conservation laws for conformally invariant variational problems we succeed in writing 2 dimensional conformally invariant non linear elliptic pde harmonic map equation prescribed mean curvature equations etc in divergence form these divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces from this form we can recover without the use of moving frame all the classical regularity results known for 2 dimensional conformally invariant non linear elliptic pde see hel it enables us also to establish new results in particular we solve a conjecture by e heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by s hildebrandt hil1 claiming that critical points of continuously differentiable elliptic conformally invariant lagrangian in two dimensions are continuous

a weak compactness result for critical elliptic systems of n harmonic type abstract the paper considers systems of the form div u n 2 u u n 2 u on a bounded domain in r n with u w 1 n a matrix l n depending on u and some additional structural assumptions on we prove that if a sequence of solutions of the above system converges weakly the limit itself is also a solution the class of systems considered includes the n harmonic system and the presented reasoning is a generalization of c wang s proof for n harmonic maps

embedded surfaces of arbitrary genus minimizing the willmore energy under isoperimetric constraint the isoperimetric ratio of an embedded surface in mathbb r 3 is defined as the ratio of the area of the surface to power three to the squared enclosed volume the aim of the present work is to study the minimization of the willmore energy under fixed isoperimetric ratio when the underlying abstract surface has fixed genus g geqq 0 the corresponding problem in the case of spherical surfaces that is g 0 was recently solved by schygulla see schygulla arch ration mech anal 203 901 941 2012 with different methods

the resolving power of seismic amplitude data an anisotropic inversion migration approach a description of the theory and numerical implementation of a 3 d linearized asymptotic anisotropic inversion method based on the generalized radon transform is given we discuss implementation aspects including 1 the use of various coordinate systems 2 regularization by both spectral and bayesian statistical techniques and 3 the effects of limited acquisition apertures on inversion we give applications of the theory in which well resolved parameter combinations are determined for particular experimental geometries and illustrate the interdependence of parameter and spatial resolutions procedures for evaluating uncertainties in the parameter estimates that result from the inversion are derived and demonstrated

embedding formulations and complexity for unions of polyhedra it is well known that selecting a good mixed integer programming mip formulation is crucial for effectively obtaining a solution with state of the art solvers although best practices and guidelines for constructing good formulations abound there is rarely a single systematic construction approach that leads to the best possible formulation here we introduce embedding formulations and complexity as a new mip formulation paradigm for systematically constructing formulations for disjunctive constraints that are optimal with respect to size more specifically they yield the smallest possible ideal formulation i e one whose lp relaxation has integral extreme points among all formulations that only use 0 1 auxiliary variables we use the paradigm to characterize optimal formulations for special ordered sets of type 2 and certain piecewise linear functions of two variables we also show that the resultant formulations can provide a significant computational advantage over all known formulations for pi

regularity of area minimizing currents iii blow up this is the last of a series of three papers in which we give a new shorter proof of a slightly improved version of almgren s partial regularity of area minimizing currents in riemannian manifolds here we perform a blow up analysis deducing the regularity of area minimizing currents from that of dir minimizing multiple valued functions

degree theory on oriented infinite dimensional varieties and the morse number of minimal surfaces spanning a curve in ℝn the algebraic number of disc minimal surfaces spanning a wire in 3 is defined and shown to be equal to one

three dimensional frequency domain full waveform inversion with an iterative solver with the acquisition of wide aperture seismic data sets full waveform inversion is an attractive method for deriving velocity models three dimensional implementations require an efficient solver for the wave equation computing 3d time harmonic responses with a frequency domain solver is complicated because a large linear system with negative and positive eigenvalues must be solved time domain schemes are an alternative nevertheless existing frequency domain iterative solvers with an efficient preconditioner are a viable option when full waveform inversion is formulated in the frequency domain an iterative solver with a multigrid preconditioner is competitive because of a high order spatial discretization numerical examples illustrated the efficiency of the iterative solvers three dimensional full waveform inversion was then studied in the context of deep water ocean bottom seismometer acquisition three dimensional synthetic data inversion results showed the behavior of full waveform inversion with respect to the initial model and the minimum frequency available in the data set results on a 3d real ocean bottom seismometer data set demonstrated the relevance of full waveform inversion especially to image the shallow part of the model

a mathematical programming approach to clusterwise regression model and its extensions abstract the clusterwise regression model is used to perform cluster analysis within a regression framework while the traditional regression model assumes the regression coefficient to be identical for all subjects in the sample the clusterwise regression model allows to vary with subjects of different clusters since the cluster membership is unknown the estimation of the clusterwise regression is a tough combinatorial optimization problem in this research we propose a generalized clusterwise regression model which is formulated as a mathematical programming mp problem a nonlinear programming procedure with linear constraints is proposed to solve the combinatorial problem and to estimate the cluster membership and simultaneously moreover by integrating the cluster analysis with the discriminant analysis a clusterwise discriminant model is developed to incorporate parameter heterogeneity into the traditional discriminant analysis the cluster membership and discriminant parameters are estimated simultaneously by another nonlinear programming model

partial regularity for mass minimizing currents in hilbert spaces recently the theory of currents and the existence theory for plateau s problem have been extended to the case of finite dimensional currents in infinite dimensional manifolds or even metric spaces see acta math 185 2000 1 80 and also proc lond math soc 3 106 2013 1121 1142 adv calc var 7 2014 227 240 for the most recent developments in this paper in the case when the ambient space is hilbert we provide the first partial regularity result in a dense open set of the support for n dimensional integral currents which locally minimize the mass our proof follows with minor variants indiana univ math j 31 1982 415 434 implementing lipschitz approximation and harmonic approximation without indirect arguments and with estimates which depend only on the dimension n and not on codimension or dimension of the target space

multiple solutions of h systems and rellich s conjecture soit x y r 2 x 2 y 2 1 on considere une fonction u r 3 satisfaisant le systeme h u 2hu x u y sur avec l une des conditions suivantes soit dirichlet u sur soit plateau u x 2 u y 2 u x u y 0 sur u et u non decroissante sur

extensions of the mountain pass theorem on considere une fonctionnelle c 1 i x r ou x est un espace de banach reel et ou i satisfait la condition de compacite de palais smale on presente une serie de theoremes qui generalisent le theoreme du col

improved estimate of the singular set of dir minimizing q valued functions via an abstract regularity result abstract in this note we prove an abstract version of a recent quantitative stratification principle introduced by cheeger and naber 2013 6 7 using this general regularity result paired with an e regularity theorem we provide a new estimate of the minkowski dimension of the set of higher multiplicity points of a dir minimizing q valued function the abstract principle is applicable to several other problems we recover recent results in the literature and we obtain also some improvements in more classical contexts

fast 2 d ray born migration inversion in complex media in this paper we evaluate the capacity of a fast 2 d ray born migration inversion algorithm to recover the true amplitude of the model parameters in 2 d complex media the method is based on a quasi newtonian linearized inversion of the scattered wavefield asymptotic green s functions are computed in a smooth reference model with a dynamic ray tracing based on the wavefront construction method the model is described by velocity perturbations associated with diffractor points both the first traveltime and the strongest arrivals can be inverted the algorithm is implemented with several numerical approximations such as interpolations and aperture limitation around common midpoints to speed the algorithm both theoritical and numerical aspects of the algorithm are assessed with three synthetic and real data examples including the 2 d marmousi example comparison between logs extracted from the exact marmousi perturbation model and the computed images shows that the amplitude of the velocity perturbations

optimization problems in statistical learning duality and optimality conditions abstract regularization methods are techniques for learning functions from given data we consider regularization problems the objective function of which consisting of a cost function and a regularization term with the aim of selecting a prediction function f with a finite representation f i 1 n c i k x i which minimizes the error of prediction here the role of the regularizer is to avoid overfitting in general these are convex optimization problems with not necessarily differentiable objective functions thus in order to provide optimality conditions for this class of problems one needs to appeal on some specific techniques from the convex analysis in this paper we provide a general approach for deriving necessary and sufficient optimality conditions for the regularized problem via the so called conjugate duality theory afterwards we employ the obtained results to the support vector machines problem and support vector regression problem formulated for different cost functions

inversion of reflection traveltimes and amplitudes a bayesian nonlinear inversion scheme is used to invert traveltime and amplitude data from supercritical reflections reflection amplitudes are determined using a spherical wave model where the amplitude of a reflected spherical wave is given by a sommerfeld integral this integral is evaluated numerically by integrating over a contour in the complex plane the spherical wave model differs from plane wave models in the regions around the critical points for p and s refraction in these regions the spherical wave amplitudes are sensitive to frequency with a dependance on the ratio h between the reflector depth h and the seismic wave length the accuracies of the inversion scheme and the spherical reflection model are tested using control data from the university of houston physical model tank the solution obtained by inversion agrees with the known solution to within 3 percent for all parameters and the amplitudes predicted by the spherical wave model agree remarkably well with the observed data

nonparametric quantile frontier estimation under shape restriction abstract this paper proposes a shape restricted nonparametric quantile regression to estimate the frontier which acts as a benchmark for whether a decision making unit achieves top efficiency this method adopts a two step strategy first identifying fitted values that minimize an asymmetric absolute loss under the nondecreasing and concave shape restriction second constructing a nondecreasing and concave estimator that links these fitted values this method makes no assumption on the error distribution and the functional form experimental results on some artificial data sets clearly demonstrate its superiority over the classical linear quantile regression we also discuss how to enforce constraints to avoid quantile crossings between multiple estimated frontiers with different values of finally this paper shows that this method can be applied to estimate the production function when one has some prior knowledge about the error term

on regularity theory for n p harmonic maps into manifolds abstract in this paper we continue the investigation started in the paper da lio and schikorra 2014 of the regularity of the so called weak n p harmonic maps in the critical case these are critical points of the following nonlocal energy l s u r n s 2 u x p d x where u h s p r n n and n r n is a closed k dimensional smooth manifold and s n p we prove holder continuity for such critical points for p 2 for p 2 we obtain the same under an additional lorentz space assumption the regularity theory is in the two cases based on regularity results for nonlocal schrodinger systems with an antisymmetric potential

bayesian regression with multivariate linear splines summary we present a bayesian analysis of a piecewise linear model constructed using basis functions which generalizes the univariate linear spline to higher dimensions prior distributions are adopted on both the number and the locations of the splines which leads to a model averaging approach to prediction with predictive distributions that take into account model uncertainty conditioning on the data produces a bayes local linear model with distributions on both predictions and local linear parameters the method is spatially adaptive and covariate selection is achieved by using splines of lower dimension than the data

numerical analysis of a finite element scheme for the approximation of harmonic maps into surfaces this article studies the numerical approximation of harmonic maps mto surfaces i e critical pomts for the dirichlet energy among weakly differentiable vector fields that are constrained to attain their pointwise values in a given manifold an iterative algorithm that is based on a linearization of the constraint about the current iterate at the nodes of a triangulation is devised and its global convergence to a discrete harmonic map is proved under general conditions weak accumulation of discrete harmonic maps at harmonic maps as discretization parameters tend to zero is established in two dimensions under certain assumptions on the underlying sequence of triangulations numerical simulations illustrate the performance of the algorithm for curved domains

a general regularity theory for stable codimension 1 integral varifolds we give a necessary and sufficient geometric structural condition which we call the a structural hypothesis for a stable codimension 1 integral varifold on a smooth riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities the a structural hypothesis says that no point of the support of the varifold has a neighborhood in which the support is the union of three or more embedded c 1 a hypersurfaces with boundary meeting only along their common boundary we establish that whenever a stable integral n varifold on a smooth n 1 dimensional riemannian manifold satisfies the a structural hypothesis for some a 0 1 2 its singular set is empty if n 6 discrete if n 7 and has hausdorff dimension n 7 if n 8 in view of well known examples this is the best possible general dimension estimate on the singular set of a varifold satisfying our hypotheses we also establish compactness of mass bounded subsets of the class of stable codimension 1 integral varifolds satisfying the a structural hypothesis for some a 0 1 2 the a structural hypothesis on an n varifold for any a 0 1 2 is readily implied by either of the following two hypotheses i the varifold corresponds to an absolutely area minimizing rectifiable current with no boundary ii the singular set of the varifold has vanishing n 1 dimensional hausdorff measure thus our theory subsumes the well known regularity theory for codimension 1 area minimizing rectifiable currents and settles the long standing question as to which weakest size hypothesis on the singular set of a stable minimal hypersurface guarantees the validity of the above regularity conclusions an optimal strong maximum principle for stationary codimension 1 integral varifolds follows from our regularity and compactness theorems

harmonic maps from degenerating riemann surfaces we study harmonic maps from degenerating riemann surfaces with uniformly bounded energy and show the so called generalized energy identity we find conditions that are both necessary and sufficient for the compactness in w 1 2 and c 0 modulo bubbles of sequences of such maps

almgren s type regularity for semicalibrated currents abstract in analogy with almgren s theorem for integral area minimizing currents of general dimension and codimension we prove that an m dimensional semicalibrated integral current in a n m dimensional c 3 e 0 manifold semicalibrated by a c 2 e 0 m form has singular set of hausdorff dimension at most m 2

existence of min max free boundary disks realizing the width of a manifold abstract we perform a replacement procedure in order to produce a free boundary minimal surface whose area achieves the min max value over all disk sweepouts of a manifold whose boundary lies in a submanifold our result is based on a proof of the convexity of the energy for free boundary harmonic maps and a generalization of colding minicozzi replacement procedure

weak solutions to the heat flow for surfaces of prescribed mean curvature in this talk we establish the existence of global weak solutions to the heat flow for surfaces of prescribed mean curvature i e the existence for the cauchy dirichlet problem to parabolic systems of the type tu u 2 h u d1u d2u in b 0 u uo on par b 0 where h r3 r is a bounded continuous function satisfying an isoperimetric condition b the unit ball in r2 and u b 0 r3 as one of the possible applications we show that the problem has a solution with values inbr r3 whenever uo b br and furthermore there holds br h 3 2r h d 9 2 h a 1 r for a br the results that will be presented in the talk are joint work with verena bogelein und chrsitoph scheven from erlangen frank duzaar department of mathematics university erlangen nurnberg cauerstr 11 91056 erlangen germany e mail address frank duzaar zuv uni erlangen de date december 29 2011 1

the qualitative behavior at the free boundary for approximate harmonic maps from surfaces let u n be a sequence of maps from a compact riemann surface m with smooth boundary to a general compact riemannian manifold n with free boundary on a smooth submanifold k subset n satisfying begin aligned sup n left vert nabla u n vert l 2 m vert tau u n vert l 2 m right le lambda end aligned where tau u n is the tension field of the map u n we show that the energy identity and the no neck property hold during a blow up process the assumptions are such that this result also applies to the harmonic map heat flow with free boundary to prove the energy identity at finite singular time as well as at infinity time also the no neck property holds at infinity time

partial regularity for stationary harmonic maps into spheres in an interesting recent paper 12 f hi lein has shown that any weakly harmonic mapping from a two dimensional surface into a sphere is smooth i present here a kind of generalization to higher dimensions asserting in effect that a stationary harmonic mapping from an open subset of nn n 3 into a sphere is smooth except possibly for a closed singular set of n 2 dimensional hausdorff measure zero my proof crucially depends upon several of h l in s observations as streamlined by p l lions to state the result precisely let us suppose that m n 2 u is a smooth open subset of nn and s m 1 denotes the unit sphere in nm a function u in the sobolev space hi u rm u u 1 urn belongs to h i u s ml provided t u 1 a e in u

decay estimates for riviere s equation with applications to regularity and compactness we derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in 2 introduced by riviere applications include sharp regularity results and compactness theorems which generalise a large amount of previous geometric pde theory including some of the theory of harmonic and almost harmonic maps from surfaces

the existence of embedded minimal surfaces and the problem of uniqueness it was in the early nineteen thirties that douglas and radd solved the plateau problem for a rectifiable jordan curve in euclidean space the minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk in 1948 morrey generalized the theorem of douglas and radd to minimal surfaces in a homogeneously regular riemannian manifold in both cases the solutions are defective as they may possess branch points it was not until 1968 that osserman 24 was able to prove that interior branch points do not exist on the douglas radd morrey solution osserman s theorem in this full generality was proved by alt 2 3 and gulliver 8 if the jordan curve is real analytic gulliver and lesley 9 showed that the solution surface is free of branch points even on its boundary hence at least in this case we know that the douglas radd morrey solution is an immersion and represents a classical surface however the douglas radd morrey solution may in general have selfintersections it was then a classical problem to find conditions on the jordan curve to guarantee that the solution is an embedded disk in 1932 radd was the first to produce a general condition of this type he proved that if the curve can be projected onto a convex curve in the plane then it bounds a unique embedded minimal disk which is a graph over this plane it was not until the seventies that there was further progress on the problem of embeddedness osserman had conjectured that if a jordan curve is extreme that is lies on the boundary of its convex hull then the douglas rad6 solution is embedded gulliver and spruck 10 proved this conjecture under the additional assumption that the absolute total curvature of the jordan curve is less than 4 z right after this work tomi and tromba 32 showed that an extreme curve must bound an embedded minimal disk which need not be stable in 1976 almgren and simon 1 established the existence for such a curve of an embedded stable minimal disk which need not be a douglas radd solution finally in the same year the authors 18 proved that any douglas rad6 solution must be embedded

analysis aspects of willmore surfaces a new formulation for the euler lagrange equation of the willmore functional for immersed surfaces in m is given as a nonlinear elliptic equation in divergence form with non linearities comprising only jacobians letting vec h be the mean curvature vector of the surface our new formulation reads mathcal l vec h 0 where mathcal l is a well defined locally invertible self adjoint elliptic operator several consequences are studied in particular the long standing open problem asking for a meaning to the willmore euler lagrange equation for immersions having only l 2 bounded second fundamental form is now solved the regularity of weak willmore immersions with l 2 bounded second fundamental form is also established its proof relies on the discovery of conservation laws which are preserved under weak convergence a weak compactness result for willmore surfaces with energy less than 8 the li yau condition ensuring the surface is embedded is proved via a point removability result established for wilmore surfaces in m thereby extending to arbitrary codimension the main result in ks3 finally from this point removability result the strong compactness of willmore tori below the energy level 8 is proved both in dimension 3 this had already been settled in ks3 and in dimension 4

morse conley theory for minimal surfaces of varying topological type after the plateau problem had been solved by douglas and rad6 by finding a possibly branched disk type minimal immersion bounded by a given jordan curve in n soon more general problems were proposed on one hand the existence of minimal surfaces of higher topological structure was investigated by douglas i d courant c and shiffman sh 1 on the other hand the problem of finding unstable minimal surfaces and more generally developing a morse theory for minimal surfaces of disk type was attacked by morse tompkins mt1 and shiffman sh2 a natural question then was to develop a morse theory for minimal surfaces of arbitrary topological structure while morse tompkins in their paper mt2 only treated a very special case namely annulus type surfaces of what their title unstable minimal surfaces of higher topological structure claims shiffman confronted the case of genus 0 and arbitrary connectivity sh3 these classical papers were not in every respect satisfactory the investigations of douglas and courant on the higher genus problem were recently critisized by tromba t 1 however luckhaus i l was able to carry out a systematic reworking of the arguments of courant and shiffman the original approaches of morse tompkins shiffman moreover severely suffer from the fact that these authors work in the c instead of the more natural h l 2 topology in the c dirichlet s integral whose critical points parametrize the sought after minimal surfaces is not differentiable and no intrinsic notion i e depending only on the surface of morse index can be defined this makes it impossible to decide in the work of morse tompkins and shiffman whether the morse relations for minimal surfaces reflect a property of the surfaces

elastic least squares reverse time migration via linearized elastic full waveform inversion with pseudo hessian preconditioning abstracttime domain elastic least squares reverse time migration lsrtm is formulated as a linearized elastic full waveform inversion problem the elastic born approximation and elastic reverse time migration rtm operators are derived from the time domain continuous adjoint state method our approach defines p and s wave impedance perturbations as unknown elastic images our algorithm is obtained using continuous functional analysis in which the problem is discretized at the final stage optimize before discretize approach the discretized numerical versions of the elastic born operator and its adjoint elastic rtm operator pass the dot product test the conjugate gradient least squares method is used to solve the least squares migration quadratic optimization problem in other words the hessian operator for elastic lsrtm is implicitly inverted via a matrix free algorithm that only requires the action of forward and adjoint operators on vectors the diagonal of the pseudo hessian operator is used t

min max theory and the willmore conjecture in 1965 t j willmore conjectured that the integral of the square of the mean curvature of a torus immersed in r 3 is at least 2 2 we prove this conjecture using the min max theory of minimal surfaces

regularity of area minimizing currents ii center manifold this is the second paper of a series of three on the regularity of higher codimension area minimizing integral currents here we perform the second main step in the analysis of the singularities namely the construction of a center manifold i e an approximate average of the sheets of an almost flat area minimizing current such a center manifold is accompanied by a lipschitz multivalued map on its normal bundle which approximates the current with a high degree of accuracy in the third and final paper these objects are used to conclude the proof of almgren s celebrated dimension bound on the singular set

a strategy for nonlinear elastic inversion of seismic reflection data the problem of interpretation of seismic reflection data can be posed with sufficient generality using the concepts of inverse theory in its roughest formulation the inverse problem consists of obtaining the earth model for which the predicted data best fit the observed data if an adequate forward model is used this best model will give the best images of the earth s interior three parameters are needed for describing a perfectly elastic isotropic earth the density x and the lame parameters x and x or the density x and the p wave and s wave velocities x and x the choice of parameters is not neutral in the sense that although theoretically equivalent if they are not adequately chosen the numerical algorithms in the inversion can be inefficient in the long spatial wavelengths of the model adequate parameters are the p wave and s wave velocities while in the short spatial wavelengths p wave impedance s wave impedance and density are adequate the problem of inversion o

regularity for dirac harmonic maps into certain pseudo riemannian manifolds abstract we show the smoothness of weakly dirac harmonic maps from a closed spin riemann surface into stationary lorentzian manifolds and obtain a regularity theorem for a class of critical elliptic systems without anti symmetry structures

the regularity of a semilinear elliptic system with quadratic growth of gradient abstract in this paper we study semilinear elliptic systems with critical nonlinearity of the form 0 1 u q x u u for u r n r k q has quadratic growth in u our work is motivated by elliptic systems for harmonic map and biharmonic map when n 2 such a system does not have smooth regularity in general for w 1 2 weak solutions by a well known example of j frehse classical results of harmonic map proved by f helein for n 2 and f bethuel for n 3 assert that a w 1 n weak solution of harmonic map is always smooth we extend bethuel s result to general system 0 1 that a w 1 n weak solution of the system is smooth for n 3 for a fourth order semilinear elliptic system with critical nonlinearity which extends biharmonic map we prove a similar result that a w 2 n 2 weak solution of such system is always smooth for n 5 we also construct various examples and these examples show that our regularity results are optimal in various sense

regularity of n 2 harmonic maps into spheres we prove holder continuity for n 2 harmonic maps from arbitrary subsets of rn into a sphere this extends a recent one dimensional result by f da lio and t riviere to arbitrary dimensions the proof relies on compensation effects which we quantify adapting an approach for wente s inequality by l tartar instead of besov space arguments which were used in the one dimensional case moreover fractional analogues of hodge decomposition and higher order poincare inequalities as well as several localization effects for nonlocal operators similar to the fractional laplacian are developed and applied this work was the author s phd thesis written in march 2010

approximation algorithms for training one node relu neural networks training a one node neural network with the relu activation function via optimization which we refer to as the on relu problem is a fundamental problem in machine learning in this paper we begin by proving the np hardness of the on relu problem we then present an approximation algorithm to solve the on relu problem whose running time is mathcal o n k where n is the number of samples and k is a predefined integral constant as an algorithm parameter we analyze the performance of this algorithm under two regimes and show that 1 given any arbitrary set of training samples the algorithm guarantees an n k approximation for the on relu problem to the best of our knowledge this is the first time that an algorithm guarantees an approximation ratio for arbitrary data scenario thus in the ideal case i e when the training error is zero the approximation algorithm achieves the globally optimal solution for the on relu problem and 2 given training sample with gaussian noise the same approximation algorithm achieves a much better asymptotic approximation ratio which is independent of the number of samples n extensive numerical studies show that our approximation algorithm can perform better than the gradient descent algorithm our numerical results also show that the solution of the approximation algorithm can provide a good initialization for gradient descent which can significantly improve the performance

nonlinear waveform inversion of plane wave seismograms in stratified elastic media we present a method for determining the elastic parameters of a horizontally stratified medium from its plane wave reflectivity the nonlinear inverse problem is iteratively solved by using a generalized least squares formalism the proposed method uses the relatively fast convergence properties of the conjugate gradient algorithm and achieves computational efficiency through analytical solutions for calculating the reference and perturbational wavefields the solution method is implemented in the frequency wave slowness domain and can be readily adapted to various source receiver configurations the behavior of the algorithm conforms to the predictions of generalized least squares inverse the ory the inversion scheme yields satisfactory results as long as the correct velocity trends are introduced in the starting model in practice the inversion algorithm should be applied first in the precritical region because of the strong nonlinear behavior of postcritical data with respect to velocity perturbations the suggested inversion strategy consists of first inverting for the density and p wave velocity or p wave impedance by considering plane waves in the low slowness region near normal angles of incidence then in optimizing for the s wave velocity by progressively including contributions from the high slowness region steep angles of incidence numerical experiments performed with noise free synthetic data prove that the proposed inversion method satisfactorally reconstructs the elastic properties of a stratified medium from a limited set of plane wave components at a reasonable computing cost

a variational characterization of complex submanifolds in this note we generalize our results in arezzo and sun reine angew math doi 10 1515 crelle 2013 0097 2012 to integer p currents of any degree we prove that if the mass of a current as a functional of the ambient metric has a critical or stable point in some special directions then the current is complex this holds for any dimension and codimension we also study a natural functional on the space of currents representing a fixed homology class closely related to the first derivative of the mass in our new approach detecting the deviation of a surface from being holomorphic

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

eigenvalues and eigenvectors of linearized elastic inversion small contrasts in the parameters allow the linearization of elastic modeling and inversion although this assumption simplifies parameter estimation it also impoverishes information a linearized model besides being an approximation requires shrinking the data space to precritical angles this work investigates the two basic questions regarding the type of information that can be retrieved and how to retrieve it the method is the singular value decomposition of the transfer function between data and parameters this tool allows a simple interpretation of the information contained in reflections and it outlines a general estimation procedure a medium characterized by a uniform background is considered reflections are linearized according to the born approximation p p reflections are analyzed the simplicity of the model allows a theoretical study of the effects of velocity errors in the overburden the analysis is performed for two different sets of elastic parameters in both cases the conclusion

rectifiability of line defects in liquid crystals with variable degree of orientation in 2 hardt lin and the author proved that the defect set of minimizers of the modified ericksen energy for nematic liquid crystals consists locally of a finite union of isolated points and holder continuous curves with finitely many crossings in this article we show that each holder continuous curve in the defect set is of finite length hence locally the defect set is rectifiable for the most part the proof closely follows the work of de lellis et al rectifiability and upper minkowski bounds for singularities of harmonic q valued maps arxiv 1612 01813 2016 on harmonic mathcal q valued maps the blow up analysis in alper et al calc var partial differ equ 56 5 128 2017 allows us to simplify the covering arguments in 11 and locally estimate the length of line defects in a geometric fashion

linearized inversion of multioffset seismic reflection data in the ω k domain in the acoustic approximation the earth is described using only density and bulk modulus assuming smooth density variations reflections can be described using a single function the velocity of compressional waves if a reference model which is close enough to the actual earth is known the problem of estimating the medium velocity from the observed data can be linearized using a least squares formulation and working in the k domain the linearized inverse problem for a homogeneous reference medium can be solved by a noniterative algorithm which is economically competitive with prestack migration numerical tests with synthetic and real data demonstrate the feasibility and the numerical stability of the method the numerical results compare well with those obtained by migration of unstacked data although superior results will only be obtained when the physics of the problem including elastic versus acoustic effects three dimensional propagation and accurate source estimation will realistically b

fitting piecewise linear continuous functions abstract we consider the problem of fitting a continuous piecewise linear function to a finite set of data points modeled as a mathematical program with convex objective we review some fitting problems that can be modeled as convex programs and then introduce mixed binary generalizations that allow variability in the regions defining the best fit function s domain we also study the additional constraints required to impose convexity on the best fit function

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

partial regularity for harmonic maps and related problems via gauge theory we give a new proof of partial regularity for harmonic maps in dimensions m 3 into arbitrary targets this proof avoids the use of adapted frames and permits to consider targets of minimal c 2 regularity the proof we present moreover extends to a large class of elliptic systems of quadratic growth

sub criticality of non local schrodinger systems with antisymmetric potentials and applications to half harmonic maps abstract we consider non local linear schrodinger type critical systems of the type 1 1 4 v v in r where is antisymmetric potential in l 2 r so m v is an r m valued map and v denotes the matrix multiplication we show that every solution v l 2 r r m of 1 is in fact in l loc p r r m for every 2 p in other words we prove that the system 1 which is a priori only critical in l 2 happens to have a subcritical behavior for antisymmetric potentials as an application we obtain the c loc 0 regularity of weak 1 2 harmonic maps into c 2 compact sub manifolds without boundary

an l p regularity theory for harmonic maps motivated by the harmonic map heat flow we consider maps between riemannian manifolds such that the tension field belongs to an lp space under an appropriate smallness condition a certain degree of regularity follows for suitable solutions of the harmonic map heat flow we have a partial regularity result as a consequence

classification and regression via integer optimization motivated by the significant advances in integer optimization in the past decade we introduce mixed integer optimization methods to the classical statistical problems of classification and regression and construct a software package called crio classification and regression via integer optimization crio separates data points into different polyhedral regions in classification each region is assigned a class while in regression each region has its own distinct regression coefficients computational experimentations with generated and real data sets show that crio is comparable to and often outperforms the current leading methods in classification and regression we hope that these results illustrate the potential for significant impact of integer optimization methods on computational statistics and data mining

mixed integer models for nonseparable piecewise linear optimization unifying framework and extensions we study the modeling of nonconvex piecewise linear functions as mixed integer programming mip problems we review several new and existing mip formulations for continuous piecewise linear functions with special attention paid to multivariate nonseparable functions we compare these formulations with respect to their theoretical properties and their relative computational performance in addition we study the extension of these formulations to lower semicontinuous piecewise linear functions

regularity theory for 2 dimensional almost minimal currents i lipschitz approximation we construct lipschitz q valued functions which carefully approximate integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense this result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of 2 dimensional integral currents area minimizing in riemannian manifolds semicalibrated and spherical cross sections of 3 dimensional area minimizing cones

boundary equations and regularity theory for geometric variational systems with neumann data we study boundary regularity of maps from two dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which at the boundary intersect perpendicularly with a support manifold for example harmonic maps or h surfaces with a partially free boundary condition in the interior it is known by the celebrated work of riviere that these maps satisfy a system with an antisymmetric potential from which one can derive the interior regularity of the solution avoiding a reflection argument we show that these maps satisfy along the boundary a system of equations which also exhibits a nonlocal antisymmetric potential that combines information from the interior potential and the geometric neumann boundary condition we then proceed to show boundary regularity for solutions to such systems