semilocal and local convergence of a fifth order iteration with frechet derivative satisfying holder condition the semilocal and local convergence in banach spaces is described for a fifth order iteration for the solutions of nonlinear equations when the frechet derivative satisfies the holder condition the holder condition generalizes the lipschtiz condition the importance of our work lies in the fact that many examples are available which fail to satisfy the lipschtiz condition but satisfy the holder condition the existence and uniqueness theorem is established with error bounds for the solution the convergence analysis is finally worked out on different examples and convergence balls for each of them are obtained these examples include nonlinear hammerstein and fredholm integral equations and a boundary value problem it is found that the larger radius of convergence balls is obtained for all the examples in comparison to existing methods using stronger conditions

hyperdisk based large margin classifier we introduce a large margin linear binary classification framework that approximates each class with a hyperdisk the intersection of the affine support and the bounding hypersphere of its training samples in feature space and then finds the linear classifier that maximizes the margin separating the two hyperdisks we contrast this with support vector machines svms which find the maximum margin separator of the pointwise convex hulls of the training samples arguing that replacing convex hulls with looser convex class models such as hyperdisks provides safer margin estimates that improve the accuracy on some problems both the hyperdisks and their separators are found by solving simple quadratic programs the method is extended to nonlinear feature spaces using the kernel trick and multi class problems are dealt with by combining binary classifiers in the same ways as for svms experiments on a range of data sets show that the method compares favourably with other popular large margin classifiers

convergence of newton like iterative methods newton like methods in which the intermediate systems of linear equations are solved by iterative techniques are examined by applying the theory of inexact newton methods radius of convergence and rate of convergence results are easily obtained the analysis is carried out in affine invariant terms the results are applicable to cases where the underlying newton like method is for example a difference newton like or update newton method

exotic subfactors of finite depth with jones indices 5 13 2 and 5 17 2 we prove existence of subfactors of finite depth of the hyperfinite ii1 factor with indices and the existence of the former was announced by the second named author in 1993 and that of the latter has been conjectured since then these are the only known subfactors with finite depth which do not arise from classical groups quantum groups or rational conformal field theory

on interpolation variants of newton s method for functions of several variables a generalization of the variants of newton s method based on interpolation rules of quadrature is obtained in order to solve systems of nonlinear equations under certain conditions convergence order is proved to be 2d 1 where d is the order of the partial derivatives needed to be zero in the solution moreover different numerical tests confirm the theoretical results and allow us to compare these variants with newton s classical method whose convergence order is d 1 under the same conditions

dynamics of a family of chebyshev halley type methods in this paper the dynamics of the chebyshev halley family is studied on quadratic polynomials a singular set that we call cat set appears in the parameter space associated to the family this set has interesting similarities with the mandelbrot set the parameter space has allowed us to find different elements of the family which have bad convergence properties since periodic orbits and attractive strange fixed points appear in the dynamical plane of the corresponding method

a fast chebyshev s method for quadratic equations a multiresolution transform corresponding to interpolatory technique is used for fast application of chebyshev s method in designing this algorithm we apply data compression to the linear and the bilinear forms that appear on the method a convergence theorem is performed finally some numerical results are presented

solving wiener hopf problems via an efficient iterative scheme abstract in the analysis of the fluid queues it is necessary to obtain the nonnegative solution of a nonsymmetric algebraic riccati matrix equation under suitable conditions this solution can be obtained transforming algebraic riccati equations into unilateral quadratic matrix equations in this paper we use an efficient iterative scheme to approximate a solution of this quadratic matrix equation we improve the efficiency and the accuracy of newton s method widely used in the literature moreover a local convergence result is proved finally we apply this efficient method to approximate the solution of a particular noisy wiener hopf problem and we compare it with newton s method moreover a predictor corrector iterative scheme is constructed that improve the accessibility of the aforesaid method

efficient high order methods based on golden ratio for nonlinear systems abstract we derive new iterative methods with order of convergence four or higher for solving nonlinear systems by composing iteratively golden ratio methods with a modified newton s method we use different efficiency indices in order to compare the new methods with other ones and present several numerical tests which confirm the theoretical results

iterative methods improving newton s method by the decomposition method in this paper we present a sequence of iterative methods improving newton s method for solving nonlinear equations the adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses the orders of convergence are derived analytically and then rederived by applying symbolic computation of maple some numerical illustrations are given

non cyclotomic fusion categories etingof nikshych and ostrik ask in 7 2 if every fusion category can be completely defined over a cyclotomic field we show that this is not the case in particular one of the fusion categories coming from the haagerup subfactor 1 and one coming from the newly constructed extended haagerup subfactor 2 can not be completely defined over a cyclotomic field on the other hand we show that the drinfel d center of the even part of the haagerup subfactor is completely defined over a cyclotomic field we identify the minimal field of definition for each of these fusion categories compute the galois groups and identify their galois conjugates ams classification 18d10 46l37

a modified chebyshev s iterative method with at least sixth order of convergence this paper is devoted to the construction and analysis of a high order variant of the classical chebyshev method the method has order of convergence at least six for simple roots the extension to system of equations and its semilocal convergence for nonlinear equations are presented finally an application to well known algebraic riccati equation is considered

twin support vector machines for pattern classification we propose twin svm a binary svm classifier that determines two nonparallel planes by solving two related svm type problems each of which is smaller than in a conventional svm the twin svm formulation is in the spirit of proximal svms via generalized eigenvalues on several benchmark data sets twin svm is not only fast but shows good generalization twin svm is also useful for automatically discovering two dimensional projections of the data

extending the applicability of newton s method for a class of boundary value problems using the shooting method abstract we use newton s method to approximate locally unique solutions for a class of boundary value problems by applying the shooting method the utilized operator is frechet differentiable between banach spaces these conditions are more general than those that appear in previous works in particular we show that the old semilocal and local convergence criteria for newton s method involving banach space value operators can be replaced by weaker ones hence extending the applicability of the method several numerical examples are developed to test the new convergence criteria and also compare them to the old ones

a note on halley s method we introduce the degree of logarithmic convexity which provides a measure of the convexity of a function at each point making use of this concept we obtain a new theorem of global convergence for halley s method

discriminative common vectors for face recognition in face recognition tasks the dimension of the sample space is typically larger than the number of the samples in the training set as a consequence the within class scatter matrix is singular and the linear discriminant analysis lda method cannot be applied directly this problem is known as the small sample size problem in this paper we propose a new face recognition method called the discriminative common vector method based on a variation of fisher s linear discriminant analysis for the small sample size case two different algorithms are given to extract the discriminative common vectors representing each person in the training set of the face database one algorithm uses the within class scatter matrix of the samples in the training set while the other uses the subspace methods and the gram schmidt orthogonalization procedure to obtain the discriminative common vectors then the discriminative common vectors are used for classification of new faces the proposed method yields an optimal solution for maximizing the modified fisher s linear discriminant criterion given in the paper our test results show that the discriminative common vector method is superior to other methods in terms of recognition accuracy efficiency and numerical stability

subfactors of index less than 5 part 3 quadruple points one major obstacle in extending the classification of small index subfactors beyond 3 sqrt 3 is the appearance of infinite families of candidate principal graphs with 4 valent vertices in particular the weeds mathcal q and mathcal q from part 1 morrison and snyder in commun math phys doi 10 1007 s00220 012 1426 y 2012 thus instead of using triple point obstructions to eliminate candidate graphs we need to develop new quadruple point obstructions in this paper we prove two quadruple point obstructions the first uses quadratic tangles techniques and eliminates the weed mathcal q immediately the second uses connections and when combined with an additional number theoretic argument it eliminates both weeds mathcal q and mathcal q finally we prove the uniqueness up to taking duals of the 3311 goodman de la harpe jones subfactor using a combination of planar algebra techniques and connections

an optimal reconstruction of chebyshev halley type methods for nonlinear equations having multiple zeros abstract establishment of a new optimal higher order iterative scheme for multiple zeros with known multiplicity m 1 of univariate function is one of the hard and demanding task in the area of computational mathematics and numerical analysis we write this paper with the aim to propose a new higher order reconstruction of chebyshev halley type iteration functions for multiple zeros in a simple way minimum order of convergence of our scheme is seven and maximum reaches to eight without using any extra functional evaluation with regard to computational cost each member of our scheme needs four functional evaluations at each step so the maximum efficiency index of our scheme is 1 6818 for 1 in addition we thoroughly study the convergence analysis which confirms the theoretical order of convergence moreover we check the efficiency and compare them to the existing ones on four real life problems namely fractional conversion chemical engineering continuous stirred tank reactor and eigen value problems and two well known academic problems finally we confirm on the basis of obtained results that our methods have better residual error errors between two consecutive iterations and stable computational order of convergence as compared to the other existing ones

improvements on twin support vector machines for classification problems the generalized eigenvalue proximal support vector machine gepsvm and twin support vector machine twsvm are regarded as milestones in the development of the powerful svms as they use the nonparallel hyperplane classifiers in this brief we propose an improved version named twin bounded support vector machines tbsvm based on twsvm the significant advantage of our tbsvm over twsvm is that the structural risk minimization principle is implemented by introducing the regularization term this embodies the marrow of statistical learning theory so this modification can improve the performance of classification in addition the successive overrelaxation technique is used to solve the optimization problems to speed up the training procedure experimental results show the effectiveness of our method in both computation time and classification accuracy and therefore confirm the above conclusion further

ball convergence comparison between three iterative methods in banach space under hypothese only on the first derivative we present a convergence ball comparison between three iterative methods for approximating a locally unique solution of a nonlinear equation in a banach space setting the convergence ball and error estimates are given for these methods under hypotheses only on the first frechet derivative in contrast to earlier studies such as adomian 1994 1 babajee et al 2008 13 cordero and torregrosa 2007 17 cordero et al 18 darvishi and barati 2007 19 using hypotheses reaching up to the fourth frechet derivative although only the first derivative appears in these methods this way we expand the applicability of these methods numerical examples are also presented in this study

tpmsvm a novel twin parametric margin support vector machine for pattern recognition a novel twin parametric margin support vector machine tpmsvm for classification is proposed in this paper this tpmsvm in the spirit of the twin support vector machine twsvm determines indirectly the separating hyperplane through a pair of nonparallel parametric margin hyperplanes solved by two smaller sized support vector machine svm type problems similar to the parametric margin email protected support vector machine email protected svm this tpmsvm is suitable for many cases especially when the data has heteroscedastic error structure that is the noise strongly depends on the input value but there is an advantage in the learning speed compared with the email protected svm the experimental results on several artificial and benchmark datasets indicate that the tpmsvm not only obtains fast learning speed but also shows good generalization

newton like methods for the computation of fixed points abstract the celebrated banach fixed point theorem provides conditions which assure that the method of successive substitution is convergent the convergence however may take place very slowly so that it may be desirable to use a newton like method for the computation of the fixed point if newton s method itself is applied one ignores the additional information that the problem arises from a frixed point problem with a contraction mapping in the present note some variants of newton s method are discussed which make use of this contraction information it turns out that the convergence of newton s method can be accelerated without any relevant additional computational labour

nonconvex online support vector machines in this paper we propose a nonconvex online support vector machine svm algorithm lasvm nc based on the ramp loss which has the strong ability of suppressing the influence of outliers then again in the online learning setting we propose an outlier filtering mechanism lasvm i based on approximating nonconvex behavior in convex optimization these two algorithms are built upon another novel svm algorithm lasvm g that is capable of generating accurate intermediate models in its iterative steps by leveraging the duality gap we present experimental results that demonstrate the merit of our frameworks in achieving significant robustness to outliers in noisy data classification where mislabeled training instances are in abundance experimental evaluation shows that the proposed approaches yield a more scalable online svm algorithm with sparser models and less computational running time both in the training and recognition phases without sacrificing generalization performance we also point out the relation between nonconvex optimization and min margin active learning

a robust minimax approach to classification when constructing a classifier the probability of correct classification of future data points should be maximized we consider a binary classification problem where the mean and covariance matrix of each class are assumed to be known no further assumptions are made with respect to the class conditional distributions misclassification probabilities are then controlled in a worst case setting that is under all possible choices of class conditional densities with given mean and covariance matrix we minimize the worst case maximum probability of misclassification of future data points for a linear decision boundary this desideratum is translated in a very direct way into a convex second order cone optimization problem with complexity similar to a support vector machine problem the minimax problem can be interpreted geometrically as minimizing the maximum of the mahalanobis distances to the two classes we address the issue of robustness with respect to estimation errors in the means and covariances of the classes via a simple modification of the input data we also show how to exploit mercer kernels in this setting to obtain nonlinear decision boundaries yielding a classifier which proves to be competitive with current methods including support vector machines an important feature of this method is that a worst case bound on the probability of misclassification of future data is always obtained explicitly

image classification with the fisher vector theory and practice a standard approach to describe an image for classification and retrieval purposes is to extract a set of local patch descriptors encode them into a high dimensional vector and pool them into an image level signature the most common patch encoding strategy consists in quantizing the local descriptors into a finite set of prototypical elements this leads to the popular bag of visual words representation in this work we propose to use the fisher kernel framework as an alternative patch encoding strategy we describe patches by their deviation from an universal generative gaussian mixture model this representation which we call fisher vector has many advantages it is efficient to compute it leads to excellent results even with efficient linear classifiers and it can be compressed with a minimal loss of accuracy using product quantization we report experimental results on five standard datasets pascal voc 2007 caltech 256 sun 397 ilsvrc 2010 and imagenet10k with up to 9m images and 10k classes showing that the fv framework is a state of the art patch encoding technique

quantum subgroups of the haagerup fusion categories we answer three related questions concerning the haagerup subfactor and its even parts the haagerup fusion categories namely we find all simple module categories over each of the haagerup fusion categories in other words we find the quantum subgroups in the sense of ocneanu we find all irreducible subfactors whose principal even part is one of the haagerup fusion categories and we compute the brauer picard groupoid of morita equivalences of the haagerup fusion categories in addition to the two even parts of the haagerup subfactor there is exactly one more fusion category which is morita equivalent to each of them this third fusion category has six simple objects and the same fusion rules as one of the even parts of the haagerup subfactor but has not previously appeared in the literature we also find the full lattice of intermediate subfactors for every irreducible subfactor whose even part is one of these three fusion categories and we discuss how our results generalize to izumi subfactors

on a characterization of some newton like methods of r order at least three in this paper we extend to banach spaces the result given by gander in amer math monthly 92 1985 131 to obtain a characterization of newton like iterative process with r order of convergence at least three to do this we consider the construction and the study of the semilocal convergence of a multiparametric family of iterative processes in banach spaces for solving the nonlinear equation f x 0

on fusion categories using a variety of methods developed in the literature in particular the theory of weak hopf algebras we prove a number of general results about fusion categories in characteristic zero we show that the global dimension of a fusion category is always positive and that the s matrix of any not necessarily hermitian modular category is unitary we also show that the category of module functors between two module categories over a fusion category is semisimple and that fusion categories and tensor functors between them are undeformable generalized ocneanu rigidity in particular the number of such categories functors realizing a given fusion datum is finite finally we develop the theory of frobenius perron dimensions in an arbitrary fusion category at the end of the paper we generalize some of these results to positive characteristic

modular data for the extended haagerup subfactor we compute the modular data that is the s and t matrices for the centre of the extended haagerup subfactor bmps12 the full structure i e the associativity data also known as 6 j symbols or f matrices still appears to be inaccessible nevertheless starting with just the number of simple objects and their dimensions obtained by a combinatorial argument in mw14 we find that it is surprisingly easy to leverage knowledge of the representation theory of sl 2 mathbb z into a complete description of the modular data we also investigate the possible character vectors associated with this modular data this is the published version of arxiv 1606 07165

best fitting hyperplanes for classification in this paper we propose novel methods that are more suitable than classical large margin classifiers for open set recognition and object detection tasks the proposed methods use the best fitting hyperplanes approach and the main idea is to find the best fitting hyperplanes such that each hyperplane is close to the samples of one of the classes and is as far as possible from the other class samples to this end we propose two different classifiers the first classifier solves a convex quadratic optimization problem but negative samples can lie on one side of the best fitting hyperplane the second classifier however allows the negative samples to lie on both sides of the fitting hyperplane by using concave convex procedure both methods are extended to the nonlinear case by using the kernel trick in contrast to the existing hyperplane fitting classifiers in the literature our proposed methods are suitable for large scale problems and they return sparse solutions the experiments on several databases show that the proposed methods typically outperform other hyperplane fitting classifiers and they work as good as the svm classifier in classical recognition tasks however the proposed methods significantly outperform svm in open set recognition and object detection tasks

multisurface proximal support vector machine classification via generalized eigenvalues a new approach to support vector machine svm classification is proposed wherein each of two data sets are proximal to one of two distinct planes that are not parallel to each other each plane is generated such that it is closest to one of the two data sets and as far as possible from the other data set each of the two nonparallel proximal planes is obtained by a single matlab command as the eigenvector corresponding to a smallest eigenvalue of a generalized eigenvalue problem classification by proximity to two distinct nonlinear surfaces generated by a nonlinear kernel also leads to two simple generalized eigenvalue problems the effectiveness of the proposed method is demonstrated by tests on simple examples as well as on a number of public data sets these examples show the advantages of the proposed approach in both computation time and test set correctness

on the convergence of a higher order family of methods and its dynamics in this paper we present the study of the local convergence of a higher order family of methods moreover the dynamical behavior of this family of iterative methods applied to quadratic polynomials is studied some anomalies are found in this family by means of studying the associated rational function parameter spaces are shown and the study of the stability of all the fixed points is presented

an iterative method with cubic convergence 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 this new method new method can be viewed as an improvement of the previously known iterative methods

enlarging the convergence domain in local convergence studies for iterative methods in banach spaces in this work we introduce a new form of setting the general assumptions for the local convergence studies of iterative methods in banach spaces that allows us to improve the convergence domains specifically a local convergence result for a family of higher order iterative methods for solving nonlinear equations in banach spaces is established under the assumption that the frechet derivative satisfies the lipschitz continuity condition for some values of the parameter these iterative methods are of fifth order the importance of our work is that it avoids the usual practice of boundedness conditions of higher order derivatives which is a drawback for solving some practical problems the existence and uniqueness theorem that establishes the convergence balls of these methods is obtained we have considered a number of numerical examples including a nonlinear hammerstein equation and computed the radii of the convergence balls it is found that the radius of convergence ball obtained by our approach is much larger when compared with some other existing methods the global convergence properties of the family are explored by analyzing the dynamics of the corresponding operator on complex quadratic polynomials

accelerated methods of order 2p for systems of nonlinear equations we present a new iterative method of order of convergence 5 for solving nonlinear systems by composing the midpoint method with newton s method and using an approximation for the jacobian matrix in order to reduce the required number of functional evaluations per iteration in addition we compare the efficiency index of these methods with that of newton s method and present several numerical tests which confirm the theoretical results and allow us to compare these variants with newton s method

near group fusion categories and their doubles a near group fusion category is a fusion category c where all but 1 simple objects are invertible examples of these include the tambara yamagami categories and the even sectors of the e6 and affine d5 subfactors though there are infinitely many others we classify the near group fusion categories and compute their doubles and the modular data relevant to conformal field theory among other things we explicitly construct over 40 new finite depth subfactors with jones index ranging from around 6 85 to around 14 93 we expect all of these doubles to be realised by rational conformal field theories

monoidal 2 structure of bimodule categories abstract we define a notion of tensor product of bimodule categories and prove that with this product the 2 category of c bimodule categories for fixed tensor c is a monoidal 2 category in the sense of kapranov and voevodsky 1991 1 we then provide a monoidal structure preserving 2 equivalence between the 2 category of c bimodule categories and z c module categories module categories over the center of c for a finite group g we show that de equivariantization is equivalent to the tensor product over rep g we derive rep g module fusion rules and show that the group of invertible rep g module categories is isomorphic to h 2 g k extending results in etingof et al 2

higher order iterative methods for approximating zeros of analytic functions abstract iterative methods with extremely rapid convergence in floating point arithmetic and circular arithmetic for simultaneously approximating simple zeros of analytic functions inside a simple smooth closed contour in the complex plane are presented the r order of convergence of the basic total step and single step methods as well as their improvements which use newton s and halley s corrections is given some hybrid algorithms that combine the efficiency of ordinary floating point iterative methods with the accuracy control provided by interval arithmetic are also considered

chebyshev type methods and preconditioning techniques recently a newton s iterative method is attracting more and more attention from various fields of science and engineering this method is generally quadratically convergent in this paper some chebyshev type methods with the third order convergence are analyzed in detail and used to compute approximate inverse preconditioners for solving the linear system ax b theoretic analysis and numerical experiments show that chebyshev s method is more effective than newton s one in the case of constructing approximate inverse preconditioners

discriminative common vector method with kernels in some pattern recognition tasks the dimension of the sample space is larger than the number of samples in the training set this is known as the small sample size problem linear discriminant analysis lda techniques cannot be applied directly to the small sample size case the small sample size problem is also encountered when kernel approaches are used for recognition in this paper we attempt to answer the question of how should one choose the optimal projection vectors for feature extraction in the small sample size case based on our findings we propose a new method called the kernel discriminative common vector method in this method we first nonlinearly map the original input space to an implicit higher dimensional feature space in which the data are hoped to be linearly separable then the optimal projection vectors are computed in this transformed space the proposed method yields an optimal solution for maximizing a modified fisher s linear discriminant criterion discussed in the paper thus under certain conditions a 100 recognition rate is guaranteed for the training set samples experiments on test data also show that in many situations the generalization performance of the proposed method compares favorably with other kernel approaches

core vector machines fast svm training on very large data sets standard svm training has o m3 time and o m2 space complexities where m is the training set size it is thus computationally infeasible on very large data sets by observing that practical svm implementations only approximate the optimal solution by an iterative strategy we scale up kernel methods by exploiting such approximateness in this paper we first show that many kernel methods can be equivalently formulated as minimum enclosing ball meb problems in computational geometry then by adopting an efficient approximate meb algorithm we obtain provably approximately optimal solutions with the idea of core sets our proposed core vector machine cvm algorithm can be used with nonlinear kernels and has a time complexity that is linear in m and a space complexity that is independent of m experiments on large toy and real world data sets demonstrate that the cvm is as accurate as existing svm implementations but is much faster and can handle much larger data sets than existing scale up methods for example cvm with the gaussian kernel produces superior results on the kddcup 99 intrusion detection data which has about five million training patterns in only 1 4 seconds on a 3 2ghz pentium 4 pc