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

bell s inequalities and quantum field theory ii bell s inequalities are maximally violated in the vacuum in the context of the study of bell s inequalities carried out in paper i j math phys 28 2440 1987 it is proven that bell s inequalities are maximally violated in the vacuum state by suitable spacelike separated observables for both bose and fermi free quantum field theories

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

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

the generally covariant locality principle a new paradigm for local quantum field theory a new approach to the model independent description of quantum field theories will be introduced in the present work the main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity thus giving rise to the concept of a locally covariant quantum field theory such locally covariant quantum field theories will be described mathematically in terms of covariant functors between the categories on one side of globally hyperbolic spacetimes with isometric embeddings as morphisms and on the other side of algebras with unital injective monomorphisms as morphisms moreover locally covariant quantum fields can be described in this framework as natural transformations between certain functors the usual haag kastler framework of nets of operator algebras over a fixed spacetime background manifold together with covariant automorphic actions of the isometry group of the background spacetime can be re gained from this new approach as a special case examples of this new approach are also outlined in case that a locally covariant quantum field theory obeys the time slice axiom one can naturally associate to it certain automorphic actions called relative cauchy evolutions which describe the dynamical reaction of the quantum field theory to a local change of spacetime background metrics the functional derivative of a relative cauchy evolution with respect to the spacetime metric is found to be a divergence free quantity which has as will be demonstrated in an example the significance of an energy momentum tensor up to addition of scalar functions for the locally covariant quantum field theory furthermore we discuss the functorial properties of state spaces of locally covariant quantum field theories that entail the validity of the principle of local definiteness

on the modular structure of local algebras of observables an asymptotic expression for the modular operators of local algebras of observables in quantum field theory is given in an asymptotically scale invariant theory this leads to an identification of the spectra of all modular operators with so in this case the local algebras are algebras of type iii and only factors of type iii1 can occur in a central decomposition

on the connection between quantum fields and von neumann algebras of local operators the relationship between a standard local quantum field and a net of local von neumann algebras is discussed two natural possibilities for such an association are identified and conditions for these to obtain are found it is shown that the local net can naturally be so chosen that it satisfies the special condition of duality the notion of an intrinsically local field operator is introduced and it is shown that such an operator defines a local net with which the field is locally associated a regularity condition on the field is formulated and it is shown that if this condition holds then there exists a unique local net with which the field is locally associated if and only if the field algebra contains at least one intrinsically local operator conditions under which a field and other fields in its borchers class are associated with the same local net are found in terms of the regularity condition mentioned

global boundary flattening transforms for acoustic propagation under rough sea surfaces this paper introduces a conformal transform of an acoustic domain under a one dimensional rough sea surface onto a domain with a flat top this non perturbative transform can include many hundreds of wavelengths of the surface variation the resulting two dimensional flat topped domain allows direct application of any existing acoustic propagation model of the helmholtz or wave equation using transformed sound speeds such a transform model combination applies where the surface particle velocity is much slower than sound speed such that the boundary motion can be neglected once the acoustic field is computed the bijective one to one and onto mapping permits the field interpolation in terms of the original coordinates the bergstrom method for inverse riemann maps determines the transform by iterated solution of an integral equation for a surface matching term rough sea surface forward scatter test cases provide verification of the method using a particular parabolic equation model of the helmholt

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

propagation modeling over terrain using the parabolic wave equation we address the numerical solution of the parabolic wave equation over terrain using the fourier split step approach the method referred to as a shift map generalizes that of beilis and tappert 1979 who introduced a coordinate transformation technique to flatten the boundary this technique is extended to a wide angle form allowing larger propagation angles with respect to the horizon a new impedance boundary condition is derived for electromagnetic waves incident on a finitely conducting surface that enables solution of the parabolic wave pwe using the previously developed mixed fourier transform it is also shown by example that in many cases of interest the boundary may be approximated by discrete piecewise linear segments without affecting the field solution a more accurate shift map solution of the pwe for a piecewise linear boundary is therefore developed for modeling propagation over digitally sampled terrain data the shift map solution is applied to various surface types including ramps wedges curved obstacles and actual terrain where possible comparisons are made between the numerical solution and an exact analytical form the examples demonstrate that the shift map performs well for surface slopes as large as 10 15 spl deg and discontinuous slope changes on the order of 15 20 spl deg to accommodate a larger range of slopes it is suggested that the most viable solution for general terrain modeling is a hybrid of the shift map with the well known terrain masking knife edge diffraction approximation

causal independence and the energy level density of states in local quantum field theory within the general framework of local quantum field theory a physically motivated condition on the energy level density of well localized states is proposed and discussed it is shown that any model satisfying this condition obeys a strong form of the principle of causal statistical independence which manifests itself in a specific algebraic structure of the local algebras split property it is also shown that the proposed condition holds in a free field theory

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

nuclear maps and modular structures ii applications to quantum field theory a correspondence between spectral properties of modular operators appearing in quantum field theory and the hamiltonian is established it allows to prove the distal split property for a wide class of models conversely any model having this property is shown to satisfy the haag swieca compactness criterion the results lead to a new type of nuclearity condition which can be applied to quantum field theories on arbitrary space time manifolds

normal product states for fermions and twisted duality for ccr and car type algebras with application to the yukawa2 quantum field model we present sufficient conditions that imply duality for the algebras of local observables in all abelian sectors of all locally normal irreducible representations of a field algebra if twisted duality obtains in one of these representations it is verified that the yukawa2 model satisfies these conditions yielding the first proof of duality for the observable algebra in all coherent charge sectors in this model this paper also constitutes the first verification of the assumptions of the axiomatic study of the structure of superselection sectors by doplicher haag and roberts in an interacting model with nontrivial sectors the existence of normal product states for the free fermi field algebra and thus the verification of the funnel property for the associated net of local algebras are demonstrated

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 proof of haag swieca s compactness property for elastic scattering states it is proved that in any massive relativistic quantum field theory satisfying two particle asymptotic completeness all the bounded energy components in the elastic two particle range of all subsets of states which are excitations of the vacuum state by uniformly bounded observables localized in a given finite region of spacetime are compact in the hilbert space of states this result which is in agreement with haag swieca s conjecture is also given a more precise form in terms of the rate of decrease of the n dimensional thickness or approximation number of such sets of states when n tends to infinity a similar computation valid at arbitrarily high energies is also given for the massive free field case

maximal violation of bell s inequalities is generic in quantum field theory under weak technical assumptions on a net of local von neumann algebras a o in a hilbert space which are fulfilled by any net associated to a quantum field satisfying the standard axioms it is shown that for every vector state in there exist observables localized in complementary wedge shaped regions in minkowski space time that maximally violate bell s inequalities in the state if in addition the algebras corresponding to wedge shaped regions are injective which is known to be true in many examples then the maximal violation occurs in any state on given by a density matrix

twisted duality of the car algebra we give a complete proof of the twisted duality property m q zm q z of the self dual car algebra in any fock representation the proof is based on the natural halmos decomposition of the reference hilbert space when two suitable closed subspaces have been distinguished we use modular theory and techniques developed by kato concerning pairs of projections in some essential steps of the proof as a byproduct of the proof we obtain an explicit and simple formula for the graph of the modular operator this formula can be also applied to fermionic free nets hence giving a formula of the modular operator for any double cone

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

hidden variables joint probability and the bell inequalities it is shown that the following statements about a quantum correlation experiment are mutually equivalent 1 there is a deterministic hidden variables model for the experiment 2 there is a factorizable stochastic model 3 there is one joint distribution for all observables of the experiment returning the experimental probabilities 4 there are well defined compatible joint distributions for all pairs and triples of commuting and noncommuting observables 5 the bell inequalities hold