wave reflection from the free end of a cylinder with an arbitrary cross section elastic wave scattering at the free end of a cylinder due to an incident monochromatic wave is investigated the cross section may have an arbitrary geometry with any number of distinct elastic rectilinear anisotropic materials comprising its planar profile the governing equations are based on a semi analytical finite element method in which the cross sectional behavior is modeled by general two dimensional finite elements with the axial dependence and time left unspecified at the outset first all the modal data for the cylinder are established two eigenproblems are posed for this purpose that are obtained by inserting a wavelike solution form into the governing equations these eigenproblems allow all propagating waves and end modes for the cylinder to be determined propagating modes are traveling waves with energy transport capabilities while the end modes are standing vibrations which in contrast do not transport any energy into the interior of the cylinder these eigendata are the basis for representing the wave reflection phenomenon at the free end the amplitudes of the traveling waves and end modes that satisfy traction free end conditions may be determined by least squares minimization or by a virtual work method four cylinders with different cross sectional geometries were considered to illustrate the analysis procedure and reveal some physical insight into the frequency dependent wave reflection phenomena in them

end reflections in a layered piezoelectric circular cylinder abstract end reflection phenomenon in a semi infinitely long layered piezoelectric circular cylinder is constructed with modal data from a spectral decomposition of the differential operator governing its natural vibrations these modal data consist of all propagating modes and edge vibrations and they constitute the basis for a wave function expansion of the reflection of waves arriving at the traction free end of the cylinder without any other external stimulus a passive reflection event occurs this traction free end condition is enforced at the gaussian integration points over the end cross section on the combination of incoming and reflected wave fields reflections due to monochromatic incoming axisymmetric m 0 and flexural m 1 waves are studied and two numerical examples are presented for an incoming axisymmetric wave there is a particular frequency that induces an end resonance which is characterized by high but finite amplitudes of end displacements vis a vis those of neighboring i e slightly different frequencies this phenomenon is illustrated in the two cylinder examples it is possible to modify the passive reflection event by imposing some voltage distribution over the free end for an oscillating end voltage that is out of phase with the incoming wave it is possible to extract electrical energy from it i e energy harvesting examples of such an oscillating voltage with a particular radial distribution are given that illustrate the amount of extracted energy as a function of the frequency of the incident monochromatic wave

the branching random walk and contact process on galton watson and nonhomogeneous trees we show that the branching random walk on a galton watson tree may have one or two phase transitions depending on the relative sizes of the mean degree and the maximum degree we show that there are some galton watson trees on which the branching random walk has one phase transition while the contact process has two this contradicts a conjecture of madras and schinazi we show that the contact process has only one phase transition on some trees of uniformly exponential growth and bounded degree contradicting a conjecture of pemantle

much ado about shear correction factors in timoshenko beam theory abstract many shear correction factors have appeared since the inception of timoshenko beam theory in 1921 while rational bases for them have been offered there continues to be some reluctance to their full acceptance because the explanations are not totally convincing and their efficacies have not been comprehensively evaluated over a range of application herein three dimensional static and dynamic information and results for a beam of general both symmetric and non symmetric cross section are brought to bear on these issues only homogeneous isotropic beams are considered semi analytical finite element safe computer codes provide static and dynamic response data for our purposes greater clarification of issues relating to the bases for shear correction factors can be seen also comparisons of numerical results with timoshenko beam data will show the effectiveness of these factors beyond the range of application of elementary bernoulli euler theory an issue concerning principal shear axes arose in the definition of shear correction factors for non symmetric cross sections in this method expressions for the shear energies of two transverse forces applied on the cross section by beam and three dimensional elasticity theories are equated to determine the shear correction factors this led to the necessity for principal shear axes we will argue against this concept and show that when two forces are applied simultaneously to a cross section it leads to an inconsistency only one force should be used at a time and two sets of calculations are needed to establish the shear correction factors for a non symmetrical cross section

rumor processes in random environment on mathbb n and on galton watson trees the aim of this paper is to study rumor processes in random environment in a rumor process a signal starts from the stations of a fixed vertex the root and travels on a graph from vertex to vertex we consider two rumor processes in the firework process each station when reached by the signal transmits it up to a random distance in the reverse firework process on the other hand stations do not send any signal but they listen for it up to a random distance the first random environment that we consider is the deterministic 1 dimensional tree mathbb n with a random number of stations on each vertex in this case the root is the origin of mathbb n we give conditions for the survival extinction on almost every realization of the sequence of stations later on we study the processes on galton watson trees with random number of stations on each vertex we show that if the probability of survival is positive then there is survival on almost every realization of the infinite tree such that there is at least one station at the root we characterize the survival of the process in some cases and we give sufficient conditions for survival extinction

an adaptive multi element generalized polynomial chaos method for stochastic differential equations we formulate a multi element generalized polynomial chaos me gpc method to deal with long term integration and discontinuities in stochastic differential equations we first present this method for legendre chaos corresponding to uniform random inputs and subsequently we generalize it to other random inputs the main idea of me gpc is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value in each subdomain or random element we then employ a generalized polynomial chaos expansion we develop a criterion to perform such a decomposition adaptively and demonstrate its effectiveness for odes including the kraichnan orszag three mode problem as well as advection diffusion problems the new method is similar to spectral element method for deterministic problems but with h p discretization of the random space

a generalized dimension reduction method for multidimensional integration in stochastic mechanics summary a new generalized multivariate dimension reduction method is presented for calculating statistical moments of the response of mechanical systems subject to uncertainties in loads material properties and geometry the method involves an additive decomposition of an n dimensional response function into at most s dimensional functions where s n an approximation of response moments by moments of input random variables and a moment based quadrature rule for numerical integration a new theorem is presented which provides a convenient means to represent the taylor series up to a specific dimension without involving any partial derivatives a complete proof of the theorem is given using two lemmas also proved in this paper the proposed method requires neither the calculation of partial derivatives of response as in commonly used taylor expansion perturbation methods nor the inversion of random matrices as in the neumann expansion method eight numerical examples involving elementary mathematical functions and solid mechanics problems illustrate the proposed method results indicate that the multivariate dimension reduction method generates convergent solutions and provides more accurate estimates of statistical moments or multidimensional integration than existing methods such as first and second order taylor expansion methods statistically equivalent solutions quasi monte carlo simulation and the fully symmetric interpolatory rule while the accuracy of the dimension reduction method is comparable to that of the fourth order neumann expansion method a comparison of cpu time suggests that the former is computationally far more efficient than the latter copyright 2004 john wiley sons ltd

rotation invariant moment problems an important theorem of marcel riesz cf 14 states that the polynomials are dense in l2 x when x is a determinate measure on the real line in the indeterminate case riesz also characterized the measures x for which the polynomials are dense in l2 they are the so called nevanlinna extremal measures introduced in nevanlinna 11 it does not seem to be known whether the polynomials are dense in l2 g when x is a determinate measure on r d d 1 cf the expository paper by fuglede 7 as well as the research problems book 8 p 529 where devinatz poses the problem as question 1 and ascribes it to the physicist john challifour 1978 in this paper we shall settle the question in the negative there exist rotation invariant measures on r d d 1 which are determinate but for which the polynomials are not dense in l2 p such measures x are necessarily of the following very special form

the d variate vector hermite polynomial of order k abstract the multivariate hermite polynomials occur in edgeworth and saddle point approximations to the distributions of random vector variables explicit expressions in vector notation for the multivariate hermite polynomials of arbitrary order are given and shown to be a special case of a general addition property of hermite polynomials general properties of these polynomials are given in vector notation including an integral representation the orthogonality property differential properties and recurrence relations as well as inversion formulas various expansions of a multivariate normal density are given in terms of a multivariate normal density with different mean different covariance or both in special cases these results give mehler s formula

rumor processes on mathbb n and discrete renewal processes we study two rumor processes on mathbb n the dynamics of which are related to an si epidemic model with long range transmission both models start with one spreader at site 0 and ignorants at all the other sites of mathbb n but differ by the transmission mechanism in one model the spreaders transmit the information within a random distance on their right and in the other the ignorants take the information from a spreader within a random distance on their left we obtain the probability of survival information on the distribution of the range of the rumor and limit theorems for the proportion of spreaders the key step of our proofs is to show that in each model the position of the spreaders on mathbb n can be related to a suitably chosen discrete renewal process

approximate determination of dispersion relations and displacement fields associated with elastic waves in bars method based on matrix formulation of hamiltons principle a method for approximate determination of dispersion relations and displacement fields associated with elastic waves in long prismatic bars is established the method is based on a matrix formulation of hamilton s principle and utilizes co ordinate functions to model the displacements in the bar the method which involves the solution of an eigenvalue problem is suitable for use in a mathematical toolbox it is applied to wave propagation in bars with square and circular cross sections the results obtained agree well with the known exact and approximate solutions the convergence rate is illustrated by systematically changing the order of the co ordinate functions

uncertainty quantification under dependent random variables by a generalized polynomial dimensional decomposition abstract this paper is concerned with uncertainty quantification analysis of complex systems subject to dependent input random variables the analysis focuses on a new generalized version of polynomial dimensional decomposition pdd referred to as gpdd entailing hierarchically ordered measure consistent multivariate orthogonal polynomials in dependent variables under a few prescribed assumptions gpdd exists for any square integrable output random variable and converges in mean square to the correct limit new analytical formulae are proposed to calculate the mean and variance of a gpdd approximation of a general output variable in terms of the expansion coefficients and second moment properties of multivariate orthogonal polynomials however unlike in pdd calculating the coefficients of gpdd requires solving a coupled system of linear equations besides the variance formula of gpdd contains extra terms due to statistical dependence among input variables the extra terms disappear when the input variables are statistically independent reverting gpdd to pdd two numerical examples the one derived from a stochastic boundary value problem and the other entailing a random eigenvalue problem illustrate second moment error analysis and estimation of the probabilistic characteristics of eigensolutions

a method of analysis for end and transitional effects in anisotropic cylinders abstract we present a rigorous verification study and an extension to an existing semi analytical finite element formulation for analysis of end and transition effects in prismatic cylinders end and transition effects in stressed cylinders are phenomena associated with the difference between results that are predicted by the saint venant solutions and the actual point wise conditions these differences manifest themselves as self equilibrated stress states notwithstanding certain well known exceptions e g restrained torsion of open thin walled sections such effects in isotropic cylinders are usually confined to a very small neighborhood of a terminal boundary or transition zone and are typically neglected for anisotropy as in the case of most smart active and composite material systems they can persist much further into the interior of the structure and need to be quantified to design geometry transition zones and to fully understand the delamination effects in the semi analytical approach we first discretize the governing equations within the cross sectional plane of the cylinder the end solution fields satisfy the homogeneous form of the resulting semi analytical system of ordinary differential equations this leads to an algebraic eigenvalue problem and an eigenfunction expansion of the stress and displacement fields due to end effects unique to the present study we formulate a procedure to quantify the transitional effects for end to end connected cylinders for which the displacement and stress continuity along the transition interface need to be enforced the semi analytical approach has several distinct advantages i it is computationally efficient as only the cross sectional geometry is discretized ii it can be applied to arbitrary cross sectional geometries and the most general form of anisotropy and iii it yields direct measures for the decay lengths or decay rates of any end or transition solution field analytical solutions to end effect problems are scarce those that exist are for simple geometry and material constitution we use these analytical solutions as well as solutions obtained using three dimensional finite element models to verify our approach and to assess its efficiency

wiener hermite polynomial expansion for multivariate gaussian probability measures abstract this paper introduces a new transformation free generalized polynomial chaos expansion pce comprising multivariate hermite orthogonal polynomials in dependent gaussian random variables the second moment properties of hermite polynomials reveal a weakly orthogonal system when obtained for a general gaussian probability measure still the exponential integrability of norm allows the hermite polynomials to constitute a complete set and hence a basis in a hilbert space the completeness is vitally important for the convergence of the generalized pce to the correct limit the optimality of the generalized pce and the approximation quality due to truncation are discussed new analytical formulae are proposed to calculate the mean and variance of a generalized pce approximation of a general output variable in terms of the expansion coefficients and statistical properties of hermite polynomials however unlike in the classical pce calculating the coefficients of the generalized pce requires solving a coupled system of linear equations besides the variance formula of the generalized pce contains additional terms due to statistical dependence among gaussian variables the additional terms vanish when the gaussian variables are statistically independent reverting the generalized pce to the classical pce numerical examples illustrate the generalized pce approximation in estimating the statistical properties of various output variables

discrete one dimensional coverage process on a renewal process consider the following coverage model on mathbb n for each site i in mathbb n associate a pair xi i r i where xi i i ge 0 is a 1 dimensional undelayed discrete renewal point process and r i i ge 0 is an i i d sequence of mathbb n valued random variables at each site where xi i 1 start an interval of length r i coverage occurs if every site of mathbb n is covered by some interval we obtain sharp conditions for both positive and null probability of coverage as corollaries we extend results of the literature of rumor processes and discrete one dimensional boolean percolation

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

concentration inequalities for random fields via coupling we present a new and simple approach to concentration inequalities in the context of dependent random processes and random fields our method is based on coupling and does not use information inequalities in case one has a uniform control on the coupling one obtains exponential concentration inequalities if such a uniform control is no more possible then one obtains polynomial or stretched exponential concentration inequalities our abstract results apply to gibbs random fields both at high and low temperatures and in particular to the low temperature ising model which is a concrete example of non uniformity of the coupling

a polynomial chaos expansion in dependent random variables abstract this paper introduces a new generalized polynomial chaos expansion pce comprising measure consistent multivariate orthonormal polynomials in dependent random variables unlike existing pces whether classical or generalized no tensor product structure is assumed or required important mathematical properties of the generalized pce are studied by constructing orthogonal decomposition of polynomial spaces explaining completeness of orthogonal polynomials for prescribed assumptions exploiting whitening transformation for generating orthonormal polynomial bases and demonstrating mean square convergence to the correct limit analytical formulae are proposed to calculate the mean and variance of a truncated generalized pce for a general output variable in terms of the expansion coefficients an example derived from a stochastic boundary value problem illustrates the generalized pce approximation in estimating the statistical properties of an output variable for 12 distinct non product type probability measures of input variables

wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics in this paper we propose a numerical method based on wiener chaos expansion and apply it to solve the stochastic burgers and navier stokes equations driven by brownian motion the main advantage of the wiener chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner the separation principle effectively reduces a stochastic equation to its associated propagator a system of deterministic equations for the coefficients of the wiener chaos expansion simple formulas for statistical moments of the stochastic solution are presented these formulas only involve the solutions of the propagator we demonstrate that for short time solutions the numerical methods based on the wiener chaos expansion are more efficient and accurate than those based on the monte carlo simulations