dwork s conjecture on unit root zeta functions annals of mathematics 150 1999 867 927 arxiv math 9911270v1 math nt 1 nov 1999 dwork s conjecture on unit root zeta functions by daqing wan 1 introduction in this article we introduce a systematic new method to investigate the conjectural p adic meromorphic continuation of professor bernard dwork s unit root zeta function attached to an ordinary family of algebraic varieties defined over a finite field of characteristic p after his pioneer p adic investigation of the weil conjectures on the zeta function of an algebraic variety over a finite field dwork went on to study the p adic analytic variation of a family of such zeta functions when the variety moves through an algebraic family in the course of doing so he was led to a new zeta function called the unit root zeta function which goes beyond the reach of the existing theory he conjectured 8 that such a unit root zeta function is p adic meromorphic everywhere these unit root zeta functions contain important arithmetic information about a family of algebraic varieties they are truly p adic in nature and are transcendental functions sometimes seeming quite mysterious in fact no single nontrivial example has been proved to be true about this conjecture other than the trivial overconvergent or log convergent case for which dwork s classical p adic theory already applies see 6 11 26 for various attempts in this article we introduce a systematic new method to study such unit root zeta functions our method can be used to prove the conjecture in the case when the involved unit root f crystal has rank one in particular this settles the first nontrivial case the rank one unit root f crystal coming from the family of higher dimensional kloosterman sums our method further allows us to understand reasonably well about analytic variation of an arithmetic family of such rank one unit root zeta functions motivated by the gouv ea mazur conjecture about dimension variation of classical and p adic modular forms we shall introduce another this work was partially supported by nsf the author wishes to thank p deligne and n katz for valuable discussions at one stage of this work during 1993 and 1994 the author would also like to thank s sperber for his careful reading of the manuscript and for his many detailed comments which led to a significant improvement of the exposition

density based multiscale data condensation a problem gaining interest in pattern recognition applied to data mining is that of selecting a small representative subset from a very large data set in this article a nonparametric data reduction scheme is suggested it attempts to represent the density underlying the data the algorithm selects representative points in a multiscale fashion which is novel from existing density based approaches the accuracy of representation by the condensed set is measured in terms of the error in density estimates of the original and reduced sets experimental studies on several real life data sets show that the multiscale approach is superior to several related condensation methods both in terms of condensation ratio and estimation error the condensed set obtained was also experimentally shown to be effective for some important data mining tasks like classification clustering and rule generation on large data sets moreover it is empirically found that the algorithm is efficient in terms of sample complexity

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

scalable tsk fuzzy modeling for very large datasets using minimal enclosing ball approximation in order to overcome the difficulty in takagi sugeno kang tsk fuzzy modeling for large datasets scalable tsk stsk fuzzy model training is investigated in this study based on the core set based minimal enclosing ball meb approximation technique the specified l2 norm penalty based insensitive criterion is first proposed for tsk model training and it is found that such tsk fuzzy model training can be equivalently expressed as a center constrained meb problem with this finding an stsk fuzzy model training algorithm which is called stsk for large or very large datasets is then proposed by using the core set based meb approximation technique the proposed algorithm has two distinctive advantages over classical tsk fuzzy model training algorithms the maximum space complexity for training is not reliant on the size of the training dataset and the maximum time complexity for training is linear with the size of the training dataset as confirmed by extensive experiments on both synthetic and real world regression datasets

fast parzen density estimation using clustering based branch and bound this correspondence proposes a fast parzen density estimation algorithm that would be especially useful in nonparametric discriminant analysis problems by preclustering the data and applying a simple branch and bound procedure to the clusters significant numbers of data samples that would contribute little to the density estimate can be excluded without detriment to actual evaluation via the kernel functions this technique is especially helpful in the multivariant case and does not require a uniform sampling grid the proposed algorithm may also be used in conjunction with the data reduction technique of fukunaga and hayes 1989 to further reduce the computational load experimental results are presented to verify the effectiveness of this algorithm

slopes of eigencurves over boundary disks let p be a prime number we study the slopes of u p eigenvalues on the subspace of modular forms that can be transferred to a definite quaternion algebra we give a sharp lower bound of the corresponding newton polygon the computation happens over a definite quaternion algebra by jacquet langlands correspondence it generalizes a prior work of jacobs slopes of compact hecke operators thesis university of london imperial college 2004 who treated the case of p 3 with a particular level in case when the modular forms have a finite character of conductor highly divisible by p we improve the lower bound to show that the slopes of u p eigenvalues grow roughly like arithmetic progressions as the weight k increases this is the first very positive evidence for buzzard kilford s conjecture on the behavior of the eigencurve near the boundary of the weight space that is proved for arbitrary p and general level we give the exact formula of a fraction of the slope sequence

density estimation by wavelet thresholding density estimation is a commonly used test case for nonparametric estimation methods we explore the asymptotic properties of estimators based on thresholding of empirical wavelet coefficients minimax rates of convergence are studied over a large range of besov function classes b pq and for a range of global l p error measures 1 p p some form of nonlinearity is essential since the minimax linear estimators are suboptimal by polynomial powers of n a second approach using an approximation of a gaussian white noise model in a mallows metric is used to attain exactly optimal rates of convergence for quadratic error p 2

the l functions of witt coverings results on l functions of artin schreier coverings by dwork bombieri and adolphson sperber are generalized to l functions of witt coverings

slopes for higher rank artin schreier witt towers author s ren rufei wan daqing xiao liang yu myungjun abstract we fix a monic polynomial bar f x in mathbb f q x over a finite field of characteristic p and consider the mathbb z p ell artin schreier witt tower defined by bar f x this is a tower of curves cdots to c m to c m 1 to cdots to c 0 mathbb a 1 whose galois group is canonically isomorphic to mathbb z p ell the degree ell unramified extension of mathbb z p which is abstractly isomorphic to mathbb z p ell as a topological group we study the newton slopes of zeta functions of this tower of curves this reduces to the study of the newton slopes of l functions associated to characters of the galois group of this tower we prove that when the conductor of the character is large enough the newton slopes of the l function asymptotically form a finite union of arithmetic progressions as a corollary we prove the spectral halo property of the spectral variety associated to the mathbb z p ell artin schreier witt tower this extends the main result in dwx from rank one case ell 1 to the higher rank case ell geq 1

discretisation of an infinite delay equation in this paper a banach phase space containing bc 0 and contained in c 0 is defined with which existence of a solution and convergence of a discrete scheme are proved for an infinite delay differential equation

the eigencurve over the boundary of weight space author s liu ruochuan wan daqing xiao liang abstract we prove that the eigencurve associated to a definite quaternion algebra over qq satisfies the following properties as conjectured by coleman mazur and buzzard kilford a over the boundary annuli of weight space the eigencurve is a disjoint union of countably infinitely many connected components each finite and flat over the weight annuli b the u p slopes of points on each fixed connected component are proportional to the p adic valuations of the parameter on weight space and c the sequence of the slope ratios form a union of finitely many arithmetic progressions with the same common difference in particular as a point moves towards the boundary on an irreducible connected component of the eigencurve the slope converges to zero

frsde fast reduced set density estimator using minimal enclosing ball approximation reduced set density estimator rsde is an important technique that can be used to replace the classical parzen window estimator pw for saving the computational cost though rsde demonstrates a nicer performance in the density accuracy and the computational time compared with several existing methods it still faces the critical challenge for practical applications because of its high time complexity no less than o n 2 and space complexity o n 2 in training the model weighting coefficients on large data sets in order to overcome this shortcoming a fast reduced set density estimator algorithm frsde is proposed in this study first the relationship between rsde and the minimal enclosing ball problems meb in computational geometry is revealed then the finding that rsde is equivalent to a special meb problem is derived with this finding the fast core set based meb approximation algorithm is introduced to develop the proposed algorithm frsde compared with rsde frsde has the following distinctive advantage it can guarantee that the upper bound of the time complexity is linear with the size n of a large data set and the upper bound of the space complexity is independent of n our experimental results show that the proposed frsde has a competitive performance in the density accuracy and an overwhelming advantage over rsde for large data sets in the data condensation rate and the training time for the weighting coefficients

liapunov razumikhin pairs and the location of positive limit sets for precompact functional differential equations with infinite delay in a recent paper l haddock and terjeki addressed the problem of locating positive limit sets for autonomous functional differential equations with infinite delay the authors obtained invariance principles and other results concerning asymptotic behavior of solutions by employing liapunov razumikhin techniques the autonomous nature of the equations they considered was central to their work the purpose of this paper is to extend their results to a class of nonautonomous equations such a task was undertaken for finite delay equations in 2 here as in 2 the assumption of autonomy is replaced by a continuity condition and our invariance principles involve limiting equations of our original equation we feel that the setting for our theory is quite natural and that our methods illustrate the role of limiting equations in extending results from the autonomous theory to a significantly larger class of equations several examples are given to illustrate our techniques

image segmentation based on adaptive cluster prototype estimation an image segmentation algorithm based on adaptive fuzzy c means fcm clustering is presented in this paper in the conventional fcm clustering algorithm cluster assignment is based solely on the distribution of pixel attributes in the feature space and does not take into consideration the spatial distribution of pixels in an image by introducing a novel dissimilarity index in the modified fcm objective function the new adaptive fuzzy clustering algorithm is capable of utilizing local contextual information to impose local spatial continuity thus exploiting the high inter pixel correlation inherent in most real world images the incorporation of local spatial continuity allows the suppression of noise and helps to resolve classification ambiguity to account for smooth intensity variation within each homogenous region in an image a multiplicative field is introduced to each of the fixed fcm cluster prototype the multiplicative field effectively makes the fixed cluster prototype adaptive to slow smooth within cluster intensity variation and allows homogenous regions with slow smooth intensity variation to be segmented as a whole experimental results with synthetic and real color images have shown the effectiveness of the proposed algorithm

probability density estimation from optimally condensed data samples the requirement to reduce the computational cost of evaluating a point probability density estimate when employing a parzen window estimator is a well known problem this paper presents the reduced set density estimator that provides a kernel based density estimator which employs a small percentage of the available data sample and is optimal in the l sub 2 sense while only requiring spl oscr n sup 2 optimization routines to estimate the required kernel weighting coefficients the proposed method provides similar levels of performance accuracy and sparseness of representation as support vector machine density estimation which requires spl oscr n sup 3 optimization routines and which has previously been shown to consistently outperform gaussian mixture models it is also demonstrated that the proposed density estimator consistently provides superior density estimates for similar levels of data reduction to that provided by the recently proposed density based multiscale data condensation algorithm and in addition has comparable computational scaling the additional advantage of the proposed method is that no extra free parameters are introduced such as regularization bin width or condensation ratios making this method a very simple and straightforward approach to providing a reduced set density estimator with comparable accuracy to that of the full sample parzen density estimator

newton slopes for artin schreier witt towers we fix a monic polynomial f x in mathbb f q x over a finite field and consider the artin schreier witt tower defined by f x this is a tower of curves cdots rightarrow c m rightarrow c m 1 rightarrow cdots rightarrow c 0 mathbb a 1 with total galois group mathbb z p we study the newton slopes of zeta functions of this tower of curves this reduces to the study of the newton slopes of l functions associated to characters of the galois group of this tower we prove that when the conductor of the character is large enough the newton slopes of the l function form arithmetic progressions which are independent of the conductor of the character as a corollary we obtain a result on the behavior of the slopes of the eigencurve associated to the artin schreier witt tower analogous to the result of buzzard and kilford

on the location of positive limit sets for autonomous functional differential equations with infinite delay in 1983 the authors published a paper lo on an invariance principle for finite delay functional differential equations fdes in 9 t krisztin and the authors extended many of the ideas of lo to include fdes with infinite delay the basic results in both 9 and lo centered around the use of liapunov razumikhin techniques and the location of positive limit sets of precompact orbits the main purpose of this paper is to extend the results of 9 and to provide several examples to illustrate the main theorems of particular interest will be to establish asymptotic constancy of solutions of equations for which each constant function is itself a solution this idea was one of the main themes in lo but was not examined in 9 likewise we will see how a family of spaces as opposed to a single phase space can be useful in examining certain equations cf example 3 4 in considering the use of limit sets in conjunction with razumikhin techniques and invariance principles for infinite delay equations there are several difficulties that are encountered these include i choice of space s ii precompactness of orbits and iii appropriate comparison principles these topics are discussed individually in 11 7 and 8

weighted parzen windows for pattern classification this paper introduces the weighted parzen window classifier the proposed technique uses a clustering procedure to find a set of reference vectors and weights which are used to approximate the parzen window kernel estimator classifier the weighted parzen window classifier requires less computation and storage than the full parzen window classifier experimental results showed that significant savings could be achieved with only minimal if any error rate degradation for synthetic and real data sets

generalized core vector machines kernel methods such as the support vector machine svm are often formulated as quadratic programming qp problems however given m training patterns a naive implementation of the qp solver takes o m 3 training time and at least o m2 space hence scaling up these qps is a major stumbling block in applying kernel methods on very large data sets and a replacement of the naive method for finding the qp solutions is highly desirable recently by using approximation algorithms for the minimum enclosing ball meb problem we proposed the core vector machine cvm algorithm that is much faster and can handle much larger data sets than existing svm implementations however the cvm can only be used with certain kernel functions and kernel methods for example the very popular support vector regression svr cannot be used with the cvm in this paper we introduce the center constrained meb problem and subsequently extend the cvm algorithm the generalized cvm algorithm can now be used with any linear nonlinear kernel and can also be applied to kernel methods such as svr and the ranking svm moreover like the original cvm its asymptotic time complexity is again linear in m and its space complexity is independent of m experiments show that the generalized cvm has comparable performance with state of the art svm and svr implementations but is faster and produces fewer support vectors on very large data sets

dimension variation of classical and p adic modular forms a quadratic bound is obtained for a conjecture of gouvea mazur on arithmetic variation of dimensions of classical and p adic modular forms

estimating the support of a high dimensional distribution suppose you are given some data set drawn from an underlying probability distribution p and you want to estimate a simple subset s of input space such that the probability that a test point drawn from p lies outside of s equals some a priori specified value between 0 and 1 we propose a method to approach this problem by trying to estimate a function f that is positive on s and negative on the complement the functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data it is regularized by controlling the length of the weight vector in an associated feature space the expansion coefficients are found by solving a quadratic programming problem which we do by carrying out sequential optimization over pairs of input patterns we also provide a theoretical analysis of the statistical performance of our algorithm the algorithm is a natural extension of the support vector algorithm to the case of unlabeled data

semi supervised classification method through oversampling and common hidden space semi supervised classification methods attempt to improve classification performance based on a small amount of labeled data through full use of abundant unlabeled data although existing semi supervised classification methods have exhibited promising results in many applications they still have drawbacks including performance degeneration due to the introduction of unlabeled data and partially false labels in a small amount of labeled data to circumvent such drawbacks a new semi supervised classification method ochs ssc through oversampling and a common hidden space is proposed in the paper the primary characteristics of the proposed method include two aspects one is that unlabeled data are only used to generate new synthetic data to extend the minimal amount of labeled data the other is that the final classifier is learned in the extended feature space which is composed of the original feature space and the common hidden space found between labeled data and the synthetic data instead of the original feature space extensive experiments on 23 datasets indicate the effectiveness of the proposed method

multivariate online kernel density estimation with gaussian kernels we propose a novel approach to online estimation of probability density functions which is based on kernel density estimation kde the method maintains and updates a non parametric model of the observed data from which the kde can be calculated we propose an online bandwidth estimation approach and a compression revitalization scheme which maintains the kde s complexity low we compare the proposed online kde to the state of the art approaches on examples of estimating stationary and non stationary distributions and on examples of classification the results show that the online kde outperforms or achieves a comparable performance to the state of the art and produces models with a significantly lower complexity while allowing online adaptation

robust relief feature weighting margin maximization and fuzzy optimization a latest advance in relief feature weighting techniques is that the iterative procedure of relief can be approximately expressed as a margin maximization problem and therefore its distinctive properties can be investigated with the help of optimization theory being motivated by this advance the relief feature weighting algorithm is investigated for the first time within a fuzzy optimization framework a new margin based objective function that incorporates three fuzzy concepts namely fuzzy difference measure fuzzy feature weighting and fuzzy instance force coefficient is introduced by the application of fuzzy optimization to this new margin based objective function several useful theoretical results are derived based upon which a set of robust relief feature weighting algorithms are proposed for two class data multiclass data and then online data as demonstrated by extensive experiments in synthetic datasets the university of california at irvine uci benchmark datasets cancer gene expression datasets and face image datasets the proposed algorithms were found to be competitive with the state of the art algorithms and robust for datasets with noise and or outliers

t2fela type 2 fuzzy extreme learning algorithm for fast training of interval type 2 tsk fuzzy logic system a challenge in modeling type 2 fuzzy logic systems is the development of efficient learning algorithms to cope with the ever increasing size of real world data sets in this paper the extreme learning strategy is introduced to develop a fast training algorithm for interval type 2 takagi sugeno kang fuzzy logic systems the proposed algorithm called type 2 fuzzy extreme learning algorithm t2fela has two distinctive characteristics first the parameters of the antecedents are randomly generated and parameters of the consequents are obtained by a fast learning method according to the extreme learning mechanism in addition because the obtained parameters are optimal in the sense of minimizing the norm the resulting fuzzy systems exhibit better generalization performance the experimental results clearly demonstrate that the training speed of the proposed t2fela algorithm is superior to that of the existing state of the art algorithms the proposed algorithm also shows competitive performance in generalization abilities

core vector machines fast svm training on very large data sets standard svm training has o m3 time and o m2 space complexities where m is the training set size it is thus computationally infeasible on very large data sets by observing that practical svm implementations only approximate the optimal solution by an iterative strategy we scale up kernel methods by exploiting such approximateness in this paper we first show that many kernel methods can be equivalently formulated as minimum enclosing ball meb problems in computational geometry then by adopting an efficient approximate meb algorithm we obtain provably approximately optimal solutions with the idea of core sets our proposed core vector machine cvm algorithm can be used with nonlinear kernels and has a time complexity that is linear in m and a space complexity that is independent of m experiments on large toy and real world data sets demonstrate that the cvm is as accurate as existing svm implementations but is much faster and can handle much larger data sets than existing scale up methods for example cvm with the gaussian kernel produces superior results on the kddcup 99 intrusion detection data which has about five million training patterns in only 1 4 seconds on a 3 2ghz pentium 4 pc