on the number of homomorphisms from a finite group to a general linear group abstract we study the number of homomorphisms from a finite group to a general linear group over a finite field in particular we give a generating function of such numbers then the rogers ramanujan identities are applicable

third order complex time step methods for transient analysis abstract using complex time steps and the method for the first order differential equations third order accurate a stable step by step time integration algorithms with controllable numerical dissipation are constructed for transient analysis the results at two sub step locations are evaluated and combined linearly with the corresponding weighting factors to give higher order accurate results at the end of a time step the sub step locations and the weighting factors are complex numbers the amplification factors at infinity are controlled by another algorithmic parameter a fourth order accurate algorithm is obtained when 1 the algorithmic characteristics and the computational procedure for the present third order algorithms are discussed equivalent algorithms and comparisons with other algorithms are given the present algorithms are found to be competitive with the other third and fourth order accurate algorithms

a unified set of single step algorithms part 2 theory part 1 of this paper1 presents a very general single step algorithm sspj for the numerical integration of first and second order time dependent differential equations in part 2 we present and discuss results of the accuracy and stability analysis for ss11 ss21 ss31 ss22 and ss32 there is also a detailed comparison of ss22 and the newmark algorithm

precise integration methods based on lagrange piecewise interpolation polynomials this paper introduces two new types of precise integration methods for dynamic response analysis of structures namely the integral formula method and the homogenized initial system method the applied loading vectors in the two algorithms are simulated by the lagrange piecewise interpolation polynomials based on the zeros of the first chebyshev polynomial developed on the basis of the integral formula and the lagrange piecewise interpolation polynomial and combined with the recurrence relationship of some key parameters in the integral computation suggested in this paper with the solving process of linear algebraic equations the integral formula method has been set up on the basis of the lagrange piecewise interpolation polynomial and transforming the non homogenous initial system into the homogeneous dynamic system the homogenized initial system method without dimensional expanding is presented this homogenized initial system method avoids the matrix inversion operation and is a general homogenized high precision direct integration scheme the accuracy of the presented time integration schemes is studied and is compared with those of other commonly used schemes the presented time integration schemes have arbitrary order of accuracy wider application and are less time consuming two numerical examples are also presented to demonstrate the applicability of these new methods copyright 2008 john wiley sons ltd

mixed finite element formulation for the solution of parabolic problems a mixed formulation of the finite element method is used to establish a higher order incremental method for the solution of parabolic equations the displacement and velocity fields in the terminology of solid mechanics are approximated independently in time using a hierarchical adaptive basis three time bases are tested namely polynomial radial and wavelet bases the time approximation criterion ensures stability and preserves parabolicity the resulting discretization in the time domain leads to an uncoupled governing system in the space dimension this system can be subsequently solved using the alternative formulations for the finite element and boundary element methods the application of the conventional conform displacement formulation of the finite element method is illustrated particular attention is given to the extension of pian and trefftz type hybrid stress finite elements for transient analysis in the time domain the performance of the coupled time space integration method is assessed and the results obtained are compared with the results reported in the literature for alternative time integration procedures

design spaces measures and metrics for evaluating quality of time operators and consequences leading to improved algorithms by design illustration to structural dynamics for the first time for time discretized operators we describe and articulate the importance and notion of design spaces and algorithmic measures that not only can provide new avenues for improved algorithms by design but also can distinguish in general the quality of computational algorithms for time dependent problems the particular emphasis is on structural dynamics applications for the purpose of illustration and demonstration of the basic concepts the underlying concepts can be extended to other disciplines as well for further developments in time discretized operators and or for evaluating existing methods from the established measures for computational algorithms the conclusion that the most effective in the sense of convergence namely the stability and accuracy and complexity namely the algorithmic formulation and algorithmic structure computational algorithm should appear in a certain algorithmic structure of the design space amongst comparable algorithms is drawn with this conclusion and also with the notion of providing new avenues leading to improved algorithms by design as an illustration a novel computational algorithm which departs from the traditional paradigm in the sense of lms methods with which we are mostly familiar with and widely used in commercial software is particularly designed into the perspective design space representation of comparable algorithms and is termed here as the forward displacement non linearly explicit l stable fdel algorithm which is unconditionally consistent and does not require non linear iterations within each time step from the established measures for comparable algorithms simply for illustration purposes the resulting design of the fdel formulation is then compared with the commonly advocated explicit central difference method and the implicit newmark average acceleration method alternately the same conclusion holds true against controllable numerically dissipative algorithms which pertain to the class of linear multi step lms methods for assessing both linear and non linear dynamic cases the conclusions that the proposed new design of the fdel algorithm which is a direct consequence of the present notion of design spaces and measures is the most effective algorithm to date to our knowledge in comparison to the class of second order accurate algorithms pertaining to lms methods for routine and general non linear dynamic situations is finally drawn through rigorous numerical experiments

higher order boundary element methods for transient diffusion problems part ii singular flux formulation despite the significant number of publications on boundary element methods bem for time dependent problems of heat diffusion there still remain issues that need to be addressed most importantly accuracy of the numerical modelling although very precise for steady state problems the common boundary element methods applied to transient problems do not yield highly accurate numerical solutions this paper investigates the reasons that prohibit achievement of a high level of accuracy for transient heat diffusion problems with continuous temperature and bounded heat flux solutions in order to greatly enhance the commonly used boundary element formulations we propose higher order time interpolation functions including quadratic and quartic approximations we show that the use of higher order time functions greatly reduces the numerical error concentrated in the corner regions and results in very good uniformity of the flux and temperature distributions along the boundaries for problems where uniform distributions are expected in order to highlight the importance of proper resolution both in time and space for the transient problems we consider one and two dimensional formulations in this paper high order boundary elements using quartic shape functions as well as high order bi quartic volume cells are used to attain mesh independent numerical solutions we consider four transient heat diffusion problems that possess exact solutions to investigate the convergence rate and accuracy of the higher order boundary element formulations a very high level of accuracy is possible for both one and two dimensional formulations additionally we show that the accuracy of a commercially available finite element code is far less than that of the boundary element methods for a given spatial and temporal discretization copyright 2002 john wiley sons ltd

higher derivative explicit one step methods for non linear dynamic problems part i design and theory a new higher order explicit one step scheme is presented for use in the solution of a certain class of non linear problems in structural dynamics the generalized scheme covers some recently developed multistep algorithms as well as some new one step methods with improved numerical properties a weighted residual approach is used which contains new aspects for design of single step algorithms accuracy and stability are investigated for linear and non linear systems algorithms with higher accurary than the central difference method are developed and optimized to minimize the reduction in the stability limit

the number of subgroups of a finite group let a be a finite group and let p be a prime suppose that ps is the highest power of p dividing a a where a is the commutator subgroup of a and that the type 1 2 with 1 2 of a

saddle point principles and numerical integration methods for second order hyperbolic equations abstract this work describes a family of functionals whose stationarity often saddle point condition leads to well known so called variational formulations for structural dynamics such as the weak hamilton ritz formulation and the continuous discontinuous galerkin formulation and in turn to methods for the numerical integration of the equations of motion it is shown that all the time integration methods based on variational formulations do descend from such functionals moreover starting from the described family of functionals it is possible to construct new families of time integration methods which might exhibit computational advantages over the corresponding ones derived from variational formulations only

use of post processing to increase the order of accuracy of the trapezoidal rule at time integration of linear elastodynamics problems in this paper we suggest a very simple and effective post processing procedure to increase the order of accuracy in time for numerical results obtained by the trapezoidal rule we first derive a new exact closed form a priori error estimator for time integration of linear elastodynamics equations by the trapezoidal rule with non uniform time increments based on this error estimator we suggest a new post processing procedure containing additional time integration of elastodynamics equations by the trapezoidal rule with few time increments that systematically improves the order of accuracy of numerical results with the increase in the number of additional time increments used for post processing for example the use of just one additional time increment for post processing after time integration with any number of uniform time increments renders the order of accuracy of numerical results equal to 10 3 numerical examples of the application of the new techniques to a system with a single degree of freedom and to a multi degree system confirm the corresponding increase in the order of convergence of numerical results after post processing because the same trapezoidal rule is used for basic computations and post processing the new technique retains all of the properties of the trapezoidal rule requires no writing of a new computer program for its implementation and can be easily used with any existing commercial and research codes for elastodynamics

spurious resonances in numerical time integration methods for linear dynamics this paper deals with the accuracy of time integration methods for linear dynamics when applied near the resonance condition an approach for the analysis is considered which allows spurious resonance conditions to be detected the analysis of newmark methods shows the existence of such conditions which can adversely affect the quality of numerical computations as an alternative a higher order algorithm which can be viewed as a generalization of the trapezoidal rule is investigated the analysis reveals that the spurious disturbance near the resonance condition is greatly reduced the reported numerical tests confirm the theoretical predictions and demonstrate that high quality simulations can be obtained by means of higher order algorithms

time discretized operators part 1 towards the theoretical design of a new generation of a generalized family of unconditionally stable implicit and explicit representations of arbitrary order for computational dynamics the new generation of a generalized family of time discretized operators encompassing implicit and explicit representations that are unconditionally stable and which theoretically inherit nth order time accurate features developed in part 1 are restricted here in part 2 of the exposition to second order time accurate operators as such unconditionally stable implicit representations are first described followed by unconditionally stable explicit representations the theoretical design leading to computational algorithms with excellent algorithmic attributes for applicability to practical situations are also addressed for both the implicit and explicit unconditionally stable representations of time discretized operators attention is first focused on linear problems and extensions to nonlinear situations are subsequently briefly addressed

minimum time step size for diffusion problem in fem analysis in this paper oscillation of results arising from the use of a time step size which is less than the minimum required is examined the phenomenon is summarised to give two requirements that the solution of boundary problems has to satisfy an approach to derive the minimum time step size for various kinds of elements is developed a theoretical verification is presented for the one dimensional two noded element case and a numerical examination of the criteria explored for the two dimensional eight noded element case 1997 john wiley sons ltd

variational based energy momentum schemes of higher order for elastic fiber reinforced continua abstract this paper contributes to the improvement of recently published higher order accurate energy momentum schemes for an anisotropic hyperelastic material formulation based on the concept of structural tensors these galerkin based energy momentum schemes are developed by means of time finite elements with the focus on improving numerical stability and robustness in the presence of stiffness and large rotations they show advantages over conventional time stepping schemes in combination with anisotropic materials formulated with polyconvex strain energy density functions in this paper higher order energy momentum schemes are developed by using a differential variational principle in order to remedy the drawbacks of these galerkin based schemes namely a variationally inconsistent time approximation of the strains in the higher order case no coordinate free superimposed stress tensor and an unseparated time approximation of the fibers and the matrix material these new structure preserving time integrators preserve all conservation laws of a free flying hyperelastic anisotropic continuum namely the total linear and the total angular momentum conservation as well as the total energy conservation numerical examples demonstrate dynamic behavior of anisotropic continua discrete conservation properties of flying continua and the effect of higher order accuracy subject to transient loads

extended framework of hamiltons principle for thermoelastic continua based upon the extended framework of hamiltons principle a variational formulation for fully coupled thermoelasticity is presented the resulting formulation can properly account for all the governing differential equations as well as initial boundary conditions thus it provides the basis for a class of unified spacetime finite element methods by employing bar elements in one dimensional space along with linear shape functions temporally the simplest spacetime finite element method is presented herein with representative examples for its validity

population coding with correlation and an unfaithful model this study investigates a population decoding paradigm in which the maximum likelihood inference is based on an unfaithful decoding model umli this is usually the case for neural population decoding because the encoding process of the brain is not exactly known or because a simplified decoding model is preferred for saving computational cost we consider an unfaithful decoding model that neglects the pair wise correlation between neuronal activities and prove that umli is asymptotically efficient when the neuronal correlation is uniform or of limited range the performance of umli is compared with that of the maximum likelihood inference based on the faithful model and that of the center of mass decoding method it turns out that umli has advantages of decreasing the computational complexity remarkably and maintaining high level decoding accuracy moreover it can be implemented by a biologically feasible recurrent network pouget zhang deneve latham 1998 the effect of correlation on the decoding accuracy is also discussed

conservative fourth order time integration of non linear dynamic systems abstract an energy conserving time integration algorithm with fourth order accuracy is developed for dynamic systems with nonlinear stiffness the discrete formulation is derived by integrating the differential state space equations of motion over the integration time increment and then evaluating the resulting time integrals of the inertia and stiffness terms via integration by parts this process introduces the time derivatives of the state space variables and these are then substituted from the original state space differential equations the resulting discrete form of the state space equations is a direct fourth order accurate representation of the original differential equations this fourth order form is energy conserving for systems with force potential in the form of a quartic polynomial in the displacement components energy conservation for a force potential of general form is obtained by addition of a higher order secant type correction term the formulation leads to a consistent representation of the motion within a time increment corresponding to cubic hermite interpolation in time this in turn leads to excellent phase representation with only a small fourth order error permitting integration of oscillatory systems with only a few integration points per period three numerical examples demonstrate the high accuracy of the algorithm

hom a g abstract in this paper we prove that for any finite group g and any finite abelian group a the number of homomorphism from a to g is divisible by the greatest common divisor of a and g this result is a generalization of an old theorem of frobenius on the number of solutions of the equation x n 1 on a finite group

evaluation of high order single step integrators for structural response calculation implicit single step integration methods which are absolutely stable and have fourth and sixth order accuracy are applied to linear structural dynamics equations test cases for which the discretized problems are solvable exactly are employed to determine the number of floating point operations needed to achieve specified accuracy levels examples involving longitudinal vibration of a bar and flexural vibration of a simply supported beam subjected to harmonic and impact loading are analyzed the central concern in this paper is efficiently to integrate a discretized system which may have been derived by using finite element or finite difference methods it is found that high order methods may permit sufficiently increased integration step size to reduce by a factor as high as six the total computation effort needed to achieve similar accuracy from traditional lower order methods

hom a g iv abstract in this paper we prove that for any finite groups a and g the number of homomorphisms from a to g is divisible by the greatest common divisor of a a a a and g where a is the commutator subgroup of a and a a is the frattini subgroup of a a

hybrid laplace transform finite difference method for transient heat conduction problems the new method involving the combined use of the laplace transform and the finite difference method is applicable to the problem of time dependent heat flow systems the present method removes the time derivatives from the governing differential equation using the laplace transform and then solves the associated equation with the finite difference method the transformed temperature is inverted numerically by the method of honig and hirdes to obtain the result in the physical quantities the present results are compared in tables with exact solutions and those obtained from the combined use of the laplace transform and the finite element method it is found that the present method is reliable and efficient

an explicit conditionally stable finite difference equation for heat conduction problems this paper reports a new explicit and conditionally stable finite difference equation for heat conduction it predicts results with an accuracy comparable with or better than that obtainable by other methods stability of operation can be extended to any desired degree by subdividing the basic time step and increasing the number of nodes some existing difference equations are special cases of the new equation reported in this paper the new solution has been tested by calculating the response of a slab to transient and progressive waves whose analytical solutions are known

comparison of some single step methods for the numerical solution of the structural dynamic equation this paper compares the performance of the ss22 and ss23 references 1 and 2 single step algorithms for the numerical solution of the second order structural dynamic equation and a related new algorithm ss32b applied to the equivalent first order system with sine and step forcing functions various aspects of stability relevant to these equations are discussed

a new energy and momentum conserving algorithm for the non linear dynamics of shells a numerical time integration scheme for the dynamics of non linear elastic shells is presented that simultaneously and independent of the time step size inherits exactly the conservation laws of total linear total angular momentum as well as total energy the proposed technique generalizes to non linear shells recent work of the authors on non linear elastodynamics and is ideally suited for long term large scale simulations the algorithm is second order accurate and can be immediately extended with no modification to a fourth order accurate scheme the property of exact energy conservation induces a strong notion of non linear numerical stability which manifests itself in actual simulations

a robust self starting explicit computational methodology for structural dynamic applications architecture and representations conventional approaches for computational structural dynamics relevant to time integration methods involve first employing the classical galerkin formulations for the spatial discretization to yield a set of ordinary differential equations in time and then employing finite difference approximations for deriving the appropriate step by step algorithms we introduce new representations and architecture towards providing not only direct self starting features with the elimination of acceleration computations but also for enhancing the computational architecture itself via several other inherent distinguishing characteristics

non linear dynamics of three dimensional rods exact energy and momentum conserving algorithms the long term dynamic response of non linear geometrically exact rods under going finite extension shear and bending accompanied by large overall motions is addressed in detail the central objective is the design of unconditionally stable time stepping algorithms which exactly preserve fundamental constants of the motion such as the total linear momentum the total angular momentum and for the hamiltonian case the total energy this objective is accomplished in two steps first a class of algorithms is introduced which conserves linear and angular momentum this result holds independently of the definition of the algorithmic stress resultants second an algorithmic counterpart of the elastic constitutive equations is developed such that the law of conservation of total energy is exactly preserved conventional schemes exhibiting no numerical dissipation symplectic algorithms in particular are shown to lead to unstable solutions when the high frequencies are not resolved compared to conventional schemes there is little if any additional computational cost involved in the proposed class of energy momentum methods the excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several numerical simulations

weighting parameters for unconditionally stable higher order accurate time step integration algorithms part 2 second order equations in this paper unconditionally stable higher order accurate time step integration algorithms suitable for linear second order differential equations based on the weighted residual method are presented the second order equations are manipulated directly as in part 1 of this paper instead of specifying the weighting functions the weighting parameters are used to control the algorithm characteristics the algorithms are at least nth order accurate if the numerical solution for displacement is approximated by a polynomial of degree n 1 with n undetermined coefficients by choosing the weighting parameters carefully the order of accuracy can be improved the generalized pade approximations for the second order equations are considered the ultimate spectral radius is an algorithmic parameter by relating the approximate solutions to the equivalent formulations presented in part 1 of this paper the required weighting parameters are found explicitly any set of linearly independent functions can be used to construct the corresponding weighting functions from the weighting parameters the stabilizing weighting functions for the weighted residual method are found explicitly to ensure higher order accuracy in the general solution the accuracy of the particular solution due to excitation is also examined copyright 1999 john wiley sons ltd

crossed homomorphisms from rank 2 abelian to exceptional p groups abstract in this paper we shall verify the conjecture z 1 a g 0 mod gcd a a g of t asai t yoshida j algebra 160 1993 273 285 in the case where a is an abelian group of rank 2 and g is an exceptional p group for a prime p namely one of the following groups for p 2 a cyclic 2 group a dihedral 2 group a generalized quaternion 2 group and a semidihedral 2 group while for p 2 a cyclic p group

high order taylor galerkin and adaptive h p methods for second order hyperbolic systems application to elastodynamics a new family of high order taylor galerkin schemes is presented for the analysis of second order hyperbolic systems the use of such schemes within an adaptive h p finite element strategy for spatial approximation in two dimensional problems is described numerical results are presented to demonstrate performance of these methods

unconditionally stable higher order newmark methods by sub stepping procedure abstract in this paper a procedure to construct unconditionally stable higher order step by step time integration algorithms based on the newmark method is presented the second order accurate newmark method is extended to third fourth and even higher order by linearly combining the numerical results evaluated at some specific sub step locations the weighting factors and the sub step locations are algorithmic parameters they are chosen to eliminate the leading truncation error terms and to control the stability characteristics the ranges of parameters for unconditionally stable third and fourth order algorithms are given besides by using complex time steps a third order l stable and a fourth order a stable non dissipative algorithms are derived furthermore the accuracy of the excitation responses is shown to be improved by the present procedure as well to maintain high order accuracy the excitation may need to be modified the computational procedure is discussed it is found that there are no changes in the implementation of the underlying newmark method and no extra computations of high order gradients as the numerical results at the sub step locations are obtained independent of one another the evaluations can be computed in parallel comparisons with other higher order algorithms are presented

conservation properties of a time fe method part iv higher order energy and momentum conserving schemes in the present paper a systematic development of higher order accurate time stepping schemes which exactly conserve total energy as well as momentum maps of underlying finite dimensional hamiltonian systems with symmetry is shown the result of this development is the enhanced galerkin eg finite element method in time the conservation of the eg method is generally related to its collocation property total energy conservation in particular is obtained by a new projection technique the eg method is moreover based on objective time discretization of the used strain measure this paper is concerned with particle dynamics and semi discrete non linear elastodynamics the related numerical examples show good performance in presence of stiffness as well as for calculating large strain motions copyright 2005 john wiley sons ltd

a taylor galerkin method for convective transport problems a method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions to produce accurate temporal differencing the method employs forward time taylor series expansions including time derivatives of second and third order which are evaluated from the governing partial differential equation this yields a generalized time discretized equation which is successively discretized in space by means of the standard bubnov galerkin finite element method the technique is illustrated first in one space dimension with linear elements and euler leap frog and crank nicolson time stepping several interesting relations with standard galerkin and recently developed petrov galerkin methods emerge and the new taylor galerkin schemes are found to exhibit particularly high phase accuracy with minimal numerical damping the method is successively extended to deal with variable coefficient problems and multi dimensional situations

a unified set of single step algorithms part 1 general formulation and applications a general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented this algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities in many cases it presents a computationally advantageous form over conventional procedures this is particularly so when compared with the newmark algorighm and its variants

population coding and decoding in a neural field a computational study this study uses a neural field model to investigate computational aspects of population coding and decoding when the stimulus is a single variable a general prototype model for the encoding process is proposed in which neural responses are correlated with strength specified by a gaussian function of their difference in preferred stimuli based on the model we study the effect of correlation on the fisher information compare the performances of three decoding methods that differ in the amount of encoding information being used and investigate the implementation of the three methods by using a recurrent network this study not only rediscovers main results in existing literatures in a unified way but also reveals important new features especially when the neural correlation is strong as the neural correlation of firing becomes larger the fisher information decreases drastically we confirm that as the width of correlation increases the fisher information saturates and no longer increases in proportion to the number of neurons however we prove that as the width increases further wider than 2 times the effective width of the turning function the fisher information increases again and it increases without limit in proportion to the number of neurons furthermore we clarify the asymptotic efficiency of the maximum likelihood inference mli type of decoding methods for correlated neural signals it shows that when the correlation covers a nonlocal range of population excepting the uniform correlation and when the noise is extremely small the mli type of method whose decoding error satisfies the cauchy type distribution is not asymptotically efficient this implies that the variance is no longer adequate to measure decoding accuracy

high order taylor galerkin methods for linear hyperbolic systems a new family of high order taylor galerkin schemes is presented for the analysis of first order linear hyperbolic systems the schemes are unconditionally stable which makes them very attractive to use in conjunction with adaptive hp finite element methods for spatial approximation

computational algorithms for transient analysis the burden of weight and consequences towards formalizing discrete numerically assigned dna algorithmic markers wp family abstract different from the way we have been looking in the past at developments encompassing modal type and a wide class of step by step time stepping approaches for transient and dynamic field problems and significantly different from the way these have been developed and described in standard text books over the years this paper outlines the theoretical ideas and basis towards providing a generalized methodology and formulations leading to integral and a wide class of integration operators for the resulting solution of these transient equation systems it is herein postulated that integral operators and a wide class of integration operators pertain to and emanate from the same family with the burden which is being carried by a virtual field or weighted function field specifically introduced for the time discretization is strictly enacted in a mathematically consistent manner so as to first permit obtaining the adjoint operator of the original semi discretized equation system subsequently the selection or burden carried by the virtual or weighted function fields originally introduced to facilitate the time discretization process determines the formal development and outcome of exact integral operators approximate integral operators and a wide class of integration operators thus the present developments not only serve as a prelude towards the formal developments for exact integral operators but also demonstrate that the resulting approximate integral operators and a wide class of integration operators and known methods are simply subsets of the generalizations of a standardized w p family and emanate from the principles presented herein the burden of weight and the resulting consequences towards subsequently enabling a formal basis for the selection of discrete numerically assigned dna algorimmic markers to particularly identitfy a wide class of algorithms is described for transient field problems and dynamic field problems respectively

weighting parameters for time step integration algorithms with predetermined coefficients in this paper the effect of using the predetermined coefficients in constructing time step integration algorithms is investigated both first and second order equations are considered the approximate solution is assumed to be in a form of polynomial in the time domain it can be related to the truncated taylor s series expansion of the exact solution therefore some of the coefficients can be predetermined from the known initial conditions if there are m predetermined coefficients and r unknown coefficients in the approximate solution the unknowns can be solved by the weighted residual method the weighting parameter method is used to investigate the resultant algorithm characteristics it is shown that the formulation is consistent with a minimum order of accuracy m r the maximum order of accuracy achievable is m 2r unconditionally stable algorithms exist if m r for first order equations and m 1 r for second order equations hence the dahlquist s theorem is generalized algorithms equivalent to the pade approximations and unconditionally stable algorithms equivalent to the generalized pade approximations are constructed the corresponding weighting parameters and weighting functions for the pade and generalized pade approximations are given explicitly copyright 2000 john wiley sons ltd

a unified set of single step algorithms part 4 backward error analysis applied to the solution of the dynamic vibration equation this paper applies the method of backward error analysis to the numerical solution of the dynamic vibration equation by the single step algorithms proposed in references 1 3 and obtains some general rules to help with the choice of parameters

how to render second order accurate time stepping algorithms fourth order accurate while retaining the stability and conservation properties abstract for a fairly general class of second order accurate algorithms a sub stepping procedure is described which achieves fourth order accuracy while retaining the stability conservation properties and implementation of the underlying second order method two features render the proposed approach especially attractive from a computational perspective over conventional fourth order methods first the improved fourth order accuracy is attained merely by two additional residual evaluations with no changes in the implementation of the underlying second order method in particular no additional storage or extra computations of high order gradients are required second for hamiltonian systems the stability properties and possible conservation properties of the underlying second order method such as energy momentum and symplectic character are preserved the new technique is applied to a number of standard algorithms for elastodynamics structural and rigid body dynamics in particular when applied to the recently proposed second order accurate energy momentum method this approach results in a fourth order accurate energy linear and angular momentum conserving algorithm for general hamiltonian systems

transfinite element methodology for non liner linear transient thermal modelling analysis progress and recent advances recent progress and advances in the development and applicability of a novel transfinite element computational methodology is presented for general non linear linear transient thermal problems the proposed methodology and concepts are new and unique and demonstrate the applicability to general transient non linear linear thermal analysis situations by combining classical galerkin schemes and transform approaches with contemporary finite element methods to preserve the modelling versatility and numerical features thereby a hybrid computational methodology is proposed characteristic features and pertinent details of the approach are described for non linear linear transient thermal problems wherein non linearities due temperature dependence of thermophysical properties and or general non linear bound ary conditions to include radiation effects due to phase change etc are considered in addition the use of high continuity formulations in conjuction with the proposed methodology to furnish accurate temperature distributions and temperature gradients making use of a relatively smaller number of degrees of freedom is also demonstrated numerical test cases are presented for a variety of problems to demonstrate the fundamental features and applicability of the proposed formulations the proposed hybrid transfinite element methodology and concepts offer significant potential for extension to several areas of mathematical physics and engineering and to interdisciplinary research areas

modified newmark family for non linear dynamic analysis a simple modification of the newmark scheme is suggested which keeps the total energy of mechanical systems constant and thereby enhances the unconditional stability in non linear dynamic analysis numerical damping which is formulated on an energy basis is also introduced and the dissipative character of the algorithm is guaranteed in the non linear regime representative examples in support of such a modified newmark family are presented copyright 2004 john wiley sons ltd

on zeta functions of modular representations of a discrete group abstract it is proved that the generating function defined by the numbers of isomorphism classes of absolutely irreducible representations of a finitely generated group g into gl d f q n for n 1 is a rational function it is also proved that the generating function defined by weighted sums over isomorphism classes of representations of g into gl d f q n for n 1 is meromorphic over both the complex numbers and the p adic complex numbers

on practical integration of semi discretized nonlinear equations of motion part 1 reasons for probable instability and improper convergence time integration is the most versatile method for analyzing the general case of nonlinear semi discretized equations of motion however the approximate responses of such analyses generally do not converge properly and might even display numerical instability this is a very significant shortcoming especially in practical time integration herein after illustrating that this phenomenon is viable even for very simple nonlinear dynamic models sources of the shortcoming are discussed in detail the conclusion is that in time integration of nonlinear dynamic mathematical models of physically stable structural systems responses may converge improperly for three major reasons these reasons are 1 inadequate number of iterations before terminating nonlinearity solutions 2 deficiencies in the formulation of some time integration methods and 3 the inherent behaviour of the models of some special dynamic systems in addition limitations on computational facilities and improper consideration of these limitations may impair the numerical stability and convergence of the computed responses the differences between static and dynamic analyses are also discussed from the viewpoint of the numerical errors induced by nonlinearity

simulation of anisotropic wave propagation based upon a spectral element method we introduce a numerical approach for modeling elastic wave propagation in 2 d and 3 d fully anisotropic media based upon a spectral element method the technique solves a weak formulation of the wave equation which is discretized using a high order polynomial representation on a finite element mesh for isotropic media the spectral element method is known for its high degree of accuracy its ability to handle complex model geometries and its low computational cost we show that the method can be extended to fully anisotropic media the mass matrix obtained is diagonal by construction which leads to a very efficient fully explicit solver we demonstrate the accuracy of the method by comparing it against a known analytical solution for a 2 d transversely isotropic test case and by comparing its predictions against those based upon a finite difference method for a 2 d heterogeneous anisotropic medium we show its generality and its flexibility by modeling wave propagation in a 3 d transversely isotropic medium with a symmetry axis tilted relative to the axes of the grid

nonlinear population codes theoretical and experimental studies of distributed neuronal representations of sensory and behavioral variables usually assume that the tuning of the mean firing rates is the main source of information however recent theoretical studies have investigated the effect of crosscorrelations in the trial to trial fluctuations of the neuronal responses on the accuracy of the representation assuming that only the first order statistics of the neuronal responses are tuned to the stimulus these studies have shown that in the presence of correlations similar to those observed experimentally in cortical ensembles of neurons the amount of information in the population is limited yielding nonzero error levels even in the limit of infinitely large populations of neurons in this letter we study correlated neuronal populations whose higher order statistics and in particular response variances are also modulated by the stimulus we ask two questions does the correlated noise limit the accuracy of the neuronal representation of the stimulus and how can a biological mechanism extract most of the information embedded in the higher order statistics of the neuronal responses specifically we address these questions in the context of a population of neurons coding an angular variable we show that the information embedded in the variances grows linearly with the population size despite the presence of strong correlated noise this information cannot be extracted by linear readout schemes including the linear population vector instead we propose a bilinear readout scheme that involves spatial decorrelation quadratic non linearity and population vector summation we show that this nonlinear population vector scheme yields accurate estimates of stimulus parameters with an efficiency that grows linearly with the population size this code can be implemented using biologically plausible neurons

time accurate solution of advection diffusion problems by finite elements abstract the taylor galerkin method recently proposed for the spatial and temporal discretization of hyperbolic equations is employed to derive accurate and efficient numerical schemes for the solution of time dependent advection diffusion problems two distinct numerical strategies are discussed the first is suitable for evolutionary problems while the second is appropriate for situations in which a steady state is eventually reached for purely evolutionary problems the taylor galerkin method is applied to the complete advection diffusion equation and through modifications of the standard mass matrix is shown to generate an incremental form of the crank nicolson time stepping method such an incremental form also lends itself to the derivation of computationally simple but accurate explicit algorithms to deal with transient situations which evolve towards a highly convective steady state the global taylor galerkin method is found to be ineffective since it reduces to the standard galerkin formulation as the temporal term vanishes for such cases we suggest the use of a splitting up method in which advection and diffusion are treated separately by appropriate taylor galerkin methods the proposed methods are illustrated first for linear constant coefficient equations they are successively extended to deal with nonlinear and multi dimensional problems numerical applications indicate their effectiveness

the spectral element method for elastic wave equations application to 2 d and 3 d seismic problems a spectral element method for the approximate solution of linear elastodynamic equations set in a weak form is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two and three dimensional geometries the computational domain is discretized into quadrangles or hexahedra defined with respect to a reference unit domain by an invertible local mapping inside each reference element the numerical integration is based on the tensor product of a gauss lobatto legendre 1 d quadrature and the solution is expanded onto a discrete polynomial basis using lagrange interpolants as a result the mass matrix is always diagonal which drastically reduces the computational cost and allows an efficient parallel implementation absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains the time discretization is based on an energy momentum conserving scheme that can be put into a classical explicit implicit predictor multicorrector format long term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions the accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two dimensional problems obtained by other methods the potentiality of the method is then illustrated by studying a simple three dimensional model very accurate modelling of rayleigh wave propagation and surface diffraction is obtained at a low computational cost the method is shown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three dimensional surface topographies copyright 1999 john wiley sons ltd

a general framework for conservative single step time integration schemes with higher order accuracy for a central force system a general framework for algorithms that conserve angular momentum for single body central force problems is presented it is shown that any family of momentum conserving algorithms can have at most three free parameters one of which may be used to ensure energy conservation and hence will be configuration dependent further restrictions can be made that enable the algorithms to recover the orbits of relative equilibria of the underlying physical problem in addition the algorithms can be made time reversible whilst still leaving two parameters unspecified the order of accuracy of a general momentum conserving family is analysed and it is shown that energy momentum algorithms that preserve the underlying physical relative equilibria can have unlimited accuracy if the two remaining parameters are appropriately chosen functions of the configuration and the time step this does not require any additional degrees of freedom extra stages of calculation or information from past solutions numerical examples are given that show the performance of some representative higher order schemes when applied to stiff and non stiff problems and the issue of newton raphson convergence is discussed

unconditionally stable higher order accurate collocation time step integration algorithms for first order equations abstract in this paper unconditionally stable higher order accurate time step integration algorithms for linear first order differential equations based on the collocation method are presented the amplification factor at the end of the spectrum is a controllable algorithmic parameter the collocation parameters for unconditionally stable higher order accurate algorithms are found to be given by the roots of a polynomial in terms of the ultimate amplification factor in general when the numerical solution is approximated by a polynomial of degree n this approximation is at least n th order accurate however by using the above collocation parameters the order of accuracy can be improved to 2 n 1 or 2 n the approximate solutions are found to be equivalent to the generalized pade approximations furthermore it is shown that the accuracy of the particular solution due to excitation given by the present method is compatible with the homogeneous solutions no modification of the collocation parameters is required

complex time step newmark methods with controllable numerical dissipation in this paper unconditionally stable higher order accurate time step integration algorithms with controllable numerical dissipation are presented the algorithms are based on the newmark method with complex time steps the ultimate spectral radius the sub step locations j and the weighting factors j are the algorithmic parameters for an algorithm that is 2n 1 th order accurate the sub step locations which may be complex are shown to be the roots of an nth degree polynomial the polynomial is given explicitly in terms of n and the weighting factors are then obtained by solving a system of n simultaneous equations it is further shown that the order of accuracy is increased by one for the non dissipative algorithms with 1 the stability properties of the present algorithms are studied it is shown that if the ultimate spectral radius is set between 1 and 1 the eigenvalues of the numerical amplification matrix are complex with magnitude less than or equal to unity the algorithms are therefore unconditionally c stable when the ultimate spectral radius is set to 0 or 1 the algorithms are found to be equivalent to the first sub diagonal and diagonal pade approximations respectively the present algorithms are more general as the numerical dissipation is controllable and are very suitable for parallel computers the accuracy of the excitation responses is found to be enhanced by the present complex time step procedure to maintain high order accuracy the excitation may need some modifications 1998 john wiley sons ltd

asymptotic upper bounds for the errors of richardson extrapolation with practical application in approximate computations the results produced by richardson extrapolation though in general very accurate are inexact numerical evaluation of this inexactness and implementation of the evaluation in practice are the objectives of this paper first considering linear changes of errors in the convergence plots asymptotic upper bounds are proposed for the errors then the achievement is extended to the results produced by richardson extrapolation and finally an error controlling procedure is proposed and successfully implemented in approximate computations originated in science and engineering copyright 2009 john wiley sons ltd

the method of fundamental solutions and quasi monte carlo method for diffusion equations the laplace transform is applied to remove the time dependent variable in the diffusion equation for non harmonic initial conditions this gives rise to a non homogeneous modified helmholtz equation which we solve by the method of fundamental solutions to do this a particular solution must be obtained which we find through a method suggested by atkinson to avoid costly gaussian quadratures we approximate the particular solution using quasi monte carlo integration which has the advantage of ignoring the singularity in the integrand the approximate transformed solution is then inverted numerically using stehfest s algorithm two numerical examples are given to illustrate the simplicity and effectiveness of our approach to solving diffusion equations in 2 d and 3 d 1998 john wiley sons ltd

a modified euler method for dynamic analyses a modified version of the well known euler method for numerical integration is presented and its application in analyses of the dynamic response of structures is discussed this modified method was first introduced in 1981 and it has been reported that it was discovered by accident when first introduced however valuable information related to the accuracy stability and merits of the method were not provided it is shown that the modified euler method has features that make it an attractive approach for applications in dynamic analyses of structures accuracy and stability analyses are made for the method by considering free vibrational responses of linear undamped single degree of freedom systems the method is explicit and extremely easy to use however it is conditionally stable applications of the method are also made which illustrate its usefulness and inherent simplicity furthermore comparisons of the modified euler method and the newmark beta method are reported which elucidate the relative merits of these methods

population coding with motion energy filters the impact of correlations the codes obtained from the responses of large populations of neurons are known as population codes several studies have shown that the amount of information conveyed by such codes and the format of this information is highly dependent on the pattern of correlations however very little is known about the impact of response correlations as found in actual cortical circuits on neural coding to address this problem we investigated the properties of population codes obtained from motion energy filters which provide one of the best models for motion selectivity in early visual areas it is therefore likely that the correlations that arise among energy filters also arise among motion selective neurons we adopted an ideal observer approach to analyze filter responses to three sets of images noisy sine gratings random dots kinematograms and images of natural scenes we report that in our model the structure of the population code varies with the type of image we also show that for all sets of images correlations convey a large fraction of the information 40 to 90 of the total information moreover ignoring those correlations when decoding leads to considerable information loss from 50 to 93 depending on the image type finally we show that it is important to consider a large population of motion energy filters in order to see the impact of correlations study of pairs of neurons as is often done experimentally can underestimate the effect of correlations

thermodynamically consistent time stepping algorithms for non linear thermomechanical systems we present the basic theory for developing novel monolithic and staggered time stepping algorithms for general non linear coupled thermomechanical problems the proposed methods are thermodynamically consistent in the sense that their solutions rigorously comply with the two laws of thermodynamics for isolated systems they preserve the total energy and the entropy never decreases furthermore if the governing equations of the problem have symmetries the proposed integrators preserve them too the formulation of such methods is based on two ideas expressing the evolution equation in the so called general equations for non equilibrium reversible irreversible coupling format and enforcing from their inception certain directionality and degeneracy conditions on the discrete vector fields the new methods can be considered as an extension of the energy momentum integration algorithms to coupled thermomechanical problems to which they reduce in the purely hamiltonian case in the article the new ideas are applied to a simple coupled problem a double thermoelastic pendulum with symmetry numerical simulations verify the qualitative features of the proposed methods and illustrate their excellent numerical stability which stems precisely from their ability to preserve the structure of the evolution equations they discretize copyright 2009 john wiley sons ltd

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells

a comprehensive unified set of single step algorithms with controllable dissipation for dynamics part i formulation a new class of time integration algorithms for dynamics is presented the algorithms are based on finite element in time the equations of motion are expressed in first order form and the variables are interpolated via time polynomials the unknown nodal variables are solved by the generalized galerkin method algorithms of any desired order of time accuracy can be achieved by proper choice of the finite element interpolation in time the salient feature of the algorithms is that it can be either asymptotic annihilating or non dissipative besides any desired algorithmic damping between these two extreme cases could be obtained the algorithms are analysed in detail the spectral characteristics equivalent to the upper diagonal diagonal of the pade approximations and other rational approximations are given