on centers of blocks with one simple module abstract let g be a finite group and let b be a non nilpotent block of g with respect to an algebraically closed field of characteristic 2 suppose that b has an elementary abelian defect group of order 16 and only one simple module the main result of this paper describes the algebra structure of the center of b this is motivated by a similar analysis of a certain 3 block of defect 2 in kessar 2012 15

degeneration of surfaces with trivial canonical bundle the object of this note is to prove the theorem stated below the main idea of the proof and much of the detail are already to be found in v i kulikov s important paper k our contribution to the proof and the differences with kulikov s paper are discussed later in this introduction theorem let wl x d be a semistable degeneration of surfaces such that

solving xxtr a over the integers abstract the paper describes an algorithm to find all k n and all x z n k with xxtr a for given symmetric positive definite a z n n

0 1 completely positive matrices abstract let s be a subset of the set of real numbers r a is called s factorizable if it can be factorized as a bbt with bij s the smallest possible number of columns of b in such factorization is called the s rank of a and is denoted by ranksa if s is a set of nonnegative numbers then a is called s cp the aim of this work is to study 0 1 cp matrices we characterize 0 1 cp matrices of order less than 4 and give a necessary and sufficient condition for a matrix of order 4 with some zero entries to be 0 1 cp we show that a nonnegative integral jacobi matrix is 0 1 cp if and only if it is diagonally dominant and obtain a necessary condition for a 2 banded symmetric nonnegative integral matrix to be 0 1 cp we give formulae for the exact value of the 0 1 rank of integral symmetric nonnegative diagonally dominant matrices and some other 0 1 cp matrices

the noether lefschetz conjecture and generalizations we prove the noether lefschetz conjecture on the moduli space of quasi polarized k3 surfaces this is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal type are dual to the classes of special cycles i e sub arithmetic manifolds of the same type for compact manifolds this was proved in 3 here we extend the results of 3 to non compact manifolds this allows us to apply our results to the moduli spaces of quasi polarized k3 surfaces

on the shafarevich and tate conjectures for hyperkahler varieties 1 problems and results 2 polarized hyperk ihler varieties and cubic fourfoids 3 the period mapping 4 the kuga satake construction 5 the kuga satake construction in the relative setting 6 proof of theorem 1 5 1 7 proof of theorem 1 6 1 8 descent 9 proof of theorem 1 3 1 appendix 1 spinorial shimura varieties appendix 2 a short proof of the hodge conjecture for cubic fourfolds 205 210 212 216 220 224 229 233 240 244 245

optical flow using spatiotemporal filters a model is presented consonant with current views regarding the neurophysiology and psychophysics of motion perception that combines the outputs of a set of spatiotemporal motion energy filters to estimate image velocity a parallel implementation computes a distributed representation of image velocity a measure of image flow uncertainty is formulated preliminary results indicate that this uncertainty measure may be used to recognize ambiguity due to the aperture problem the model appears to deal with the aperture problem as well as the human visual system since it extracts the correct velocity for some patterns that have large differences in contrast at different spatial orientations

low dimensional lattices v integral coordinates for integral lattices we say that an n dimensional classically integral lattice lambda is s integrable for an integer s if it can be described by vectors s frac 1 2 x 1 x k with all x i in mathbb z in a euclidean space of dimension k geqslant n equivalently lambda is s integrable if and only if any quadratic form f x corresponding to lambda can be written as s 1 times a sum of k squares of linear forms with integral coefficients or again if and only if the dual lattice lambda contains a eutactic star of scale s this paper gives many techniques for s integrating low dimensional lattices such as e 8 and the leech lattice a particular result is that any one dimensional lattice can be 1 integrated with k 4 this is lagrange s four squares theorem let phi s be the smallest dimension n in which there is an integral lattice that is not s integrable in 1937 ko and mordell showed that phi 1 6 we prove that phi 2 12 phi 3 14 21 leqslant phi 4 leqslant 25 16 leqslant phi 5 leqslant 22 phi s 4 leqslant 4s 2 s odd phi s leqslant 2 pi es 1 o 1 s even and phi s geqslant 2 ln ln s ln ln ln s 1 o 1

the tate conjecture for k3 surfaces in odd characteristic we show that the classical kuga satake construction gives rise away from characteristic 2 to an open immersion from the moduli of primitively polarized k3 surfaces of any fixed degree to a certain regular integral model for a shimura variety of orthogonal type this allows us to attach to every polarized k3 surface in odd characteristic an abelian variety such that divisors on the surface can be identified with certain endomorphisms of the attached abelian variety in turn this reduces the tate conjecture for k3 surfaces over finitely generated fields of odd characteristic to a version of the tate conjecture for certain endomorphisms on the attached kuga satake abelian variety which we prove as a by product of our methods we also show that the moduli stack of primitively polarized k3 surfaces of degree 2d is quasi projective and when d is not divisible by p 2 is geometrically irreducible in characteristic p we indicate how the same method applies to prove the tate conjecture for co dimension 2 cycles on cubic fourfolds

good reduction criterion for k3 surfaces we prove a neron ogg shafarevich type criterion for good reduction of k3 surfaces which states that a k3 surface over a complete discrete valuation field has potential good reduction if its l adic cohomology group is unramified we also prove a p adic version of the criterion these are analogues of the criteria for good reduction of abelian varieties the model of the surface will be in general not a scheme but an algebraic space as a corollary of the criterion we obtain the surjectivity of the period map of k3 surfaces in positive characteristic

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

the tate conjecture for k3 surfaces over finite fields artin s conjecture states that supersingular k3 surfaces over finite fields have picard number 22 in this paper we prove artin s conjecture over fields of characteristic p 5 this implies tate s conjecture for k3 surfaces over finite fields of characteristic p 5 our results also yield the tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields with some restrictions on the characteristic as a consequence we prove the tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p 5

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

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