the exterior algebra λkrn and area minimization abstract the structure of the exterior algebra k r n is studied in low dimensions and consequences are drawn for k dimensional area minimizing surfaces in r n for a general form o 2 r 4 section 3 gives explicit formulas for the comass o and the face of the grassmannian exposed by o section 4 classifies the faces of the grassmannian of the 3 planes in r 6 and hence the associated geometries of area minimizing surfaces there are four types section 5 establishes an equality involving the comass norm in low dimension and draws implications on when the cartesian product of area minimizing surfaces is area minimizing new examples of area minimizing integral currents with singularities follow

on hamiltonian stationary lagrangian spheres in non einstein kahler surfaces hamiltonian stationary lagrangian spheres in kahler einstein surfaces are minimal we prove that in the family of non einstein kahler surfaces given by the product 1 2 of two complete orientable riemannian surfaces of different constant gauss curvatures there is only a non minimal hamiltonian stationary lagrangian sphere this example defined when the surfaces 1 and 2 are spheres is unstable

the structured sensitivity of vandermonde like systems we consider a general class of structured matrices that includes possibly confluent vandermonde and vandermonde like matrices here the entries in the matrix depend nonlinearly upon a vector of parameters we define condition numbers that measure the componentwise sensitivity of the associated primal and dual solutions to small componentwise perturbations in the parameters and in the right hand side convenient expressions are derived for the infinity norm based condition numbers and order of magnitude estimates are given for condition numbers defined in terms of a general vector norm we then discuss the computation of the corresponding backward errors after linearising the constraints we derive an exact expression for the infinity norm dual backward error and show that the corresponding primal backward error is given by the minimum infinity norm solution of an underdetermined linear system exact componentwise condition numbers are also derived for matrix inversion and the least squares problem and the linearised least squares backward error is characterised

inversion error condition number and approximate inverses of uncertain matrices abstract the classical condition number is a very rough measure of the effect of perturbations on the inverse of a square matrix first it assumes that the perturbation is infinitesimally small second it does not take into account the perturbation structure e g vandermonde similarly the classical notion of the inverse of a matrix neglects the possibility of large structured perturbations we define a new quantity the structured maximal inversion error that takes into account both structure and non necessarily small perturbation size when the perturbation is infinitesimal we obtain a structured condition number we introduce the notion of approximate inverse as a matrix that best approximates the inverse of a matrix with structured perturbations when the perturbation varies in a given range for a wide class of perturbation structures we show how to use convex semidefinite programming to compute bounds on the structured maximal inversion error and structured condition number and compute an approximate inverse the results are exact when the perturbation is unstructured we then obtain an analytic expression for the approximate inverse when the perturbation is unstructured and additive we recover the classical condition number the approximate inverse is the operator related to the total least squares orthogonal regression problem

an alternating least squares method for the weighted approximation of a symmetric matrix bailey and gower examined the least squares approximationc to a symmetric matrixb when the squared discrepancies for diagonal elements receive specific nonunit weights they focussed on mathematical properties of the optimalc in constrained and unconstrained cases rather than on how to obtainc for any givenb in the present paper a computational solution is given for the case wherec is constrained to be positive semidefinite and of a fixed rankr or less the solution is based on weakly constrained linear regression analysis

area minimizing integral currents with boundaries invariant under polar actions let g be a compact connected subgroup of so n acting on rn and let the action of g be polar polar actions include the adjoint action of a lie group h on the tangent space to the symmetric space g h at the identity coset let b be an m l dimensional submanifold without boundary invariant under the action of g and lying in the union of the principal orbits of g it is shown that if s is an area minimizing integral current with boundary b then s is invariant under the action of g this result is extended to a larger class of boundaries and to a class of parametric integrands including the area integrand

on the singular structure of three dimensional area minimizing surfaces a sufficient condition is given for the union of two three dimensional planes through the origin in r to be area minimizing the condition is in terms of the three angles 0 7 their singular sets have hausdorff dimension at most m 1 larger singular sets occur in higher codimensions for example complex analytic varieties which are automatically area minimizing provide real m dimensional surfaces with 7 2 dimensional singular sets recent work of almgren 1 seems to show that the dimension of a singular set never exceeds m 2 little is known about the structure of the singular set only for two dimensional area minimizing surfaces is even the first order structure of singularities understood 5 cf also 6 this paper studies the first order structure of singularities in three dimensional area minimizing surfaces at each point in the surface there is a tangent cone not known to be unique itself area minimizing which records first order behavior if a tangent cone is a plane with multiplicity one the surface is locally a smooth manifold since the next simplest cone is a sum i e union of planes we address the following basic question question when is the sum of two oriented 3 planes through the origin in r area minimizing the geometric relationship between two oriented 3 planes in r is characterized by three angles 0 y y2 w 2 y2 y3 s m y2 see lemma 1 and remarks we prove a sufficient condition in terms of these three angles theorem 7 y3 s y y2 the sum of the planes is area minimizing we conjecture that the converse holds as well except for the trivial case of the

extremal problems on the set of nonnegative definite matrices abstract an optimization problem of minimizing a real valued function of certain elements of a symmetric matrix subject to this matrix being nonnegative definite is considered optimality conditions are proposed the duality result of olkin and pukelsheim 1982 is extended to a wide class of such problems applications are discussed

on finiteness of the number of stable minimal hypersurfaces with a fixed boundary can there be infinitely many minimal hypersurfaces with a given boundary in a riemannian manifold a number of previous results positive and negative already indicated that the answer depends on the definition of surface on orientability on stability and minimizing properties of the surface on the smoothness and geometry of the boundary and on the ambient manifold 1 finiteness for area minimizing hypersurfaces in certain manifolds several years ago r hardt and l simon hs 1979 12 following the original work of f tomi t2 1974 proved that a c 4 a jordan curve t in r 3 bounds only finitely many orientable area minimizing immersed manifolds with boundary of least area the immersed manifolds are assumed to minimize area over all immersed manifolds of finite topological type it has been more or less understood that this finiteness theorem applies as well to hypersurfaces in r n n 7 bounded by a compact connected c submanifold b of r n such finiteness for oriented area minimizing hypersurfaces fails in general ambient manifolds for example in the standard 2 dimensional sphere there is a continuum of length minimizing paths from the south pole to the north pole however we prove the following finiteness theorem for a noncompact real analytic riemannian manifold theorem a fix n 7 0 a 1 let n be an n dimensional complete connected noncompact real analytic riemannian manifold with sectional curvature bounded above and injectivity radius bounded away from 0 let b be a compact oriented n 2 dimensional c 2 a submanifold of n with a positive integer multiplicity assigned to each component let m be the space of equivalence classes under reparameterization ofc immersions of compact orientable n 1 dimensional c manifolds with boundary m into n such that f dm is a covering of b with the prescribed multiplicities let a inf area m g al then mq m g m area a is finite when the ambient manifold n is compact it is shown that infinitely many area minimizing hypersurfaces with a common smooth boundary necessarily give an open book decomposition of n cf wi g it follows that if n cp 2 for example or if n is nonorientable there are only finitely many area minimizing hypersurfaces with a common smooth boundary received by the editors april 25 1985 1980 mathematics subject classificatian primary 53c42 49f22

graph sampling for matrix completion using recurrent gershgorin disc shift matrix completion algorithms fill missing entries in a large matrix given a subset of observed samples the problem of how to pre select a subset of entries for sampling to maximize the reconstructed matrix fidelity is largely unaddressed in this paper we propose two sampling algorithms to tackle this problem i a fast base sampling algorithm on general single graphs and ii an extended sampling algorithm from the base algorithm for active matrix completion assuming the double graph smoothness prior where rows columns of the matrix signal are smooth with respect to row column factor graphs we first show that a matrix signal can be recovered from partial observations by solving a system of linear equations mathbf q mathbf x mathbf y where the sought matrix is vectorized as mathbf x and interpreted as a signal residing on a single graph on this single graph to promote small reconstruction error and stability of the linear system we maximize the smallest eigenvalue lambda min of mathbf q by greedily selecting samples corresponding to the largest magnitude entries in the first eigenvectors of mathbf q based on a corollary of the gershgorin circle theorem extending to the active matrix completion case we alternately choose entries in the rows and columns corresponding to the largest magnitude entries in the first eigenvectors of the row column coefficient matrices mathbf q r and mathbf q c our algorithm benefits computationally from warm start as the first eigenvectors are computed recurrently for more samples extensive experiments on both synthetic and real world datasets show that our extended graph sampling algorithm outperforms existing sampling schemes for matrix completion when combined with a range of modern matrix completion algorithms

approximate minimization of weighted tree automata abstract this paper studies the following approximate minimization problem given a minimal weighted tree automaton a with n states recognizing a weighted tree language f can we construct a smaller automaton a with n n states recognizing a language f that is a good approximation of f the corresponding problem for weighted automata on strings was recently studied by balle et al 16 15 where the authors introduced a new canonical form for weighted automata called singular value automata inspired by spectral methods and showed that truncating such canonical form yields a solution for the problem satisfying a certain approximation criteria in this paper we take a similar approach and show that in the tree case one can obtain an analogous canonical form that we call singular value tree automata and use it to study the approximate minimization problem for weighted tree automata we first establish the existence of this canonical form for weighted tree automata and then provide bounds on the quality of the resulting approximation method based on truncation we also study the problem of computing the canonical form given a minimal weighted tree automaton and show that in the tree case this task is considerably more complicated than in the string case in particular computing the canonical form reduces to solving a system of polynomial equations by further reducing this problem to the computation of generalized partition functions for weighted tree automata we propose and analyze two methods for computing the canonical form based on iterative methods fixed point iteration and newton s method our analysis of newton s method unveils a connection between iterative methods and sequences of sets of trees satisfying a certain technical condition that might be of independent interest

the angle criterion we complete the proof of the angle conjecture on when a pair of orientedm planes is area minimizing the nonzero sum oriented union p1 p2 is area minimizing if and only if the characterizing angles between them satisfy the inequality beta m leqq beta 1 beta m 1

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

minimal hypersurfaces of least area in this paper we study closed embedded minimal hypersurfaces in a riemannian n 1 manifold 2 n 6 that minimize area among such hypersurfaces we show they exist and arise either by minimization techniques or by min max methods they have index at most 1 we apply this to obtain a lower area bound for such minimal surfaces in some hyperbolic 3 manifolds

complete embedded minimal n dimensional submanifolds in ℂn a classical problem in the theory of minimal submanifolds of euclidean spaces is to understand whether a minimal submanifold exists with a prescribed behavior at infinity or to determine from the asymptotes the geometry of the whole submanifold beyond the intrinsic interest of these questions they are also of crucial importance when studying the possible singularities of minimal submanifolds in general riemannian manifolds for minimal surfaces i e two dimensional submanifolds the standard tool to solve these problems is given by the weierstrass representation formula which relates the geometry of the minimal surface to complex analytic properties of holomorphic 1 forms on riemann surfaces recently gluing techniques have been developed that provide an abundant number of new examples of minimal hypersurfaces in euclidean space for higher dimensional minimal submanifolds such a link clearly disappears and no complex analysis can be put into play although gluing techniques have been extensively used in the study of minimal hypersurfaces they have not been adapted to handle higher codimensional submanifolds the aim of this paper is to use a gluing technique for minimal submanifolds to make a step toward understanding these questions in higher codimension more precisely we will restrict ourselves to the case of real n dimensional submanifolds of cn there are two main reasons for doing so the first one comes from the fact that when trying to desingularize the intersection for example of a pair of n planes that give the desired asymptotic behavior one needs a model of a minimal submanifold with this behavior at infinity to rescale and to glue into the pair of planes where a neighborhood of the intersection is removed this local model needs to be sufficiently simple to allow a detailed study of the linearized mean curvature operator in our situation this is provided by a generalization of an area minimizing submanifold found by lawlor in 12 while in more general cases such an example is not known the second reason is that among minimal n submanifolds ofcn there is a

min max theory for free boundary minimal hypersurfaces ii general morse index bounds and applications for any smooth riemannian metric on an n 1 dimensional compact manifold with boundary m partial m where 3 le n 1 le 7 we establish general upper bounds for the morse index of free boundary minimal hypersurfaces produced by min max theory in the almgren pitts setting we apply our morse index estimates to prove that for almost every in the c infty baire sense riemannan metric the union of all compact properly embedded free boundary minimal hypersurfaces is dense in m if partial m is further assumed to have a strictly mean convex point we show the existence of infinitely many compact properly embedded free boundary minimal hypersurfaces whose boundaries are non empty our results prove a conjecture of yau for generic metrics in the free boundary setting

models and methods for the analysis of correlation coefficients abstract the normal distribution plays a very prominant role in statistics and in multivariate analysis the multinormal model is even more dominant than in the classical univariate or bivariate situations in this paper we show that the assumption of continuous multinormal observations can be relaxed in various ways for continuous multivariate analysis most of the existing large sample theory can be applied without assuming multivariate normality we also develop several discrete multinormal models the first one is the classical pearson model which results from discreticizing multinormal variables the second one is inspired by correspondence analysis and the third one by loglinear analysis the models are presented in terms of correlation coefficients and applied to the spearman model of factor analyses using an example from political science

a regularity theorem for minimizing hypersurfaces modulo it is proved that an n 1 dimensional area minimizing flat chain modulo v in rw with smooth extremal boundary of at most v 2 components has an interior singular set of hausdorff dimension at most n 8 similar results hold for more general integrands

a min max theorem for complex symmetric matrices abstract we optimize the form re x t tx to obtain the singular values of a complex symmetric matrix t we prove that for 0 k n 2 min codim v k max x v x 1 re x t tx 2 k 1 where t is an n n complex symmetric matrix having singular values 1 n we also show that the singular values missed in this theorem i e 2 4 are obtained by a similar optimization over real subspaces

indefinite trust region subproblems and nonsymmetric eigenvalue perturbations this paper extends the theory of trust region subproblems in two ways i it allows indefinite inner products in the quadratic constraint and ii it uses a two sided upper and lower bound quadratic constraint characterizations of optimality are presented that have no gap between necessity and sufficiency conditions for the existence of solutions are given in terms of the definiteness of a matrix pencil a simple dual program is introduced that involves the maximization of a strictly concave function on an interval this dual program simplifies the theory and algorithms for trust region subproblems it also illustrates that the trust region subproblems are implicit convex programming problems and thus explains why they are so tractable the duality theory also provides connections to eigenvalue perturbation theory trust region subproblems with zero linear term in the objective function correspond to eigenvalue problems and adding a linear term in the objective function is seen to correspond to a p

size minimizing rectifiable currents 1 introduct ion 333 2 on the existence and regularity of size minimizers 336 3 isoperimetric inequalities and size bounds 345 4 bernstein theorems 346 references 348

least squares approximation of an improper correlation matrix by a proper one an algorithm is presented for the best least squares fitting correlation matrix approximating a given missing value or improper correlation matrix the proposed algorithm is based upon a solution for mosier s oblique procrustes rotation problem offered by ten berge and nevels a necessary and sufficient condition is given for a solution to yield the unique global minimum of the least squares function empirical verification of the condition indicates that the occurrence of non optimal solutions with the proposed algorithm is very unlikely a possible drawback of the optimal solution is that it is a singular matrix of necessity in cases where singularity is undesirable one may impose the additional nonsingularity constraint that the smallest eigenvalue of the solution be where is an arbitrary small positive constant finally it may be desirable to weight the squared errors of estimation differentially a generalized solution is derived which satisfies the additional nonsingularity constraint and also allows for weighting the generalized solution can readily be obtained from the standard unweighted singular solution by transforming the observed improper correlation matrix in a suitable way

random walk graph laplacian based smoothness prior for soft decoding of jpeg images given the prevalence of joint photographic experts group jpeg compressed images optimizing image reconstruction from the compressed format remains an important problem instead of simply reconstructing a pixel block from the centers of indexed discrete cosine transform dct coefficient quantization bins hard decoding soft decoding reconstructs a block by selecting appropriate coefficient values within the indexed bins with the help of signal priors the challenge thus lies in how to define suitable priors and apply them effectively in this paper we combine three image priors laplacian prior for dct coefficients sparsity prior and graph signal smoothness prior for image patches to construct an efficient jpeg soft decoding algorithm specifically we first use the laplacian prior to compute a minimum mean square error initial solution for each code block next we show that while the sparsity prior can reduce block artifacts limiting the size of the overcomplete dictionary to lower computation would lead to poor recovery of high dct frequencies to alleviate this problem we design a new graph signal smoothness prior desired signal has mainly low graph frequencies based on the left eigenvectors of the random walk graph laplacian matrix lerag compared with the previous graph signal smoothness priors lerag has desirable image filtering properties with low computation overhead we demonstrate how lerag can facilitate recovery of high dct frequencies of a piecewise smooth signal via an interpretation of low graph frequency components as relaxed solutions to normalized cut in spectral clustering finally we construct a soft decoding algorithm using the three signal priors with appropriate prior weights experimental results show that our proposal outperforms the state of the art soft decoding algorithms in both objective and subjective evaluations noticeably

calibrations modulo v abstract the theory of calibrations is extended to show that certain surfaces are area minimizing modulo v for example a complex algebraic variety in c n of degree d is area minimizing modulo v for all v 2 d

embeddedness of minimal surfaces with total boundary curvature at most 4 pi this paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 47r must be smoothly embedded related results are proved for varifolds and for soap film surfaces

fast gaussian elimination with partial pivoting for matrices with displacement structure fast o n 2 implementation of gaussian elimination with partial pivoting is designed for matrices possessing cauchy like displacement structure we show how toeplitz like toeplitz plus hankel like and vandermonde like matrices can be transformed into cauchy like matrices by using discrete fourier cosine or sine transform matrices in particular this allows us to propose a new fast o n 2 toeplitz solver gko which incorporates partial pivoting a large set of numerical examples showed that gko demonstrated stable numerical behavior and can be recommended for solving linear systems especially with nonsymmetric indefinite and ill conditioned positive definite toeplitz matrices it is also useful for block toeplitz and mosaic toeplitz toeplitz block matrices the algorithms proposed in this paper suggest an attractive alternative to look ahead approaches where one has to jump over ill conditioned leading submatrices which in the worst case requires o n 3 operations

the unit mass ball of three vectors in r6 abstract we study the unit ball in 3 r 6 we give a test for distinguishing the types of mass decompositions establish an algorithm for finding mass decompositions and provide computational formulae for the mass norm

systems of linear interval equations this paper presents theory and methods for computing the exact bounds on the solution of a system of n linear equations in n variables whose coefficients and right hand sides vary in some real intervals finite and iterative methods are given based on results from linear complementarity theory also regularity conditions regularity testing and computing the exact inverse of an interval matrix are dealt with

testing for inequality constraints in singular models by trimming or winsorizing the variance matrix there are many applications in which a statistic follows at least asymptotically a normal distribution with a singular or nearly singular variance matrix a classic example occurs in linear regre

examples of unoriented area minimizing surfaces a comprehensive study is made of constructions of area minimizing flat chains modulo two many have singularities for instance any bounded area mini mizing submanifold of rn occurs as the singular set of some area minimizing flat chain modulo two in some rn introduction given a k l dimensional boundary in r one can seek to minimize area in some class of surfaces we want to take a large class that includes nonorientable as well as orientable surfaces without restrictions on the singularities or topological type such a class appears in geometric measure theory as the flat chains modulo two the existence and regularity of area minimizing flat chains modulo two are already well established cf 7 4 2 26 9 unfortunately examples have been rare this paper presents several methods for constructing area minimizing flat chains modulo two the first five theorems include the following five examples 1 a mobius strip 2 the graph over a convex region of any function satisfying the minimal surface equation

the constrained total least squares technique and its applications to harmonic superresolution the constrained total least squares ctls method is a natural extension of tls to the case when the noise components of the coefficients are algebraically related the ctls technique is developed and some of its applications to superresolution harmonic analysis are presented the ctls problem is reduced to an unconstrained minimization problem over a small set of variables a perturbation analysis of the ctls solution is derived and from it the root mean square error rmse of the ctls solution which is valid for small noise levels is obtained in closed form the complex version of the newton method is derived and applied to determine the ctls solution it is also shown that the ctls problem is equivalent to a constrained parameter maximum likelihood problem the ctls technique is applied to estimate the frequencies of sinusoids in white noise and the angle of arrival of narrowband wavefronts at a linear uniform array in both cases the ctls method shows superior or similar accuracy to other advanced techniques

existence of infinitely many minimal hypersurfaces in positive ricci curvature in the early 1980s s t yau conjectured that any compact riemannian three manifold admits an infinite number of closed immersed minimal surfaces we use min max theory for the area functional to prove this conjecture in the positive ricci curvature setting more precisely we show that every compact riemannian manifold with positive ricci curvature and dimension at most seven contains infinitely many smooth closed embedded minimal hypersurfaces in the last section we mention some open problems related with the geometry of these minimal hypersurfaces

point cloud denoising via feature graph laplacian regularization point cloud is a collection of 3d coordinates that are discrete geometric samples of an object s 2d surfaces imperfection in the acquisition process means that point clouds are often corrupted with noise building on recent advances in graph signal processing we design local algorithms for 3d point cloud denoising specifically we design a signal dependent feature graph laplacian regularizer sdfglr that assumes surface normals computed from point coordinates are piecewise smooth with respect to a signal dependent graph laplacian matrix using sdfglr as a signal prior we formulate an optimization problem with a general ell p norm fidelity term that can explicitly remove only two types of additive noise small but non sparse noise like gaussian using ell 2 fidelity term and large but sparser noise like laplacian using ell 1 fidelity term to establish a linear relationship between normals and 3d point coordinates we first perform bipartite graph approximation to divide the point cloud into two disjoint node sets red and blue we then optimize the red and blue nodes coordinates alternately for ell 2 norm fidelity term we iteratively solve an unconstrained quadratic programming qp problem efficiently computed using conjugate gradient with a bounded condition number to ensure numerical stability for ell 1 norm fidelity term we iteratively minimize an ell 1 ell 2 cost function using accelerated proximal gradient apg where a good step size is chosen via lipschitz continuity analysis finally we propose simple mean and median filters for flat patches of a given point cloud to estimate the noise variance given the noise type which in turn is used to compute a weight parameter trading off the fidelity term and signal prior in the problem formulation extensive experiments show state of the art denoising performance among local methods using our proposed algorithms

fast graph sampling set selection using gershgorin disc alignment graph sampling set selection where a subset of nodes are chosen to collect samples to reconstruct a smooth graph signal is a fundamental problem in graph signal processing gsp previous works employ an unbiased least squares ls signal reconstruction scheme and select samples via expensive extreme eigenvector computation instead we assume a biased graph laplacian regularization glr based scheme that solves a system of linear equations for reconstruction we then choose samples to minimize the condition number of the coefficient matrix specifically maximize the smallest eigenvalue lambda min circumventing explicit eigenvalue computation we maximize instead the lower bound of lambda min designated by the smallest left end of all gershgorin discs of the matrix to achieve this efficiently we first convert the optimization to a dual problem where we minimize the number of samples needed to align all gershgorin disc left ends at a chosen lower bound target t algebraically the dual problem amounts to optimizing two disc operations i shifting of disc centers due to sampling and ii scaling of disc radii due to a similarity transformation of the matrix we further reinterpret the dual as an intuitive disc coverage problem bearing strong resemblance to the famous np hard set cover sc problem the reinterpretation enables us to derive a fast approximation scheme from a known sc error bounded approximation algorithm we find an appropriate target t efficiently via binary search extensive simulation experiments show that our disc based sampling algorithm runs substantially faster than existing sampling schemes and outperforms other eigen decomposition free sampling schemes in reconstruction error

nonsingularity under data rounding abstract for a square floating point matrix a we give a test for the existence of a singular real matrix which rounds to a

boundary rigidity and filling volume minimality of metrics close to a flat one we say that a riemannian manifold m g with a non empty boundary m is a minimal orientable filling if for every compact orientable m g with m m the inequality d g x y d g x y for all x y m implies vol m g vol m g we show that if a metric g on a region m c r n with a connected boundary is sufficiently c 2 close to a euclidean one then it is a minimal filling by studying the equality case vol m g vol m g we show that if d g x y d g x y for all x y m then m g is isometric to m g this gives the first known open class of boundary rigid manifolds in dimensions higher than two and makes a step towards a proof of michel s conjecture

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

bipartite approximation for graph wavelet signal decomposition to compactly represent a graph signal in the frequency domain critically sampled biorthogonal wavelet filterbanks have been proposed to decompose signals on bipartite graphs however in practice graph signals often reside on general graph structures that are not bipartite thus an original nonbipartite graph must be decomposed into a sequence of bipartite graph approximations so that the filterbanks can be applied successively for signal decomposition in this paper unlike previous proposals that are heuristic in nature we design new bipartite approximation strategies for model based and empirically derived probability distributions in the first case a signal prior assumes a gaussian markov random field gmrf model parametrized by the original graph in the second case beyond the gmrf signal prior empirical signal observations are available to compute a posterior probability distribution in both cases we optimize for energy compaction in the bipartite subgraphs with two criteria the kullback leibler divergence metric which encourages preserving the spectral characteristics of the original graph and multiplicity of eigenvalue at frequency 1 for graph laplacian which is the frequency with minimal energy discrimination the relative importance between the two criteria is determined by a proposed measure that evaluates the degree of mismatch between the signal prior and posterior given these criteria we first design a global numerical optimization and then propose a local heuristic approach for fast implementation with simplifications leading to local metric computation experimental results show that our proposed bipartite subgraph decomposition outperforms competing proposals in terms of energy compaction

overestimations in bounding solutions of perturbed linear equations abstract it is proved that some classical bounds on solutions of perturbed systems of linear equations may yield arbitrarily large overestimations for arbitrarily narrow perturbations the proofs are constructive

image denoising by sparse 3 d transform domain collaborative filtering we propose a novel image denoising strategy based on an enhanced sparse representation in transform domain the enhancement of the sparsity is achieved by grouping similar 2d image fragments e g blocks into 3d data arrays which we call groups collaborative altering is a special procedure developed to deal with these 3d groups we realize it using the three successive steps 3d transformation of a group shrinkage of the transform spectrum and inverse 3d transformation the result is a 3d estimate that consists of the jointly filtered grouped image blocks by attenuating the noise the collaborative filtering reveals even the finest details shared by grouped blocks and at the same time it preserves the essential unique features of each individual block the filtered blocks are then returned to their original positions because these blocks are overlapping for each pixel we obtain many different estimates which need to be combined aggregation is a particular averaging procedure which is exploited to take advantage of this redundancy a significant improvement is obtained by a specially developed collaborative wiener filtering an algorithm based on this novel denoising strategy and its efficient implementation are presented in full detail an extension to color image denoising is also developed the experimental results demonstrate that this computationally scalable algorithm achieves state of the art denoising performance in terms of both peak signal to noise ratio and subjective visual quality

self similar solutions and translating solitons for lagrangian mean curvature flow we construct many self similar and translating solitons for lagrangian mean curvature flow including self expanders and translating solitons with arbitrarily small oscillation on the lagrangian angle our translating solitons play the same role as cigar solitons in ricci flow and are important in studying the regularity of lagrangian mean curvature flow given two transverse lagrangian planes wn in cn with sum of characteristic angles less than a we show there exists a lagrangian self expander asymptotic to this pair of planes the maslov class of these self expanders is zero thus they can serve as local models for surgeries on lagrangian mean curvature flow families of self shrinkers and self expanders with different topologies are also constructed this paper generalizes the work of anciaux i joyce 12 lawlor 15 and lee and wang 17 18

positive definiteness via off diagonal scaling of a symmetric indefinite matrix indefinite symmetric matrices that are estimates of positive definite population matrices occur in a variety of contexts such as correlation matrices computed from pairwise present missing data and multinormal based theory for discretized variables this note describes a methodology for scaling selected off diagonal rows and columns of such a matrix to achieve positive definiteness as a contrast to recently developed ridge procedures the proposed method does not need variables to contain measurement errors when minimum trace factor analysis is used to implement the theory only correlations that are associated with heywood cases are shrunk

special lagrangian cones with higher genus links for every odd natural number g 2d 1 we prove the existence of a countably infinite family of special lagrangian cones in mathbb c 3 over a closed riemann surface of genus g using a geometric pde gluing method

robust estimation of dispersion matrices and principal components abstract this paper uses monte carlo methods to compare the performances of several robust procedures for estimating a correlation matrix and its principal components the estimators are formed either from separate bivariate analyses or by simultaneous manipulation of all variables by using techniques such as multivariate trimming and m estimation the m estimators stand up exceptionally well they and the multivariate trimming procedure are especially effective at estimating the principal components including a near singularity however the m estimators can break down relatively easily when the dimensionality is large and the outliers are asymmetric with missing data the element wise approach becomes more attractive

a comprehensive review of stability analysis of continuous time recurrent neural networks stability problems of continuous time recurrent neural networks have been extensively studied and many papers have been published in the literature the purpose of this paper is to provide a comprehensive review of the research on stability of continuous time recurrent neural networks including hopfield neural networks cohen grossberg neural networks and related models since time delay is inevitable in practice stability results of recurrent neural networks with different classes of time delays are reviewed in detail for the case of delay dependent stability the results on how to deal with the constant variable delay in recurrent neural networks are summarized the relationship among stability results in different forms such as algebraic inequality forms m matrix forms linear matrix inequality forms and lyapunov diagonal stability forms is discussed and compared some necessary and sufficient stability conditions for recurrent neural networks without time delays are also discussed concluding remarks and future directions of stability analysis of recurrent neural networks are given

special lagrangian submanifolds with isolated conical singularities v survey and applications this is the last in a series of five papers studying compact special lagrangian submanifolds sl m folds x in almost calabi yau m folds m with singularities x1 x n locally modelled on special lagrangian cones c1 c n in c m with isolated singularities at 0 readers are advised to begin with this paper we survey the major results of the previous four papers giving brief explanations of the proofs we apply the results to describe the boundary of a moduli space of compact nonsingular sl m folds n in m we prove the existence of special lagrangian connected sums n1 nk of sl mfolds n1 n k in m we also study sl 3 folds with t 2 cone singularities proving results related to ideas of the author on invariants of calabi yau 3 folds and the syz conjecture let x be a compact sl m fold with isolated conical singularities xi and cones ci for i 1 n the first paper studied the regularity of x near its singular points and the the second the moduli space of deformations of x the third and fourth papers construct desingularizations of x realizing x as a limit of a family of compact nonsingular sl m folds n t in m for small t 0 let li be an asymptotically conical sl m fold in cm asymptotic to ci at infinity we make n t by gluing tli into x at xi for i 1 n

an interior point method for generalized linear fractional programming we develop an interior point polynomial time algorithm for a generalized linear fractional problem the latter problem can be regarded as a nonpolyhedral extension of the usual linear fractional programming typical example which is of interest for control theory is the minimization of the generalized eigenvalue of a pair of symmetric matrices linearly depending on the decision variables

selected topics in robust convex optimization robust optimization is a rapidly developing methodology for handling optimization problems affected by non stochastic uncertain but bounded data perturbations in this paper we overview several selected topics in this popular area specifically 1 recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data 2 tractability of robust counterparts 3 links between ro and traditional chance constrained settings of problems with stochastic data and 4 a novel generic application of the ro methodology in robust linear control

minimal submanifolds of ⁿ arising from degenerate 3 orbits on the grassmannian we give new examples of minimal submanifolds of e 2n characterised by having their gauss map s image lie in degenerate s0 3 orbits of gp2n 1 the grassmannian of p planes in e where the action on 2n 1 n gp 2n 1 is induced from the irreducible s0 3 action on r2n these submanifolds are all given explicitly in terms of holomorphic data and are linearly 2n l1 full in e

constructing special lagrangian m folds in c m by evolving quadrics this is the second in a series of papers constructing explicit examples of special lagrangian submanifolds sl m folds in c the first paper of the series 7 studied sl m folds with large symmetry groups and subsequent papers 8 9 10 11 construct examples of sl 3 folds in c using evolution equations symmetries ruled submanifolds and integrable systems the principal motivation for these papers is to lay the foundations for a study of the singularities of compact special lagrangian m folds in calabi yau m folds particularly in low dimensions such as m 3 special lagrangian m folds in c and especially special lagrangian cones should provide local models for singularities of sl m folds in calabi yau m folds understanding such singularities will be essential in making rigorous the explanation of mirror symmetry of calabi yau 3 folds x x proposed by strominger yau and zaslow 13 which involves dual fibrations of x x by special lagrangian 3 tori with some singular fibres it will also be important in resolving conjectures made by the author 6 which attempt to define an invariant of calabi yau 3 folds by counting special lagrangian homology 3 spheres the paper falls into three parts the first this section and 2 is introductory the second part 3 4 describes a general construction of special lagrangianm folds n in c depending on a set of evolution data p where p is an m 1 submanifold in r then n is the subset of c swept out by the image of p under a 1 parameter family of linear or affine maps t r n c which satisfy a first order nonlinear o d e in t examples of sets of evolution data will be given in 4 together with some progress towards a classification of such data the simplest interesting sets of evolution data occur when n m and p is a nondegenerate quadric in r in the third part 5 7 we apply the construction to these examples in this case t r must be a lagrangian plane in c for each t thus n is fibred by quadrics in lagrangian planes r in c the construction of 3 4 will also be used in the sequel to this paper 8 with different evolution data to construct families of sl 3 folds in c the construction has both a linear and an affine version in the linear version