a new algorithm for minimizing a linear objective function with fuzzy relation equation constraints in this paper we study the problem of minimizing a linear objective function subject to a fuzzy system constraint by utilizing the fuzzy system s compact system we define k form chained solutions and spaces of k form chained solutions then we prove that any minimal solution of the fuzzy system is a k form chained solution by using an operation method expressed in tables we sieve out basic solutions from the spaces of k form chained solutions finally we obtain optimal solutions of the studied problem from the obtained basic solutions examples are provided to show that our algorithm is simple and convenient

consistency of the group lasso and multiple kernel learning we consider the least square regression problem with regularization by a block l1 norm that is a sum of euclidean norms over spaces of dimensions larger than one this problem referred to as the group lasso extends the usual regularization by the l1 norm where all spaces have dimension one where it is commonly referred to as the lasso in this paper we study the asymptotic group selection consistency of the group lasso we derive necessary and sufficient conditions for the consistency of group lasso under practical assumptions such as model mis specification when the linear predictors and euclidean norms are replaced by functions and reproducing kernel hilbert norms the problem is usually referred to as multiple kernel learning and is commonly used for learning from heterogeneous data sources and for non linear variable selection using tools from functional analysis and in particular covar iance operators we extend the consistency results to this infinite dimensional case and also propose an adaptive scheme to obtain a consistent model estimate even when the necessary condition required for the non adaptive scheme is not satisfied

on the minimal solutions of max min fuzzy relational equations in this paper the minimal solutions of max min fuzzy relational equations are investigated a sufficient and necessary condition for discriminating whether a given solution is minimal or not is shown furthermore we propose a new algorithm for computing all minimal solutions less than or equal to a given one

free vibration analysis of a uniform multi span beam carrying multiple spring mass systems abstract the literature regarding the free vibration analysis of single span beams carrying a number of spring mass systems is plenty but that of multi span beams carrying multiple spring mass systems is fewer thus this paper aims at determining the exact solutions for the natural frequencies and mode shapes of a uniform multi span beam carrying multiple spring mass systems firstly the coefficient matrices for an intermediate pinned support an intermediate spring mass system left end support and right end support of a uniform beam are derived next the numerical assembly technique for the conventional finite element method is used to establish the overall coefficient matrix for the whole vibrating system finally equating the last overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes in this paper the natural frequencies and associated mode shapes of the vibrating system are obtained directly from the differential equation of motion of the continuous beam and no other assumptions are made thus the last solutions are the exact ones the effects of attached spring mass systems on the free vibration characteristics of the 1 4 span beams are studied

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

an efficient algorithm for solving general periodic toeplitz systems an efficient algorithm is presented for inverting matrices which are periodically toeplitz i e whose diagonal and subdiagonal entries exhibit periodic repetitions such matrices are not per symmetric and thus cannot be inverted by trench s 1964 method an alternative approach based on appropriate matrix factorization and partitioning is suggested the algorithm provides certain insight on the formation of the inverse matrix is implementable on a set of circularly pipelined processors and as a special case can be used for inverting a set of block toeplitz matrices without requiring any matrix operation

dsc ritz method for the free vibration analysis of mindlin plates this paper introduces a novel method for the free vibration analysis of mindlin plates the proposed method takes the advantage of both the local bases of the discrete singular convolution dsc algorithm and the pb 2 ritz boundary functions to arrive at a new approach called dsc ritz method two basis functions are constructed by using dsc delta sequence kernels of the positive type the energy functional of the mindlin plate is represented by the newly constructed basis functions and is minimized under the ritz variational principle extensive numerical experiments are considered by different combinations of boundary conditions of mindlin plates of rectangular and triangular shapes the performance of the proposed method is carefully validated by convergence analysis the frequency parameters agree very well with those in the literature numerical experiments indicate that the proposed dsc ritz method is a very promising new method for vibration analysis of mindlin plates copyright 2004 john wiley sons ltd

numerical and applicational aspects of fuzzy relational equations we discuss numerical methods leading to a resolution of fuzzy relational equations which create a formal description of ill defined systems an applicability of the numerical approach is shown in solving some problems in fuzzy systems such as identification and control an idea of fuzzy discretization fuzzy chopping as a meaningful tool for handling various problems is also investigated

optimal rates of convergence for sparse covariance matrix estimation this paper considers estimation of sparse covariance matrices and establishes the optimal rate of convergence under a range of matrix operator norm and bregman divergence losses a major focus is on the derivation of a rate sharp minimax lower bound the problem exhibits new features that are significantly different from those that occur in the conventional nonparametric function estimation problems standard techniques fail to yield good results and new tools are thus needed we first develop a lower bound technique that is particularly well suited for treating two directional problems such as estimating sparse covariance matrices the result can be viewed as a generalization of le cam s method in one direction and assouad s lemma in another this lower bound technique is of independent interest and can be used for other matrix estimation problems we then establish a rate sharp minimax lower bound for estimating sparse covariance matrices under the spectral norm by applying the general lower bound technique a thresholding estimator is shown to attain the optimal rate of convergence under the spectral norm the results are then extended to

exact solution of free vibration of a uniform tensioned beam combined with both lateral and rotational linear sub systems abstract a general exact solution of free vibration of a uniform tensioned beam with any number of both lateral and rotational sub systems is addressed this study is completed in complex domain because of the presence of the non proportional damping caused by the dampers utilized in sub systems the sub systems constructed herein could degenerate to the conventional supports and sub systems and be arbitrarily placed accordingly the solution is also suitable for multi span beams the argument principle method apm which does not need initial values and iteration is employed to solve the characteristic equation in complex domain nevertheless the presence of multi valued square root functions due to the axial loading renders the function non analytic in complex domain a variable substitution method in which the main branch of logarithmic function is chosen is proposed to solve this problem the results obtained are compared with those in literature furthermore several typical case studies were completed and the root loci of the frequency parameters attained are compared with those obtained by the finite element method fem a good agreement is observed which confirms the validity of the presented methodologies in this paper

asymptotic equivalence of spectral density estimation and gaussian white noise we consider the statistical experiment given by a sample y 1 y n of a stationary gaussian process with an unknown smooth spectral density f asymptotic equivalence in the sense of le cam s deficiency distance to two gaussian experiments with simpler structure is established the first one is given by independent zero mean gaussians with variance approximately f i where i is a uniform grid of points in nonparametric gaussian scale regression this approximation is closely related to well known asymptotic independence results for the periodogram and corresponding inference methods the second asymptotic equivalence is to a gaussian white noise model where the drift function is the log spectral density this represents the step from a gaussian scale model to a location model and also has a counterpart in established inference methods that is log periodogram regression the problem of simple explicit equivalence maps markov kernels allowing to directly carry over inference appears in this context but is not solved here

an optimization problem on the image set of a max min fuzzy operator abstract the paper deals with the unsolvability of max min linear equation systems with coefficients in the unit interval for fuzzy systems or in an arbitrary real interval general case if the system has no solution then the nearest vector to the right hand side of the system for which the system is solvable is computed a polynomial algorithm for the problem is presented the method is illustrated by numerical examples

on the use of orthogonal polynomials in the rayleigh ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates series composed of recently proposed orthogonal polynomial functions are used in the rayleigh ritz method to generate results for a number of flexural vibration and buckling problems for rectangular isotropic and orthotropic plates it is demonstrated that the approach yields excellent results not only for natural frequencies and buckling loads but also for the more sensitive mode shape bending moment and shear force distribution problem

operator norm consistent estimation of large dimensional sparse covariance matrices estimating covariance matrices is a problem of fundamental importance in multivariate statistics in practice it is increasingly frequent to work with data matrices x of dimension n x p where p and n are both large results from random matrix theory show very clearly that in this setting standard estimators like the sample covariance matrix perform in general very poorly in this large n large p setting it is sometimes the case that practitioners are willing to assume that many elements of the population covariance matrix are equal to 0 and hence this matrix is sparse we develop an estimator to handle this situation the estimator is shown to be consistent in operator norm when for instance we have p n as n oo in other words the largest singular value of the difference between the estimator and the population covariance matrix goes to zero this implies consistency of all the eigenvalues and consistency of eigenspaces associated to isolated eigenvalues we also propose a notion of sparsity for matrices that is compatible with spectral analysis and is independent of the ordering of the variables

duality of optimization problems with generalized fuzzy relation equation and inequality constraints abstract a generalization of fuzzy relation equations and inequalities is introduced an explicit method for solving optimization problems the feasible set of which is described by a system of generalized fuzzy relation equations and inequalities is considered the objective function of the problems is equal to the maximum of continuous increasing functions of one variable a duality concept for such problems is proposed the problems are considered in a finitely dimensional space

free vibration analysis of a uniform cantilever beam with point masses by an analytical and numerical combined method abstract the natural frequencies and mode shapes of a uniform cantilever beam carrying any number of concentrated masses were determined by using an analytical and numerical combined method anc method the eigenvalue equation was first derived analytically by using the expansion theorem and then the eigenvalues and eigenvectors were calculated numerically in comparison with the general finite element method fem the anc method has the advantage that there is no necessity to derive the property matrices of each beam element and then to develop the overall ones of the entire beam in comparison with the transfer matrix method tmm the anc method requires no tedious matrix multiplication so that some computer time may be saved the anc method is also better than the pure analytical closed form method since instead of a few special cases a uniform beam carrying any number of point masses of various magnitudes or distributions along the beam length can be easily solved besides by using the anc method there is no difficulty in taking more modes in the mode superposition equation and hence more higher mode natural frequencies of better accuracy may be obtained

reduce and boost recovering arbitrary sets of jointly sparse vectors the rapid developing area of compressed sensing suggests that a sparse vector lying in a high dimensional space can be accurately and efficiently recovered from only a small set of nonadaptive linear measurements under appropriate conditions on the measurement matrix the vector model has been extended both theoretically and practically to a finite set of sparse vectors sharing a common sparsity pattern in this paper we treat a broader framework in which the goal is to recover a possibly infinite set of jointly sparse vectors extending existing algorithms to this model is difficult due to the infinite structure of the sparse vector set instead we prove that the entire infinite set of sparse vectors can be recovered by solving a single reduced size finite dimensional problem corresponding to recovery of a finite set of sparse vectors we then show that the problem can be further reduced to the basic model of a single sparse vector by randomly combining the measurements our approach is exact for both countable and uncountable sets as it does not rely on discretization or heuristic techniques to efficiently find the single sparse vector produced by the last reduction step we suggest an empirical boosting strategy that improves the recovery ability of any given suboptimal method for recovering a sparse vector numerical experiments on random data demonstrate that when applied to infinite sets our strategy outperforms discretization techniques in terms of both run time and empirical recovery rate in the finite model our boosting algorithm has fast run time and much higher recovery rate than known popular methods

minimizing a linear function under a fuzzy max min relational equation constraint in this paper we investigate the problem of minimizing a linear objective function subject to a fuzzy relational equation constraint a necessary condition for optimal solution is proposed based on this necessary condition we propose three rules to simplify the work of computing an optimal solution numerical examples are provided to illustrate the procedure experimental results are reported showing that our new procedure systematically outperforms our previous work

an algorithm for solving optimization problems with fuzzy relational inequality constraints an algorithm for solving a kind of optimization problems with fuzzy relational inequalities was proposed by guo and xia 6 however it is too expensive to verify the optimal condition in this paper some rules for reducing these problems are proposed and the relationship between minimal solutions and fri paths is also given these lead to a new algorithm for solving this kind of problems numerical experiments are presented for illustrating the efficiency of the proposed algorithm

numerical convergence of simple and orthogonal polynomials for the unilateral plate buckling problem using the rayleigh ritz method unilateral buckling is a contact problem whereby buckling is confined to take place in only one lateral direction for plate structures this can occur when a thin steel plate is juxtaposed with a rigid concrete medium and the steel may only buckle locally away from the concrete core this paper investigates the use of simple and orthogonal polynomials in the rayleigh ritz method for unilateral plate buckling the orthogonal polynomials used are the classical chebyshev types 1 and 2 legrende hermite and laguerre the study presents a comparison between the efficiency of the polynomial based displacement functions with regard to elastic bilateral and unilateral plate buckling where efficiency is measured as a function of their convergence characteristics some buckling solutions for plates with varying boundary conditions and in plane shear loads are also provided as an illustration copyright 1999 john wiley sons ltd

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

nonstationary spectral analysis based on time frequency operator symbols and underspread approximations we present a unified framework for time varying or time frequency tf spectra of nonstationary random processes in terms of tf operator symbols we provide axiomatic definitions and tf operator symbol formulations for two broad classes of tf spectra one of which is new these classes contain all major existing tf spectra such as the wigner ville evolutionary instantaneous power and physical spectrum our subsequent analysis focuses on the practically important case of nonstationary processes with negligible high lag tf correlations so called underspread processes we demonstrate that for underspread processes all tf spectra yield effectively identical results and satisfy several desirable properties at least approximately we also show that gabor frames provide approximate karhunen loeve kl functions of underspread processes and tf spectra provide a corresponding approximate kl spectrum finally we formulate simple approximate input output relations for the tf spectra of underspread processes that are passed through underspread linear time varying systems all approximations are substantiated mathematically by upper bounds on the associated approximation errors our results establish a tf calculus for the second order analysis and time varying filtering of underspread processes that is as simple as the conventional spectral calculus for stationary processes

a note on the vibrations of restrained beams and rods with point masses abstract this paper is concerned with the approximate determination of the fundamental frequency and first mode shape of a beam or rod to which springs and point masses are attached after formulating the general eigenvalue problem which forms the basis of the analysis the theory is applied to three examples and it is shown that the results are quite satisfactory

dynamic analysis of a multi span uniform beam carrying a number of various concentrated elements abstract this paper employs the numerical assembly method nam to determine the exact frequency response amplitudes of a multiple span beam carrying a number of various concentrated elements and subjected to a harmonic force and the exact natural frequencies and mode shapes of the beam for the case of zero harmonic force first the coefficient matrices for the intermediate concentrated elements pinned support applied force left end support and right end support of a beam are derived next the overall coefficient matrix for the whole vibrating system is obtained using the numerical assembly technique of the conventional finite element method fem finally the exact dynamic response amplitude of the forced vibrating system corresponding to each specified exciting frequency of the harmonic force is determined by solving the simultaneous equations associated with the last overall coefficient matrix the graph of dynamic response amplitudes versus various exciting frequencies gives the frequency response curve for any point of a multiple span beam carrying a number of various concentrated elements for the case of zero harmonic force the above mentioned simultaneous equations reduce to an eigenvalue problem so that natural frequencies and mode shapes of the beam can also be obtained

information theoretic limits of selecting binary graphical models in high dimensions the problem of graphical model selection is to estimate the graph structure of a markov random field given samples from it we analyze the information theoretic limitations of the problem of graph selection for binary markov random fields under high dimensional scaling in which the graph size and the number of edges k and or the maximal node degree d are allowed to increase to infinity as a function of the sample size n for pair wise binary markov random fields we derive both necessary and sufficient conditions for correct graph selection over the class gp k of graphs on vertices with at most k edges and over the class gp d of graphs on p vertices with maximum degree at most d for the class gp k we establish the existence of constants c and c such that if n c k2 log p similarly for the class gp d we exhibit constants c and c such that for n c d3 log p

unilateral buckling of thin elastic plates by the boundary integral equation method the unilateral buckling of thin elastic plates according to kirchhoff s theory is studied by using a boundary integral method a representation for the second member of the equation is given in the matrix formulatiea boundary unknowns are eliminated therefore the unilateral buckling problem reduces to compute the eigenvalues and the eigenvectors of a matrix depending on the contact zone with the rigid foundation an iterative process allows this zone and the buckling load to be computed the capacities of the proposed method are illustrated by four examples

minimizing a linear objective function under a max t norm fuzzy relational equation constraint this paper aims to solve the problem of minimizing a linear objective function subject to a max t fuzzy relational equation constraint with t a continuous t norm this study makes two contributions first two extensions are given based on the results in fang et al where the max min or max product fuzzy relational equation constraint is considered second a procedure for solving optimization problems with the max archimedean t norm fuzzy relational equation constraint is provided a numerical example is provided to illustrate the procedure

on the numerical convergence of discretization methods for the free vibrations of beams with attached inertia elements abstract this paper is concerned with a numerical convergence study of three discretization methods rayleigh ritz galerkin and finite element as applied to the analysis of free bending linear vibrations of a uniform bernoulli euler beam carrying inertia elements at intermediate points a cantilever beam carrying a lumped mass with rotary inertia at an arbitrary intermediate point and another at the beam tip is used as a case study the first five frequency parameters of this beam obtained by using the rayleigh ritz method in conjunction with broadly admissible base beam eigenfunctions galerkin s method in conjunction with strictly admissible base beam eigenfunctions and the classical finite element method are compared with the exact ones over a wide range of values of the system parameters conclusions are drawn regarding the effects of the various system parameters on the accuracy and convergence characteristics of the above three discrete methods and on the mode shapes and the first five frequency parameters of the beam system

adaptive thresholding for sparse covariance matrix estimation in this article we consider estimation of sparse covariance matrices and propose a thresholding procedure that is adaptive to the variability of individual entries the estimators are fully data driven and demonstrate excellent performance both theoretically and numerically it is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm in contrast the commonly used universal thresholding estimators are shown to be suboptimal over the same parameter spaces support recovery is discussed as well the adaptive thresholding estimators are easy to implement the numerical performance of the estimators is studied using both simulated and real data simulation results demonstrate that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators the method is also illustrated in an analysis on a dataset from a small round blue cell tumor microarray experiment a supplement to this ar

high dimensional ising model selection using l1 regularized logistic regression we consider the problem of estimating the graph associated with a binary ising markov random field we describe a method based on l1 regularized logistic regression in which the neighborhood of any given node is estimated by performing logistic regression subject to an l1 constraint the method is analyzed under high dimensional scaling in which both the number of nodes p and maximum neighborhood size d are allowed to grow as a function of the number of observations n our main results provide sufficient conditions on the triple n p d and the model parameters for the method to succeed in consistently estimating the neighborhood of every node in the graph simultaneously with coherence conditions imposed on the population fisher information matrix we prove that consistent neighborhood selection can be obtained for sample sizes n d3log p with exponentially decaying error when these same conditions are imposed directly on the sample matrices we show that a reduced sample size of n d2log p suffices for the method to estimate neighborhoods consistently although this paper focuses on the binary graphical models we indicate how a generalization of the method of the paper would apply to general discrete markov random fields

an exact modal analysis approach to vibration analysis of structures with mass spring subsystems and rotational joints abstract a novel exact modal analysis approach is presented for vibration analysis of plane continuous structures which are coupled with discrete mass spring subsystems and include elastic rotational joints modelling local flexibility using the theory of generalised functions to handle the discontinuities of the response variables every continuous member with any number of mass spring subsystems and joints is treated as a two node element for which a 6 6 exact dynamic stiffness matrix is obtained in closed form as a result the global dynamic stiffness matrix is built by a standard finite element assembling procedure with size depending only on the number of nonzero nodal degrees of freedom of member to member nodes upon deriving pertinent orthogonality conditions for the modes the system response under arbitrary loads is obtained by modal impulse and modal frequency response functions under the assumption of proportional damping the solutions are exact and can be used as benchmark for classical finite element solutions the approach is formulated for various mass spring subsystems acting in transverse and axial directions relative to every member

optimal rates of convergence for covariance matrix estimation covariance matrix plays a central role in multivariate statistical analysis significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices however a minimax theory has yet been developed in this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and frobenius norm it is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation the minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties a key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound the technical analysis requires new ideas that are quite different from those used in the more conventional function sequence estimation problems

vibration frequencies of a rotating flexible arm carrying a moving mass abstract a clamped free flexible beam rotating in a horizontal plane and carrying a moving mass is modelled by the euler bernoulli beam theory the equation of motion is derived by hamilton s principle including the effects of centrifugal stiffening arising from the rotation of the beam the motion of the moving mass and the beam is coupled the equation of motion is a coupled non linear partial differential equation where the coupling terms have to be evaluated at the position of the moving mass in order to obtain the mode shapes which account for the motion of the moving mass the solution is discretized into space and time functions and the beam is divided into two separate regions with respect to the moving mass this results in two non homogeneous linear mode shape ordinary differential equations with four boundary one discontinuity and three continuity conditions the power series method is used to solve for the mode shape differential equations a frequency equation is derived giving the relationship between the non dimensional modal frequencies and the four non dimensional parameters i e the moving mass position the moving mass the beam angular velocity and the total moment of inertia about the hub the numerical bisection method is used to solve for the vibration frequencies under different parameters results are presented for the first three modes of vibration

edge exclusion tests for graphical model selection complex gaussian vectors and time series we consider the problem of inferring the conditional independence graph cig of a stationary multivariate gaussian time series a p variate gaussian time series graphical model associated with an undirected graph with p vertices is defined as the family of time series that obey the conditional independence restrictions implied by the edge set of the graph in some existing methods partial coherence has been used as a test statistic for graphical model selection to test inclusion exclusion of a given edge in the graph the test is applied at distinct frequencies requiring multiple tests and leading to a loss in test power the nonparametric method of matsuda uses the kullback leibler divergence measure to define a test statistic which does not need multiple testing to test between two competing models in this paper we propose a generalized likelihood ratio test glrt based edge exclusion test statistic that also does not need multiple testing it is computationally significantly faster than the method of matsuda and simulations show that we achieve comparable power levels our proposed approach is based on a novel formulation of a glrt based edge exclusion test for p variate complex gaussian graphical models cggms this result is also of independent interest the computational complexity of the proposed statistic is mathcal o p 3 compared to mathcal o p 5 for the existing result we also apply our time series graphical model selection method to a foreign exchange data set consisting of monthly trends of foreign exchange rates of 10 countries

a constrained l1 minimization approach to sparse precision matrix estimation this article proposes a constrained l1 minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p variate random variables the resulting estimator is shown to have a number of desirable properties in particular the rate of convergence between the estimator and the true s sparse precision matrix under the spectral norm is when the population distribution has either exponential type tails or polynomial type tails we present convergence rates under the elementwise l norm and frobenius norm in addition we consider graphical model selection the procedure is easily implemented by linear programming numerical performance of the estimator is investigated using both simulated and real data in particular the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods

on fuzzy relational equations and the covering problem previous studies have shown that fuzzy relational equations fres based on either the max continuous archimedean t norm or the max arithmetic mean composition can be transformed into the covering problem which is an np hard problem exploiting the properties common to the continuous archimedean t norm and the arithmetic mean this study proposes a generalization of them as the u norm enabling fres that are based on the max continuous u norm composition also to be transformed into the covering problem this study also proposes a procedure for transforming the covering problem into max product fres consequently max continuous u norm fres can be solved by extending any procedure for solving either the covering problem or max product fres

on the eigenfrequencies of a cantilever beam with attached tip mass and a spring mass system this paper deals with the derivation of the frequency equation of a special combined dynamic system it consists of a clamped free bernoulli euler beam with a tip mass where a spring mass system is attached to it the derivation of the frequency equation is essentially carried out by means of the lagrange multipliers method frequency equations of some simpler systems are also obtained from the general expression by using limiting processes these equations are then numerically solved for various combinations of physical parameters the comparison with the results from the solution of the corresponding bernoulli euler frequency equations shows that the frequency equations established are very accurate

topology selection in graphical models of autoregressive processes an algorithm is presented for topology selection in graphical models of autoregressive gaussian time series the graph topology of the model represents the sparsity pattern of the inverse spectrum of the time series and characterizes conditional independence relations between the variables the method proposed in the paper is based on an l1 type nonsmooth regularization of the conditional maximum likelihood estimation problem we show that this reduces to a convex optimization problem and describe a large scale algorithm that solves the dual problem via the gradient projection method results of experiments with randomly generated and real data sets are also included

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

application of two dimensional orthogonal plate function to flexural vibration of skew plates abstract a set of two dimensional orthogonal plate functions is used as an admissible deflection function for the study of the flexural vibration of skew plates by the rayleigh ritz method from a prescribed starting function satisfying the boundary conditions the higher terms in the two dimensional orthogonal plate functions are generated by using the gram schmidt orthogonalization process the method is applied to obtain results for a few particular rhombic plates with different support conditions at various skew angles where possible the numerical results presented are compared with the values obtained by other numerical methods in the literature the results should be of considerable value and practical significance in design applications

on the frequency response function of a damped cantilever simply supported in span and carrying a tip mass abstract this paper deals with the determination of the frequency response function of a cantilevered bernoulli euler beam which is viscously damped by a single damper the beam is simply supported in span and carries a tip mass the frequency response function is obtained through a formula that was established for the receptance matrix of discrete linear systems subjected to linear constraint equations by considering the simple support as a linear constraint imposed on generalized co ordinates the comparison of the numerical results obtained via a boundary value problem formulation justifies the approach used here

optimal rates of convergence for estimating toeplitz covariance matrices toeplitz covariance matrices are used in the analysis of stationary stochastic processes and a wide range of applications including radar imaging target detection speech recognition and communications systems in this paper we consider optimal estimation of large toeplitz covariance matrices and establish the minimax rate of convergence for two commonly used parameter spaces under the spectral norm the properties of the tapering and banding estimators are studied in detail and are used to obtain the minimax upper bound the results also reveal a fundamental difference between the tapering and banding estimators over certain parameter spaces the minimax lower bound is derived through a novel construction of a more informative experiment for which the minimax lower bound is obtained through an equivalent gaussian scale model and through a careful selection of a finite collection of least favorable parameters in addition optimal rate of convergence for estimating the inverse of a toeplitz covariance matrix is also established

compressive buckling analysis of plates in unilateral contact the unilateral contact buckling behaviour of delaminated plates in a composite member is studied in this paper where the two dimensional mechanical problem is simplified to a one dimensional mathematical model following the assumption of a buckling mode function in terms of the lateral coordinates after the governing differential equations for the plate parts in the contact and non contact regions have been solved the problem is reduced to just two nonlinear algebraic equations allowing the buckling coefficient to be obtained through an iteration method a simplified method is also deduced based on an assumed zero length of contact the numerical results of the buckling coefficients based on both methods show good agreement

on the linear convergence of the alternating direction method of multipliers we analyze the convergence rate of the alternating direction method of multipliers admm for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints previous analysis of the admm typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks moreover there has been no rate of convergence analysis for the admm without strong convexity in the objective function in this paper we establish the global r linear convergence of the admm for minimizing the sum of any number of convex separable functions assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping and a nonsmooth ell 1 l1 regularizer this result implies the linear convergence of the admm for contemporary applications such as lasso without assuming strong convexity of the objective function

a new approach for determining the natural frequencies and mode shapes of a uniform beam carrying any number of sprung masses abstract in theory one may obtain five equations from each attaching point of a spring mass system and two boundary equations from each end of the uniform beam hence for a uniform beam carrying n spring mass systems simultaneous equations of the form b c 0 will be obtained the solutions of b 0 where represents a determinant give the natural frequencies of the constrained beam and the substitution of each corresponding values of c j j 1 4 into the eigenfunction will define the associated mode shapes while the foregoing theory is simple the lengthy explicit mathematical expressions become impractical if the total number of spring mass systems is larger than two for this reason it was applied to do the free vibration analysis of a uniform beam carrying one spring mass system only in the existing literature the purpose of this paper is to present a numerical technique to apply the foregoing theory to obtain the exact solutions for the lowest several natural frequencies and mode shapes of a uniform beam carrying any number of spring mass systems with various boundary conditions to this end each integration constant c vi and each mode displacement z v v 1 n i 1 4 at the attaching point and the two ends of the beam are considered as nodal displacements of a finite beam element and are assigned an appropriate degree of freedom dof hence each associated coefficient matrix will be equivalent to the stiffness matrix of a beam element and the conventional numerical assembly technique for the finite element method fem may be used to determine the overall coefficient matrix b therefore the eigenvalue equation b c 0 is easily obtained

learning the conditional independence structure of stationary time series a multitask learning approach we propose a method for inferring the conditional independence graph cig of a high dimensional gaussian vector time series discrete time process from a finite length observation by contrast to existing approaches we do not rely on a parametric process model such as e g an autoregressive model for the observed random process instead we only require certain smoothness properties in the fourier domain of the process the proposed inference scheme works even for sample sizes much smaller than the number of scalar process components if the true underlying cig is sufficiently sparse a theoretical performance analysis provides sufficient conditions on the sample size such that the new method is consistent asymptotically some numerical experiments validate our theoretical performance analysis and demonstrate superior performance of our scheme compared to an existing parametric approach in case of model mismatch

nonparametric estimation of large covariance matrices of longitudinal data summary estimation of an unstructured covariance matrix is difficult because of its positivedefiniteness constraint this obstacle is removed by regressing each variable on its predecessors so that estimation of a covariance matrix is shown to be equivalent to that of estimating a sequence of varying coefficient and varying order regression models our framework is similar to the use of increasing order autoregressive models in approximating the covariance matrix or the spectrum of a stationary time series as an illustration we adopt fan zhang s 2000 two step estimation of functional linear models and propose nonparametric estimators of covariance matrices which are guaranteed to be positive definite for parsimony a suitable order for the sequence of auto regression models is found using penalised likelihood criteria like aic and bic some asymptotic results for the local polynomial estimators of components of a covariance matrix are established two longitudinal datasets are analysed to illustrate the methodology a simulation study reveals the advantage of the nonparametric covariance estimator over the sample covariance matrix for large covariance matrices

on the natural frequencies and mode shapes of a multispan timoshenko beam carrying a number of various concentrated elements abstract the purpose of this paper is to utilize the numerical assembly method nam to determine the exact natural frequencies and mode shapes of the multispan timoshenko beam carrying a number of various concentrated elements including point masses rotary inertias linear springs rotational springs and spring mass systems first the coefficient matrices for an intermediate pinned support an intermediate concentrated element left and right end support of a timoshenko beam are derived next the overall coefficient matrix for the whole structural system is obtained using the numerical assembly technique of the finite element method finally the exact natural frequencies and the associated mode shapes of the vibrating system are determined by equating the determinant of the last overall coefficient matrix to zero and substituting the corresponding values of integration constants into the associated eigenfunctions respectively the effects of distribution of in span pinned supports and various concentrated elements on the dynamic characteristics of the timoshenko beam are also studied

flexural vibration of polygonal plates using characteristic orthogonal polynomials in two variables a set of characteristic orthogonal polynomials in two variables is used as deflection functions to obtain the natural frequencies and mode shapes of polygonal plates by the rayleigh ritz method the set of orthogonal polynomials is constructed by employing the gram schmidt orthogonalization procedure the first six natural frequencies are numerically evaluated for triangular plates of different configurations by using these orthogonal functions and the corresponding mode shapes are plotted results are compared with those obtained previously by other methods