the calderon problem with partial data in this paper we improve an earlier result by bukhgeim and uhlmann 1 by showing that in dimension n 3 the knowledge of the cauchy data for the schr nodinger equation measured on possibly very small subsets of the boundary determines uniquely the potential we follow the general strategy of 1 but use a richer set of solutions to the dirichlet problem this implies a similar result for the problem of electrical impedance tomography which consists in determining the conductivity of a body by making voltage and current measurements at the boundary

stability estimate for the aligned magnetic field in a periodic quantum waveguide from dirichlet to neumann map in this article we study the boundary inverse problem of determining the aligned magnetic field appearing in the magnetic schrodinger equation in a periodic quantum cylindrical waveguide by knowledge of the dirichlet to neumann map we prove a holder stability estimate with respect to the dirichlet to neumann map by means of the geometrical optics solutions of the magnetic schrodinger equation

compactications defined by arrangements ii locally symmetric varieties of type iv we define a new class of completions of locally symmetric varieties of type iv which interpolates between the baily borel compactification and mumford s toric compactifications an arithmetic arrangement in a locally symmetric variety of type iv determines such a completion canonically this completion admits a natural contraction that leaves the complement of the arrangement untouched the resulting completion of the arrangement complement is very much like a baily borel compactification it is the proj of an algebra of meromorphic automorphic forms when that complement has a moduli space interpretation then what we get is often a compactification obtained by means of geometric invariant theory we illustrate this with several examples moduli spaces of polarized k3 and enriques surfaces and the semi universal deformation of a triangle singularity we also discuss the question when a type iv arrangement is definable by an automorphic form

a sparsity regularization and total variation based computational framework for the inverse medium problem in scattering abstract we present a fast computational framework for the inverse medium problem in scattering i e we look at discretization reconstruction and numerical performance the helmholtz equation in two and three dimensions is used as a physical model of scattering including point sources and plane waves as incident fields as well as near and far field measurements for the reconstruction of the medium we set up a rapid variational regularization scheme and indicate favorable choices of the various parameters the underlying paradigm is roughly speaking to minimize the discrepancy between the reconstruction and measured data while at the same time taking into account various structural a priori information via suitable penalty terms in particular the involved penalty terms are designed to promote information expected in real world environments to this end a combination of sparsity promoting terms total variation and physical bounds of the inhomogeneous medium e g positivity constraints is employed in the regularization penalty a primal dual algorithm is used to solve the minimization problem related to the variational regularization the computational feasibility performance and efficiency of the proposed approach is demonstrated for synthetic as well as experimentally measured data

lattices and k3 surfaces of degree 6 we relate a class of degenerations of k3 surfaces of degree 6 to certain lattices via hodge theory

the maximum number of lines lying on a k3 quartic surface we show that there cannot be more than 64 lines on a quartic surface with isolated rational double points over an algebraically closed field of characteristic p ne 2 3 thus extending segre rams schutt theorem our proof offers a deeper insight into the triangle free case and takes advantage of a special configuration of lines thereby avoiding the technique of the flecnodal divisor we provide several examples of non smooth k3 quartic surfaces with many lines

a fast randomized algorithm for overdetermined linear least squares regression we introduce a randomized algorithm for overdetermined linear least squares regression given an arbitrary full rank m n matrix a with m n any m 1 vector b and any positive real number e the procedure computes an n 1 vector x such that x minimizes the euclidean norm ax b to relative precision e the algorithm typically requires log n log 1 e mn n 3 floating point operations this cost is less than the mn 2 required by the classical schemes based on qr decompositions or bidiagonalization we present several numerical examples illustrating the performance of the algorithm

uniqueness in the cauchy problem for operators with partially holomorphic coefficients the problem of the uniqueness in the cauchy problem for linear di erential operators has been widely investigated during the last years see z for references it is now well understood in the analytic framework with holmgren s theorem where uniqueness always holds at least for non characteristic surfaces and in the c1 case with hoe rmander s theorem h1 iv chap 28 where the uniqueness is governed by principal normality and pseudo convexity the purpose of this work is to ll the gap between these two theorems by considering operators with c1 and partly analytic coe cients in particular one of our results will contain both the theorems mentioned above let us be more precise let na nb be two non negative integers with n na nb 1 we shall set rn rna rnb and for x or n in rn x xa xb n na nb let p p x d p xa xb dxa dxb be a linear di erential operator of arbitrary order m with principal symbol pm we shall assume that

stability of the calderon problem for less regular conductivities abstract in these notes we prove log type stability for the calderon problem with conductivities in c 1 e we follow the lines of a recent work by haberman and tataru in which they prove uniqueness for c 1

optimal subsampling for large sample logistic regression for massive data the family of subsampling algorithms is popular to downsize the data volume and reduce computational burden existing studies focus on approximating the ordinary least square esti

stability estimates for the calderon problem with partial data this is a follow up of a previous article where we proved local stability estimates for a potential in a schr odinger equation on an open bounded set in dimension n 3 from the dirichlet to neumann map with partial data the region under control was the penumbra delimited by a source of light outside of the convex hull of the open set these local estimates provided stability of log log type corresponding to the uniqueness results in calder on s inverse problem with partial data proved by kenig sj ostrand and uhlmann in this article we prove the corresponding global estimates in all dimensions higher than three the estimates are based on the construction of solutions of the schr odinger equation by complex geometrical optics developed in the anisotropic setting by dos santos ferreira kenig salo and uhlmann to solve the calder on problem in certain admissible geometries

reflection groups in algebraic geometry after a brief exposition of the theory of discrete reflection groups in spherical euclidean and hyperbolic geometry as well as their analogs in complex spaces we present a survey of appearances of these groups in various areas of algebraic geometry

leech roots and vinberg groups this chapter investigates the remarkable properties of the leech roots which are the fundamental roots for the even unimodular lattice in lorentzian space r 25 1 and correspond one for one with the points of the leech lattice we give an extensive table of the leech roots in both euclidean and hyperbolic coordinates we also provide the first of what promise to be many applications by showing that the leech roots simplify and explain the remarkable results of vinberg kaplinskaja and meyer on the reflection groups of unimodular lorentzian lattices in dimensions below 20 they also enable us to make some progress on the study of these groups in the next few dimensions

recent developments in inverse acoustic scattering theory we survey some of the highlights of inverse scattering theory as it has developed over the last 15 years with emphasis on uniqueness theorems and reconstruction algorithms for time harmonic acoustic waves included in our presentation are numerical experiments using real data and numerical examples of the use of inverse scattering methods to detect buried objects

optimal stability estimate in the inverse boundary value problem for periodic potentials with partial data abstract we consider the inverse boundary value problem for operators of the form q in an infinite domain r r 1 n n 3 with a periodic potential q for dirichlet to neumann data localized on a portion of the boundary of the form 1 r 1 with 1 being the complement either of a flat or spherical portion of we prove that a log type stability estimate holds

on the most algebraic k3 surfaces and the most extremal log enriques surfaces we shall characterize the unique k3 surface of discriminant 3 or 4 called the most algebraic k3 surfaces by vinberg in terms of the fixed locus of an automorphism on it based on this result we show that there is up to isomorphisms only one rational log eriques surface of type d 19 and one of type a 19

inverse scattering problem for a two dimensional random potential we study an inverse problem for the two dimensional random schrodinger equation q k 2 u 0 the potential q x is assumed to be a gaussian random function whose covariance operator is a classical pseudodifferential operator we show that the backscattered field obtained from a single realization of the random potential q determines uniquely the principal symbol of the covariance operator of q the analysis is carried out by combining harmonic and microlocal analysis with stochastic methods

on a representation of the matching polytope via semidefinite liftings we consider the relaxation of the matching polytope defined by the non negativity and degree constraints we prove that given an undirected graph on n nodes and the corresponding relaxation of the matching polytope n 2 iterations of the lovasz schrijver semidefinite lifting procedure are needed to obtain the matching polytope in the worst case we show that n 2 iterations of the procedure always suffice

calderon inverse problem with partial data on riemann surfaces on a fixed smooth compact riemann surface with boundary m0 g we show that for the schrodinger operator v with potential v c m0 for some 0 the dirichlet to neumann map n measured on an open set m0 determines uniquely the potential v we also discuss briefly the corresponding consequences for potential scattering at 0 frequency on riemann surfaces with asymptotically euclidean or asymptotically hyperbolic ends

determining conductivity by boundary measurements a p calderon pose la question est il possible de determiner la conductivite thermique d un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite on demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

fast approximation of matrix coherence and statistical leverage the statistical leverage scores of a matrix a are the squared row norms of the matrix containing its top left singular vectors and the coherence is the largest leverage score these quantities are of interest in recently popular problems such as matrix completion and nystrom based low rank matrix approximation as well as in large scale statistical data analysis applications more generally moreover they are of interest since they define the key structural nonuniformity that must be dealt with in developing fast randomized matrix algorithms our main result is a randomized algorithm that takes as input an arbitrary n d matrix a with n d and that returns as output relative error approximations to all n of the statistical leverage scores the proposed algorithm runs under assumptions on the precise values of n and d in o nd logn time as opposed to the o nd2 time required by the naive algorithm that involves computing an orthogonal basis for the range of a our analysis may be viewed in terms of computing a relative error approximation to an underconstrained least squares approximation problem or relatedly it may be viewed as an application of johnson lindenstrauss type ideas several practically important extensions of our basic result are also described including the approximation of so called cross leverage scores the extension of these ideas to matrices with n d and the extension to streaming environments

stability estimate for an inverse problem for the magnetic schrodinger equation from the dirichlet to neumann map abstract we consider the problem of stability estimate of the inverse problem of determining the magnetic field entering the magnetic schrodinger equation in a bounded smooth domain of r n with input dirichlet data from measured neumann boundary observations this information is enclosed in the dynamical dirichlet to neumann map associated to the solutions of the magnetic schrodinger equation we prove in dimension n 2 that the knowledge of the dirichlet to neumann map for the magnetic schrodinger equation measured on the boundary determines uniquely the magnetic field and we prove a holder type stability in determining the magnetic field induced by the magnetic potential

holder stable determination of a quantum scalar potential in unbounded cylindrical domains we consider the inverse problem of determining the time independent scalar potential of the dynamic schrodinger equation in an infinite cylindrical domain from one boundary neumann observation of the solution we prove holder stability by choosing the dirichlet boundary condition suitably

the calderon problem with partial data in two dimensions we prove for a two dimensional bounded domain that the cauchy data for the schrodinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential this implies for the conductivity equation that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset we can determine uniquely the conductivity we use carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results

a periodic faddeev type solution operator abstract we construct periodic solution operators for the equation u 2i u fin a bounded domain with the help of fourier series we prove that thel2 norms of these operators converge to zero if the parameter im goes to infinity then we apply these operators to show that functionsu c20 rd satisfying an inequality u x m u x inrdmust vanish everywhere we extend this result to other second order elliptic differential operators with constant coefficients replacing the laplacian finally we use the solution operators to derive that the span of products of solutions to differential equations is dense inl1

carleman estimate for the schrodinger equation and application to magnetic inverse problems abstract in this paper we prove lipschitz stable determination of the complex valued electric potential and the direction of the magnetic field appearing in the dynamic schrodinger equation with static coefficients by finitely many partial boundary measurements of the solution this is by means of the bukhgeim klibanov method based on an appropriate carleman estimate since the time symmetrization of the static magnetic schrodinger equation around t 0 is not possible we preliminarily establish a carleman inequality specifically designed for this problem

the alternating group of degree 6 in the geometry of the leech lattice and k3 surfaces the alternating group of degree 6 is located at the junction of three series of simple non commutative groups simple sporadic groups alternating groups and simple groups of lie type it plays a very special role in the theory of finite groups we shall study its new roles both in a finite geometry of a certain pentagon in the leech lattice and also in the complex algebraic geometry of k3 surfaces

an inverse problem of the wave equation this paper is organized as follows in sect 1 the statement of the inverse problem is formulated the basic results are formulated and proved in sect 2 some auxiliary material on the x ray transform is given in a self contained way for convenience of the reader this material is used in section one

fitting logistic models under case control or choice based sampling on compare deux approches dans le cas important de l ajustement des modeles de regression logistique

the automorphism group of the 26 dimensional even unimodular lorentzian lattice abstract the group mentioned in the title is shown to be a certain infinitely generated coxeter group extended by the negative of the identity operation and the group of all automorphisms of the notorious leech lattice

stability estimates for the radon transform with restricted data and applications in this article we prove a stability estimate going from the radon transform of a function with limited angle distance data to the l p norm of the function itself under some conditions on the support of the function we apply this theorem to obtain stability estimates for an inverse boundary value problem with partial data

an inverse stability result for non compactly supported potentials by one arbitrary lateral neumann observation in this paper we investigate the inverse problem of determining the time independent scalar potential of the dynamic schrodinger equation in an infinite cylindrical domain from partial measurement of the solution on the boundary namely if the potential is known in a neighborhood of the boundary of the spatial domain we prove that it can be logarithmic stably determined in the whole waveguide from a single observation of the solution on any arbitrary strip shaped subset of the boundary

stability estimates for the inverse boundary value problem for the biharmonic operator with bounded potentials abstract in this article stability estimates are given for the determination of the zeroth order bounded perturbations of the biharmonic operator when the boundary neumann measurements are made on the whole boundary and on slightly more than half the boundary respectively for the case of measurements on the whole boundary the stability estimates are of ln type and for the case of measurements on slightly more than half of the boundary we derive estimates that are of ln ln type

limiting carleman weights and anisotropic inverse problems in this article we consider the anisotropic calderon problem and related inverse problems the approach is based on limiting carleman weights introduced in kenig et al ann math 165 567 591 2007 in the euclidean case we characterize those riemannian manifolds which admit limiting carleman weights and give a complex geometrical optics construction for a class of such manifolds this is used to prove uniqueness results for anisotropic inverse problems via the attenuated geodesic ray transform earlier results in dimension n 3 were restricted to real analytic metrics

inverse problems for the schrodinger equations with time dependent electromagnetic potentials and the aharonov bohm effect we consider the inverse boundary value problem for the schrodinger operator with time dependent electromagnetic potentials in domains with obstacles we extend the resuls of the author s works inverse probl 19 49 2003 19 985 2003 20 1497 2004 to the case of time dependent potentials we relate our results to the aharonov bohm effect caused by magnetic and electric fluxes

the attenuated ray transform on simple surfaces we show that the attenuated geodesic ray transform on two dimensional simple surfaces is injective moreover we give a stability estimate and develop a reconstruction procedure

on the existence of maximum likelihood estimators for the binomial response models summary necessary and sufficient conditions are given for the existence of maximum likelihood estimators of the linear regression parameter in binomial response this includes logit and probit models the question of existence of maximum likelihood estimators mle for logit models arose in an analysis of the relationship of psychiatric caseness to scores on a psychiatric screening questionnaire tennant 1977 administered the ghq general health questionnaire goldberg 1972 to 120 patients attending a general practitioner s surgery and also gave each one a standardized psychiatric interview from the interview patients were classified as psychiatric case non case in a secondary analysis of tennant s data duncan jones and henderson 1978 fitted a logit regression of caseness on ghq score and obtained a good fit with the model

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

local case control sampling efficient subsampling in imbalanced data sets for classification problems with significant class imbalance subsampling can reduce computational costs at the price of inflated variance in estimating model parameters we propose a method for subsampling efficiently for logistic regression by adjusting the class balance locally in feature space via an accept reject scheme our method generalizes standard case control sampling using a pilot estimate to preferentially select examples whose responses are conditionally rare given their features the biased subsampling is corrected by a post hoc analytic adjustment to the parameters the method is simple and requires one parallelizable scan over the full data set standard case control sampling is inconsistent under model misspecification for the population risk minimizing coefficients by contrast our estimator is consistent for provided that the pilot estimate is moreover under correct specification and with a consistent independent pilot estimate our estimator has exactly twice the asymptotic variance of the full sample mle even if the selected subsample comprises a miniscule fraction of the full data set as happens when the original data are severely imbalanced the factor of two improves to formula see text if we multiply the baseline acceptance probabilities by c 1 and weight points with acceptance probability greater than 1 taking roughly formula see text times as many data points into the subsample experiments on simulated and real data show that our method can substantially outperform standard case control subsampling

infinitely imbalanced logistic regression in binary classification problems it is common for the two classes to be imbalanced one case is very rare compared to the other in this paper we consider the infinitely imbalanced case where one class has a finite sample size and the other class s sample size grows without bound for logistic regression the infinitely imbalanced case often has a useful solution under mild conditions the intercept diverges as expected but the rest of the coefficient vector approaches a non trivial and useful limit that limit can be expressed in terms of exponential tilting and is the minimum of a convex objective function the limiting form of logistic regression suggests a computational shortcut for fraud detection problems

the period map for cubic fourfolds the period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20 we determine the image of this period map thus confirming a conjecture of hassett and give at the same time a new proof of the theorem of voisin that asserts that this period map is an open embedding an algebraic version of our main result is an identification of the algebra of sl 6 invariant polynomials on the representation space sym 3 6 with a certain algebra of meromorphic automorphic forms on a symmetric domain of orthogonal type of dimension 20 we also describe the stratification of the moduli space of semistable cubic fourfolds in terms of a vinberg dynkin diagram

stability of the inverse conductivity problem in the plane for less regular conductivities abstract the dirichlet to neumann map or voltage to current map takes dirichlet data u f to the conormal derivatives u at the boundary where u is a solution to the elliptic equation div u 0 on a plane lipchitz domain r2 we consider the inverse conductivity problem that consists of the recovery of in the interior of from the knowledge of we prove stability for this inverse map assuming an isotropic conductivity c1 e we reduce the equation div u 0 to an equivalent system and use the scattering transform this approach relaxes the number of derivatives of required in former works

observability and control of schrodinger equations we propose an exact controllability result for schrodinger equations in bounded domains under the bardos lebeau rauch geometric control condition with an estimate of the control which is explicit with respect to the time of controllability also we prove an explicit in time logarithmic observability estimate for the schrodinger equation where no geometrical conditions are supposed on the domain

faster least squares approximation least squares approximation is a technique to find an approximate solution to a system of linear equations that has no exact solution in a typical setting one lets n be the number of constraints and d be the number of variables with n gg d then existing exact methods find a solution vector in o nd 2 time we present two randomized algorithms that provide accurate relative error approximations to the optimal value and the solution vector of a least squares approximation problem more rapidly than existing exact algorithms both of our algorithms preprocess the data with the randomized hadamard transform one then uniformly randomly samples constraints and solves the smaller problem on those constraints and the other performs a sparse random projection and solves the smaller problem on those projected coordinates in both cases solving the smaller problem provides relative error approximations and if n is sufficiently larger than d the approximate solution can be computed in o nd ln d time

determining a magnetic schrodinger operator from partial cauchy data in this paper we show in dimension n 3 that knowledge of the cauchy data for the schrodinger equation in the presence of a magnetic potential measured on possibly very small subsets of the boundary determines uniquely the magnetic field and the electric potential we follow the general strategy of 7 using a richer set of solutions to the dirichlet problem that has been used in previous works on this problem

lipschitz stability in determining density and two lame coefficients we consider an inverse problem of determining spatially varying density and two lame coefficients in a non stationary isotropic elastic equation by a single measurement of data on the whole lateral boundary we prove the lipschitz stability provided that initial data are suitably chosen the proof is based on a carleman estimate which can be obtained by the decomposition of the lame system into the rotation and the divergence components