new estimates for solutions of lyapunov equations results for estimating the solution of differential and algebraic lyapunov matrix equations are obtained and some of the well known results are generalized

gradient based iterative algorithm for solving coupled matrix equations abstract this paper is concerned with iterative methods for solving a class of coupled matrix equations including the well known coupled markovian jump lyapunov matrix equations as special cases the proposed method is developed from an optimization point of view and contains the well known jacobi iteration gauss seidel iteration and some recently reported iterative algorithms by using the hierarchical identification principle as special cases we have provided analytically the necessary and sufficient condition for the convergence of the proposed iterative algorithm simultaneously the optimal step size such that the convergence rate of the algorithm is maximized is also established in explicit form the proposed approach requires less computation and is numerically reliable as only matrix manipulation is required some other existing results require either matrix inversion or special matrix products numerical examples show the effectiveness of the proposed algorithm

a finite element method for the simulation of strong and weak discontinuities in solid mechanics in this paper we introduce and analyze a finite element method for elasticity problems with interfaces the method allows for discontinuities internal to the elements in the approximation across the interface we propose a general approach that can handle both perfectly and imperfectly bonded interfaces without modifications of the code for the case of linear elasticity we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh we present numerical examples for the linear case as well as for contact and crack propagation model problems

a note on trapezoidal approximations of fuzzy numbers in this paper we propose some generalized and new properties of the trapezoidal approximations of fuzzy numbers first we claim that the trapezoidal approximation operator is linear with respect to the operations of fuzzy numbers second we show a new property concerning the centroid points of fuzzy numbers furthermore a comment for computing the trapezoidal approximations is provided finally some properties about the distance between a fuzzy number and its trapezoidal approximation are proved

simultaneous eigenvalue lower bounds for the lyapunov matrix equation simultaneous eigenvalue lower bounds for the solution of the lyapunov equation are presented the n summation and the n product bounds generalize existing bounds such as those for the trace and the determinant these bounds are stronger than the majority of the relevant bounds shown in the literature

bounds in algebraic riccati and lyapunov equations a survey and some new results this paper summarizes and investigates the relations of a number of bounds for the solutions of the algebraic riccati and lyapunov equations that have been reported during the last two decades also presented are bounds for the unified riccati equation using the delta operator and it is shown that some bounds for the continuous and discrete riccati equations can be unified by them

on 2d elliptic discontinuous galerkin methods we discuss the discretization using discontinuous galerkin dg formulation of an elliptic poisson problem two commonly used dg schemes are investigated the original average flux proposed by bassi and rebay j comput phys 1997 131 267 and the local discontinuous galerkin ldg siam j numer anal 1998 35 2440 2463 scheme in this paper we expand on previous expositions discontinuous galerkin methods theory computation and applications springer berlin 2000 135 146 siam j sci comput 2002 24 2 524 547 int j numer meth engng 2003 58 2 1119 1148 by adopting a matrix based notation with a view to highlighting the steps required in a numerical implementation of the dg method through consideration of standard c0 type expansion bases as opposed to elementally orthogonal expansions with the matrix formulation we are able to apply static condensation techniques to improve efficiency of the direct solver when high order expansions are adopted the use of c0 type expansions also permits the direct enforcement of dirichlet boundary conditions through a lifting approach where the ldg flux does not require further stabilization in our construction we also adopt a formulation of the continuous dg fluxes that permits a more general interpretation of their numerical implementation in particular it allows us to determine the conditions under which the ldg method provides a near local stencil finally a study of the conditioning and the size of the null space of the matrix systems resulting from the dg discretization of the elliptic problem is undertaken copyright 2005 john wiley sons ltd

inferring a possibility distribution from empirical data several transformations from probabilities to possibilities have been proposed in particular dubois and prade s procedure produces the most specific possibility distribution among the ones dominating a given probability distribution in this paper this method is generalized to the case where the probabilities are unknown the only information being a data sample represented by a histogram it is proposed to characterize the probabilities of the different classes by simultaneous confidence intervals with a given confidence level email protected from this imprecise specification a procedure for constructing a possibility distribution is described insuring that the resulting possibility distribution will dominate the true probability distribution in at least 100 email protected of the cases finally a simple efficient algorithm is given which makes the computations tractable even if the number of classes is high

on the numerical modelling of quasi static crack growth in linear elastic fracture mechanics in this paper we present a strategy for the simulation of a propagating crack under mixed mode linear elastic conditions using a discontinuous finite element method a key issue is to accurately compute the incremental change in the kink angle of the propagating crack during subsequent steps we have chosen to work with a domain formulation of the material force vector as a criteria for the propagation direction we describe the theoretical background together with the numerical implementation in detail and show some results for different loading conditions copyright 2005 john wiley sons ltd

simulation of fuzzy random variables this work deals with the simulation of fuzzy random variables which can be used to model various realistic situations where uncertainty is not only present in form of randomness but also in form of imprecision described by means of fuzzy sets utilizing the common arithmetics in the space of all fuzzy sets only induces a conical structure as a consequence it is difficult to directly apply the usual simulation techniques for functional data in order to overcome this difficulty two different approaches based on the concept of support functions are presented the first one makes use of techniques for simulating hilbert space valued random elements and afterwards projects on the cone of all fuzzy sets it is shown by empirical results that the practicability of this approach is limited the second approach imitates the representation of every element of a separable hilbert space in terms of an orthonormal basis directly on the space of fuzzy sets in this way a new approximation of fuzzy sets useful to approximate and simulate fuzzy random variables is developed this second approach is adequate to model various realistic situations

trapezoidal approximations of fuzzy numbers the problem of the trapezoidal approximation of fuzzy numbers is discussed a set of criteria for approximation operators is formulated these constraints can be used for direct operator derivation a new nearest trapezoidal approximation operator preserving expected interval is suggested

on discontinuous galerkin methods discontinuous galerkin methods have received considerable attention in recent years for problems in which advection and diffusion terms are present several alternatives for treating the diffusion and advective fluxes have been introduced this report summarizes some of the methods that have been proposed several numerical examples are included in the paper these present discontinuous galerkin solutions of one dimensional problems with a scalar variable results are presented for diffusion reaction problems and advection diffusion problems we discuss the performance of various formulations with respect to accuracy as well as stability of the method copyright 2003 john wiley sons ltd

on the concept of possibility probability consistency both probability and possibility may be seen as information about an experiment it is conceivable to have at some time these two forms of information about a same experiment and then the question of the relation between them arises at once in this paper some aspects of the concept of possibility probability consistency are studied the consistency is considered as a fuzzy property relative to the coherence between possibilistic and probabilistic information we analyse several measures of the degree of consistency and introduce an axiomatic to characterize them

fuzzy random renewal reward process and its applications this paper studies a renewal reward process with fuzzy random interarrival times and rewards under the independence associated with any continuous archimedean t norm the interarrival times and rewards of the renewal reward process are assumed to be positive fuzzy random variables whose fuzzy realizations are independent fuzzy variables under these conditions some limit theorems in mean chance measure are derived for fuzzy random renewal rewards in the sequel a fuzzy random renewal reward theorem is proved for the long run expected reward per unit time of the renewal reward process the renewal reward theorem obtained in this paper can degenerate to that of stochastic renewal theory finally some application examples are provided to illustrate the utility of the result

two dimensional discrete fuzzy numbers and applications in this paper the concept of two dimensional discrete fuzzy numbers is presented based on the idea of representation theorem of one dimensional discrete fuzzy numbers then a sufficient condition is proposed which makes a fuzzy set on r2 become a two dimensional discrete fuzzy number and the representations of joint membership function and the two edge membership functions of two dimensional discrete fuzzy number are given some special two dimensional discrete fuzzy numbers are also defined and then some weak orders in the two dimensional discrete fuzzy number space are set up based on the mass centers and the degree of ambiguities of two dimensional discrete fuzzy numbers and their properties are also investigated at last some practical examples are included to demonstrate the effectiveness and potential of the theoretic results obtained

gradient based iterative algorithms for generalized coupled sylvester conjugate matrix equations abstract by applying the hierarchical identification principle the gradient based iterative algorithm is suggested to solve a class of complex matrix equations with the real representation of a complex matrix as a tool the sufficient and necessary conditions for the convergence factor are determined to guarantee that the iterative solutions given by the proposed algorithm converge to the exact solution for any initial matrices also we solve the problem which is proposed by wu et al 2010 finally some numerical examples are provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper

on the lyapunov matrix differential equation a lower bound for the determinant of the solution to the lyapunov matrix differential equation is derived it is shown that this bound is obtained as a solution to a simple scalar differential equation in the limiting case where the solution to the lyapunov differential equation becomes stationary the result reduces to one of the existing bounds for the algebraic equation

fuzzy number approximation via shadowed sets interval and trapezoidal approximation methods for fuzzy numbers have been intensively developed and examined recently in this paper another approach to simplify fuzzy numbers called shadowed set approximation is suggested some basic properties of the proposed method is also discussed it seems that the proposed approximation might be useful for granular computing as a tool that removes excessive precision in describing imprecise phenomena

temporal granulation and its application to signal analysis in this study we elaborate on the role of information granulation and the ensuing information granules in description of time series and signal analysis in general information granules are entities of elements quite commonly numeric data that are combined together aggregated owing to their vicinity similarity and alike proceeding with a given window of granulation that is an initial collection of numeric data we propose an algorithm that produces a complete information granule fuzzy set the principle supported by the method leads to the formation of fuzzy sets that are legitimate in terms of experimental data being at the same time maximized with regard to their specificity compactness it has been shown that information granules can be are regarded as generic conceptual entities contributing to the description of numeric time series in this capacity they are used as building blocks aimed at achieving high level compact and comprehensible models of signals more importantly the phase of information granulation could be viewed as a prerequisite to more synthetic and abstract processing such as the one witnessed in syntactic pattern recognition

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

fuzzy equalization in the construction of fuzzy sets in this note we introduce and study a concept of fuzzy equalization fuzzy equalization concerns a process of building information granules that are semantically and experimentally meaningful the experimental relevance of a given information granule fuzzy set is directly linked with an encapsulation of a certain experimental evidence conveyed by the respective probability density function of available data we establish a detailed equalization algorithm developed for triangular fuzzy sets the study elaborates on the role of the fuzzy equalization in system design

eigenvalue bounds for algebraic riccati and lyapunov equations the majorization result of wimmer 2 relating the eigenvalues of the matrices involved in the lyapunov equation is extended to the algebraic riccati equation this result coupled with certain results on the eigenvalue bounds for sum and product of matrices yields several lower and upper bounds for the eigenvalues of the solution matrix of the algebraic riccati and lyapunov equations

new results for the bounds of the solution for the continuous riccati and lyapunov equations this paper presents upper and lower matrix bounds for the solution of the continuous algebraic matrix riccati equation furthermore a new lower matrix bound for the solution of the continuous algebraic lyapunov equation is also developed these are new results

expected value of fuzzy variable and fuzzy expected value models this paper will present a novel concept of expected values of fuzzy variables which is essentially a type of choquet integral and coincides with that of random variables in order to calculate the expected value of general fuzzy variable a fuzzy simulation technique is also designed finally we construct a spectrum of fuzzy expected value models and integrate fuzzy simulation neural network and genetic algorithms to produce a hybrid intelligent algorithm for solving general fuzzy expected value models

bounds for the solution of the lyapunov matrix equation a unified approach abstract in recent years several bounds have been reported for the solution of the continuous and the discrete lyapunov equations using the unified lyapunov equation we give in this paper bounds for the solution of this equation in the limiting cases the bounds reduce to existing bounds for both the continuous and discrete lyapunov equations

trapezoidal approximations of fuzzy numbers revisited fuzzy number approximation by trapezoidal fuzzy numbers which preserve expected interval is discussed the previously proposed approximation operator is improved so as to always produce a well formed trapezoidal fuzzy number

constructing membership functions using statistical data abstract determination of the membership functions is vital in practical applications of the fuzzy set theory this paper presents a guideline to construct the membership functions for fuzzy sets whose elements have a defining feature with a known probability density function pdf in the universe of discourse the method finds the smallest fuzzy set which assigns high average membership values to those objects with the defining features distributed according to the given pdf it is show that for any pdf the method is capable of generating membership functions in accordance with the possibility probability consistency principle membership functions derived from some of the well known pdfs and an application in solving noise contaminated linear system of equations are presented

matrix bounds of the solutions of the continuous and discrete riccati equations a unified approach in this paper a new scheme is introduced to measure the matrix bounds of the continuous and discrete riccati equations by estimating upper and lower matrix bounds of the solution of the unified algebraic riccati equation uare the same measurements for the solutions of the continuous and discrete riccati equations respectively can be obtained in limiting cases according to these obtained matrix bounds several eigenvalue bounds are also defined all the proposed results for the uare are new and more general than previous work some obtained results are compared with those of the literature via numerical examples it is shown that in some cases the presented results are tighter than the existing ones

a note on the discrete lyapunov and riccati matrix equations abstract inequalities which are satisfied by the determinants of the positive definite solutions of the discrete algebraic riccati and lyapunov matrix equations are presented the results give lower bounds for the product of the eigenvalues of the matrix solutions also for a discrete lyapunov equation an algorithm is presented to determine under what conditions a positive diagonal solution will exist if all the conditions are satisfied the algorithm also provides such a diagonal solution

a note on eigenvalue bounds in algebraic riccati equation in this note an upper bound on the maximum eigenvalue of the solution matrix k of the algebraic riccati equation is established the approach outlined also results in several lower bounds which are more general than those derived in 1 for some of the largest eigenvalues of k

trapezoidal and triangular approximations preserving the expected interval in this paper an improvement of the nearest trapezoidal approximation operator preserving the expected interval is studied which is proposed by grzegorzewski and mrowka a formula for computing the improved approximation is presented moreover the nearest triangular approximation operator preserving the expected interval is also investigated

estimating the expected value of fuzzy random variables in the stratified random sampling from finite populations abstract in this paper we consider the problem of estimating the expected value of a fuzzy valued random element in the stratified random sampling from finite populations to this purpose we quantify the associated sampling error by means of a generalized measure introduced in a previous paper we also suggest a way to compare different variates for stratification as well as to test the adequacy of a certain one

bounds in the lyapunov matrix differential equation upper and lower bounds for the trace of the solution of the lyapunov matrix differential equation are derived it is shown that they are obtained as solutions to simple scalar differential equations as a special case the bounds for the stationary solution give ones for the solution to the lyapunov algebraic equation

a fuzziness measure for fuzzy numbers applications abstract extending the ideas of delgado et al 1995 to include the notion of fuzziness we obtain canonical representations of fuzzy numbers in particular we show that any fuzzy number can be represented by either a trapezoidal or a quasitrapezoidal number having the same basic attributes as the original fuzzy number

unified type non stationary lyapunov matrix equation simultaneous eigenvalue bounds abstract simultaneous eigenvalue bounds for the solution of the unified non stationary lyapunov matrix equation are presented when the solution becomes stationary the results reduce to bounds of the unified type algebraic lyapunov equation in the limiting cases the results reduce to bounds for the solution of the differential and difference lyapunov equations the bounds given in this paper are a generalization of some existing bounds obtained separately for the continuous and discrete type stationary and non stationary lyapunov equations

on bounds of the riccati and lyapunov matrix equations various bounds for the traces of the solutions of the algebraic riccati and lyapunov matrix equations are established in the continuous and discrete domain respectively the presented results give lower bounds for the sum of the eigenvalue of the matrix solutions

improved bounds for the eigenvalues of solutions of lyapunov equations bounds on the solution and on the eigenvalues of the solution of the discrete and of the continuous matrix lyapunov equations are presented these bounds are nontrivial in the case of system matrices having multiple eigenvalues especially in the latter case these bounds are better than bounds found in the literature further these bounds are sharp i e there exist systems such that the lower and upper bounds coincide yielding the solution or its eigenvalues respectively

linear finite elements with orthogonal discontinuous basis functions for explicit earthquake ground motion modeling the dynamic explicit finite element method is commonly used in earthquake ground motion modeling in this method the element mass matrix is approximately lumped which may lead to numerical dispersion on the other hand the orthogonal finite element method based on orthogonal polynomial basis functions naturally derives a lumped diagonal mass matrix and can be applied to dynamic explicit finite element analysis in this paper we propose finite elements based on orthogonal discontinuous basis functions the element mass matrices of which are lumped without approximation orthogonal discontinuous basis functions are used to improve the accuracy and reduce the numerical dispersion in earthquake ground motion modeling we present a detailed formulation of the 4 node tetrahedral and 8 node hexahedral elements the relationship between the proposed finite elements and conventional finite elements is investigated and the solutions obtained from the conventional explicit finite element method are compared with analytical solutions to verify the numerical dispersion caused by the lumping approximation comparison of solutions obtained with the proposed finite elements to analytical solutions demonstrates the usefulness of the technique examples are also presented to illustrate the effectiveness of the proposed method in earthquake ground motion modeling in the actual three dimensional crust structure

maximum entropy membership functions for discrete fuzzy variables due to the deficiency of information the membership function of a fuzzy variable cannot be obtained explicitly it is a challenging work to find an appropriate membership function when certain partial information about a fuzzy variable is given such as expected value or moments this paper solves such problems for discrete fuzzy variables via maximum entropy principle and proves some maximum entropy theorems with certain constraints a genetic algorithm is designed to solve the general maximum entropy model for discrete fuzzy variables which is illustrated by some numerical experiments

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

trace bounds on the solution of the algebraic matrix riccati and lyapunov equation lower and upper bounds on the trace of the positive semidefinite solution of the algebraic matrix riccati and lyapunov equation are derived the upper trace bound obtained in this note in many cases results in a tighter bound as compared to the upper bound for the maximal eigenvalue proposed in 1 and 2

best lipschitz constant of the trapezoidal approximation operator preserving the expected interval in this paper we prove new distance properties between a fuzzy number and its trapezoidal approximation preserving the expected interval then we find the best lipschitz constant of the trapezoidal approximation operator preserving the expected interval finally we use this result in finding within a reasonable error the trapezoidal approximation of a fuzzy number preserving the expected interval in the case when the direct formula leads to complex calculations

why triangular membership functions abstract in this note we look at a certain theoretically sound motivation behind the common use of triangular and trapezoidal membership functions the studies are completed within a conceptual framework of fuzzy modelling whose structure comprises of input and output interfaces linked with a single transformation module aimed at processing linguistic information it is shown that under some additional mild assumptions these triangular fuzzy sets comply with a request for a uniformly excited codebook in the case of the input interfaces and a satisfaction of a zero error reconstruction criterion being formulated for the output interface

eigenvalue bounds for the discrete lyapunov matrix equation upper and lower bounds for every eigenvalue of the positive definite solution to the discrete lyapunov matrix equation are provided the obtained bounds have an advantage that they can give nontrivial lower bounds even when some existing results yield trivial ones

on the discrete lyapunov matrix equation some bounds for the arithmetic and the geometric means of the characteristic roots of the positive semidefinite solution to the discrete lyapunov matrix equation are derived

a comparative study on finite elements for capturing strong discontinuities e fem vs x fem a comparative study on finite elements for capturing strong discontinuities by means of elemental e fem or nodal enrichments x fem is presented based on the same constitutive model continuum damage and linear elements triangles and tetrahedra optimized implementations of both types of enrichments in the same non linear code are tested for a representative set of 2d and 3d crack propagation examples it is shown that both methods provide the same qualitative and quantitative results for enough refined meshes for the performed tests e fem exhibited in general a higher accuracy mostly for coarse meshes whereas convergence rate with mesh refinement which is super linear showed slightly higher for x fem as for the computational costs for single crack modelling x fem showed depending on the case from 1 1 to about 2 5 times more expensive than e fem for multiple cracks the computational cost of e fem keeps constant whereas the cost associated to x fem increases linearly with the number of modelled cracks

3d crack propagation in unreinforced concrete a two step algorithm for tracking 3d crack paths tensile failure of unreinforced concrete involves progressive micro cracking and the related strain softening can coalesce into geometrical discontinuities which separate the material advanced mechanical theories and numerical schemes are required to efficiently and adequately represent crack propagation in 3d in this paper we use the concept of strong discontinuities to model concrete failure we introduce a cohesive fracture process zone which is characterized by a transversely isotropic traction separation law we combine the cohesive crack concept with the partition of unity finite element method where the finite element space is enhanced by the heaviside function the concept is implemented for tetrahedral elements and the failure initialization is based on the simple non local rankine criterion for each element we assume the embedded discontinuity to be flat in the reference configuration which leads to a non smooth crack surfaces approximation in 3d in general different concepts for tracking non planar cracks in 3d are reviewed in addition we propose a two step algorithm for tracking the crack path where a predictor step defines discontinuities according to the non local failure criterion and a corrector step draws in non local information of the existing discontinuities in order to predict a closed 3d crack surface implementation details are provided the proposed framework is used to analyze the predictability of concrete failure by two benchmark examples i e the nooru moharned test and the brokenshire test we compare our numerical results which are mesh independent with experimental data and numerical results adopted from the literature

upper bounds for the eigenvalues of the solution of the lyapunov matrix equation eigenvalue upper bounds for the solution of the lyapunov matrix equation are presented for single eigenvalues summations of eigenvalues including the trace and products of eigenvalues including the determinant these bounds are compared to those for the trace and the extreme eigenvalues

a brief summary of the bounds on the solution of the algebraic matrix equations in control theory abstract in recent years several bounds have been reported for different measures of the extent or size of the solution of the algebraic matrix equations arising in control theory such as the riccati equation and the lyapunov equation this paper collects the bounds that have been presented up to now and summarizes them in an unified form this will prove particularly convenient for those wishing to get a ready estimate of the solution while solving the equations numerically or to develop theoretical results that rely on these bounds

fuzzy sets and statistical data abstract specific features of probability and possibility theories are discussed with emphasis on semantical aspects instead of putting forward the acknowledged usefulness of possibility theory for the non statistical modelling of subjective categories we try to figure out how statistical data and possibility theory could be matched as a result procedures for constructing weak possibilistic substitutes of probability measures and for processing imprecise statistical data are outlined they provide new insights on relationship between fuzzy sets and probability theories

fuzzy numbers from raw discrete data using linear regression this paper focuses on modelling fuzzy numbers with meaningful membership functions more precisely it proposes a method to construct trapezoidal fuzzy number approximations from raw discrete data in many applications input information is numerical and therefore particular fuzzy sets such as fuzzy numbers hold great interest and relevance in managing data imprecision and vagueness the proposed technique provides an efficient way to obtain trapezoidal numbers using linear regression the technique is simple fast and effective preliminary tests are performed using different types of input data a gaussian function a sigmoidal function three datasets of synthetic discrete data and an histogram obtained from a colour satellite image

regranulation a granular algorithm enabling communication between granular worlds abstract in this paper we describe a granular algorithm for translating information between two granular worlds represented as fuzzy rulebases these granular worlds are defined on the same universe of discourse but employ different granulations of this universe in order to translate information from one granular world to the other we must regranulate the information so that it matches the information granularity of the target world this is accomplished through the use of a first order interpolation algorithm implemented using linguistic arithmetic a set of elementary granular computing operations we first demonstrate this algorithm by studying the common fuzzy pd rulebase at several different granularities and conclude that the 3 3 granulation may be too coarse for this objective we then examine the question of what the natural granularity of a system might be this is studied through a 10 fold cross validation experiment involving three different granulations of the same underlying mapping for the problem under consideration we find that a 7 7 granulation appears to be the minimum necessary precision

on solution bounds for three types of lyapunov matrix equations continuous discrete and unified equations an approach is proposed to obtain a series of solution bounds for the unified algebraic lyapunov matrix equation by making use of its feature i e a mediator between the continuous time and discrete time lyapunov equations due to the approach every existing solution bound for both the continuous and discrete equations becomes intact in the sense that it yields a corresponding bound for the unified equation the approach is endorsed by this simple fact the three types of lyapunov equations share the solution provided certain relationships are maintained among corresponding coefficient matrices of the equations some examples are presented to show that the proposed method actually produces new bounds and builds bridges over existing bounds

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

on some bounds in the algebraic riccati and lyanupov equations some bounds for the trace and determinant of the solution to the algebraic riccati and lyapunov matrix equations are presented these bounds give information on mean size of the solution to the equations in a simple way the obtained determinant bound for the lyapunov equation shows the improvement over the previously reported one

canonical representations of discrete fuzzy numbers in this paper we obtain two canonical representations of discrete fuzzy numbers in particular it is shown that any discrete fuzzy number can be represented by a triangular or a block discrete fuzzy number having the same value ambiguity and fuzziness as the original number

on the discrete riccati equation several bounds have been reported recently for the trace of the solution to the discrete algebraic matrix riccati equation this note adds an alternative one to them

approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the ambiguity and value value and ambiguity are two parameters which were introduced to represent fuzzy numbers in this paper we find the nearest trapezoidal approximation and the nearest symmetric trapezoidal approximation to a given fuzzy number with respect to the average euclidean distance preserving the value and ambiguity to avoid the laborious calculus associated with the karush kuhn tucker theorem the working tool in some recent papers a less sophisticated method is proposed algorithms for computing the approximations many examples proofs of continuity and two applications to ranking of fuzzy numbers and estimations of the defect of additivity for approximations are given

non planar 3d crack growth by the extended finite element and level sets part i mechanical model a methodology for solving three dimensional crack problems with geometries that are independent of the mesh is described the method is based on the extended finite element method in which the crack discontinuity is introduced as a heaviside step function via a partition of unity in addition branch functions are introduced for all elements containing the crack front the branch functions include asymptotic near tip fields that improve the accuracy of the method the crack geometry is described by two signed distance functions which in turn can be defined by nodal values consequently no explicit representation of the crack is needed examples for three dimensional elastostatic problems are given and compared to analytic and benchmark solutions the method is readily extendable to inelastic fracture problems copyright 2002 john wiley sons ltd

a note on the lyapunov matrix equation in 5 bound for the determinant of the solution to the lyapunov matrix equation was reported this note gives an another bound for this value

numerical analysis of growing crack problems using particle discretization scheme this paper presents the particle discretization scheme pds to analyze brittle failure of solids the scheme uses characteristic functions of voronoi and delaunay tessellations to discretize a function and its derivatives respectively a discretized function has numerous discontinuities so that these discontinuities are utilized as a candidate of crack path segment in modeling propagating cracks without making any extra computation to accommodate new displacement discontinuities when the scheme is implemented to a finite element method fem the resulting stiffness matrix coincides with the one that is obtained by using linear elements the accuracy of computing a stress intensity factor at a crack tip is examined it is shown that the accuracy is better than that of a fem with linear elements when the rotational degree of freedom is included in discretizing displacement functions three three dimensional growing crack problems are solved by means of the pds and the results are presented copyright 2009 john wiley sons ltd

bounds for the eigenvalues of the solution of the unified algebraic riccati matrix equation abstract in recent years several eigenvalues bounds have been investigated separately for the solutions of the continuous and the discrete riccati and lyapunov matrix equations in this paper lower bounds for the eigenvalues of the solution of the unified riccati equation relatively to continuous and discrete cases are presented in the limiting cases the results reduce to some new bounds for both the continuous and discrete riccati equation

approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval the problem to find the nearest trapezoidal approximation of a fuzzy number with respect to a well known metric which preserves the expected interval of the fuzzy number is completely solved the previously proposed approximation operators are improved so as to always obtain a trapezoidal fuzzy number properties of this new trapezoidal approximation operator are studied

lower bounds on the solution of lyapunov matrix and algebraic riccati equations in this note various lower bounds for all the eigenvalues of the solution matrix k of the lyapunov matrix equation are established a special case of this result is a generalization of that presented in 1 3 where lower bounds for the maximum and minimum eigenvalues of k are given moreover the approach used here enables one to establish various lower bounds for some of the largest eigenvalues of the solution matrix of the algebraic riccati equation

on a canonical representation of fuzzy numbers abstract fuzzy numbers and more generally linguistic values are approximate assessments given by experts and accepted by decision makers when obtaining more accurate values is impossible or unnecessary to simplify the task of representing and handling fuzzy numbers several authors have introduced real indices in order to capture the information contained in a fuzzy number in this paper we propose two parameters value and ambiguity for this purpose we use these parameters to obtain canonical representations and to deal with fuzzy numbers in decision making problems several examples illustrate these ideas

non planar 3d crack growth by the extended finite element and level sets part ii level set update we present a level set method for treating the growth of non planar three dimensional cracks the crack is defined by two almost orthogonal level sets signed distance functions one of them describes the crack as a two dimensional surface in a three dimensional space and the second is used to describe the one dimensional crack front which is the intersection of the two level sets a hamilton jacobi equation is used to update the level sets a velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface the technique is coupled with the extended finite element method which approximates the displacement field with a discontinuous partition of unity this displacement field is constructed directly in terms of the level sets so the discretization by finite elements requires no explicit representation of the crack surface numerical experiments show the robustness of the method both in accuracy and in treating cracks with significant changes in topology copyright 2002 john wiley sons ltd

modelling strong discontinuities in solid mechanics via strain softening constitutive equations part 2 numerical simulation on the basis of the strong discontinuity analysis of standard local stress strain constitutive equations a finite element framework for the simulation of strong discontinuities which belongs to the family of assumed enhanced strain methods is presented taking the standard linear triangle as the underlying element an additional incompatible mode leads to the formulation of an enriched strain field which is shown to be able to appropriately capture strong discontinuities the presented numerical simulations show that mesh size and mesh alignment dependencies can be completely removed

identification of fuzzy systems by means of an auto tuning algorithm and its application to nonlinear systems abstract the study concerns a design procedure of rule based systems the proposed rule based fuzzy modeling implements system structure and parameter identification in the efficient form of if then statements and exploits the theory of system optimization and fuzzy implication rules two types of methods for rule based fuzzy modeling are studied this classification concerns the form of the conclusion part of the rules that can be either constant or formed by some linear functions both triangular and gaussian like membership function are studied the optimization hinges on an auto tuning algorithm that covers a modified constrained optimization method known as a complex method the study introduces a weighted performance index objective function that helps achieve a sound balance between the quality of results produced for the training and testing set this methodology sheds light on the role and impact of different parameters of the model on its performance in particular the mapping and predicting capabilities of the rule based computing the study is illustrated with the aid of several representative numerical examples

the expected value of a fuzzy number abstract the notions of expected interval and expected value of fuzzy number are introduced the expected interval is defined as the expected value of an interval random set generated by the fuzzy number and the expected value of this number is defined as the centre of the expected interval the linearity of the expected interval is studied

a motion compression reconstruction method based on max t norm composite fuzzy relational equations a motion compression reconstruction method based on max t norm composite fuzzy relational equations mcf is proposed where into intra pictures i pictures and predictive pictures p pictures of the original motion are compressed by uniform and non uniform coders respectively the non uniform coders of the proposed method can preserve edge information of p pictures on the compressed image to perform an effective compression reconstruction of the p pictures a design method of non uniform coders is proposed based on an overlap level of fuzzy sets and a fuzzy equalization an experiment using 10 p pictures confirms that the root mean square error of the reconstructed images obtained by the proposed non uniform coders is decreased to 89 4 of that one of the uniform coders under the condition that compression rate the ratio between the file size of compressed image and original one is 0 0057 two test motions tennis and woman 100 frames are compressed and reconstructed by the proposed mcf

ranking of multidimensional uncertain information based on metrics on the fuzzy ellipsoid number space on the usual high dimensional fuzzy number space the usual supremum metric d derived by the hausdorff metric between the level sets of usual n dimension fuzzy numbers and the p metric rho p are the most common metrics however due to the complexity of the level sets of usual n dimensional fuzzy numbers the two kinds of metrics not only have a tendency to be rougher but also are difficult to give concrete expression formulas this affects their theory and application research in this paper some new metrics on fuzzy ellipsoid number space are introduced which not only can better reveal the difference between two different fuzzy ellipsoid numbers but also have concrete expression formulas expressed with the level set functions of fuzzy ellipsoid numbers and the properties of the new introduced metrics and the relationships between the new metrics and the usual metrics d and rho p are studied and some results are obtained then we give the concept of supremum infimum of bounded subsets of fuzzy ellipsoid number space and obtain its concrete calculation formula and then by using the obtained results we propose a method to rank multidimensional uncertain information and give a practical example to show the application and the rationality of the proposed techniques

bounds for the eigenvalues of the solution of the discrete riccati and lyapunov equations and the continuous lyapunov equation we present some bounds for the eigenvalues and certain sums and products of the eigenvalues of the solution of the discrete riccati and lyapunov matrix equations and the continuous lyapunov matrix equation nearly all of our bounds for the discrete riccati equation are new the bounds for the discrete and continuous lyapunov equations give a completion of some known bounds for the extremal eigenvalues and the determinant and the trace of the solution of the respective equation