a linear model representing a probe image as a linear combination of class specific galleries the inverse problem is solved using the least squares method and the decision is ruled in favor of the class with the minimum reconstruction error the proposed linear regression classification lrc algorithm falls in the category of nearest subspace classification the algorithm is extensively evaluated on several standard databases under a number of exemplary evaluation protocols reported in the face recognition literature a comparative study with state of the art algorithms clearly reflects the efficacy of the proposed approach for the problem of contiguous occlusion we propose a modular lrc approach introducing a novel distance based evidence fusion def algorithm the proposed methodology achieves the best results ever reported for the challenging problem of scarf occlusion

harnack inequalities and sub gaussian estimates for random walks we show that the beta parabolic harnack inequality for random walks on graphs is equivalent on one hand to the sub gaussian estimate for the transition probability and on the other hand to the conjunction of the elliptic harnack inequality the doubling volume property and the fact that the mean exit time in any ball of radius r is of the order r beta the latter condition can be replaced by a certain estimate of the resistance of annuli

hardy littlewood theory for semigroups soit t t t 0 un semi groupe symetrique sousmarkovien et soit n 2 alors on a les proprietes equivalentes a dim t t n b il existe c 2 0 s t t t f c 2 t n 2 f 1 f l 1 x t 0 en fait a b meme dans le cas n 2

unique solvability of max min fuzzy equations and strong regularity of matrices over fuzzy algebra a matrix is said to have strongly linearly independent columns or in the case of square matrices to be strongly regular if for some vector b the system a x b has a unique solution we formulate a necessary and sufficient condition for a linear system of equations over a fuzzy algebra to have a unique solution and prove the equivalence of strong regularity and trapezoidal property moreover an algorithm for testing these properties is reviewed

energy inequalities for cutoff functions and some applications let x d m be a metric measure space with a local regular dirichlet form we establish necessary and sufficient conditions for upper heat kernel bounds with subdiffusive space time exponent to hold this characterization is stable under rough isometries that is it is preserved under bounded perturbations of the dirichlet form further we give a criterion for stochastic completeness in terms of a sobolev inequality for cutoff functions as an example we show that this criterion applies to an anomalous diffusion on a geodesically incomplete fractal space where the well established criterion in terms of volume growth fails

random walks on supercritical percolation clusters we obtain gaussian upper and lower bounds on the transition density qt x y of the continuous time simple random walk on a supercritical percolation cluster c1 in the euclidean lattice the bounds analogous to aronsen s bounds for uniformly elliptic divergence form diusions hold with constants ci depending only on p the percolation probability and d the irregular nature of the medium means that the bound for qt x only holds for t sx where the constant sx depends on the percolation congura tion

solvability and unique solvability of max min fuzzy equations abstract the problem of solvability and the problem of unique solvability of a fuzzy relation equation in an arbitrary max min algebra are considered and corresponding necessary and sufficient conditions are presented the results allow to solve both problems by an o mnp algorithm where m n p are the dimensions of the corresponding relations in the equation the existence of the greatest and of the least solution is also considered

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

sparse regression with output correlation for cardiac ejection fraction estimation traditional regression methods minimize the sum of errors of samples with various regularization terms such as the 1 norm and 2 norm for the diagnosis of cardiovascular diseases the cardiac ejection fraction ef represents an essential measure however existing regularization terms do not consider the output correlation the correlation between ground truth volumes and estimated volumes w r t each subject which is beneficial in estimating the cardiac ef in this paper we first propose a sparse regression with two regularization terms of the 1 norm and output correlation sroc then we propose a one dimensional solution path algorithm for quickly finding two good regulation parameters in the formulation of sroc the solution path algorithm can effectively handle singularities and infinities in the key matrix finally we conduct experiments on a clinical cardiac image dataset with 100 subjects the experimental results show that our method produces competitive and strong results for estimating the cardiac ef based on quantitative and qualitative analyses

heat kernel fluctuations for a resistance form with non uniform volume growth in this article we consider the problem of estimating the heat kernel on measure metric spaces equipped with a resistance form such spaces admit a corresponding resistance metric that reflects the conductivity properties of the set in this situation it has been proved that when there is uniform polynomial volume growth with respect to the resistance metric the behaviour of the on diagonal part of the heat kernel is completely determined by this rate of volume growth however recent results have shown that for certain random fractal sets there are global and local point wise fluctuations in the volume as r 0 and so these uniform results do not apply motivated by these examples we present global and local on diagonal heat kernel estimates when the volume growth is not uniform and demonstrate that when the volume fluctuations are non trivial there will be non trivial fluctuations of the same order up to exponents in the short time heat kernel asymptotics we also provide bounds for the off diagonal part of the heat kernel these results apply to deterministic and random self similar fractals and metric space dendrites the topological analogues of graph trees

the einstein relation for random walks on graphs this paper investigates the einstein relation the connection between the volume growth the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph the einstein relation is proved under different set of conditions in the simplest case it is shown under the volume doubling and time comparison principles this and the other set of conditions provide the basic framework for the study of sub diffusive behavior of the random walks on weighted graphs

manifolds and graphs with slow heat kernel decay we give upper estimates on the long time behavior of the heat kernel on noncompact riemannian manifolds and infinite graphs which only depend on a lower bound of the volume growth we also show that these estimates are sharp

weighted sparse coding regularized nonconvex matrix regression for robust face recognition most existing regression based classification methods for robust face recognition usually characterize the representation error using l1 norm or frobenius norm for the pixel level noise or nuclear norm for the image level noise and code the coefficients vector by l1 norm or l2 norm to our best knowledge nuclear norm can be used to describe the low rank structural information but may lead to the suboptimal solution while l1 norm or l2 norm can promote the sparsity or cooperativity but may neglect the prior information e g the locality and similarity relationship among data to solve these drawbacks we propose two weighted sparse coding regularized nonconvex matrix regression models including weighted sparse coding regularized matrix norm based matrix regression wsmr for the structural noise and weighted sparse coding regularized matrix norm plus minimax concave plus mcp function based matrix regression wsm2r for the mixed noise e g structural noise plus sparse noise the mcp induced nonconvex function can overcome the imbalanced penalization of different singular values and entries of the error image matrix and the weighted sparse coding can consider the prior information by borrowing a novel distance metric the variants of inexact augmented lagrange multiplier ialm algorithm including nonconvex ialm ncialm and majorization minimization ialm mmialm are developed to solve the proposed models respectively the matrix norm based classifier is devised for classification finally experiments on four popular face image databases can validate the superiority of our methods compared with the state of the art regression methods

power dissipation in fractal feynman sierpinski ac circuits this paper studies the concept of power dissipation in infinite graphs and fractals associated with passive linear networks consisting of non dissipative elements in particular we analyze the so called feynman sierpinski ladder a fractal ac circuit motivated by feynman s infinite ladder that exhibits power dissipation and wave propagation for some frequencies power dissipation in this circuit is obtained as a limit of quadratic forms and the corresponding power dissipation measure associated with harmonic potentials is constructed the latter measure is proved to be continuous and singular with respect to an appropriate hausdorff measure defined on the fractal dust of nodes of the network

brownian motion on the sierpinski gasket we construct a brownian motion taking values in the sierpinski gasket a fractal subset of 2 and study its properties this is a diffusion process characterized by local isotropy and homogeneity properties we show for example that the process has a continuous symmetric transition density p t x y with respect to an appropriate hausdorff measure and obtain estimates on p t x y

stability of parabolic harnack inequalities let g e be a graph with weights a xy for which a parabolic harnack inequality holds with space time scaling exponent 2 suppose xy is another set of weights that are comparable to a xy we prove that this parabolic harnack inequality also holds for g e with the weights a xy we also give stable necessary and sufficient conditions for this parabolic harnack inequality to hold

strong regularity of matrices in a discrete bounded bottleneck algebra abstract the results concerning strong regularity of matrices over bottleneck algebras are reviewed we extend the known conditions to the discrete bounded case and modify the known algorithms for testing strong regularity

quenched invariance principles for walks on clusters of percolation or among random conductances in this work we principally study random walk on the supercritical infinite cluster for bond percolation on d we prove a quenched functional central limit theorem for the walk when d 4 we also prove a similar result for random walk among i i d random conductances along nearest neighbor edges of d when d 1

strong regularity of matrices in general max min algebra abstract the problem of the strong regularity of a square matrix in a general max min algebra is considered and a necessary and sufficient condition using the trapezoidal property is described the results are valid without any restrictions on the underlying max min algebra concerning the density or the boundedness previous results on this topic are special cases of the theorems presented in this paper

weak uncertainty principle for fractals graphs and metric measure spaces we develop a new approach to formulate and prove the weak uncertainty inequality which was recently introduced by okoudjou and strichartz we assume either an appropriate measure growth condition with respect to the effective resistance metric or in the absence of such a metric we assume the poincare inequality and reverse volume doubling property we also consider the weak uncertainty inequality in the context of nash type inequalities our results can be applied to a wide variety of metric measure spaces including graphs fractals and manifolds

random walk on the range of random walk we study the random walk x on the range of a simple random walk on d in dimensions d 4 when d 5 we establish quenched and annealed scaling limits for the process x which show that the intersections of the original simple random walk path are essentially unimportant for d 4 our results are less precise but we are able to show that any scaling limit for x will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions furthermore we demonstrate that when d 4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of x to the origin

strong regularity of matrices in a discrete bottleneck algebra abstract systems of linear equations of the form a x b over a discrete bottleneck algebra b where max and min are studied a square matrix a is said to be strongly regular if for some vector b the system a x b is uniquely solvable a necessary and sufficient condition for strong regularity is proved together with an o n 2 log n method for testing this property

heat kernel estimates and parabolic harnack inequalities for symmetric dirichlet forms abstract in this paper we consider the following symmetric dirichlet forms on a metric measure space m d e f g e c f g m m f x f y g x g y j d x d y where e c is a strongly local symmetric bilinear form and j d x d y is a symmetric radon measure on m m under general volume doubling condition on m d and some mild assumptions on scaling functions we establish stability results for upper bounds of heat kernel resp two sided heat kernel estimates in terms of the jumping kernels the cut off sobolev inequalities and the faber krahn inequalities resp the poincare inequalities we also obtain characterizations of parabolic harnack inequalities our results apply to symmetric diffusions with jumps even when the underlying spaces have walk dimensions larger than 2

robust nuclear norm regularized regression for face recognition with occlusion recently regression analysis based classification methods are popular for robust face recognition these methods use a pixel based error model which assumes that errors of pixels are independent this assumption does not hold in the case of contiguous occlusion where the errors are spatially correlated furthermore these methods ignore the whole structure of the error image nuclear norm as a matrix norm can describe the structural information well based on this point we propose a nuclear norm regularized regression model and use the alternating direction method of multipliers admm to solve it we thus introduce a novel robust nuclear norm regularized regression rnr method for face recognition with occlusion compared with the existing structured sparse error coding models which perform error detection and error support separately our method integrates error detection and error support into one regression model experiments on benchmark face databases demonstrate the effectiveness and robustness of our method which outperforms state of the art methods we propose a nuclear norm regularized regression model nr and use admm to solve it we also provide the complexity analysis and convergence analysis of nr the robust nr based classification rnr is introduced for face recognition experimental results demonstrate the advantages of our method

minorations pour les chaines de markov unidimensionnelles we prove a space time lower estimate for reversible random walks on graphs having linear growth this estimate complements a similar upper bound that follows from a work of hebisch and saloff coste

volume and time doubling of graphs and random walks the strongly recurrent case this paper proves upper and lower off diagonal sub gaussian transition probability estimates for strongly recurrent random walks under sufficient and necessary conditions besides the known conditions volume doubling and the elliptic harnack inequality a new property is introduced time doubling 2001 john wiley sons inc

recurrence and transience of branching diffusion processes on riemannian manifolds we relate the recurrence and transience of a branching diffusion process on a riemannian manifold m to some properties of a linear elliptic operator onm including spectral properties there is a trade off between the tendency of the transient brownian motion to escape and the birth process of the new particles if the latter has a high enough intensity then it may override the transience of the brownian motion leading to the recurrence of the branching process and vice versa in the case of a spherically symmetric manifold the critical intensity of the population growth can be found explicitly

transition density estimates for diffusion processes on post critically finite self similar fractals the recent development of analysis on fractal spaces is physically motivated by the study of diffusion in disordered media the natural questions that arise concern the existence and uniqueness of a suitable laplace operator and the behaviour of the associated heat semigroup on a space which is fractal the classes of fractals for which these questions were rst answered were classes of exactly self similar fractals with strong spatial symmetry such as nested or af ne nested fractals see for example 2 8 the existence of a laplacian and estimates on the heat kernel were obtained by considering the associated diffusion process and using the symmetry of the space the uniqueness of the laplacian for nested and af ne nested fractals has recently been solved through consideration of their dirichlet forms 23 in 15 the framework of post critically nite which we abbreviate to p c f self similar sets was introduced in order to capture the notion of exactly selfsimilar nitely rami ed fractals as used in the physics literature finitely rami ed fractals have the property that the intersection of any connected subset of the fractal with the rest of the set should occur only at a nite number of points this makes these structures much easier to analyse than in nitely rami ed sets such as the sierpinski carpet 5 the p c f self similar sets do not have spatial symmetry in general and have provided a mathematical test bed for analysis on fractals in this paper we will obtain uniform short time estimates on the heat kernel associated with a natural laplacian on the fractal a laplacian can be constructed on a p c f self similar fractal as a limit of discrete laplacians on graph approximations to the fractal based on the rami cation points we construct these operators via their dirichlet forms which can be set on any l space with a full measure n in 23 criteria are given which give a partial answer to the existence and uniqueness of laplace type operators on p c f self similar sets in a series of papers 16 17 18 20 25 7 some of the interesting spectral properties of p c f sets have been elucidated it has been shown that there can exist localized eigenfunctions if there is a high degree of symmetry in the set and corresponding laplace operator this corresponds to oscillation in the leading order term for the asymptotics of the eigenvalue counting function such behaviour occurs for nested fractals 7 and p c f sets with strong harmonic

characterization of sub gaussian heat kernel estimates on strongly recurrent graphs sub gaussian estimates for random walks are typical of fractal graphs we characterize them in the strongly recurrent case in terms of resistance estimates only without assuming elliptic harnack inequalities

spectra of graphs and fractal dimensions i in this paper we consider the nearest neighbour random walk on infinite graphs we discuss the connection between the two smallest eigenvalues of the laplacian of the graph and the diffusion speed of the rw

off diagonal upper estimates for the heat kernel of the dirichlet forms on metric spaces we give equivalent characterizations for off diagonal upper bounds of the heat kernel of a regular dirichlet form on the metric measure space in two settings for the upper bounds with the polynomial tail typical for jump processes and for the upper bounds with the exponential tail for diffusions our proofs are purely analytic and do not use the associated hunt process

linear independence in bottleneck algebras abstract let r be a linearly ordered set a bottleneck algebra is the set r together with two operations a b max a b and a b min a b in this paper we suggest various definitions of linear independence of sets of n tuples over r study their characteristics and interrelations

strong linear independence in bottleneck algebra abstract let b be a dense linearly ordered set without maximum and minimum and max min we say that a matrix a has strongly linearly independent sli columns if for some b the system a x b is uniquely solvable an n n matrix a a ij is said a to be strongly regular if it has sli columns b to have a strong permanent if the equality per a i 1 n a i i holds for unique p n per a is p n n i 1 a i i and p n is the set all permutations of the set 1 2 n we prove i that an m n matrix has sli columns if and only if it contains an n n submatrix which is strongly regular we derive an o mn log n algorithm for checking this property ii that every matrix with strong permanent is strongly regular and iii that a solution to the bottleneck assignment problem for strongly regular n n matrices can be found using o n 2 log n operations

iterative re constrained group sparse face recognition with adaptive weights learning in this paper we consider the robust face recognition problem via iterative re constrained group sparse classifier irgsc with adaptive weights learning specifically we propose a group sparse representation classification gsrc approach in which weighted features and groups are collaboratively adopted to encode more structure information and discriminative information than other regression based methods in addition we derive an efficient algorithm to optimize the proposed objective function and theoretically prove the convergence there are several appealing aspects associated with irgsc first adaptively learned weights can be seamlessly incorporated into the gsrc framework this integrates the locality structure of the data and validity information of the features into l 2 p norm regularization to form a unified formulation second irgsc is very flexible to different size of training set as well as feature dimension thanks to the l 2 p norm regularization third the derived solution is proved to be a stationary point globally optimal if p geq 1 comprehensive experiments on representative data sets demonstrate that irgsc is a robust discriminative classifier which significantly improves the performance and efficiency compared with the state of the art methods in dealing with face occlusion corruption and illumination changes and so on

harmonic calculus on p c f self similar sets the main object of this paper is the laplace operator on a class of fractals first we establish the concept of the renormalization of difference operators on post critically finite p c f for short self similar sets which are large enough to include finitely ramified self similar sets and extend the results for sierpinski gasket given in 10 to this class under each invariant operator for renormalization the laplace operator green function dirichlet form and neumann derivatives are explicitly constructed as the natural limits of those on finite pre self similar sets which approximate the p c f self similar sets also harmonic functions are shown to be finite dimensional and they are characterized by the solution of an infinite system of finite difference equations

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

analysis on nilpotent groups soit g un groupe de lie nilpotent reel simplement connexe et soit l son algebre de lie soient l 1 l et l k 1 l k l k 1 2 soit d d g d l k 1 kdim r l k l k 1 d est la dimension de dirichlet de g soit h un groupe nilpotent discret finiment genere on demontre des theoremes d analyse pour g

explicit formulas for heat kernels on diamond fractals this paper provides explicit pointwise formulas for the heat kernel on compact metric measure spaces that belong to a mathbb n times mathbb n parameter family of fractals which are regarded as projective limits of metric measure graphs and do not satisfy the volume doubling property the formulas are applied to obtain uniform continuity estimates of the heat kernel and to derive an expression of the fundamental solution of the free schrodinger equation the results also open up the possibility to approach infinite dimensional spaces based on this model

acquiring linear subspaces for face recognition under variable lighting previous work has demonstrated that the image variation of many objects human faces in particular under variable lighting can be effectively modeled by low dimensional linear spaces even when there are multiple light sources and shadowing basis images spanning this space are usually obtained in one of three ways a large set of images of the object under different lighting conditions is acquired and principal component analysis pca is used to estimate a subspace alternatively synthetic images are rendered from a 3d model perhaps reconstructed from images under point sources and again pca is used to estimate a subspace finally images rendered from a 3d model under diffuse lighting based on spherical harmonics are directly used as basis images in this paper we show how to arrange physical lighting so that the acquired images of each object can be directly used as the basis vectors of a low dimensional linear space and that this subspace is close to those acquired by the other methods more specifically there exist configurations of k point light source directions with k typically ranging from 5 to 9 such that by taking k images of an object under these single sources the resulting subspace is an effective representation for recognition under a wide range of lighting conditions since the subspace is generated directly from real images potentially complex and or brittle intermediate steps such as 3d reconstruction can be completely avoided nor is it necessary to acquire large numbers of training images or to physically construct complex diffuse harmonic light fields we validate the use of subspaces constructed in this fashion within the context of face recognition

diffusion on the scaling limit of the critical percolation cluster in the diamond hierarchical lattice we construct critical percolation clusters on the diamond hierarchical lattice and show that the scaling limit is a graph directed random recursive fractal a dirichlet form can be constructed on the limit set and we consider the properties of the associated laplace operator and diffusion process in particular we contrast and compare the behaviour of the high frequency asymptotics of the spectrum and the short time behaviour of the on diagonal heat kernel for the percolation clusters and for the underlying lattice in this setting a number of features of the lattice are inherited by the critical cluster

harmonic calculus on limits of networks and its application to dendrites abstract the main purpose of this paper is to give a general framework to analysis on fractals including finitely ramified self similar sets random fractals canter sets dendrites and the recurrent case of infinitely ramified self similar sets we study a limit of a sequence of electrical networks satisfying certain compatibility which is a generalization of the decimation invariance of random walks playing an important role in constructing diffusion processes on self similar sets as a limit we propose notions of finite resistance forms and effective resistance metrics also it is shown that a finite resistance form induces an effective resistance metric and vice versa a finite resistance form becomes a regular dirichlet form if the domain is dense in l 2 space in such a case the effective resistance metric is thought of as the intrinsic metric for the dirichlet form in the latter half of the paper we apply the general framework to dendrites which are topological spaces containing no loops a shortest path metric where all arcs are geodesics on a dendrite is shown to be an effective resistance metric and the corresponding finite resistance form turns out to be a local regular dirichlet form furthermore we establish a simple relation between the hausdorff dimensions and the spectral dimensions of dendrites with shortest path metrics

dirichlet forms on fractals poincare constant and resistance we study dirichlet forms associated with random walks on fractal like finite grahs we consider related poincare constants and resistance and study their asymptotic behaviour we construct a markov semi group on fractals as a subsequence of random walks and study its properties finally we construct self similar diffusion processes on fractals which have a certain recurrence property and plenty of symmetries

parabolic harnack inequality and heat kernel estimates for random walks with long range jumps we investigate the relationships between the parabolic harnack inequality heat kernel estimates some geometric conditions and some analytic conditions for random walks with long range jumps unlike the case of diffusion processes the parabolic harnack inequality does not in general imply the corresponding heat kernel estimates

transition densities for brownian motion on the sierpinski carpet upper and lower bounds are obtained for the transition densitiesp t x y of brownian motion on the sierpinski carpet these are of the same form as those which hold for the sierpinski gasket in addition the joint continuity ofp t x y is proved the existence of the spectral dimension is established and the einstein relation connecting the spectral dimension the hausdorff dimension and the resistance exponent is shown to hold

weyl s problem for the spectral distribution of laplacians on p c f self similar fractals we establish an analogue of weyl s classical theorem for the asymptotics of eigenvalues of laplacians on a finitely ramified i e p c f self similar fractalk such as for example the sierpinski gasket we consider both dirichlet and neumann boundary conditions as well as laplacians associated with bernoulli type multifractal measures onk from a physical point of view we study the density of states for diffusions or for wave propagation in fractal media more precisely let x be the number of eigenvalues less thanx then we show that x is of the order ofxds 2 asx where the spectral exponent ds is computed in terms of the geometric as well as analytic structures ofk further we give an effective condition that guarantees the existence of the limit ofx ds 2 x asx this condition is in some sense generic in addition we define in terms of the above spectral exponents and calculate explicitly the spectral dimension ofk

the general trapezoidal algorithm for strongly regular max min matrices abstract the problem of the strong regularity for square matrices over a general max min algebra is considered an o n 2 log n algorithm for recognition of the strong regularity of a given n n matrix is proposed the algorithm works without any restrictions on the underlying max min algebra concerning the density or the boundedness

scaling limit for the ant in a simple high dimensional labyrinth we prove that after suitable rescaling the simple random walk on the trace of a large critical branching random walk in mathbb z d converges to the brownian motion on the integrated super brownian excursion when d 14