nearest neighbor classification with dependent training sequences the asymptotic classification risk for nearest neighbor procedures is well understood in the case of i i d training sequences in this article we generalize these results to a class of dependent models including hidden markov models in the case where the observed patterns have lebesgue densities the asymptotic risk takes the same expression as in the i i d case for discrete distributions we show that the asymptotic risk depends on the rule used for breaking ties of equal distances

uniqueness of initial boundary value problems in nonlocal elasticity abstract in this study we have dealt with the uniqueness of the solutions of a class of initial boundary value problems in linear isotropic homogeneous nonlocal elasticity the proof of uniqueness is based on the positive definiteness of total strain energy after giving the sufficient conditions under which total strain energy is positive definite it has been shown that the solutions of the initial boundary value problems which are considered in this study are unique existence of the solutions of these problems is an a priori assumption

rates of convergence of nearest neighbor estimation under arbitrary sampling rates of convergence for nearest neighbor estimation are established in a general framework in terms of metric covering numbers of the underlying space the first result is to find explicit finite sample upper bounds for the classical independent and identically distributed i i d random sampling problem in a separable metric space setting the convergence rate is a function of the covering numbers of the support of the distribution for example for bounded subsets of r sup r the convergence rate is o 1 n sup 2 r the main result is to extend the problem to allow samples drawn from a completely arbitrary random process in a separable metric space and to examine the performance in terms of the individual sample sequences the authors show that for every sequence of samples the asymptotic time average of nearest neighbor risks equals twice the time average of the conditional bayes risks of the sequence finite sample upper bounds under arbitrary sampling are again obtained in terms of the covering numbers of the underlying space in particular for bounded subsets of r sup r the convergence rate of the time averaged risk is o 1 n sup 2 r the authors then establish a consistency result for k sub n nearest neighbor estimation under arbitrary sampling and prove a convergence rate matching established rates for i i d sampling finally they show how their arbitrary sampling results lead to some classical i i d sampling results and in fact extend them to stationary sampling the framework and results are quite general while the proof techniques are surprisingly elementary

a note on the representations for the drazin inverse of 2 2 block matrices abstract in 1979 campbell and meyer proposed the problem of finding a formula for the drazin inverse of a 2 2 matrix m a b c d in terms of its various blocks where the blocks a and d are required to be square matrices special cases of the problems have been studied in particular a representation of the drazin inverse of m denoted by md has recently been obtained under the assumptions that c i aad b o and a i aad b o together with the condition that the generalized schur complement d cadb be either nonsingular or zero we derive an alternative representation for md under the same assumptions but with the condition on the schur complement in the hypothesis replaced by the condition that r caa d n b n d where r and n are the range and null space of a matrix

nonparametric output prediction for nonlinear fading memory systems the authors construct a class of elementary nonparametric output predictors of an unknown discrete time nonlinear fading memory system their algorithms predict asymptotically well for every bounded input sequence every disturbance sequence in certain classes and every linear or nonlinear system that is continuous and asymptotically time invariant causal and with fading memory the predictor is based on k sub n nearest neighbor estimators from nonparametric statistics it uses only previous input and noisy output data of the system without any knowledge of the structure of the unknown system the bounds on the input or the properties of noise under additional smoothness conditions the authors provide rates of convergence for the time average errors of their scheme finally they apply their results to the special case of stable linear time invariant lti systems

nonparametric identification of hammerstein systems a discrete time nonlinear hammerstein system is identified and the correlation and frequency domain methods for identification of its linear subsystem are presented the main results concern the estimation of the nonlinear memoryless subsystem no conditions concerning the functional form of the transform characteristic of the subsystem are made and an algorithm for estimation of the characteristic is given the algorithm is simply a nonparametric kernel estimate of the regression function calculated from dependent data it is shown that the algorithm converges to the characteristic of the subsystem regardless of the probability distribution of the input variable pointwise as well as global consistencies are established for lipschitz characteristics the rate of the convergence in probability is o n sup 1 3

representation for the generalized drazin inverse of block matrices in banach algebras abstract several representations of the generalized drazin inverse of a block matrix with a group invertible generalized schur complement in banach algebra are presented

minimax nonparametric classification i rates of convergence this paper studies minimax aspects of nonparametric classification we first study minimax estimation of the conditional probability of a class label given the feature variable this function say f is assumed to be in a general nonparametric class we show the minimax rate of convergence under square l sub 2 loss is determined by the massiveness of the class as measured by metric entropy the second part of the paper studies minimax classification the loss of interest is the difference between the probability of misclassification of a classifier and that of the bayes decision as is well known an upper bound on risk for estimating f gives an upper bound on the risk for classification but the rate is known to be suboptimal for the class of monotone functions this suggests that one does not have to estimate f well in order to classify well however we show that the two problems are in fact of the same difficulty in terms of rates of convergence under a sufficient condition which is satisfied by many function classes including besov sobolev lipschitz and bounded variation this is somewhat surprising in view of a result of devroye gorfi and lugosi see a probabilistic theory of pattern recognition new york springer verlag 1996

relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices application of perturbation theory for simple invariant subspaces perturbation bounds for the relative error in the eigenvalues of diagonalizable and singular matrices are derived by using perturbation theory for simple invariant subspaces of a matrix and the group inverse of a matrix these upper bounds are supplements to the related perturbation bounds for the eigenvalues of diagonalizable and nonsingular matrices

bayes error estimation using parzen and k nn procedures the use of k nearest neighbor k nn and parzen density estimates to obtain estimates of the bayes error is investigated under limited design set conditions by drawing analogies between the k nn and parzen procedures new procedures are suggested and experimental results are given which indicate that these procedures yield a significant improvement over the conventional k nn and parzen procedures we show that by varying the decision threshold many of the biases associated with the k nn or parzen density estimates may be compensated and successful error estimation may be performed in spite of these biases experimental results are given which demonstrate the effect of kernel size and shape parzen the size of k k nn and the number of samples in the design set

expression of the drazin and mp inverse of partitioned matrix and quotient identity of generalized schur complement in this paper we give representations of the drazin and mp inverse of a 2x2 block matrix and quotient identities for the generalized schur complement of a partitioned 3x3 matrix under conditions different than those used in recent papers on the subject we present numerical examples to illustrate our results

the drazin inverse of the sum of two matrices and its applications in this paper a representation of p q d such that pqp 0 pq 2 0 is given which recovers the case pq 0 studied by hartwig et al r e hartwig g wang y wei some additive results on drazin inverse linear algebra appl 322 2001 207 217 furthermore we apply our results to establish the representations for the drazin inverses of a 2x2 partitioned matrix m abcd where a and d are square matrices finally two numerical examples are given to illustrate our results

some results on the generalized drazin inverse of operator matrices the generalized drazin inverse m d of a 2 2 operator matrix is considered where a x and d y are generalized drazin invertible expressions for the generalized drazin inverse m d of operator matrix m in terms of the individual blocks a b c d a d and d d are derived under some conditions

on drazin inverse of operator matrices in this short paper we offer another formula for the drazin inverse of an operator matrix for which certain products of the entries vanish we also give formula for the drazin inverse of the sum of two operators under special conditions

nonlocal integral elasticity 2d finite element based solutions abstract a finite element based method theorized in the context of nonlocal integral elasticity and founded on a nonlocal total potential energy principle is numerically implemented for solving 2d nonlocal elastic problems the key idea of the method known as nonlocal finite element method nl fem relies on the assumption that the postulated nonlocal elastic behaviour of the material is captured by a finite element endowed with a set of cross stiffness element s matrices able to interpret the nonlocality effects induced in the element itself by the other elements in the mesh an eringen type nonlocal elastic model is assumed with a constitutive stress strain law of convolutive type which governs the nonlocal material behaviour computational issues as the construction of the nonlocal element and global stiffness matrices are treated in detail few examples are presented and the relevant numerical findings discussed both to verify the reliability of the method and to prove its effectiveness

on the rate of convergence of local averaging plug in classification rules under a margin condition the rates of convergence of plug in kernel partitioning and nearest neighbors classification rules are analyzed a margin condition which measures how quickly the a posteriori probabilities cross the decision boundary smoothness conditions on the a posteriori probabilities and boundedness of the feature vector are imposed the rates of convergence of the plug in classifiers shown in this paper are faster than previously known

optimal reference subset selection for nearest neighbor classification by tabu search this paper presents an approach to select the optimal reference subset ors for nearest neighbor classifier the optimal reference subset which has minimum sample size and satisfies a certain resubstitution error rate threshold is obtained through a tabu search ts algorithm when the error rate threshold is set to zero the algorithm obtains a near minimal consistent subset of a given training set while the threshold is set to a small appropriate value the obtained reference subset may have reasonably good generalization capacity a neighborhood exploration method and an aspiration criterion are proposed to improve the efficiency of ts experimental results based on a number of typical data sets are presented and analyzed to illustrate the benefits of the proposed method the performances of the result consistent and non consistent reference subsets are evaluated

bias of nearest neighbor error estimates the bias of the finite sample nearest neighbor nn error from its asymptotic value is examined expressions are obtained which relate the bias of the nn and 2 nn errors to sample size dimensionality metric and distributions these expressions isolate the effect of sample size from that of the distributions giving an explicit relation showing how the bias changes as the sample size is increased experimental results are given which suggest that the expressions accurately predict the bias it is shown that when the dimensionality of the data is high it may not be possible to estimate the asymptotic error simply by increasing the sample size a new procedure is suggested to alleviate this problem this procedure involves measuring the mean nn errors at several sample sizes and using our derived relationship between the bias and the sample size to extrapolate an estimate of the asymptotic nn error the results are extended to the multiclass problem the choice of an optimal metric to minimize the bias is also discussed

closed form solution for a nonlocal elastic bar in tension a simple mechanical one dimensional problem in the context of nonlocal integral elasticity is solved analytically the nonlocal elastic material behaviour is described by the eringen model whose nonlocality features all reside in the constitutive relation this relation of integral type contains an attenuation function usually assumed symmetric aimed to capture the diffusion process of the nonlocality effects it also exhibits a convolution format the governing equation is a fredholm integral equation of second kind whose analytical treatment even for the usual choice of a symmetric kernel is not easy to develop in the present paper assuming a specific shape for the attenuation function a closed form solution in terms of strains is alternatively obtained by solving a volterra integral equation of second kind the latter can be easily solved with standard techniques at least for the adopted kernel taking also advantage from the symmetry of the solution such a closed form solution is an essential result to validate the effectiveness of numerical procedures aimed to solve more complex mechanical problems in the context of nonlocal elasticity 2002 elsevier science ltd all rights reserved

explicit group inverse of an innovative patterned matrix in this paper we present an innovative patterned matrix rfpl toeplitz matrix is neither the extension of toeplitz matrix nor its special case we show that the group inverse of this new patterned matrix can be represented as the sum of products of lower and upper triangular toeplitz matrices first the explicit expression of the group inverse of an rfpl toeplitz matrix is obtained second the decomposition of the group inverse is given finally an example demonstrates availability of the two methods for the group inverse

bounds on the bayes classification error based on pairwise risk functions upper and lower bounds on the bayes risk for multiple composite hypothesis classification are obtained bounds on the bayes risk for m simple classes are derived in terms of the risk functions for m 1 classes and so on until the desired result depends only on the pairwise m 2 bayes risks a method of computing upper and lower bounds on the pairwise bayes risk for composite classes is developed algorithms for computing the upper and lower bounds for the general m class case and for composite hypothesis classes are presented numerical examples of the application of the bounding techniques to a problem involving the classification of aircraft are discussed results for the bounds and other performance measures are compared for the most interesting cases

pattern classification using projection pursuit abstract this article discusses the adaptation of recently developed regression techniques to classifier design apart from finite sample effects projection pursuit pp regression may be used to model a desired response class as a sum of ridge functions according to a minimum expected squared error criterion this approach can be shown to furnish an optimal discriminant function which can satisfy the neyman pearson criterion over all possible thresholds basis function expansions are used instead of smoothed histograms to reduce computation since good approximation of a discriminant by a linear combination of moderate number of ridge functions may not be easy we introduce an improved method utilizing a nonlinear weighting function

further results on the representation of the drazin inverse of a 2 2 block matrix abstract the aim of this paper is to establish an explicit representation of the drazin inverse of a 2 2 block matrix m a b c d where a and d are square matrices in terms of the drazin inverses of a and d first we give a formula for the drazin inverse of m with the blocks satisfying the conditions bc 0 bdc 0 and bd 2 0 further we derive representations of the drazin inverse of m under less restrictive conditions than those found in the current literature

on the drazin inverse of the sum of two operators and its application to operator matrices abstract given two bounded linear operators f g on a banach space x such that g 2 f g f 2 0 we derive an explicit expression for the drazin inverse of f g for this purpose firstly we obtain a formula for the resolvent of an auxiliary operator matrix in the form m f i g f g from the provided representation of f g d several special cases are considered in particular we recover the case g f 0 studied by hartwig et al r e hartwig g wang y wei some additive results on drazin inverse linear algebra appl 322 2001 207 217 for matrices and by djordjevic and wei d s djordjevic y wei additive results for the generalized drazin inverse j aust math soc 73 1 2002 115 126 for operators finally we apply our results to obtain representations for the drazin inverse of operator matrices in the form m a b c d which are extensions of some cases given in the literature

a test of the gaussian ness of a data set using clustering the properties of the magnifying glass method of clustering are discussed these properties which include unbiased and consistent estimation of the mean for gaussian distributions and biased and inconsistent estimation of the mean for non gaussian distributions lead to the development of a technique for testing data to determine whether or not it is gaussian an example of a non gaussian distribution is given to show the sensitivity of the proposed gaussian test

the 2 2 0 drazin inverse problem abstract we consider the additive drazin problem and we study the existence of the drazin inverse of a two by two matrix with zero 2 2 entry

an optimal global nearest neighbor metric a quadratic metric dao x y x y t ao x y is proposed which minimizes the mean squared error between the nearest neighbor asymptotic risk and the finite sample risk under linearity assumptions a heuristic argument is given which indicates that this metric produces lower mean squared error than the euclidean metric a nonparametric estimate of ao is developed if samples appear to come from a gaussian mixture an alternative parametrically directed distance measure is suggested for nearness decisions within a limited region of space examples of some two class gaussian mixture distributions are included

a strain difference based nonlocal elasticity model abstract a two component local nonlocal constitutive model for macroscopically inhomogeneous linear elastic materials but constant internal length is proposed in which the stress is the sum of the local stress and a nonlocal type stress expressed in terms of the strain difference field hence identically vanishing in the case of uniform strain attention is focused upon the particular case of piecewise homogeneous material the proposed model is thermodynamically consistent with a suitable free energy potential it constitutes an improved form of the vermeer and brinkgreve a new effective nonlocal strain measure for softening plasticity in chambon r desrues j vardulakis i eds localization and bifurcation theory for soils and rocks balkema rotterdam 1994 pp 89 100 model and can also be considered derivable from the eringen nonlocal elasticity model through a suitable enhancement technique based on the concept of redistribution of the local stress the concept of equivalent distance is introduced to macroscopically account for the further attenuation effects produced by the inhomogeneity upon the long distance interaction forces with the aid of a piecewise homogeneous bar in tension a portion of which degrades progressively till failure it is shown that under a suitable choice of a material constant the solution procedure exhibits no pathological features numerical instability mesh sensitivity in every degraded bar condition including the limit idealized stress free condition of the failed bar

benchmarks in nonlocal elasticity defined by eringen s integral model abstract in this paper we present low residual approximate solutions for nonlocal 1d and 2d elasticity problems defined according to eringen s integral model the benchmarks in the 1d cases are defined by prescribing the stress field while the unknown fields are the strains or the displacements for the 2d cases we define problems with equilibrated tractions and evaluate the approximate displacement field meanwhile a fourier series as well as a set of chebyshev polynomials are used as the basis functions for the main unknown fields we increase the number of the approximation functions to decrease the norm of the residuals and repeat the procedure until reasonable accuracy is obtained for the final solution since the procedure is very time consuming in some benchmark problems we present the calculated coefficients and in some other we give some point wise values for further use the results presented in this paper are particularly useful for the validation and convergence studies when numerical methods are to be used for the solution of the nonlocal elasticity problems

the optimal distance measure for nearest neighbor classification a local distance measure is shown to optimize the performance of the nearest neighbor two class classifier for a finite number of samples the difference between the finite sample error and the asymptotic error is used as the criterion of improvement this new distance measure is compared to the well known euclidean distance an algorithm for practical implementation is introduced this algorithm is shown to be computationally competitive with the present nearest neighbor procedures and is illustrated experimentally a closed form for the corresponding second order moment of this criterion is found finally the above results are extended to

some additive results on drazin inverse some additive perturbation results for drazin inverses are given in particular a formula is given for the drazin inverse of a sum of two matrices when one of the products of these matrices vanishes some special applications of this are also considered

neighborhood size selection in the k nearest neighbor rule using statistical confidence the k nearest neighbor rule is one of the most attractive pattern classification algorithms in practice the choice of k is determined by the cross validation method in this work we propose a new method for neighborhood size selection that is based on the concept of statistical confidence we define the confidence associated with a decision that is made by the majority rule from a finite number of observations and use it as a criterion to determine the number of nearest neighbors needed the new algorithm is tested on several real world datasets and yields results comparable to the k nearest neighbor rule however in contrast to the k nearest neighbor rule that uses a fixed number of nearest neighbors throughout the feature space our method locally adjusts the number of nearest neighbors until a satisfactory level of confidence is reached in addition the statistical confidence provides a natural way to balance the trade off between the reject rate and the error rate by excluding patterns that have low confidence levels we believe that this property of our method can be of great importance in applications where the confidence with which a decision is made is equally or more important than the overall error rate

on the drazin inverse of block matrices and generalized schur complement in this paper we give an additive result for the drazin inverse with its applications we obtain representations for the drazin inverse of a 2x2 complex block matrix having generalized schur complement s d ca db equal to zero or nonsingular several situations are analyzed and recent results are generalized r e hartwig x li y wei representations for the drazin inverse of a 2x2 block matrix siam j matrix anal appl 27 3 2006 757 771

lower bounds for bayes error estimation we give a short proof of the following result let x y be any distribution on n spl times 0 1 and let x sub 1 y sub 1 x sub n y sub n be an i i d sample drawn from this distribution in discrimination the bayes error l inf sub g p g x spl ne y is of crucial importance here we show that without further conditions on the distribution of x y no rate of convergence results can be obtained let spl phi sub n x sub 1 y sub 1 x sub n y sub n be an estimate of the bayes error and let spl phi sub n be a sequence of such estimates for any sequence a sub n of positive numbers converging to zero a distribution of x y may be found such that e l spl phi sub n x sub 1 y sub 1 x sub n y sub n spl ges a sub n often converges infinitely

classification error for a very large number of classes classification error is analyzed for a situation where the number of possible classes may be on the order of a hundred or more the error associated with classifying to a single class is shown to depend mainly on average nearest neighbor distance between class means noise level and effective dimensionality of the class mean distribution and not much on other aspects of the distribution noise correlation or number of classes since single class error is large separation of classes into groups is also explored group classification error has the same properties as single class error but the size of the error is moderated by the bayes overlap between groups standard curves are provided to predict single class and group error also discussed are the effect of pattern blurring on classification error and the nearest neighbor distance statistics throughout a distribution

representation of the drazin inverse for special block matrix abstract for a c n n the matrix x c n n is said to be the drazin inverse of a if it holds that xax x ax xa a r xa a r where r is the smallest nonnegative integer such that rank a r rank a r 1 in this paper we give the representation for the drazin inverse of the block matrix a a b 0 a b c n n a 2 a

a note on the drazin inverses with banachiewicz schur forms let h be a hilbert space m the closed subspace of h with orthocomplement m according to the orthogonal decomposition email protected m every operator email protected b h can be written in a block form m abcd in this note we give necessary and sufficient conditions for a partitioned operator matrix m to have the drazin inverse with banachiewicz schur form in addition this paper investigates the relations among the drazin inverse the moore penrose inverse and the group inverse when they can be expressed in the banachiewicz schur forms

additive results for the drazin inverse of block matrices and applications in this paper we consider the drazin inverse of a sum of two matrices and derive additive formulas under conditions weaker than those used in some recent papers on the subject as a corollary we get the main results from the paper of yang and liu h yang x liu the drazin inverse of the sum of two matrices and its applications j comput appl math 235 2011 1412 1417 as an application we give some new representations for the drazin inverse of a block matrix

formulas for the drazin inverse of special block matrices properties of the drazin inverse of the matrix f iie0 with e square are investigated based on this approach it is obtained an explicit formula for the drazin inverse of matrices of the form a ip tquv t where u v p and q are nxk the representation for a d is given in terms of kxk matrices involving the individual blocks under some conditions some special cases are also analyzed

green s functions in non local three dimensional linear elasticity in this paper we compare small deformations in an infinite linear elastic body due to the presence of point loads within the classical local formulation to the corresponding deformations in the peridynamic non local formulation owing to the linearity of the problem the response to a point load can be used to obtain the response to general body force loading functions by superposition using laplace and fourier transforms we thus obtain an integral representation for the three dimensional peridynamic solution with the help of green s functions we illustrate this new theoretical result by dynamic and static examples in one and three dimensions in addition to this main result we also derive the non local three dimensional jump conditions as well as the weak formulation of peridynamics together with the associated finite element discretization

expressions for the g drazin inverse of additive perturbed elements in a banach algebra abstract we study additive properties of the g drazin inverse in a banach algebra a in our development we derive a representation of the resolvent of a 2 2 matrix with entries in a which is then used to find explicit expressions for the g drazin inverse of the sum a b under new conditions on a b a as an application of our results we obtain a representation for the drazin inverse of a 2 2 complex block matrix in terms of the individual blocks under certain conditions

rectangular rigid stamp on a nonlocal elastic half plane first the solution of the problem of a rectangular rigid stamp uniformly moving on an elastic half plane is given secondly the solution is obtained for a motionless rectangular stamp on a nonlocal elastic half space the finite distribution of the stress under the stamp has been compared with the unbounded stresses of local case

a bootstrap technique for nearest neighbor classifier design a bootstrap technique for nearest neighbor classifier design is proposed our primary interest in designing a classifier is in small training sample size situations conventional bootstrapping techniques sample the training samples with replacement on the other hand our technique generates bootstrap samples by locally combining original training samples the nearest neighbor classifier is designed on the bootstrap samples and is tested on the test samples independent of training samples the performance of the proposed classifier is demonstrated on three artificial data sets and one real data set experimental results show that the nearest neighbor classifier designed on the bootstrap samples outperforms the conventional k nn classifiers as well as the edited 1 nn classifiers particularly in high dimensions

is the k nn classifier in high dimensions affected by the curse of dimensionality abstract there is an increasing body of evidence suggesting that exact nearest neighbour search in high dimensional spaces is affected by the curse of dimensionality at a fundamental level does it necessarily mean that the same is true for k nearest neighbours based learning algorithms such as the k nn classifier we analyse this question at a number of levels and show that the answer is different at each of them as our first main observation we show the consistency of a k approximate nearest neighbour classifier however the performance of the classifier in very high dimensions is provably unstable as our second main observation we point out that the existing model for statistical learning is oblivious of dimension of the domain and so every learning problem admits a universally consistent deterministic reduction to the one dimensional case by means of a borel isomorphism

the rate of convergence of k_n nn regression estimates and classification rules corresp the rate of convergence of k n nn regression estimates and the corresponding multiple classification error are calculated without assuming the existence of the density of the observations

the rates of convergence of kernel regression estimates and classification rules both nonrecursive and recursive nonparametric regression estimates are studied the rates of weak and strong convergence of kernel estimates as well as corresponding multiple classification errors are derived without assuming the existence of the density of the measurements an application of the obtained results to a nonparametric bayes predication is presented

group inverse for the block matrices with an invertible subblock let m abcd a is square be a square block matrix with an invertible subblock over a skew field k in this paper we give the necessary and sufficient conditions for the existence as well as the expressions of the group inverse for m under some conditions

further results on the group inverses and drazin inverses of anti triangular block matrices abstract in this paper the sufficient and necessary conditions for the existence of the group inverse for a class of 2 2 anti triangular block matrices m with some certain conditions satisfied and representations of the group inverse of m are obtained which can be regarded as the extension of bu et al c bu j zhao j zheng group inverse for a class 2 2 block matrices over skew fields appl math comput 204 2008 45 49 and c bu j zhao k zhang some results on group inverses of block matrices over skew fields electron j linear algebra 18 2009 117 125 further we use these results to determine the expressions of the drazin inverse of m finally two numerical examples are given to illustrate our results

on some theorems in the nonlocal theory of micropolar elasticity abstract this paper is concerned with the linear nonlocal theory of micropolar elasticity we have established a work and energy theorem and a uniqueness theorem without making any definiteness assumptions about the elastic moduli a reciprocity theorem is also established

a representation for the drazin inverse of block matrices with a singular generalized schur complement abstract consider a 2 2 block complex square matrix m a b c d where a and d are square matrices suppose that i aa d b o and c i aa d o where a d is the drazin inverse of a the representations of the drazin inverse m d have been studied in the case where the generalized schur complement s a ca d b is either zero or nonsingular in this paper we develop a representation under certain conditions for m d when s is singular and group invertible moreover this formula includes the case where s o or nonsingular a numerical example is given to illustrate the result

on exponential bounds on the bayes risk of the kernel classification rule the exponential distribution free bounds for the kernel classification rule are derived the equivalence of all modes of the global convergence of the rule is established under optimal assumptions on the smoothing sequence also derived is the optimal global rate of convergence of the kernel regression estimate within the class of lipschitz distributions the rate is optimal for the nonparametric regression but not for classifications it is shown using the martingale device that weak strong and complete l sub 1 bayes risk consistencies are equivalent consequently the conditions on the smoothing sequence h sub n to 0 and nh sub n to infinity are necessary and sufficient for bayes risk consistency of the kernel classification rule the rate of convergence of the kernel classification rule is also given

choice of the smoothing parameter and efficiency of k nearest neighbor classification abstract a simulation study was performed to investigate the sensitivity of the k nearest neighbor nn k rule of classification to the choice of k the optimal choice of k was found to be a function of the dimension of the sample space the size of the space the covariance structure and the sample proportions the nearest neighbor rules chosen using the k suggested by the simulations had correct classification rates at least as high as those rates for the linear discriminant function and the logistic regression method in particular the rule became more efficient as the difference in the covariance matrices increased and also when the difference in sample proportion was large an adaptive rule which selects k by iteratively maximizing the local mahalanobis distance is shown to be efficient thus abrogating the need to know the underlying population variance covariance structure

a different approach to eringen s nonlocal integral stress model with applications for beams abstract the nonlocal continuum theory either integral or differential form has been formulated evolved and widely used in our era to explain size effect phenomena in micro and nano structures in the case of euler bernoulli beam theory ebbt the nonlocal integral form produces energy consistent formulas unlike the nonlocal differential form revealing inconsistencies and giving rise to paradoxes in this work our overall research objective mainly focuses on nonlocal integral elasticity analysis of beams by employing a normalized symmetric kernel this kernel corresponds to a finite domain and will be called modified kernel from now on that satisfies all the properties of a probability density function as well as successfully handles the physical inconsistencies of classic types of kernel our concern is to investigate the static response of a beam with various types of loading and boundary conditions bcs by making use the modified kernel and the kernel corresponding to the two phase nonlocal integral tpni model carrying out numerical methods to our problems the deducing results appear more flexible behavior than those of classic local and the common nonlocal differential respectively what is more a comparison is made between the results of the aforementioned kernels to demonstrate the advantages of the modified kernel unlike the common nonlocal differential form the nonlocal integral forms do not give rise to paradoxes moreover it is critical that the tpni model does not raise paradoxes as has been observed in other publications presented in the literature

nonlocal elasticity and related variational principles abstract the eringen model of nonlocal elasticity is considered and its implications in solid mechanics studied the model is refined by assuming an attenuation function depending on the geodetical distance between material particles such that in the diffusion processes of the nonlocality effects certain obstacles as holes or cracks existing in the domain can be circumvented a suitable thermodynamic framework with nonlocality is also envisaged as a firm basis of the model the nonlocal elasticity boundary value problem for infinitesimal displacements and quasi static loads is addressed and the conditions for the solution uniqueness are established three variational principles nonlocal counterparts of classical ones i e the total potential energy the complementary energy and the mixed hu washizu principles are provided the former is used to derive a nonlocal type fem nl fem in which the symmetric global stiffness matrix reflects the nonlocality features of the problem an alternative standard fem based solution method is also provided which consists in an iterative procedure of the type local prediction nonlocal correction in which the nonlocality is simulated by an imposed like correction strain the potentialities of these analysis methods are pointed out their numerical implementations being the object of an ongoing research work

on the asymptotic probability of error in nonparametric discrimination let x y x 1 y 1 cdots x n y n be independent identically distributed random vectors from r d times 0 1 and let hat y be the k nearest neighbor estimate of y from x and the x i y i s we show that for all distributions of x y the limit of l n p hat y neq y exists and satisfies lim n rightarrow infty l n leq 1 a k r ast a k frac alpha sqrt k k 3 25 big 1 frac beta sqrt k 3 big k text odd k geq 5 where r ast is the bayes probability of error and alpha 0 3399 cdots and beta 0 9749 cdots are universal constants this bound is shown to be best possible in a certain sense

representations for the drazin inverse of 2 2 block operator matrix with singular schur complement abstract in this paper we present formulae for the drazin inverse of 2 2 block operator matrix with generalized schur complement being drazin invertible moreover necessary and sufficient conditions for the existence as well as the expressions for the group inverse are obtained

data dependent k sub n nn and kernel estimators consistent for arbitrary processes let x sub 1 x sub 2 be an arbitrary random process taking values in a totally bounded subset of a separable metric space associated with x sub i we observe y sub i drawn from an unknown conditional distribution f y x sub i x with continuous regression function m x e y x x the problem of interest is to estimate y sub n based on x sub n and the data x sub i y sub i sub i 1 sup n 1 we construct appropriate data dependent nearest neighbor and kernel estimators and show with a very elementary proof that these are consistent for every process x sub 1 x sub 2

nonlocal elasticity defined by eringen s integral model introduction of a boundary layer method abstract in this paper we consider a nonlocal elasticity theory defined by eringen s integral model and introduce for the first time a boundary layer method by presenting the exponential basis functions ebfs for such a class of problems the ebfs playing the role of the fundamental solutions are found so that they satisfy the governing equations on an unbounded domain some insight to the theory is given by showing that the ebfs satisfying the navier equations in the classical elasticity theory also satisfy the governing equations in the nonlocal theory some additional ebfs are particularly obtained for the nonlocal theory in order to use the ebfs on bounded domains the effects of the boundary conditions are taken into account by truncating the kernel attenuation function in the constitutive equations this leads to some residuals in the governing equations which appear near the boundaries a weighted residual approach is employed to minimize the residuals near the boundaries the method presented in this paper has much in common with trefftz methods especially when the influence area of the kernel function is much smaller than the main computational domain several one two dimensional problems are solved to demonstrate the way in which the ebfs can be used through the proposed boundary layer method

continuous and discontinuous finite element methods for a peridynamics model of mechanics in contrast to classical partial differential equation models the recently developed peridynamic nonlocal continuum model for solid mechanics is an integro differential equation that does not involve spatial derivatives of the displacement field as a result the peridynamic model admits solutions having jump discontinuities so that it has been successfully applied to fracture problems the peridynamic model features a horizon which is a length scale that determines the extent of the nonlocal interactions based on a variational formulation continuous and discontinuous galerkin finite element methods are developed for the peridynamic model discontinuous discretizations are conforming for the model without the need to account for fluxes across element edges through a series of simple one dimensional computational experiments we investigate the convergence behavior of the finite element approximations and compare the results with theoretical estimates one issue addressed is the effect of the relative sizes of the horizon and the grid for problems with smooth solutions we find that continuous and discontinuous piecewise linear approximations result in the same accuracy as that obtained by continuous piecewise linear approximations for classical models piecewise constant approximations are less robust and require the grid size to be small with respect to the horizon we then study problems having solutions containing jump discontinuities for which we find that continuous approximations are not appropriate whereas discontinuous approximations can result in the same convergence behavior as that seen for smooth solutions in case a grid point is placed at the locations of the jump discontinuities such results are directly obtained in the general case we show that such results can be obtained through a simple automated abrupt local refinement of elements containing the discontinuity in order to reduce the number of degrees of freedom while preserving accuracy we also briefly consider a hybrid discretization which combines continuous discretizations in regions where the solution is smooth with discontinuous discretizations in small regions surrounding the jump discontinuities

a note on the drazin inverse of an anti triangular matrix abstract in this paper we give formulae for the generalized drazin inverse m d of an anti triangular matrix m under some conditions moreover some particular cases of these results are also considered

the mechanically based approach to 3d non local linear elasticity theory long range central interactions abstract this paper presents the generalization to a three dimensional 3d case of a mechanically based approach to non local elasticity theory recently proposed by the authors in a one dimensional 1d case the proposed model assumes that the equilibrium of a volume element is attained by contact forces between adjacent elements and by long range forces exerted by non adjacent elements specifically the long range forces are modelled as central body forces depending on the relative displacement between the centroids of the volume elements measured along the line connecting the centroids further the long range forces are assumed to be proportional to a proper material dependent distance decaying function and to the products of the interacting volumes consistently with the modelling of the long range forces as central body forces the static boundary conditions enforced on the free surface of the solid involve only local stress due to contact forces the proposed 3d formulation is developed both in a mechanical and in a variational context for this the elastic energy functionals of the solid with long range interactions are introduced based on the principle of virtual work to set the proper correspondence between the mechanical and the kinematic variables of the model numerical applications are reported for 2d solids under plane stress conditions

stabilized nearest neighbor classifier and its statistical properties abstractthe stability of statistical analysis is an important indicator for reproducibility which is one main principle of the scientific method it entails that similar statistical conclusions can be reached based on independent samples from the same underlying population in this article we introduce a general measure of classification instability cis to quantify the sampling variability of the prediction made by a classification method interestingly the asymptotic cis of any weighted nearest neighbor classifier turns out to be proportional to the euclidean norm of its weight vector based on this concise form we propose a stabilized nearest neighbor snn classifier which distinguishes itself from other nearest neighbor classifiers by taking the stability into consideration in theory we prove that snn attains the minimax optimal convergence rate in risk and a sharp convergence rate in cis the latter rate result is established for general plug in classifiers under a low noise condition ext

a note on the group inverse of some 2 2 block matrices over skew fields abstract let k be a skew field and km n be the set of all m n matrices over k in this paper we give the necessary and sufficient conditions to the existence and the representations of the group inverse for block matrix ax yb a b 0 under some conditions where a y km n x b kn m

several expressions for the generalized drazin inverse of a block matrix in a banach algebra we present some new representations for the generalized drazin inverse of a block matrix with generalized schur complement being generalized drazin invertible in a banach algebra under conditions weaker than those used in recent papers on the subject

drazin inverse of partitioned matrices in terms of banachiewicz schur forms abstract let m a b c d be a partitioned matrix where a and d are square matrices denote the drazin inverse of a by a d the purpose of this paper is twofold firstly we develop conditions under which the drazin inverse of m having generalized schur complement s d ca d b group invertible can be expressed in terms of a matrix in the banachiewicz schur form and its powers secondly we deal with partitioned matrices satisfying rank m rank a d rank s d and give conditions under which the group inverse of m exists and a formula for its computation

a note on the representation for the drazin inverse of 2 2 block matrices abstract in this note we consider representations of the drazin inverse of 2 2 block matrices under conditions weaker than those used in recent papers on the subject in particular in d s djordjevic p s stanimirovic on the generalized drazin inverse and generalized resolvent czechoslovak math j 51 126 2001 617 634 r hartwig x li y wei representations for the drazin inverse of 2 2 block matrix siam j matrix anal appl 27 2006 757 771

on the inequality of cover and hart in nearest neighbor discrimination when x1 1 xn n are independent identically distributed random vectors from ird x 0 1 distributed as x and when is estimated by its nearest neighbor estimate 1 then cover and hart have shown that p 1 n 2e x 1 x 2r 1 r where r is the bayes probability of error and x p 1 x x they have conditions on the distribution of x we give two proofs one due to stone and a short original one of the same result for all distributions of x if ties are carefully taken care of we also show that p 1 x1 1 xn n converges in probability to a constant for all distributions of x thereby strengthening results of wagner and fritz

the group inverse of a triangular matrix block conditions are given for the group inverse of a triangular matrix over a field to exist these involve the zero nonzero schur complement applications are given to idempotent matrices

relationships between dmp relation and some partial orders let h be an infinite dimensional complex hilbert space and let b h be the algebra of all bounded linear operators on h the equivalent definition of the dmp relation on b h using operator block matrices is introduced we present a new interpretation of this relation which allows to generalize many known results for matrices to general hilbert spaces regarding the dmp relation and its relationship with the star order and minus partial order also some properties of dmp relation defined with the help of idempotent operators are investigated

almost everywhere convergence of a recursive regression function estimate and classification corresp it is shown that the recursive kernel estimate of the regression function e y x x is consistent at almost every x mu regardless of the distribution mu of x thus the result is distribution free from this we show that the risk for a suitable classification rule derived from the estimate converges to bayes risk no matter what the class distributions are

on the rate of convergence of the bagged nearest neighbor estimate bagging is a simple way to combine estimates in order to improve their performance this method suggested by breiman in 1996 proceeds by resampling from the original data set constructing a predictor from each subsample and decide by combining by bagging an n sample the crude nearest neighbor regression estimate is turned into a consistent weighted nearest neighbor regression estimate which is amenable to statistical analysis letting the resampling size kn grows appropriately with n it is shown that this estimate may achieve optimal rate of convergence independently from the fact that resampling is done with or without replacement since the estimate with the optimal rate of convergence depends on the unknown distribution of the observations adaptation results by data splitting are presented

on the asymptotic properties of smoothed estimators of the classifiction error rate abstract a smoothed correction of the classification error rate counting estimate is examined the consistency of the estimate under very general conditions is established for normal distributions asymptotic theory reveals that the relative deficiency of the counting estimate with respect to the smoothed one can tend to infinity as the sample size increases

properties of bagged nearest neighbour classifiers summary it is shown that bagging a computationally intensive method asymptotically improves the performance of nearest neighbour classifiers provided that the resample size is less than 69 of the actual sample size in the case of with replacement bagging or less than 50 of the sample size for without replacement bagging however for larger sampling fractions there is no asymptotic difference between the risk of the regular nearest neighbour classifier and its bagged version in particular neither achieves the large sample performance of the bayes classifier in contrast when the sampling fractions converge to 0 but the resample sizes diverge to 1 the bagged classifier converges to the optimal bayes rule and its risk converges to the risk of the latter these results are most readily seen when the two populations have well defined densities but they may also be derived in other cases where densities exist in only a relative sense cross validation can be used effectively to choose the sampling fraction numerical calculation is used to illustrate these theoretical properties

a note on the formulas for the drazin inverse of the sum of two matrices abstract in this paper we give the explicit formulas for the drazin inverse of p q under some conditions furthermore we give the representations for the drazin inverse of some block matrices a b c d a and d are square under some conditions

generalized drazin inverses of anti triangular block matrices abstract in this paper we give formulae for the generalized drazin inverse m d of an anti triangular matrix m a b c 0 in two different ways one is to express m d in terms of a d with arbitrary b and c the other is to express m d in terms of b d and c d with arbitrary a moreover the results are applied to obtain generalized drazin inverses of various structured matrices and some special cases are analyzed

hammerstein system identification by non parametric regression estimation a discrete time multiple input non linear hammerstein system is identified the dynamical subsystem is recovered using the standard correlation method the main results concern estimation of the non linear memoryless subsystem no conditions concerning the functional form of the transform characteristic of the subsystem are made and an algorithm for estimation of the characteristic is given the algorithm is simply a non parametric kernel estimate of the regression function calculated from the dependent data it is shown that the algorithm converges to the characteristic of the subsystem in the pointwise as well as the global sense for sufficiently smooth characteristics the rate of convergence is o n 1 2 d in probability where d is the dimension of the input variable

on almost sure convergence of conditional empirical distribution functions on etudie la convergence presque sure d une distribution empirique conditionnelle de type noyau a la fois en norme sup et en norme sup ponderee

representations for the drazin inverse of the sum p q r s and its applications abstract let p q r and s be complex square matrices and m p q r s a quadruple p q r s is called a pseudo block decomposition of m if pq qp 0 ps sq qr rp 0 and r d s d 0 where r d and s d are the drazin inverses of r and s respectively we investigate the problem of finding formulae for the drazin inverse of m the explicit representations for the drazin inverses of m and p q r are developed under some assumptions as its application some representations are presented for 2 2 block matrices a b 0 c and a b d c where the blocks a and c are square matrices several results of this paper extend the well known representation for the drazin inverse of a b 0 c given by hartwig and shoaf meyer and rose in 1977 an illustrative example is given to verify our new representations

representations for the drazin inverses of 2 2 block matrices abstract let m denote a 2 2 block complex matrix a b c d where a and d are square matrices not necessarily with the same orders in this paper explicit representations for the drazin inverse of m are presented under the condition that bd i c 0 for i 0 1 n 1 where n is the order of d