generic smoothness of the moduli spaces of rank two stable vector bundles over algebraic surfaces we denote by m k the moduli space of all isomorphic classes of h stable rank 2 vector bundles with fixed determinant bundle d and second chern class k it is well known that m k is a quasi projective variety and is not empty if k is sufficiently large see g and m the local structure of m k is described precisely by the kuranishi deformation theorem in the following way suppose that v m k and let r h 1 endo v hz endo v be the kuranishi map then the stalk of the structure sheaf gm at v is naturally isomorphic to the germs of holomorphic functions at 0 h endo v divided by the ideal generated by the components of in particular if hz endo v o then m k is smooth at v and the tangent space t m k v is identified with h 1 endo v the hirzebruch riemann roch theorem gives the dimension formula

optimization landscape of tucker decomposition tucker decomposition is a popular technique for many data analysis and machine learning applications finding a tucker decomposition is a nonconvex optimization problem as the scale of the problems increases local search algorithms such as stochastic gradient descent have become popular in practice in this paper we characterize the optimization landscape of the tucker decomposition problem in particular we show that if the tensor has an exact tucker decomposition for a standard nonconvex objective of tucker decomposition all local minima are also globally optimal we also give a local search algorithm that can find an approximate local and global optimal solution in polynomial time

moduli spaces of holomorphic triples over compact riemann surfaces a holomorphic triple over a compact riemann surface consists of two holomorphic vector bundles and a holomorphic map between them after fixing the topological types of the bundles and a real parameter there exist moduli spaces of stable holomorphic triples in this paper we study non emptiness irreducibility smoothness and birational descriptions of these moduli spaces for a certain range of the parameter our results have important applications to the study of the moduli space of representations of the fundamental group of the surface into unitary lie groups of indefinite signature 5 7 another application that we study in this paper is to the existence of stable bundles on the product of the surface by the complex projective line

handwritten digit classification using higher order singular value decomposition in this paper we present two algorithms for handwritten digit classification based on the higher order singular value decomposition hosvd the first algorithm uses hosvd for construction of the class models and achieves classification results with error rate lower than 6 the second algorithm uses the hosvd for tensor approximation simultaneously in two modes classification results for the second algorithm are almost down at 5 even though the approximation reduces the original training data with more than 98 before the construction of the class models the actual classification in the test phase for both algorithms is conducted by solving a series least squares problems considering computational amount for the test presented the second algorithm is twice as efficient as the first one

moduli of high rank vector bundles over surfaces the purpose of this work is to apply the degeneration theory developed in gl to study the moduli space of stable vector bundles of arbitrary rank on any smooth algebraic surface over c we will show that most of the recent progress in understanding moduli of rank two vector bundles can be carried over to high rank cases after introducing the notion of stable vector bundles the first author constructed the moduli schemes of vector bundles on surfaces he showed that for any smooth algebraic surface x with ample divisor h and line bundle i on x there is a coarse moduli scheme m x i h parameterizing modulo equivalence relation the set of all h semistable rank r torsion free sheaves e on x with dete i and c2 e d since then many mathematicians have studied the geometry of this moduli space especially for the rank two case to cite a few maruyama taubes and the first author showed that the moduli space m x m r d x i h is non empty when d is large moduli spaces of vector bundles of some special surfaces have been studied also the deep understanding of m x for arbitrary x and r 2 begins with donaldson s generic smoothness result roughly speaking donaldson do later generalized by friedman fr and k zhu zh showed that when d is large enough then the singular locus sing m x of m x is a proper subset of m 2 d x and its codimension in m x increases linearly in d this theorem indicates that the moduli m x behaves as expected when the second chern class d is large later using general deformation theory the second author proved that m x is normal and has local complete intersection l c i singularities at stable sheaves provided d is large l2 he also showed that when x is a surface of general type satisfying some mild technical conditions then m x is of general type for d 0 l2 in our paper gl we also proved that m x is irreducible if d is large in this and subsequent papers we shall show that the geometry of m x and the geometry of m x r 3 is rather similar the main obstacle in doing so is the lack of an analogy of the generic smoothness result in high rank case in this paper we will use the degeneration of moduli developed in gl to establish the following main technical theorem

event location for ordinary differential equations abstract an initial value problem for y f t y may have an associated event function g t y an event is said to occur at t when g t y t 0 we consider problems for which the definition of f t y changes at the time of an event a number of solvers locate events and restart the integration there so as to deal with the changes in f but there is little theoretical support for what is done here we prove that with reasonable assumptions about the problem and the solver the error of the numerical solution is qualitatively the same whether or not events occur numerical results obtained with a wide range of solvers confirm the theory developed here

strong approximations of stochastic differential equations with jumps this paper is a survey of strong discrete time approximations of jump diffusion processes described by stochastic differential equations sdes it also presents new results on strong discrete time approximations for the specific case of pure jump sdes strong approximations based on jump adapted time discretizations which produce no discretization error in the case of pure jump processes are analyzed the computational complexity of these approximations is proportional to the jump intensity by exploiting a stochastic expansion for pure jump processes higher order discrete time approximations whose computational complexity is not dependent on the jump intensity are proposed for the specific case of pure jump sdes the strong order of convergence of strong taylor schemes is established under weaker conditions than those currently known in the literature

complete dictionary recovery over the sphere i overview and the geometric picture we consider the problem of recovering a complete i e square and invertible matrix a 0 from y in mathbb r n times p with y a 0 x 0 provided x 0 is sufficiently sparse this recovery problem is central to theoretical understanding of dictionary learning which seeks a sparse representation for a collection of input signals and finds numerous applications in modern signal processing and machine learning we give the first efficient algorithm that provably recovers a 0 when x 0 has o left n right nonzeros per column under suitable probability model for x 0 in contrast prior results based on efficient algorithms either only guarantee recovery when x 0 has o sqrt n zeros per column or require multiple rounds of semidefinite programming relaxation to work when x 0 has o n nonzeros per column our algorithmic pipeline centers around solving a certain nonconvex optimization problem with a spherical constraint in this paper we provide a geometric characterization of the objective landscape in particular we show that the problem is highly structured with high probability 1 there are no spurious local minimizers and 2 around all saddle points the objective has a negative directional curvature this distinctive structure makes the problem amenable to efficient optimization algorithms in a companion paper we design a second order trust region algorithm over the sphere that provably converges to a local minimizer from arbitrary initializations despite the presence of saddle points

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

cubic regularization of newton method and its global performance in this paper we provide theoretical analysis for a cubic regularization of newton method as applied to unconstrained minimization problem for this scheme we prove general local convergence results however the main contribution of the paper is related to global worst case complexity bounds for different problem classes including some nonconvex cases it is shown that the search direction can be computed by standard linear algebra technique

pathwise error bounds in multiscale variable splitting methods for spatial stochastic kinetics stochastic computational models in the form of pure jump processes occur frequently in the description of chemical reactive processes of ion channel dynamics and of the spread of infections in po

the orientation of yang mills moduli spaces and 4 manifold topology si x est une 4 variete lisse orientee fermee dont la forme intersection q h 2 x z torsion z est definie negative alors la forme est equivalente sur les entiers a la forme standard 1 1 1 soit x une 4 variete lisse orientee fermee avec les trois proprietes suivantes 1 h 1 x z n a pas de 2 torsion 2 la forme intersection q sur h 2 x torsion a une partie positive de rang 1 ou 2 3 la forme intersection est paire alors q est equivalente sur les entiers a une des formes 01 10 01 10 01 10

almost sure convergence of numerical approximations for piecewise deterministic markov processes hybrid systems and piecewise deterministic markov processes in particular are widely used to model and numerically study systems exhibiting multiple time scales in biochemical reaction kinetics and related areas in this paper an almost sure convergence analysis for numerical simulation algorithms for piecewise deterministic markov processes is presented the discussed numerical methods arise through discretising a constructive method defining these processes the stochastic problem of simulating the random path dependent jump times of such processes is reformulated as a hitting time problem for a system of ordinary differential equations with random threshold then deterministic continuous methods methods with dense output are serially employed to solve these problems numerically we show that the almost sure convergence rate of the stochastic algorithm is identical to the order of the embedded deterministic method we illustrate our theoretical findings by numerical examples from mathematical neuroscience piecewise deterministic markov processes are used as biophysically accurate stochastic models of neuronal membranes

runge kutta methods for jump diffusion differential equations in this paper we consider runge kutta methods for jump diffusion differential equations we present a study of their mean square convergence properties for problems with multiplicative noise we are concerned with two classes of runge kutta methods first we analyse schemes where the drift is approximated by a runge kutta ansatz and the diffusion and jump part by a maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the runge kutta ansatz for the drift the second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise we present results showing when the implicit stochastic equations defining the stage values of the runge kutta methods are uniquely solvable finally simulation results illustrate the theoretical findings

operator norm inequalities between tensor unfoldings on the partition lattice interest in higher order tensors has recently surged in data intensive fields with a wide range of applications including image processing blind source separation community detection and feature extraction a common paradigm in tensor related algorithms advocates unfolding or flattening the tensor into a matrix and applying classical methods developed for matrices despite the popularity of such techniques how the functional properties of a tensor changes upon unfolding is currently not well understood in contrast to the body of existing work which has focused almost exclusively on matricizations we here consider all possible unfoldings of an order k tensor which are in one to one correspondence with the set of partitions of 1 k we derive general inequalities between the lp norms of arbitrary unfoldings defined on the partition lattice in particular we demonstrate how the spectral norm p 2 of a tensor is bounded by that of its unfoldings and obtain an improved upper bound on the ratio of the frobenius norm to the spectral norm of an arbitrary tensor for specially structured tensors satisfying a generalized definition of orthogonal decomposability we prove that the spectral norm remains invariant under specific subsets of unfolding operations

theta functions and hodge numbers of moduli spaces of sheaves on rational surfaces we compute generating functions for the hodge numbers of the moduli spaces of h stable rank 2 sheaves on a rational surface s in terms of theta functions for indefinite lattices if h lies in the closure of the ample cone and has self intersection 0 it follows that the generating functions are jacobi forms in particular the generating functions for the euler numbers can be expressed in terms of modular forms and their transformation behaviour is compatible with the predictions of s duality we also express the generating functions for the signatures in terms of modular forms it turns out that these generating functions are also with respect to another developing parameter the generating function for the donaldson invariants of s evaluated on all powers of the point class