vibration analysis of annular like plates abstract the existence of eccentricity of the central hole for an annular plate results in a significant change in the natural frequencies and mode shapes of the structure in this paper the vibration analysis of annular like plates is presented based on numerical and experimental approaches using the finite element analysis code nastran the effects of the eccentricity hole size and boundary condition on vibration modes are investigated systematically through both global and local analyses the results show that analyses for perfect symmetric conditions can still roughly predict the mode shapes of recessive modes of the plate with a slightly eccentric hole they will however lead to erroneous results for dominant modes in addition the residual displacement mode shape is verified as an effective parameter for identifying damage occurring in plate like structures experimental modal analysis on a clamped free annular like plate is performed and the results obtained reveal good agreement with those obtained by numerical analysis this study provides guidance on modal analysis vibration measurement and damage detection of plate like structures

solving second order initial value problems by a hybrid multistep method without predictors abstract a three step seventh order hybrid linear multistep method hlmm with three non step points is proposed for the direct solution of the special second order initial value problems ivps of the form y f x y with an extension to y f x y y the main method and additional methods are obtained from the same continuous scheme derived via interpolation and collocation procedures the stability properties of the methods are discussed by expressing them as a one step method in higher dimension the methods are then applied in block form as simultaneous numerical integrators over non overlapping intervals numerical results obtained using the proposed block form reveal that it is highly competitive with existing methods in the literature

variable stepsize implementation of multistep methods for y f x y y the falkner method is a multistep scheme intended for the numerical solution of second order initial value problems where the first derivative does appear explicitly in this paper we develop a procedure to obtain k step falkner methods explicit and implicit in their variable step size versions providing recurrence formulas to compute the coefficients efficiently considering a pair of explicit and implicit formulae these may be implemented in predictor corrector mode

quasiregular representations of the infinite dimensional nilpotent group the notion of quasiregular representation of lie groups nauka moscow 1983 or geometric grundlehren der mathematischen wissenschaften band 220 springer berlin new york 1976 encyclopaedia of mathematical science vol 22 springer berlin 1994 pp 1 156 representation is well known for locally compact groups in the present work an analog of the quasiregular representation for the solvable infinite dimensional borel group g bor0n is constructed and a criterion of irreducibility of the constructed representations is presented this construction uses g quasi invariant gaussian measures on some g spaces x and extends the method used in kosyak funktsional anal i prilo hen 37 2003 78 81 for the construction of the quasiregular representations as applied to the nilpotent infinite dimensional group b0n

on the free vibration analysis of circular plates with stepped thickness over a concentric region by the differential quadrature element method it can be seen the dqem retains all advantages of the differential quadrature method extends the applicability range of the dq method and is more convenient for treating various boundary conditions

third derivative modification of k step block falkner methods for the numerical solution of second order initial value problems abstract this paper is devoted to the development and analysis of a modified family of falkner type methods for solving differential systems of second order initial value problems the approaches of collocation and interpolation are adopted to derive the new methods these modified methods are implemented in block form to obtain the numerical solutions to the considered problems the study of the properties of the proposed block falkner type methods reveals that they are consistent and zero stable and thus convergent from the stability analysis it could be seen that the proposed falkner methods have non empty stability regions for k 2 3 4 some numerical test are presented to illustrate the efficiency of the proposed family

mixed collocation methods for y f x y inline equatio abstract the collocation methods introduced here are based on linear combinations of trigonometric functions and powers the motivation is to provide better approximations for oscillatory solutions of initial value problems for differential equations of the special form y f x y the resulting methods for two or more collocation points are implicit runge kutta nystrom methods with coefficients which depend on both the fitted angular frequency and the steplength algebraic and trigonometric order conditions are considered and the stability properties of some methods are examined particular mixed collocation methods and other methods for the same class of problems are compared by applying them to a variety of test problems

induced representations of infinite dimensional groups i abstract induced representations ind h g s were introduced and studied by f g frobenius 8 for finite groups and developed by g w mackey 22 23 for locally compact groups we generalize the mackey construction for infinite dimensional groups to do this we construct some g quasi invariant measures on an appropriate completion x h g of the initial space x h g since the haar measure on g does not exist and extend the representation s of the subgroup h to the representation s of the corresponding completion h kirillov s orbit method 9 describes all irreducible unitary representations of the finite dimensional nilpotent group g n in terms of induced representations associated with orbits in coadjoint action of the group g n in a dual space g n of the lie algebra g n the induced representation defined in such a way allows us to start to develop an analog of the orbit method for the infinite dimensional nilpotent group b 0 z lim n g 2 n 1 of infinite in both directions matrices

the natural vibration of a conical shell with an annular end plate in this paper natural frequencies and mode shapes for a conical shell with an annular end plate or a round end plate are investigated in detail by combining the vibration theory with the transfer matrix method the governing equations of vibration for this structure are expressed in terms of a matrix differential equation and a novel recurrence formula method that can close in on analytical solution of transfer matrix is presented once the transfer matrix of single component has been determined and the product of each component matrix and the joining matrix can obtain entire structure matrix the frequency equations and mode shape functions are represented in terms of the elements of the structural transfer matrices the 3d finite element numerical simulations have verified the present formulas of natural frequencies and mode shapes the conclusions show that the transfer matrix method can accurately obtain natural vibration characteristics of the conical shell with an annular end plate

the fit of a sum of exponentials to noisy data abstract given a set of noisy data y0 ym measured at equidistant points on the time axis find n n 1 2 m exponents bi and coefficients ci such that i ci exp bit yields a good approximation of the data we study two algorithms for the reconstruction of the exponents and coefficients we show that the error in the reconstructed exponents is of the order of the gap n 1 n n 1 between the smallest accepted singular value n and the largest rejected singular value n 1 of a hankel matrix constructed from the data moreover we provide a statistical analysis which gives some insight in the optimal choice of the dimensions of the hankel matrix to be used given a fixed number of datapoints and we illuminate this by several clarifying examples

irreducible regular gaussian representations of the groups of the interval and circle diffeomorphisms abstract we define the analogue of the regular representations for infinite dimensional groups using quasi invariant measures on such groups and present the criterium of irreducibility and equivalence of constructed representations for groups of the interval and circle diffeomorphisms in a v kosyak funktsional anal i prilozhen 24 1990 82 a similar construction was done for a certain infinite dimensional matrix group

a continuous two step method of order 8 with a block extension for y f x y y abstract we derive a piecewise continuous hybrid third derivative approximation chtda that is defined for all values of the independent variable on the range of interest this continuous approximation has the ability to provide a continuous solution between all the grid points with a uniform accuracy comparable to that obtained at the grid points furthermore the chtda is used to produce several implicit hybrid third derivative formulas ihtdfs which are expressed in block form and applied as a block hybrid third derivative algorithm bhtda for the numerical solution of second order initial value problems the stability of the block extension is discussed numerical examples are given to illustrate the accuracy and efficiency of the bhtda

harmonic analysis on the infinite symmetric group the infinite symmetric group s whose elements are finite permutations of 1 2 3 is a model example of a big group by virtue of an old result of murray von neumann the one sided regular representation of s in the hilbert space l2 s generates a type ii1 von neumann factor while the two sided regular representation is irreducible this shows that the conventional scheme of harmonic analysis is not applicable to s for the former representation decomposition into irreducibles is highly non unique and for the latter representation there is no need of any decomposition at all we start with constructing a compactification mathfrak s supset s infty which we call the space of virtual permutations although mathfrak s is no longer a group it still admits a natural two sided action of s thus mathfrak s is a g space where g stands for the product of two copies of s on mathfrak s there exists a unique g invariant probability measure 1 which has to be viewed as a true haar measure for s more generally we include 1 into a family t t 0 of distinguished g quasiinvariant probability measures on virtual permutations by making use of these measures we construct a family tz z of unitary representations of g called generalized regular representations each representation tz with z 0 can be realized in the hilbert space l 2 mathfrak s mu t where t z 2 as z the generalized regular representations tz approach in a suitable sense the naive two sided regular representation of the group g in the space l2 s in contrast with the latter representation the generalized regular representations tz are highly reducible and have a rich structure we prove that any tz admits a unique decomposition into a multiplicity free continuous integral of irreducible representations of g for any two distinct and not conjugate complex numbers z1 z2 the spectral types of the representations t z 1 and t z 2 are shown to be disjoint in the case z a complete description of the spectral type is obtained further work on the case z reveals a remarkable link with stochastic point processes and random matrix theory

infinite dimensional schur weyl duality and the coxeter laplace operator we extend the classical schur weyl duality between representations of the groups sl n mathbb c and mathfrak s n to the case of sl n mathbb c and the infinite symmetric group mathfrak s mathbb n our construction is based on a dynamic or inductive scheme of schur weyl dualities it leads to a new class of representations of the infinite symmetric group which has not appeared earlier we describe these representations and in particular find their spectral types with respect to the gelfand tsetlin algebra the main example of such a representation acts in an incomplete infinite tensor product as an important application we consider the weak limit of the so called coxeter laplace operator which is essentially the hamiltonian of the xxx heisenberg model in these representations

harmonic analysis on the infinite dimensional unitary group and determinantal point processes the infinite dimensional unitary group u is the inductive limit of growing compact unitary groups u n in this paper we solve a problem of harmonic analysis on u stated in ol3 the problem consists in computing spectral decomposition for a remarkable 4 parameter family of characters of u these characters generate representations which should be viewed as analogs of nonexisting regular representation of u the spectral decomposition of a character of u is described by the spectral measure which lives on an infinite dimensional space of indecomposable characters the key idea which allows us to solve the problem is to embed into the space of point configurations on the real line without two points this turns the spectral measure into a stochastic point process on the real line the main result of the paper is a complete description of the processes corresponding to our concrete family of characters we prove that each of the processes is a determinantal point process that is its correlation functions have determinantal form with a certain kernel our kernels have a special integrable form and are expressed through the gauss hypergeometric function from the analytic point of view the problem of computing the correlation kernels can be reduced to a problem of evaluating uniform asymptotics of certain discrete orthogonal polynomials studied earlier by richard askey and peter lesky one difficulty lies in the fact that we need to compute the asymptotics in the oscillatory regime with the period of oscillations tending to 0 we do this by expressing the polynomials in terms of a solution of a discrete riemann hilbert problem and computing the nonoscillatory asymptotics of this solution from the point of view of statistical physics we study thermodynamic limit of a discrete log gas system an interesting feature of this log gas is that its density function is asymptotically equal to the characteristic function of an interval our point processes describe how different the random particle configuration is from the typical densely packed configuration in simpler situations of harmonic analysis on infinite symmetric groups and harmonic analysis of unitarily invariant measures on infinite hermitian matrices similar results were obtained in our papers bo1 bo2 bo4

classical jacobi polynomials closed form solutions for transverse vibrations this paper deals with transverse vibrations of nonuniform homogeneous beams and plates classes of beams and axisymmetrical circular plates whose boundary value problems of free transverse vibrations and free transverse axisymmetrical vibrations respectively can be reduced to an eigenvalue singular problem singularities occur at both ends of orthogonal polynomials are reported exact natural frequencies and jacobi polynomials as exact mode shapes which result directly from eigenvalues and eigenfunctions of eigenvalue singular problems of classical orthogonal polynomials are reported for these classes the above classes of beams and plates hereafter called jacobi classes are given by geometry and boundary conditions the geometry consists of parabolic thickness variation with respect to the axial coordinate for beams and with respect to the radius for plates beams belonging to this class have either one or two sharp ends singularities along with certain boundary conditions plates have zero thickness at zero and outer radii the boundary value problems associated with plates and beams of two sharp ends are free boundary problems two other boundary value problems hinged free and sliding free are reported for beams with one sharp end also exact natural frequencies and mode shapes for uniformly rotating beams with hinged free boundary are reported

block hybrid method for the numerical solution of fourth order boundary value problems abstract a linear multistep block hybrid method with four off grid points is presented for the direct approximation of the solution of fourth order boundary value problems multiple finite difference formulas are derived and grouped into a unique block to form a numerical integrator to solve directly the fourth order problem without the need to reduce it previously to a first order system the convergence of the proposed method is discussed the superiority of this method over existing methods is established numerically considering different problems that have appeared in the literature

solving directly special fourth order ordinary differential equations using runge kutta type method in this paper an explicit runge kutta method for solving directly fourth order ordinary differential equations odes is constructed and denoted as rkfd we present a relevant colored tree theory and the associated b series theory for the order conditions based on the order conditions a three stage fourth order rkfd method and a three stage fifth order rkfd method are constructed numerical illustrations are presented to show the efficiency of the new rkfd methods by comparing them with other existing runge kutta nystrom rkn and runge kutta rk methods in the scientific literature after converting the problem into a system of second order odes and a system of first order odes respectively

natural frequencies of transverse vibrations of non uniform circular and annular plates abstract this paper presents a method for the free transverse vibration analysis of thin elastic isotropic uniform and non uniform circular and annular plates the circumferential mode numbers n 0 and n 1 are dealt with in this paper the method is a hybrid of plate theory and finite element analysis the plate is subdivided into one circular and many annular finite elements two new finite elements were developed the first type being a circular plate and the second an annular plate the displacement functions of the finite element model are the classical solution shape functions of plate theory mass and stiffness matrices are determined by precise analytical integration the free vibrations of uniform circular and annular plates are studied by this method as well as non uniform plates the results obtained reveal that the frequencies calculated by this method are in good agreement with those obtained by other authors this method combines the advantages of the standard finite element analysis and the high accuracy formulation provided by the use of displacement functions derived from plate theory instead of the usual low order polynomials the present method is remarkable for the fact that it enables us to determine with equal precision both low and high natural frequencies

multiderivative methods for periodic initial value problems a family of two step multiderivative methods is developed for the numerical solution of periodic initial value problems of the form bf y t bf f t bf y with initial conditions bf y t 0 bf y 0 bf y t 0 bf y 0 given the methods are developed by showing that the solution of the single test equation y t lambda 2 y t lambda real satisfies a recurrence relation in which exponential functions are involved the methods of the family are then derived by replacing the exponential terms by pade approximants the methods are analyzed for accuracy and periodicity properties the error constants and intervals of periodicity are contained in an appendix those pade approximants for which the degree of the numerator do not exceed the degree of the denominator are seen to be p stable suitable predictor corrector combinations of the methods are analyzed for use in pece mode the methods are compared with existing methods and they are tested on four problems from the

criteria of irreducibility of the koopman representations for the group gl0 2 r abstract our aim is to find the irreducibility criteria for the koopman representation when the group acts on some space with a measure problem 1 5 some general necessary conditions of the irreducibility of this representation are established in the particular case of the group gl 0 2 r lim n gl 2 n 1 r the inductive limit of the general linear groups we prove that these conditions are also the necessary ones the corresponding measure is infinite tensor products of one dimensional arbitrary gaussian non centered measures the corresponding g space x m is a subspace of the space mat 2 r of infinite in both directions real matrices in fact x m is a collection of m infinite in both directions rows this result was announced in 22 we give the proof only for m 2 the general case will be studied later

influence of the poisson ratio on the natural frequencies of stepped thickness circular plate the natural frequencies of simply supported and clamped stepped thickness plates are determined using classical plate solutions with exact continuity conditions at the step the effect of incorporating the poisson ratio in the continuity conditions on the natural frequencies for nodal diameters 0 1 and nodal interior circle numbers 0 1 2 is thoroughly investigated for engineering applications a design criterion is proposed for simply supported and clamped plates based on an approximate linear model for the natural frequencies the literature lacks experimental results on this type of plates hence in this paper experimental results are presented for four models with two poisson s ratios and prove their consistency with the proposed criterion

embedded explicit runge kutta type methods for directly solving special third order differential equations y f x y abstract in this paper three pairs of embedded runge kutta type methods for directly solving special third order ordinary differential equations odes of the form y f x y denoted as rkd methods are presented the first is the rkd4 3 pair which is third order embedded in fourth order method has the property first same as last fsal whereby the last row of the coefficient matrix is equal to the vector output the second method is the rkd5 4 pair followed by the rkd6 5 pair the methods are derived with the strategies such that the higher order methods are very accurate and the lower order methods will give the best error estimates variables stepsize codes are developed based on the methods and used to solve a set of special third order problems numerical results are compared with the existing embedded runge kutta pairs which require the problems to be reduced into a system of first order odes numerical results have clearly shown the advantage and the efficiency of the new rkd pairs