weil etale cohomology of curves over p adic fields abstract recent research has demonstrated a connection between weil etale cohomology and special values of zeta functions in particular lichtenbaum has shown that the order of vanishing and leading coefficient of the zeta function of a smooth projective variety over a finite field has a weil etale cohomological interpretation in terms of certain secondary euler characteristics these results rely on a duality theorem stated in terms of cup product in weil etale cohomology we define weil etale cohomology for varieties over p adic fields and prove a duality theorem for the cohomology of g m on a smooth proper geometrically connected curve of index 1 this duality theorem is a p adic analogue of lichtenbaum s weil etale duality theorem for curves over finite fields as well as a weil etale analogue of his classical duality theorem for curves over p adic fields finally we show that our duality theorem implies this latter classical duality theorem for index 1 curves

robust algebraic segmentation of mixed rigid body and planar motions from two views this paper studies segmentation of multiple rigid body motions in a 3 d dynamic scene under perspective camera projection we consider dynamic scenes that contain both 3 d rigid body structures and 2 d planar structures based on the well known epipolar and homography constraints between two views we propose a hybrid perspective constraint hpc to unify the representation of rigid body and planar motions given a mixture of k hybrid perspective constraints we propose an algebraic process to partition image correspondences to the individual 3 d motions called robust algebraic segmentation ras particularly we prove that the joint distribution of image correspondences is uniquely determined by a set of 2k th degree polynomials a global signature for the union of k motions of possibly mixed type the first and second derivatives of these polynomials provide a means to recover the association of the individual image samples to their respective motions finally using robust statistics we show that the polynomials can be robustly estimated in the presence of moderate image noise and outliers we conduct extensive simulations and real experiments to validate the performance of the new algorithm the results demonstrate that ras achieves notably higher accuracy than most existing robust motion segmentation methods including random sample consensus ransac and its variations the implementation of the algorithm is also two to three times faster than the existing methods the implementation of the algorithm and the benchmark scripts are available at http perception csl illinois edu ras

a cocycle model for topological and lie group cohomology we propose a unified framework in which the different constructions of cohomology groups for topological and lie groups can all be treated on equal footings in particular we show that the cohomology of locally continuous cochains respectively locally smooth in the case of lie groups fits into this framework which provides an easily accessible cocycle model for topological and lie group cohomology we illustrate the use of this unified framework and the relation between the different models in various applications this includes the construction of cohomology classes characterizing the string group and a direct connection to lie algebra cohomology

zeta functions of regular arithmetic schemes at s 0 lichtenbaum conjectured in 22 the existence of a weil etale cohomology in order to describe the vanishing order and the special value of the zeta function of an arithmetic scheme x at s 0 in terms of euler poincare characteristics assuming the conjectured finite generation of some motivic cohomology groups we construct such a cohomology theory for regular schemes proper over spec z in particular we compute uncon ditionally the right weil etale cohomology of number rings and projective spaces over number rings we state a precise version of lichtenbaum s con jecture which expresses the vanishing order resp the special value of the zeta function x s at s 0 as the rank resp the determinant of a single perfect complex of abelian groups r w c x z then we relate this conjecture to soule s conjecture and to the tamagawa number conjecture lichtenbaum s conjecture for projective spaces over the ring of integers of an abelian number field follows

on the weil etale cohomology of number fields we give a direct description of the category of sheaves on lichtenbaum s weil etale site of a number ring then we apply this result to define a spectral sequence relating weil etale cohomology to artin verdier etale cohomology finally we construct complexes of etale sheaves computing the expected weil etale cohomology

continuity properties of measurable group cohomology a version of group cohomology for locally compact groups and polish modules has previously been developed using a bar resolution restricted to measurable cochains that theory was shown to enjoy analogs of most of the standard algebraic properties of group cohomology but various analytic features of those cohomology groups were only partially understood this paper re examines some of those issues at its heart is a simple dimension shifting argument which enables one to regularize measurable cocycles leading to some simplifications in the description of the cohomology groups a range of consequences are then derived from this argument first we prove that for target modules that are frechet spaces the cohomology groups agree with those defined using continuous cocycles and hence they vanish in positive degrees when the acting group is compact using this we then show that for frechet discrete or toral modules the cohomology groups are continuous under forming inverse limits of compact base groups and also under forming direct limits of discrete target modules lastly these results together enable us to establish various circumstances under which the measurable cochains cohomology groups coincide with others defined using sheaves on a semi simplicial space associated to the underlying group or sheaves on a classifying space for that group we also prove in some cases that the natural quotient topologies on the measurable cochains cohomology groups are hausdorff

random matrix theory and l functions at s 1 2 recent results of katz and sarnak 8 9 suggest that the low lying zeros of families of l functions display the statistics of the eigenvalues of one of the compact groups of matrices u n o n or usp 2n we here explore the link between the value distributions of the l functions within these families at the central point s 1 2 and those of the characteristic polynomials z u of matrices u with respect to averages over so 2n and usp 2n at the corresponding point 0 using techniques previously developed for u n in 10 for any matrix size n we find exact expressions for the moments of z u 0 for each ensemble and hence calculate the asymptotic large n value distributions for z u 0 and log z u 0 the asymptotic results for the integer moments agree precisely with the few corresponding values known for l functions the value distributions suggest consequences for the non vanishing of l functions at the central point

n dimensional tensor voting and application to epipolar geometry estimation we address the problem of epipolar geometry estimation by formulating it as one of hyperplane inference from a sparse and noisy point set in an 8d space given a set of noisy point correspondences in two images of a static scene without correspondences even in the presence of moving objects our method extracts good matches and rejects outliers the methodology is novel and unconventional since unlike most other methods optimizing certain scalar objective functions our approach does not involve initialization or any iterative search in the parameter space therefore it is free of the problem of local optima or poor convergence further since no search is involved it is unnecessary to impose simplifying assumption to the scene being analyzed for reducing the search complexity subject to the general epipolar constraint only we detect wrong matches by a computation scheme 8d tensor voting which is an instance of the more general n dimensional tensor voting framework in essence the input set of matches is first transformed into a sparse 8d point set dense 8d tensor kernels are then used to vote for the most salient hyperplane that captures all inliers inherent in the input with this filtered set of matches the normalized eight point algorithm can be used to estimate the fundamental matrix accurately by making use of efficient data structure and locality our method is both time and space efficient despite the higher dimensionality we demonstrate the general usefulness of our method using example image pairs for aerial image analysis with widely different views and from nonstatic 3d scenes each example contains a considerable number of wrong matches

estimation of subspace arrangements with applications in modeling and segmenting mixed data recently many scientific and engineering applications have involved the challenging task of analyzing large amounts of unsorted high dimensional data that have very complicated structures from both geometric and statistical points of view such unsorted data are considered mixed as different parts of the data have significantly different structures which cannot be described by a single model in this paper we propose to use subspace arrangements a union of multiple subspaces for modeling mixed data each subspace in the arrangement is used to model just a homogeneous subset of the data thus multiple subspaces together can capture the heterogeneous structures within the data set in this paper we give a comprehensive introduction to a new approach for the estimation of subspace arrangements this is known as generalized principal component analysis gpca in particular we provide a comprehensive summary of important algebraic properties and statistical facts that are crucial for making the inference of subspace arrangements both efficient and robust even when the given data are corrupted by noise or contaminated with outliers this new method in many ways improves and generalizes extant methods for modeling or clustering mixed data there have been successful applications of this new method to many real world problems in computer vision image processing and system identification in this paper we will examine several of those representative applications this paper is intended to be expository in nature however in order that this may serve as a more complete reference for both theoreticians and practitioners we take the liberty of filling in several gaps between the theory and the practice in the existing literature

on the p adic local invariant cycle theorem for a proper flat generically smooth scheme x over a complete discrete valuation ring with finite residue field of characteristic p we construct a specialization morphism from the rigid cohomology of the geometric special fibre to d cris of the p adic etale cohomology of the geometric generic fibre and we make a conjecture p adic local invariant cycle theorem that describes the behavior of this map for regular x analogous to the situation in ell adic etale cohomology for ell ne p our main result is that if x has semistable reduction this specialization map induces an isomorphism on the slope 0 1 part

the weil etale topology for number rings there should be a grothendieck topology for an arithmetic scheme x such that the euler characteristic of the cohomology groups of the constant sheaf z with compact support at infinity gives up to sign the leading term of the zeta function of x at s 0 we construct a topology the weil etale topology for the ring of integers in a number field whose cohomology groups hl z determine such an euler characterstic if we restrict to 3

a closed form solution to tensor voting theory and applications we prove a closed form solution to tensor voting cftv given a point set in any dimensions our closed form solution provides an exact continuous and efficient algorithm for computing a structure aware tensor that simultaneously achieves salient structure detection and outlier attenuation using cftv we prove the convergence of tensor voting on a markov random field mrf thus termed as mrftv where the structure aware tensor at each input site reaches a stationary state upon convergence in structure propagation we then embed structure aware tensor into expectation maximization em for optimizing a single linear structure to achieve efficient and robust parameter estimation specifically our emtv algorithm optimizes both the tensor and fitting parameters and does not require random sampling consensus typically used in existing robust statistical techniques we performed quantitative evaluation on its accuracy and robustness showing that emtv performs better than the original tv and other state of the art techniques in fundamental matrix estimation for multiview stereo matching the extensions of cftv and emtv for extracting multiple and nonlinear structures are underway

on the torsion in the cohomology of arithmetic hyperbolic 3 manifolds in this paper we consider the cohomology of a closed arithmetic hyperbolic 3 manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of sl 2 c the cohomology is defined over the integers and is a finite abelian group we show that the order of the 2nd cohomology grows exponentially as the local system grows we also consider the twisted ruelle zeta function of a closed arithmetic hyperbolic 3 manifold and we express the leading coefficient of its laurent expansion at the origin in terms of the orders of the torsion subgroups of the cohomology

a voting based computational framework for visual motion analysis and interpretation most approaches for motion analysis and interpretation rely on restrictive parametric models and involve iterative methods which depend heavily on initial conditions and are subject to instability further difficulties are encountered in image regions where motion is not smooth typically around motion boundaries this work addresses the problem of visual motion analysis and interpretation by formulating it as an inference of motion layers from a noisy and possibly sparse point set in a 4d space the core of the method is based on a layered 4d representation of data and a voting scheme for affinity propagation the inherent problem caused by the ambiguity of 2d to 3d interpretation is usually handled by adding additional constraints such as rigidity however enforcing such a global constraint has been problematic in the combined presence of noise and multiple independent motions by decoupling the processes of matching outlier rejection segmentation and interpretation we extract accurate motion layers based on the smoothness of image motion and then locally enforce rigidity for each layer in order to infer its 3d structure and motion the proposed framework is noniterative and consistently handles both smooth moving regions and motion discontinuities without using any prior knowledge of the motion model

moments and distribution of central l values of quadratic twists of elliptic curves we show that if one can compute a little more than a particular moment for some family of l functions then one has upper bounds of the conjectured order of magnitude for all smaller positive real moments and a one sided central limit theorem holds we illustrate our method for the family of quadratic twists of an elliptic curve obtaining sharp upper bounds for all moments below the first we also establish a one sided central limit theorem supporting a conjecture of keating and snaith our work leads to a conjecture on the distribution of the order of the tate shafarevich group for rank zero quadratic twists of an elliptic curve and establishes the upper bound part of this conjecture assuming the birch swinnerton dyer conjecture

the fundamental matrix theory algorithms and stability analysis in this paper we analyze in some detail the geometry of a pair of cameras i e a stereo rig contrarily to what has been done in the past and is still done currently for example in stereo or motion analysis we do not assume that the intrinsic parameters of the cameras are known coordinates of the principal points pixels aspect ratio and focal lengths this is important for two reasons first it is more realistic in applications where these parameters may vary according to the task active vision second the general case considered here captures all the relevant information that is necessary for establishing correspondences between two pairs of images this information is fundamentally projective and is hidden in a confusing manner in the commonlyused formalism of the essential matrix introduced by longuet higgins 1981 this paper clarifies the projective nature of the correspondence problem in stereo and shows that the epipolar geometry can be summarized in one 3 x 3 matrix of rank 2 which we propose to call the fundamental matrix after this theoretical analysis we embark on the task of estimating the fundamental matrix from point corre spondences a task which is of practical importance we analyze theoretically and compare experimentally using synthetic and real data several methods of estimation the problem of the stability of the estimation is studied from two complementary viewpoints first we show that there is an interesting relationship between the fundamental matrix and three dimensional planes which induce homographies between the images and create unstabilities in the estimation procedures second we point to a deep relation between the unstability of the estimation procedure and the presence in the scene of so called critical surfaces which have been studied in the context of motion analysis finally we conclude by stressing the fact that we believe that the fundamental matrix will play a crucial role in future applications of three dimensional computer vision by greatly increasing its versatility robustness and hence applicability to real difficult problems

in defense of the eight point algorithm the fundamental matrix is a basic tool in the analysis of scenes taken with two uncalibrated cameras and the eight point algorithm is a frequently cited method for computing the fundamental matrix from a set of eight or more point matches it has the advantage of simplicity of implementation the prevailing view is however that it is extremely susceptible to noise and hence virtually useless for most purposes this paper challenges that view by showing that by preceding the algorithm with a very simple normalization translation and scaling of the coordinates of the matched points results are obtained comparable with the best iterative algorithms this improved performance is justified by theory and verified by extensive experiments on real images

simultaneous two view epipolar geometry estimation and motion segmentation by 4d tensor voting we address the problem of simultaneous two view epipolar geometry estimation and motion segmentation from nonstatic scenes given a set of noisy image pairs containing matches of n objects we propose an unconventional efficient and robust method 4d tensor voting for estimating the unknown n epipolar geometries and segmenting the static and motion matching pairs into n independent motions by considering the 4d isotropic and orthogonal joint image space only two tensor voting passes are needed and a very high noise to signal ratio up to five can be tolerated epipolar geometries corresponding to multiple rigid motions are extracted in succession only two uncalibrated frames are needed and no simplifying assumption such as affine camera model or homographic model between images other than the pin hole camera model is made our novel approach consists of propagating a local geometric smoothness constraint in the 4d joint image space followed by global consistency enforcement for extracting the fundamental matrices corresponding to independent motions we have performed extensive experiments to compare our method with some representative algorithms to show that better performance on nonstatic scenes are achieved results on challenging data sets are presented

the development and comparison of robust methodsfor estimating the fundamental matrix this paper has two goals the first is to develop a variety of robust methods for the computation of the fundamental matrix the calibration free representation of camera motion the methods are drawn from the principal categories of robust estimators viz case deletion diagnostics m estimators and random sampling and the paper develops the theory required to apply them to non linear orthogonal regression problems although a considerable amount of interest has focussed on the application of robust estimation in computer vision the relative merits of the many individual methods are unknown leaving the potential practitioner to guess at their value the second goal is therefore to compare and judge the methods comparative tests are carried out using correspondences generated both synthetically in a statistically controlled fashion and from feature matching in real imagery in contrast with previously reported methods the goodness of fit to the synthetic observations is judged not in terms of the fit to the observations per se but in terms of fit to the ground truth a variety of error measures are examined the experiments allow a statistically satisfying and quasi optimal method to be synthesized which is shown to be stable with up to 50 percent outlier contamination and may still be used if there are more than 50 percent outliers performance bounds are established for the method and a variety of robust methods to estimate the standard deviation of the error and covariance matrix of the parameters are examined the results of the comparison have broad applicability to vision algorithms where the input data are corrupted not only by noise but also by gross outliers

the eighth moment of dirichlet l functions abstract we prove an asymptotic for the eighth moment of dirichlet l functions averaged over primitive characters modulo q over all moduli q q and with a short average on the critical line conditionally on grh we derive the analogous result for the fourth moment of dirichlet twists of gl 2 l functions our results match the moment conjectures in the literature in particular the constant 24 024 appears as a factor in the leading order term of the eighth moment

ror rejection of outliers by rotations we address the problem of rejecting false matches of points between two perspective views the two views are taken from two arbitrary unknown positions and orientations we present an algorithm for identification of the false matches between the views the algorithm exploits the possibility of rotating one of the images to achieve some common behavior of the correct matches those matches that deviate from this common behavior turn out to be false matches our algorithm does not in any way use the image characteristics of the matched features in particular it avoids problems that cause the false matches in the first place the algorithm works even in cases where the percentage of false matches is as high as 85 percent the algorithm may be run as a post processing step on output from any point matching algorithm use of the algorithm may significantly improve the ratio of correct matches to incorrect matches we present the algorithm identify the conditions under which it works and present results of the test

moments of the riemann zeta function assuming the riemann hypothesis we obtain an upper bound for the moments of the riemann zeta function on the critical line our bound is nearly as sharp as the conjectured asymptotic formulae for these moments the method extends to moments in other families of l functions

determining the epipolar geometry and its uncertainty a review two images of a single scene object are related by the epipolar geometry which can be described by a 3 3 singular matrix called the essential matrix if images internal parameters are known or the fundamental matrix otherwise it captures all geometric information contained in two images and its determination is very important in many applications such as scene modeling and vehicle navigation this paper gives an introduction to the epipolar geometry and provides a complete review of the current techniques for estimating the fundamental matrix and its uncertainty a well founded measure is proposed to compare these techniques projective reconstruction is also reviewed the software which we have developed for this review is available on the internet

shape from focus by tensor voting in this correspondence we address the task of recovering shape from focus sff as a perceptual organization problem in 3 d using tensor voting depth hypotheses from different focus operators are validated based on their likelihood to be part of a coherent 3 d surface thereby exploiting scene geometry and focus information to generate reliable depth estimates the proposed method is fast and yields significantly better results compared with existing sff methods

two view multibody structure and motion with outliers through model selection multibody structure and motion msam is the problem in establishing the multiple view geometry of several views of a 3d scene taken at different times where the scene consists of multiple rigid objects moving relative to each other we examine the case of two views the setting is the following given are a set of corresponding image points in two images which originate from an unknown number of moving scene objects each giving rise to a motion model furthermore the measurement noise is unknown and there are a number of gross errors which are outliers to all models the task is to find an optimal set of motion models for the measurements it is solved through monte carlo sampling careful statistical analysis of the sampled set of motion models and simultaneous selection of multiple motion models to best explain the measurements the framework is not restricted to any particular model selection mechanism because it is developed from a bayesian viewpoint different model selection criteria are seen as different priors for the set of moving objects which allow one to bias the selection procedure for different purposes

breaking the 12 barrier for the twisted second moment of dirichlet l functions abstract we study the second moment of dirichlet l functions to a large prime modulus q twisted by the square of an arbitrary dirichlet polynomial we break the 1 2 barrier in this problem and obtain an asymptotic formula provided that the length of the dirichlet polynomial is less than q 51 101 q 1 2 1 202 as an application we obtain an upper bound of the correct order of magnitude for the third moment of dirichlet l functions we give further results when the coefficients of the dirichlet polynomial are more specialized

extreme values of zeta and l functions we introduce a resonance method to produce large values of the riemann zeta function on the critical line and large and small central values of l functions

lower bounds for moments of l functions the moments of central values of families of l functions have recently attracted much attention and with the work of keating and snaith 2000 commun math phys 214 57 89 and 91 110 there are now precise conjectures for their limiting values we develop a simple method to establish lower bounds of the conjectured order of magnitude for several such families of l functions as an example we work out the case of the family of all dirichlet l functions to a prime modulus

random matrix theory and ζ 1 2 it we study the characteristic polynomials z u of matrices u in the circular unitary ensemble cue of random matrix theory exact expressions for any matrix size n are derived for the moments of z and z z and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log z as n in the limit we show that these two distributions are independent and gaussian costin and lebowitz 15 previously found the gaussian limit distribution for im log z using a different approach and our result for the cumulants proves a conjecture made by them in this case we also calculate the leading order n asymptotics of the moments of z and z z these cue results are then compared with what is known about the riemann zeta function s on its critical line re s 1 2 assuming the riemann hypothesis equating the mean density of the non trivial zeros of the zeta function at a height t up the critical line with the mean density of the matrix eigenvalues gives a connection between n and t invoking this connection our cue results coincide with a theorem of selberg for the value distribution of log 1 2 it in the limit t they are also in close agreement with numerical data computed by odlyzko 29 for large but finite t this leads us to a conjecture for the moments of 1 2 it finally we generalize our random matrix results to the circular orthogonal coe and circular symplectic cse ensembles