compact expression for lowdin s alpha function a simple analytical expression for lowdin s alpha function is derived this expression is expected to be more convenient with the expansion of a general function about a displaced center than any other available in the literature

positive steady states of the holling tanner prey predator model with diffusion this paper is concerned with the holling tanner prey predator model with diffusion subject to the homogeneous neumann boundary condition we obtain the existence and non existence of positive non constant steady states

neural network based reliability analysis a comparative study abstract a study on the applicability of different kinds of neural networks for the probabilistic analysis of structures when the sources of randomness can be modeled as random variables is summarized the networks are employed as numerical devices for substituting the finite element code needed by monte carlo simulation the comparison comprehends two network types multi layer perceptrons and radial basis functions classifiers cost functions sum of square errors and cross entropy optimization algorithms back propagation gauss newton newton raphson sampling methods for generating the training population using uniform and actual distributions of the variables and purposes of neural network use as functional approximators and data classifiers the comparative study is performed over four examples corresponding to different types of the limit state function and structural behaviors the analysis indicates some recommended ways of employing neural networks in this field

tracking pattern evolution through extended center manifold reduction and singular perturbations in this paper we develop an extended center manifold reduction method a methodology to analyze the formation and bifurcations of small amplitude patterns in certain classes of multi component singularly perturbed systems of partial differential equations we specifically consider systems with a spatially homogeneous state whose stability spectrum partitions into eigenvalue groups with distinct asymptotic properties one group of successive eigenvalues in the bifurcating group are widely interspaced while the eigenvalues in the other are stable and cluster asymptotically close to the origin along the stable semi axis the classical center manifold reduction provides a rigorous framework to analyze destabilizations of the trivial state as long as there is a spectral gap of sufficient width when the bifurcating eigenvalue becomes commensurate to the stable eigenvalues clustering close to the origin the center manifold reduction breaks down moreover it cannot capture subsequent bifurcations of the bifurcating pattern through our methodology we formally derive expressions for low dimensional manifolds exponentially attracting the full flow for parameter combinations that go beyond those allowed for the classical center manifold reduction i e to cases in which the spectral gap condition no longer can be satisfied our method also provides an explicit description of the flow on these manifolds and thus provides an analytical tool to study subsequent bifurcations our analysis centers around primary bifurcations of transcritical type that can be either of co dimension 1 or 2 in two and three component pde systems we employ our method to study bifurcation scenarios of small amplitude patterns and the possible appearance of low dimensional spatio temporal chaos we also exemplify our analysis by a number of characteristic reaction diffusion systems with disparate diffusivities

mesa type patterns in the one dimensional brusselator and their stability the brusselator is a generic reaction diffusion model for a tri molecular chemical reaction we consider the case when the input and output reactions are slow in this limit we show the existence of k periodic spatially bi stable structures mesas and study their stability using singular perturbation techniques we find a threshold for the stability of k mesas this threshold occurs in the regime where the exponentially small tails of the localized structures start to interact by comparing our results with turing analysis we show that in the generic case a turing instability is followed by a slow coarsening process whereby logarithmically many mesas are annihilated before the system reaches a steady equilibrium state we also study a breather type instability of a mesa which occurs due to a hopf bifurcation full numerical simulations are shown to confirm the analytical results in 1952 turing proposed that the formation of spatial patterns during morphogenesis could be explained in terms of the instability of a homogeneous steady state solution of a reaction diffusion network describing the evolution of a set of morphogens 31 turing himself illustrated his ideas on two chemical models turing s original work is primarily concerned with the stability analysis of the homogeneous steady state solution of the rate equations for the interacting morphogens 22 the main point of biological interest however is whether stable spatial structures may be generated beyond the instability i e whether the rate equations admit stable and positive inhomogeneous solutions exhibiting the most characteristic features of morphogenetic patterns this point has been taken up seriously in the early seventies more systematic numerical studies of turing s model have been performed showing irregular spatial structures 4 this led to serious reservations about the relevance of turing s theory in developmental biology particularly its ability to generate regular patterns but these criticisms originate from a somewhat unfortunate choice of turing s example and they do not touch the essential points of turing s theory more specifically the obvious requirement that the rate equations must admit positive and bounded solutions is not satisfied in turing s example 11

turing instability for a semi discrete gierer meinhardt system abstract in this paper we investigate the spatial patterns of a gierer meinhardt system where the space is discrete in two dimensions with the periodic boundary condition and time is continuous in contrast to the continuum models the conditions of turing instability are obtained by linear analysis and a series of numerical simulations are performed in the instability region we have shown that this system can produce a number of different patterns such as stripes and snowflake pattern other than ubiquitous fix spotted patterns as mentioned the results are substantiated only by means of snapshots of the apatial grid however we also give some analysis by using the time series at three random grids and of the average value of states that is the stable state patterns can be observed on the other hand the effects of varying parameters on pattern formation are also discussed moreover simulations for fixed parameters and special initial conditions indicate that the initial conditions can alter the structure of patterns the patterns can form as a consequence of cellular interaction so patterns arising from a semi discrete model can present simulations on a geometrically accurate representation in biology as a result our work is interesting and important in ecology

slowly varying control parameters delayed bifurcations and the stability of spikes in reaction diffusion systems we present three examples of delayed bifurcations for spike solutions of reaction diffusion systems the delay effect results as the system passes slowly from a stable to an unstable reg ime and was previously analyzed in the context of ode s in p mandel and t erneux j stat phys 48 5 6 pp 1059 1070 1987 it was found that the instability would not be fully realized until the system had entered well into the unstab le regime the bifurcation is said to have been delayed relative to the threshold value computed directly from a linear stability analysis in contrast to the study of mandel and erneux we analyze the delay effect in systems of partial differential equations pde s in particular for spike solutions of singularly perturbed generalized giere r meinhardt and gray scott models we analyze three examples of delay resulting from slow passage into regimes of oscillatory and competition instability in the first example for the gierer meinhardt model on the infinite real line we analyze the delay resulting from slowly tuning a control parameter through a hopf bifurcation in the second example we consider a hopf bifurcation of the gierer meinhardt model on a finite one dimensional domain in this s cenario as opposed to the extrinsic tuning of a system parameter through a bifurcation value we analyze the delay of a bifurcation triggered by slow intrinsic dynamics of the pde system in the third example we consider competition instabilities triggered by the extrinsic tuning of a feed rate parameter in all three cases we find that the syste m must pass well into the unstable regime before the onset of instability is fully observed indicating delay we also fin d that delay has an important effect on the eventual dynamics of the system in the unstable regime we give analytic predictions for the magnitude of the delays as obtained through the analysis of certain explicitly solvable nonloca l eigenvalue problems nlep s the theory is confirmed by numerical solutions of the full pde systems

stochastic differential equations an approach to the generation of continuous non gaussian processes the generation of continuous random processes with jointly specified probability density and covariation functions is considered the proposed approach is based on the interpretation of the simulated process as a stationary output of a nonlinear dynamic system excited by white gaussian noise and described by a system of a first order stochastic differential equations sde the authors explore how the statistical characteristics of the equation s solution depends on the form of its operator and on the intensity of the input noise some aspects of the approximate synthesis of stochastic differential equations and examples of their application to the generation of non gaussian continuous processes are considered the approach should be useful in signal processing when it is necessary to translate the available a priori information on the real random process into the language of its markov model as well as in simulation of continuous correlated processes with the known probability density function

a comprehensive overview of a non parametric probabilistic approach of model uncertainties for predictive models in structural dynamics abstract in structural dynamics a predictive model is constructed by developing a mathematical mechanical model of a designed system in order to predict the response of the real system which is the manufactured system realized from the designed system the mathematical mechanical modelling process of the designed system introduces two fundamental types of uncertainties the data uncertainties and the model uncertainties uncertainties have to be taken into account for improving the predictability of the model model uncertainties cannot be modelled by using the usual parametric probabilistic approach recently a general non parametric probabilistic approach of model uncertainties for dynamical systems has been proposed using the random matrix theory this paper gives a comprehensive overview of this approach in developing its foundations in simple terms and in illustrating all the concepts and the tools introduced in the general theory by using a simple example this paper deals with 1 notions of designed systems real systems mean models as predictive models errors and uncertainties 2 the definition of a simple example in linear elastodynamics 3 a comprehensive overview of the non parametric probabilistic approach of model uncertainties for predictive models in structural dynamics 4 a summary of the random matrix ensembles which are necessary for the non parametric modelling of random uncertainties 5 the estimation of the dispersion parameters of the non parametric probabilistic model using experimental data 6 the method to solve the stochastic equation of the dynamical system with non parametric probabilistic model of random uncertainties 7 a numerical simulation and the validation for the simple example

analysis of discrete vibratory systems with parameter uncertainties part ii impulse response abstract this paper is a continuation of its companion part i in which a new analytical method was proposed to estimate the first and second moments of eigenvalues for a linear time invariant proportionally damped discrete vibratory system with uncertain parameters in this part the force amplitude is also considered as a random variable but its time history is assumed to be deterministic based on certain simplifying assumptions and given probabilistic eigenvalue solutions the first two moments of response in both modal and physical domains are estimated first an impulse excitation is considered and single and two degree of freedom system examples are used to illustrate the proposed method unlike the commonly used first order perturbation technique our method does not include any secular terms it is verified by comparing predictions with the benchmark monte carlo simulation second a convolution integral formulation is developed for harmonic and other excitations one example case is considered to illustrate and validate the proposed approach overall our method overcomes the deficiencies of first order perturbation technique and is reasonably accurate and computationally inexpensive

dynamics of pulse solutions in gierer meinhardt model with time dependent diffusivity abstract dispersive processes with a time dependent diffusivity appear in a plethora of physical systems most often a solution is attained for a predefined form of diffusion coefficient d t here existence of pulse solutions with an arbitrary time dependence thereof is proved for the gierer meinhardt model with three types of transport regular diffusion sub diffusion and levy flights admission of a solution of the classical pulse shape but for an unencumbered form of d t is a valuable property that allows to study phenomena of the ilk observed in various ostensibly unrelated applications closed form solutions are obtained for some pulse constellations transitions between periods of nearly constant diffusivities trigger respective cross over between counterpart solutions known for a constant diffusivity thereupon exhibiting otherwise unattainable behaviour qualitatively reconstructing observable evolution peculiarities of tagged molecular structures such as essential slowing down or speeding up during various stages of motion inexplicable with a single constant diffusion coefficient

addition theorem of slater type orbitals a numerical evaluation of barnett coulson lowdin functions when using the one centre two range expansion method to evaluate multicentre integrals over slater type orbitals stos it may become necessary to compute numerical values of the corresponding fourier coefficients also known as barnett coulson lowdin functions bclfs bouferguene and jones 1998 j chem phys 109 5718 to carry out this task it is crucial to not only have a stable numerical procedure but also a fast algorithm in previous work bouferguene and rinaldi 1994 int j quantum chem 50 21 bclfs were represented by a double integral which led to a numerically stable algorithm but this turned out to be disappointingly time consuming the present work aims at exploring another path in which bclfs are represented either by an infinite series involving modified bessel functions or by an integral whose integrand is a smooth function both of these representations have the advantage of being symmetrical with respect to the cusp parameter a and the radial variable r as a consequence it is no longer necessary to split the integrals over r 0 into several components with a different analytical form in each of these a numerical study is also carried out to help select the most appropriate method to be used in practice

strategy and stationary pattern in a three species predator prey model abstract in this paper we study a strongly coupled system of partial differential equations which models the dynamics of a two predator one prey ecosystem in which the prey exercises a defense switching mechanism and the predators collaboratively take advantage of the prey s strategy we demonstrate the emergence of stationary patterns for this system and show that it is due to the cross diffusion that arises naturally in the model as far as the authors are aware this is the first example of stationary patterns in a predator prey system arising solely from the effect of cross diffusion

a simple derivation of the addition theorems of the irregular solid harmonics the helmholtz harmonics and the modified helmholtz harmonics in this article a simple derivation of the addition theorems of the irregular solid harmonics the helmholtz harmonics and the modified helmholtz harmonics is presented our derivation is based upon properties of the differential operator yml which is obtained from the regular solid harmonic yml r by replacing the cartesian components of r by the cartesian components of with the help of this differential operator yml which is an irreducible spherical tensor of rank l the addition theorems of the anisotropic functions are obtained by differentiating the addition theorems of the isotropic functions the performance of the necessary differentiations is greatly facilitated by a systematic exploitation of the tensorial nature of the differential operator yml

homoclinic bifurcations at the onset of pulse self replication we establish a series of properties of symmetric n pulse homoclinic solutions of the reduced gray scott system u uv 2 v v uv 2 which play a pivotal role in questions concerning the existence and self replication of pulse solutions of the full gray scott model specifically we establish the existence and study properties of solution branches in the plane that represent multi pulse homoclinic orbits where and are the central values of u x and v x respectively we prove bounds for these solution branches study their behavior as and establish a series of geometric properties of these branches which are valid throughout the plane we also establish qualitative properties of multi pulse solutions and study how they bifurcate i e how they change along the solution branches

qualitative analysis of a prey predator model with stage structure for the predator in this paper we propose a diffusive prey predator model with stage structure for the predator we first analyze the stability of the nonnegative steady states for the reduced ode system and then study the same question for the corresponding reaction diffusion system with homogeneous neumann boundary conditions we find that a hopf bifurcation occurs in the ode system but no turing pattern happens in the reaction diffusion system however when a natural cross diffusion term is included in the model we can prove the emergence of stationary patterns i e nonconstant positive stationary solutions for this system moreover these stationary patterns do not exist in the considered parameter regime when there is no cross diffusion

pattern formation for a two dimensional reaction diffusion model with chemotaxis abstract this paper is devoted to study the formation of stationary patterns for a chemotaxis model with nonlinear diffusion and volume filling effect over a bounded rectangular domain by using linear stability analysis around the homogeneous steady states we establish conditions for the existence of unstable mode bands that lead to the formation of spatial patterns we derive the stuart landau equations for the pattern amplitudes by means of weakly nonlinear multiple scales analysis and fredholm theory in particular we find asymptotic expressions for a wide range of patterns sustained by the system these patterns include mixed mode square hexagonal and roll stationary configurations our analytical results are corroborated by direct simulations of the underlying chemotaxis system

stationary patterns for a prey predator model with prey dependent and ratio dependent functional responses and diffusion abstract we study a reaction diffusion system arising from a three species prey predator model with prey dependent and ratio dependent functional responses the main aim of this paper is to study the global existence and bifurcation of non constant positive steady states in particular we will show that even though the unique positive constant steady state is stable for the ode dynamics non constant positive steady states exist for the reaction diffusion system this demonstrates that stationary patterns arise as a result of diffusion

importance sampling for randomly excited dynamical systems an importance sampling technique for linear and non linear dynamical systems subjected to random excitations is presented applying a transformation of probability measures controls are introduced in the system of ito stochastic differential equations such that the sample trajectories can be influenced in a predetermined way as is shown there exist controls resulting in unbiased zero variance estimators however these optimal controls are in general not accessible and have to be replaced by sub optimal ones derived from an optimization procedure analogous to the first order reliability method known from time invariant problems the efficiency of the proposed monte carlo simulation technique is demonstrated by estimating first passage probabilities of typical oscillators under external white noise excitation

positive clustered layered solutions for the gierer meinhardt system abstract we consider the stationary gierer meinhardt system in a ball of r n e 2 u u u p v q 0 in v v u m v s 0 in u v 0 and u v 0 on where b r is a ball of r n n 2 with radius r e 0 is a small parameter and p q m s satisfy the following condition p 1 q 0 m 1 s 0 q m p 1 1 s 1 assume 0 p 1 q a if n 2 and 0 p 1 q 1 if n 3 where a 1 whose numerical value is a 1 06119 we prove that there exists a unique r a 0 such that for r r a r corresponds to r n case and for any fixed integer k 1 this system has at least one sometimes two radially symmetric positive solution u e k v e k which concentrate at k spheres j 1 k x r e j where r e 1 r e 2 r e k are such that r 0 r e 1 e log 1 e r e j 1 r e j e log 1 e j 2 k where r 0 r is a root of some function m r r this generalizes the results in w m ni j wei on positive solutions concentrating on spheres for the gierer meinhardt system j differential equations 221 2006 158 189 where a special case k 1 and n 2 n 1 p 1 q 1 was considered

stationary patterns caused by cross diffusion for a three species prey predator model we study a system arising from a three component prey predator model with prey dependent and ratio dependent functional responses where cross diffusion is included in such a way that predator chases the prey and the prey runs away from the predator we prove that the cross diffusion can generate the stationary patterns nonconstant positive steady states

generation of a random sequence having a jointly specified marginal distribution and autocovariance we consider the problem of generating a random sequence with a specified marginal distribution and autocovariance the proposed scheme for generating such a sequence consists of a white gaussian noise source input to a linear digital filter followed by a zero memory nonlinearity zmnl the zmnl is chosen so that the desired distribution is exactly realized and the digital filter is designed so that the desired autocovariance is closely approximated both analytic results and examples are included the proposed scheme should prove useful in simulations involving non gaussian processes

random field finite elements the probabilistic finite element method pfem is formulated for linear and non linear continua with inhomogeneous random fields analogous to the discretization of the displacement field in finite element methods the random field is also discretized the formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization furthermore it is shown that a reduced set of the uncorrelated variables is sufficient for the second moment analysis based on the linear formulation of the pfem the method is then extended to transient analysis in non linear continua the accuracy and efficiency of the method is demonstrated by application to a one dimensional elastic plastic wave propagation problem and a two dimensional plane stress beam bending problem the moments calculated compare favourably with those obtained by monte carlo simulation also the procedure is amenable to implementation in deterministic fem based computer programs

stochastic isogeometric analysis of free vibration of functionally graded plates considering material randomness abstract the paper is to develop stochastic isogeometric analysis for the free vibration of functionally graded plates with spatially varying random material properties the isogeometric analysis method is employed in the new stochastic analysis scheme which is called stochastic isogeometric analysis siga the elastic modulus and mass density are modeled as homogeneous gaussian random fields along the plane of structure the governing equation of stochastic isogeometric analysis for free vibration of functionally graded plates is derived in conjunction with perturbation expansions to predict the first and second moments of eigenvalues in order to verify proposed method the brute force monte carlo simulation is employed the mean variance and cov coefficient of variation predicted by siga and those predicted by monte carlo simulation show good agreement the numerical examples demonstrate that the randomness of material properties affects significantly the structural responses of the functionally graded plates the correlation between elastic modulus and mass density is also observed to have significant effect on the response cov of eigenvalue

a study of linear combination of load effects abstract the performance of a combination formula extensively used in stochastic dynamics for estimating extreme values of linear combination of load effects is investigated it is shown that this combination formula is nearly optimal in the case of independent gaussian load effect components for non gaussian load effects the performance is investigated by studying specific examples and it is shown that the accuracy deteriorates significantly

stochastic finite elements for a beam on a random foundation with uncertain damping under a moving force abstract an infinitely long beam on an elastic foundation is subjected to a constant force which is moving with a constant speed along it the beam rests on a random foundation the stiffness of which is a random function of the length co ordinate the coefficient of viscous damping is also a random variable the steady state vibration is analyzed it appears that a stochastic stationary process occurs viewed by an observer moving together with the moving force a stochastic finite element analysis by means of the first order perturbation and first order second moment method provides an evaluation of the variance of the deflection and of the bending moment of the beam the effects of several parameters and of various types of correlation functions are investigated it is shown that the randomness of the foundation is of greater importance than the uncertainty in the damping the dynamic and stochastic effects increase with increasing speed of the movement of the force in the subcritical domain but decrease in the supercritical domain the coefficient of variation of the deflection of the beam is larger than that of the bending moment at the point of application of the moving force

parallel solution methods for stochastic finite element analysis using monte carlo simulation in the present paper innovative solution strategies for parallel computer implementation have been developed in connection with the monte carlo simulation mcs and the weighted integral method to produce efficient numerical handling of stochastic finite element analysis for 2d plane stress strain problems furthermore mcs in conjunction with the local average method is also used to extend the stochastic finite element analysis to 3d solid structures although mcs approaches have the major advantage that accurate solutions can be obtained for any type of problem whose deterministic solution is known either numerically or analytically their applicability is hindered by the high computational effort that is required the implementation of innovative parallel solution techniques in this study resulted in cost effective treatment of these highly computationally demanding problems one and two level domain decomposition methods have been implemented numerical results revealed that the proposed approaches permit an efficient treatment of stochastic finite element analysis for real scale 2d plane stress strain and 3d solid structures

probabilistic analysis of non self adjoint mechanical systems with uncertain parameters a new approach to analyze mechanical systems that have an uncertain distribution of material properties and loadings for the variability of their dynamic response parameters is developed the methodology is demonstrated through solving for the probabilistic moments of eigenvalues of a non self adjoint structural system material properties are modelled using random fields and uncertain loadings are modelled using random variables by treating the random fluctuations of vibratory response to be the stochastic perturbations to mean response second order moments of eigenvalues are obtained to circumvent the practical difficulty of obtaining the exact correlation models for parameter variability second order moments of eigenvalues are evaluated in terms of variance functions both the sensitivity of eigenvalues to the variability in system parameters and the effects on the eigenvalue variability of correlation properties of uncertain parameters are demonstrated

the role of eigenparameter gradients in the detection of perturbations in discrete linear systems abstract continuous mechanical systems can be characterized through analysis of discrete linear models such models can provide approximations for the eigenvalues and eigenvectors collectively eigenparameters although the eigenparameters do not qualify as measures for the state of the system in response to specific loading and boundary conditions they do reflect the identity of the system and this in itself has important applications in particular the eigenparameters can be used to study the sensitivity of a system to perturbations due perhaps to damage incurred by one or more discrete elements these studies can rationalize the choice and weighting of eigenparameters for system identification strategies damage detection algorithms and damage assessment methods to this end this paper develops a set of sensitivity coefficients based on gradients of the eigenparameters sensitivities are normalized with respect to that of the harmonic oscillator and generalized to include the mode vectors through the definition of a figure of merit analytical and numerical examples based on appropriate elements are used to illustrate the utility of the approach

a response surface approach for the static analysis of stochastic structures with geometrical nonlinearities a response surface approach for the finite element analysis of uncertain structures undergoing large displacements is presented this method is based on the use of ad hoc response surface functions built up by ratios of polynomials as opposite to commonly used linear or quadratic polynomials such functions are insensitive to the sampling point positions once the response surface form is defined response statistics can be approximated by analytical relationships or statistical simulation taking full advantage of sensitivity analysis numerical investigations demonstrated that a remarkable accuracy is achieved in the evaluation of both statistical moments and probability density functions of the response

response of uncertain dynamic systems i this paper and its sequel deal with the characterization of the response of a structure modeled by a single degree of freedom system in which the mass stiffness and damping parameters are only imprecisely known using an informationtheoretic framework closed form expressions are developed for the probability densities of the parameters that control the characteristics of the dynamic response namely the natural frequency of vibration the damped natural frequency and the percentage of critical damping the manner in which these closed form expressions depend on the extent and nature of our knowledge about the uncertain parameters is highlighted some comparisons with perturbation solutions are also presented

the dynamics and pinning of a spike for a reaction diffusion system the motion of a one spike solution to a simplified form of the gierer meinhardt activator inhibitor model is studied in both a one dimensional and a two dimensional domain the pinning effect on the spike motion associated with the presence of spatially varying coefficients in the differential operator referred to as precursor gradients is studied in detail in the one dimensional case we derive a differential equation for the trajectory of the spike in the limit e 0 where e is the activator diffusivity a similar differential equation is derived for the two dimensional problem in the limit for which e 1 and d 1 where d is the inhibitor diffusivity a numerical finite element method is presented to track the motion of the spike for the full problem in both one and two dimensions finally the numerical results for the spike motion are compared with corresponding asymptotic results for various examples

expansion of a function about a displaced center an extremely simple closed expression is obtained for the coefficients which appear in the expansion of a function of a special type about a displaced center a conjecture about the vanishing of a certain coefficient which appear in the expansion of a slater type orbital about a displaced center is also proved

free vibration analysis of stochastic strings the free vibration of strings with randomly varying mass and stiffness is considered the joint probability density functions of the eigenvalues and eigenfunctions are characterized in terms of the solution of a pair of stochastic non linear initial value problems analytical solutions of these equations based on the method of stochastic averaging are obtained the effects of the mean and autocorrelation of the mass process are included in the analysis numerical results for the marginal probability density functions of eigenvalues and eigenfunctions are obtained and are found to compare well with monte carlo simulation results the random eigenvalues when normalized with respect to their corresponding deterministic values are observed to tend to become first order stochastically stationary with respect to the mode count

construction of a probabilistic model for impedance matrices impedance matrices allow for the coupling of domains with very different properties and possibly modeled with different methods this paper presents the construction of a probabilistic model for such matrices using a nonparametric method that allows for the consideration of data errors as well as model errors to enable the application of this method in the case of a domain for which no finite element model is available the identification of a hidden state variables model from the knowledge of the impedance matrix at a discrete set of frequencies is also described finally the construction of the probabilistic model of the impedance of a pile foundation on a layered unbounded soil illustrates the capabilities of the method

crime modeling with levy flights the ucla burglary hotspot model introduced in m b short m r d orsogna v b pasour g e tita p j brantingham a l bertozzi and l b chayes math models methods appl sci 18 2008 pp 1249 1267 models the formation of hotspots of criminal activity in this paper we extend the ucla model to incorporate a more realistic model of human locomotion the movement of the criminal agents follows a biased levy flight with step sizes distributed according to a power law distribution the biased brownian motion of the original model is then derived as a special case starting with an agent based model we derive its continuum limit this consists of two equations and involves the fractional laplacian operator a numerical method based on the fast fourier transform is used to simulate the continuum model these simulations compare favorably with the direct numerical simulations of the agent based model a turing type analysis is performed to estimate how the instability of the homogeneous ste

simple expressions for radial functions appearing in the expansions of ym1l1 f m2l2 r and 2nym1l1 f m2l2 r in this paper simple analytic expressions for the radial functions r ll1l2 f r and glnl1l2 f r which appear in the expressions for ym1l1 fl2 r ym2l2 r and 2nym1l1 fl2 r ym2l2 r when expressed as linear combinations of yml r where ym1l1 is obtained from the solid harmonic ym1l1 r r l1ym1l1 r are derived by replacing x y and z in its representation as a polynomial of degree l1 by x y and z respectively writing r ll1l2 f r r ll1l20 r f r and glnl1l2 f r glnl1l20 r f r the expression for r ll1l20 r glnl1l20 r is found to be 2 r l1 2 r l1 2n times a product of commuting factors containing the single operator 1 2 r d dr and not two noncommuting operators r and 1 r d dr in the many equivalent previously obtained results which are thus synthesized in this approach also expressions for these operators manifest the symmetries in the problem in addition a simple connection glnl1l20 r r ll1 2nl20 r between these two operators and the c

reliability analysis of randomly vibrating structures with parameter uncertainties the problem of reliability analysis of randomly driven stochastically parametered vibrating structures is considered the excitation components are assumed to be jointly stationary gaussian random processes the randomness in structural parameters is modelled through a vector of mutually correlated non gaussian random variables the structure is defined to be safe if a set of response quantities remain within prescribed thresholds over a given duration of time an approximation for the failure probability conditioned on structure randomness is first formulated in terms of recently developed multivariate extreme value distributions an estimate of the unconditional failure probability is subsequently obtained using a method based on taylor series expansion this has involved the evaluation of gradients of the multivariate extreme value distribution conditioned on structure random variables with respect to the basic variables analytical and numerical techniques for computing these gradients have been discussed issues related to the accuracy of the proposed method have been examined the proposed method has applications in reliability analysis of vibrating structural series systems where different failure modes are mutually dependent illustrative examples presented include the seismic fragility analysis of a piping network in a nuclear power plant

frequency response of linear systems with parameter uncertainties abstract the primary objective of the present study is to develop a new analytical method for estimating the frequency response characteristics of a linear time variant discrete vibratory system when the mass stiffness and damping parameters are uncertain the excitation amplitude is also considered to be random but the frequency is deterministic given several simplifying assumptions a direct product technique is proposed to estimate the mean and standard deviation of the steady state displacement response at the excitation frequency application of this theory is demonstrated by several single and multi degree of freedom examples the distinction between off and near resonance regime characteristics is clearly pointed out in order to verify the proposed analytical technique predictions are compared with the results obtained by the monte carlo simulation and or perturbation methods it is seen that unlike the perturbation method the proposed technique works well in the vicinity of a resonance our methodology can be implemented easily and is reasonably accurate and computationally inexpensive when compared with the existing methods

dynamics and stability of spike type solutions to a one dimensional gierer meinhardt model with sub diffusion abstract the dynamics and stability of spike type patterns to a sub diffusive gierer meinhardt reaction diffusion system is studied in a one dimensional spatial domain a differential algebraic system dae is derived to characterise the dynamics of an n spike quasi equilibrium pattern in the presence of sub diffusion with sub diffusive effects it is shown that quasi equilibrium spike patterns exist for diffusivity ratios asymptotically smaller than for the case of regular diffusion and that the spikes approach their equilibrium locations at an algebraic rather than exponential rate in time a new non local eigenvalue problem nlep is derived to examine the stability of an n spike pattern for a two spike pattern sub diffusion has little effect on the competition instability threshold whereas the threshold associated with an oscillatory instability of the spike profile increases significantly furthermore for a multi spike pattern it is shown that an asynchronous oscillatory instability of the spike profile rather than a synchronous oscillatory instability characteristic of the case of regular diffusion is the dominant instability when the anomaly index is below a certain threshold detailed numerical results are presented for the two spike case

simple calculation of lowdin s alpha function iv procedure for evaluating the coefficients in expansion of a slater type atomic orbital by using the explicit expression for the coefficients bkk lm l eq 16 in j math phys 26 3193 1985 which is referred to hereafter as part ii a closed expression for the coefficients cij nlm l appearing in lowdin s alpha function 1 r l nlm a r for a slater type atomic orbital is obtained by manipulating its expression mathematically it was manifested that cij nlm l vanishes if i n l m l for the specific cases that each of i and j takes 0 or 1 cij nlm l take the same value from the symmetry relation on bkk lm l eq 6 in part ii it was proved that cij nlm l cji nlm l provided m l through some manipulation the recursion formula for cij nlm l in n was derived while that about m was obtained by introducing the recursion formula for bkk lm l in m eq 24 in part ii into the expression for cij nlm l for the special cases where n and m both are equal to l the expression for cij nlm l is given in a compact and symmetric form the use of this expression leads to a recursion

new representations for the spherical tensor gradient and the spherical delta function in this article we analyze representations for the product ym1l1 fm2l2 r with yml specifying a solid harmonic whose argument is the nabla operator r instead of the vector r since both ym1l1 and fm2l2 r are irreducible spherical tensors we can use angular momentum algebra for evaluating the product accordingly the problem of finding a representation for the product is reduced to the determination of the radial functions generated by the product analytical expressions for these radial functions are derived by direct differentiation and with the help of fourier transforms closely related to the spherical tensor gradient yml is the spherical delta function ml r we derive new representations for ml by considering convolution integrals involving b functions these functions are closely related to the modified bessel functions and also to the yukawa potential e r r we show that the definition of the b functions can be extended to include a large class of derivatives of the delta func

parametric instability of stochastic columns columns which have stochastically distributed young s modulus and mass density and are subjected to deterministic periodic axial loadings are considered the general case of a column supported on a winkler elastic foundation of random stiffness and also on discrete elastic supports which are also random is considered material property fluctuations are modeled as independent one dimensional univariate homogeneous real random fields in space in addition to autocorrelation functions or their equivalent power spectral density functions the input random fields are characterized by scale of fluctuations or variance functions for their second order properties the foundation stiffness coefficient and the stiffnesses of discrete elastic supports are treated to constitute independent random variables the system equations of boundary frequencies are obtained using bolotin s method for deterministic systems stochastic fem is used to obtain the discrete system with random as well as periodic coefficients statistical properties of boundary frequencies are derived in terms of input parameter statistics a complete covariance structure is obtained the equations developed are illustrated using a numerical example employing a practical correlation structure

basic mechanisms driving complex spike dynamics in a chemotaxis model with logistic growth recently wang and hillen chaos 17 3 2007 037108 037108 introduced a modified keller segel model with logistic growth that exhibits complex spatiotemporal dynamics of spikes these dynamics are driven by merging of spikes on one hand and spike insertion on the other in this paper we analyse the basic mechanisms that initiate and sustain these events we identify two distinguished regimes in the first regime a single interior spike drifts towards a boundary this instability is responsible for spike merging the same regime further exhibits spike insertion we identify a fold point bifurcation which is precursor to spike insertion event in the second regime we show that it is possible to stabilize a single interior spike and we compute analytically a critical threshold which is responsible for spike stabilization in particular our calculation characterizes a stable spike in the keller segel model with logistic growth this is in contrast to the classical keller segel model where the interior spike is known to be unstable

analysis of discrete vibratory systems with parameter uncertainties part i eigensolution abstract a new analytical method is proposed for the estimation of eigensolutions for undamped and proportionally damped discrete vibratory systems when the system parameters are uncertain or random variables given several simplifying assumptions a direct product technique is used to estimate the first and second moments of eigensolutions in terms of the moments of system parameter matrices the proposed methodology is reasonably accurate and computationally fast when compared with existing methods such as the first perturbation method and the monte carlo simulation application of the new method is demonstrated through several single and multi degree of freedom systems it is seen that the standard deviation predictions match well with the results yielded by the monte carlo simulation unlike the first order perturbation method especially when moderately large random fluctuations are considered the impulse response issue is addressed in the companion paper part ii

on a new class of gradient formulas in the angular momentum theory recently b f bayman has derived the formula for ylm r ylm r where ylm r is a solid spherical harmonic in terms of the derivatives of the function r by the scalar parameter r this result is clarified and essentially generalized for the case of the tensor product ynl r yl r where ynlm nylm in this paper applications to the taylor series in the three dimensional euclidean space as well as some other expansions are discussed briefly

generating differential operators for the basis functions in the variational lcao type methods the analytical expressions for operators which allow us to generate the basis functions from the simplest ground states by means of differentiating by parameters are obtained some important properties of these operators are indicated this approach allows us to simplify the structure of molecular matrix elements whose calculation presents considerable difficulties in the variational lcao type methods

random matrix theory and non parametric model of random uncertainties in vibration analysis recently a new approach called a non parametric model of random uncertainties has been introduced for modelling random uncertainties in linear and non linear elastodynamics in the low frequency range this non parametric approach differs from the parametric methods for random uncertainties modelling and has been developed in introducing a new ensemble of random matrices constituted of symmetric positive definite real random matrices this ensemble differs from the gaussian orthogonal ensemble goe and from the other known ensembles of the random matrix theory the present paper has three main objectives the first one is to study the statistics of the random eigenvalues of random matrices belonging to this new ensemble and to compare with the goe the second one is to compare this new ensemble of random matrices with the goe in the context of the non parametric approach of random uncertainties in structural dynamics for the low frequency range the last objective is to give a new validation for the non parametric model of random uncertainties in structural dynamics in comparing in the low frequency range the dynamical response of a simple system having random uncertainties modelled by the parametric and the non parametric methods these three objectives will allow us to conclude about the validity of the different theories

statistical tolerancing using a modification of taguchi s method the expanding use of experimental design techniques for statistical tolerancing is primarily due to their simplicity they can be understood easily and implemented by engineers and scientists having only a limited knowledge of statistics and experimental design the method is generally attributed to taguchi 1978 in this article we describe taguchi s method and why it works both intuitively and mathematically our results show that taguchi s method although giving good results for many applications is not optimal we propose alternative tolerancing procedures that are uniformly better than taguchi s method with little sacrifice in simplicity we illustrate the use of these methods first with some simple examples and then with a partial differential equation model

global existence and finite time blow up of solutions of a gierer meinhardt system abstract we are concerned with the gierer meinhardt system with zero neumann boundary condition u t d 1 u a 1 u u p v q 1 x x t 0 v t d 2 v a 2 v u r v s 2 x x t 0 u x 0 u 0 x v x 0 v 0 x x where p 1 s 1 q r d 1 d 2 a 1 a 2 are positive constants 1 2 u 0 v 0 are nonnegative smooth functions r d d 1 is a bounded smooth domain we obtain new sufficient conditions for global existence and finite time blow up of solutions especially in the critical exponent cases p 1 r and q r p 1 s 1

simple calculation of lowdin s alpha function ii easier procedure for evaluating bk k l m l and vanishing of hn 2n i l m l for special values of i and n this paper subsequent to the one j math phys 25 1133 1984 referred to as part i presents the following new results it is found out that for m l and l 1 the coefficients bk k l m l in lowdin s function have properties other than manifested in part i the expression for bk k l m l is shown to be equivalent to the one into which sharma s expression obtained in a different manner from that in part i is simplified by rashid the use of rashid s expression leads to the recurrence formula for bk k l m l with respect to m only this formula and the expression for the bk k l m l with m l provide an easier procedure for successively evaluating bk k l m l than in part i furthermore it is proved that the coefficients hn 2n i l m l in the asymptotic form of the function vanish for i l m and for n l

mutually exclusive spiky pattern and segmentation modeled by the five component meinhardt gierer system we consider the five component meinhardt gierer model for mutually exclusive patterns and segmentation which was proposed in h meinhardt and a gierer j theoret biol 85 1980 pp 429 450 we prove rigorous results on the existence and stability of mutually exclusive spikes which are located in different positions for the two activators sufficient conditions for existence and stability are derived which depend in particular on the relative size of the various diffusion constants our main analytical methods are the liapunov schmidt reduction and nonlocal eigenvalue problems the analytical results are confirmed by numerical simulations

stationary patterns created by cross diffusion for the competitor competitor mutualist model abstract in this paper we consider a competitor competitor mutualist model with cross diffusion we prove some existence and non existence results concerning non constant positive steady states patterns in particular we demonstrate that the cross diffusion can create patterns when the corresponding model without cross diffusion fails

free vibration of laminated spherical panels with random material properties randomness in the material properties is inherent in all engineering materials composites exhibit a greater scatter compared to conventional materials because of the larger number of parameters associated with its fabrication and manufacturing for accurate prediction of its behavior the composite material properties have been modelled as random variables in the present study higher order shear deformation theory including rotatory inertia effects has been employed in developing the system equations and first order perturbation technique has been adopted for the solution an approach has been presented for obtaining the analytical solution for generalized eigenvalue problem associated with free vibrations mean and variance of the natural frequency have been obtained for cross ply spherical laminates projected in rectangular plan form with different boundary conditions

pattern formation of a diffusive predator prey model with strong allee effect and nonconstant death rate abstract this paper is concerned with the pattern formation of a diffusive predator prey model with strong allee effect in the prey population and nonconstant death rate for the predator the necessary and sufficient conditions of turing instability are explicitly obtained by regarding the diffusion coefficient d 2 of the predator as bifurcation parameter we demonstrate that at the critical value of the bifurcation parameter d 2 a turing bifurcation occurs i e a pattern arises the conditions for the stability of the pattern are also derived in detail our results show that strong allee effect in our model plays a crucial role in the formation of stationary pattern which is a strong contrast to the case without strong allee effect

structural dynamics challenge problem summary the six papers in this special issue that develop solutions to the structural dynamics challenge problem are summarized herein the goal is to emphasize different tools and approaches applied to various parts of the structural dynamics problem specifically the following issues are considered 1 development of a mathematical framework for uncertainty quantification of a substructure 2 calibration of a mathematical model for the substructure 3 validation of the substructure mathematical model 4 use of the mathematical model of the substructure in conjunction with another structure to make a prediction of the probability that a regulatory limit is surpassed and discussion of the uncertainty of the prediction different methodologies are presented and specific results vary however conclusions regarding satisfaction of the regulatory requirement match

stability of spiky solutions in a competition model with cross diffusion we consider the shigesada kawasaki teramoto model of species segregation in the limit of high cross diffusion rate of one species and small diffusion rate of another recently steady states in the shape of an inverted spike were constructed in this limit in one dimension y lou w m ni and s yotsutani discrete contin dyn syst 10 2004 pp 435 458 y wu and q xu discrete contin dyn syst 29 2011 pp 367 385 in this paper we consider the stability properties of such spiky states we show that k symmetric spikes are stable if the domain length is sufficiently large more precisely we derive a sequence of thresholds 0 l 1 l k when k 2 the instability of a small eigenvalue is triggered first resulting in a very slow drift of the two spikes with eventual absorption of one by the other when k geq3 the primary instability is due to a large eigenvalue resulting in a quick death

simple calculation of lowdin s alpha function iii procedure for calculating hn 2n i lm l 2n i successively when r a 0 lowdin s function 1 r l flm a r is expressed as 1 r l flm a r 2 lm l al 1 n l imaxi l m hn 2n i lm l 2n i a2n ig 2n i a r a 2n l where a is a separation between a new center placed on the origin and the old center located at a given point on the z axis that is the origin of the coordinate system defining a function f r yml to be expanded around the new center f r is an arbitrary radial function and yml a complex spherical harmonics and r is a distance from the new center here imax min 2n 2 l l and the function g j a is expressed as g j a d dr j f ir rl 1 r a the closed form of the coefficients hn 2n i lm l is given by eq 27 in j math phys 26 3193 1985 referred to as part ii since they are expressed in terms of the different coefficients bkk lm l independent of the form of f r hn 2n i lm l constitute a set of universal constants as do bkk lm l the explicit expression for the factor lm l in front of the summa

vibration of an oscillator with random damping analytical expression for the probability density function this study deals with the impact of probabilistic viscous and hysteretic damping upon the dynamic response of a linear single degree of freedom oscillator assuming that damping is governed by an unspecified probability density function pdf an analytical expression of the output pdf of the oscillator s transfer function is provided in terms of the input damping pdf the instance of a uniform pdf is thoroughly treated for both viscous and hysteretic damping cases the methodology also yields to analytical expressions of all moments of the output law of probability a definition of a new concept of envelopes is then introduced and their analytical expression is derived exact analytical results are compared with monte carlo simulations for an academic case study an identification procedure of a predefined input damping pdf is proposed allowing one to obtain probabilistic parameters from experimental frequency response data

maximum entropy approach for modeling random uncertainties in transient elastodynamics a new approach is presented for analyzing random uncertainties in dynamical systems this approach consists of modeling random uncertainties by a nonparametric model allowing transient responses of mechanical systems submitted to impulsive loads to be predicted in the context of linear structural dynamics the information used does not require the description of the local parameters of the mechanical model the probability model is deduced from the use of the entropy optimization principle whose available information is constituted of the algebraic properties related to the generalized mass damping and stiffness matrices which have to be positive definite symmetric matrices and the knowledge of these matrices for the mean reduced matrix model an explicit construction and representation of the probability model have been obtained and are very well suited to algebraic calculus and to monte carlo numerical simulation in order to compute the transient responses of structures submitted to impulsive loads the fundamental properties related to the convergence of the stochastic solution with respect to the dimension of the random reduced matrix model are analyzed finally an example is presented

transitions and heteroclinic cycles in the general gierer meinhardt equation and cardiovascular calcification model abstract the article is concerned with transitions and pattern selection analysis of the inhibitor activator system proposed in connection with recent studies of cardiovascular calcification patterns explicit criteria are derived to enable us to distinguish between stable and metastable patterns by deriving a reduced system of equations the existence of certain complicated structures is discussed in particular heteroclinic cycles are identified and their properties are studied it is also discussed that the change of boundary conditions can affect the transitions of the system in this connection we will also study asymptotic behavior of patterns after transitions and will compare the results with numerical simulations

the stability of spike solutions to the one dimensional gierer meinhardt model abstract the stability properties of an n spike equilibrium solution to a simplified form of the gierer meinhardt activator inhibitor model in a one dimensional domain is studied asymptotically in the limit of small activator diffusivity e the equilibrium solution consists of a sequence of spikes of equal height the two classes of eigenvalues that must be considered are the o 1 eigenvalues and the o e2 eigenvalues which are referred to as the large and small eigenvalues respectively the spike pattern is stable when the parameters in the gierer meinhardt model are such that both sets of eigenvalues lie in the left half plane for a certain range of these parameters and for n 2 and e 0 it is shown the o 1 eigenvalues are in the left half plane only when d d n where d n is another critical value of d which satisfies d n n thus when n 2 and e 1 the spike pattern is stable only when d n an explicit formula for d n is given for the special case n 1 it is shown that a one spike equilibrium solution is stable when d d1 e an asymptotic formula for d1 e is given finally the dynamics of a one spike solution is studied by deriving a differential equation for the trajectory of the center of the spike

dynamic stiffness method for circular stochastic timoshenko beams response variability and reliability analyses the problem of characterizing response variability and assessing reliability of vibrating skeletal structures made up of randomly inhomogeneous curved straight timoshenko beams is considered the excitation is taken to be random in nature a frequency domain stochastic finite element method is developed in terms of dynamic stiffness coefficients of the constituent stochastic beam elements the displacement fields are discretized by using frequency and damping dependent shape functions questions related to discretizing the inherently non gaussian random fields that characterize beam elastic mass and damping properties are considered analytical methods combined analytical and simulation based methods direct monte carlo simulations and simulation procedures that employ importance sampling strategies are brought to bear on analyzing dynamic response variability and assessment of reliability satisfactory performance of approximate solution procedures outlined in the study is demonstrated using limited monte carlo simulations

modal probabilistic analysis in this paper we first recall a method for estimating the numerical probability density function pdf with parzen rosenblatt estimators of experimental data like the sound absorption coefficient with the kundt s tube then we focus on the model of mikki which is used as a predictive tool we are interested then in inversing the model and in identifying the 3 parameters q of mikki s model but we want to go further and to identify from the estimated pdf directly the pdf of the 3 parameters we explain first when it is possible then we give some examples to illustrate the method this work allows for evaluating the robustness of a model from experimental data

weakly convergent expansions of a plane wave and their use in fourier integrals the fourier transform of an irreducible spherical tensor is normally computed with the help of the rayleigh expansion of a plane wave in terms of spherical bessel functions and spherical harmonics the angular integrations are then trivial however the remaining radial integral containing a spherical bessel function may be so complicated that the applicability of fourier transformation is severely restricted as an alternative the use of weakly convergent expansions of a plane wave in terms of complete orthonormal sets of functions is suggested the weakly convergent expansions of a plane wave are constructed in such a way that their application in fourier integrals leads to expansions of the fourier or inverse fourier transform that converge with respect to the norm of either the hilbert space l2 r3 or the sobolev space w 1 2 r3 accordingly these weakly convergent expansions may be viewed as distributions that are defined on either l2 r3 or w 1 2 r3 the properties of some complete orthonormal se

calibration of the damping dispersion parameter in a non parametric probabilistic approach abstract response predictions in structural dynamics are in general very sensitive to random uncertainties associated with the underlying predictive mathematical model one way to capture parametric and model uncertainties consists in the construction of a non parametric probabilistic model in which the reduced structural matrices are replaced by random matrices the uncertainties are introduced at a global level and controlled through one dispersion parameter each for the mass damping and stiffness matrix in the present paper the role of the damping dispersion parameter is investigated for the case in which the adopted non parametric model is calibrated with respect to an existing parametric model it is shown that in the non parametric model the influence of the damping dispersion on the scatter in the frequency response is relatively small if a calibration criterion based on matrix norms is used a novel method for calibrating this non parametric model is proposed which enforces the same scatter of the frequency response function frf at the first resonance frequency in the parametric and the non parametric model a case study involving a satellite fe model shows that with this approach the frf scatter of the non parametric model reaches a level similar to that of the parametric model used for its calibration

comment on compact expression for lowdin s alpha function j math phys 26 940 1985 recently antone j math phys 26 940 1985 has presented a compact expression for lowdin s function and expected that in calculation on the electronic properties of molecules and solids it is the most convenient of all the expressions available in literature but in the present paper it is shown that the expression for the function obtained from the expansion formula of silverstone and moats phys rev a 16 1731 1977 is remarkably more compact and therefore more convenient than antone s expression though not visibly it is proved with the aid of some manipulation that the formula of silverstone and moats can be reduced to the well known expansion formula of a solid harmonic in the special case where the center of the function to be expanded and that of the function both are located on the z axis some mistakes are found that lead to antone s expression for the function

random eigenvalues algebraic methods and structural dynamic models a review of the literature of the study of structural vibration problems with random parameters is provided there have been several approaches to this problem these have been encapsulated in this paper for the benefit of those who need to assess such possibilities in particular the algebraic theory of random variables is delineated here with the application in mind being the determination or estimation of the statistics of the eigenvalues of linear dynamical systems various transformation techniques are summarized and discussed with simple examples

simple calculation of lowdin s alpha function a function of the type f r yml around a specific center b is expanded about another center a as f r yml l m 1 r l flm a r y ml where yml and yml are the complex spherical harmonics around the respective centers which have been assumed to be quantized in the direction of a to b the distances from the respective centers are r and r and the separation between the two centers is a the expansion coefficient l flm a r which is usually called lowdin s function can be expressed in a remarkably simple form by using silverstone and moats formula owing to the simplicity the properties of the coefficients bkk lm l appearing in the function are manifested and the recurrence formulas for evaluating bkk lm l are derived furthermore when r 0 the asymptotic expression for the function is given in a compact form

positive steady state solutions of the noyes field model for belousov zhabotinskii reaction abstract in the paper we investigate the noyes field model for belousov zhabotinskii reaction and study positive steady state solutions of this model with the homogeneous neumann boundary condition we obtain the existence and non existence of non constant positive steady state solutions

addition theorems for b functions and other exponentially declining functions in this paper addition theorems are derived for a special class of exponentially declining functions the so called b functions bmn l r e filter and e o steinborn phys rev a 18 1 1978 although these b functions have a relatively complicated analytical structure they nevertheless have some mathematical properties that are particularly advantageous in connection with multicenter problems also all the commonly occurring exponentially declining functions like for instance bound state hydrogen eigenfunctions and slater type functions can be expressed by simple finite sums of b functions consequently addition theorems for these functions can also be written down immediately the various addition theorems for b functions are derived by applying suitable generating differential operators to the well known addition theorem of the special b function bm l l which is that solution of the modified helmholtz equation that is irregular at the origin and regular at infinity e j weniger and e o

a new gauss quadrature for multicentre integrals over stos in the gaussian integral transform approach when computing multicentre integrals over slater type orbitals stos by means of the shavitt and karplus gaussian integral transforms shavitt and karplus 1962 j chem phys 36 550 one usually ends up with a multiple integral of the form shavitt and karplus 1965 j chem phys 43 398 in which all the integrals are inter related the most widely used approach for computing such an integral is to apply a product of gauss legendre quadratures for the integrals over 0 1 while the semi infinite term is evaluated by a special procedure although numerous approaches have been developed to accurately perform the integration over 0 efficiently it is the aim of this work to add a new tool that could be of some benefit in carrying out the hard task of multicentre integrals over stos the new approach relies on a special gauss quadrature referred to as gauss bessel to accurately evaluate the semi infinite integral of interest in this work emphasis is put on accuracy rather than efficiency since its aim is essentially to bring a proof of concept showing that gauss bessel quadrature can successfully be applied in the context of multicentre integrals over stos the obtained accuracy is comparable to that obtained with other methods available in the literature

dynamics of structural systems with spatial randomness the objective of this paper is the analysis of the dynamics of structural systems with randomly varying parameters the formulation and analysis are based on the theory of random integral equations the general analysis including error estimation is applied to the problems of harmonic vibration of i a beam with randomly varying density and ii a uniform beam resting on a randomly inhomogeneous winkler foundation for these applications the analytical formulae for the mean and correlation function of the response are illustrated numerically and graphically

the existence and stability of spike equilibria in the one dimensional gray scott model the pulse splitting regime abstract the existence stability and pulse splitting behavior of spike patterns in the one dimensional gray scott model on a finite domain is analyzed in the semi strong spike interaction regime this regime is characterized by a localization of one of the components of the reaction near certain spike locations while the other component exhibits a more global spatial variation across the domain the method of matched asymptotic expansions is then used to construct k spike equilibria in terms of a certain core problem this core problem is studied numerically and asymptotically for each integer k 1 it is shown that there are two branches of k spike equilibria that meet at a saddle node bifurcation value for small values of the diffusivity d of the second component these saddle node bifurcation points occur at approximately the same value a combination of asymptotic and numerical methods is used to analyze the stability of these branches of k spike equilibria with respect to both drift instabilities associated with the small eigenvalues and oscillatory instabilities of the spike profile in this way the key bifurcation and spectral conditions of ei et al s ei y nishiura k ueda 2 n splitting or edge splitting a manner of splitting in dissipative systems jpn j ind appl math 18 2001 181 205 believed to be essential for pulse splitting behavior in a reaction diffusion system are verified by having verified these conditions a simple analytical criterion for the occurrence of pulse splitting is then formulated and confirmed with full numerical simulations of the gray scott model this criterion verifies a conjecture based on numerics and topological arguments reported in a doelman r a gardner t j kaper stability analysis of singular patterns in the 1d gray scott model a matched asymptotics approach physica d 122 1 4 1998 1 36 the analytical results are compared with previously obtained results for pulse splitting behavior

a computational procedure to estimate the stochastic dynamic response of large non linear fe models abstract an algorithm for the computation of the stochastic non stationary non linear response of large fe models is presented in stochastic analysis the response is described by the mean and covariance function and possibly by the probability distribution of the non linear response to estimate the covariance matrix the method of equivalent statistical linearization is applied for linearizing all non linear elements the large fully populated and symmetric covariance matrix of dimension 2 n is described by the so called karhunen loeve expansion representation which allows one to employ a feasible description in this context an efficient procedure is suggested to determine the minimal number of karhunen loeve vectors necessary to assure a sufficiently accurate representation this method in fact allows one to employ any available deterministic integration scheme to compute the karhunen loeve vectors to increase the computational efficiency modal coordinates are used to represent the linear sub system special attention is given to the effects of mode truncation which are generally not negligible for large fe systems moreover the suggested approach has no limitation w r t the size of the model in terms of degrees of freedom dofs or the number of non linear elements the feasibility of the proposed procedure is demonstrated in the numerical examples where the methodology is applied to an office building with approximately 25 50 and 100 thousand dofs containing hysteretic damping elements

global bifurcations at the onset of pulse self replication in this work we carried out an extensive numerical exploration of the delicate global dynamics of pulse self replication and analyzed the stability of singular homoclinic stationary solutions and their bifurcations in the one dimensional gray scott model this stability analysis has several implications for understanding the recently discovered phenomena of self replicating pulses the solutions of the ordinary differential equation are organized around a codimension 2 global bifurcation from which two or n branches of homoclinic orbits originate corresponding to solitary pulse solutions in the partial differential equation a careful analysis of the bifurcation scenarios in the global bifurcation diagram suggests that the dynamics of the self replicating system are related to a hierarchy structure of folding bifurcation branches in parameter space the numerical simulation suggests that the bogdanov takens points together with the presence of critical points emanating from a particular codimension 2 homoclinic orbit play a central role for the global bifurcation of periodic orbits the homoclinic solutions and the complex chaotic dynamics the numerical simulation also reveals the existence of a modulating two pulse or multipulse which accompanies the procedure of pulse self replication in reaction diffusion systems

analysis of mechanical systems using interval computations applied to finite element methods this paper addresses the problem of mechanical systems in which parameters are uncertain and bounded interval calculation is used to find an envelope of transfer functions for mechanical systems modelled with finite elements within this context a new formulation has been developed for finite element problems involving bounded parameters to avoid the problems of overestimation an iterative algorithm is introduced which leads to a conservative solution for linear mechanical problems a method to ensure the convergence of this algorithm is also proposed this new algorithm has been tested on simple mechanical systems and leads to a conservative envelope of the transfer functions