global existence of harmonic maps in minkowski space on considere la construction d une classe d applications harmoniques classiques globales de l espace de minkowski m 4 dans une variete de riemann complete a n dimensions

on global existence and scattering for the wave maps equation we prove global existence and scattering for the wave maps equation in n 1 dimensions n 2 3 for initial data which is small in the scale invariant homogeneous besov space 2 1 n 2 x 2 1 n 2 1 this result was proved in an earlier paper by the author for n 4

noise tolerant parallel learning of geometric concepts we present several efficient parallel algorithms for pac learning geometric concepts in a constant dimensional space the algorithms are robust even against malicious classification noise of any rate less than 1 2 we first give an efficient noise tolerant parallel algorithm to pac learn the class of geometric concepts defined by a polynomial number of d 1 dimensional hyperplanes against an arbitrary distribution where each hyperplane has a slope from a set of known slopes we then describe how boosting techniques can be used so that our algorithms dependence on and does not depend ond next we give an efficient noise tolerant parallel algorithm to pac learn the class of geometric concepts defined by a polynomial number of d 1 dimensional hyperplanes of unrestricted slopes against a uniform distribution we then show how to extend our algorithm to learn this class against any unknown product distribution next we define a complexity measure of any setsof d 1 dimensional surfaces that we call thevariantofsand prove that the class of geometric concepts defined by surfaces of polynomial variant can be efficiently learned in parallel under a product distribution even under malicious classification noise furthermore we show that the vc dimension of the class of geometric concepts defined by a single surface of variant one is finally we give an efficient parallel noise tolerant algorithm to pac learn any 2 dimensional geometric concept defined by a setsof 1 dimensional surfaces of polynomial length under a uniform distribution

on the formation of singularities in the critical o 3 sigma model we study the phenomena of energy concentration for the critical o 3 sigma model also known as the wave map flow from 2 1 minkowski space into the sphere s 2 we establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities the construction and analysis are done in the context of the k equivariant symmetry reduction and we restrict to maps with homotopy class k 4 the concentration mechanism we uncover is essentially due to a resonant self focusing shrinking of a corresponding harmonic map we show that the phenomenon is generic e g in certain sobolev spaces in that it persists under small perturbations of initial data while the resulting blowup is bounded by a log modified self similar asymptotic

cauchy problems of the gauged sigma model we study the gauged sigma model in r1 1 the existence of the global smooth solution will be proved furthermore we show that the global weak solutions exist in rn 1 n 2 3

smooth optimization with approximate gradient we show that the optimal complexity of nesterov s smooth first order optimization algorithm is preserved when the gradient is computed only up to a small uniformly bounded error in applications of this method to semidefinite programs this means in some instances computing only a few leading eigenvalues of the current iterate instead of a full matrix exponential which significantly reduces the method s computational cost this also allows sparse problems to be solved efficiently using sparse maximum eigenvalue packages

construction of type ii blowup solutions for the 1 corotational energy supercritical wave maps abstract we consider the energy supercritical wave maps from r d into the d sphere s d with d 7 under an additional assumption of 1 corotational symmetry the problem reduces to the one dimensional semilinear wave equation t 2 u r 2 u d 1 r r u d 1 2 r 2 sin 2 u we construct for this equation a family of c solutions which blow up in finite time via concentration of the universal profile u r t q r t where q is the stationary solution of the equation and the speed is given by the quantized rates t c u t t l l n l d 1 2 the construction relies on two arguments the reduction of the problem to a finite dimensional one thanks to a robust universal energy method and modulation techniques developed by merle raphael and rodnianski 49 for the energy supercritical nonlinear schrodinger equation then we proceed by contradiction to solve the finite dimensional problem and conclude using the brouwer fixed point theorem

global well posedness for the three dimensional simplified inertial ericksen leslie systems near equilibrium abstract we study a simplified inertial ericksen leslie system for the nematic liquid crystal flow which can be viewed as a system coupling navier stokes equations and wave map equations we prove the global existence of classical solution with initial data near equilibrium

characterization of large energy solutions of the equivariant wave map problem ii we consider 1 equivariant wave maps from bbb r 1 2 to bbb s 2 of finite energy we establish a classification of all degree one global solutions whose energies are less than three times the energy of the harmonic map q in particular for each global energy solution of topological degree one we show that the solution asymptotically decouples into a rescaled harmonic map plus a radiation term together with a companion article part i where we consider the case of finite time blow up this gives a characterization of all 1 equivariant degree 1 wave maps in the energy regime e q 3e q

the cauchy problem for the schrodinger kdv system abstract in this paper we prove that in the general case i e not necessarily vanishing the cauchy problem for the schrodinger korteweg de vries system is locally well posed in l 2 h 3 4 and if 0 then it is locally well posed in h s h 3 4 with 3 16 s 1 4 these results improve the corresponding results of corcho and linares 2007 5 idea of the proof is to establish some bilinear and trilinear estimates in the space g s f s where g s and f s are dyadic bourgain type spaces related to the schrodinger operator i t x 2 and the airy operator t x 3 respectively but with a modification on f s in low frequency part of functions with a weaker structure related to the maximal function estimate of the airy operator

global schrodinger maps in dimensions d 2 small data in the critical sobolev spaces we consider the schrodinger map initial value problem t on r d r 0 0 where r d r s 2 r 3 is a smooth function in all dimensions d 2 we prove that the schrodinger map initial value problem admits a unique global smooth solution 2 c r h 1 q q2 s 2 provided that the data 02 h 1 q is smooth and satises the smallness condition

almost optimal local well posedness for the 3 1 dimensional maxwell klein gordon equations this paper contains a detailed study of the local in time regularity properties of the maxwell klein gordon mkg equations the mkg equations represent a physical model for the interaction of a spin 0 particle sometimes referred to as a higgs or scalar field with the classical equations of electrodynamics i e maxwell s equations in terms of the calculus of variations the mkg system consists of the euler lagrange equations associated to the density l 4 f f 1 2 d d where f a a is the electro magnetic field associated to the set a of real valued potentials on the minkowski space d 1a is the corresponding covariant derivative with respect to those potentials and r c is a complex scalar field in terms of the field f the euler lagrange equations read f d 1a d d 0 1b

scattering for the quartic generalised korteweg de vries equation abstract we show that the quartic generalised kdv equation u t u x x x u 4 x 0 is globally well posed for data in the critical scale invariant space h x 1 6 r with small norm and locally well posed for large norm improving a result of grunrock a grunrock a bilinear airy estimate with application to gkdv 3 differential integral equations 18 12 2005 1333 1339 as an application we obtain scattering results in h x 1 r h x 1 6 r for the radiation component of a perturbed soliton for this equation improving the asymptotic stability results of martel and merle y martel f merle asymptotic stability of solitons for subcritical generalized kdv equations arch ration mech anal 157 3 2001 219 254

excessive gap technique in nonsmooth convex minimization in this paper we introduce a new primal dual technique for convergence analysis of gradient schemes for nonsmooth convex optimization as an example of its application we derive a primal dual gradient method for a special class of structured nonsmooth optimization problems which ensures a rate of convergence of order o 1 k where k is the iteration count another example is a gradient scheme which minimizes a nonsmooth strongly convex function with known structure with rate of convergence o 1 k2 in both cases the efficiency of the methods is higher than the corresponding black box lower complexity bounds by an order of magnitude

remarks on the cauchy problem for the one dimensional quadratic fractional heat equation abstract we prove that the cauchy problem associated with the one dimensional quadratic fractional heat equation u t u u 2 t 0 t x r or t with 0 1 is well posed in h s for s max 1 2 2 except in the case 1 2 where it is shown to be well posed for s 1 2 and ill posed for s 1 2 as a by product we improve the known well posedness results for the heat equation 1 by reaching the end point sobolev index s 1 finally in the case 1 2 1 we also prove optimal results in the besov spaces b 2 s q

on schrodinger maps we study the local well posedness theory for the schr odinger maps equation we work in n 1 dimensions for n geq 2 and prove a local well posedness for small initial data in h frac n 2 varepsilon

parallel linear programming in fixed dimension almost surely in constant time for any fixed dimension d thelinear programming problem with n inequality constraints can be solved on a probabilistic crcw pram with o n processors almost surely in constant time the algorithm always findsthe correct solution with nd log 2 d processors the probability that the algorithm will not finish within o d 2 log 2 d time tends to zero exponentially with n authors abstract

global well posedness for the yang mills equation in 4 1 dimensions small energy author s krieger joachim tataru daniel abstract we consider the hyperbolic yang mills equation on the minkowski space r 4 1 our main result asserts that this problem is globally well posed for all initial data whose energy is sufficiently small this solves a longstanding open problem

asymptotic decomposition for semilinear wave and equivariant wave map equations in this paper we give a unified proof to the soliton resolution conjecture along a sequence of times for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations including the four dimensional radial yang mills equation without using outer energy type inequalities such inequalities have played a crucial role in previous works with similar results roughly speaking we prove that along a sequence of times t n to t the maximal time of existence the solution decouples to a sum of rescaled solitons and a term vanishing in the energy space plus a free radiation term in the global case or a regular part in the finite time blow up case the main difficulty is that in general for instance for the radial four dimensional yang mills case and the radial six dimensional semilinear wave case we do not have a favorable outer energy inequality for the associated linear wave equations our main new input is the simultaneous use of two virial identities

absence of self similar blow up and local well posedness for the constant mean curvature wave equation abstract in this note we consider the constant mean curvature wave equation in 1 2 dimensions we show that it does not admit any self similar blow up we also remark that the equation is locally well posed for initial data in h 3 2

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

log depth circuits for division and related problems we present optimal depth boolean circuits depth o log n for integer division powering and multiple products we also show that these three problems are of equivalent uniform depth and space complexity in addition we describe an algorithm for testing divisibility that is optimal for both depth and space

global existence of smooth solutions for wave maps in de sitter spacetime abstract in this paper we study the problem of a harmonic map from de sitter spacetime into a geodesic on an n dimensional complete riemannian manifold we show that the global smooth solution exists for such wave map when the solution is a small perturbation along the geodesic solution by the method of weighted energy estimates

algorithms and hardness results for parallel large margin learning we consider the problem of learning an unknown large margin halfspace in the context of parallel computation giving both positive and negative results as our main positive result we give a parallel algorithm for learning a large margin half space based on an algorithm of nesterov s that performs gradient descent with a momentum term we show that this algorithm can learn an unknown margin halfspace over n dimensions using n poly 1 processors and running in time o 1 o log n in contrast naive parallel algorithms that learn a margin halfspace in time that depends polylogarithmically on n have an inverse quadratic running time dependence on the margin parameter our negative result deals with boosting which is a standard approach to learning large margin halfspaces we prove that in the original pac framework in which a weak learning algorithm is provided as an oracle that is called by the booster boosting cannot be parallelized more precisely we show that if the algorithm is allowed to call the weak learner multiple times in parallel within a single boosting stage this ability does not reduce the overall number of successive stages of boosting needed for learning by even a single stage our proof is information theoretic and does not rely on unproven assumptions

weak solutions and development of singularities of the su 2 σ model pour toutes donnees initiales d energie finie le modele sigma o k a des solutions faibles globales en chaque dimension d espace temps

renormalization and blow up for wave maps from s 2 times mathbb r to s 2 we construct a one parameter family of finite time blow ups to the co rotational wave maps problem from s 2 x r to s 2 parameterized by nu is an element of 1 2 1 the longitudinal function u t alpha which is the main object of study will be obtained as a perturbation of a rescaled harmonic map of rotation index one from r 2 to s 2 the domain of this harmonic map is identified with a neighborhood of the north pole in the domain s 2 via the exponential coordinates alpha theta in these coordinates u t alpha q lambda t alpha r t alpha where q r 2arctan r is the standard co rotational harmonic map to the sphere lambda t t 1 nu and r t alpha is the error with local energy going to zero as t 0 blow up will occur at t alpha 0 0 due to energy concentration and up to this point the solution will have regularity h1 nu

global regularity of wave maps from r 2 1 to h 2 small energy we demonstrate that wave maps with smooth initial data and small energy from r2 1 to the lobatchevsky plane stay smooth globally in time our method is similar to the one employed in 18 however the multilinear estimates required are considerably more involved and present novel technical challenges in particular we shall have to work with a modification of the functional analytic framework used in 30 33 18

an algorithm reconstructing convex lattice sets in this paper we study the problem of reconstructing special lattice sets from x rays in a finite set of prescribed directions we present the class of q convex sets which is a new class of subsets of z2 having a certain kind of weak connectedness the main result of this paper is a polynomial time algorithm solving the reconstruction problem for the q convex sets these sets are uniquely determined by certain finite sets of directions as a result this algorithm can be used for reconstructing convex subsets of z2 from their x rays in some suitable sets of four lattice directions or in any set of seven mutually nonparallel lattice directions

the wave maps equation the wave maps equation has become a very popular topic in recent years the aim of these expository notes is to present a non technical survey of the ideas and methods which have proved useful in the study of wave maps leading up to the latest results the remaining open problems are also stated and explained

on the regularity of spherically symmetric wave maps wave maps are critical points u m n of the lagrangian u m du 2 where m is an einsteinian manifold and n a riemannian one for the case m 2 1 and u a spherically symmetric map it is shown that the solution to the cauchy problem for u with smooth initial data of arbitrary size is smooth for all time provided the target manifold n satisfies the two conditions that 1 it is either compact or there exists an orthonormal frame of smooth vectorfields on n whose structure functions are bounded and 2 there are two constants c and c such that the smallest eigenvalue and the largest eigenvalue of the second fundamental form kab of any geodesic sphere p s of radius s centered at p n satisfy s c and s a c 1 s this is proved by first analyzing the energy momentum tensor and using the second condition to show that near the first possible singularity the energy of the solution cannot concentrate and hence is small one then proves that for targets satisfying the first condition initial data of small energy imply global regularity of the solution 1993 john wiley sons inc