ranks of elliptic curves this paper gives a general survey of ranks of elliptic curves over the field of rational numbers the rank is a measure of the size of the set of rational points the paper includes discussions of the birch and swinnertondyer conjecture the parity conjecture ranks in families of quadratic twists and ways to search for elliptic curves of large rank

on minimum class locality preserving variance support vector machine in this paper a so called minimum class locality preserving variance support machine mclpv svm algorithm is presented by introducing the basic idea of the locality preserving projections lpp which can be seen as a modified class of support machine svm and or minimum class variance support machine mcvsvm mclpv svm in contrast to svm and mcvsvm takes the intrinsic manifold structure of the data space into full consideration and inherits the characteristics of svm and mcvsvm we discuss in the paper the linear case the small sample size case and the nonlinear case of the mclpv svm similar to mcvsvm the mclpv svm optimization problem in the small sample size case is solved by using dimensionality reduction through principal component analysis pca and one in the nonlinear case is transformed into an equivalent linear mclpv svm problem under kernel pca kpca experimental results on real datasets indicate the effectiveness of the mclpv svm by comparing it with svm and mcvsvm

torsion points on families of squares of elliptic curves in a recent paper we proved that there are at most finitely many complex numbers 0 1 such that the points 2 sqrt 2 2 lambda and 3 sqrt 6 3 lambda are both torsion on the elliptic curve defined by y 2 x x 1 x here we give a generalization to any two points with coordinates algebraic over the field q and even over c this implies a special case of a variant of pink s conjecture for a variety inside a semiabelian scheme namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes

cross domain soft partition clustering with diversity measure and knowledge reference conventional soft partition clustering approaches such as fuzzy c means fcm maximum entropy clustering mec and fuzzy clustering by quadratic regularization fc qr are usually incompetent in those situations where the data are quite insufficient or much polluted by underlying noise or outliers in order to address this challenge the quadratic weights and gini simpson diversity based fuzzy clustering model qwgsd fc is first proposed as a basis of our work based on qwgsd fc and inspired by transfer learning two types of cross domain soft partition clustering frameworks and their corresponding algorithms referred to as type i type ii knowledge transfer oriented c means ti kt cm and tii kt cm are subsequently presented respectively the primary contributions of our work are four fold 1 the delicate qwgsd fc model inherits the most merits of fcm mec and fc qr with the weight factors in the form of quadratic memberships similar to fcm it can more effectively calculate the total intra cluster deviation than the linear form recruited in mec and fc qr meanwhile via gini simpson diversity index like shannon entropy in mec and equivalent to the quadratic regularization in fc qr qwgsd fc is prone to achieving the unbiased probability assignments 2 owing to the reference knowledge from the source domain both ti kt cm and tii kt cm demonstrate high clustering effectiveness as well as strong parameter robustness in the target domain 3 ti kt cm refers merely to the historical cluster centroids whereas tii kt cm simultaneously uses the historical cluster centroids and their associated fuzzy memberships as the reference this indicates that tii kt cm features more comprehensive knowledge learning capability than ti kt cm and tii kt cm consequently exhibits more perfect cross domain clustering performance and 4 neither the historical cluster centroids nor the historical cluster centroid based fuzzy memberships involved in ti kt cm or tii kt cm can be inversely mapped into the raw data this means that both ti kt cm and tii kt cm can work without disclosing the original data in the source domain i e they are of good privacy protection for the source domain in addition the convergence analyses regarding both ti kt cm and tii kt cm are conducted in our research the experimental studies thoroughly evaluated and demonstrated our contributions on both synthetic and real life data scenarios the novel qwgsd fc model is designed for soft partition clustering two cross domain clustering methods ti tii kt cm are subsequently proposed ti tii kt cm are superior to or competitive with other existing methods the convergences of both ti kt cm and tii kt cm are proved

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

kernel methods in machine learning we review machine learning methods employing positive definite kernels these methods formulate learning and estimation problems in a reproducing kernel hilbert space rkhs of functions defined on the data domain expanded in terms of a kernel working in linear spaces of function has the benefit of facilitating the construction and analysis of learning algorithms while at the same time allowing large classes of functions the latter include nonlinear functions as well as functions defined on nonvectorial data we cover a wide range of methods ranging from binary classifiers to sophisticated methods for estimation with structured data

training ν support vector classifiers theory and algorithms the support vector machine svm for classification proposed by scholkopf smola williamson and bartlett 2000 has the advantage of using a parameter on controlling the number of support vectors in this article we investigate the relation between svm and c svm in detail we show that in general they are two different problems with the same optimal solution set hence we may expect that many numerical aspects of solving them are similar however compared to regular c svm the formulation of svm is more complicated so up to now there have been no effective methods for solving large scale svm we propose a decomposition method for svm that is competitive with existing methods for c svm we also discuss the behavior of svm by some numerical experiments

prime specialization in genus 0 for a prime polynomial f t z t a classical conjecture predicts how often f has prime values for a finite field k and a prime polynomial f t k u t the natural analogue of this conjecture a prediction for how often f takes prime values on k u is not generally true when f t is a polynomial in t p p the characteristic of k the explanation rests on a new global obstruction which can be measured by an appropriate average of the nonzero mobius values f g as g varies we prove the surprising fact that this mobius average which can be defined without reference to any conjectures has a periodic behavior governed by the geometry of the plane curve f 0 the periodic mobius average behavior implies in specific examples that a polynomial in u r does not take prime values as often as analogies with z t suggest and it leads to a modified conjecture for how often prime values occur

sub domain adaptation learning methodology regarded as global methods maximum mean discrepancy mmd based transfer learning frameworks only reflect the global distribution discrepancy and structural differences between domains they can reflect neither the inner local distribution discrepancy nor the structural differences between domains to address this problem a novel transfer learning framework with local learning ability a sub domain adaptation learning framework sdal is proposed in this framework a projected maximum local weighted mean discrepancy pmlmd is constructed by integrating the theory and method of local weighted mean lwm into mmd pmlmd reflects global distribution discrepancy between domains through accumulating local distribution discrepancies between the local sub domains in domains in particular we formulate in theory that pmlmd is one of the generalized measures of mmd on the basis of sdal two novel methods are proposed by using multi label classifiers mlc and support vector machine svm finally tests on artificial data sets high dimensional text data sets and face data sets show the sdal based transfer learning methods are superior to or at least comparable with benchmarking methods

hilbert space embeddings and metrics on probability measures a hilbert space embedding for probability measures has recently been proposed with applications including dimensionality reduction homogeneity testing and independence testing this embedding represents any probability measure as a mean element in a reproducing kernel hilbert space rkhs a pseudometric on the space of probability measures can be defined as the distance between distribution embeddings we denote this as k indexed by the kernel function k that defines the inner product in the rkhs we present three theoretical properties of k first we consider the question of determining the conditions on the kernel k for which k is a metric such k are denoted characteristic kernels unlike pseudometrics a metric is zero only when two distributions coincide thus ensuring the rkhs embedding maps all distributions uniquely i e the embedding is injective while previously published conditions may apply only in restricted circumstances e g on compact domains and are difficult to check our conditions are straightforward and intuitive integrally strictly positive definite kernels are characteristic alternatively if a bounded continuous kernel is translation invariant on d then it is characteristic if and only if the support of its fourier transform is the entire d second we show that the distance between distributions under k results from an interplay between the properties of the kernel and the distributions by demonstrating that distributions are close in the embedding space when their differences occur at higher frequencies third to understand the nature of the topology induced by k we relate k to other popular metrics on probability measures and present conditions on the kernel k under which k metrizes the weak topology

exact solutions to the two dimensional korteweg de vries burgers equation two methods are described for obtaining exact travelling monotone shock wave solutions to the two dimensional korteweg de vries burgers equation they are compared with two methods that have been reported recently by other authors

root numbers and ranks in positive characteristic abstract for a global field k and an elliptic curve e over k t silverman s specialization theorem implies rank e k t rank e t k for all but finitely many t p 1 k if this inequality is strict for all but finitely many t the elliptic curve e is said to have elevated rank all known examples of elevated rank for k q rest on the parity conjecture for elliptic curves over q and the examples are all isotrivial some additional standard conjectures over q imply that there does not exist a non isotrivial elliptic curve over q t with elevated rank in positive characteristic an analogue of one of these additional conjectures is false inspired by this for the rational function field k u over any finite field with characteristic 2 we construct an explicit 2 parameter family e c d of non isotrivial elliptic curves over k t depending on arbitrary c d such that under the parity conjecture each e c d has elevated rank

on minimum distribution discrepancy support vector machine for domain adaptation domain adaptation learning dal is a novel and effective technique to address pattern classification problems where the prior information for training is unavailable or insufficient its effectiveness depends on the discrepancy between the two distributions that respectively generate the training data for the source domain and the testing data for the target domain however dal may not work so well when only the distribution mean discrepancy between source and target domains is considered and minimized in this paper we first construct a generalized projected maximum distribution discrepancy gpmdd metric for dal on reproducing kernel hilbert space rkhs based domain distributions by simultaneously considering both the projected maximum distribution mean and the projected maximum distribution scatter discrepancy between the source and the target domain in the sequel based on both the structure risk and the gpmdd minimization principle we propose a novel domain adaptation kernelized support vector machine daksvm with respect to the classical svm and its two extensions called ls daksvm and m daksvm with respect to the least square svm and the v svm respectively moreover our theoretical analysis justified that the proposed gpmdd metric could effectively measure the consistency not only between the rkhs embedding domain distributions but also between the scatter information of source and target domains hence the proposed methods are distinctive in that the more consistency between the scatter information of source and target domains can be achieved by tuning the kernel bandwidth the better the convergence of gpmdd metric minimization is and thus improving the scalability and generalization capability of the proposed methods for dal experimental results on artificial and real world problems indicate that the performance of the proposed methods is superior to or at least comparable with existing benchmarking methods

the difference between the weil height and the canonical height on elliptic curves estimates for the difference of the weil height and the canonical height of points on elliptic curves are used for many purposes both theoretical and computational in this note we give an explicit estimate for this difference in terms of the j invariant and discriminant of the elliptic curve the method of proof suggested by serge lang is to use the decomposition of the canonical height into a sum of local heights we illustrate one use for our estimate by computing generators for the mordell weil group in three examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 x ax b with a and b in the ring of integers of k the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 10 chapter viii 9 or 6 chapter vi the canonical height is determined by this property together with the fact that the difference

generalized tanh method with symbolic computation and generalized shallow water wave equation abstract we report that symbolic computation with the generalized tanh method leads to new soliton like solutions for a 2 1 dimensional generalization of the shallow water wave equations

special points on fibered powers of elliptic surfaces consider a fibered power of an elliptic surface we characterize its subvarieties that contain a zariski dense set of points that are torsion points in fibers with complex multiplication this result can be viewed as a mix of the manin mumford and andre oort conjecture and is related to a conjecture of pink the main technical tool is a new height inequality we also use it to give another proof of a case of gubler s result on the bogomolov conjecture over function fields

prime specialization in higher genus i a classical conjecture predicts how often a polynomial in z t takes prime values the natural analogous conjecture for prime values of a polynomial f t u t where is a finite field is false the conjecture over u was modified in earlier work by introducing a correction factor that encodes unexpected periodicity of the mobius function at the values of f on u when f u t where p is the characteristic of in this paper for p 6 2 we extend the mobius periodicity results for u the affine line to the case when f has coefficients in the coordinate ring a of any higher genus smooth affine curve with one geometric point at infinity the basic strategy is to pull up results from the genus 0 case by means of well chosen projections to the affine line our techniques can also be used to prove nontrivial properties of a correction factor in the conjecture on primality statistics for values of f a t on a even as f and vary

existence of simple jacobian varieties of genus g with rank at least 4g 5 we prove the existence of infinitely many curves of genus g over the rational number field which have absolutely simple jacobian varieties with mordell weil rank at least 4 g 5

new support vector algorithms we propose a new class of support vector algorithms for regression and classification in these algorithms a parameter lets one effectively control the number of support vectors while this can be useful in its own right the parameterization has the additional benefit of enabling us to eliminate one of the other free parameters of the algorithm the accuracy parameter epsilon in the regression case and the regularization constant c in the classification case we describe the algorithms give some theoretical results concerning the meaning and the choice of and report experimental results

the tanh method a tool for solving certain classes of nonlinear evolution and wave equations the tanh or hyperbolic tangent method is a powerful technique to search for travelling waves coming out from one dimensional nonlinear wave and evolution equations in particular in those problems where dispersive effects reaction diffusion and or convection play an important role to show the strength of the method an overview is given to find out which kind of problems are solved with this technique and how in some nontrivial cases this method adapted to the problem at hand still can be applied single as well as coupled equations arising from wave phenomena which appear in different scientific domains such as physics chemical kinetics geochemistry and mathematical biology will be treated next attention is focussed towards approximate solutions as a result solitary and shock wave profiles are derived together with the associated width and velocity the same method can be easily extended so that difference differential equations can be similarly solved finally some extensions of the method are discussed

orthogonal laplacianfaces for face recognition following the intuition that the naturally occurring face data may be generated by sampling a probability distribution that has support on or near a submanifold of ambient space we propose an appearance based face recognition method called orthogonal laplacianface our algorithm is based on the locality preserving projection lpp algorithm which aims at finding a linear approximation to the eigenfunctions of the laplace beltrami operator on the face manifold however lpp is nonorthogonal and this makes it difficult to reconstruct the data the orthogonal locality preserving projection olpp method produces orthogonal basis functions and can have more locality preserving power than lpp since the locality preserving power is potentially related to the discriminating power the olpp is expected to have more discriminating power than lpp experimental results on three face databases demonstrate the effectiveness of our proposed algorithm

torsion points pell s equation and integration in elementary terms the main results of this paper involve general algebraic differentials omega on a general pencil of algebraic curves we show how to determine if omega is integrable in elementary terms for infinitely many members of the pencil in particular this corrects an assertion of james davenport from 1981 and provides the first proof even in rather strengthened form we also indicate analogies with work of andre and hrushovski and with the grothendieck katz conjecture to reach this goal we first provide proofs of independent results which extend conclusions of relative manin mumford type allied to the zilber pink conjectures we characterize torsion points lying on a general curve in a general abelian scheme of arbitrary relative dimension at least 2 in turn we present yet another application of the latter results to a rather general pencil of pell equations a 2 db 2 1 over a polynomial ring we determine whether the pell equation with squarefree d is solvable for infinitely many members of the pencil

robust truncated hinge loss support vector machines the support vector machine svm has been widely applied for classification problems in both machine learning and statistics despite its popularity however svm has some drawbacks in certain situations in particular the svm classifier can be very sensitive to outliers in the training sample moreover the number of support vectors svs can be very large in many applications to circumvent these drawbacks we propose the robust truncated hinge loss svm rsvm which uses a truncated hinge loss the rsvm is shown to be more robust to outliers and to deliver more accurate classifiers using a smaller set of svs than the standard svm our theoretical results show that the rsvm is fisher consistent even when there is no dominating class a scenario that is particularly challenging for multicategory classification similar results are obtained for a class of margin based classifiers

on the rank of an elliptic surface nagao has recently given a conjectural limit formula for the rank of an elliptic surface e in terms of a weighted average of fibral frobenius trace values we show that tate s conjecture on the order of vanishing of l 2 e s essentially implies nagao s formula in particular we prove nagao s formula for rational elliptic surfaces in the case that e is a twist we reduce nagao s and tate s conjectures to the case of products of curves and we verify the conjectures for many new classes of elliptic surfaces of kodaira dimension 0 and 1

nonlinear periodic solutions for isothermal magnetostatic atmospheres the equations of magnetohydrostatic equilibria for a plasma in a gravitational field are investigated analytically for equilibria with one ignorable spatial coordinate the equations reduce to a single nonlinear elliptic equation for the magnetic potential known as the grad shafranov equation specifying the arbitrary function in the latter equation yields a nonlinear elliptic equation analytical nonlinear periodic solutions of this elliptic equation are obtained for the case of an isothermal atmosphere in a uniform gravitational field e g a model for the solar atmosphere we obtained several classes of exact solutions of five nonlinear evolution equations liouville sinh poisson double sinh poisson sine poisson and double sine poisson using the generalized tanh method moreover the backlund transformations are used to generate further new classes of solutions the final results may be used to investigate some models in solar physics

robust multi source adaptation visual classification using supervised low rank representation abstract how to guarantee the robustness of multi source adaptation visual classification is an important challenge in current visual learning community to this end we address in this paper the problem of robust visual classification with few labeled samples from the target domain of interest by leveraging multiple prior source models motivated by the recent success of low rank representation we formulate this problem as a robust multi source adaptation visual classification rmavc model with supervised low rank representation by combining the strength of discriminative information from the target domain and the prior models from multiple source domains specifically we propose a joint supervised low rank representation and multi source adaptation visual classification framework which achieves dual goals of finding the most discriminative low rank representation and multi source adaptation classifier parameters for the target domain while it is showed in this paper that the proposed rmavc framework is effective and can produce high accuracy on several tasks of multi source adaptation visual classification this framework fails to consider the local geometrical structure of the target data and the heterogeneousness among multiple source domains hence under this framework we further present two effective extensions or variants i e rmavck and rmavc fm by exploiting multiple kernel trick and flexible manifold regularization respectively the proposed framework and its variants are robust for classifying visual objects accurately and the experimental results demonstrate the effectiveness of our methods on several types of image and video datasets

travelling wave solutions of some classes of nonlinear evolution equations in 1 1 and 2 1 dimensions 1 the tanh method is proposed to find travelling wave solutions in 1 1 and 2 1 dimensional wave equations it can be extended to solve a whole family of modified korteweg de vries type of equations higher dimensional wave equations and nonlinear evolution equations

a least squares approach to direct importance estimation we address the problem of estimating the ratio of two probability density functions which is often referred to as the importance the importance values can be used for various succeeding tasks such as covariate shift adaptation or outlier detection in this paper we propose a new importance estimation method that has a closed form solution the leave one out cross validation score can also be computed analytically therefore the proposed method is computationally highly efficient and simple to implement we also elucidate theoretical properties of the proposed method such as the convergence rate and approximation error bounds numerical experiments show that the proposed method is comparable to the best existing method in accuracy while it is computationally more efficient than competing approaches

the square free sieve and the rank of elliptic curves let e be an elliptic curve over q a celebrated theorem of mordell asserts that e q the abelian group of rational points of e is finitely generated by the rank of e we mean the rank of e q thus the rank of e is positive if and only if e possesses an infinity of rational points relatively few general qualitative assertions can be made about the rank as e varies how large can the rank get although we expect that there are elliptic curves over q with arbitrarily high ranks this is presently unknown what is the average size of the rank recently brumer and mcguinness have reported on their study of 310 716 elliptic curves of prime conductor less than 108 where they have found that 20 06 of those curves have even rank 2 cf bm also see forthcoming work of brumer where subject to a number of standard conjectures he shows that 2 3 is an upper bound for the average rank for all elliptic curves over q ordered in terms of their faltings height what is the behavior of the rank over the family of twists of a given elliptic curve here one has three natural kinds of families of twists 1 quadratic twists one can take any elliptic curve and systematically twist it by all quadratic characters this is the type of family of elliptic curves we are concerned with in this paper specifically if e is an elliptic curve over q given by the weierstrass equation y2 x a x b and d is any squarefree integer the quadratic twist of e by d ed is given by the equation

minimum class variance support vector machines in this paper a modified class of support vector machines svms inspired from the optimization of fisher s discriminant ratio is presented the so called minimum class variance svms mcvsvms the mcvsvms optimization problem is solved in cases in which the training set contains less samples that the dimensionality of the training vectors using dimensionality reduction through principal component analysis pca afterward the mcvsvms are extended in order to find nonlinear decision surfaces by solving the optimization problem in arbitrary hilbert spaces defined by mercer s kernels in that case it is shown that under kernel pca the nonlinear optimization problem is transformed into an equivalent linear mcvsvms problem the effectiveness of the proposed approach is demonstrated by comparing it with the standard svms and other classifiers like kernel fisher discriminant analysis in facial image characterization problems like gender determination eyeglass and neutral facial expression detection

neron models of jacobians over base schemes of dimension greater than 1 we investigate to what extent the theory of neron models of jacobians and of abel jacobi maps extends to relative curves over base schemes of dimension greater than 1 we give a necessary and sufficient criterion for the existence of a neron model we use this to show that in general neron models do not exist even after making a modification or even alteration of the base on the other hand we show that neron models do exist outside some codimension 2 locus

bounded height in pencils of finitely generated subgroups in this paper we prove a general bounded height result for specializations in finitely generated subgroups varying in families which complements and sharpens the toric mordell lang theorem by replacing finiteness by emptyness for the intersection of varieties and subgroups all moving in a pencil except for bounded height values of the parameters and excluding identical relations more precisely an instance of the results is as follows consider the torus scheme g r m c over a curve c defined over q and let be a subgroup scheme generated by finitely many sections satisfying some necessary conditions further let v be any subscheme then there is a bound for the height of the points p c q such that for some which does not generically lie in v p lies in the fiber vp we further offer some direct diophantine applications to illustrate once again that the results implicitly contain information absent from the previous bounds in this context

backlund transformations and exact solutions for some nonlinear evolution equations in solar magnetostatic models the backlund transformations for some nonlinear evolution equations the liouville the sine and sinh poisson equations are constructed through the akns system in unified manner the obtained backlund transformations are used to generate new classes of solutions the latter is employed to obtain an infinite sequence of solutions moreover we generate an infinite sequence of additional solutions by employing the permutability theorem and give a general expression for the nth order solution it turns out that this general expression can be employed to obtain solutions for the sine and sinh poisson equations the final results are used to investigate some models in solar plasma physics conclusions and comments are given

domain adaptation via transfer component analysis domain adaptation allows knowledge from a source domain to be transferred to a different but related target domain intuitively discovering a good feature representation across domains is crucial in this paper we first propose to find such a representation through a new learning method transfer component analysis tca for domain adaptation tca tries to learn some transfer components across domains in a reproducing kernel hilbert space using maximum mean miscrepancy in the subspace spanned by these transfer components data properties are preserved and data distributions in different domains are close to each other as a result with the new representations in this subspace we can apply standard machine learning methods to train classifiers or regression models in the source domain for use in the target domain furthermore in order to uncover the knowledge hidden in the relations between the data labels from the source and target domains we extend tca in a semisupervised learning setting which encodes label information into transfer components learning we call this extension semisupervised tca the main contribution of our work is that we propose a novel dimensionality reduction framework for reducing the distance between domains in a latent space for domain adaptation we propose both unsupervised and semisupervised feature extraction approaches which can dramatically reduce the distance between domain distributions by projecting data onto the learned transfer components finally our approach can handle large datasets and naturally lead to out of sample generalization the effectiveness and efficiency of our approach are verified by experiments on five toy datasets and two real world applications cross domain indoor wifi localization and cross domain text classification