harmonic measure versus hausdorff measures on repellers for holomorphic maps this paper is a continuation of a joint paper of the author with f przytycki and m urbanski we study a harmonic measure on a boundary of so called repelling boundary domain an important example is a basin of a sink for a rational map using the results of the above mentioned paper we prove that either the boundary of the domain is an analytically embedded circle or interval or else the harmonic measure is singular with respect to the hausdors measure corresponding to the function xc t t exp c log t logloglog 1 for some c 0 0 introduction and statement of results let q be a simply connected domain in the riemann sphere card q 2 let r d q be the riemann map from the unit disc d onto q since r has nontangential limits almost everywhere on s1 ad d the image r l of the length measure i on s1 can be considered this measure denoted usually by co and called a harmonic measure on aq has been studied a long time see mk2 and puz for the corresponding references with the most interesting results obtained recently in two remarkable papers mkl and mk2 for an increasing function 0 di di 0 0 o we define an outer measure for a c c a a lim inf diambj where the infimum is taken over all coverings of a with balls of a diameter smaller than a the diameter is computed with respect to the spherical metrics on in particular for 0 t ta we obtain the ol dimensional hausdorff measure ata will be denoted by aa let u be a probability measure on borel subsets of hausdorff dimension of u is defined as

thermodynamics dimension and the weil petersson metric in this paper we study branched coverings of metrized simplicial trees f t t which arise from polynomial maps f c c with disconnected julia sets we show that the collection of all such trees up to scale forms a contractible space ptd compactifying the moduli space of polynomials of degree d that f records the asymptotic behavior of the multipliers of f and that any meromorphic family of polynomials over can be completed by a unique tree at its central fiber in the cubic case we give a combinatorial enumeration of the trees that arise and show that pt3 is itself a tree resume dans ce travail nous etudions des revetements ramifies d arbres metriques simpliciaux f t t qui sont obtenus a partir d applications polynomiales f c c possedant un ensemble de julia non connexe nous montrons que la collection de tous ces arbres a un facteur d echelle pres forme un espace contractile ptd qui compactifier l espace des modules des polynomes de degre d nous montrons aussi que f enrigistre le comportement asymptotique des multiplicateurs de f et que tout famille meromorphe de polynomes definis sur peut etre completee par un unique arbre comme sa fibre centrale dans le cas cubique nous donnons une enumeration combinatoire des arbres ainsi obtenus et montrons que pt3 est lui meme un arbre

manifold regularization a geometric framework for learning from labeled and unlabeled examples we propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution we focus on a semi supervised framework that incorporates labeled and unlabeled data in a general purpose learner some transductive graph learning algorithms and standard methods including support vector machines and regularized least squares can be obtained as special cases we use properties of reproducing kernel hilbert spaces to prove new representer theorems that provide theoretical basis for the algorithms as a result in contrast to purely graph based approaches we obtain a natural out of sample extension to novel examples and so are able to handle both transductive and truly semi supervised settings we present experimental evidence suggesting that our semi supervised algorithms are able to use unlabeled data effectively finally we have a brief discussion of unsupervised and fully supervised learning within our general framework

riemann map and holomorphic dynamics f f r a f r a and f u c c l a f r a i e fra repels to the side of a n o assume that u is always sufficiently small so that f has no singularities in u c a let r d z a be a riemann map a conformal homeomorphism of the unit disc onto a then there exists a holomorphic extension g of r l o f o r to a neighbourhood of s 1 and gls happens to be an expanding map i e there exists n 0 such that for every z s l l g z l although the proofs are straightforward we include them for sceptics in the last section sect 7 together with a remark about examples of a and f satisfying the above assumptions denote by gthe non tangential limit of r and by k s 1 the domain where it exists denote by ijl g the space of all borel probability g invariant ergodic positive entropy measures on s 1 it is known see p that for every egji g k exists almost everywhere so the f invariant measure r on fra can be considered the functions log g log lf are z respectively r 0 integrable see 1 5 denote these integrals by u g g f in this paper we consider derivatives with respect to the riemannian metric on 2 define the hausdorff dimension of any probability measure v

locally linear discriminant analysis for multimodally distributed classes for face recognition with a single model image we present a novel method of nonlinear discriminant analysis involving a set of locally linear transformations called locally linear discriminant analysis llda the underlying idea is that global nonlinear data structures are locally linear and local structures can be linearly aligned input vectors are projected into each local feature space by linear transformations found to yield locally linearly transformed classes that maximize the between class covariance while minimizing the within class covariance in face recognition linear discriminant analysis lida has been widely adopted owing to its efficiency but it does not capture nonlinear manifolds of faces which exhibit pose variations conventional nonlinear classification methods based on kernels such as generalized discriminant analysis gda and support vector machine svm have been developed to overcome the shortcomings of the linear method but they have the drawback of high computational cost of classification and overfitting our method is for multiclass nonlinear discrimination and it is computationally highly efficient as compared to gda the method does not suffer from overfitting by virtue of the linear base structure of the solution a novel gradient based learning algorithm is proposed for finding the optimal set of local linear bases the optimization does not exhibit a local maxima problem the transformation functions facilitate robust face recognition in a low dimensional subspace under pose variations using a single model image the classification results are given for both synthetic and real face data

l1 norm kernel discriminant analysis via bayes error bound optimization for robust feature extraction a novel discriminant analysis criterion is derived in this paper under the theoretical framework of bayes optimality in contrast to the conventional fisher s discriminant criterion the major novelty of the proposed one is the use of l1 norm rather than l2 norm which makes it less sensitive to the outliers with the l1 norm discriminant criterion we propose a new linear discriminant analysis l1 lda method for linear feature extraction problem to solve the l1 lda optimization problem we propose an efficient iterative algorithm in which a novel surrogate convex function is introduced such that the optimization problem in each iteration is to simply solve a convex programming problem and a close form solution is guaranteed to this problem moreover we also generalize the l1 lda method to deal with the nonlinear robust feature extraction problems via the use of kernel trick and hereafter proposed the l1 norm kernel discriminant analysis l1 kda method extensive experiments on simulated and real data sets are conducted to evaluate the effectiveness of the proposed method in comparing with the state of the art methods

analytical representations of the spread harmonic measure density we study the spread harmonic measure that characterizes the spatial distribution of reaction events on a partially reactive surface for euclidean domains in which brownian motion can be split into independent lateral and transverse displacements we derive analytical formulas for the spread harmonic measure density and analyze its asymptotic behavior this analysis is applicable to slab domains general cylindrical domains and a half space we investigate the spreading effect due to multiple reflections on the surface and the underlying role of finite reactivity we discuss further extensions and applications of analytical results to describe laplacian transfer phenomena such as permeation through semipermeable membranes secondary current distribution on partially blocking electrodes and surface relaxation in nuclear magnetic resonance

some remarks on normal maps with applications to eigenvalues and singular values of matrices abstract in this paper by using normal maps originated by lewis a s lewis group invariance and convex matrix analysis siam j matrix anal appl 17 1996 927 949 we present a method for producing majorization inequalities on eigenvalues and singular values of matrices some classical and new inequalities are derived

discriminative semi supervised feature selection via manifold regularization feature selection has attracted a huge amount of interest in both research and application communities of data mining we consider the problem of semi supervised feature selection where we are given a small amount of labeled examples and a large amount of unlabeled examples since a small number of labeled samples are usually insufficient for identifying the relevant features the critical problem arising from semi supervised feature selection is how to take advantage of the information underneath the unlabeled data to address this problem we propose a novel discriminative semi supervised feature selection method based on the idea of manifold regularization the proposed approach selects features through maximizing the classification margin between different classes and simultaneously exploiting the geometry of the probability distribution that generates both labeled and unlabeled data in comparison with previous semi supervised feature selection algorithms our proposed semi supervised feature selection method is an embedded feature selection method and is able to find more discriminative features we formulate the proposed feature selection method into a convex concave optimization problem where the saddle point corresponds to the optimal solution to find the optimal solution the level method a fairly recent optimization method is employed we also present a theoretic proof of the convergence rate for the application of the level method to our problem empirical evaluation on several benchmark data sets demonstrates the effectiveness of the proposed semi supervised feature selection method

dahlberg s theorem in higher co dimension abstract in 1977 the celebrated theorem of b dahlberg established that the harmonic measure is absolutely continuous with respect to the hausdorff measure on a lipschitz graph of dimension n 1 in r n and later this result has been extended to more general non tangentially accessible domains and beyond in the present paper we prove the first analogue of dahlberg s theorem in higher co dimension on a lipschitz graph of dimension d in r n d n 1 with a small lipschitz constant we construct a linear degenerate elliptic operator l such that the corresponding harmonic measure l is absolutely continuous with respect to the hausdorff measure on more generally we provide sufficient conditions on the matrix of coefficients of l which guarantee the mutual absolute continuity of l and the hausdorff measure

σ finiteness of elliptic measures for quasilinear elliptic pde in space abstract in this paper we study the hausdorff dimension of a elliptic measure f in space associated to a positive weak solution to a certain quasilinear elliptic pde in an open subset and vanishing on a portion of the boundary of that open set we show that this measure is concentrated on a set of finite n 1 dimensional hausdorff measure for p n and the same result holds for p n with an assumption on the boundary we also construct an example of a domain in space for which the corresponding measure has hausdorff dimension n 1 for p n for some which depends on various constants including p the first result generalizes the authors previous work in 3 when the pde is the p laplacian and the second result generalizes the well known theorem of wolff in 24 when p 2 and n 2

the geometry of the weil petersson metric in complex dynamics in this work we study an analogue of the weil petersson metric on the space of blaschke products of degree 2 proposed by mcmullen we show that the weilpetersson metric is incomplete and study its metric completion our work parallels known results for the teichmuller space of a punctured torus

graph embedding and extensions a general framework for dimensionality reduction a large family of algorithms supervised or unsupervised stemming from statistics or geometry theory has been designed to provide different solutions to the problem of dimensionality reduction despite the different motivations of these algorithms we present in this paper a general formulation known as graph embedding to unify them within a common framework in graph embedding each algorithm can be considered as the direct graph embedding or its linear kernel tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided furthermore the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms by utilizing this framework as a tool we propose a new supervised dimensionality reduction algorithm called marginal fisher analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class while the penalty graph connects the marginal points and characterizes the interclass separability we show that mfa effectively overcomes the limitations of the traditional linear discriminant analysis algorithm due to data distribution assumptions and available projection directions real face recognition experiments show the superiority of our proposed mfa in comparison to lda also for corresponding kernel and tensor extensions

a unified framework for semi supervised dimensionality reduction in practice many applications require a dimensionality reduction method to deal with the partially labeled problem in this paper we propose a semi supervised dimensionality reduction framework which can efficiently handle the unlabeled data under the framework several classical methods such as principal component analysis pca linear discriminant analysis lda maximum margin criterion mmc locality preserving projections lpp and their corresponding kernel versions can be seen as special cases for high dimensional data we can give a low dimensional embedding result for both discriminating multi class sub manifolds and preserving local manifold structure experiments show that our algorithms can significantly improve the accuracy rates of the corresponding supervised and unsupervised approaches

semisupervised generalized discriminant analysis generalized discriminant analysis gda is a commonly used method for dimensionality reduction in its general form it seeks a nonlinear projection that simultaneously maximizes the between class dissimilarity and minimizes the within class dissimilarity to increase class separability in real world applications where labeled data are scarce gda may not work very well however unlabeled data are often available in large quantities at very low cost in this paper we propose a novel gda algorithm which is abbreviated as semisupervised generalized discriminant analysis ssgda we utilize unlabeled data to maximize an optimality criterion of gda and formulate the problem as an optimization problem that is solved using the constrained concave convex procedure the optimization procedure leads to estimation of the class labels for the unlabeled data we propose a novel confidence measure and a method for selecting those unlabeled data points whose labels are estimated with high confidence the selected unlabeled data can then be used to augment the original labeled dataset for performing gda we also propose a variant of ssgda called m ssgda which adopts the manifold assumption to utilize the unlabeled data extensive experiments on many benchmark datasets demonstrate the effectiveness of our proposed methods

on the hausdorff dimension of harmonic measure in higher dimension assume a i n e a domain in f n where e is a compact set denote o a a x the harmonic measure for a of a evaluated at x r d according to 0ksendal s theorem o o e o a x is singular with respect to d dimensional lebesgue measure for d 2 and general domains this result seemed to be so far the only known localization property recently for d 2 it has been shown by p jones and t wolff j w that s o9 has dimension at most 1 this result completes previous work due to n g makarov and l carleson see i m and c2 let g be the green s function of a r with pole at some point of c r o9 in both the carleson and jones wolff arguments the integral g g nn log nn as boundary

one hundred years of complex dynamics holomorphic dynamics has been a key area of study for 100 years in this survey we focus on the dynamics of rational functions

flexible manifold embedding a framework for semi supervised and unsupervised dimension reduction we propose a unified manifold learning framework for semi supervised and unsupervised dimension reduction by employing a simple but effective linear regression function to map the new data points for semi supervised dimension reduction we aim to find the optimal prediction labels f for all the training samples x the linear regression function h x and the regression residue f0 f h x simultaneously our new objective function integrates two terms related to label fitness and manifold smoothness as well as a flexible penalty term defined on the residue f0 our semi supervised learning framework referred to as flexible manifold embedding fme can effectively utilize label information from labeled data as well as a manifold structure from both labeled and unlabeled data by modeling the mismatch between h x and f we show that fme relaxes the hard linear constraint f h x in manifold regularization mr making it better cope with the data sampled from a nonlinear manifold in addition we propose a simplified version referred to as fme u for unsupervised dimension reduction we also show that our proposed framework provides a unified view to explain and understand many semi supervised supervised and unsupervised dimension reduction techniques comprehensive experiments on several benchmark databases demonstrate the significant improvement over existing dimension reduction algorithms

conformal distortion of boundary sets conformal maps f of the disk into itself have the property that dim f 1 f dim f for any set f on the unit circle

null sets of harmonic measure on nta domains lipschitz approximation revisited we show the david jerison construction of big pieces of lipschitz graphs inside a corkscrew domain does not require surface measure be upper ahlfors regular thus we can study absolute continuity of harmonic measure and surface measure on nta domains of locally finite perimeter using lipschitz approximations a partial analogue of the f and m riesz theorem for simply connected planar domains is obtained for nta domains in space as one consequence every wolff snowflake has infinite surface measure

p harmonic measure in simply connected domains revisited let be a bounded simply connected domain in the complex plane c let n be a neighborhood of let p be xed 1 p 1 and let be a positive weak solution to the p laplace equation in n assume that u has zero boundary values on

boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions in this work we introduce the use of powerful tools from geometric measure theory gmt to study problems related to the size and structure of sets of mutual absolute continuity for the harmonic measure omega of a domain omega omega subset mathbb r n and the harmonic measure omega of omega omega int omega c in dimension n ge 3

complex projective structures lyapunov exponent degree and harmonic measure we study several new invariants associated to a holomorphic projective struc ture on a riemann surface of finite analytic type the lyapunov exponent of its holonomy which is of probabilistic dynamical nature and was introduced in our previous work the degree which measures the asymptotic covering rate of the developing map and a family of harmonic measures on the riemann sphere previously introduced by hussenot we show that the degree and the lyapunov exponent are related by a simple formula and give estimates for the hausdorff dimension of the harmonic measures in terms of the lyapunov exponent in accordance with the famous sullivan dictionary this leads to a descrip tion of the space of such projective structures that is reminiscent of that of the space of polynomials in holomorphic dynamics

improved kinetic renormalisation group approach to diffusion limited aggregation an improved kinetic renormalisation group approach to diffusion limited aggregation is presented this approach is based on the growth process itself and accounts for the dispersity in the growth probabilities it yields the multifractal spectrum d q with better values at smaller q on the 2d square lattice the authors find df 1 694 for the fractal dimensions of the cluster and that of its interface and d 1 1 01 for the information dimension the former agrees with the simulation results df approximately 1 70 and the latter compares very well with the exact value d 1 1

parallel algorithm and dynamic exponent for diffusion limited aggregation a parallel algorithm for diffusion limited aggregation dla is described and analyzed from the perspective of computational complexity the dynamic exponent z of the algorithm is defined with respect to the probabilistic parallel random access machine pram model of parallel computation according to t sim l z where l is the cluster size t is the running time and the algorithm uses a number of processors polynomial in l it is argued that z d d 2 2 where d is the fractal dimension and d 2 is the second generalized dimension simulations of dla are carried out to measure d 2 and to test scaling assumptions employed in the complexity analysis of the parallel algorithm it is plausible that the parallel algorithm attains the minimum possible value of the dynamic exponent in which case z characterizes the intrinsic history dependence of dla

julia sets and complex singularities in hierarchical ising models we study the analytical continuation in the complex plane of free energy of the ising model on diamond like hierarchical lattices it is known 12 13 that the singularities of free energy of this model lie on the julia set of some rational endomorphismf related to the action of the migdal kadanoff renorm group we study the asymptotics of free energy when temperature goes along hyperbolic geodesics to the boundary of an attractive basin off we prove that for almost all with respect to the harmonic measure geodesics the complex critical exponent is common and compute it

fine structure of connectedness loci we study distributions of the harmonic and lebesgue measures on the boundary of the connectedness locus mathcal m d for uni critical polynomials z d c and dynamical properties which are typical with respect to the harmonic measure one result is a conformal similarity between mathcal m d and the corresponding julia set for almost every point of mathcal m d with respect to the harmonic measure it is also shown that the harmonic measure is supported on lebesgue density points of the complement of mathcal m d which are not accessible from outside within john angles and at which the boundary of mathcal m d spirals infinitely often in both directions in the case of quadratic polynomials we prove that every parameter c from the boundary of the mandelbrot set mathcal m that is recurrent under iterates of z 2 c but not infinitely tunable is a lebesgue weak density point of the complement of mathcal m a direct consequence of our theory is the yoccoz local connectivity theorem a new technical ingredient in the paper is the method of amplification which is based on promoting the small scale geometry of the parameter space at a typical point in the sense of harmonic measure to uniform geometry of the corresponding julia set the pointwise conformality of the similarity map is derived through inducing techniques and a classical twb theory

embedding new observations via sparse coding for non linear manifold learning non linear dimensionality reduction techniques are affected by two critical aspects i the design of the adjacency graphs and ii the embedding of new test data the out of sample problem for the first aspect the proposed solutions in general were heuristically driven for the second aspect the difficulty resides in finding an accurate mapping that transfers unseen data samples into an existing manifold past works addressing these two aspects were heavily parametric in the sense that the optimal performance is only achieved for a suitable parameter choice that should be known in advance in this paper we demonstrate that the sparse representation theory not only serves for automatic graph construction as shown in recent works but also represents an accurate alternative for out of sample embedding considering for a case study the laplacian eigenmaps we applied our method to the face recognition problem to evaluate the effectiveness of the proposed out of sample embedding experiments are conducted using the k nearest neighbor knn and kernel support vector machines ksvm classifiers on six public face datasets the experimental results show that the proposed model is able to achieve high categorization effectiveness as well as high consistency with non linear embeddings manifolds obtained in batch modes highlightsa framework based on sparse coding is proposed for out of sample embedding the locality preserving property is jointly used with sparse coding classification performance and embedding consistency with batch modes are assessed experiments are conducted on six public face databases k nearest neighbor and kernel support vector machines classifiers are used

wavelet analysis of fractal boundaries part 2 multifractal analysis this second part deals with the global analysis of the boundary of domains omega rd we develop methods for determining the dimensions of the sets where the local behaviors introduced in part 1 occur these methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains

semi supervised classification based on random subspace dimensionality reduction graph structure is vital to graph based semi supervised learning however the problem of constructing a graph that reflects the underlying data distribution has been seldom investigated in semi supervised learning especially for high dimensional data in this paper we focus on graph construction for semi supervised learning and propose a novel method called semi supervised classification based on random subspace dimensionality reduction ssc rsdr in short different from traditional methods that perform graph based dimensionality reduction and classification in the original space ssc rsdr performs these tasks in subspaces more specifically ssc rsdr generates several random subspaces of the original space and applies graph based semi supervised dimensionality reduction in these random subspaces it then constructs graphs in these processed random subspaces and trains semi supervised classifiers on the graphs finally it combines the resulting base classifiers into an ensemble classifier experimental results on face recognition tasks demonstrate that ssc rsdr not only has superior recognition performance with respect to competitive methods but also is robust against a wide range of values of input parameters

two representations in multifractal analysis two representations in multifractal analysis the so called q and tau representations are discussed theoretically and computed practically complementary to the standard q representation the so called tau representation is especially suited to resolving the most rarified subsets of the distributed measure moreover these two representations are especially adapted respectively to the well known fixed size and fixed mass box counting algorithms both strategies are first applied to iteratively constructed mathematical measures once tested in this way we use them to analyse the mass distribution and the growth probability distribution of an experimental electrodeposited pattern

carleson sobolev measures for weighted bloch spaces abstract carleson sobolev measures for weighted bloch spaces on the unit ball of c n are described the classical carleson measures for growth spaces on special circular domains are characterized

some inequalities for sum and product of positive semidefinite matrices abstract the purpose of this paper is to present some inequalities on majorization unitarily invariant norm trace and eigenvalue for sum and product of positive semidefinite hermitian matrices some open questions proposed by marshall and olkin are resolved

the harmonic measure of balls in random trees we study properties of the harmonic measure of balls in typical large discrete trees for a ball of radius nn centered at the root we prove that although the size of the boundary is of order nn most of the harmonic measure is supported on a boundary set of size approximately equal to n n where 0 78 0 78 is a universal constant to derive such results we interpret harmonic measure as the exit distribution of the ball by simple random walk on the tree and we first deal with the case of critical galton watson trees conditioned to have height greater than nn an important ingredient of our approach is the analogous continuous model related to aldous continuum random tree where the dimension of harmonic measure of a level set of the tree is equal to whereas the dimension of the level set itself is equal to 11 the constant is expressed in terms of the asymptotic distribution of the conductance of large critical galton watson trees

efficient and robust feature extraction by maximum margin criterion in pattern recognition feature extraction techniques are widely employed to reduce the dimensionality of data and to enhance the discriminatory information principal component analysis pca and linear discriminant analysis lda are the two most popular linear dimensionality reduction methods however pca is not very effective for the extraction of the most discriminant features and lda is not stable due to the small sample size problem in this paper we propose some new linear and nonlinear feature extractors based on maximum margin criterion mmc geometrically feature extractors based on mmc maximize the average margin between classes after dimensionality reduction it is shown that mmc can represent class separability better than pca as a connection to lda we may also derive lda from mmc by incorporating some constraints by using some other constraints we establish a new linear feature extractor that does not suffer from the small sample size problem which is known to cause serious stability problems for lda the kernelized nonlinear counterpart of this linear feature extractor is also established in the paper our extensive experiments demonstrate that the new feature extractors are effective stable and efficient

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

inequalities theory of majorization and its applications by albert w marshall and ingram olkin abstract in the years since the appearance of marshall and olkin s masterful survey the influence of this work has been so great that some comments however incomplete and idiosyncratic on subsequent developments will be of value figure 1 an appendix lists minor errata to the first printing of the book

heat transport through rough channels abstract we investigate the two dimensional transport of heat through viscous flow between two parallel rough interfaces with a given fractal geometry the flow and heat transport equations are solved through direct numerical simulations and for different conduction convection conditions compared with the behavior of a channel with smooth interfaces the results for the rough channel at low and moderate values of the peclet number indicate that the effect of roughness is almost negligible on the efficiency of the heat transport system this is explained here in terms of makarov s theorem using the notion of active zone in laplacian transport at sufficiently high peclet numbers where convection becomes the dominant mechanism of heat transport the role of the interface roughness is to generally increase both the heat flux across the wall as well as the active length of heat exchange when compared with the smooth channel finally we show that this last behavior is closely related with the presence of recirculation zones in the re entrant regions of the fractal geometry

on the existence of conformal measures a general notion of conformal measure is introduced and some basic properties are studied sufficient conditions for the existence of these measures are obtained using a general construction principle the geometric properties of conformal measures relate equilibrium states and hausdorff measures this is shown for invariant subsets of s under expanding maps

bloch functions and asymptotic tail variance let p denote the bergman projection on the unit disk dd p z d w 1 zw 2da w z d where dada is normalized area measure we prove that if z 1 z 1 on dd then the integral i a r 02 exp ar4 p rei 2log 11 r2 d 2 0 r 1 has the bound i a r c a 10 1 a 3 2i a r c a 10 1 a 3 2 for 0 1a 1 no such uniform bound is possible we interpret the theorem in terms the asymptotic tail variance of such a bergman projection p p by the way the asymptotic tail variance induces a seminorm on the bloch space this improves upon earlier work of makarov which covers the range 0 a 264 0 1542 we then apply the theorem to obtain an estimate of the universal integral means spectrum for conformal mappings with a k quasiconformal extension for 0 k 10 k 1 the estimate reads for t ct c and 0 k 10 k 1 b k t 14k2 t 2 1 7k 2 for t 2k 1 7k 2 k t 1 1 7k 2 for t 2k 1 7k 2 which should be compared with the conjecture by prause and smirnov to the effect that for real t with t 2 k t 2 k we should have b k t 14k2t2

a supervised non linear dimensionality reduction approach for manifold learning in this paper we introduce a novel supervised manifold learning technique called supervised laplacian eigenmaps s le which makes use of class label information to guide the procedure of non linear dimensionality reduction by adopting the large margin concept the graph laplacian is split into two components within class graph and between class graph to better characterize the discriminant property of the data our approach has two important characteristics i it adaptively estimates the local neighborhood surrounding each sample based on data density and similarity and ii the objective function simultaneously maximizes the local margin between heterogeneous samples and pushes the homogeneous samples closer to each other our approach has been tested on several challenging face databases and it has been conveniently compared with other linear and non linear techniques demonstrating its superiority although we have concentrated in this paper on the face recognition problem the proposed approach could also be applied to other category of objects characterized by large variations in their appearance such as hand or body pose for instance

hausdorff dimension and σ finiteness of p harmonic measures in space when p n abstract in this paper we study a measure associated with a positive p harmonic function u defined in an open set o r n and vanishing on a portion of o if p n we show is concentrated on a set of finite h n 1 measure while if p n the same conclusion holds provided is uniformly fat in the sense of n capacity our work nearly answers in the affirmative a conjecture in lewis 2015 and also appears to be the natural extension of jones and wolff 1988 wolff 1993 to higher dimensions

bloch functions asymptotic variance and geometric zero packing motivated by a problem in quasiconformal mapping we introduce a problem in complex analysis with its roots in the mathematical physics of the bose einstein condensates in superconductivity the p

harmonic measure and polynomial julia sets there is a natural conjecture that the universal bounds for the dimension spectrum of harmonic measure are the same for simply connected and for non simply connected domains in the plane because of the close relation to conformal mapping theory the simply connected case is much better understood and proving the above statement would give new results concerning the properties of harmonic measure in the general case we establish the conjecture in the category of domains bounded by polynomial julia sets the idea is to consider the coefficients of the dynamical zeta function as subharmonic functions on a slice of teichmuller s space of the polynomial and then to apply the maximum principle 1 dimension spectrum of harmonic measure in this paper we discuss some properties of harmonic measure in the complex plane for a domain and a point a let a denote the harmonic measure of evaluated at a the measure a can be defined for instance as the hitting distribution of a brownian motion started at a if e then a e is the probability that a random brownian path first hits the boundary at a point of e much work has been devoted to describing dimensional properties of when the domain is as general as possible in particular jones and wolff 7 proved that no matter what the domain is harmonic measure is concentrated on a borel set of hausdorff dimension at most one in other words dim 1 for all plane domains 1 1 we are interested in finding similar but stronger universal estimates involving the dimension spectrum of 1 1 universal spectrum for every positive we denote f dim z where z is the lower pointwise dimension of z lim inf 0 log b z log b z is a general notation for the disc with center z and radius the universal dimension spectrum is the function sup f 1 2 where the supremum is taken over harmonic measures of all planar domains the first author is supported by n s f grant dms 9970283 the second author is supported by n s f grant dms 9800714

inductive and flexible feature extraction for semi supervised pattern categorization this paper proposes a novel discriminant semi supervised feature extraction method for generic classification and recognition tasks this method called inductive flexible semi supervised feature extraction is a graph based embedding method that seeks a linear subspace close to a non linear one it is based on a criterion that simultaneously exploits the discrimination information provided by the labeled samples maintains the graph based smoothness associated with all samples regularizes the complexity of the linear transform and minimizes the discrepancy between the unknown linear regression and the unknown non linear projection we extend the proposed method to the case of non linear feature extraction through the use of kernel trick this latter allows to obtain a nonlinear regression function with an output subspace closer to the learned manifold than that of the linear one extensive experiments are conducted on ten benchmark databases in order to study the performance of the proposed methods obtained results demonstrate a significant improvement over state of the art algorithms that are based on label propagation or semi supervised graph based embedding highlightsa flexible graph based semi supervised embedding is proposed a kernel version of the linear semi supervised algorithm is also proposed they simultaneously estimate a non linear embedding and its out of sample extension classification performance after embedding is assessed on ten benchmark datasets we use knn svm and two phase test sample sparse representation as classifiers

dissipation in laplacian fields across irregular boundaries the entropy production associated to a laplacian field distributed across irregular boundaries is studied in the context of the active zone approximation an explicit expression is given for the entropy production in terms of geometry whose relation to the variational formulation is discussed it is shown that the entropy production diminishes for successive prefractal generations of the same fractal generator so that the final fractal object is expected to dissipate less than all previous ones the relevance of this result in the abundance of fractal surfaces or interfaces observed in nature is discussed