a unifying study between modal like operators topologies and fuzzy sets the paper presents the essential connections between modal like operators topologies and fuzzy sets we show for example that each fuzzy set determines a preorder and an alexandrov topology and that similar correspondences hold also for the other direction further a category for preorder based fuzzy sets is defined and it is shown that its equivalent subcategory of representatives is isomorphic to the categories of preordered sets and alexandrov spaces moreover joins meets and complements for the objects in this category of representatives are determined this suggests how to define for fuzzy subsets of a certain universe the lattice operations in a canonical way

axiomatics for fuzzy rough sets abstract a fuzzy t rough set consists of a set x and a t similarity relation r on x where t is a lower semi continuous triangular norm we generalize the farinas prade definition for the upper approximation operator a i x i x of a fuzzy t rough set x r given originally for the special case t min to the case of arbitrary t we propose a new definition for the lower approximation operator a i x i x of x r our definition satisfies the two important identities a a a and a a a as well as a number of other interesting properties we provide axiomatics to fully characterize those upper and lower approximations

topological structure of generalized rough sets in this paper we investigate some properties of x t q such as compactness separate property lindelof property and connectedness where x t q is induced by a reflexive relation q on x which may be infinite moreover we introduce the concept of approximating spaces and research their characterizations and properties particularly we obtain a sufficient and necessary condition that topological spaces are approximating spaces these will be not only conducive to better understanding basic concepts and properties of a rough set but also have theory and actual significance to topology

tests for high dimensional covariance matrices we propose tests for sphericity and identity of high dimensional covariance matrices the tests are nonparametric without assuming a specific parametric distribution for the data they can accommodate situations where the data dimension is much larger than the sample size namely the large p small n situations we demonstrate by both theoretical and empirical studies that the tests have good properties for a wide range of dimensions and sample sizes we applied the proposed test on a microarray dataset on yorkshire gilts and tested for the covariance structure for the expression levels for sets of genes

clt for linear spectral statistics of large dimensional sample covariance matrices let bn 1 n t1 2nxnx nt1 2n where xn xij is n n with i i d complex standardized entries having finite fourth moment and t1 2n is a hermitian square root of the nonnegative definite hermitian matrix tn the limiting behavior as n with n n approaching a positive constant of functionals of the eigenvalues of bn where each is given equal weight is studied due to the limiting behavior of the empirical spectral distribution of bn it is known that these linear spectral statistics converges a s to a nonrandom quantity this paper shows their rate of convergence to be 1 n by proving after proper scaling that they form a tight sequence moreover if exppx211 0 and expp x11 4 2 or if x11 and tn are real and exppx411 3 they are shown to have gaussian limits

stress intensity factors by enriched mixed finite elements an enriched finite element model for linear elastic fracture mechanics is developed for a mixed variational statement the independent approximations for the displacement and stress components are enriched by adding the near field analytic expressions for a cracked body to the polynomial approximations of a conventional element this allows for an accurate representation of the stress and displacement fields near the crack tip and also results in the direct calculation of the appropriate stress intensity factors the accuracy of this formulation is demonstrated through several numerical examples

model based gaussian and non gaussian clustering abstract the classification maximum likelihood approach is sufficiently general to encompass many current clustering algorithms including those based on the sum of squares criterion and on the criterion of friedman and rubin 1967 however as currently implemented it does not allow the specification of which features orientation size and shape are to be common to all clusters and which may differ between clusters also it is restricted to gaussian distributions and it does not allow for noise we propose ways of overcoming these limitations a reparameterization of the covariance matrix allows us to specify that some features but not all be the same for all clusters a practical framework for non gaussian clustering is outlined and a means of incorporating noise in the form of a poisson process is described an approximate bayesian method for choosing the number of clusters is given the performance of the proposed methods is studied by simulation with encouraging results the methods are applied to the analysis of a data set arising in the study of diabetes and the results seem better than those of previous analyses rh

a comprehensive study of implicator conjunctor based and noise tolerant fuzzy rough sets definitions properties and robustness analysis abstract both rough and fuzzy set theories offer interesting tools for dealing with imperfect data while the former allows us to work with uncertain and incomplete information the latter provides a formal setting for vague concepts the two theories are highly compatible and since the late 1980s many researchers have studied their hybridization in this paper we critically evaluate most relevant fuzzy rough set models proposed in the literature to this end we establish a formally correct and unified mathematical framework for them both implicator conjunctor based definitions and noise tolerant models are studied we evaluate these models on two different fronts firstly we discuss which properties of the original rough set model can be maintained and secondly we examine how robust they are against both class and attribute noise by highlighting the benefits and drawbacks of the different fuzzy rough set models this study appears a necessary first step to propose and develop new models in future research

on t congruence l relations on groups and rings in this paper we further study t congruence l relations on groups and rings and give the formulas for calculating the t congruence l relations generated by l relations where t is an arbitrary infinitely distributive t norm on a given complete brouwerian lattice l

limit of the smallest eigenvalue of a large dimensional sample covariance matrix in this paper the authors show that the smallest if p leq n or the p n 1 th smallest if p n eigenvalue of a sample covariance matrix of the form 1 n xx tends almost surely to the limit 1 sqrt y 2 as n rightarrow infty and p n rightarrow y in 0 infty where x is a p times n matrix with iid entries with mean zero variance 1 and fourth moment finite also as a by product it is shown that the almost sure limit of the largest eigenvalue is 1 sqrt y 2 a known result obtained by yin bai and krishnaiah the present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices

optimization algorithms for the solution of the frictionless normal contact between rough surfaces this paper revisits the fundamental equations for the solution of the frictionless unilateral normal contact problem between a rough rigid surface and a linear elastic half plane using the boundary element method bem after recasting the resulting linear complementarity problem lcp as a convex quadratic program qp with nonnegative constraints different optimization algorithms are compared for its solution i a greedy method based on different solvers for the unconstrained linear system conjugate gradient cg gauss seidel cholesky factorization ii a constrained cg algorithm iii the alternating direction method of multipliers admm and iv the non negative least squares nnls algorithm possibly warm started by accelerated gradient projection steps or taking advantage of a loading history the latter method is two orders of magnitude faster than the greedy cg method and one order of magnitude faster than the constrained cg algorithm finally we propose another type of warm start based on a refined criterion for the identification of the initial trial contact domain that can be used in conjunction with all the previous optimization algorithms this method called cascade multi resolution cmr takes advantage of physical considerations regarding the scaling of the contact predictions by changing the surface resolution the method is very efficient and accurate when applied to real or numerically generated rough surfaces provided that their power spectral density function is of power law type as in case of self affine fractal surfaces

relationship between generalized rough sets based on binary relation and covering rough set theory is a powerful tool for dealing with uncertainty granularity and incompleteness of knowledge in information systems this paper systematically studies a type of generalized rough sets based on covering and the relationship between this type of covering based rough sets and the generalized rough sets based on binary relation firstly we present basic concepts and properties of this kind of rough sets then we investigate the relationships between this type of generalized rough sets and other five types of covering based rough sets the major contribution in this paper is that we establish the equivalency between this type of covering based rough sets and a type of binary relation based rough sets through existing results in binary relation based rough sets we present axiomatic systems for this type of covering based lower and upper approximation operations in addition we explore the relationships among several important concepts such as minimal description reduction representative covering exact covering and unary covering in covering based rough sets investigation of this type of covering based will benefit to our understanding of other types of rough sets based on covering and binary relation

contact between representative rough surfaces a numerical analysis of mechanical frictionless contact between rough self affine elastic manifolds was carried out it is shown that the lower cutoff wave number in surface spectra is a key parameter controlling the representativity of the numerical model using this notion we demonstrate that for representative surfaces the evolution of the real contact area with load is universal and independent of the hurst roughness exponent by introducing a universal law containing three constants we extend the study of this evolution beyond the limit of infinitesimal area fractions

representative contact element size determination for micromechanical contact analysis of self affine topographies abstract a generic and straightforward numerical strategy for the determination of the size of representative contact elements rces which are employed in finite element contact homogenization procedures is proposed in this contribution the approach enables the determination of the length height and mesh discretisation of rces that provide a good statistical representation of the contact interface described by a given set of topography properties the case of gaussian self affine and elastic rough profiles under normal frictionless and non adhesive contact is analysed in detail a collection of conditions has been derived for two dimensional problems based on a trade off between the convergence of the real contact area and the computational cost with the increasing width of the roughness power spectrum the restrictions imposed on the length and mesh size can be relaxed allowing to reduce the size of numerical models the corresponding class of problems in three dimensions has also been studied in this case the influence of the numerical scheme adopted for the evaluation of the contact area has been analysed leading to the identification of two bounds which converge for the same value with progressively finer meshes state of the art numerical results fall within the bounded region and the application of the area correction technique proposed by yastrebov et al to the upper node based bound accelerates the convergence of the mesh and renders a good agreement with reference data

on the topological properties of fuzzy rough sets this paper is devoted to the discussion of the relationship between fuzzy rough set models and fuzzy topologies on a finite universe the tc axiom for fuzzy topology is proposed it is proved that the set of all lower approximation sets based on a reflexive and transitive fuzzy relation consists of a fuzzy topology which satisfies tc axiom and conversely a fuzzy topology which obey tc axiom is just the set of all lower approximation sets under a reflexive and transitive fuzzy relation that is to say there exists a one to one correspondence between the set of all reflexive and transitive fuzzy relations and the set of all fuzzy topologies which satisfy tc axiom

topological characterizations of generalized fuzzy rough sets in this paper topological characterizations of generalized fuzzy rough sets are investigated in the context of basic rough equalities various fuzzy topologies induced by different fuzzy relations are studied with respect to the lower fuzzy rough approximation operator determined by a fuzzy implicator which is left continuous in the first argument and right continuous in the second argument the relationships among fuzzy topologies induced by fuzzy relations are discussed the algebraic structures of t similarity set of fuzzy relations are investigated with respect to different fuzzy implicators where t is a left continuous t norm moreover the t transitive subset of t similarity set is proven to be a complete distributive lattice

regularized gaussian discriminant analysis through eigenvalue decomposition abstract friedman proposed a regularization technique rda of discriminant analysis in the gaussian framework rda uses two regularization parameters to design an intermediate classifier between the linear the quadratic the nearest means classifiers in this article we propose an alternative approach called edda that is based on the reparameterization of the covariance matrix k of a group gk in terms of its eigenvalue decomposition k k d k a k d k where k specifies the volume of density contours of gk the diagonal matrix of eigenvalues specifies its shape the eigenvectors specify its orientation variations on constraints concerning volumes shapes orientations k a k and d k lead to 14 discrimination models of interest for each model we derived the normal theory maximum likelihood parameter estimates our approach consists of selecting a model by minimizing the sample based estimate of future misclassification risk by cross validation numerical experiments on simulated and rea

fuzzy rough sets fuzzy preorders and fuzzy topologies this paper shows that observations made by different authors at different times regarding one to one correspondence between the family of fuzzy preorders on a nonempty set and the family of all fuzzy topologies on this set satisfying certain extra conditions are essentially equivalent

assessing a mixture model for clustering with the integrated completed likelihood we propose an assessing method of mixture model in a cluster analysis setting with integrated completed likelihood for this purpose the observed data are assigned to unknown clusters using a maximum a posteriori operator then the integrated completed likelihood icl is approximated using the bayesian information criterion bic numerical experiments on simulated and real data of the resulting icl criterion show that it performs well both for choosing a mixture model and a relevant number of clusters in particular icl appears to be more robust than bic to violation of some of the mixture model assumptions and it can select a number of dusters leading to a sensible partitioning of the data

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

on relationship between modified sets topological spaces and rough sets abstract in this paper we define modifiers by relations especially weakening and substantiating modifiers are defined by a so called accessibility relation which is a reflexive relation on a non empty set x after presenting some main results we prove that this type of modifiers will satisfy the kuratowski closure axioms this means that modifiers in fact induce topological spaces also rough sets are considered to be a special case of modified sets

determination of the size of the representative volume element for random composites statistical and numerical approach the representative volume element rve plays a central role in the mechanics and physics of random heterogeneous materials with a view to predicting their effective properties a quantitative definition of its size is proposed in this work a rve size can be associated with a given precision of the estimation of the wanted overall property and the number of realizations of a given volume v of microstructure that one is able to consider it is shown to depend on the investigated morphological or physical property the contrast in the properties of the constituents and their volume fractions the methodology is applied to a specific random microstructure namely a two phase three dimensional voronoi mosaic finite element simulations of volumes of different sizes are performed in the case of linear elasticity and thermal conductivity the volumes are subjected to homogeneous strain stress or periodic boundary conditions the effective properties can be determined for large volumes and a small number of realizations conversely smaller volumes can be used providing that a sufficient number of realizations are considered a bias in the estimation of the effective properties is observed for too small volumes for all types of boundary conditions the variance of computed apparent properties for each volume size is used to define the precision of the estimation the key notion of integral range is introduced to relate this error estimation and the definition of the rve size for given wanted precision and number of realizations one is able to provide a minimal volume size for the computation of effective properties the results can also be used to predict the minimal number of realizations that must be considered for a given volume size in order to estimate the effective property for a given precision the rve sizes found for elastic and thermal properties but also for a geometrical property like volume fraction are compared

on the topological properties of generalized rough sets in this paper we consider some topological properties of generalized rough sets induced by binary relations and show that1 any serial binary relation can induce a topology 2 let r be a binary relation on a universe u t r and e r denote the transitive closure and the equivalence closure of r respectively if r is a reflexive relation on u then r and t r induce the same topology i e t r t t r the interior and closure operators of the topology t r induced by r are the lower and upper approximation operators t r and t r respectively moreover r t r t r where r t r is the relation induced by the topology t r 3 when r is a reflexive and symmetric relation r and e r induce the same topology i e t r t e r the interior and closure operators of the topology t r induced by r are the lower and upper approximation operators e r and e r respectively moreover r t r e r 4 based on the above conclusions the notion of topological reduction of incomplete information systems is proposed and characterizations of reduction of consistent incomplete decision tables are obtained

granular variable precision fuzzy rough sets with general fuzzy relations the variable precision fuzzy rough sets were proposed to remedy the defects of preexisting fuzzy rough set models however the variable precision fuzzy rough sets were only defined and investigated on fuzzy similarity relations in this paper the granular variable precision fuzzy rough sets with general fuzzy relations are proposed on arbitrary fuzzy relations the equivalent expressions of the approximation operators are given with fuzzy co implications on arbitrary fuzzy relations which can calculate efficiently the approximation operators the granular variable precision fuzzy rough sets are characterized from the constructive approach which are investigated on different fuzzy relations the conclusions on the variable precision fuzzy rough sets are generalized into the granular variable precision fuzzy rough sets

finite element analysis of contact between elastic self affine surfaces finite element methods are used to study nonadhesive frictionless contact between elastic solids with self affine surfaces we find that the total contact area rises linearly with the load at small loads the mean pressure in the contact regions is independent of load and proportional to the root mean square slope of the surface the constant of proportionality is nearly independent of the poisson ratio and roughness exponent and lies between previous analytic predictions the contact morphology is also analyzed connected contact regions have a fractal area and perimeter the probability of finding a cluster of area a c drops as a tau c where tau increases with a decrease in roughness exponent the distribution of pressures shows an exponential tail that is also found in many jammed systems these results are contrasted to simpler models and experiments

on the generalization of fuzzy rough sets rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets the existing generalizations of fuzzy rough sets are all based on special fuzzy relations fuzzy similarity relations t similarity relations it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches in this paper from the viewpoint of constructive approach we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined in axiomatic approach we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms the lattice and topological structures of fuzzy rough sets are also proposed in order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets a special lower approximation operator is applied to a fuzzy reasoning system which coincides with the mamdani algorithm

new results on single axioms for l fuzzy rough approximation operators abstract this paper further studies axiomatic characterizations of l fuzzy rough sets where l denotes a residuated lattice single axioms for upper l fuzzy rough approximation operators are investigated in two different approaches which describe upper l fuzzy approximation operators with ordinary l fuzzy operations and l fuzzy product operation respectively we characterize upper l fuzzy rough approximation operators by only one axiom with ordinary l fuzzy operations when the l fuzzy relation is euclidean as well as the composition of euclidean with serial resp reflexive we also discuss the single axioms for upper l fuzzy rough approximation operators with l fuzzy product operation on arbitrary l fuzzy relations that are not limited as either general l fuzzy relations or symmetric l fuzzy relations the single axioms for lower l fuzzy rough approximation operators are discussed with ordinary l fuzzy operations regardless of the regularity of residuated lattice l

generalized rough sets based on reflexive and transitive relations in this paper we investigate the relationship between generalized rough sets induced by reflexive and transitive relations and the topologies on the universe which is not restricted to be finite it is proved that there exists a one to one correspondence between the set of all reflexive and transitive relations and the set of all topologies which satisfy a certain kind of compactness condition

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

t similarity of fuzzy relations and related algebraic structures in this paper we show that fuzzy alexandrov topologies can be induced by fuzzy relations the concept of t similarity of fuzzy relations is introduced variations of the same fuzzy relation are investigated the fact that every fuzzy relation is solely t similar to some t preorder fuzzy relation is proved a characteristic condition for fuzzy relations to be t transitive is established these results illustrate that fuzzy relations can be studied by means of topology moreover algebraic structures based on t similarity of fuzzy relations are obtained

generalized rough sets based on relations rough set theory has been proposed by pawlak as a tool for dealing with the vagueness and granularity in information systems the core concepts of classical rough sets are lower and upper approximations based on equivalence relations this paper studies arbitrary binary relation based generalized rough sets in this setting a binary relation can generate a lower approximation operation and an upper approximation operation but some of common properties of classical lower and upper approximation operations are no longer satisfied we investigate conditions for a relation under which these properties hold for the relation based lower and upper approximation operations

topological properties of generalized approximation spaces rough set theory is a powerful mathematical tool for dealing with inexact uncertain or vague information the core concepts of rough set theory are information systems and approximation operators of approximation spaces approximation operators draw close links between rough set theory and topology this paper concerns generalized approximation spaces via topological methods and studies topological properties of rough sets classical separation axioms compactness and connectedness for topological spaces are extended to generalized approximation spaces relationships among separation axioms for generalized approximation spaces and relationships between topological spaces and their induced generalized approximation spaces are investigated an example is given to illustrate a new approach to recover missing values for incomplete information systems by regularity of generalized approximation spaces

topological and lattice structures of l fuzzy rough sets determined by lower and upper sets this paper builds the topological and lattice structures of l fuzzy rough sets by introducing lower and upper sets in particular it is shown that when the l relation is reflexive the upper resp lower set is equivalent to the lower resp upper l fuzzy approximation set then by the upper resp lower set it is indicated that an l preorder is the equivalence condition under which the set of all the lower resp upper l fuzzy approximation sets and the alexandrov l topology are identical however associating with an l preorder the equivalence condition that l interior resp closure operator accords with the lower resp upper l fuzzy approximation operator is investigated at last it is proven that the set of all the lower resp upper l fuzzy approximation sets forms a complete lattice when the l relation is reflexive

constructive and algebraic methods of the theory of rough sets this paper reviews and compares constructive and algebraic approaches in the study of rough sets in the constructive approach one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation different classes of rough set algebras are obtained from different types of binary relations in the algebraic approach one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators various classes of rough set algebras are characterized by different sets of axioms axioms of approximation operators guarantee the existence of certain types of binary relations producing the same operators

on characterizations of i t fuzzy rough approximation operators in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties various fuzzy generalizations of rough approximations have been made in the literature this paper proposes a general framework for the study of i t fuzzy rough approximation operators within which both constructive and axiomatic approaches are used in the constructive approach a pair of lower and upper generalized fuzzy rough approximation operators determined by an implicator i and a triangular norm t is first defined basic properties of i t fuzzy rough approximation operators are investigated the connections between fuzzy relations and fuzzy rough approximation operators are further established in the axiomatic approach an operator oriented characterization of rough sets is proposed that is i t fuzzy approximation operators are defined by axioms different axiom sets of t upper and i lower fuzzy set theoretic operators guarantee the existence of different types of fuzzy relations which produce the same operators finally an open problem proposed by radzikowska and kerre in fuzzy sets and systems 126 2002 137 is solved

fuzzy preorder and fuzzy topology this paper presents by means of alexandrov topology for fuzzy preordered sets and specialization order for fuzzy topological spaces a systematic investigation of the interrelationship between fuzzy preordered sets topological spaces and fuzzy topological spaces

some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size this paper analyzes whether standard covariance matrix tests work when dimensionality is large and in particular larger than sample size in the latter case the singularity of the sample covariance matrix makes likelihood ratio tests degenerate but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well defined we study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together with their ratio converging to a finite nonzero limit we find that the existing test for sphericity is robust against high dimensionality but not the test for equality of the covariance matrix to a given matrix for the latter test we develop a new correction to the existing test statistic that makes it robust against high dimensionality

the central limit theorem for local linear statistics in classical compact groups and related combinatorial identities author s soshnikov alexander abstract we discuss clt for the global and local linear statistics of random matrices from classical compact groups the main part of our proofs are certain combinatorial identities much in the spirit of works by kac and spohn

similarity of fuzzy relations based on fuzzy topologies induced by fuzzy rough approximation operators this paper studies similarity of fuzzy relations which is based on fuzzy topologies induced by fuzzy rough approximation operators first fuzzy topologies induced by fuzzy rough approximation operators are investigated and the concept of similarity of fuzzy relations is introduced next it is proved that every fuzzy relation is uniquely similar to some fuzzy preorder relation and a characteristic condition for fuzzy relations to be transitive is established finally algebraic structures on similarity of fuzzy relations are obtained

from infinitesimal to full contact between rough surfaces evolution of the contact area we carry out a statistically meaningful study on self affine rough surfaces in elastic frictionless non adhesive contact we study the evolution of the true contact area under increasing squeezing pressure from zero up to full contact which enables us to compare the numerical results both with asperity based models at light pressures and with persson s contact model for the entire range of pressures a good agreement of numerical results with persson s model is obtained for the shape of the area pressure curve especially near full contact however we obtain qualitatively different results for its derivative at light pressures we investigate the effects of the longest and shortest wavelengths in surface spectrum which control the surface gaussianity and spectrum breadth nayak s parameter we revisit the influence of nayak s parameter which is frequently assumed to play an important role in mechanics of rough contact

exact separation of eigenvalues of large dimensional sample covariance matrices let b n 1 n t n 1 2 x n x n t n 1 2 where x n is n times n with i i d complex standardized entries having finite fourth moment and t n 1 2 is a hermitian square root of the nonnegative definite hermitian matrix t n it was shown in an earlier paper by the authors that under certain conditions on the eigenvalues of t n with probability 1 no eigenvalues lie in any interval which is outside the support of the limiting empirical distribution known to exist for all large n for these n the interval corresponds to one that separates the eigenvalues of t n the aim of the present paper is to prove exact separation of eigenvalues that is with probability 1 the number of eigenvalues of b n and t n lying on one side of their respective intervals are identical for all large n

a new efficient numerical method for contact mechanics of rough surfaces abstract in this work a numerical method has been developed to investigate the adhesionless contact mechanics between rough surfaces to solve the elastic problem a boundary elements approach is used with self equilibrated square elements the domain of integration is discretized developing an intelligent adaptive mesh and obtaining a considerable memory saving the numerical convergence of the method has been verified by comparing the results with the hertzian solution and by checking the stress probability distribution at the contact interface the methodology has been then utilized to analyse the contact between an elastic flat substrate and a periodic numerically generated self affine fractal rigid surface the fractal surface has been generated by employing spectral methods the results of our investigation supports the findings of some analytical theories persson 2001 and numerical findings yang et al 2006 hyun et al 2004 carbone and bottiglione 2008 campana and muser 2007 in terms of linearity between contact area and load and stress probability distributions

overhanging of membranes and filaments adhering to periodic graph substrates abstract this paper mathematically studies membranes and filaments adhering to periodic patterned substrates in a one dimensional model the problem is formulated by the minimizing problem of an elastic energy with a contact potential on graph substrates global minimizers ground states are mainly considered in view of their graph representations our main results exhibit sufficient conditions for the graph representation and examples of situations where any global minimizer must overhang

substitution principle for clt of linear spectral statistics of high dimensional sample covariance matrices with applications to hypothesis testing abstract sample covariance matrices are widely used in multivariate statistical analysis the central limit theorems clt s for linear spectral statistics of high dimensional non centralized sample covariance matrices have received considerable attention in random matrix theory and have been applied to many high dimensional statistical problems however known population mean vectors are assumed for non centralized sample covariance matrices some of which even assume gaussian like moment conditions in fact there are still another two most frequently used sample covariance matrices the me moment estimator constructed by subtracting the sample mean vector from each sample vector and the unbiased sample covariance matrix by changing the denominator n as n n 1 in the me without depending on unknown population mean vectors in this paper we not only establish the new clt s for non centralized sample covariance matrices when the gaussian like moment conditions do not hold but also characterize the non negligible differences among the clt s for the three classes of high dimensional sample covariance matrices by establishing a substitution principle by substituting the adjusted sample size n n 1 for the actual sample size n in the centering term of the new clt s we obtain the clt of the unbiased sample covariance matrices moreover it is found that the difference between the clt s for the me and unbiased sample covariance matrix is non negligible in the centering term although the only difference between two sample covariance matrices is a normalization by n and n 1 respectively the new results are applied to two testing problems for high dimensional covariance matrices

corrections to lrt on large dimensional covariance matrix by rmt in this paper we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from gaussian populations when the dimension p is large compared to the sample size n next using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random f matrices we propose necessary corrections for these lr tests to cope with high dimensional effects the asymptotic distributions of these corrected tests under the null are given simulations demonstrate that the corrected lr tests yield a realized size close to nominal level for both moderate p around 20 and high dimension while the traditional lr tests with 2 approximation fails another contribution from the paper is that for testing the equality between two covariance matrices the proposed correction applies equally for non gaussian populations yielding a valid pseudo likelihood ratio test

central limit theorem for mutual information of large mimo systems with elliptically correlated channels by considering elliptical correlations in the channel matrix of a large mimo system this paper investigates how the non linear dependency affects the asymptotic behaviors of shannon s mutual information as the numbers of antennas at transmitter and receiver both tend to infinity at the same rate beyond the almost sure convergence of the information per transmit antenna we also propose several central limit theorems for the information statistics under various scenarios the results show that the non linear correlation has a little impact on the almost sure convergence but has a tremendous effect on the fluctuation of the information statistics in particular the centralized information statistics could experience a phase transition in convergence rate when the strength of the non linear correlation increases as extensions several central limit theorems are established as well for general linear spectral statistics of large sample covariance matrices in elliptical distributions

on structure testing for component covariance matrices of a high dimensional mixture summary by studying the family of p dimensional scale mixtures the paper shows for the first time a non trivial example where the eigenvalue distribution of the corresponding sample covariance matrix does not converge to the celebrated marcenko pastur law a different and new limit is found and characterized the reasons for failure of the marcenko pastur limit in this situation are found to be a strong dependence between the p co ordinates of the mixture next we address the problem of testing whether the mixture has a spherical covariance matrix to analyse the traditional john s type test we establish a novel and general central limit theorem for linear statistics of eigenvalues of the sample covariance matrix it is shown that john s test and its recent high dimensional extensions both fail for high dimensional mixtures precisely because of the different spectral limit above as a remedy a new test procedure is constructed afterwards for the sphericity hypothesis this test is then applied to identify the covariance structure in model based clustering it is shown that the test has much higher power than the widely used integrated classification likelihood and bayesian information criteria in detecting non spherical component covariance matrices of a high dimensional mixture

gaussian parsimonious clustering models gaussian clustering models are useful both for understanding and suggesting powerful criteria banfield and raftery 1993 have considered a parametrization of the variance matrix ek of a cluster pk in terms of its eigenvalue decomposition ek lkdkakd k lk where lk defines the volume of pk dk is an orthogonal matrix which defines its orientation and ak is a diagonal matrix with determinant 1 which defines its shape this parametrization allows us to propose many general clustering criteria from the simplest one spherical cluster with equal volumes which leads to the classical k means criterion to the most complex one unknown and different volumes orientations and shapes for all clusters methods of optimization to derive the maximum likelihood estimates as well as the practical usefulness of these models are discussed we especially analyze the influence of the volumes of clusters we report monte carlo simulations and an application on stellar data which dramatically illustrated the relevance of allowing clusters to have different volumes

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

the relationship between l fuzzy rough set and l topology various fuzzy generalizations of rough approximations have been proposed in the literature this paper is devoted to the discussion of the relationship between l fuzzy rough sets and l topologies on an arbitrary universe finally one to one correspondence between the set of all reflexive transitive l relations and the set of all alexandrov l topologies is obtained

using green s function molecular dynamics to rationalize the success of asperity models when describing the contact between self affine surfaces we use green s function molecular dynamics to evaluate the effectiveness of asperity models when describing the contact mechanics of elastic solids with self affine surfaces surfaces are created with the help of a fourier filtering algorithm and the interactions between the solids are modeled via hard wall potentials we illustrate how the real area of contact a real is formed by a set of contact clusters two different regimes are identified when the normal force per cluster l c is plotted as a function of its area a c small clusters satisfy a hertzian type law l c approximately a c 32 while large clusters display a linear l c approximately a c behavior it is shown how the area a c where the crossover between the two regimes takes place depends only on the roughness at the smallest length scale if the longitudinal dimension of the surface remains unaltered moreover our results display a distribution of cluster sizes p a c remaining nearly constant for areas smaller than a c while showing power law decay above such a critical value furthermore we found the heights of the contacting atoms to be normally distributed with width inversely proportional to the surface roughness

generalized fuzzy rough sets determined by a triangular norm the theory of rough sets has become well established as an approach for uncertainty management in a wide variety of applications various fuzzy generalizations of rough approximations have been made over the years this paper presents a general framework for the study of t fuzzy rough approximation operators in which both the constructive and axiomatic approaches are used by using a pair of dual triangular norms in the constructive approach some definitions of the upper and lower approximation operators of fuzzy sets are proposed and analyzed by means of arbitrary fuzzy relations the connections between special fuzzy relations and the t upper and t lower approximation operators of fuzzy sets are also examined in the axiomatic approach an operator oriented characterization of rough sets is proposed that is t fuzzy approximation operators are defined by axioms different axiom sets of t upper and t lower fuzzy set theoretic operators guarantee the existence of different types of fuzzy relations producing the same operators the independence of axioms characterizing the t fuzzy rough approximation operators is examined then the minimal sets of axioms for the characterization of the t fuzzy approximation operators are presented based on information theory the entropy of the generalized fuzzy approximation space which is similar to shannon s entropy is formulated to measure uncertainty in t generalized fuzzy rough sets a notion of fuzziness is introduced some basic properties of this measure are examined for a special triangular norm t min it is proved that the measure of fuzziness of the generalized fuzzy rough set is equal to zero if and only if the set is crisp and definable

high dimensionality effects in the markowitz problem and other quadratic programs with linear constraints risk underestimation we study the properties of solutions of quadratic programs with linear equality constraints whose parameters are estimated from data in the high dimensional setting where p the number of variables in the problem is of the same order of magnitude as n the number of observations used to estimate the parameters the markowitz problem in finance is a subcase of our study assuming normality and independence of the observations we relate the ecient frontier computed empirically to the true ecient frontier our computations show that there is a separation of the errors induced by estimating the mean of the observations and estimating the covariance matrix in particular the price paid for estimating the covariance matrix is an underestimation of the variance by a factor roughly equal to 1 p n therefore the risk of the optimal population solution is underestimated when we estimate it by solving a similar quadratic program with estimated parameters we also characterize the statistical behavior of linear functionals of the empirical optimal vector and show that they are biased estimators of the corresponding population quantities we investigate the robustness of our gaussian results by extending the study to certain elliptical models and models where our n observations are correlated in time we show a lack of robustness of the gaussian results but are still able to get results concerning rst order properties of the quantities of interest even in the case of relatively heavy tailed data we require two moments risk underestimation is still present in the elliptical case and more pronounced that in the gaussian case we discuss properties of the non parametric and parametric bootstrap in this context we show several results including the interesting fact that standard applications of the bootstrap generally yields inconsistent estimates of bias finally we propose some strategies to correct these problems and practically validate them in some simulations in all the paper we will assume that p n and n p tend to innity and p n