localization of solutions of anisotropic parabolic equations abstract we study the localization properties of solutions of the dirichlet problem for the anisotropic parabolic equations u t i 1 n d i a i z u d i u p i 2 d i u f z z x t 0 t with constant exponents p i 1 and x r n n 2 such equations arise from the mathematical description of diffusion processes it is shown that if the equation combines the directions of slow diffusion for which p i 2 and the directions of fast or linear diffusion corresponding to p i 1 2 or p 2 then the solutions may simultaneously display the properties intrinsic for the solutions of isotropic equations of fast or slow diffusion under the assumptions that f 0 for t t f and u 0 0 f 0 for x 1 s we show on the one hand that the solution vanishes in a finite time if n 2 i 1 n 1 p i 1 n 2 and on the other hand that the support of the same solution never reaches the plane x 1 s provided that 1 n 1 1 n 1 i 2 n 1 p i 1 p 1

t prefilter theory abstract in the first part of this paper the notion of t prefilter is introduced and studied as are the applications t and t their properties and relations between them we characterize the minimal prime t prefilters finer than a given t prefilter by using ultrafilters on x and we give equivalents conditions of the t maximality in the second part we define t convergence and we obtain a characterization of t compactness gantner steinlage by means of the maximal t prefilters finally some applications of these concepts related to the continuity separation axioms and fuzzy compactness are studied

a variable kinematic ritz formulation for vibration study of quadrilateral plates with arbitrary thickness a new variable kinematic ritz method applied to free vibration analysis of arbitrary quadrilateral thin and thick isotropic plates is presented carrera s unified formulation and the versatile pb 2 ritz method are properly combined to build a powerful yet simple modeling and solution framework the proposed technique allows to generate arbitrarily accurate ritz solutions from a large variety of refined two dimensional plate theories by expanding so called ritz fundamental nuclei of the plate mass and stiffness matrices theoretical development of the present methodology is described in detail convergence and accuracy of the method are examined through several examples on thin moderately thick and very thick plates of rectangular skew trapezoidal and general quadrilateral shapes with an arbitrary combination of clamped free and simply supported edges present results are compared with existing three dimensional solutions from open literature maximum and average differences of various higher order plate theories and three dimensional results are also presented with the aim of providing useful guidelines on the choice of appropriate plate theory to get a desired accuracy on frequency parameters

a new approach to fuzzy bitopological spaces abstract given a fuzzy bitopological space x 1 2 we have used the operator c12 ix ix defined by c 12 1 cl 2 cl i x which is a supra fuzzy closure operator fuzzy sets and systems 74 1995 353 to generate a family s which is a supra fuzzy topology loc cit 1995 so the space x s is a supra fuzzy topological space associated to the fuzzy bitopological space x 1 2 we extend the notions of fti strong fti and ultra fti due to fuzzy sets and system 43 1991 95 to the space x s the properties of fuzzy bitopological spaces have been studied by using their associated supra fuzzy topological spaces we introduce the notions of fp ti s fp ti and u fp ti by using the level supratopology s of s we investigate the relationship between these notions the notions fp ti loc cit 1995 i 1 2 3 4 and the notions fpti resp s fpti u fpti fuzzy sets and systems 105 1999 459 we give a number of examples which illustrate that these concepts are not equivalent also we extend the notions of f continuous open strong f continuous open and ultra f continuous open due to fuzzy sets and systems 38 1990 115 to the class of supra fuzzy topological spaces associated to the class of fuzzy bitopological spaces we study some properties of fp ti resp s fp ti u fp ti under these mappings

fµ strongly semiopen sets fµ strong semicontinuity and fµ strongly semiretracts throughout the literature if x d is an f ts and y x the induced f topological subspace y d y is defined so that d y n g y n d macho stadler and de prada vicente fuzzy sets and systems 58 1993 365 introduced a new concept of f topological subspaces which coincides with the usual definition in the case that m g y also they introduced the concepts of f m open sets and f m continuity in this paper using the previous concepts we introduce weaker forms of f m open sets and f m continuity the notion of an f retract was introduced by rodabough j math anal appl 79 1981 273 here we introduce the notion of a f m retract and weaker forms of it the basic properties of these notions are given many results are obtained

modal probabilistic analysis in this paper we first recall a method for estimating the numerical probability density function pdf with parzen rosenblatt estimators of experimental data like the sound absorption coefficient with the kundt s tube then we focus on the model of mikki which is used as a predictive tool we are interested then in inversing the model and in identifying the 3 parameters q of mikki s model but we want to go further and to identify from the estimated pdf directly the pdf of the 3 parameters we explain first when it is possible then we give some examples to illustrate the method this work allows for evaluating the robustness of a model from experimental data

from euler ritz and galerkin to modern computing the so called ritz galerkin method is one of the most fundamental tools of modern computing its origins lie in hilbert s direct approach to the variational calculus of euler lagrange and in the thesis of walther ritz who died 100 years ago at the age of 31 after a long battle with tuberculosis the thesis was submitted in 1902 in gottingen during a period of dramatic developments in physics ritz tried to explain the phenomenon of balmer series in spectroscopy using eigenvalue problems of partial differential equations on rectangular domains while this physical model quickly turned out to be completely obsolete his mathematics later enabled him to solve difficult problems in applied sciences he thereby revolutionized the variational calculus and became one of the fathers of modern computational mathematics we will see in this article that the path leading to modern computational methods and theory involved a long struggle over three centuries requiring the efforts of many great mathematicians

the historical bases of the rayleigh and ritz methods abstract rayleigh s classical book theory of sound was first published in 1877 in it are many examples of calculating fundamental natural frequencies of free vibration of continuum systems strings bars beams membranes plates by assuming the mode shape and setting the maximum values of potential and kinetic energy in a cycle of motion equal to each other this procedure is well known as rayleigh s method in 1908 ritz laid out his famous method for determining frequencies and mode shapes choosing multiple admissible displacement functions and minimizing a functional involving both potential and kinetic energies he then demonstrated it in detail in 1909 for the completely free square plate in 1911 rayleigh wrote a paper congratulating ritz on his work but stating that he himself had used ritz s method in many places in his book and in another publication subsequently hundreds of research articles and many books have appeared which use the method some calling it the ritz method and others the rayleigh ritz method the present article examines the method in detail as ritz presented it and as rayleigh claimed to have used it it concludes that although rayleigh did solve a few problems which involved minimization of a frequency these solutions were not by the straightforward direct method presented by ritz and used subsequently by others therefore rayleigh s name should not be attached to the method

some strong separation in fuzzy topological spaces abstract recently macho stadler et al fuzzy sets and systems 43 1991 95 116 introduced the notion of fuzzy separation using the family of its level topologies lt t t i1 the present work deals with introducing and studying the operations on ri fuzzy topological spaces i 0 1 moreover the notion of fuzzy separation which is well defined as a fuzzy topological concept is described a comparison between the above notions with the analogous fuzzy ones are discussed an acceptable clue which indicates that these notions are equivalent is offered

minimum stiffness location of point support for control of fundamental natural frequency of rectangular plate by rayleigh ritz method abstract the rayleigh ritz method is employed to determine the minimum stiffness location of the elastic point support for raising the fundamental natural frequency of a rectangular plate to the second frequency of the unsupported plate which usually is the upper limit of the first frequency for a single support based on the optimal design of the support position the minimum stiffness can be obtained numerically by solving a characteristic eigenvalue sub problem in the rayleigh ritz procedure the boundary characteristic orthogonal polynomials are used for the admissible functions several typical examples of plate structures with the additional point supports are analyzed in detail and the results prove that the proposed method is very effective in the solution to the optimal design of the supports it will be shown that elastic supports can be designed like rigid ones for the purpose of increasing the fundamental natural frequency of a rectangular plate

strong and ultra separation axioms of fuzzy bitopological spaces given a fuzzy bitopological space x 1 2 we introduce a new notion of fuzzy pairwise separation axioms by using the family of its level bitopologies 1 2 0 1 we prove that these concepts are good extension and we compare them with its corresponding fpt kandil and ei shafee 1991 and fpt1 abu safiya et al 1994 i 0 1 2 3 4 respectively we show that these notions are not equivalent and we give a number of examples which illustrate this fact

continuity of the flows and upper semicontinuity of global attractors for ps x laplacian parabolic problems abstract in this work we prove continuity of solutions with respect to initial conditions and exponent parameters and we prove upper semicontinuity of a family of global attractors for one dimensional problems of the form u s t x u s x p s x 2 u s x b u s where b is a globally lipschitz map p s p in l i i c d and p 2 constant as s goes to infinity

the critical forms of the attractors for semigroups and the existence of critical attractors the critical forms of the minimal closed global attractor m and the minimal closed global b attractor m for operator semigroups are given i e if the set x x x x attracts each point x in the phase space x then it must be the unique minimal closed global attractor m and if the set b b x x attracts each bounded set in x then it must be the unique minimal closed global b attractor m on this basis the sufficient conditions for the existence of m and m interrelation between m and m are discussed we introduce some classes of semigroups and study a variety of relations of them and the different classes of semigroups considered by ladyzhenskaya hale and temam when a semigroup9s continuity is not required it is proved that the semigroups with b acp introduced by us coincide with the asymptotically smooth semigroups the semigroups of class ak or b ak respectively and that if these semigroups are b bounded and x x x b then each of them possesses the unique non empty minimal invariant compact global attractor x x x x and the unique non empty minimal invariant compact global b attractor m b b x x m omega o where 0is a small number furthermore the sufficient conditions for m and m are given it is further proved that these conditions which guarantee the existence of the compact invariant global b attractor m are not only sufficient but also necessary so in order to guarantee a semigroup to have such attractor m the conditions which are put on the semigroup cannot be weakened again these results will be useful in studying the existence and construction of attractors

stiff string approximations in rayleigh ritz method for rotating beams the governing differential equation of the rotating beam reduces to that of a stiff string when the centrifugal force is assumed as constant the solution of the static homogeneous part of this equation is enhanced with a polynomial term and used in the rayleighs method numerical experiments show better agreement with converged finite element solutions compared to polynomials using this as an estimate for the first mode shape higher mode shape approximations are obtained using gram schmidt orthogonalization estimates for the first five natural frequencies of uniform and tapered beams are obtained accurately using a very low order rayleigh ritz approximation

on separation axioms in fuzzy topological spaces fuzzy neighborhood spaces and fuzzy uniform spaces submitted by l zadeh a certain number of separation axioms for fuzzy topological spaces are provided all of which are good extensions of the topological t t or t2 all valid implications between the different axioms are studied and counterexamples are given for the nonvalid ones in this paper we introduce and study a number of separation properties of fuzzy topological fuzzy neighborhood and fuzzy uniform spaces all of which are extensions of t t and t in topological spaces we started from the viewpoint that separation axioms at least t originated in the need to guarantee in some sense uniqueness of limits and we tried to generalize the topological t axiom in that direction it appears that a whole range of axioms from a fairly weak condition wt up to a rather strong one tj all give rise to acceptable extensions of the topological property moreover it seems likely that it will not be possible to single out one axiom in fuzzy topology as the best extension of topological t2 we showed in a preceding paper 181 on completions of fuzzy uniform spaces that tj as well as wt collapsing with wt in uniform spaces have a specific role to play in the same sense wt has already shown its usefulness in the study of hyperspaces 131 and t2 in its relation with compactness 191 as to our other axioms from t to wt they all seem to be natural generalizations of the corresponding properties in topological spaces and of the different t notions in fuzzy topological spaces they are moreover hereditary and t and ty excepted also productive as to the relations with other work in this field we first point out that our wt axiom is in fact the 1 hausforff separation property of hiihle 2 31 hutton and reilly in 4 and rodabaugh in 161 also studied separation current address is rijksuniversitair centrum 2020 antwerpen belgium

the equations of lagrange for a continuous deformable body with rigid body degrees of freedom written in a momentum based formulation abstract the present paper is concerned with lagrange s equations applied to a deformable body in the presence of rigid body degrees of freedom the lagrange description of continuum mechanics is used an exact version of the equations is derived first this version which represents an identical extension of the fundamental law of dynamics does involve the idea of virtual motions the virtual motion is described in the framework of the ritz ansatz but our derivation does not make use of d alemberts principle the principle of virtual work or variational principles from the exact version by involving arguments related to the galerkin approximation technique we derive an approximate ritz type version of lagrange s equations this approximate version coincides with the traditional one which is based on the notion of kinetic energy however since our derivation stems from the fundamental law of dynamics we have at our disposal an alternative formulation which is based on the notion of local momentum this momentum based version which is the main topic of the present contribution can be used for the purpose of performing independent checks of the energy based version of lagrange s equations the momentum based version also clarifies that and how certain terms in the energy based version do cancel out the momentum based version is worked out in the framework of the floating frame of reference formulation of multibody dynamics explicit formulas for the single terms of lagrange s equations are derived for the translational rotational and flexible degrees of freedom of the deformable body respectively corresponding lagrange s equations are explained in the light of the relations of balance of total momentum balance of total moment of momentum of the mean stress theorem and the notion of virial of forces an embedding into the literature is given

strong separation and strong countability in fuzzy topological spaces given a fuzzy topological space x we introduce new notions of fuzzy separation and fuzzy countability using the family of its level topologies t t 0 1 we check that these are well defined fuzzy topological concepts and we compare them with the analogous fuzzy ones introduced in the literature we verify that these notions are not equivalent and we give a large number of examples which illustrate this fact

compactness in fuzzy fuzzy supra topological spaces the notion of q compact fuzzy topological spaces was first introduced by zhongfu kexue tongbao 29 5 1984 582 585 in this paper we extend this notion to fuzzy supra topological spaces and study some of their properties we also introduce the notion of fuzzy q almost compact q almost supracompact this notion is defined for any fuzzy subset and is a good extension finally we try to extend such a notion to l fuzzy supra topological spaces and discuss some of their properties

on fuzzy hausdorffness concepts we continue our earlier work on fuzzy hausdorffness 11 by relating our fuzzy hausdorffness notion with those due to other authors

an efficient general approach to modal analysis of frame resonators with applications to support loss in microelectromechanical systems abstract the design of high q resonators such as xylophone bar resonators xbrs capable of being fabricated using micro electro mechanical systems mems processes is of considerable interest in light of the widespread and rapidly growing use of systems dependent on their availability and performance this paper is concerned with vibration analysis and q optimisation of an xbr with the method extending directly to other planar frames and straightforwardly to more complex structures the rayleigh ritz method is discussed in some detail first treating the discrete case followed by developing and applying a kinematical procedure to an l frame structure attention is given to geometric interpretation of the rayleigh ritz procedure and to developing an intuitive understanding the method before turning to the xbr case having developed an approximation for system dynamics the results are used in conjunction with an analytical model of elastic wave propagation in the substrate to obtain an estimate for the support q factor natural frequencies mode shapes and support q values are presented and compared to finite element models of the same problem with excellent agreement observed at substantially lower computational cost for the first time in the literature the geometric impedance tuning principle underlying the xbr design is validated and quantified including sensitivity to manufacturing error

a short note on fuzzy neighbourhood spaces abstract lowen introduced the concept of fuzzy neighbourhood spaces fns s and proved that the family of level topologies of a fns is an antichain wuyts characterized fns s as being those fully stratified fuzzy topological spaces fts s whose level topologies form antichains and whose fuzzy topologies are the finest possible for thier level topologies mashhour et al introduced the axiom of structure a on fts s and derived for it several characterizations and fuzzy topological consequences we use those characterizations of structure a and wuyts characterization of fns s to prove that a fts is a fns iff it is of structure a and fully stratified

free and forced flexural vibration analysis of cantilever plates with attached point mass abstract in this paper the method of superposition or the gorman method is employed to obtain an analytical solution for free and forced vibrations of cantilever plates carrying point masses the free lateral vibration analysis is carried out by employing five building blocks four for the cantilever plate and one for the point mass computed eigenvalues show that the proposed scheme is accurate and convergent for free vibration analysis once the eigen analysis is completed the modal summation method is then used to deal with the base induced lateral vibration of cantilever plates carrying a point mass effects of mass ratios and locations of the point mass on eigenvalues and modal participation factors are investigated for square and rectangular plates

on fuzzy bitopological spaces abstract in this paper we have used the fuzzy topologies 1 and 2 to generate a family s which is a supra fuzzy topology on x using the family s we introduce and study the concepts of separation axioms continuity resp openness closedness of a mapping and compactness for a fuzzy bitopological space x 1 2 our definitions preserve much of the correspondence between concepts of fuzzy bitopological spaces and the associated fuzzy topological spaces we then investigate the relationship between these concepts and their correspondence with the fuzzy bitopological spaces kandil and el shafee 1991

strong and ultra separation axioms on l fuzzy topological spaces given an l fuzzy topological space x d we introduce new notions of l fuzzy separation axioms using the family of its level topology i a d a pr l we check that these are well defined l fuzzy topological concepts and we compare them with the analogous l fuzzy ones introduced in the literature we verify that these notations are not equivalent and we give many examples which illustrate this fact furthermore some results concerning the new concepts have been established