shape preserving local histogram modification a novel approach for shape preserving contrast enhancement is presented in this paper contrast enhancement is achieved by means of a local histogram equalization algorithm which preserves the level sets of the image this basic property is violated by common local schemes thereby introducing spurious objects and modifying the image information the scheme is based on equalizing the histogram in all the connected components of the image which are defined based both on the grey values and spatial relations between pixels in the image and following mathematical morphology constitute the basic objects in the scene we give examples for both grey value and color images

optimal mock jacobi theta functions abstract we classify the optimal mock jacobi forms of weight one with rational coefficients the space they span is thirty four dimensional and admits a distinguished basis parameterized by genus zero groups of isometries of the hyperbolic plane we show that their fourier coefficients can be expressed explicitly in terms of singular moduli and obtain positivity conditions which distinguish the optimal mock jacobi forms that appear in umbral moonshine we find that many of ramanujan s mock theta functions can be expressed simply in terms of the optimal mock jacobi forms with rational coefficients

exact histogram specification while in the continuous case statistical models of histogram equalization specification would yield exact results their discrete counterparts fail this is due to the fact that the cumulative distribution functions one deals with are not exactly invertible otherwise stated exact histogram specification for discrete images is an ill posed problem invertible cumulative distribution functions are obtained by translating the problem in a k dimensional space and further inducing a strict ordering among image pixels the proposed ordering refines the natural one experimental results and statistical models of the induced ordering are presented and several applications are discussed image enhancement normalization watermarking etc

borcherds lifts of harmonic maass forms and modular integrals we extend borcherds singular theta lift in signature 1 2 to harmonic maass forms of weight 1 2 whose non holomorphic part is allowed to be of exponential growth at i infty we determine the singularities of the lift and compute its fourier expansion it turns out that the lift is continuous but not differentiable along certain geodesics in the upper half plane corresponding to the non holomorphic principal part of the input as an application we obtain a generalization to higher level of the weight 2 modular integral of duke imamoglu and toth further we construct automorphic products associated to harmonic maass forms

formulas for the coefficients of half integral weight harmonic maas forms recently bruinier and ono proved that the coefficients of certain weight 1 2 harmonic weak maas forms are given as traces of singular moduli for harmonic weak maas forms here we prove that similar results hold for the coefficients of harmonic weak maas forms of weight 3 2 k k even and weight 1 2 k k odd by extending the theta lift of bruinier funke and bruinier ono moreover we generalize these results to include twisted traces of singular moduli using earlier work of the author and ehlen on the twisted bruinier funke lift employing a general duality result between weight k and 2 k we obtain formulas for all half integral weights we also show that the non holomorphic part of the theta lift in weight 1 2 k k odd is connected to the vanishing of the special value of the l function of a certain derivative of the lifted function

an energy based model for the image edge histogram specification problem in this correspondence we present an original energy based model that achieves the edge histogram specification of a real input image and thus extends the exact specification method of the image luminance or gray level distribution recently proposed by coltuc et al our edge histogram specification approach is stated as an optimization problem in which each edge of a real input image will tend iteratively toward some specified gradient magnitude values given by a target edge distribution or a normalized edge histogram possibly estimated from a target image to this end a hybrid optimization scheme combining a global and deterministic conjugate gradient based procedure and a local stochastic search using the metropolis criterion is proposed herein to find a reliable solution to our energy based model experimental results are presented and several applications follow from this procedure

the f q mock theta function conjecture and partition ranks in 1944 freeman dyson initiated the study of ranks of integer partitions here we solve the classical problem of obtaining formulas for ne n resp no n the number of partitions of n with even resp odd rank thanks to rademacher s celebrated formula for the partition function this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f q a problem with its own long history dating to ramanujan s last letter to hardy little was known about this problem until dragonette in 1952 obtained asymptotic results in 1966 g e andrews refined dragonette s results and conjectured an exact formula for the coefficients of f q by constructing a weak maass poincare series whose holomorphic part is q 1f q24 we prove the andrews dragonette conjecture and as a consequence obtain the desired formulas for ne n and no n

eichler shimura theory for mock modular forms we use mock modular forms to compute generating functions for the critical values of modular l functions and we answer a generalized form of a question of kohnen and zagier by deriving the extra relation that is satisfied by even periods of weakly holomorphic cusp forms to obtain these results we derive an eichler shimura theory for weakly holomorphic modular forms and mock modular forms this includes two eichler shimura isomorphisms a multiplicity two hecke theory a correspondence between mock modular periods and classical periods and a haberland type formula which expresses petersson s inner product and a related antisymmetric inner product on m k in terms of periods

sobolev bounds on functions with scattered zeros with applications to radial basis function surface fitting in this paper we discuss sobolev bounds on functions that vanish at scattered points in a bounded lipschitz domain that satisfies a uniform interior cone condition the sobolev spaces involved may have fractional as well as integer order we then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions rbfs these estimates include situations in which the target function does not belong to the native space of the rbf

borcherds products and arithmetic intersection theory on hilbert modular surfaces we prove an arithmetic version of a theorem of hirzebruch and zagier saying that hirzebruch zagier divisors on a hilbert modular surface are the coecients of an elliptic modular form of weight two moreover we determine the arithmetic selfintersection number of the line bundle of modular forms equipped with its petersson metric on a regular model of a hilbert modular surface and study faltings heights of arithmetic hirzebruch zagier divisors

image enhancement by entropy maximization and quantization resolution upconversion this article introduces a new contrast enhancement algorithm of tone preserving entropy maximization its design objective is to present the maximal amount of information content in the enhanced image or being optimal in an information theoretical sense while preventing the loss of tone continuity the resulting optimization problem can be graph theoretically modeled as the construction of the k edges maximum weight path and it can be solved efficiently by dynamic programming moreover the proposed algorithm is made more effective by being combined with a preprocess of image restoration that aims to correct quantization errors caused by the analog to digital conversion of image signals empirical evidences are provided to demonstrate the superior visual quality obtained by the new image enhancement algorithm

a novel 3 d color histogram equalization method with uniform 1 d gray scale histogram the majority of color histogram equalization methods do not yield uniform histogram in gray scale after converting a color histogram equalized image into gray scale the contrast of the converted image is worse than that of an 1 d gray scale histogram equalized image we propose a novel 3 d color histogram equalization method that produces uniform distribution in gray scale histogram by defining a new cumulative probability density function in 3 d color space test results with natural and synthetic images are presented to compare and analyze various color histogram equalization algorithms based upon 3 d color histograms we also present theoretical analysis for nonideal performance of existing methods

harmonic weak maass forms of integral weight a geometric approach the purpose of the present work is to provide a geometric framework for the study of the fourier coefficients of harmonic weak maass forms a space of smooth modular forms first introduced by bruinier and funke in the context of singular theta lifts in this geometric framework harmonic weak maass forms arise from the construction of differentials whose classes are exact in certain de rham cohomology groups attached to modular forms we show how this new interpretation naturally leads to strengthenings of the theorems of bruinier ono and rhoades by answering in the affirmative conjectures about the field of definitions of fourier coefficients of harmonic weak maass forms moreover as part of our geometric framework we describe a geometric interpretation for the shimura maass lowering operator analogous to the description of the shimura maass raising operator given by katz we also produce eichler shimura style isomorphisms for the de rham cohomology attached to modular forms generalizing results of bringmann guerzhoy kent and ono to any level and field of definition

theta lifts for lorentzian lattices and coefficients of mock theta functions we evaluate regularized theta lifts for lorentzian lattices in three different ways in particular we obtain formulas for their values at special points involving coefficients of mock theta functions by comparing the different evaluations we derive recurrences for the coefficients of mock theta functions such as hurwitz class numbers andrews spt function and ramanujan s mock theta functions

the runge phenomenon and spatially variable shape parameters in rbf interpolation many studies mostly empirical have been devoted to finding an optimal shape parameter for radial basis functions rbf when exploring the underlying factors that determine what is a good such choice we are led to consider the runge phenomenon rp best known in cases of high order polynomial interpolation as a key error mechanism this observation suggests that it can be advantageous to let the shape parameter vary spatially rather than assigning a single value to it benefits typically include improvements in both accuracy and numerical conditioning still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas this idea is routinely used when working with splines or finite element methods however local refinement with rbfs may cause rp type errors unless we use a spatially variable shape paremeter with this enhancement rbf approximations combine freedom from meshes with spectral accuracy on irregular domains and furthermore permit local node clustering to improve the resolution wherever this might be needed

error estimates for scattered data interpolation on spheres we study sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n sphere the interpolation knots are scattered our approach partly follows the general theory of golomb and weinberger and related estimates these error estimates are then based on series expansions of smooth functions in terms of spherical harmonics the markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics

nonvanishing of quadratic twists of modular l functions and applications to elliptic curves if d is square free or is a fundamental discriminant then let d d denote the kronecker character for the quadratic field q d whose fundamental discriminant is d throughout d shall denote a fundamental discriminant the d quadratic twist of f denoted f d is the newform corresponding to the twist of f by the character d in particular if gcd m d 1 then f d z n 1 d n a n q n and

a standard test set for numerical approximations to the shallow water equations in spherical geometry a suite of seven test cases is proposed for the evaluation of numerical methods intended for the solution of the shallow water equations in spherical geometry the shallow water equations exhibit the major difficulties associated with the horizontal dynamical aspects of atmospheric modeling on the spherical earth these cases are designed for use in the evaluation of numerical methods proposed for climate modeling and to identify the potential trade offs which must always be made in numerical modeling before a proposed scheme is applied to a full baroclinic atmospheric model it must perform well on these problems in comparison with other currently accepted numerical methods the cases are presented in order of complexity they consist of advection across the poles steady state geostrophically balanced flow of both global and local scales forced nonlinear advection of an isolated low zonal flow impinging on an isolated mountain rossby haurwitz waves and observed atmospheric states one of the cases is also identified as a computer performance algorithm efficiency benchmark for assessing the performance of algorithms adapted to massively parallel computers

adaptive image contrast enhancement using generalizations of histogram equalization this paper proposes a scheme for adaptive image contrast enhancement based on a generalization of histogram equalization he he is a useful technique for improving image contrast but its effect is too severe for many purposes however dramatically different results can be obtained with relatively minor modifications a concise description of adaptive he is set out and this framework is used in a discussion of past suggestions for variations on he a key feature of this formalism is a cumulation function which is used to generate a grey level mapping from the local histogram by choosing alternative forms of cumulation function one can achieve a wide variety of effects a specific form is proposed through the variation of one or two parameters the resulting process can produce a range of degrees of contrast enhancement at one extreme leaving the image unchanged at another yielding full adaptive equalization

fast ordering algorithm for exact histogram specification this paper provides a fast algorithm to order in a meaningful strict way the integer gray values in digital quantized images it can be used in any exact histogram specification based application our algorithm relies on the ordering procedure based on the specialized variational approach this variational method was shown to be superior to all other state of the art ordering algorithms in terms of faithful total strict ordering but not in speed indeed the relevant functionals are in general difficult to minimize because their gradient is nearly flat over vast regions in this paper we propose a simple and fast fixed point algorithm to minimize these functionals the fast convergence of our algorithm results from known analytical properties of the model our algorithm is equivalent to an iterative nonlinear filtering furthermore we show that a particular form of the variational model gives rise to much faster convergence than other alternative forms we demonstrate that only a few iterations of this filter yield almost the same pixel ordering as the minimizer thus we apply only few iteration steps to obtain images whose pixels can be ordered in a strict and faithful way numerical experiments confirm that our algorithm outperforms by far its main competitors

shimura lifts of weakly holomorphic modular forms we show how to realize the shimura lift of arbitrary level and character using the vector valued theta lifts of borcherds using the regularization of borcherds lift we extend the shimura lift to take weakly holomorphic modular forms of half integral weight to meromorphic modular forms of even integral weight having poles at cm points

a study and modification of the local histogram equalization algorithm abstract an analysis of the local histogram equalization algorithm is presented an adaptation of the algorithm is suggested that involves varying the window size over different regions of the image this enables each region to be enhanced equally the algorithm has a parameter s to control the amount of stretching required some results are presented and analysed

contrast enhancement based on layered difference representation of 2d histograms a novel contrast enhancement algorithm based on the layered difference representation of 2d histograms is proposed in this paper we attempt to enhance image contrast by amplifying the gray level differences between adjacent pixels to this end we obtain the 2d histogram h k k l from an input image which counts the pairs of adjacent pixels with gray levels k and k l and represent the gray level differences in a tree like layered structure then we formulate a constrained optimization problem based on the observation that the gray level differences occurring more frequently in the input image should be more emphasized in the output image we first solve the optimization problem to derive the transformation function at each layer we then combine the transformation functions at all layers into the unified transformation function which is used to map input gray levels to output gray levels experimental results demonstrate that the proposed algorithm enhances images efficiently in terms of both objective quality and subjective quality

harmonic maass forms of weight 1 the object of this paper is to initiate a study of the fourier coefficients of a weight one harmonic maass form and relate them to the complex galois representation associated to a weight one newform which is the form s image under a certain differential operator in this paper our focus will be on weight one dihedral newforms of prime level p 3 mod 4 in this case we give properties of the fourier coefficients that are similar to and sometimes reduce to cases of stark s conjectures on derivatives of l functions we also give a new modular interpretation of certain products of differences of singular moduli studied by gross and zagier finally we provide some numerical evidence that the fourier coefficients of a mock modular form whose shadow is exotic are similarly related to the associated complex galois representation

error and stability estimates for surface divergence free rbf interpolants on the sphere recently a new class of surface divergence free radial basis function interpolants has been developed for surfaces in 3 in this paper several approximation results for this class of interpolants will be derived in the case of the sphere s 2 in particular sobolev type error estimates are obtained as well as optimal stability estimates for the associated interpolation matrices in addition a bernstein estimate and an inverse theorem are also derived numerical validation of the theoretical results is also given

overconvergent generalised eigenforms of weight one and class fields of real quadratic fields abstract this article examines the fourier expansions of certain non classical p adic modular forms of weight one the eponymous generalised eigenforms of the title so called because they lie in a generalised eigenspace for the hecke operators when this generalised eigenspace contains the theta series attached to a character of a real quadratic field k in which the prime p splits the coefficients of the attendant generalised eigenform are expressed as p adic logarithms of algebraic numbers belonging to an idoneous ring class field of k this suggests an approach to explicit class field theory for real quadratic fields which is simpler than the one based on stark s conjecture or its p adic variants and is perhaps closer in spirit to the classical theory of singular moduli

cycle integrals of the j function and mock modular forms in this paper we construct certain mock modular forms of weight 1 2 whose fourier coecients are given in terms of cycle integrals of the modular j function their shadows are weakly holomorphic forms of weight 3 2 these new mock modular forms occur as holomorphic parts of weakly harmonic maass forms we also construct a generalized mock modular form of weight 1 2 having a real quadratic class number times a regulator as a fourier coecient as an application of these forms we study holomorphic modular integrals of weight 2 whose rational period functions have poles at certain real quadratic integers the fourier coecients of these modular integrals are given in terms of cycle integrals of modular functions such a modular integral can be interpreted in terms of a shimura type lift of a mock modular form of weight 1 2 and yields a real quadratic analogue of a borcherds product

the rationality of vector valued modular forms associated with the weil representation in a recent paper duke math j 97 219 233 borcherds asks whether or not the spaces of vector valued modular forms associated to the weil representation have bases of modular forms whose fourier expansions have only integer coefficients we give an affirmative answer to borcherds question this strengthens and simplifies borcherds main theorem which is a generalization of a theorem of gross kohnen and zagier math ann 278 497 562

dyson s ranks and maass forms which were defined by ramanujan and watson decades ago in his last letter to hardy dated january 1920 see pages 127 131 of 27 ramanujan lists 17 such functions and he gives 2 more in his lost notebook 27 in his paper the final problem an account of the mock theta functions 32 watson defines 3 further functions surprisingly much remains unknown about these enigmatic series ramanujan s claims about their analytic properties remain open and there is even debate concerning the rigorous definition of such a function despite these seemingly problematic issues ramanujan s mock theta functions indeed possess many striking properties and they have been the subject of an astonishing number of important works for example see 5 6 7 8 12 13 14 18 19 20 23 27 28 32 33 35 36 to name a few watson predicted this high level of activity in his 1936 presidential address to the london mathematical society with his prophetic words see page 80 of 32

on two geometric theta lifts the theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type in this paper we establish for the orthogonal group o p 2 an adjointness result between borcherds s singular theta lift and the kudla millson lift we extend this result to arbitrary signature by introducing a new singular theta lift for o p q on the geometric side this lift can be interpreted as a differential character in the sense of cheeger and simons for the cycles under consideration

parametric oriented fitting for local contrast enhancement local image contrast enhancement methods extract more informative details comparing with the global methods yet the traditional schemes require to access all of the pixels in a local region to obtain the cumulative distribution function cdf for further enhancement i e adaptive histogram equalization ahe and thus the time complexity drastically increases accordingly when the block size increases consequently some speed up ingenuities such as block wise strategy and simple proportionally averaging manner are employed to cope with this issue yet serious blocking artifact or low contrast issues are accompanied in this study a local enhancement method namely corrected parametric oriented histogram equalization cpohe is proposed to effectively yield enhanced results with high contrast and artifact free texture using the concept of integral image and further correct the accompanied distortion as documented in the experimental results the proposed method provides a high practical value for applied in visual perception biometric and tracking detection applications

illumination insensitive image representation via synergistic weighted center surround receptive field model and weber law a novel illumination insensitive texture feature descriptor weber synergistic center surround pattern is presented we model differential synergistic excitation pattern synergistic straight orientation pattern and synergistic diagonal orientation pattern we discuss different scale and direction weighted coefficients in detail we mine higher order spatial information of weighted wscp image representation under complex illumination condition is a great challenge in computer vision field motivated by weber law and synergistic center surround receipt field model we propose an illumination insensitive feature descriptor named as weber synergistic center surround pattern wscp including three components differential synergistic excitation pattern dsep synergistic straight orientation pattern ssop and synergistic diagonal orientation pattern sdop to further enhance the discriminative power of wscp we present a scale and orientation weighted wscp wwscp which fully considers the inner and outer layer pixels excitation distributions as well as their orientation information importance to acquire more discriminative and rich features we utilize the spatiograms to describe patterns instead of the conventional histograms which can obtain higher order spatial information in the final classification stage we present a novel distance measurement model called weighted similarity measurement model wsmm to improve classification accuracy by sufficiently utilizing the information contents and orientation distributions of each pattern extensive experimental comparisons between our methods and other state of the art methods are conducted on cmupie yale b yale b ext feret photex alot and rawfoot databases and performance results have verified the effectiveness and efficiency of our proposed methods on the robustness to illumination variations

image editing with spatiograms transfer histogram equalization is a well known method for image contrast enhancement nevertheless as histograms do not include any information on the spatial repartition of colors their application to local image editing problems remains limited to cope with this lack of spatial information spatiograms have been recently proposed for tracking purposes a spatiogram is an image descriptor that combines a histogram with the mean and the variance of the position of each color in this paper we address the problem of local retouching of images by proposing a variational method for spatiogram transfer more precisely a reference spatiogram is used to modify the color value of a given region of interest of the processed image experiments on shadow removal and inpainting demonstrate the strength of the proposed approach

differential operators for harmonic weak maass forms and the vanishing of hecke eigenvalues for integers k 2 we study two differential operators on harmonic weak maass forms of weight 2 k the operator 2 k resp d k 1 defines a map to the space of weight k cusp forms resp weakly holomorphic modular forms we leverage these operators to study coefficients of harmonic weak maass forms although generic harmonic weak maass forms are expected to have transcendental coefficients we show that those forms which are dual under 2 k to newforms with vanishing hecke eigenvalues such as cm forms have algebraic coefficients using regularized inner products we also characterize the image of d k 1

generalized hermite interpolation via matrix valued conditionally positive definite functions in this paper we consider a broad class of interpolation problems for both scalarand vector valued multivariate functions subject to linear side conditions such as being divergence free where the data are generated via integration against compactly supported distributions we show that by using certain families of matrix valued conditionally positive definite functions such interpolation problems are well poised that is the interpolation matrices are invertible as a sample result we show that a divergence free vector field can be interpolated by a linear combination of convolutions of the data generating distributions with a divergence free 3 x 3 matrix valued conditionally positive definite function in addition we obtain norm estimates for inverses of interpolation matrices that arise in a class of multivariate hermite interpolation problems

heegner divisors l functions and harmonic weak maass forms recent works mostly related to ramanujan s mock theta functions make use of the fact that harmonic weak maass forms can be combinatorial generating functions generalizing works of waldspurger kohnen and zagier we prove that such forms also serve as generating functions for central values and derivatives of quadratic twists of weight 2 modular l functions to obtain these results we construct differentials of the third kind with twisted heegner divisor by suitably generalizing the borcherds lift to harmonic weak maass forms the connection with periods fourier coefficients derivatives of l functions and points in the jacobian of modular curves is obtained by analyzing the properties of these differentials using works of scholl waldschmidt and gross and zagier

fourier coefficients of harmonic weak maass forms and the partition function in a recent paper bruinier and ono proved that certain harmonic weak maass forms have the property that the fourier coefficients of their holomorphic parts are algebraic traces of weak maass forms evaluated on heegner points as a special case they obtained a remarkable finite algebraic formula for the hardy ramanujan partition function p n which counts the number of partitions of a positive integer n we establish an asymptotic formula with a power saving error term for the fourier coefficients in the bruinier ono formula as a consequence we obtain a new asymptotic formula for p n one interesting feature of this formula is that the main term contains essentially 3 cdot h 24n 1 fewer terms than the truncated main term in rademacher s exact formula for p n where h 24n 1 is the class number of the imaginary quadratic field bbb q sqrt 24n 1

the approximation of vector functions and their derivatives on the sphere a vector basis is defined for vector functions that are given on the sphere it is shown that the basis is orthogonal that it is complete and that the convergence properties of the spectral representation are determined by the smoothness of the vector function in cartesian coordinates the method for analyzing a vector function in terms of this basis uses the methods for analyzing scalar functions in terms of surface harmonics a vector function that is smooth in cartesian coordinates and nonzero at the pole will be discontinuous in spherical coordinates this is a result of the discontinuous spherical coordinate system rather than the vector function itself this in turn induces singularities in certain surface derivatives on the sphere even though the function is bounded and differentiable in cartesian coordinates many of the individual terms in a partial differential equation on the sphere are unbounded at the poles however the cancellation between these terms is sufficient to yield a bounded resu

sobolev type approximation rates for divergence free and curl free rbf interpolants recently error estimates have been made available for divergence free radial basis function rbf interpolants however these results are only valid for functions within the associated reproducing kernel hilbert space rkhs of the matrix valued rbf functions within the associated rkhs also known as the native space of the rbf can be characterized as vector fields having a specific smoothness making the native space quite small in this paper we develop sobolev type error estimates when the target function is less smooth than functions in the native space

automatic exact histogram specification for contrast enhancement and visual system based quantitative evaluation histogram equalization which aims at information maximization is widely used in different ways to perform contrast enhancement in images in this paper an automatic exact histogram specification technique is proposed and used for global and local contrast enhancement of images the desired histogram is obtained by first subjecting the image histogram to a modification process and then by maximizing a measure that represents increase in information and decrease in ambiguity a new method of measuring image contrast based upon local band limited approach and center surround retinal receptive field model is also devised in this paper this method works at multiple scales frequency bands and combines the contrast measures obtained at different scales using lp norm in comparison to a few existing methods the effectiveness of the proposed automatic exact histogram specification technique in enhancing contrasts of images is demonstrated through qualitative analysis and the proposed image contrast measure based quantitative analysis

niebur poincare series and traces of singular moduli abstract we compute the fourier coefficients of analogues of kohnen and zagier s modular forms f k of weight 2 and negative discriminant these functions can also be written as twisted traces of certain weight 2 poincare series with evaluations of niebur poincare series as fourier coefficients this allows us to study twisted traces of singular moduli in an integral weight setting in particular we recover explicit series expressions for twisted traces of singular moduli and extend algebraicity results by bengoechea to the weight 2 case we also compute regularized inner products of these functions which in the higher weight case have been related to evaluations of higher green s functions at cm points

a linear programming approach for optimal contrast tone mapping this paper proposes a novel algorithmic approach of image enhancement via optimal contrast tone mapping in a fundamental departure from the current practice of histogram equalization for contrast enhancement the proposed approach maximizes expected contrast gain subject to an upper limit on tone distortion and optionally to other constraints that suppress artifacts the underlying contrast tone optimization problem can be solved efficiently by linear programming this new constrained optimization approach for image enhancement is general and the user can add and fine tune the constraints to achieve desired visual effects experimental results demonstrate clearly superior performance of the new approach over histogram equalization and its variants

non vanishing of quadratic twists of modular l functions k have been the subject of much study both because of their intrinsicinterest and because of the prominent role they have played inkolyvagin s work on the birch and swinnerton dyer conjecture see b f h i j ko ma m m m1 o s p p waldspurger proved a fundamental theorem the ore me 1 w1 relating these central critical values to the fourier coe cients of half integral weight cusp forms for notational convenience if d is afundamental discriminant of a quadratic number eld then de ne d

algebraic formulas for the coefficients of mock theta functions and weyl vectors of borcherds products abstract we present some applications of the kudla millson and the millson theta lift the two lifts map weakly holomorphic modular functions to vector valued harmonic maass forms of weight 3 2 and 1 2 respectively we give finite algebraic formulas for the coefficients of ramanujan s mock theta functions f q and q in terms of traces of cm values of a weakly holomorphic modular function further we construct vector valued harmonic maass forms whose shadows are unary theta functions and whose holomorphic parts have rational coefficients this yields a rationality result for the coefficients of mock theta functions i e harmonic maass forms whose shadows lie in the space of unary theta functions moreover the harmonic maass forms we construct can be used to evaluate the petersson inner products of unary theta functions with harmonic maass forms giving formulas and rationality results for the weyl vectors of borcherds products

the distribution of free path lengths in the periodic lorentz gas and related lattice point problems the periodic lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers and is one of the fundamental models for chaotic diffusion in the present paper we investigate the boltzmann grad limit where the radius of each scatterer tends to zero and prove the existence of a limiting distribution for the free path length we also discuss related problems such as the statistical distribution of directions of lattice points that are visible from a fixed position

exact order based feature descriptor for illumination robust image matching we present a novel method for a feature descriptor called an exact order based descriptor eod the proposed method consists of three steps first to resolve ordering ambiguity for pixels of the same intensity an exact order image is created by changing the discrete intensity into a k dimensional continuous value second exact order based features are generated globally and locally finally the eod is constructed by combining the global and local exact order features using the discrete cosine transform experimental results show that the proposed method outperforms other state of the art descriptors over a number of images

the weil representation and hecke operators for vector valued modular forms we define hecke operators for vector valued modular forms transforming with the weil representation associated to a discriminant form we describe the properties of the corresponding algebra of hecke operators and study the action on modular forms

spatial mutual information and pagerank based contrast enhancement and quality aware relative contrast measure this paper proposes a novel algorithm for global contrast enhancement using a new definition of spatial mutual information smi of gray levels of an input image and pagerank algorithm the gray levels are used to represent nodes in pagerank algorithm and the weights between the nodes are computed according to their dependence and spatial spread over the image which is quantified by using smi the rank vector of gray levels resulted from pagerank algorithm is used in mapping input gray levels to output the damping factor of the pagerank algorithm is utilized to control the level of perceived global contrast on the output image furthermore a new metric is proposed for image quality aware relative contrast measurement between input and output images experimental results show that the proposed algorithm consistently produces good results

on the converse theorem for borcherds products abstract we prove a new converse theorem for borcherds multiplicative theta lift which improves the previously known results to this end we develop a newform theory for vector valued modular forms for the weil representation which might be of independent interest we also derive lower bounds for the ranks of the picard groups and the spaces of holomorphic top degree differential forms of modular varieties associated to orthogonal groups

joint exact histogram specification and image enhancement through the wavelet transform histogram specification equalization is an important tool for tasks such as image enhancement and normalization although this problem has exact solution for continuous images in the case of digital images it is ill posed recently an exact pixel ordering method based solely on local image intensity was proposed yet still with some limitations especially since it ignores the important image edge information in this paper we present a wavelet based method that simultaneously achieves the exact histogram specification and good image enhancement performance it does so through a carefully designed strict pixel ordering process during which the wavelet coefficients are fine tuned for the image enhancement purpose compared to previous work this approach takes into account not only local mean intensity values but also local edge information other advantages include fast pixel ordering good statistical models and better image enhancement performance experimental results and comparison with state of the art methods are presented

extrinsic meshless collocation methods for pdes on manifolds we proposed ways to implement meshless collocation methods extrinsically for solving elliptic pdes on smooth closed connected and complete riemannian manifolds with arbitrary codimensions our m

fourier transform and the global gan gross prasad conjecture for unitary groups by the relative trace formula approach of jacquet rallis we prove the global gan gross prasad conjecture for unitary groups under some local restrictions for the automorphic representations

parity of the partition function abstract although much is known about the partition function little is known about its parity for the polynomials d x d x 2 1 24 where d 23 mod 24 we show that there are infinitely many m resp n for which p d m is even resp p d n is odd if there is at least one such m resp n we bound the first m and n if any in terms of the class number h d for prime d we show that there are indeed infinitely many even values to this end we construct new modular generating functions using generalized borcherds products and we employ galois representations and locally nilpotent hecke algebras

cm values of regularized theta lifts and harmonic weak maas forms of weight 1 we study special values of regularized theta lifts at complex multiplication cm points in particular we show that cm values of borcherds products can be expressed in terms of finitely many fourier coefficients of certain harmonic weak maas forms of weight 1 as it turns out these coefficients are logarithms of algebraic integers whose prime ideal factorization is determined by special cycles on an arithmetic curve our results imply a conjecture of duke and li and give a new proof of the modularity of a certain arithmetic generating series of weight 1 studied by kudla rapoport and yang

harmonic maass forms and periods according to waldspurger s theorem the coefficients of half integral weight eigenforms are given by central critical values of twisted hecke l functions and therefore by periods here we prove that the coefficients of the holomorphic parts of weight 1 2 harmonic maass forms are determined by periods of algebraic differentials of the third kind on modular and elliptic curves

improved error bounds for scattered data interpolation by radial basis functions if additional smoothness requirements and boundary conditions are met the well known approximation orders of scattered data interpolants by radial functions can roughly be doubled

faltings heights of cm cycles and derivatives of l functions we study the faltings height pairing of arithmetic special divisors and cm cycles on shimura varieties associated to orthogonal groups we compute the archimedean contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a rankin l function we prove the conjecture in certain cases where the shimura variety has dimension 0 1 or 2 in particular we obtain a new proof of the gross zagier formula

transform based image enhancement algorithms with performance measure this paper presents a new class of the frequency domain based signal image enhancement algorithms including magnitude reduction log magnitude reduction iterative magnitude and a log reduction zonal magnitude technique these algorithms are described and applied for detection and visualization of objects within an image the new technique is based on the so called sequency ordered orthogonal transforms which include the well known fourier hartley cosine and hadamard transforms as well as new enhancement parametric operators a wide range of image characteristics can be obtained from a single transform by varying the parameters of the operators we also introduce a quantifying method to measure signal image enhancement called eme this helps choose the best parameters and transform for each enhancement a number of experimental results are presented to illustrate the performance of the proposed algorithms

l adic properties of smallest parts functions abstract we prove explicit congruences modulo powers of arbitrary primes for three smallest parts functions one for partitions one for overpartitions and one for partitions without repeated odd parts the proofs depend on l adic properties of certain modular forms and mock modular forms of weight 3 2 with respect to the hecke operators t l 2 m

eichler integrals period relations and jacobi forms this paper contains three main results the first one is to derive two period relations and the second one is a complete characterization of period functions of jacobi forms in terms of period relations these are done by introducing a concept of jacobi integrals on the full jacobi group the last one is to show for the given holomorphic function p z having two period relations there exists a unique jacobi integral up to jacobi forms with a given function p z as its period function this is done by constructing a generalized jacobi poincare series explicitly this is to say that every holomorphic function with period relations is coming from a jacobi integral it is an analogy of eichler cohomology theory studied in knopp bull am math soc 80 607 632 1974 for the functions with elliptic and modular variables it explains the functional equations satisfied by the mordell integrals associated with the lerch sums zwegers in mock theta functions phd thesis universiteit utrecht 2002 or more generally with the higher appell functions semikhatov et al in commun math phys 255 2 469 512 2005 developing theories of jacobi integrals with elliptic and modular variables in this paper is a natural extension of the eichler integral with modular variable period functions can be explained in terms of the parabolic cohomology group as well

naturalness preserved enhancement algorithm for non uniform illumination images image enhancement plays an important role in image processing and analysis among various enhancement algorithms retinex based algorithms can efficiently enhance details and have been widely adopted since retinex based algorithms regard illumination removal as a default preference and fail to limit the range of reflectance the naturalness of non uniform illumination images cannot be effectively preserved however naturalness is essential for image enhancement to achieve pleasing perceptual quality in order to preserve naturalness while enhancing details we propose an enhancement algorithm for non uniform illumination images in general this paper makes the following three major contributions first a lightness order error measure is proposed to access naturalness preservation objectively second a bright pass filter is proposed to decompose an image into reflectance and illumination which respectively determine the details and the naturalness of the image third we propose a bi log transformation which is utilized to map the illumination to make a balance between details and naturalness experimental results demonstrate that the proposed algorithm can not only enhance the details but also preserve the naturalness for non uniform illumination images

on a theta lift related to the shintani lift abstract we study the millson theta lift which maps weight 2 k to weight 1 2 k harmonic weak maass forms for k z k 0 and which is closely related to the classical shintani lift from weight 2 k 2 to weight k 3 2 cusp forms we compute the fourier expansion of the theta lift and show that it involves twisted traces of cm values and geodesic cycle integrals of the input function as an application we recover zagier s generating series of twisted traces of singular moduli of weight 1 2 as millson lifts

eulerian series as modular forms in 1988 hickerson proved the celebrated mock theta conjectures a collection of ten identities from ramanujan s lost notebook which express certain modular forms as linear combinations of mock theta functions in the context of maass forms these identities arise from the peculiar phenomenon that two different harmonic maass forms may have the same non holomorphic parts using this perspective we construct several infinite families of modular forms which are differences of mock theta functions

boundary behavior and the martin boundary problem for p harmonic functions in lipschitz domains boundary behavior and the martin boundary problem for p harmonic functions in lipschitz domains