on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

dynamical spectral rigidity among mathbb z _2 symmetric strictly convex domains close to a circle appendix b coauthored with h hezari

statistical mechanics of classical particles with logarithmic interactions the inhomogeneous mean field thermodynamic limit is constructed and evaluated for both the canonical thermodynamic functions and the states of systems of classical point particles with logarithmic interactions in two space dimensions the results apply to various physical models of translation invariant plasmas gravitating systems as well as to planar fluid vortex motion for attractive interactions a critical behavior occurs which can be classified as an extreme case of a second order phase transition to include in particular attractive interactions a new inequality for configurational integrals is derived from the arithmetic geometric mean inequality the method developed in this paper is easily seen to apply as well to systems with fairly general interactions in all space dimensions in addition it also provides us with a new proof of the trudinger moser inequality known from differential geometry in its sharp form

an algorithm for the computation of eigenvalues spectral zeta functions and zeta determinants on hyperbolic surfaces we present a rigorous scheme that makes it possible to compute eigenvalues of the laplace operator on hyperbolic surfaces within a given precision the method is based on an adaptation of the method of particular solutions to the case of locally symmetric spaces and on explicit estimates for the approximation of eigenfunctions on hyperbolic surfaces by certain basis functions it can be applied to check whether or not there is an eigenvalue in an e neighborhood of a given number 0 this makes it possible to find all the eigenvalues in a specified interval up to a given precision with rigorous error estimates the method converges exponentially fast with the number of basis functions used combining the knowledge of the eigenvalues with the selberg trace formula we are able to compute values and derivatives of the spectral zeta function again with error bounds as an example we calculate the spectral determinant and the casimir energy of the bolza surface and other surfaces

modular forms of weight 8 for γg 1 2 we complete the program indicated by the ansatz of d hoker and phong in genus 4 by proving the uniqueness of the restriction to jacobians of the weight 8 siegel cusp forms satisfying the ansatz we prove dim 4 1 2 8 0 2 and dim 4 1 2 8 7 in each genus we classify the linear relations among the self dual lattices of rank 16 we extend the program to genus 5 by constructing the unique linear combination of theta series that satisfies the ansatz

principal curve algorithms for partitioning high dimensional data spaces most partitioning algorithms iteratively partition a space into cells that contain underlying linear or nonlinear structures using linear partitioning strategies the compactness of each cell depends on how well the locally linear partitioning strategy approximates the intrinsic structure to partition a compact structure for complex data in a nonlinear context this paper proposes a nonlinear partition strategy this is a principal curve tree pc tree which is implemented iteratively given that a pc passes through the middle of the data distribution it allows for partitioning based on the arc length of the pc to enhance the partitioning of a given space a residual version of the pc tree algorithm is developed denoted here as the principal component analysis tree pcr tree algorithm because of its residual property the pcr tree can yield the intrinsic dimension of high dimensional data comparisons presented in this paper confirm that the proposed pc tree and pcr tree approaches show a better performance than several other competing partitioning algorithms in terms of vector quantization error and nearest neighbor search the comparison also shows that the proposed algorithms outperform competing linear methods in total average coverage which measures the nonlinear compactness of partitioning algorithms

can one hear the shape of a drum revisited in a landmark paper mark kac in 1966 amer math monthly 73 pp 1 23 showed that geometric properties of regions in r 2 can be obtained by studying the asymptotic properties of the spectrum of the laplacian he also conjectured that the shape of a region might be completely determined by the spectrum we describe recent developments in this field and discuss the status of the kac conjecture as well as similar conjectures for operators other than the laplacian

pullbacks of saito kurokawa lifts pullbacks of siegel eisenstein series have been studied by garrett 17 18 bocherer 6 7 heim 26 and play a key role in the proof of the algebraicity of critical values of certain automorphic l functions more generally one might consider pullbacks of siegel cusp forms for example ikeda 30 gave a conjectural formula for pullbacks of ikeda lifts 29 in terms of critical values of l functions for spn gl2 also the gross prasad conjecture 22 23 8 27 8 would relate pullbacks of siegel cusp forms of degree 2 to central critical values of l functions for gsp2 gl2 gl2 indeed bocherer furusawa and schulze pillot 8 gave an explicit formula for pullbacks of yoshida lifts 57 in this paper we give an explicit formula for pullbacks of saito kurokawa lifts and prove the algebraicity of central critical values of certain l functions for sp1 gl2 to be precise let be an odd positive integer let f s2 sl2 z be a normalized hecke eigenform and h s 1 2 0 4 a hecke eigenform associated to f by the shimura correspondence let f s 1 sp2 z be the saito kurokawa lift of h for each normalized hecke eigenform g s 1 sl2 z we consider the period integral f h h g g given by f h h g g

the nonexistence of certain level structures on abelian varieties over complex function fields let a be a principally polarized abelian variety of dimension g over a number field k the mordell weil theorem tells us that the group a k of k rational points of a is finitely generated in particular the torsion subgroup of a k is finite for the important special case of an elliptic curve e q mazur mal 2 has proved that the torsion subgroup of e q has order 16 in general one would expect the torsion subgroup of a k to have order c g k a constant depending only on the number field k and the dimension g but not

on the holomorphic differential forms of the siegel modular variety ii in this paper we follow our investigation about the construction of alternating differential forms on the siegel s modular variety let a represent the quotient of the siegel s upper half space hg by the full integral symplectic group sp 29 z fg it is a well known fact that ag is a quasi projective algebraic variety we shall denote by 4g its satake s compactification it is a normal projective but singular variety we shall denote by a0 the regular part of ag and by t 0 a desingularization of ag without loss of generality we can assume ao contained in ao for any complex manifold m we shall denote by or m the sections of the sheaf of alternating holomorphic differential forms on m of degree v in 3 it has been shown that for g 2 the injection

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

extremal properties of the determinant of the laplacian in the bergman metric on the moduli space of genus two riemann surfaces we study extremal properties of the determinant of the laplacian in the bergman metric on the moduli space of compact genus two riemann surfaces by a combination of analytical and numerical methods we identify four non degenerate critical points of this function and compute the signature of the hessian at these points the curve with the maximal number of automorphisms the burnside curve turns out to be the point of the absolute maximum our results agree with the mass formula for virtual euler characteristics of the moduli space a similar analysis is performed for bolza s strata of symmetric riemann surfaces of genus two

similarity preserving principal curve an optimal 1 d feature extractor for data representation this paper discusses the problem of what kind of learning model is suitable for the tasks of feature extraction for data representation and suggests two evaluation criteria for nonlinear feature extractors reconstruction error minimization and similarity preservation based on the suggested evaluation criteria a new type of principal curve similarity preserving principal curve sppc is proposed sppcs minimize the reconstruction error under the condition that the similarity between similar samples are preserved in the extracted features thus giving researchers effective and reliable cognition of the inner structure of data sets the existence and properties of sppcs are analyzed a practical learning algorithm is proposed and high dimensional extensions of sppcs are also discussed experimental results show the virtues of sppcs in preserving inner structures of data sets and discovering manifolds with high nonlinearity

some mathematical constants the theory of multiple gamma functions was studied in about 1900 and has been revived in the study of determinants of laplacians in the middle or so of 1980s there are some classes of mathematical constants involved naturally in the gamma and multiple gamma functions here we introduce those known mathematical constants associated with the gamma and multiple gamma functions and will show how they are involved if possible

the planar k means problem is np hard in the k means problem we are given a finite set s of points in m and integer k 1 and we want to find k points centers so as to minimize the sum of the square of the euclidean distance of each point in s to its nearest center we show that this well known problem is np hard even for instances in the plane answering an open question posed by dasgupta 2007 7

superstring scattering amplitudes in higher genus in this paper we continue the program pioneered by d hoker and phong and recently advanced by cacciatori dalla piazza and van geemen of finding the chiral superstring measure by constructing modular forms satisfying certain factorization constraints we give new expressions for their proposed ansatze in genera 2 and 3 respectively which admit a straightforward generalization we then propose an ansatz in genus 4 and verify that it satisfies the factorization constraints and gives a vanishing cosmological constant we further conjecture a possible formula for the superstring amplitudes in any genus subject to the condition that certain modular forms admit holomorphic roots

on the kodaira dimensions of hilbert modular varieties f o r m e e r where e l e 9 are of general type let us give the summary of this paper our argument is based on the criterion on the extendability of pluri canonical differential forms developed by tai 13 in w 1 we mainly introduce our notations and reproduce well known facts in w 2 we introduce tai s criterion lemma 2 and proposition 1 which assures that elliptic fixed points almost always satisfy tai s criterion and discuss the extendability of pluri canonical differential forms over such points in w we discuss the extendability over the cusp singularities by means of toroidal compactification 1 and tai s criterion and give some criterion for a

on the kodaira dimension of orthogonal modular varieties we prove that up to scaling there are only finitely many integral lattices l of signature 2 n with n ge 21 or n 17 such that the modular variety defined by the orthogonal group of l is not of general type in particular when n ge 108 every modular variety defined by an arithmetic group for a rational quadratic form of signature 2 n is of general type we also obtain similar finiteness in n ge 9 for the stable orthogonal groups as a byproduct we derive finiteness of lattices admitting reflective modular form of bounded vanishing order which proves a conjecture of gritsenko and nikulin

a three dimensional ball quotient in this paper we determine a very particular example of a picard modular variety of general type on its non singular models there exist many holomorphic differential forms in a forthcoming paper we will show that one can construct calabi yau manifolds by considering quotients of this variety and resolving singularities

nonlinear dimensionality reduction by locally linear inlaying high dimensional data is involved in many fields of information processing however sometimes the intrinsic structures of these data can be described by a few degrees of freedom to discover these degrees of freedom or the low dimensional nonlinear manifold underlying a high dimensional space many manifold learning algorithms have been proposed here we describe a novel algorithm locally linear inlaying lli which combines simple geometric intuitions and rigorously established optimality to compute the global embedding of a nonlinear manifold using a divide and conquer strategy lli gains some advantages in itself first its time complexity is linear in the number of data points and hence lli can be implemented efficiently second lli overcomes problems caused by the nonuniform sample distribution third unlike existing algorithms such as isometric feature mapping isomap local tangent space alignment ltsa and locally linear coordination llc lli is robust to noise in addition to evaluate the embedding results quantitatively two criteria based on information theory and kolmogorov complexity theory respectively are proposed furthermore we demonstrated the efficiency and effectiveness of our proposal by synthetic and real world data sets

face recognition based on the uncorrelated discriminant transformation abstract the extraction of discriminant features is the most fundamental and important problem in face recognition this paper presents a method to extract optimal discriminant features for face images by using the uncorrelated discriminant transformation and kl expansion experiments on the orl database and the nust603 database have been performed experimental results show that the uncorrelated discriminant transformation is superior to the foley sammon discriminant transformation and the new method to extract uncorrelated discriminant features for face images is very effective an error rate of 2 5 is obtained with the experiments on the orl database an average error rate of 1 2 is obtained with the experiments on the nust603 database experiments show that by extracting uncorrelated discriminant features face recognition could be performed with higher accuracy on lower than 16 16 resolution mosaic images it is suggested that for the uncorrelated discriminant transformation the optimal face image resolution can be regarded as the resolution m n which makes the dimensionality n mn of the original image vector space be larger and closer to the number of known face classes

on the lifting of elliptic cusp forms to siegel cusp forms of degree 2n i 1 cf 8 in this paper we will prove that their conjecture is true moreover we obtain a simple fourier coefficient formula for f z ibukiyama 23 also made a similar conjecture which was formulated in terms of the koecher maass series which we will not consider here let us describe our results throughout this paper we fix positive integers k and n such that k n mod 2 and put e 1 k 1 n for any positive rational number n we denote the absolute value of the discriminant of q vn q by dn the primitive dirichlet character corresponding to q v n q is denoted by xn let fn be the positive rational number such that n nf2nlet b e m2n q be a positive definite half integral symmetric matrix put db det 2b db d db fb f db and xb x db note that edb 0 1 mod 4 and that fb is an integer for each prime p let p qp cx be the unique additive character such that bp x exp 27ir ltx for x e z p 1 recall that the siegel series for b is defined by

moser trudinger and beckner onofri s inequalities on the cr sphere we derive sharp moser trudinger inequalities on the cr sphere the rst type is in the adams form for powers of the sublaplacian and for general spectrally dened operators on the space of cr pluriharmonic functions we will then obtain the sharp beckner onofri inequality for crpluriharmonic functions on the sphere and as a consequence a sharp logarithmic hardy littlewood sobolev inequality in the form given by carlen and loss

congruence of siegel modular forms and special values of their standard zeta functions in this paper we consider the relationship between the congruence of cuspidal hecke eigenforms with respect to sp n z and the special values of their standard zeta functions in particular we propose a conjecture concerning the congruence between saito kurokawa lifts and non saito kurokawa lifts and prove it under certain condition

the kodaira dimension of the moduli of k3 surfaces the global torelli theorem for projective k3 surfaces was first proved by piatetskii shapiro and shafarevich 35 years ago opening the way to treating moduli problems for k3 surfaces the moduli space of polarised k3 surfaces of degree 2d is a quasi projective variety of dimension 19 for general d very little has been known hitherto about the kodaira dimension of these varieties in this paper we present an almost complete solution to this problem our main result says that this moduli space is of general type for d 61 and for d 46 50 54 57 58 60

asymptotics of the spectrum and the selberg zeta function on the space of riemann surfaces letz s r be the selberg zeta function of a compact riemann surfacer we study the behavior ofz s r asr tends to infinity in the moduli space of stable curves the main result is an estimate forz s r valid fors in a neighborhood depending only on the genus ofs 1 our analysis gives an alternate proof of the belavin knizhnik double pole result 5

motives and siegel modular forms for a motive m over q we define the fundamental periods of m using invariant theory our definition generalizes deligne s periods we show that if a motive m is constructed from motives m 1 m 2 m n by a standard algebraic operation then the fundamental periods of m can be expressed as monomials of the fundamental periods of m 1 m 2 m n applying this theory we discuss two hypothetical motives attached to a siegel modular form we show that a siegel modular form of degree m has at most m 1 period invariants

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

the nef cone of toroidal compactifications of cal a _4 the moduli space of principally polarised abelian 4 folds can be compactified in several different ways by toroidal methods here we consider in detail the igusa compactification and the second voronoi compactification we describe in both cases the cone of nef cartier divisors the proof depends on a detailed description of the voronoi compactification which makes it possible to proceed by induction using the known description of the nef cone for compactifications of mathcal a 3 the igusa compactification has a non mathbb q factorial singularity which is resolved by a single blow up this resolution is the voronoi compactification the exceptional divisor e is a toric fano variety of dimension 9 the other boundary divisor d corresponds to degenerations with corank 1 after imposing a level structure in order to avoid certain technical complications we show that the closure of d in the voronoi compactification maps to the voronoi compactification of mathcal a 3 the toric description of the exceptional divisor allows us to describe the map in sufficient detail to estimate the intersection numbers needed this inductive process is only valid for the voronoi compactification the result for the igusa compactification is deduced from the voronoi compactification

petersson norms of not necessarily cuspidal jacobi modular forms and applications abstract we extend the usual notion of petersson inner product on the space of cuspidal jacobi forms to include non cuspidal forms as well this is done by examining carefully the relation between certain growth killing invariant differential operators on h 2 and those on h 1 c here h n denotes the siegel upper half space of degree n as applications we can understand better the growth of petersson norms of fourier jacobi coefficients of klingen eisenstein series which in turn has applications to finer issues about representation numbers of quadratic forms and as a by product we also show that any siegel modular form of degree 2 is determined by its fundamental fourier coefficients

clustering to minimize the maximum intercluster distance the problem of clustering a set of points so as to minimize the maximum intercluster distance is studied an o kn approximation algorithm where n is the number of points and k is the number of clusters that guarantees solutions with an objective function value within two times the optimal solution value is presented this approximation algorithm succeeds as long as the set of points satisfies the triangular inequality we also show that our approximation algorithm is best possible with respect to the approximation bound if pz np

supervised optimal locality preserving projection in the past few years the computer vision and pattern recognition community has witnessed a rapid growth of a new kind of feature extraction method the manifold learning methods which attempt to project the original data into a lower dimensional feature space by preserving the local neighborhood structure among these methods locality preserving projection lpp is one of the most promising feature extraction techniques unlike the unsupervised learning scheme of lpp this paper follows the supervised learning scheme i e it uses both local information and class information to model the similarity of the data based on novel similarity we propose two feature extraction algorithms supervised optimal locality preserving projection solpp and normalized laplacian based supervised optimal locality preserving projection nl solpp optimal here means that the extracted features via solpp or nl solpp are statistically uncorrelated and orthogonal we compare the proposed solpp and nl solpp with lpp orthogonal locality preserving projection olpp and uncorrelated locality preserving projection ulpp on publicly available data sets experimental results show that the proposed solpp and nl solpp achieve much higher recognition accuracy

face recognition using laplacianfaces we propose an appearance based face recognition method called the laplacianface approach by using locality preserving projections lpp the face images are mapped into a face subspace for analysis different from principal component analysis pca and linear discriminant analysis lda which effectively see only the euclidean structure of face space lpp finds an embedding that preserves local information and obtains a face subspace that best detects the essential face manifold structure the laplacianfaces are the optimal linear approximations to the eigenfunctions of the laplace beltrami operator on the face manifold in this way the unwanted variations resulting from changes in lighting facial expression and pose may be eliminated or reduced theoretical analysis shows that pca lda and lpp can be obtained from different graph models we compare the proposed laplacianface approach with eigenface and fisherface methods on three different face data sets experimental results suggest that the proposed laplacianface approach provides a better representation and achieves lower error rates in face recognition

optimization of spectral functions of dirichlet laplacian eigenvalues we consider the shape optimization of spectral functions of dirichlet laplacian eigenvalues over the set of star shaped symmetric bounded planar regions with smooth boundary the regions are represented using fourier cosine coefficients and the optimization problem is solved numerically using a quasi newton method the method is applied to maximizing two particular nonsmooth spectral functions the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue both of which are generalizations of the payne polya weinberger ratio the optimal values and attaining regions for n 13 are presented and interpreted as a study of the range of the dirichlet laplacian eigenvalues for both spectral functions and each n the optimal attaining region has multiplicity two nth eigenvalue

on the best constant in the moser onofri aubin inequality let s2 be the 2 dimensional unit sphere and let j denote the nonlinear functional on the sobolev space h1 s2 defined by j alpha u frac alpha 16 pi int s 2 nabla u 2 d mu 0 frac 1 4 pi int s 2 u d mu 0 rm ln int s 2 e u frac d mu 0 4 pi where d mu 0 sin theta d theta wedge d phi onofri had established that j is non negative on h1 s2 provided 1 in this note we show that if j is restricted to those u in h 1 s 2 that satisfies the aubin condition int s 2 e u x j d mu 0 0 quad rm for all 1 leq j leq 3 then the same inequality continues to hold i e j u 0 whenever alpha geq frac 2 3 epsilon 0 for some epsilon 0 0 the question of chang yang on whether this remains true for all alpha geq frac 1 2 remains open

linear dependence among siegel modular forms theorems are given which describe when high enough vanishing at the cusps implies that a siegel modular cusp form is zero formerly impractical computations become practical and examples are given in degree four vanishing order is described by kernels a type of polyhedral convex hull

the uniformization of the moduli space of principally polarized abelian 6 folds author s alexeev valery donagi ron farkas gavril izadi elham ortega angela abstract abstractstarting from a beautiful idea of kanev we construct a uniformization of the moduli space mathcal a 6 of principally polarized abelian 6 folds in terms of curves and monodromy data we show that the general principally polarized abelian variety of dimension 6 is a prym tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the weyl group of the e 6 lattice along the way we establish numerous facts concerning the geometry of the hurwitz space of such e 6 covers including 1 a proof that the canonical class of the hurwitz space is big 2 a concrete geometric description of the hodge hurwitz eigenbundles with respect to the kanev correspondence and 3 a description of the ramification divisor of the prym tyurin map from the hurwitz space to mathcal a 6 in the terms of syzygies of the abel prym tyurin curve

hybrid manifold embedding in this brief we present a novel supervised manifold learning framework dubbed hybrid manifold embedding hyme unlike most of the existing supervised manifold learning algorithms that give linear explicit mapping functions the hyme aims to provide a more general nonlinear explicit mapping function by performing a two layer learning procedure in the first layer a new clustering strategy called geodesic clustering is proposed to divide the original data set into several subsets with minimum nonlinearity in the second layer a supervised dimensionality reduction scheme called locally conjugate discriminant projection is performed on each subset for maximizing the discriminant information and minimizing the dimension redundancy simultaneously in the reduced low dimensional space by integrating these two layers in a unified mapping function a supervised manifold embedding framework is established to describe both global and local manifold structure as well as to preserve the discriminative ability in the learned subspace experiments on various data sets validate the effectiveness of the proposed method

nonvanishing of koecher maass series attached to siegel cusp forms we prove a nonvanishing result for koecher maass series attached to siegel cusp forms of weight k and degree n in certain strips on the complex plane when n 2 we prove such a result for forms orthogonal to the space of the saito kurokawa lifts up to finitely many exceptions in bounded regions c 2015 elsevier inc all rights reserved

local linear discriminant analysis framework using sample neighbors the linear discriminant analysis lda is a very popular linear feature extraction approach the algorithms of lda usually perform well under the following two assumptions the first assumption is that the global data structure is consistent with the local data structure the second assumption is that the input data classes are gaussian distributions however in real world applications these assumptions are not always satisfied in this paper we propose an improved lda framework the local lda llda which can perform well without needing to satisfy the above two assumptions our llda framework can effectively capture the local structure of samples according to different types of local data structure our llda framework incorporates several different forms of linear feature extraction approaches such as the classical lda and principal component analysis the proposed framework includes two llda algorithms a vector based llda algorithm and a matrix based llda mllda algorithm mllda is directly applicable to image recognition such as face recognition our algorithms need to train only a small portion of the whole training set before testing a sample they are suitable for learning large scale databases especially when the input data dimensions are very high and can achieve high classification accuracy extensive experiments show that the proposed algorithms can obtain good classification results

locality preserved maximum information projection dimensionality reduction is usually involved in the domains of artificial intelligence and machine learning linear projection of features is of particular interest for dimensionality reduction since it is simple to calculate and analytically analyze in this paper we propose an essentially linear projection technique called locality preserved maximum information projection lpmip to identify the underlying manifold structure of a data set lpmip considers both the within locality and the between locality in the processing of manifold learning equivalently the goal of lpmip is to preserve the local structure while maximize the out of locality global information of the samples simultaneously different from principal component analysis pca that aims to preserve the global information and locality preserving projections lpps that is in favor of preserving the local structure of the data set lpmip seeks a tradeoff between the global and local structures which is adjusted by a parameter alpha so as to find a sub space that detects the intrinsic manifold structure for classification tasks computationally by constructing the adjacency matrix lpmip is formulated as an eigenvalue problem lpmip yields orthogonal basis functions and completely avoids the singularity problem as it exists in lpp further we develop an efficient and stable lpmip qr algorithm for implementing lpmip especially on high dimensional data set theoretical analysis shows that conventional linear projection methods such as weighted pca maximum margin criterion mmc linear discriminant analysis lda and lpp could be derived from the lpmip framework by setting different graph models and constraints extensive experiments on face digit and facial expression recognition show the effectiveness of the proposed lpmip method

pullback of the lifting of elliptic cusp forms and miyawaki s conjecture we construct a lifting from siegel cusp forms of degree r to siegel cusp forms of degree r 2n for r n 1 our result is a partial solution of a conjecture made by miyawaki 27 page 307 in 1992 in particular we can calculate the standard l function of a cusp form of degree 3 and weight 12 which is in accordance with miyawaki s conjecture we give a conjecture on the petersson inner product of the lifting in terms of certain l values

critical values of l functions on gsp 2 gl 2 this paper deals with deligne s conjecture on the critical values of l functions let z g h s denote the tensor product l function attached to a siegel modular form g of weight k and an elliptic cusp form h of weight l we assume that the first fourier jacobi coefficient of g is not identically zero then deligne s conjecture is fully proven for z g h s when l 2k 2 and partly for the remaining case

local coordinates alignment with global preservation for dimensionality reduction dimensionality reduction is vital in many fields and alignment based methods for nonlinear dimensionality reduction have become popular recently because they can map the high dimensional data into a low dimensional subspace with the property of local isometry however the relationships between patches in original high dimensional space cannot be ensured to be fully preserved during the alignment process in this paper we propose a novel method for nonlinear dimensionality reduction called local coordinates alignment with global preservation we first introduce a reasonable definition of topology preserving landmarks tpls which not only contribute to preserving the global structure of datasets and constructing a collection of overlapping linear patches but they also ensure that the right landmark is allocated to the new test point then an existing method for dimensionality reduction that has good performance in preserving the global structure is used to derive the low dimensional coordinates of tpls local coordinates of each patch are derived using tangent space of the manifold at the corresponding landmark and then these local coordinates are aligned into a global coordinate space with the set of landmarks in low dimensional space as reference points the proposed alignment method called landmarks based alignment can produce a closed form solution without any constraints while most previous alignment based methods impose the unit covariance constraint which will result in the deficiency of global metrics and undesired rescaling of the manifold experiments on both synthetic and real world datasets demonstrate the effectiveness of the proposed algorithm

graph optimized locality preserving projections locality preserving projections lpp is a typical graph based dimensionality reduction dr method and has been successfully applied in many practical problems such as face recognition however lpp depends mainly on its underlying neighborhood graph whose construction suffers from the following issues 1 such neighborhood graph is artificially defined in advance and thus does not necessary benefit subsequent dr task 2 such graph is constructed using the nearest neighbor criterion which tends to work poorly due to the high dimensionality of original space 3 it is generally uneasy to assign appropriate values for the neighborhood size and heat kernel parameter involved in graph construction to address these problems we develop a novel dr algorithm called graph optimized locality preserving projections golpp the idea is to integrate graph construction with specific dr process into a unified framework which results in an optimized graph rather than predefined one moreover an entropy regularization term is incorporated into the objective function for controlling the uniformity level of the edge weights in graph so that a principled graph updating formula naturally corresponding to conventional heat kernel weights can be obtained finally the experiments on several publicly available uci and face data sets show the feasibility and effectiveness of the proposed method with encouraging results