ensemble kalman filter with the unscented transform abstract a modification scheme to the ensemble kalman filter enkf is introduced based on the concept of the unscented transform s julier j uhlmann h durrant whyte a new method for the nonlinear transformation of means and covariances in filters and estimators ieee trans automat control 45 2000 477 482 s j julier j k uhlmann unscented filtering and nonlinear estimation proc ieee 92 2004 401 422 which therefore will be called the ensemble unscented kalman filter enukf in this work when the error distribution of the analysis is symmetric not necessarily gaussian it can be shown that compared with the ordinary enkf the enukf has more accurate estimations of the ensemble mean and covariance of the background by examining the multidimensional taylor series expansion term by term this implies that the enukf may have better performance in state estimation than the ordinary enkf in the sense that the deviations from the true states are smaller for verification some numerical experiments are conducted on a 40 dimensional system due to lorenz and emanuel e n lorenz k a emanuel optimal sites for supplementary weather observations simulation with a small model j atmos sci 55 1998 399 414 simulation results support our argument

a central limit theorem for critical first passage percolation in two dimensions consider independent first passage percolation on the edges of 2 denote the passage time of the edge e in 2 by t e and assume that p t e 0 1 2 p 0 t e c 0 0 for some constant c 0 0 and that e t e 4 denote by b 0 n the passage time from 0 to the halfplane x y x n and by t 0 nu the passage time from 0 to the nearest lattice point to nu for u a unit vector we prove that there exist constants 0 c 1 c 2 and n such that c 1 log n 1 2 n c 2 log n 1 2 and such that n 1 b 0 n eb 0 n and 2 n 1 t 0 nu et 0 nu converge in distribution to a standard normal variable as n u fixed a similar result holds for the site version of first passage percolation on 2 when the common distribution of the passage times t v of the vertices satisfies p t v 0 1 p t v c 0 p c 2 site critical probability of site percolation on 2 and e t u 4

a nonlinear conjugate gradient method with a strong global convergence property conjugate gradient methods are widely used for unconstrained optimization especially large scale problems the strong wolfe conditions are usually used in the analyses and implementations of conjugate gradient methods this paper presents a new version of the conjugate gradient method which converges globally provided the line search satisfies the standard wolfe conditions the conditions on the objective function are also weak being similar to those required by the zoutendijk condition

a decomposition theorem for singular spaces with trivial canonical class of dimension at most five in this paper we partly extend the beauville bogomolov decomposition theorem to the singular setting we show that any complex projective variety of dimension at most five with canonical singularities and numerically trivial canonical class admits a finite cover etale in codimension one that decomposes as a product of an abelian variety and singular analogues of irreducible calabi yau and irreducible holomorphic symplectic varieties

asymptotics for 2d critical and near critical first passage percolation we study bernoulli first passage percolation fpp on the triangular lattice in which sites have 0 and 1 passage times with probability p and 1 p respectively denote by mathcal c infty the infinite cluster with 0 time sites when p p c where p c 1 2 is the critical probability denote by t 0 mathcal c infty the passage time from the origin 0 to mathcal c infty first we obtain explicit limit theorem for t 0 mathcal c infty as p searrow p c the proof relies on the limit theorem in the critical case the critical exponent for correlation length and kesten s scaling relations next for the usual point to point passage time a 0 n in the critical case we construct subsequences of sites with different growth rate along the axis the main tool involves the large deviation estimates on the nesting of cle 6 loops derived by miller et al ann probab 44 1013 1052 2016 finally we apply the limit theorem for critical bernoulli fpp to a random graph called cluster graph obtaining explicit strong law of large numbers for graph distance

semistable sheaves on projective varieties and their restriction to curves let x be a nonsingular projective variety of dimension n over an algebraically closed field k let h be a very ample line bundle on x if v is a torsion free coherent sheaf on x we define deg v to be cl v c h and v deg v rk v we call v sernistable resp stable if for all proper subsheaves w of v we have w v resp w v cf 7 14 in this paper we prove that if v is semistable on x then its restriction to a general complete intersection curve of sufficiently high degree is semistable theorem 6 1 to give an idea of the proof assume x is a surface and v a vector bundle of rank 2 the restriction of v to a general curve c of degree m is not semistable if and only if it is not semistable on the generic curve ym defined over the function field of iph h let s be the line bundle on y contradicting the semistability of vi y cf sects 4 1 and 4 2 first we show that l extends uniquely to a line bundle l m on x proposition 2 1 if we can get l as a subbundle of v we are through for then l would contradict the semistability of v so we would like the restriction map h hom lm v h hom l m v to be surjective now for fixed l it follows from the lemma of enriques severi proposition 3 2 6 corollary 7 8 that h horn l v h horn l v is surjective for large m therefore it is enough if the l remain the same line bundle l for infinitely many m to prove that l l we construct a degenerating family of curves d f s x x s 3 d p x such that the generic fibre is a curve ct 1 of degree 2 1 and the special fibre is a reduced curve with two nonsingular components ci of degree 2 cf sect 5 let m denote 2 extending the subbundle lt ct to a subsheaf of p v on d and restricting the extension to ci gives a lower bound for the maximal degree of a line subbundle of v cim in terms of that for v ctm i proposition 4 3 this implies that degl is bounded lemma 6 5 1 so that for an infinite subsequence of m degl is constant if degltm r deglt by refining the above argument with the degenerating family one can prove that lt ci

singular foliations with trivial canonical class this paper describes the structure of singular codimension one foliations with numerically trivial canonical bundle on complex projective manifolds

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

infinite ab percolation clusters exist on the triangular lattice infinite ab percolation clusters are shown to exist almost surely on the triangular lattice for an interval of parameter values it was not previously known that infinite ab percolation occurred with positive probability on any planar lattice

semistable sheaves in positive characteristic we prove maruyama s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field this result implies that in mixed characteristic the moduli spaces of gieseker semistable sheaves are projective schemes of finite type the proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant this inequality also leads to effective restriction theorems in all characteristics improving earlier results in characteristic zero

monge ampere equations in big cohomology classes we define non pluripolar products of an arbitrary number of closed positive 1 1 currents on a compact kahler manifold x given a big 1 1 cohomology class on x i e a class that can be represented by a strictly positive current and a positive measure on x of total mass equal to the volume of and putting no mass on pluripolar sets we show that can be written in a unique way as the top degree self intersection in the non pluripolar sense of a closed positive current in we then extend kolodziedj s approach to sup norm estimates to show that the solution has minimal singularities in the sense of demailly if has l 1 e density with respect to lebesgue measure if is smooth and positive everywhere we prove that t is smooth on the ample locus of provided is nef using a fixed point theorem we finally explain how to construct singular kahler einstein volume forms with minimal singularities on varieties of general type

equidistribution and convergence speed for zeros of holomorphic sections of singular hermitian line bundles abstract we establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular hermitian line bundle with an estimate of the convergence speed

a numerical criterion for very ample line bundles let x be a projective algebraic manifold of dimension n and let l be an ample line bundle over x we give a numerical criterion ensuring that the adjoint bundle kx l is very ample the sufficient conditions are expressed in terms of lower bounds for the intersection numbers l y over subvarieties y of x in the case of surfaces our criterion gives universal bounds and is only slightly weaker than i reider s criterion when dimx 3 and codimy 2 the lower bounds for l y involve a numerical constant which depends on the geometry of x by means of an iteration process it is finally shown that 2kx ml is very ample form 12n our approach is mostly analytic and based on a combination of hormander s l estimates for the operator lelong number theory and the aubin calabi yau theorem

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

algebraic integrability of foliations with numerically trivial canonical bundle given a reflexive sheaf on a mildly singular projective variety we prove a flatness criterion under certain stability conditions this implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second chern class does not vanish combined with the recent works of druel and greb guenancia kebekus this establishes the beauville bogomolov decomposition for minimal models with trivial canonical class

unscented ensemble transform based variational filter abstract to deal with high dimensional nonlinear filtering problems a hybrid scheme called unscented ensemble transform variational filter uevf is introduced this is the combination of an unscented transform ut an ensemble transform et and a rank reduction method to compute the background covariance error matrices as well as a variational minimization to conduct the mean correction the proposed uevf is more efficient than the unscented kalman filter ukf to estimate the ensemble mean and covariance by the blending of a variational optimization instead of a kalman linear correction as well as the et like covariance estimation into the update step moreover in order to tackle the high dimension dynamics truncated singular value decomposition is applied to provide a size reduction of a sigma points set with an adaptive fashion for performance verifications we present two numerical experiments with different dynamics the first system is the chaotic and high dimensional lorenz 95 model we show the performance of different filters including the uevf as the increasing of dimensionality or noise level the second simulation is a model based on the 2d shallow water equation the same tests are provided on this hydrodynamical system all the numerical experiments confirm that the uevf outperforms the widely applied kalman like filters explicitly

unbiased ensemble square root filters there is an established framework that describes a class of ensemble kalman filter algorithms as square root filters srfs these algorithms produce analyses by updating a state estimate and a square root of the ensemble covariance matrix the matrix square root of the forecast covariance is post multiplied by another matrix to give a matrix square root of the analysis covariance the choice of post multiplier is not unique and can be multiplied by any orthogonal matrix to give another scheme that is also a srf not all filters of this type bear the desired relationship to the forecast ensemble the analysis ensemble mean may not be equal to the analysis state estimate and consequently there may be an accompanying shortfall in the spread of the analysis ensemble as expressed by the ensemble covariance matrix this points to the need for a restricted version of the notion of an ensemble srf which we call an unbiased ensemble srf this paper provides a generic set of necessary and sufficient conditions for the scheme to be centred on the analysis state estimate unbiased a few of these results have already been published elsewhere in the literature this paper brings these together with new results and provides simple proofs and examples as well as a mathematical description of the set of unbiased ensemble srfs many but not all published ensemble srf algorithms satisfy the criteria that we establish while these conditions are not a cure all and cannot deal with independent sources of bias such as model and observation errors they should be useful to designers of ensemble srfs in the future

on monochromatic arm exponents for 2d critical percolation we investigate the so called monochromatic arm exponents for critical percolation in two dimensions these exponents describing the probability of observing j disjoint macroscopic paths are shown to exist and to form a different family from the now well understood polychromatic exponents more specifically our main result is that the monochromatic j arm exponent is strictly between the polychromatic j arm and j 1 arm exponents

global convergence properties of conjugate gradient methods for optimization this paper explores the convergence of nonlinear conjugate gradient methods without restarts and with practical line searches the analysis covers two classes of methods that are globally convergent on smooth nonconvex functions some properties of the fletcher reeves method play an important role in the first family whereas the second family shares an important property with the polak ribiere method numerical experiments are presented

subvarieties with partially ample normal bundle we show that local complete intersection subvarieties of smooth projective varieties which have partially ample normal bundle possess the g2 property this generalizes results of hartshorne and b descu schneider

relative critical exponents non vanishing and metrics with minimal singularities in this article we establish a very general non vanishing statement as well as several properties of metrics with minimal singularities of adjoint bundles our arguments combine the proof of the classical non vanishing theorem due to v shokurov as well as many ideas from y t siu s analytic proof of the finite generation of the canonical ring an important tool is the notion of relative critical exponent of two closed positive currents with respect to a measure

the complex monge ampere type equation on compact hermitian manifolds and applications abstract we prove the existence and uniqueness of continuous solutions to the complex monge ampere type equation with the right hand side in l p p 1 on compact hermitian manifolds next we generalise results of eyssidieux guedj and zeriahi 17 18 to compact hermitian manifolds which a priori are not in the fujiki class these generalisations lead to a number of applications we obtain partial results on a conjecture of tosatti and weinkove 40 and on a weak form of a conjecture of demailly and paun 11

a new method for the nonlinear transformation of means and covariances in filters and estimators this paper describes a new approach for generalizing the kalman filter to nonlinear systems a set of samples are used to parametrize the mean and covariance of a not necessarily gaussian probability distribution the method yields a filter that is more accurate than an extended kalman filter ekf and easier to implement than an ekf or a gauss second order filter its effectiveness is demonstrated using an example

asymptotics for 2d critical first passage percolation we consider first passage percolation on z2z2 with i i d weights whose distribution function satisfies f 0 pc 1 2f 0 pc 1 2 this is sometimes known as the critical case because large clusters of zero weight edges force passage times to grow at most logarithmically giving zero time constant denote t 0 b n t 0 b n as the passage time from the origin to the boundary of the box n n n n n n n n we characterize the limit behavior of t 0 b n t 0 b n by conditions on the distribution function ff we also give exact conditions under which t 0 b n t 0 b n will have uniformly bounded mean or variance these results answer several questions of kesten and zhang from the 1990s and in particular disprove a conjecture of zhang from 1999 in the case when both the mean and the variance go to infinity as n n we prove a clt under a minimal moment assumption the main tool involves a new relation between first passage percolation and invasion percolation up to a constant factor the passage time in critical first passage percolation has the same first order behavior as the passage time of an optimal path constrained to lie in an embedded invasion cluster

comment on ensemble kalman filter with the unscented transform the results of numerical experiments with the ensemble unscented kalman filter and 40 dimensional model of lorentz and emanuel in luo and moroz 2009 2 are inconclusive

etale fundamental groups of kawamata log terminal spaces flat sheaves and quotients of abelian varieties given a quasiprojective variety x with only kawamata log terminal singularities we study the obstructions to extending finite etale covers from the smooth locus xreg of x to x itself a simplified version of our main results states that there exists a galois cover y x ramified only over the singularities of x such that the etale fundamental groups of y and of yreg agree in particular every etale cover of yreg extends to an etale cover of y as a first major application we show that every flat holomorphic bundle defined on yreg extends to a flat bundle that is defined on all of y as a consequence we generalize a classical result of yau to the singular case every variety with at worst terminal singularities and with vanishing first and second chern class is a finite quotient of an abelian variety as a further application we verify a conjecture of nakayama and zhang describing the structure of varieties that admit polarized endomorphisms

multiplier ideal sheaves and kahler einstein metrics of positive scalar curvature we present a method for proving the existence of kiihler einstein metrics of positive scalar curvature on certain compact complex manifolds and use the method to produce a large class of examples of compact kiihler einstein manifolds of positive scalar curvature suppose that m is a compact complex manifold of positive first chern class as is well known the existence of a kiihler einstein metric on m is equivalent to the existence of a solution to a certain complex monge ampere equation on m to solve this complex mongeampere equation by the method of continuity one needs only to establish the appropriate zeroth order a priori estimate suppose now that m does not admit a kiihler einstein metric so that the zeroth order a priori estimate fails to hold from this lack of an estimate we extract various global algebro geometric properties of m by introducing a coherent sheaf of ideals j on m called the multiplier ideal sheaf which carefully measures the extent to which the estimate fails the sheaf y is analogous to the subelliptic multiplier ideal sheaf that j j kohn introduced over a decade ago to obtain sufficient conditions for subellipticity of the d neumann problem now j is a global algebro geometric object on m and it so happens that j satisfies a number of highly nontrivial global algebro geometric conditions including a cohomology vanishing theorem in particular the complex analytic subspace v c m cut out by j is nonempty connected and has arithmetic genus zero if v is zero dimensional then it is a single reduced point while if v is one dimensional then its support is a tree of smooth rational curves the logarithmic geometric genus of m v always vanishes these considerations place nontrivial global algebro geometric restric

extreme nesting in the conformal loop ensemble the conformal loop ensemble cle with parameter 8 3 8 is the canonical conformally invariant measure on countably infinite collections of noncrossing loops in a simply connected domain given and we compute the almost sure hausdorff dimension of the set of points z for which the number of cle loops surrounding the disk of radius e centered at z has asymptotic growth log 1 e as e 0 by extending these results to a setting in which the loops are given i i d weights we give a cle based treatment of the extremes of the gaussian free field

injectivity radius and gonality of a compact riemann surface we obtain a sharp lower bound for the volumes of purely 1 dimensional complex analytic subvarieties in a geodesic tubular neighborhood of the diagonal of the cartesian product of a compact riemann surface with itself this leads to a lower bound of the seshadri number of the canonical line bundle of the cartesian product with respect to the diagonal as a consequence we obtain an upper bound for the hyperbolic injectivity radii of compact riemann surfaces of a fixed gonality in particular we obtain the limiting behavior of the gonalities of a tower of compact riemann surfaces we also give an application of our results to an invariant related to the ample cone of the symmetric product of a riemann surface

rational curves on fibered varieties let x be a complex projective variety with log terminal singularities and vanishing augmented irregularity in this paper we prove that if x admits a relatively minimal genus one fibration then it contains a subvariety of codimension one covered by rational curves contracted by the fibration we then focus on the case of varieties with numerically trivial canonical bundle and we discuss several consequences of this result