on bivariate classical orthogonal polynomials abstract we deduce new characterizations of bivariate classical orthogonal polynomials associated with a quasi definite moment functional and we revise old properties for these polynomials more precisely new characterizations of classical bivariate orthogonal polynomials satisfying a diagonal pearson type equation are proved they are solutions of two separate partial differential equations one for every partial derivative their partial derivatives are again orthogonal and every vector polynomial can be expressed in terms of its partial derivatives by means of a linear relation involving only three terms of consecutive total degree moreover we study general solutions of the matrix second order partial differential equation satisfied by classical orthogonal polynomials and we deduce the explicit expressions for the matrix coefficients of the structure relation finally some illustrative examples are given

the spectrum of the chebyshev collocation operator for the heat equation it is shown that the eigenvalues of the pseudospectral chebyshev second derivative operator with separated boundary conditions are real negative and distinct an application to the full potential equation is also presented

on a class of bivariate second order linear partial difference equations and their monic orthogonal polynomial solutions abstract in this paper we classify the bivariate second order linear partial difference equations which are admissible potentially self adjoint and of hypergeometric type using vector matrix notation explicit expressions for the coefficients of the three term recurrence relations satisfied by monic orthogonal polynomial solutions are obtained in terms of the coefficients of the partial difference equation finally we make a compilation of the examples existing in the literature belonging to the class analyzed in this paper namely bivariate charlier meixner kravchuk and hahn orthogonal polynomials

pseudospectra of the convection diffusion operator the spectrum of the simplest 1d convection diffusion operator is a discrete subset of the negative real axis but the pseudospectra are regions in the complex plane that approximate parabolas put another way the norm of the resolvent is exponentially large as a function of the peclet number throughout a certain parabolic region these observations have a simple physical basis and suggest that conventional spectral analysis for convection diffusion operators may be of limited value in some applications

identification for critical flutter load and boundary conditions of a beam using neural networks with the increasing size and complexity of machines and vessels the inverse problems of continuous bodies are becoming important in this paper the possibility of using a multilayer perceptron network trained with the backpropagation algorithm for detecting the critical flutter load and boundary conditions of tapered beams is studied the beam model considered is a timoshenko beam which with the use of the transfer matrix method estimates the changes in various modal parameters caused by the shape parameters and boundary conditions of beams

stability of a timoshenko beam resting on a winkler elastic foundation the influences of a winkler elastic foundation modulus slenderness ratio and elastically restrained boundary conditions on the critical load of a timoshenko beam subjected to an end follower force are investigated the characteristic equation for elastic stability is derived it is found that the critical flutter load for the cantilever timoshenko beam will first decrease as the elastic foundation modulus is increased and when the elastic foundation modulus is greater than the corresponding critical value which corresponds to the lowest critical load it will increase instead in particular if the elastic foundation modulus is large enough the critical flutter load for the cantilever timoshenko beam can be greater than that of the bernoulli euler beam for a clamped translational or clamped rotational elastic spring supported beam resting on an elastic foundation there exists a critical value of the spring constant for each beam at this critical point the critical load jumps and the type of instability mechanism changes the jump mechanisms for beams resting on elastic foundations of different modulus values are different

spectral method and high order finite differences for the nonlinear cable equation we use high order approximation schemes for the space derivatives in the nonlinear cable equation and investigate the behavior of numerical solution errors by using exact solutions where available and grid convergence the space derivatives are numerically approximated by means of differentiation matrices nonlinearity in the equation arises from the hodgkin huxley dynamics of the gating variables for ion channels we have investigated in particular the effects of synaptic current distribution and compared the accuracy of the spectral solutions with that of finite differencing a flexible form for the injected current is used that can be adjusted smoothly from a very broad to a narrow peak which furthermore leads for the passive cable to a simple exact solution we have used three distinct approaches to assess the numerical solutions comparison with exact solutions in an unbranched passive cable the convergence of solutions with progressive refinement of the grid in an active cable and the simulation of spike initiation in a biophysically realistic single neuron model the spectral method provides good numerical solutions for passive cables comparable in accuracy to those from the second order finite difference method and far greater accuracy in the case of a simulated system driven by inputs that are smoothly distributed in space it provides faster convergence in active cables and in a realistic neuron model due to better approximation of propagating spikes

classroom note calculation of weights in finite difference formulas the classical techniques for determining weights in finite difference formulas were either computationally slow or very limited in their scope e g specialized recursions for centered and staggered approximations for adams bashforth adams moulton and bdf formulas for odes etc two recent algorithms overcome these problems for equispaced grids such weights can be found very conveniently with a two line algorithm when using a symbolic language such as mathematica reducing to one line in the case of explicit approximations for arbitrarily spaced grids we describe a computationally very inexpensive numerical algorithm

non conservative instability of non uniform beams resting on an elastic foundation abstract the influence of a winkler elastic foundation and the slenderness ratio on the non conservative instability of cantilever non uniform beams of rectangular cross section with constant height and linearly varied breadth t1 constant breadth and linearly varied height t2 and double taper t3 subjected to an end concentrated follower force is investigated it is found that without the elastic foundation the critical flutter load of the non uniform beam decreases as the taper ratio of the beam is increased however when the elastic foundation modulus is greater than a critical value the critical flutter load of the taper beams is always greater than that of uniform beams within the domain considered when the taper ratio of the beam lies in a certain range several critical turning points including a jump phenomenon may exist for the critical flutter load the jump mechanism and the influence of the elastic foundation modulus and the slenderness ratio on the jump phenomenon of timoshenko beams is explored

identification technique for nonlinear boundary conditions of a circular plate as a basic study for developing an identification technique for boundary conditions of machines and structures a new technique for a circular plate is proposed this technique has features that do not require data measured on the boundary and is applicable to nonlinear boundary conditions in the proposed technique the boundary is modelled by springs and dampers as well as effective mass and moment of inertia then their characteristics are determined by using the analytical solution together with the experimental data since the technique is based on the analytical solution it is applicable to any structure provided that its analytical solution can be derived numerical simulation is conducted to show that the procedure determines the boundary conditions accurately

bivariate orthogonal polynomials in the lyskova class classical orthogonal polynomials in two variables can be characterized as the polynomial solutions of a matrix second order partial differential equation involving matrix polynomial coefficients in this work we study classical orthogonal polynomials in two variables whose partial derivatives satisfy again a second order partial differential equation of the same type

neural identification of non linear dynamic structures neural networks are applied to the identification of non linear structural dynamic systems two complementary problems inspired from customer surveys are successively considered each of them calls for a different neural approach first the mass of the system is identified based on acceleration recordings statistical experiments are carried out to simultaneously characterize optimal pre processing of the accelerations and optimal neural network models it is found that key features for mass identification are the fourth statistical moment and the normalized power spectral density of the acceleration second two architectures of recurrent neural networks an autoregressive and a state space model are derived and tested for dynamic simulations showing higher robustness of the autoregressive form discussion is first based on a non linear two degree of freedom problem neural identification is then used to calculate the load from seven acceleration measurements on a car eighty three per cent of network estimations show below 5 error

numerical solution of calcium mediated dendritic branch model the standard algorithms for spatial discretizations of calcium mediated dendritic branch models via finite difference methods are quite accurate but they are also extremely slow to improve computational efficiency we apply spatial discretization using a spectral collocation method simulations using the spectral collocation method are compared to the finite difference approach using a model for calcium mediated restructuring with spine pruning we find that the spectral collocation method is about fifteen times more efficient to achieve similar accuracy than the finite difference approach even though spectral collocation requires more steps

image representation using separable two dimensional continuous and discrete orthogonal moments this paper addresses bivariate orthogonal polynomials which are a tensor product of two different orthogonal polynomials in one variable these bivariate orthogonal polynomials are used to define several new types of continuous and discrete orthogonal moments some elementary properties of the proposed continuous chebyshev gegenbauer moments cgm gegenbauer legendre moments glm and chebyshev legendre moments clm as well as the discrete tchebichef krawtchouk moments tkm tchebichef hahn moments thm krawtchouk hahn moments khm are presented we also detail the application of the corresponding moments describing the noise free and noisy images specifically the local information of an image can be flexibly emphasized by adjusting parameters in bivariate orthogonal polynomials the global extraction capability is also demonstrated by reconstructing an image using these bivariate polynomials as the kernels for a reversible image transform comparisons with the known moments are performed and the results show that the proposed moments are useful in the field of image analysis furthermore the study investigates invariant pattern recognition using the proposed three moment invariants that are independent of rotation scale and translation and an example is given of using the proposed moment invariants as pattern features for a texture classification application

orthogonal polynomials in two variables and second order partial differential equations abstract we study the second order partial differential equations l u au xx 22 bu xy cu yy du x eu y n u which have orthogonal polynomials in two variables as solutions by using formal functional calculus on moment functionals we first give new simpler proofs and improvements of the results by krall and sheffer and littlejohn we then give a two variable version of al salam and chihara s characterization of classical orthogonal polynomials in one variable we also study in detail the case when l belongs to the basic class that is a y c x 0 in particular we characterize all such differential equations which have a product of two classical orthogonal polynomials in one variable as solutions

weak classical orthogonal polynomials in two variables orthogonal polynomials in two variables constitute a very old subject in approximation theory usually they are studied as solutions of second order partial differential equations in this work we study two variable orthogonal polynomials associated with a moment functional satisfying the two variable analogue of the pearson differential equation from this approach we derive the extension of some of the usual characterizations of the classical orthogonal polynomials in one variable

bivariate second order linear partial differential equations and orthogonal polynomial solutions in this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second order linear partial differential equations which are admissible potentially self adjoint and of hypergeometric type general formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation using vector matrix notation moreover rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given as illustration these results are applied to a two parameter monic appell polynomials finally the non monic case is briefly discussed 2011 elsevier inc

second order partial differential equations for gradients of orthogonal polynomials in two variables in this work we introduce the classical orthogonal polynomials in two variables as the solutions of a matrix second order partial differential equation involving matrix polynomial coefficients the usual gradient operator and the divergence operator here we show that the successive gradients of these polynomials also satisfy a matrix second order partial differential equation closely related to the first one

the influence of an intermediate support on the stability behaviour of cantilever beams subjected to follower forces abstract a non uniform undamped beam which is simply supported at one end resting on an elastically flexible support at an intermediate abscissa q is examined the beam is subjected to a generic set of follower forces along the axis so that beck leipholz and hauger problems can be easily recovered as particular cases in all these systems a location for the intermediate support exists which corresponds to passage from flutter to divergence

approximation by the legendre collocation method of a model problem in electrophysiology abstract we examine the polynomial approximation of the solution of a nonlinear differential problem modelling the evolution of the potential inside an electrically stimulated neuron the collocation method at the legendre gauss lobatto nodes is used for the discretization with respect to the space variable

the approach of a neuron population firing rate to a new equilibrium an exact theoretical result the response of a noninteracting population of identical neurons to a step change in steady synaptic input can be analytically calculated exactly from the dynamical equation that describes the population s evolution in time here for model integrate and fire neurons that undergo a fixed finite upward shift in voltage in response to each synaptic event we compare the theoretical prediction with the result of a direct simulation of 90 000 model neurons the degree of agreement supports the applicability of the population dynamics equation the theoretical prediction is in the form of a series convergence is rapid so that the full result is well approximated by a few terms