compressive sensing via nonlocal low rank regularization sparsity has been widely exploited for exact reconstruction of a signal from a small number of random measurements recent advances have suggested that structured or group sparsity often leads to more powerful signal reconstruction techniques in various compressed sensing cs studies in this paper we propose a nonlocal low rank regularization nlr approach toward exploiting structured sparsity and explore its application into cs of both photographic and mri images we also propose the use of a nonconvex log det x as a smooth surrogate function for the rank instead of the convex nuclear norm and justify the benefit of such a strategy using extensive experiments to further improve the computational efficiency of the proposed algorithm we have developed a fast implementation using the alternative direction multiplier method technique experimental results have shown that the proposed nlr cs algorithm can significantly outperform existing state of the art cs techniques for image recovery

universal teichmuller space and diff s 1 s 1 we point out that the coset space diffs1 s1 is a dense complex submanifold of the universal teichmuller spaces of compact riemann spaces of genus g 1 a holomorphic map ofs into the inifinite dimensional segal diskd1 is constructed this is the universal analogue of the map of teichmuller spaces into the siegel disk provided by the period matrix the kahler potential for the general homogenous metric on diffs1 s1 is computed explicitly using the map intod1 some applications to string theory are discussed

a one parameter family of quadratically convergent iteration formulae a new one parameter family of iteration methods for solving nonlinear equations is presented the convergence analysis shows that their order of convergence is two numerical examples are given to illustrate the efficiency and performance of the proposed methods

third order iterative methods free from second derivatives for nonlinear equation in this paper we suggest and analyze a new two step iterative method for solving nonlinear equations which is called the modified householder method without second derivatives for nonlinear equation we also prove that the modified method has cubic convergence several examples are given to illustrate the efficiency and the performance of the new method new method can be considered as an alternative to the present cubic convergent methods for solving nonlinear equations

geometric quantization of n 2 superstring a nonperturbative geometric formulation of the n 2 neveu schwarz superstring theory which has been recently interpreted by h ooguri and c vafa mod phys lett a5 1389 1990 as a consistent quantum theory of self dual gravity in four dimensions is constructed it is shown that the natural complex structure over the loop superspace md d associated to the n 2 neveu schwarz fermionic string is invariant under symmetry group osp 2 2 superdiffs1 2 moreover it is proved that there is a unique lorentz and osp 2 2 invariant complex structure on md d this result implies that the superspace of all admissible complex structures over md d is isomorphic to the homogeneous kahler supermanifold superdiffs1 2 osp 2 2 the ricci curvature of superdiffs1 2 osp 2 2 is calculated applying the method of geometric quantization to the n 2 neveu schwarz superstring theory along the lines suggested by m j bowick and s g rajeev nucl phys b361 469 1991 a representation is constructed of nonperturbativ

a note on some three step iterative methods for nonlinear equations abstract in a recent paper m a noor k i noor three step iterative methods for nonlinear equations applied mathematics and computation 183 2006 322 327 a three step iterative method with third order convergence for solving nonlinear equations has been presented in this paper we show that the iterative method given by the authors has quadratic convergence not the third order one suggested in their work

on a class of quadratic convergence iteration formulae without derivatives a class of iterative formulae without derivatives are proposed in this paper these methods are shown to be quadratic convergence some numerical tests are given

a period mapping in universal teichmuller space in previous work it had been shown that the remarkable homogeneous space m diff s 1 psl 2 r sits as a complex analytic and kahler submanifold of the universal teichmuller space there is a natural immersion of m into the infinite dimensional version due to segal of the siegel space of period matrices that map is proved to be injective equivariant holomorphic and kahler isometric with respect to the canonical metrics regarding a period mapping as a map describing the variation of complex structure we explain why is an infinite dimensional period mapping

nonconvex splitting for regularized low rank sparse decomposition we develop new nonconvex approaches for matrix optimization problems involving sparsity the heart of the methods is a new nonconvex penalty function that is designed for efficient minimization by means of a generalized shrinkage operation we apply this approach to the decomposition of video into low rank and sparse components which is able to separate moving objects from the stationary background better than in the convex case in the case of noisy data we add a nonconvex regularization and apply a splitting approach to decompose the optimization problem into simple parallelizable components the nonconvex regularization ameliorates contrast loss thereby allowing stronger denoising without losing more signal to the residual

modified householder iterative method free from second derivatives for nonlinear equations abstract in this paper we suggest and analyze a new two step predictor corrector type iterative method free from second derivatives for solving nonlinear equations of the type f x 0 this new method includes the two step newton method as a special case we prove that the new iterative method is of fourth order several examples are given to illustrate the efficiency of this new method and its comparison with other iterative methods this method can be considered as a significant improvement of the newton method and its variant forms

third order and fourth order iterative methods for finding multiple and distinct zeros of non linear equations in this paper we point out and analyse six two step iterative methods for finding multiple as well as distinct zeros of non linear equations we prove that the methods for multiple zeros have third order convergence where as the methods for distinct zeros have fourth order convergence the methods calculate the multiple as well as distinct zeros with high accuracy the numerical tests show their better performance in case of algebraic as well as non algebraic equations

robust recovery of corrupted low rankmatrix by implicit regularizers low rank matrix recovery algorithms aim to recover a corrupted low rank matrix with sparse errors however corrupted errors may not be sparse in real world problems and the relationship between l1 regularizer on noise and robust m estimators is still unknown this paper proposes a general robust framework for low rank matrix recovery via implicit regularizers of robust m estimators which are derived from convex conjugacy and can be used to model arbitrarily corrupted errors based on the additive form of half quadratic optimization proximity operators of implicit regularizers are developed such that both low rank structure and corrupted errors can be alternately recovered in particular the dual relationship between the absolute function in l1 regularizer and huber m estimator is studied which establishes a connection between robust low rank matrix recovery methods and m estimators based robust principal component analysis methods extensive experiments on synthetic and real world data sets corroborate our claims and verify the robustness of the proposed framework

low rank structure learning via nonconvex heuristic recovery in this paper we propose a nonconvex framework to learn the essential low rank structure from corrupted data different from traditional approaches which directly utilizes convex norms to measure the sparseness our method introduces more reasonable nonconvex measurements to enhance the sparsity in both the intrinsic low rank structure and the sparse corruptions we will respectively introduce how to combine the widely used lp norm 0 p 1 and log sum term into the framework of low rank structure learning although the proposed optimization is no longer convex it still can be effectively solved by a majorization minimization mm type algorithm with which the nonconvex objective function is iteratively replaced by its convex surrogate and the nonconvex problem finally falls into the general framework of reweighed approaches we prove that the mm type algorithm can converge to a stationary point after successive iterations the proposed model is applied to solve two typical problems robust principal component analysis and low rank representation experimental results on low rank structure learning demonstrate that our nonconvex heuristic methods especially the log sum heuristic recovery algorithm generally perform much better than the convex norm based method 0 p 1 for both data with higher rank and with denser corruptions

the ricci curvature of diff s1 sl 2 r previous calculations of the ricci curvature for the manifold diff diff s1 s1 are extended to diff s1 sl 2r these manifolds are distinguished by being coadjoint orbits of diff s1 which admit compatible symphectic and complex structures making them kahler manifolds

a fourth order iterative method for solving nonlinear equations the present paper illustrates an iterative numerical method to solve nonlinear equations of the form f x 0 especially those containing the partial and non partial involvement of transcendental terms comparative analysis shows that the present method is faster than newton raphson method hybrid iteration method new hybrid iteration method and others cost is also found to be minimum than these methods the beauty in our method can be seen because of the optimization in important effecting factors i e lesser number of iteration steps lesser number of functional evaluations and lesser value of absolute error in final as well as in individual step as compared to the other methods this work also demonstrates the higher order convergence of the present method as compared to others without going to the computation of second derivative

an improved regula falsi method for finding simple roots of nonlinear equations abstract in this note an improved regula falsi irf method based on the classical regula falsi rf method is proposed the method is verified on a number of test examples and numerical results the proposed method has faster convergence rate with respect to the classical regula falsi method

a third order newton type method for nonlinear equations based on modified homotopy perturbation method in this paper we present a new iterative method for solving nonlinear algebric equations by using modified homotopy perturbation method we also discuss the convergence criteria of the present method to assess its validity and accuracy the method is applied to solve several test problems

three step iterative methods for nonlinear equations in this paper we suggest and analyze a new three step iterative method for solving nonlinear equations we show that this new iterative method has third order convergence several numerical examples are given to illustrate the efficiency and performance of the new method

the concave convex procedure the concave convex procedure cccp is a way to construct discrete time iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically this procedure can be applied to almost any optimization problem and many existing algorithms can be interpreted in terms of it in particular we prove that all expectation maximization algorithms and classes of legendre minimization and variational bounding algorithms can be reexpressed in terms of cccp we show that many existing neural network and mean field theory algorithms are also examples of cccp the generalized iterative scaling algorithm and sinkhorn s algorithm can also be expressed as cccp by changing variables cccp can be used both as a new way to understand and prove the convergence of existing optimization algorithms and as a procedure for generating new algorithms

projections of gibbs measures may be non gibbsian consider the phase of the two dimensional nearest neighbor ferromagnetic ising model at a temperature belowt c let be the restriction of this measure to a coordinate axis we prove that there is no one dimensional translation invariant summable interaction for which is a gibbs measure this is proven by showing that if such an interaction existed would have large deviation properties different from those it actually has percolation methods are used in the proof

numerical comparison of iterative methods for solving nonlinear equations abstract in this paper we suggest some new predictor corrector type iterative method for solving nonlinear equations by combining the well known regula falsi and newton method numerical experiments show that new predictor corrector type methods perform better than the previously known methods

third order methods from quadrature formulae for solving systems of nonlinear equations we extend to p dimensional problems a modification of the newton method based on quadrature formulas of order at least one which produces iterative methods with order of convergence three a general error analysis providing the higher order of convergence is given these new methods may be more efficient then other third order methods as they do not require the use of the second order frechet derivative

newton like iterative methods for solving system of non linear equations abstract homotopy perturbation method hpm is applied to construct a new family of newton like iterative methods for solving system of non linear equations comparison of the result obtained by the present method with newton raphson method reveals that the accuracy and fast convergence of the new method

robust pca via ell _ 0 ell _ 1 regularization we study the problem of low rank and sparse decomposition from possibly noisy observations known as robust pca where the sparse component can be seen as outliers we first propose a modified objective function where the nuclear norm captures the low rank term ell 0 norm addresses the sparse outlier term and an ell 1 norm to deal with the additive noise term the associated algorithm termed sparsity regularized principal component pursuit srpcp is shown to converge under certain model and algorithm parameter settings it is shown that srpcp can recover the low rank component and sparse component exactly in the noiseless case in the noisy case we first prove that the widely used principal component pursuit pcp method which was designed for the noiseless case is actually stable to dense noise then we show that srpcp has smaller estimation error bound and can identify outlier entries without any false alarm another important byproduct of our analysis is the result that pcp with missing entries is also stable to dense noise we further propose another objective function which replaces the above nuclear norm by the log determinant the proposed algorithm termed iterative reweighted sparsity regularized principal component pursuit is also shown to converge in each iteration it solves a weighted nuclear norm regularized robust matrix completion problem we propose an alternating direction method of multipliers algorithm to solve this nonconvex subproblem which also converges empirical studies demonstrate the efficacy of the proposed ell 0 ell 1 regularization framework to deal with the outliers as well as its advantage over the existing state of the art methods

robust matrix decomposition with sparse corruptions suppose a given observation matrix can be decomposed as the sum of a low rank matrix and a sparse matrix and the goal is to recover these individual components from the observed sum such additive decompositions have applications in a variety of numerical problems including system identification latent variable graphical modeling and principal components analysis we study conditions under which recovering such a decomposition is possible via a combination of l1 norm and trace norm minimization we are specifically interested in the question of how many sparse corruptions are allowed so that convex programming can still achieve accurate recovery and we obtain stronger recovery guarantees than previous studies moreover we do not assume that the spatial pattern of corruptions is random which stands in contrast to related analyses under such assumptions via matrix completion

coadjoint orbits of the virasoro group the coadjoint orbits of the virasoro group which have been investigated by lazutkin and pankratova and by segal should according to the kirillov kostant theory be related to the unitary representations of the virasoro group in this paper the classification of orbits is reconsidered with an explicit description of the possible centralizers of coadjoint orbits the possible orbits are diff s1 itself diff s1 s1 and diff s1 sl n 2 r withsl n 2 r a certain discrete series of embeddings ofsl 2 r in diff s1 and diffs1 t wheret may be any of certain rather special one parameter subgroups of diffs1 an attempt is made to clarify the relation between orbits and representations it appears that quantization of diffs1 s1 is related to unitary representations with nondegenerate kac determinant unitary verma modules while quantization of diffs1 sl n 2 r is seemingly related to unitary representations with null vectors in leveln a better understanding of how to quantize the relevant orbits might lead to a better geometrical understanding of virasoro representation theory in the process of investigating virasoro coadjoint orbits we observe the existence of left invariant symplectic structures on the virasoro group manifold as is described in an appendix these give rise to lie bialgebra structures with the virasoro algebra as the underlying lie algebra

new high order convergence iteration methods without employing derivatives for solving nonlinear equations abstract a family of new iteration methods without employing derivatives is proposed in this paper we have proved that these new methods are quadratic convergence their efficiency is demonstrated by numerical experiments the numerical experiments show that our algorithms are comparable to well known methods of newton and steffensen furthermore combining the new method with bisection method we construct another new high order iteration method with nice asymptotic convergence properties of the diameters b n a n

fifth order iterative methods for solving nonlinear equations abstract in this paper we suggest and analyze a new two step iterative method for solving nonlinear equation f x 0 by rewriting the given nonlinear equation as a coupled system of equations and using the taylor series it is shown that this new iterative method is of fifth order several examples are given to illustrate its performance and efficiency comparison with other methods is also given this method can be viewed as an improvement of the halley s method

some classes of equations of discrete type with harmonic singular operator and convolution in this paper we study four classes of discrete type equations with harmonic singular operator and convolution such equations are turned into boundary value problems for analytic function with discontinuous coefficients by discrete fourier transform the general solutions and the conditions of solvability are obtained in class h by our method thus this paper generalizes the theory of classical equations of convolution type

geometric constructions of iterative functions to solve nonlinear equations in this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation in addition we also review the extension to general banach spaces and some computational aspects of these methods

modified householder iterative method for nonlinear equations abstract in this paper we suggest and analyze a new two step predictor corrector type iterative method for solving nonlinear equations of the type f x 0 this new method includes the two step newton method as a special case we show that this new two step method is a sixth order convergent method several examples are given to illustrate the efficiency of this new method and its comparison with other sixth order methods this method can be considered as a significant improvement of the newton method and its variant forms

weighted nuclear norm minimization and its applications to low level vision as a convex relaxation of the rank minimization model the nuclear norm minimization nnm problem has been attracting significant research interest in recent years the standard nnm regularizes each singular value equally composing an easily calculated convex norm however this restricts its capability and flexibility in dealing with many practical problems where the singular values have clear physical meanings and should be treated differently in this paper we study the weighted nuclear norm minimization wnnm problem which adaptively assigns weights on different singular values as the key step of solving general wnnm models the theoretical properties of the weighted nuclear norm proximal wnnp operator are investigated albeit nonconvex we prove that wnnp is equivalent to a standard quadratic programming problem with linear constrains which facilitates solving the original problem with off the shelf convex optimization solvers in particular when the weights are sorted in a non descending order its optimal solution can be easily obtained in closed form with wnnp the solving strategies for multiple extensions of wnnm including robust pca and matrix completion can be readily constructed under the alternating direction method of multipliers paradigm furthermore inspired by the reweighted sparse coding scheme we present an automatic weight setting method which greatly facilitates the practical implementation of wnnm the proposed wnnm methods achieve state of the art performance in typical low level vision tasks including image denoising background subtraction and image inpainting

new iterative technique for solving a system of nonlinear equations various problems of pure and applied sciences can be studied in the unified frame work of the system of nonlinear equations in this paper a new family of iterative methods for solving a system of nonlinear equations is developed by using a new decomposition technique the convergence of the new methods is proved efficiency index of the proposed methods is discussed and compared with some other well known methods the upper bounds of the error and the radius of convergence of the methods are also found for the implementation and performance of the new methods the combustion problem streering problem and van der pol equation are solved and the results are compared with some existing methods several new iterative methods are derived from the general iterative scheme using the ideas and techniques of this paper one may be able to suggest and investigate a wide class of iterative methods for solving the system of nonlinear equations this is another direction of future research

low rank matrix recovery from errors and erasures this paper considers the recovery of a low rank matrix from an observed version that simultaneously contains both 1 erasures most entries are not observed and 2 errors values at a constant fraction of unknown locations are arbitrarily corrupted we provide a new unified performance guarantee on when minimizing nuclear norm plus l1 norm succeeds in exact recovery our result allows for the simultaneous presence of random and deterministic components in both the error and erasure patterns by specializing this one single result in different ways we recover up to poly log factors as corollaries all the existing results in exact matrix completion and exact sparse and low rank matrix decomposition our unified result also provides the first guarantees for 1 recovery when we observe a vanishing fraction of entries of a corrupted matrix and 2 deterministic matrix completion

quasi invariance of brownian measures on the group of circle homeomorphisms and infinite dimensional riemannian geometry abstract a cameron martin type theorem is proved for the canonical brownian motion on the group of homeomorphisms of the circle

some higher order modifications of newton s method for solving nonlinear equations in this paper we consider constructing some higher order modifications of newton s method for solving nonlinear equations which increase the order of convergence of existing iterative methods by one or two or three units this construction can be applied to any iteration formula and per iteration the resulting methods add only one additional function evaluation to increase the order some illustrative examples are provided and several numerical results are given to show the performance of the presented methods

third and fourth order iterative methods free from second derivative for nonlinear systems in this work we develop a family of predictor corrector methods free from second derivative for solving systems of nonlinear equations in general the obtained methods have order of convergence three but in some particular cases the order is four we also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with newton s classical method and with other recently published methods

a method for solving nonlinear equations a method for finding a solution of the equation f x 0 is presented the method is based on some specially derived quadrature rules it is shown that the method can give better results than the newton method

on a class of quadratically convergent iteration formulae in this paper a new class of iterative formulae having quadratic convergence is presented furthermore these algorithms are comparable to the well known method of newton and the computed results support this theory

multi stage convex relaxation method for low rank and sparse matrix separation problem the low rank and sparse matrix separation problem recovering a low rank matrix and a sparse matrix from a small number of linear observations has arisen in a broad set of applications the desire for efficiency motivated the research on it however most of them focus on the nuclear norm plus 1 norm minimization problem which is a single convex relaxation of the low rank and sparse matrix separation problem this paper is devoted to investigate a multi stage convex relaxation method to recover the low rank and sparse components via solving a sequence of convex relaxations while the first stage of the method includes the nuclear norm plus 1 norm minimization problem numerical studies indicate that the multi stage convex relaxation outperforms the one stage convex relaxation especially when observations are incomplete and noisy we have also verified the efficiency of the proposed method by applying it to the static background extraction from surveillance video

some iterative methods free from second derivatives for nonlinear equations in a recent paper noor m aslam noor new classes of iterative methods for nonlinear equations appl math comput 2007 doi 10 1016 j amc 2007 suggested and analyzed a generalized one parameter halley method for solving nonlinear equations using in this paper we modified this method which has fourth order convergence as special cases we obtain a family of third order iterative methods for appropriate and suitable choice of the parameter we have compared this modified noor method with some other iterative methods which shows that this new iterative method is robust and efficient one several examples are given to illustrate the efficiency and the performance of this new method

on iterative methods for nonlinear equations several iterative methods have been suggested and analyzed for solving nonlinear equation f x 0 all these methods can be classified as one step and two step methods in this paper we have suggested and analyzed several two step methods for solving nonlinear equations we have also shown that these two step methods are quadratically convergent several numerical examples are given to illustrate this comparison our results can be viewed as important refinement and improvement of the previously known results

a note on three step iterative method for nonlinear equations abstract noor and noor m a noor k i noor three step iterative methods for nonlinear equations appl math comput 183 2006 322 327 proposed a three step iterative method for solving nonlinear equations and they showed that the method has third order convergence in this paper we propose a new three step iterative method for solving nonlinear equations which is a significant improvement of the method proposed by noor and noor it is shown that this iterative method has fourth order convergence not third order convergence some numerical experiments are given to illustrate the performance of the three step iterative method

new classes of iterative methods for nonlinear equations abstract in this paper we suggest and analyze some new iterative methods for solving nonlinear equations f x 0 by using the variational iteration technique these new methods includes the householder and its variant forms as special cases we also give several examples to illustrate the efficiency of these methods comparison with other similar methods is also given these new methods can be considered as an alternative to the newton method this technique can be used to suggest a wide class of new iterative methods for solving system of nonlinear equations

a geometric construction of iterative functions of order three to solve nonlinear equations in this paper we consider a geometric construction of iteration functions of order three to develop cubically convergent iterative methods for solving nonlinear equations this construction can be applied to any iteration function of order two to develop an iteration function of order three some examples are given of deriving several third order iteration methods and several numerical results follow to illustrate the performance of the derived methods