definition of the riesz derivative and its application to space fractional quantum mechanics we investigate and compare different representations of the riesz derivative which plays an important role in anomalous diffusion and space fractional quantum mechanics in particular we show that a certain representation of the riesz derivative r x that is generally given as also valid for 1 behaves no differently than the other definition given in terms of its fourier transform in the light of this we discuss the 1 limit of the space fractional quantum mechanics and its consistency

on fractional powers of generators of fractional resolvent families abstract we show that if a generates a bounded times resolvent family for some 0 2 then a generates an analytic times resolvent family for 0 2 2 and 0 2 and a generalized subordination principle is derived in particular if a generates a bounded times resolvent family for some 1 2 then a 1 generates an analytic c 0 semigroup such relations are applied to study the solutions of cauchy problems of fractional order and first order

cauchy problems for fractional differential equations with riemann liouville fractional derivatives abstract in this paper we are concerned with cauchy problems of fractional differential equations with riemann liouville fractional derivatives in infinite dimensional banach spaces we introduce the notion of fractional resolvent obtain some its properties and present a generation theorem for exponentially bounded fractional resolvents moreover we prove that a homogeneous order cauchy problem is well posed if and only if its coefficient operator is the generator of an order fractional resolvent and we give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of an inhomogeneous order cauchy problem

on the nonlocality of the fractional schrodinger equation a number of papers over the past eight years have claimed to solve the fractional schrodinger equation for systems ranging from the one dimensional infinite square well to the coulomb potential to one dimensional scattering with a rectangular barrier however some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal preventing the fractional schrodinger equation from being solved in the usual piecewise fashion we focus on the one dimensional infinite square well and show that the purported ground state which is based on a piecewise approach is definitely not a solution of the fractional schrodinger equation for the general fractional parameter on a more positive note we present a solution to the fractional schrodinger equation for the one dimensional harmonic oscillator with 1

time fractional telegraph equations and telegraph processes with brownian time we study the fundamental solutions to time fractional telegraph equations of order 2 we are able to obtain the fourier transform of the solutions for any and to give a representation of their inverse in terms of stable densities for the special case 1 2 we can show that the fundamental solution is the distribution of a telegraph process with brownian time in a special case this becomes the density of the iterated brownian motion which is therefore the fundamental solution to a fractional diffusion equation of order 1 2 with respect to time

automatic smoothness detection of the resolvent krylov subspace method for the approximation of c_0 semigroups the resolvent krylov subspace method builds approximations to operator functions f a times a vector v for the semigroup and related operator functions this method is proved to possess the favorable property that the convergence is automatically faster when the vector v is smoother the user of the method does not need to know the presented theory and alterations of the method are not necessary in order to adapt to the possibly unknown smoothness of v the findings are illustrated by numerical experiments

multiplicity and concentration results for fractional schrodinger system with steep potential wells abstract this paper is concerned with the fractional coupled schrodinger system by using the nehari manifold and fibering map we obtain the multiplicity and concentration of solutions for the given problem with steep potential wells where some new estimates will be established in particular although there exist concave convex nonlinearities in the coupled system it is not necessary to assume that the corresponding lebesgue norms of the weight functions of the convex terms need to be small enough

the fractional d alembert s formulas abstract in this paper we develop generalized d alembert s formulas for abstract fractional integro differential equations and fractional differential equations on banach spaces some examples are given to illustrate our abstract results and the probability interpretations of these fractional d alembert s formulas are also given moreover we also provide d alembert s formulas for abstract fractional telegraph equations

convergence analysis of the extended krylov subspace method for the lyapunov equation the extended krylov subspace method has recently arisen as a competitive method for solving large scale lyapunov equations using the theoretical framework of orthogonal rational functions in this paper we provide a general a priori error estimate when the known term has rank one special cases such as symmetric coefficient matrix are also treated numerical experiments confirm the proved theoretical assertions

on boundary values of fractional resolvent families abstract boundary values of analytic fractional resolvent families are deduced via the approximation of fractional powers of operators and square root reductions of fractional resolvent families are given as well

a continuum limit of the relativistic toda lattice asymptotic theory of discrete laurent orthogonal polynomials with varying recurrence coefficients we consider the continuum limit of the relativistic toda lattice in particular we propose a method in order to integrate this system of nonlinear partial differential equations for some particular initial data and boundary conditions before possible shocks first we recall the relation between the finite relativistic toda lattice and the theory of discrete laurent orthogonal polynomials our analysis is then based on some results for the asymptotic theory of discrete laurent orthogonal polynomials with varying recurrence coefficients and the connection with a constrained and weighted extremal problem for logarithmic potentials

on adaptive choice of shifts in rational krylov subspace reduction of evolutionary problems we compute u t exp ta varphi using rational krylov subspace reduction for 0 leq t infty where u t varphi in mathbf r n and 0 a a in mathbf r n times n a priori optimization of the rational krylov subspaces for this problem was considered in v druskin l knizhnerman and m zaslavsky siam j sci comput 31 2009 pp 3760 3780 there was suggested an algorithm generating sequences of equidistributed shifts which are asymptotically optimal for the cases with uniform spectral distributions here we develop a recursive greedy algorithm for choice of shifts taking into account nonuniformity of the spectrum the algorithm is based on an explicit formula for the residual in the frequency domain allowing adaptive shift optimization at negligible cost the effectiveness of the developed approach is demonstrated in an example of the three dimensional diffusion problem for maxwell s equation arising in geophysical exploration we compare our approach with the one using the above mentioned equidistributed sequences of shifts numerical examples show that our algorithm is able to adapt to the spectral density of operator a for examples with near uniform spectral distributions both algorithms show the same convergence rates but the new algorithm produces superior convergence for cases with nonuniform spectra

modulating shape features by color attention for object recognition bag of words based image representation is a successful approach for object recognition generally the subsequent stages of the process feature detection feature description vocabulary construction and image representation are performed independent of the intentioned object classes to be detected in such a framework it was found that the combination of different image cues such as shape and color often obtains below expected results this paper presents a novel method for recognizing object categories when using multiple cues by separately processing the shape and color cues and combining them by modulating the shape features by category specific color attention color is used to compute bottom up and top down attention maps subsequently these color attention maps are used to modulate the weights of the shape features in regions with higher attention shape features are given more weight than in regions with low attention we compare our approach with existing methods that combine color and shape cues on five data sets containing varied importance of both cues namely soccer color predominance flower color and shape parity pascal voc 2007 and 2009 shape predominance and caltech 101 color co interference the experiments clearly demonstrate that in all five data sets our proposed framework significantly outperforms existing methods for combining color and shape information

estimation of ergodicity limits of bag of words modeling for guaranteed stochastic convergence abstract this paper suggests an efficient dual ergodicity limits based bag of words del bow modeling technique the suggested del bow technique estimates two limits of ergodicity of a discrete random variable drv that is formed from the bow classification performance of multiple runs the first limit of ergodicity is estimated with a relatively larger ball of convergence to keep the drv shorter hence both robustness against random initialization and estimation of the optimal model order are realized with a reduced number of iterations once the optimal model order is estimated the radius of ball of convergence is reduced and a second limit of ergodicity is estimated reducing the ball of convergence enlarges the size of the considered performance drv that enhances the classification performance experiments conducted on caltech 101 caltech 256 15 scenes and flower 102 datasets resulted in classification accuracy of 86 91 72 57 90 57 and 90 86 respectively comparison with state of the art techniques shows the excellent performance of the del bow modeling process

image retrieval based on multi texton histogram this paper presents a novel image feature representation method called multi texton histogram mth for image retrieval mth integrates the advantages of co occurrence matrix and histogram by representing the attribute of co occurrence matrix using histogram it can be considered as a generalized visual attribute descriptor but without any image segmentation or model training the proposed mth method is based on julesz s textons theory and it works directly on natural images as a shape descriptor meanwhile it can be used as a color texture descriptor and leads to good performance the proposed mth method is extensively tested on the corel dataset with 15000 natural images the results demonstrate that it is much more efficient than representative image feature descriptors such as the edge orientation auto correlogram and the texton co occurrence matrix it has good discrimination power of color texture and shape features

a statistical approach to texture classification from single images we investigate texture classification from single images obtained under unknown viewpoint and illumination a statistical approach is developed where textures are modelled by the joint probability distribution of filter responses this distribution is represented by the frequency histogram of filter response cluster centres textons recognition proceeds from single uncalibrated images and the novelty here is that rotationally invariant filters are used and the filter response space is low dimensional classification performance is compared with the filter banks and methods of leung and malik ijcv 2001 schmid cvpr 2001 and cula and dana ijcv 2004 and it is demonstrated that superior performance is achieved here classification results are presented for all 61 materials in the columbia utrecht texture database we also discuss the effects of various parameters on our classification algorithm such as the choice of filter bank and rotational invariance the size of the texton dictionary as well as the number of training images used finally we present a method of reliably measuring relative orientation co occurrence statistics in a rotationally invariant manner and discuss whether incorporating such information can enhance the classifier s performance

acceleration of contour integration techniques by rational krylov subspace methods we suggest a rational krylov subspace approximation for products of matrix functions and a vector appearing in exponential integrators we consider matrices with a field of values in a sector lying in the left complex half plane the choice of the poles for our method is suggested by a fixed rational approximation based on contour integration along a hyperbola around the sector compared to the fixed approximation our rational krylov subspace method exhibits an accelerated and more stable convergence of order o e c n

convergence analysis of an extended krylov subspace method for the approximation of operator functions in exponential integrators we analyze the convergence of an extended krylov subspace method for the approximation of operator functions that appear in exponential integrators for operators the size of the polynomial part of the extended krylov subspace is restricted according to the smoothness of the initial data this restriction for the continuous operator has a significant influence on the approximation of matrix functions evaluated for matrices stemming from space discretizations of the continuous operator we prove convergence of the method for the continuous operator and in the discrete case this leads to a convergence independent of the norm of the discretized operator uniformly over all possible grids the analysis is illustrated by numerical experiments

existence of solutions for perturbed fractional p laplacian equations abstract the purpose of this paper is to investigate the existence of weak solutions for a perturbed nonlinear elliptic equation driven by the fractional p laplacian operator as follows p s u v x u p 2 u a x u r 2 u b x u q 2 u in r n where is a real parameter p s is the fractional p laplacian operator with 0 s 1 p p r min q p s and v a b r n 0 are three positive weights using variational methods we obtain nonexistence and multiplicity results for the above mentioned equations depending on and according to the integrability properties of the ratio a q p b r p our results extend the previous work of autuori and pucci 2013 5 to the fractional p laplacian setting furthermore we weaken one of the conditions used in their paper hence the results of this paper are new even in the fractional laplacian case

local binary circumferential and radial derivative pattern for texture classification motivated by gradients polar coordinate representation and its discrete approximation investigating the theoretical scheme of local differential approximation to build feature circumferential and radial derivatives pattern to capture tangential and radial information of derivatives respectively new descriptor is built based on radial plus tangential components and derivative information our approach produces high classification performance on several texture databases building discriminative and robust texture representation to deal with the changes of texture appearance is a fundamental issue in texture classification the local binary pattern lbp and its variants gain a lot of attention during the past decade and achieve great success in texture description however the current existing lbp based features which treat lbp as local differential or orientation gradient operator exploited local orientation pattern or anisotropic structure information separately in this paper we investigate the theoretical scheme of local differential approximation on the polar coordinate system in order to build a new lbp based descriptor which better takes into account both radial plus tangential components and derivative information first we present an operator called circumferential derivative cd based on the tangential information with different order of derivatives then we present an operator called radial derivative rd based on the radial information with different order of derivatives both extract complementary information locally around a central pixel a new descriptor the local binary circumferential and radial derivative pattern crdp is constructed to fuse both local circumferential and radial derivative features based on different orders as well as a global feature based on global difference gd of central pixels intensity extensive experiments on outex curet kth tips and kth tips2 a texture datasets indicate that the proposed crdp descriptor is discriminative and robust the results obtained by the proposed crdp descriptor outperforms more than twenty recent lbp based state of the art methods including the best reported results in the literature for aforementioned texture datasets to the best of our knowledge

superlinear convergence of the rational arnoldi method for the approximation of matrix functions a superlinear convergence bound for rational arnoldi approximations to functions of matrices is derived this bound generalizes the well known superlinear convergence bound for the conjugate gradient method to more general functions with finite singularities and to rational krylov spaces a constrained equilibrium problem from potential theory is used to characterize a max min quotient of a nodal rational function underlying the rational arnoldi approximation where an additional external field is required for taking into account the poles of the rational krylov space the resulting convergence bound is illustrated at several numerical examples in particular the convergence of the extended krylov method for the matrix square root

texton theory revisited a bag of words approach to combine textons the aim of this paper is to revisit an old theory of texture perception and update its computational implementation by extending it to colour with this in mind we try to capture the optimality of perceptual systems this is achieved in the proposed approach by sharing well known early stages of the visual processes and extracting low dimensional features that perfectly encode adequate properties for a large variety of textures without needing further learning stages we propose several descriptors in a bag of words framework that are derived from different quantisation models on to the feature spaces our perceptual features are directly given by the shape and colour attributes of image blobs which are the textons in this way we avoid learning visual words and directly build the vocabularies on these low dimensional texton spaces main differences between proposed descriptors rely on how co occurrence of blob attributes is represented in the vocabularies our approach overcomes current state of art in colour texture description which is proved in several experiments on large texture datasets

from potential theory to matrix iterations in six steps the theory of the convergence of krylov subspace iterations for linear systems of equations conjugate gradients biconjugate gradients gmres qmr bi cgstab and so on is reviewed for a computation of this kind an estimated asymptotic convergence factor rho le 1 can be derived by solving a problem of potential theory or conformal mapping six approximations are involved in relating the actual computation to this scalar estimate these six approximations are discussed in a systematic way and illustrated by a sequence of examples computed with tools of numerical conformal mapping and semidefinite programming

influence of normalization and color space to color texture classification abstract color texture classification has recently attracted significant attention due to its multiple applications the color texture images depend on the texture surface and its albedo the illumination the camera and its viewing position a key problem to get an acceptable performance is the ambient illumination which can vary the perceived structures in the surface given a color texture classification problem it would be desirable to know which is the best approach to solve the problem making the minimal assumptions about the illumination conditions the present work does an exhaustive evaluation of the state of the art color texture classification methods considering 5 different color spaces 12 normalization methods to achieve illumination invariances 19 texture feature vectors and 23 pure color feature vectors our experiments allow to conclude that parallel approaches are better than integrative approaches for color texture classification achieving the first positions in the friedman ranking multiresolution local binary patterns mlbp are the best intensity texture features followed by wavelet and gabor filters combined with luminance chrominance color spaces lab and lab2000hl and for pure color classification the best are first order statistics fos calculated in rgb color space for intensity texture features the learning methods work better on the four smallest datasets although they could not be tested in other four bigger datasets due to its huge computational cost nor with color texture classification normalization and color spaces slightly increase the average accuracy of color texture classification although the differences achieved using normalization are not statistically significant in a paired t test lab2000hl and rgb are the best color spaces but the former is the slowest one regarding elapsed time the best vector features mlbp for intensity texture daub4 daubechies filters using mean and variance statistics for color texture and fos for pure color are nearly the fastest or are in the middle interval of all the tested methods

what are textons textons refer to fundamental micro structures in natural images and videos and are considered as the atoms of pre attentive human visual perception julesz 1981 unfortunately the word texton remains a vague concept in the literature for lack of a good mathematical model in this article we first present a three level generative image model for learning textons from texture images in this model an image is a superposition of a number of image bases selected from an over complete dictionary including various gabor and laplacian of gaussian functions at various locations scales and orientations these image bases are in turn generated by a smaller number of texton elements selected from a dictionary of textons by analogy to the waveform phoneme word hierarchy in speech the pixel base texton hierarchy presents an increasingly abstract visual description and leads to dimension reduction and variable decoupling by fitting the generative model to observed images we can learn the texton dictionary as parameters of the generative model then the paper proceeds to study the geometric dynamic and photometric structures of the texton representation by further extending the generative model to account for motion and illumination variations 1 for the geometric structures a texton consists of a number of image bases with deformable spatial configurations the geometric structures are learned from static texture images 2 for the dynamic structures the motion of a texton is characterized by a markov chain model in time which sometimes can switch geometric configurations during the movement we call the moving textons as motons the dynamic models are learned using the trajectories of the textons inferred from video sequence 3 for photometric structures a texton represents the set of images of a 3d surface element under varying illuminations and is called a lighton in this paper we adopt an illumination cone representation where a lighton is a texton triplet for a given light source a lighton image is generated as a linear sum of the three texton bases we present a sequence of experiments for learning the geometric dynamic and photometric structures from images and videos and we also present some comparison studies with k mean clustering sparse coding independent component analysis and transformed component analysis we shall discuss how general textons can be learned from generic natural images

object localization using color texture and shape abstract we address the problem of localizing objects using color texture and shape given a handrawn sketch for querying an object shape and its color and texture the proposed algorithm automatically searches the image database for objects which meet the query attributes the database images do not need to be presegmented or annotated the proposed algorithm operates in two stages in the first stage we use local texture and color features to find a small number of candidate images in the database and identify regions in the candidate images which share similar texture and color as the query to speed up the processing the texture and color features are directly extracted from the discrete cosine transform dct compressed domain in the second stage we use a deformable template matching method to match the query shape to the image edges at the locations which possess the desired texture and color attributes this algorithm is different from other content based image retrieval algorithms in that i no presegmentation of the database images is needed and ii the color and texture features are directly extracted from the compressed images experimental results demonstrate performance of the algorithm and show that substantial computational savings can be achieved by utilizing multiple image cues

iterative exponential filtering for large discrete ill posed problems we describe a new iterative method for the solution of large very ill conditioned linear systems of equations that arise when discretizing linear ill posed problems the right hand side vector represents the given data and is assumed to be contaminated by measurement errors our method applies a filter function of the form varphi beta t 1 exp beta t 2 with the purpose of reducing the influence of the errors in the right hand side vector on the computed approximate solution of the linear system here beta is a regularization parameter the iterative method is derived by expanding varphi beta t in terms of chebyshev polynomials the method requires only little computer memory and is well suited for the solution of large scale problems we also show how a value of beta and an associated approximate solution that satisfies the morozov discrepancy principle can be computed efficiently an application to image restoration illustrates the performance of the method

on fractional duhamelʼs principle and its applications abstract the classical duhamel principle established nearly 200 years ago by jean marie constant duhamel reduces the cauchy problem for an inhomogeneous partial differential equation to the cauchy problem for the corresponding homogeneous equation in this paper we generalize this famous principle to a wide class of fractional order differential operator equations

convergence analysis of krylov subspace iterations with methods from potential theory krylov subspace iterations are among the best known and most widely used numerical methods for solving linear systems of equations and for computing eigenvalues of large matrices these methods are polynomial methods whose convergence behavior is related to the behavior of polynomials on the spectrum of the matrix this leads to an extremal problem in polynomial approximation theory how small can a monic polynomial of a given degree be on the spectrum this survey gives an introduction to a recently developed technique to analyze this extremal problem in the case of symmetric matrices it is based on global information on the spectrum in the sense that the eigenvalues are assumed to be distributed according to a certain measure then depending on the number of iterations the lanczos method for the calculation of eigenvalues finds those eigenvalues that lie in a certain region which is characterized by means of a constrained equilibrium problem from potential theory the same constrained equilibrium problem also describes the superlinear convergence of conjugate gradients and other iterative methods for solving linear systems

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

superlinear convergence of conjugate gradients we give a theoretical explanation for superlinear convergence behavior observed while solving large symmetric systems of equations using the conjugate gradient method or other krylov subspace methods we present a new bound on the relative error after n iterations this bound is valid in an asymptotic sense when the size n of the system grows together with the number of iterations the bound depends on the asymptotic eigenvalue distribution and on the ratio n n under appropriate conditions we show that the bound is asymptotically sharp our findings are related to some recent results concerning asymptotics of discrete orthogonal polynomials an important tool in our investigations is a constrained energy problem in logarithmic potential theory the new asymptotic bounds for the rate of convergence are illustrated by discussing toeplitz systems as well as a model problem stemming from the discretization of the poisson equation

eigenvalue problems for fractional differential equations with right and left fractional derivatives this paper studies the eigenvalue problem of a class of fractional differential equations with right and left fractional derivatives with the aid of the spectral theory of compact self adjoint operators in hilbert spaces we show that the spectrum of this problem consists of only countable real eigenvalues with finite multiplicity and the corresponding eigenfunctions form a complete orthogonal system furthermore the lower bound of the eigenvalues is obtained

on the boundary strichartz estimates for wave and schrodinger equations abstract we consider the l t 2 l x r estimates for the solutions to the wave and schrodinger equations in high dimensions for the homogeneous estimates we show l t 2 l x estimates fail at the critical regularity in high dimensions by using stable levy process in r d moreover we show that some spherically averaged l t 2 l x estimate holds at the critical regularity as a by product we obtain strichartz estimates with angular smoothing effect for the inhomogeneous estimates we prove double l t 2 type estimates

error estimates and evaluation of matrix functions via the faber transform the need to evaluate expressions of the form f a or f a b where f is a nonlinear function a is a large sparse n times n matrix and b is an n vector arises in many applications this paper describes how the faber transform applied to the field of values of a can be used to determine improved error bounds for popular polynomial approximation methods based on the arnoldi process applications of the faber transform to rational approximation methods and in particular to the rational arnoldi process also are discussed

rational krylov sequence methods for eigenvalue computation abstract algorithms to solve large sparse eigenvalue problems are considered a new class of algorithms which is based on rational functions of the matrix is described the lanczos method the arnoldi method the spectral transformation lanczos method and rayleigh quotient iteration all are special cases but there are also new algorithms which correspond to rational functions with several poles in the simplest case a basis of a rational krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of hessenberg matrices

convergence rates for inverse free rational approximation of matrix functions abstract this article deduces geometric convergence rates for approximating matrix functions via inverse free rational krylov methods in applications one frequently encounters matrix functions such as the matrix exponential or matrix logarithm often the matrix under consideration is too large to compute the matrix function directly and krylov subspace methods are used to determine a reduced problem if many evaluations of a matrix function of the form f a v with a large matrix a are required then it may be advantageous to determine a reduced problem using rational krylov subspaces these methods may give more accurate approximations of f a v with subspaces of smaller dimension than standard krylov subspace methods unfortunately the system solves required to construct an orthogonal basis for a rational krylov subspace may create numerical difficulties and or require excessive computing time this paper investigates a novel approach to determine an orthogonal basis of an approximation of a rational krylov subspace of small dimension from a standard orthogonal krylov subspace basis of larger dimension the approximation error will depend on properties of the matrix a and on the dimension of the original standard krylov subspace we show that our inverse free method for approximating the rational krylov subspace converges geometrically for increasing dimension of the standard krylov subspace to a rational krylov subspace the convergence rate may be used to predict the dimension of the standard krylov subspace necessary to obtain a certain accuracy in the approximation computed examples illustrate the theory developed

stability of standing waves for the fractional schrodinger hartree equation abstract in this paper we consider the stability of standing waves for the fractional schrodinger hartree equation by using the profile decomposition theory and variational methods we firstly obtain the best constant of a generalized gagliardo nirenberg inequality and then prove that the standing waves are orbitally stable