a new weighted mean filter with a two phase detector for removing impulse noise abstract this paper proposes a new weighted mean filter with a two phase noise detector for image denoising operations are carried out by the detection followed by filtering strategy for detection a two phase noise detector is presented to detect impulse noise in in the first phase a rank ordered difference of road rod road scheme is introduced for detecting noise candidates different from most of the existing in detectors the proposed detector identifies a pixel by a fuzzy rule that matches the stochastic nature of in and greatly improves the denoising performance in the second phase a local image statistic minimum edge pixels difference mepd is proposed to identify edge pixels from noise candidates this preserves edges from being wrongly detected as noise therefore improves the detection accuracy for filtering we design a new weighted mean filter wmf that is more suitable for in to suppress the detected noisy pixels finally an iterative denoising algorithm is presented by combining the proposed two phase noise detector and the new wmf the proposed method is accessible and easy to implement experimental results show that the proposed method outperforms all the tested state of the art denoising methods with respect to the visual effects and quantitative measure results

attractors of continuous difference equations abstract this paper aims at arranging the general ideas and results of the research on the nonlinear continuous difference equations and acquainting the reader with the properties of the solutions which seem unusual in the context of the theory of ordinary differential equations and have great importance in simulating sophisticated oscillatory processes such as self organization and deterministic chaos and just in those cases when it is difficult to apply ordinary differential equations

invariant and attracting sets of impulsive delay difference equations with continuous variables the aim of this paper is to study the invariant and attracting sets of impulsive delay difference equations with continuous variables some criteria for the invariant and attracting sets are obtained by using the decomposition approach and delay difference inequalities with impulsive initial conditions

circle and divisor problems and double series of bessel functions abstract in approximately 1915 ramanujan recorded two identities involving doubly infinite series of bessel functions the identities were brought to the mathematical public for the first time when his lost notebook was published in 1988 and are connected with the classical long standing circle and divisor problems respectively we provide a proof of the first identity for the first time by analytically continuing a new kind of dirichlet series delicate estimates of exponential sums are needed and the new methods we introduce may be of independent interest

nonlocal hierarchical dictionary learning using wavelets for image denoising exploiting the sparsity within representation models for images is critical for image denoising the best currently available denoising methods take advantage of the sparsity from image self similarity pre learned and fixed representations most of these methods however still have difficulties in tackling high noise levels or noise models other than gaussian in this paper the multiresolution structure and sparsity of wavelets are employed by nonlocal dictionary learning in each decomposition level of the wavelets experimental results show that our proposed method outperforms two state of the art image denoising algorithms on higher noise levels furthermore our approach is more adaptive to the less extensively researched uniform noise

dictionary learning for sparse coding algorithms and convergence analysis in recent years sparse coding has been widely used in many applications ranging from image processing to pattern recognition most existing sparse coding based applications require solving a class of challenging non smooth and non convex optimization problems despite the fact that many numerical methods have been developed for solving these problems it remains an open problem to find a numerical method which is not only empirically fast but also has mathematically guaranteed strong convergence in this paper we propose an alternating iteration scheme for solving such problems a rigorous convergence analysis shows that the proposed method satisfies the global convergence property the whole sequence of iterates is convergent and converges to a critical point besides the theoretical soundness the practical benefit of the proposed method is validated in applications including image restoration and recognition experiments show that the proposed method achieves similar results with less computation when compared to widely used methods such as k svd

mixed noise removal via laplacian scale mixture modeling and nonlocal low rank approximation recovering the image corrupted by additive white gaussian noise awgn and impulse noise is a challenging problem due to its difficulties in an accurate modeling of the distributions of the mixture noise many efforts have been made to first detect the locations of the impulse noise and then recover the clean image with image in painting techniques from an incomplete image corrupted by awgn however it is quite challenging to accurately detect the locations of the impulse noise when the mixture noise is strong in this paper we propose an effective mixture noise removal method based on laplacian scale mixture lsm modeling and nonlocal low rank regularization the impulse noise is modeled with lsm distributions and both the hidden scale parameters and the impulse noise are jointly estimated to adaptively characterize the real noise to exploit the nonlocal self similarity and low rank nature of natural image a nonlocal low rank regularization is adopted to regularize the denoising process experimental results on synthetic noisy images show that the proposed method outperforms existing mixture noise removal methods

distribution of energy levels of quantum free particle on the liouville surface and trace formulae we consider the weyl asymptotic formula left e n leqq r 2 right frac area q 4 pi r 2 n r for eigenvalues of the laplace beltrami operator on a two dimensional torusq with a liouville metric which is in a sense the most general case of an integrable metric we prove that if the surfaceq is non degenerate then the remainder termn r has the formn r r1 2 r where r is an almost periodic function of the besicovitch classb1 and the fourier amplitudes and the fourier frequencies of r can be expressed via lengths of closed geodesics onq and other simple geometric characteristics of these geodesics we prove then that if the surfaceq is generic then the limit distribution of r has a densityp t which is an entire function oft possessing an asymptotics on a real line logp t c t4 ast an explicit expression for the fourier transform ofp t via fourier amplitudes of r is also given we obtain the analogue of the guillemin duistermaat trace formula for the liouville surfaces and discuss its accuracy

image denoising via sparse and redundant representations over learned dictionaries we address the image denoising problem where zero mean white and homogeneous gaussian additive noise is to be removed from a given image the approach taken is based on sparse and redundant representations over trained dictionaries using the k svd algorithm we obtain a dictionary that describes the image content effectively two training options are considered using the corrupted image itself or training on a corpus of high quality image database since the k svd is limited in handling small image patches we extend its deployment to arbitrary image sizes by defining a global image prior that forces sparsity over patches in every location in the image we show how such bayesian treatment leads to a simple and effective denoising algorithm this leads to a state of the art denoising performance equivalent and sometimes surpassing recently published leading alternative denoising methods

aubry mather theory for conformally symplectic systems in this article we develop an analogue of aubry mather theory for a class of dissipative systems namely conformally symplectic systems and prove the existence of interesting invariant sets which in analogy to the conservative case will be called the aubry and the mather sets besides describing their structure and their dynamical significance we shall analyze their attracting repelling properties as well as their noteworthy role in driving the asymptotic dynamics of the system

the fibonacci unimodal map the fibonacci recurrence of the critical orbit appeared in the work of branner and hubbard on complex cubic polynomials bh 12 and in yoccoz s work yl y2 on quadratic ones as the worst pattern of recurrence on the other hand a real quadratic fibonacci map was suggested by hofbauer and keller hk as a possible candidate for a map having a wild attractor that is a set a which is the w limit set for lebesgue almost every orbit but is strictly smaller than the w limit set for a generic orbit the w limit set of the critical orbit in hk possesses all known topological properties of wild attractors compare bl2 in fact we will see below that the quadratic fibonacci map does not have a wild attractor however the corresponding question for a map with a degenerate critical point remains open actually the first indication of the fibonacci map appeared in the numerical work of tsuda t related to the belousova zhabotinskii reaction and also in numerical work of shibayama sh more precisely they studied the sequence of fibonacci bifurcations creating the fibonacci map this paper will study topological geometrical and measure theoretical properties of the real fibonacci map our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by sullivan s it turns out that the situation can be understood completely and is of quite regular nature in particular any fibonacci map with negative schwarzian and nondegenerate critical point has an absolutely continuous invariant measure so we deal with a regular type of chaotic dynamics it turns out also that geometrical properties of the closure of the critical orbit are quite different from those of the feigenbaum map its hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter branner and hubbard introduce the concept of a tableau in order to describe recurrence of critical orbits their fibonacci tableau is a basic example which corresponds to one particularly close and regular pattern of recurrence if a complex quadratic map z 1 z2 c realizes this fibonacci tableau then the

order and chaos in some trigonometric series curious adventures of a statistical mechanic this paper tells the story how a maple assisted quest for an interesting undergraduate problem in trigonometric series led some amateurs to the discovery that the one parameter family of deterministic trigonometric series p 1 exhibits both order and apparent chaos and how this has prompted some professionals to offer their expert insights as to order an elementary undergraduate proof is given that t with explicitly computed constant p as to chaos the seemingly erratic fluctuations about this overall trend are discussed experts commentaries are reproduced as to why the fluctuations of are presumably not gaussian inspired by a central limit type theorem of marc kac a well motivated conjecture is formulated to the effect that the fluctuations of the t 1 p 1 th partial sum of when properly scaled do converge in distribution to a standard gaussian when t though provided that p is chosen so that the frequencies n p n are rationally linear independent no conjecture has been forthcoming for rationally dependent n p n moreover following other experts tip offs the interesting relationship of the asymptotics of to properties of the riemann function is exhibited using the mellin transform

on the unfolding of a blowout bifurcation abstract suppose a chaotic attractor a in an invariant subspace loses stability on varying a parameter at the point of loss of stability the most positive lyapunov exponent of the natural measure on a crosses zero at what has been called a blowout bifurcation we introduce the notion of an essential basin of an attractor a this is the set of points x such that accumulation points of the sequence of measures 1 n n 1 k 0 f k x are supported on a we characterise supercritical and subcritical scenarios according to whether the lebesgue measure of the essential basin of a is positive or zero we study a drift diffusion model and a model class of piecewise linear mappings of the plane in the supercritical case we find examples where a lyapunov exponent of the branch of attractors may be positive hyperchaos or negative depending purely on the dynamics far from the invariant subspace for the mappings we find asymptotically linear scaling of lyapunov exponents average distance from the subspace and basin size on varying a parameter we conjecture that these are general characteristics of blowout bifurcations

slyrb measures natural invariant measures for chaotic systems in many applications it is useful to consider not only the set that constitutes an attractor but also if it exists the asymptotic distribution of a typical trajectory converging to the attractor indeed in the physics literature such a distribution is often assumed to exist when it exists it is called a natural invariant measure the results by lasota and yorke and by sinai ruelle and bowen represent two approaches both of which establish the existence of an invariant measure the goal of this paper is to relate the lasota yorke measure for chaotic attractors in one dimensional maps and the sinai ruelle bowen measure for chaotic attractors in higher dimensional dynamical systems we introduce the notion of slyrb measure we pronounce the term slyrb as a single word slurb the srb concept of measure can be motivated by asking how a trajectory from a typical initial point is distributed asymptotically similarly the slyrb concept of measure can be motivated by asking what the average distribution is for trajectories of a large collection of initial points in some region not necessarily restricted to a single basin the latter is analogous to ask where all the rain drops from a rain storm go and the former asks about where a single rain drop goes perhaps winding up distributed throughout a particular lake

nonlocally centralized sparse representation for image restoration sparse representation models code an image patch as a linear combination of a few atoms chosen out from an over complete dictionary and they have shown promising results in various image restoration applications however due to the degradation of the observed image e g noisy blurred and or down sampled the sparse representations by conventional models may not be accurate enough for a faithful reconstruction of the original image to improve the performance of sparse representation based image restoration in this paper the concept of sparse coding noise is introduced and the goal of image restoration turns to how to suppress the sparse coding noise to this end we exploit the image nonlocal self similarity to obtain good estimates of the sparse coding coefficients of the original image and then centralize the sparse coding coefficients of the observed image to those estimates the so called nonlocally centralized sparse representation ncsr model is as simple as the standard sparse representation model while our extensive experiments on various types of image restoration problems including denoising deblurring and super resolution validate the generality and state of the art performance of the proposed ncsr algorithm

mixed noise removal by weighted encoding with sparse nonlocal regularization mixed noise removal from natural images is a challenging task since the noise distribution usually does not have a parametric model and has a heavy tail one typical kind of mixed noise is additive white gaussian noise awgn coupled with impulse noise in many mixed noise removal methods are detection based methods they first detect the locations of in pixels and then remove the mixed noise however such methods tend to generate many artifacts when the mixed noise is strong in this paper we propose a simple yet effective method namely weighted encoding with sparse nonlocal regularization wesnr for mixed noise removal in wesnr there is not an explicit step of impulse pixel detection instead soft impulse pixel detection via weighted encoding is used to deal with in and awgn simultaneously meanwhile the image sparsity prior and nonlocal self similarity prior are integrated into a regularization term and introduced into the variational encoding framework experimental results show that the proposed wesnr method achieves leading mixed noise removal performance in terms of both quantitative measures and visual quality

a no reference image sharpness metric based on structural information using sparse representation as a ubiquitous image distortion blur casts non trivial influence on image visual quality many image sharpness assessment methods have been proposed in the views of edge information gradient map frequency spectrum or other natural image statistics features in this paper we propose a no reference image sharpness metric based on structural information using sparse representation sr we observe that the dictionary atoms learned by sr algorithm convey clear structural information considering the distinct sensibility of human visual system hvs to different structures we use the learned dictionary to encode the patches of the blurry image to embed the locality of the representation a multi scale spatial max pooling scheme is incorporated the final sharpness score is given by an efficient linear support vector regression svr model we evaluate our approach on three public databases i e live ii tid2008 and csiq the experiments demonstrate that the proposed method achieves competitive performance compared with the state of the art blind image sharpness assessment algorithms

a weighted dictionary learning model for denoising images corrupted by mixed noise this paper proposes a general weighted l2 l0 norms energy minimization model to remove mixed noise such as gaussian gaussian mixture impulse noise and gaussian impulse noise from the images the approach is built upon maximum likelihood estimation framework and sparse representations over a trained dictionary rather than optimizing the likelihood functional derived from a mixture distribution we present a new weighting data fidelity function which has the same minimizer as the original likelihood functional but is much easier to optimize the weighting function in the model can be determined by the algorithm itself and it plays a role of noise detection in terms of the different estimated noise parameters by incorporating the sparse regularization of small image patches the proposed method can efficiently remove a variety of mixed or single noise while preserving the image textures well in addition a modified k svd algorithm is designed to address the weighted rank one approximation the experimental results demonstrate its better performance compared with some existing methods

a new proof of the voronoi summation formula we present a short alternative proof of the voronoi summation formula which plays an important role in dirichlet s divisor problem and has recently found an application in physics as a trace formula for a schrodinger operator on a noncompact quantum graph

new pathways and connections in number theory and analysis motivated by two incorrect claims of ramanujan abstract the focus of this paper commences with an examination of three not obviously related pages in ramanujan s lost notebook pages 336 335 and 332 in decreasing order of attention on page 336 ramanujan proposes two identities but the formulas are wrong each is vitiated by divergent series we concentrate on only one of the two incorrect identities which may have been devised to attack the extended divisor problem we prove here a corrected version of ramanujan s claim which contains the convergent series appearing in it the convergent series in ramanujan s faulty claim is similar to one used by g f voronoi g h hardy and others in their study of the classical dirichlet divisor problem this now brings us to page 335 which comprises two formulas featuring doubly infinite series of bessel functions the first being conjoined with the classical circle problem initiated by gauss and the second being associated with the dirichlet divisor problem the first and fourth authors along with sun kim have written several papers providing proofs of these two difficult formulas in different interpretations in this monograph we return to these two formulas and examine them in more general settings the aforementioned convergent series in ramanujan s identity is also similar to one that appears in a curious identity found in chapter 15 in ramanujan s second notebook written in a more elegant equivalent formulation on page 332 in the lost notebook this formula may be regarded as a formula for 1 2 and in 1925 s wigert obtained a generalization giving a formula for 1 k for any even integer k 2 we extend the work of ramanujan and wigert in this paper the voronoi summation formula appears prominently in our study in particular we generalize work of j r wilton and derive an analogue involving the sum of divisors function s n the modified bessel functions k s x arise in several contexts as do lommel functions we establish here new series and integral identities involving modified bessel functions and modified lommel functions among other results we establish a modular transformation for an infinite series involving s n and modified lommel functions we also discuss certain obscure related work of n s koshliakov we define and discuss two new related classes of integral transforms which we call koshliakov transforms because he first found elegant special cases of each

weighted divisor sums and bessel function series on page 335 in his lost notebook ramanujan records without proof an identity involving a finite trigonometric sum and a doubly infinite series of ordinary bessel functions we provide the first published proof of this result the identity yields as corollaries representations of weighted divisor sums in particular the summatory function for r 2 n the number of representations of the positive integer n as a sum of two squares

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

the variance of the error function in the shifted circle problem is a wild function of the shift we prove that the variance of the error function in the shifted circle problem as a function of the shift is a continuous function which has a sharp local maximum with infinite derivatives at every rational point on a plane

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

weighted couple sparse representation with classified regularization for impulse noise removal many impulse noise in reduction methods suffer from two obstacles the improper noise detectors and imperfect filters they used to address such issue in this paper a weighted couple sparse representation model is presented to remove in in the proposed model the complicated relationships between the reconstructed and the noisy images are exploited to make the coding coefficients more appropriate to recover the noise free image moreover the image pixels are classified into clear slightly corrupted and heavily corrupted ones different data fidelity regularizations are then accordingly applied to different pixels to further improve the denoising performance in our proposed method the dictionary is directly trained on the noisy raw data by addressing a weighted rank one minimization problem which can capture more features of the original data experimental results demonstrate that the proposed method is superior to several state of the art denoising methods

strange attractors and asymptotic measures of discrete time dissipative systems we investigate asymptotic properties of certain discrete time dynamical systems in two and three dimensions with solenoidal attractor it is proved that the asymptotic measures relevant for the generalized version of the ergodic theorem all derive from one haar measure

a universal variational framework for sparsity based image inpainting in this paper we extend an existing universal variational framework for image inpainting with new numerical algorithms given certain regularization operator and denoting u the latent image the basic model is to minimize the l p p 0 1 norm of u preserving the pixel values outside the inpainting region utilizing the operator splitting technique the original problem can be approximated by a new problem with extra variable with the alternating minimization method the new problem can be decomposed as two subproblems with exact solutions there are many choices for in our approach such as gradient operator wavelet transform framelet transform or other tight frames moreover with slight modification we can decouple our framework into two relatively independent parts 1 denoising and 2 linear combination therefore we can take any denoising method including bm3d filter in the denoising step the numerical experiments on various image inpainting tasks such as scratch and text removal randomly missing pixel filling and block completion clearly demonstrate the super performance of the proposed methods furthermore the theoretical convergence of the proposed algorithms is proved

some mathematical problems in the theory of quantum chaos abstract we discuss the general problem of distribution of spacings between levels in quantum problems for the quantum kicked rotator model we formulate the results on the poisson distribution for spacings some general remarks concerning the reasons for the appearance of this distribution are given

a universal noise removal algorithm with an impulse detector we introduce a local image statistic for identifying noise pixels in images corrupted with impulse noise of random values the statistical values quantify how different in intensity the particular pixels are from their most similar neighbors we continue to demonstrate how this statistic may be incorporated into a filter designed to remove additive gaussian noise the result is a new filter capable of reducing both gaussian and impulse noises from noisy images effectively which performs remarkably well both in terms of quantitative measures of signal restoration and qualitative judgements of image quality our approach is extended to automatically remove any mix of gaussian and impulse noise

generalized lambert series and arithmetic nature of odd zeta values it is pointed out that the generalized lambert series sum nolimits n 1 infty n n 2h e n nx 1 studied by kanemitsu tanigawa and yoshimoto can be found on page 332 of ramanujan s lost notebook in a slightly more general form we extend an important transformation of this series obtained by kanemitsu tanigawa and yoshimoto by removing restrictions on the parameters n and h that they impose from our extension we deduce a beautiful new generalization of ramanujan s famous formula for odd zeta values which for n odd and m 0 gives a relation between 2 m 1 and 2 nm 1 a result complementary to the aforementioned generalization is obtained for any even n and m it generalizes a transformation of wigert and can be regarded as a formula for 2 m 1 1 n applications of these transformations include a generalization of the transformation for the logarithm of dedekind eta function z zudilin and rivoal type results on transcendence of certain values and a transcendence criterion for euler s constant

distribution of the error term for the number of lattice points inside a shifted circle define begin array 20 c n x r left n in z d x n leqslant r right x in 0 1 d end array 1 1 and f x r r frac d 1 2 n x r r d text vol d b 1 1 2 where b 1 is the unit ball in r d thus n x r is the number of lattice points in z d in the r ball centered at x and f x r is the normalized difference with the volume of the ball we are interested in the problem of the distribution of f x r as a function of r for given x

low rank prior in single patches for nonpointwise impulse noise removal this paper introduces a low rank prior in small oriented noise free image patches considering an oriented patch as a matrix a low rank matrix approximation is enough to preserve the texture details in the optimally oriented patch based on this prior we propose a single patch method within a generalized joint low rank and sparse matrix recovery framework to simultaneously detect and remove nonpointwise random valued impulse noise e g very small blobs a weighting matrix is incorporated in the framework to encode an initial estimate of the spatial noise distribution an accelerated proximal gradient method is adapted to estimate the optimal noise free image patches experiments show the effectiveness of our framework in detecting and removing nonpointwise random valued impulse noise

integral representations for the exceptional univariate lommel functions an elementary integral representation for the lommel function s 1 0 z is given and extended to other exceptional cases

the minimal center of attraction of measurable systems and their discretizations a generalization of the birkhoff minimal center of attraction is introduced and investigated for dynamical systems generated by single and multi valued borel measurable mappings in particular it is shown to be equal to the minimal closed set containing the supports of the semi invariant measures of the system its approximation under the spatial discretization of the state space is also considered

joint image denoising using adaptive principal component analysis and self similarity the non local means nlm has attracted enormous interest in image denoising problem in recent years in this paper we propose an efficient joint denoising algorithm based on adaptive principal component analysis pca and self similarity that improves the predictability of pixel intensities in reconstructed images the proposed algorithm consists of two successive steps without iteration the low rank approximation based on parallel analysis and the collaborative filtering first for a pixel and its nearest neighbors the training samples in a local search window are selected to form the similar patch group by the block matching method next it is factorized by singular value decomposition svd whose left and right orthogonal basis denote local and non local image features respectively the adaptive pca automatically chooses the local signal subspace dimensionality of the noisy similar patch group in the svd domain by the refined parallel analysis with monte carlo simulation thus image features can be well preserved after dimensionality reduction and simultaneously the noise is almost eliminated then after the inverse svd transform the denoised image is reconstructed from the aggregate filtered patches by the weighted average method finally the collaborative wiener filtering is used to further remove the noise the experimental results validate its generality and effectiveness in a wide range of the noisy images the proposed algorithm not only produces very promising denoising results that outperforms the state of the art methods in most cases but also adapts to a variety of noise levels

robust subspace segmentation via nonconvex low rank representation a nonconvex formulation to determine the low rank representation from contaminated data is proposed we provide a proximal iteratively reweighed algorithm for solving the nonconvex model the proposed nonconvex model can recover the underlying low rank structure of subspaces in spite of noisy corruptions recently low rank representation lrr has been successfully applied to explore subspace segmentation of data in this paper we propose a nonconvex formulation to determine the lrr from contaminated data unlike in traditional methods which directly utilize the nuclear norm to approximate the rank function and penalize noise using the 2 1 norm our method introduces the ky fan p k norm and the 2 q norm to better approximate the rank minimization problem and enhance the robustness against noise an efficient algorithm is derived for solving the novel objective function and this is followed by a rigorous theoretical proof of the convergence extensive experiments on face datasets clearly demonstrate that the proposed methods are more robust to illumination variations corruptions and occlusions

sensitivity of dendrite maps abstract suppose that x is a dendrite and f x x is a sensitive continuous map we show that a x f contains a bilaterally transitive subsystem with nonempty interior b the system x f satisfies only one of the following two conditions b1 x f contains a topologically transitive subsystem with nonempty interior b2 there exists an f invariant nowhere dense closed subset a of x such that the attraction basin basin a f contains a residual subset b of an open set and the strong attraction basin sbasin a f is dense in b c if x is completely regular then x f contains a relatively strongly mixing subsystem with nonempty interior dense periodic points and positive topological entropy unlike for interval maps we construct a sensitive dendrite map with zero topological entropy

a detection statistic for random valued impulse noise this paper proposes an image statistic for detecting random valued impulse noise by this statistic we can identify most of the noisy pixels in the corrupted images combining it with an edge preserving regularization we obtain a powerful two stage method for denoising random valued impulse noise even for noise levels as high as 60 simulation results show that our method is significantly better than a number of existing techniques in terms of image restoration and noise detection

nonlinear dynamics of networks the groupoid formalism a formal theory of symmetries of networks of coupled dynamical systems stated in terms of the group of permutations of the nodes that preserve the network topology has existed for some time global network symmetries impose strong constraints on the corresponding dynamical systems which affect equilibria periodic states heteroclinic cycles and even chaotic states in particular the symmetries of the network can lead to synchrony phase relations resonances and synchronous or cycling chaos symmetry is a rather restrictive assumption and a general theory of networks should be more flexible a recent generalization of the group theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network the input sets now the symmetry group becomes a groupoid which is an algebraic structure that resembles a group except that the product of two elements may not be defined the groupoid formalism makes it possible to extend group theoretic methods to more general networks and in particular it leads to a complete classification of robust patterns of synchrony in terms of the combinatorial structure of the network many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology a network of dynamical systems is not just a dynamical system with a high dimensional phase space it is also equipped with a canonical set of observables the states of the individual nodes of the network moreover the form of the underlying ode is constrained by the network topology which variables occur in which component equations and how those equations relate to each other the result is a rich and new range of phenomena only a few of which are yet properly understood

image denoising by sparse 3 d transform domain collaborative filtering we propose a novel image denoising strategy based on an enhanced sparse representation in transform domain the enhancement of the sparsity is achieved by grouping similar 2d image fragments e g blocks into 3d data arrays which we call groups collaborative altering is a special procedure developed to deal with these 3d groups we realize it using the three successive steps 3d transformation of a group shrinkage of the transform spectrum and inverse 3d transformation the result is a 3d estimate that consists of the jointly filtered grouped image blocks by attenuating the noise the collaborative filtering reveals even the finest details shared by grouped blocks and at the same time it preserves the essential unique features of each individual block the filtered blocks are then returned to their original positions because these blocks are overlapping for each pixel we obtain many different estimates which need to be combined aggregation is a particular averaging procedure which is exploited to take advantage of this redundancy a significant improvement is obtained by a specially developed collaborative wiener filtering an algorithm based on this novel denoising strategy and its efficient implementation are presented in full detail an extension to color image denoising is also developed the experimental results demonstrate that this computationally scalable algorithm achieves state of the art denoising performance in terms of both peak signal to noise ratio and subjective visual quality

dual sparsity regularized sparse representation for single image super resolution row nonlocal similarity regularization term with l1 norm constraint is explored dual sparsity regularized sparse representation model is presented for sisr iterative shrinkage algorithm is extended to solve dual l1 norm constraints model recently by exploring the column nonlocal similarity prior among the sparse representation coefficients the column nonlocal similarity sparse representation models for solving the ill posed single image super resolution sisr problem are attracting more and more attention however these conventional models consider only the prior among nonlocal similar sparse representation coefficients and fail to consider the prior among all entries or rows of the sparse representation coefficient hence the modeling capability may be limited in fact if a cluster of similar representation coefficients is rearranged into a matrix in the sparse representation coefficient space the nonlocal similarity priors exist both among columns and rows using the row nonlocal similarity prior a row nonlocal similarity regularization term with l1 norm constraint is explored by introducing it to the conventional column nonlocal similarity sparse representation model we present a dual sparsity regularized sparse representation dsrsr model a surrogate function based iterative shrinkage algorithm is introduced to effectively solve the proposed model extensive experiments on sisr demonstrate that the presented model can effectively reconstruct the edge structures and suppress the noise achieving convincing improvement over many state of the art example based methods in terms of psnr ssim and visual quality

the spatial temporal lamellar structures in the confined ideal polymer blends we consider the formation of self organized spatial temporal oscillating structures in symmetric binary polymer blends confined by two flat walls an influence of these walls on the formation of the oscillating volume structures is studied this phenomenon is simulated by an initial boundary value problem for the conserved order parameter or the concentration of one of the components in a binary mixture under a special choice the dynamical puri binder s boundary conditions these structures look like the lamellar structures the behavior of the order parameter is described by the modified cahn hilliard equation which models so called the non fickian diffusion in the symmetric binary polymer blends the nonlinear dynamical boundary conditions correspond to the process of adsorption desorption on the walls as a result these nonlinear surface processes induce into the volume the spatial temporal asymptotically periodic structures of relaxation pre turbulent or turbulent type with finite countable or non countable points of discontinuities on the period correspondingly the frequency of oscillations on the period follows a power law for the relaxation type and increases exponentially in the other cases

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

local structure based multi phase collaborative representation for face recognition with single sample per person many real world face recognition applications can only provide single sample for each person while most face recognition approaches require a large set of training samples which leads to single sample per person sspp problem in this paper we propose local structure based multi phase collaborative representation classification ls mpcrc to solve sspp problem by adopting the divide conquer aggregate strategy we successfully alleviate the dilemma of high data dimensionality and small samples where we first divide the face into local blocks and classify each local block and then integrate all the classification results by voting for each local block we further divide it into overlapped local patches and assume that these patches lie in a linear subspace this subspace assumption reflects local structure relationship of the overlapped patches and makes crc robust for sspp problem motivated by the fact that the entropy of the class probability distribution is a measure about classification confidence we further apply multi phase technique to reduce entropy where useless classes are eliminated after each phase classification this strategy finally produces a sparse class probability distribution with higher classification confidence experimental results show that the proposed method generalizes well to sspp problem and outperforms many state of the art methods it also shows strong robustness to the large variation of expression illumination little poses variation occlusion and time variation

on spectral statistics of classically integrable systems this work is an extensive study of the spectral statistics of three representative classically integrable systems namely rectangle torus and circle billiards we analyse the statistics and focus on the related level spacing distribution p s and the delta statistics the agreement with the poissonian model is typically found to be perfect up to the outer unfolded energy scale beyond which the saturation is observed in agreement with berry s dynamical theory of the spectral rigidity where as the untypical systems are those where for example p s is not a smooth distribution but a sum of the delta functions due to the granularity of the energy scale for example in the rectangle with the rational squared sides ratio however even there we find reasonable trend towards poissonian statistics for large ranges l but we describe theoretically and numerically the broadening of the delta spikes when the rectangular billiard is slightly distorted away from a rational to an irrational shape and find excellent agreement also in irrational rectangle billiards we show and explain the existence of large fluctuations by one order of magnitude bigger than the statistical ones whose origin is in the closeness to some rational billiard shape these fluctuations and their amplitude are independent of the energy if the bin size shrinks inversely with energy finally we tested the mode distribution i e distribution of the reduced mode fluctuation number w and found that it was not poissonian in agreement with steiner s conjecture and in fact follows the prediction by bleher et al that its tail behaves as the general reason for non universal behaviour of p w is that at largest energy scales we are always in the saturation regime

transcritical riddling in a system of coupled maps the transition from fully synchronized behavior to two cluster dynamics is investigated for a system of n globally coupled chaotic oscillators by means of a model of two coupled logistic maps an uneven distribution of oscillators between the two clusters causes an asymmetry to arise in the coupling of the model system while the transverse period doubling bifurcation remains essentially unaffected by this asymmetry the transverse pitchfork bifurcation is turned into a saddle node bifurcation followed by a transcritical riddling bifurcation in which a periodic orbit embedded in the synchronized chaotic state loses its transverse stability we show that the transcritical riddling transition is always hard for this we study the sequence of bifurcations that the asynchronous point cycles produced in the saddle node bifurcation undergo and show how the manifolds of these cycles control the magnitude of asynchronous bursts in the case where the system involves two subpopulations of oscillators with a small mismatch of the parameters the transcritical riddling will be replaced by two subsequent saddle node bifurcations or the saddle cycle involved in the transverse destabilization of the synchronized chaotic state may smoothly shift away from the synchronization manifold in this way the transcritical riddling bifurcation is substituted by a symmetry breaking bifurcation which is accompanied by the destruction of a thin invariant region around the symmetrical chaotic state

discrete analogues in harmonic analysis spherical averages in this paper we prove an analogue in the discrete setting of d o fthe spherical maximal theorem for d the methods used are two fold the application of certain sampling techniques and ideas arising in the study of the number of representations of an integer as a sum of d squares in particular the circle method the results we obtained are by necessity limited to d 5 and moreover the range of p for the l p estimates differs from its analogue in d

on the strength of attractors in a high dimensional system milnor attractor network robust global attraction and noise induced selection abstract strength of attractor is studied by the return rate to itself after perturbations for a multi attractor state of a globally coupled map it is found that fragile milnor attractors have a large basin volume at the partially ordered phase such dominance of fragile attractors is understood by robustness of global attraction in the phase space change of the attractor strength and basin volume against the parameter and size are studied in the partially ordered phase the dynamics is often described as milnor attractor network which leads to a new interpretation of chaotic itinerancy noise induced selection of fragile attractors is found that has a sharp dependence on the noise amplitude relevance of the observed results to neural dynamics and cell differentiation is also discussed

spectral form factors of rectangle billiards the berry tabor conjecture asserts that local statistical measures of the eigenvalues j of a generic integrable quantum system coincide with those of a poisson process we prove that in the case of a rectangle billiard with random ratio of sides the sum behaves for random and n large like a random walk in the complex plane with a non gaussian limit distribution the expectation value of the distribution is zero its variance which is essentially the average pair correlation function is one in accordance with the berry tabor conjecture but all higher moments 4 diverge the proof of the existence of the limit distribution uses the mixing property of a dynamical system defined on a product of hyperbolic surfaces the berry tabor conjecture and the existence of the limit distribution for a fixed generic rectangle are related to an equidistribution conjecture for long horocycles on this product space

cycling chaos its creation persistence and loss of stability in a model of nonlinear magnetoconvection we examine a model system where attractors may consist of a heteroclinic cycle between chaotic sets this cycling chaos manifests itself as trajectories that spend increasingly long periods lingering near chaotic invariant sets interspersed with short transitions between neighbourhoods of these sets such behaviour is robust to perturbations that preserve the symmetry of the system we examine bifurcations of this state we discuss a scenario where an attracting cycling chaotic state is created at a blowout bifurcation of a chaotic attractor in an invariant subspace this differs from the standard scenario for the blowout bifurcation in that in our case the blowout is neither subcritical nor supercritical the robust cycling chaotic state can be followed to a point where it loses stability at a resonance bifurcation and creates a series of large period attractors the model we consider is a ninth order truncated ordinary differential equation ode model of three dimensional incompressible convection in a plane layer of conducting fluid subjected to a vertical magnetic field and a vertical temperature gradient symmetries of the model lead to the existence of invariant subspaces for the dynamics in particular there are invariant subspaces that correspond to regimes of two dimensional flows with variation in the vertical but only one of the two horizontal directions stable two dimensional chaotic flow can go unstable to three dimensional flow via the cross roll instability we show how the bifurcations mentioned above can be located by examination of various transverse liapunov exponents we also consider a reduction of the ode to a map and demonstrate that the same behaviour can be found in the corresponding map this allows us to describe and predict a number of observed transitions in these models the dynamics we describe is new but nonetheless robust and so should occur in other applications

an exact trace formula and zeta functions for an infinite quantum graph with a non standard weyl asymptotics we study a quantum hamiltonian that is given by the negative laplacian and an infinite chain of like potentials with strength 0 on the half line and which is equivalent to a one parameter family of laplacians on an infinite metric graph this graph consists of an infinite chain of edges with the metric structure defined by assigning an interval in 0 ln to each edge with length we show that the one parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each 0 and prove that the dirichlet laplacian is the limit of the one parameter family in the strong resolvent sense the spectrum of the resulting dirichlet quantum graph is also purely discrete the eigenvalues are given by n n2 with multiplicities d n where d n denotes the divisor function we can thus relate the spectral problem of this infinite quantum graph to dirichlet s famous divisor problem and infer the non standard weyl asymptotics for the eigenvalue counting function based on an exact trace formula the voronoi summation formula we derive explicit formulae for the trace of the wave group the heat kernel the resolvent and for various spectral zeta functions these results enable us to establish a well defined renormalized secular equation and a selberg like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the riemann hypothesis is true

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

zeros of combinations of the riemann ξ function and the confluent hypergeometric function on bounded vertical shifts abstract in 1914 hardy proved that infinitely many non trivial zeros of the riemann zeta function lie on the critical line using the transformation formula of the jacobi theta function recently the first author obtained an integral representation involving the riemann function and the confluent hypergeometric function linked to the general theta transformation using this result we show that a series consisting of bounded vertical shifts of a product of the riemann function and the real part of a confluent hypergeometric function has infinitely many zeros on the critical line thereby generalizing a previous result due to the first and the last authors along with roy and robles the latter itself is a generalization of hardy s theorem

image super resolution via sparse representation this paper presents a new approach to single image superresolution based upon sparse signal representation research on image statistics suggests that image patches can be well represented as a sparse linear combination of elements from an appropriately chosen over complete dictionary inspired by this observation we seek a sparse representation for each patch of the low resolution input and then use the coefficients of this representation to generate the high resolution output theoretical results from compressed sensing suggest that under mild conditions the sparse representation can be correctly recovered from the downsampled signals by jointly training two dictionaries for the low and high resolution image patches we can enforce the similarity of sparse representations between the low resolution and high resolution image patch pair with respect to their own dictionaries therefore the sparse representation of a low resolution image patch can be applied with the high resolution image patch dictionary to generate a high resolution image patch the learned dictionary pair is a more compact representation of the patch pairs compared to previous approaches which simply sample a large amount of image patch pairs reducing the computational cost substantially the effectiveness of such a sparsity prior is demonstrated for both general image super resolution sr and the special case of face hallucination in both cases our algorithm generates high resolution images that are competitive or even superior in quality to images produced by other similar sr methods in addition the local sparse modeling of our approach is naturally robust to noise and therefore the proposed algorithm can handle sr with noisy inputs in a more unified framework

a dictionary learning approach for poisson image deblurring the restoration of images corrupted by blur and poisson noise is a key issue in medical and biological image processing while most existing methods are based on variational models generally derived from a maximum a posteriori map formulation recently sparse representations of images have shown to be efficient approaches for image recovery following this idea we propose in this paper a model containing three terms a patch based sparse representation prior over a learned dictionary the pixel based total variation regularization term and a data fidelity term capturing the statistics of poisson noise the resulting optimization problem can be solved by an alternating minimization technique combined with variable splitting extensive experimental results suggest that in terms of visual quality peak signal to noise ratio value and the method noise the proposed algorithm outperforms state of the art methods

a universal denoising framework with a new impulse detector and nonlocal means impulse noise detection is a critical issue when removing impulse noise and impulse gaussian mixed noise in this paper we propose a new detection mechanism for universal noise and a universal noise filtering framework based on the nonlocal means nl means the operation is carried out in two stages i e detection followed by filtering for detection first we propose the robust outlyingness ratio ror for measuring how impulselike each pixel is and then all the pixels are divided into four clusters according to the ror values second different decision rules are used to detect the impulse noise based on the absolute deviation to the median in each cluster in order to make the detection results more accurate and more robust the from coarse to fine strategy and the iterative framework are used in addition the detection procedure consists of two stages i e the coarse and fine detection stages for filtering the nl means are extended to the impulse noise by introducing a reference image then a universal denoising framework is proposed by combining the new detection mechanism with the nl means ror nlm finally extensive simulation results show that the proposed noise detector is superior to most existing detectors and the ror nlm produces excellent results and outperforms most existing filters for different noise models unlike most of the other impulse noise filters the proposed ror nlm also achieves high peak signal to noise ratio and great image quality by efficiently removing impulse gaussian mixed noise

weighted divisor sums and bessel function series ii abstract on page 335 in his lost notebook ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of bessel functions in each case there are three possible interpretations for the double series in an earlier paper two of the present authors proved the first identity under one possible interpretation in the present paper the second identity is proved under a similar interpretation with one additional assumption moreover under a second interpretation entirely different proofs of both identities depending on weighted or twisted divisor sums are offered the two identities are intimately connected with the classical circle and divisor problems respectively

a robust bi sparsity model with non local regularization for mixed noise reduction sparse representation model srm has been widely used in many image processing and computer vision tasks however the conventional srm usually neglects the prior knowledge about similar signals considering the fact that similar signals also have subtle differences in this paper we propose a robust bi sparsity model rbsm to effectively exploit the prior knowledge about the similarities and the distinctions of signals in rbsm similar signals are encouraged to be coded on the same sub dictionary but the distinctiveness of similar signals is also addressed by imposing the l0 norm regularization on the difference between each coefficient and its non local means in addition a weight vector is incorporated into the loss function to make the proposed model robust to outliers we apply rbsm for mixed noise reduction and experimental results show that our proposed model is superior to several state of the art mixed noise removal methods