correction to new obstructions to symplectic embeddings we correct here an error in the proof of theorem 1 1 of our paper new obstructions to symplectic embeddings henceforth referred to as 6

new obstructions to symplectic embeddings in this paper we establish new restrictions on the symplectic embeddings of basic shapes in symplectic vector spaces by refining an embedding technique due to guth we also show that they are sharp

a gradient algorithm locally equivalent to the em algorithm in many problems of maximum likelihood estimation it is impossible to carry out either the e step or the m step of the em algorithm the present paper introduces a gradient algorithm that is closely related to the em algorithm this em gradient algorithm approximately solves the m step of the em algorithm by one iteration of newton s method since newton s method converges quickly the local properties of the em gradient algorithm are almost identical with those of the em algorithm any strict local maximum point of the observed likelihood locally attracts the em and em gradient algorithm at the same rate of convergence and near the maximum point the em gradient algorithm always produces an increase in the likelihood with proper modification the em gradient algorithm also exhibits global convergence properties that are similar to those of the em algorithm our proof of global convergence applies and improves existing theory for the em algorithm these theoretical points are reinforced by a discussion of three realistic examples illustrating how the em gradient algorithm can succeed where the em algorithm is intractable

the embedding capacity of 4 dimensional symplectic ellipsoids this paper calculates the function c a whose value at a is the inmum of the size of a ball that contains a symplectic image of the ellipsoide 1 a here a 1 is the ratio of the area of the large axis to that of the smaller axis the structure of the graph of c a is surprisingly rich the volume constraint implies that c a is always greater than or equal to the square root of a and it is not hard to see that this is equality for large a however for a less than the fourth power 4 of the golden ratio c a is piecewise linear with graph that alternately lies on a line through the origin and is horizontal we prove this by showing that there are exceptional curves in blow ups of the complex projective plane whose homology classes are given by the continued fraction expansions of ratios of fibonacci numbers on the interval 4 7 we nd c a a 1 3 for a 7 the function c a coincides with the square root except on a nite number of intervals where it is again piecewise linear the embedding constraints coming from embedded contact homology give rise to another capacity function cech which may be computed by counting lattice points in appropriate right angled triangles according to hutchings and taubes the functorial properties of embedded contact homology imply that cech a c a for all a we show here that cech a c a for all a

recent progress on symplectic embedding problems in four dimensions we survey some recent progress on understanding when one four dimensional symplectic manifold can be symplectically embedded into another in 2010 mcduff established a number theoretic criterion for the existence of a symplectic embedding of one four dimensional ellipsoid into another this result is related to previously known criteria for when a disjoint union of balls can be symplectically embedded into a ball numerical invariants defined using embedded contact homology give general obstructions to symplectic embeddings in four dimensions which turn out to be sharp in the above cases

conjugate gradient acceleration of the em algorithm the em algorithm is a very popular and widely applicable algorithm for the computation of maximum likelihood estimates although its implementation is generally simple the em algorithm often exhibits slow convergence and is costly in some areas of application past attempts to accelerate the em algorithm have most commonly been based on some form of aitken acceleration here we propose an alternative method based on conjugate gradients the key as we show is that the em step can be viewed approximately at least as a generalized gradient making it natural to apply generalized conjugate gradient methods in an attempt to accelerate the em algorithm the proposed method is relatively simple to implement and can handle problems with a large number of parameters an important feature of most em algorithms to demonstrate the effectiveness of the proposed acceleration method we consider its application to several problems in each of the following areas estimation of a covariance matrix from incomplete mu

a fast improvement to the em algorithm on its own terms the em algorithm is a numerical technique for the evaluation of maximum likelihood estimates for parameters describing incomplete data models it is easy to apply in many problems and is stable but slow the algorithm fails to provide a consistent estimator of the standard errors of the maximum likelihood estimates unless the additional analysis required by the louis method is performed newton type or other gradient methods are faster and provide error estimates but tend to be unstable and require the analytical evaluation of likelihoods to derive expressions for the score function and at least approximations to the fisher information matrix the purpose of this paper is to expand on a result by fisher that permits a unification of em methodology and newton methods the evaluation of the individual observation by observation score functions of the incomplete data is a byproduct of the application of the e step of the em algorithm once these become available the fisher information matrix may be consistently estimated and the m step may be replaced by a fast newton type step

a reciprocity theorem in linear gradient elasticity and the corresponding saint venant principle in many practical applications of nanotechnology and in microelectromechanical devices typical structural components are in the form of beams plates shells and membranes when the scale of such components is very small the material microstructural lengths become important and strain gradient elasticity can provide useful material modelling in addition small scale beams and bars can be used as test specimens for measuring the lengths that enter the constitutive equations of gradient elasticity it is then useful to be able to apply approximate solutions for the extension shear and flexure of slender bodies such approach requires the existence of some form of the saint venant principle the present work presents a statement of the saint venant principle in the context of linear strain gradient elasticity a reciprocity theorem analogous to betti s theorem in classic elasticity is provided first together with necessary restrictions on the constitutive equations and the body forces it is shown that the order of magnitude of displacements are in accord with the sternberg s statement of the saint venant principle the cases of stretching shearing and bending of a beam were examined in detail using two dimensional finite elements the numerical examples confirmed the theoretical results 2005 elsevier ltd all rights reserved

hofer s question on intermediate symplectic capacities roughly twenty five years ago hofer asked can the cylinder b 2 1 times mathbb r 2 n 1 be symplectically embedded into b 2 n 1 r times mathbb r 2 for some r 0 we show that this is the case if r geq sqrt 2 n 1 2 n 2 2 we deduce that there are no intermediate capacities between 1 capacities first constructed by gromov in 1985 and n capacities answering another question of hofer in 2008 guth reached the same conclusion under the additional hypothesis that the intermediate capacities should satisfy the exhaustion property

symplectic topology and hamiltonian dynamics on etudie des applications symplectiques non lineaires capacites symplectiques construction d une capacite symplectique problemes de plongement problemes de rigidite

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

acceleration of the em algorithm by using quasi newton methods the em algorithm is a popular method for maximum likelihood estimation its simplicity in many applications and desirable convergence properties make it very attractive its sometimes slow convergence however has prompted researchers to propose methods to accelerate it we review these methods classifying them into three groups pure hybrid and em type accelerators we propose a new pure and a new hybrid accelerator both based on quasi newton methods and numerically compare these and two other quasi newton accelerators for this we use examples in each of three areas poisson mixtures the estimation of covariance from incomplete data and multivariate normal mixtures in these comparisons the new hybrid accelerator was fastest on most of the examples and often dramatically so in some cases it accelerated the em algorithm by factors of over 100 the new pure accelerator is very simple to implement and competed well with the other accelerators it accelerated the em algorithm in some cases by factors of over 50 to obtain standard errors we propose to approximate the inverse of the observed information matrix by using auxiliary output from the new hybrid accelerator a numerical evaluation of these approximations indicates that they may be useful at least for exploratory purposes

using em to obtain asymptotic variance covariance matrices the sem algorithm abstract the expectation maximization em algorithm is a popular and often remarkably simple method for maximum likelihood estimation in incomplete data problems one criticism of em in practice is that asymptotic variance covariance matrices for parameters e g standard errors are not automatic byproducts as they are when using some other methods such as newton raphson in this article we define and illustrate a procedure that obtains numerically stable asymptotic variance covariance matrices using only the code for computing the complete data variance covariance matrix the code for em itself and code for standard matrix operations the basic idea is to use the fact that the rate of convergence of em is governed by the fractions of missing information to find the increased variability due to missing information to add to the complete data variance covariance matrix we call this supplemented em algorithm the sem algorithm theory and particular examples reinforce the conclusion that the sem alg

irreversibility from staircases in symplectic embeddings abstract we present an argument whose goal is to trace the origin of the macroscopically irreversible behavior of hamiltonian systems of many degrees of freedom we use recent flexibility and rigidity results of symplectic embeddings quantified via the stabilized fibonacci and pell staircases to encode the underlying breadth of the possible initial conditions which alongside the multitude of degrees of freedom of the underlying system give rise to time irreversibility

the lens space realization problem we determine the lens spaces that arise by integer dehn surgery along a knot in the three sphere specically if surgery along a knot produces a lens space then there exists an equivalent surgery along a berge knot with the same knot floer homology groups this leads to sharp information about the genus of such a knot the arguments rely on tools from floer homology and lattice theory they are primarily combinatorial in nature

symplectic embeddings of polydisks if p and p are symplectic polydisks of radii r 1 r n and r 1 r n respectively then we prove that p symplectically embeds in p provided that c n r 1 r 1 and c n r 1 r n r 1 r n up to a constant factor these conditions are optimal