on the stochastic pendulum with ornstein uhlenbeck noise we study a frictionless pendulum subject to multiplicative random noise because of destructive interference between the angular displacement of the system and the noise term the energy fluctuations are reduced when the noise has a non zero correlation time we derive the long time behaviour of the pendulum in the case of ornstein uhlenbeck noise by a recursive adiabatic elimination procedure an analytical expression for the asymptotic probability distribution function of the energy is obtained and the results agree with numerical simulations lastly we compare our method with other approximation schemes

external non white noise theory and experiment abstract we consider a system under the influence of an external gaussian noise with a flat spectrum and finite cut off frequency the stationery properties of the system are studied as a function of the correlation time of the noise theoretical predictions based on an expansion in powers of the correlation time are corroborated by experimental measurements on an electric circuit with a digital noise generator the two main qualitative properties experimentally observed which can not be explained in a white noise limit theory are i appearance of relative extrema of the stationery distribution for high noise intensity when the correlation time is not too small ii dependence on the noise characteristics of the position of the maximum of the stationary distribution for low noise intensity

preference voting and project ranking using dea and cross evaluation cook and kress 1990 using data envelopment analysis dea as their starting point proposed a procedure to rank order the candidates in a preferential election notionally each candidate is permitted to choose the most favourable weights to be applied to his her standings first place second place etc votes in the usual dea manner with the additional assurance region restriction that the weight for a j place vote should be more than that for a j 1 amount we consider that this freedom to choose weights is essentially illusory when maximum discrimination between the candidates is sought in which case the weights used to evaluate and rank the candidates are as if imposed externally at the outset to avoid this we present an alternative procedure which retains cook and kress central idea but where as well as using each candidate s rating of him herself we now make use of each candidate s ratings of all the candidates we regard the so called cross evaluation matrix as the summary of a self and peer rating process in which the candidates seek to interpret the voters preferences as favourably for themselves relative to the other candidates as possible the problem then becomes one of establishing an overall rating for each candidate from these individual ratings for this for each candidate we use a weighted average of all the candidates ratings of that candidate where the weights themselves are in proportion to each candidate s overall rating the overall ratings are therefore proportional to the components of the principal left hand eigenvector of the cross evaluation matrix these ideas are then applied to the selection of r d projects to comprise an r d program thus indicating thier wider applicability

polyhedral cone ratio dea models with an illustrative application to large commercial banks abstract polyhedral cone ratio data envelopment analysis models generalizing the ccr ratio model are developed for situations with a finite number of dmus and employing polyhedral cones of virtual multipliers they provide improved definitions of efficiency over ccr models whose input output data and or numbers of dmus are inadequate to capture aspects or restrictions which should be involved the focus here is on the sum form for cones which easily provides for capturing exogenous expert opinion as well as mathematical reduction to the old form with its very powerful software transformation from the usual intersection form to it and vice versa is explicitly given thereby the advantages of either or both are available the theory is illustrated with two dimensional examples and by real banking examples for motivation

dynamical analysis in a bioeconomic phytoplankton zooplankton system with double time delays and environmental stochasticity abstract this paper presents a double delayed bioeconomic phytoplankton zooplankton system with commercial harvesting on zooplankton and environmental stochasticity maturation delay for toxin producing phytoplankton and gestation delay for zooplankton are considered environmental stochasticity is incorporated into the proposed system in form of gaussian white noises some sufficient conditions are derived to show that the proposed system has a unique global positive solution in absence of double time delays stochastic stability and existence of stochastic hopf bifurcation are studied based on invariant measure theory and singular boundary theory of diffusion process for the proposed system in presence of double time delays asymptotic behaviors of the interior equilibrium are discussed by constructing some appropriate lyapunov functions

diffusion in a symmetric bistable potential a variational approach an approximation procedure for the solution of stochastic nonlinear equations which was derived from a variational principle in a previous paper is applied to the problem of a particle that diffuses in a symmetric bistable potential starting from the point of unstable equilibrium the second moment left langle widetilde x 2 t right rangle and variance left langle widetilde x 4 t right rangle left langle widetilde x 2 t right rangle 2 for the particle s position widetilde x t are calculated as functions of the timet good agreement is found with results recently obtained by baibuzet al from an approximate evaluation of a path integral expression for the probability density

dynamic analysis of a hybrid bioeconomic plankton system with double time delays and stochastic fluctuations abstract in this paper we establish a double delayed hybrid bioeconomic plankton system with stochastic fluctuations and commercial harvesting on zooplankton where maturation delay for toxin producing phytoplankton and gestation delay for zooplankton are considered stochastic fluctuations are incorporated into the proposed system in form of gaussian white noises to depict stochastic environmental factors in plankton system for deterministic system without double time delays existence of singularity induced bifurcation is studied due to variation of economic interest of commercial harvesting and state feedback controllers are designed to eliminate singularity induced bifurcation for deterministic system with double time delays positivity and uniform persistence of solutions are studied and some sufficient conditions associated with asymptotic stability of interior equilibrium are investigated for stochastic system without double time delays stochastic stability and existence of stochastic hopf bifurcation are discussed based on singular boundary theory of diffusion process and invariant measure theory for stochastic system with double time delays existence and uniqueness of global positive solution are investigated and asymptotic behaviors of the interior equilibrium are studied by constructing appropriate lyapunov functions numerical simulations are carried out to validate theoretical analysis

influence of the noise spectrum on the anomalous diffusion in a stochastic system we use an effective markovian description to study the long time behavior of a nonlinear second order langevin equation with a gaussian noise when dissipation is neglected the energy of the system grows as with time a power law with an anomalous scaling exponent that depends both on the confining potential and on the high frequency distribution of the noise the asymptotic expression of the probability distribution function in phase space is calculated analytically the results are extended to the case where small dissipative effects are taken into account

extension of statistical replacement to systems with time correlated fluctuations nonlinear systems with correlated stochastic parameters are approximated by simpler systems this method is an extension of an earlier version of statistical replacement and statistical linearization the extended method is applicable to systems with correlated fluctuations we show how this general method reduces to the earlier methods in special cases

large cluster results for two parametric multinomial extra variation models abstract two parametric extra variation models are considered approximate closed form expressions are given for the fisher information matrices the expressions are useful in computing maximum likelihood estimates and obtaining large cluster efficiencies a simulation study shows that the approximations perform very well even in clusters of moderate size the models are applied in illustrative examples a goodness of fit test is developed that is applicable even when the cluster sizes are unequal the null distribution of the test statistic is shown to be well approximated by a chi squared distribution for the cluster size configurations in the examples the test also has high power in distinguishing between the two models considered the goodness of fit test shows that the new model provides adequate description of the data from the three experiments designed to study induced mutagenic effect

clustered binary logistic regression in teratology data using a finite mixture distribution the beta binomial distribution introduced by skellam has been applied in many teratology problems for modelling the litter effect recently morel and nagaraj proposed a new distribution for modelling cluster multinomial data when the clustering is believed to be caused by clumped sampling it turns out that the distribution is a mixture of two binomial distributions and accommodates the estimation of an additional parameter to account for intra litter effect the new distribution arises from a cluster mechanism in which some individuals within a cluster exhibit the same behaviour while the remaining individuals from the cluster react independently of each other such a mechanism is a natural model in teratology problems where typically a genetic trait is passed with a certain probability to the foetuses of the same litter in this article we use the new distribution to model binary responses with logistic regression we analyse data from a teratology experiment to demonstrate that the new model provides a useful addition to current methodology the experiment investigates the synergistic effect of the anticonvulsant phenytoin and trichloropopene oxide on the prenatal development of inbred mice in a simulation study we investigate the type i error rate and the power of the maximum likelihood ratio test when the data follow a finite mixture distribution

optimal gaussian solutions of nonlinear stochastic partial differential equations we present a linearization procedure of a stochastic partial differential equation for a vector field x i t x t 0 x r d i l n t x i t x b i x t x d x i t x i f i t x here is the laplace beltrami operator inr d and f i t x is a gaussian random field with f i t x f j t x ij t t x x the procedure is a natural extension of the equivalent linearization for stochastic ordinary differential equations the linearized solution is optimal in the sense that the distance between true and approximate solutions is minimal when it is measured by the kullback leibler entropy the procedure is applied to the scalar valued ginzburg landau model in r1 withb 1 z z vz 3 stationary values of mean variance and correlation length are calculated they almost agree with exact ones if 1 24 2 1 4 d 1 1 3 c when c there appear quasistationary states fluctuating around one of the bottoms of the potentialu z b 1 z dz the second moment at the quasistationary states almost agrees with the exact one transient phenomena are also discussed half width at half maximum of a structure function decays liket 1 2 for small t the diffusion term 2 x accelerates the relaxation from the neighborhood of an unstable initial statex 0 x 0

random oscillator with general gaussian noise we study the long time behaviour of a nonlinear oscillator subject to a random multiplicative noise with a spectral density or power spectrum that decays as a power law at high frequencies when the dissipation is negligible physical observables such as the amplitude the velocity and the energy of the oscillator grow as power laws with time we calculate the associated scaling exponents and we show that their values depend on the asymptotic behaviour of the external potential and on the high frequencies of the noise our results are generalized to include dissipative effects and additive noise

side effects of absolute weight bounds in dea models abstract this paper analyses effects of incorporating absolute weight bounds for input and output weights in the classical models of data envelopment analysis dea it is shown that a dea model with such restrictions may not maximise the relative efficiency of the assessed decision making unit dmu and may not find the set of weights representing the assessed dmu in the best light in comparison to the other dmus consequently it may produce misleading target values for an inefficient dmu and a wrong reference set of efficient peers a way of avoiding these side effects is based on a utilisation of a maximin dea model equivalent to the classical dea model if no additional restrictions are imposed

exact statistics for linear time delayed oscillators subjected to gaussian excitation abstract large classes of stochastic systems of interest to acoustics and vibrations have time delays the presence of these time delays hinders efforts to provide physical insight into the system behavior because of the difficulty of finding exact solutions this paper addresses this issue by presenting exact steady state solutions for the probability density functions pdf s for linear time delayed oscillators subjected to gaussian excitation of arbitrary correlation a key parameter influencing these pdf s is shown to be the ratio of mean generalized kinetic energy to the mean generalized potential energy in a single damped harmonic oscillator this ratio is unity which implies an equi partition of generalized energy time delays cause deviations in the equi partition of generalized energy which creates regions of preferential total phase and causes the amplitude pdf to shift from a rayleigh distribution to a hoyt distribution

theory of nonlinear langevin equation with quadratic noise a generalized langevin equation with a multiplicative linear and quadratic ornstein uhlenbeck stochastic noise is investigated an exact generalized fokker planck equation for this langevin equation is obtained this generalized fokker planck equation has the form of an integro differential equation an exact form of the integral kernel is given in terms of an operator valued matrix continued fraction in the limit of small fluctuations and short coherence time of the noise a standard fokker planck equation with drift and diffusion is derived in a systematic way

statistical replacement for systems with delta correlated fluctuations nonlinear systems with stochastic parameters are approximated by simpler systems using a method that we call statistical replacement this method is an extension of the previously developed agree which was restricted to systems with additive fluctuations statistical replacement incorporates the distinctions between globally stable thermodynamically closed systems and thermodynamically open systems that can be unstable

a variational approach to stochastic nonlinear problems a variational principle is formulated which enables the mean value and higher moments of the solution of a stochastic nonlinear differential equation to be expressed as stationary values of certain quantities approximations are generated by using suitable trial functions in this variational principle and some of these are investigated numerically for the case of a bernoulli oscillator driven by white noise comparison with exact data available for this system shows that the variational approach to such problems can be quite effective

piecewise optimal linearization method for nonlinear stochastic differential equations a method for estimating the dynamical statistical properties of the solutions of nonlinear langevin type stochastic differential equations is presented the non linear equation is linearized within a small interval of the independent variable and statistical properties are expressed analytically within the interval the linearization procedure is optimal in the sense of the chebyshev inequality long term behavior of the solution process is obtained by appropriately matching the approximate solutions at the boundaries between intervals the method is applied to a model nonlinear equation for which the exact time dependent moments can be obtained by numerical methods the calculations demonstrate that the method represents a significant improvement over the method of statistical linearization in time regimes far from equilibrium

quasilinear approximations for the propagator of the fokker planck equation abstract four novel classes of the approximate fokker planck propagators are developed by means of a decoupling operator treatment for a small time increment t these are correct at least through second order t the propagators have an ornstein uhlenbeck form thus leading to novel discrete path integrals with a linear reference system their ambiguities are characterized by an arbitrary function of the time increment and by a matrix and a vector field which may be both time and state dependent the efficiency of the treatment used is very sensitive to the choice of these functions several forms of which are proposed the power of the treatment is illustrated by a physically meaningful fokker planck equation relevant to laser physics being used as the global approximations the approximate propagators obtained are shown to provide a correct description of a bistable system in the entire time domain a scaling theory of suzuki including its unified version emerges from them in a very natural way another attractive feature of our general treatment when compared to the others known in the literature is that it allows for equations with singular diffusion matrices two of which a kramers equation and a colored noise problem are considered

suitability and redundancy of non homogeneous weight restrictions for measuring the relative efficiency in dea abstract weight restrictions are non homogeneous if they are formulated as linear less than or equal to inequalities with a non zero constant on the right hand side absolute weight bounds are typical examples it has recently been shown that in the presence of such restrictions the fractional linear data envelopment analysis dea model and its linear forms may incorrectly evaluate the maximum relative efficiency of the assessed unit this paper investigates the problem further and identifies certain types of non homogeneous restrictions that do not cause the observed error thus the relative efficiency is always assessed correctly if in the same ccr model no positive lower bounds are imposed on any of the input weights and no upper bounds are imposed on any of the output weights the redundancy of certain types of weight restrictions in dea models is also considered based on this the traditional use of small positive constants to separate weights from zero in dea models is questioned

approximate gaussian representation of evolution equations i single degree of freedom nonlinear equations we have developed a generalization of the method of statistical linearization to enable us to describe transient and other nonstationary phenomena obeying stochastic nonlinear differential equations this approximation technique provides an optimal gaussian representation with time dependent parameters the algorithm specifies a set of ordinary differential equations for the gaussian parameters in terms of the time dependent average nonlinearities we apply the general formalism developed herein for single degree of freedom dissipative systems to a particular example

a new approximate stationary probability distribution for processes driven by ornstein uhlenbeck noise a novel approximate probability distribution for a class of nonlinear processes driven by multiplicative linear ornstein uhlenbeck noise is presented a comparison is made with other approximate distributions reported in the literature

on single stage dea models with weight restrictions the literature on data envelopment analysis dea often employs multiplier models that incorporate very small theoretically infinitesimal lower bounds on the input and output weights computational problems arising from the solution of such programs are well known in this paper we identify an additional theoretical problem that may arise if such bounds are used in a multiplier model with weight restrictions namely we show that the use of small lower bounds may lead to the identification of an efficient target with negative inputs we suggest a corrected model that overcomes this problem

finite correlation time effects in nonequilibrium phase transitions i dynamic equation and steady state properties abstract we determine an approximate renormalized equation of evolution for an arbitrary nonlinear single degree of freedom system externally driven by gaussian parametric fluctuations of finite correlation time the renormalization scheme used here gives a second order equation with a time and state dependent diffusion coefficient we are able to calculate the diffusion coefficient in closed form the steady state distribution can easily be obtained from the evolution equation we are thus able to determine the parameter dependence of the steady state distribution and in particular the influence of a correlation time of the fluctuations which does not vanish on the steady state distribution

langevin like equation with colored noise consider a stochastic differential equation of the form of a langevin equation but in which the noise source is not white if it is nearly white i e its autocorrelation time is short a systematic approximation method is known it leads to a fokker planck equation with successive higher order corrections to obtain the coefficients more explicitly a secondary expansion may be employed the validity of the resulting double series approximation is discussed and confronted with the various results given in the literature in addition an alternative approximation method is obtained using the technique for eliminating fast variables it produces the same terms in a different sequence

diffusion in a bistable potential a comparative study of different methods of solution the problem of diffusion in a bistable potential is studied by considering the associated nonlinear langevin equation and its equivalent fokker planck equation two numerically exact methods of solution namely the monte carlo solution of the nonlinear langevin equation and the solution of the fokker planck equation via the finite difference technique are considered the latter method has the advantage that it directly gives the evolution of the probability distribution function approximate analyses of the fluctuations using the system size expansion the gaussian decoupling procedure and the scaling approach are also carried out these investigations are performed on a representative problem for two specific cases 1 evolution from intrinsically unstable states and 2 evolution from extensive regime the fluctuations obtained using these approximate methods are compared with those obtained via the numerically exact methods the study brings out the advantages and limitations of each of the methods considered

group testing for case identification with correlated responses this article examines group testing procedures where units within a group or pool may be correlated the expected number of tests per unit i e efficiency of hierarchical and matrix based procedures is derived based on a class of models of exchangeable binary random variables the effect on efficiency of the arrangement of correlated units within pools is then examined in general when correlated units are arranged in the same pool the expected number of tests per unit decreases sometimes substantially relative to arrangements that ignore information about correlation

unstable state dynamics treatment of colored noise the diffusion of a particle set near an unstable point in a bistable potential is considered the scaling theory of fluctuations proposed originally for onedimensional systems driven by gaussian white noise is extended to arbitrary dimensions the merits and drawbacks of the scaling theory are discussed by taking a model problem in one dimension it is shown in passing that the saddle point approximation enables one to get analytic expressions for various moments of the stochastic process the two different methods to include asymptotic fluctuations which are absent in the usual scaling solution are shown to be equivalent an alternate way of including asymptotic fluctuations is attempted by solving the associated fokker planck equation using the fer formula the reason for the failure of this method is traced after this it is argued that the unified scaling theory should be applicable for treatment of colored noise as well for the scaling assumption is independent of the statistical property of the driving noise explicit monte carlo simulation of a model onedimensional system driven by exponentially correlated gaussian noise is performed and compared with the scaling solution to bolster this point the agreement is very good

on the linearization of nonlinear langevin type stochastic differential equations we show that a logical extension of the piecewise optimal linearization procedure leads to the gaussian decoupling scheme where no iteration is required the scheme is equivalent to solving a few coupled equations the method is applied to models which represent a a single steady state b passage from an initial unstable state to a final preferred stable state by virtue of a finite displacement from the unstable state and c a bivariate case of passage from an unstable state to a final stable state the results are shown to be in very good agreement with the monte carlo calculations carried out for these cases the method should be of much value in multidimensional cases

the application of statistical linearization to the problem of decay from an unstable state abstract it is shown that the method of statistical linearization for the approximate treatment of stochastic nonlinear systems driven by gaussian white noise may be applied successfully to systems which evolve from a state of unstable equilibrium thereby developing non gaussian statistics the essential idea is the replacement of the full ensemble of realizations by subensembles according to the values taken by a small number of linear functionals of the noise the method is illustrated by applying it to two problems i strongly damped linear motion of a particle in a symmetric bistable potential ii the single mode laser in both cases it is found that low order moments can be calculated quite easily to an accuracy of a few per cent and that corrections with error estimates can be found by using the approximation to provide a control variate for numerical simulation

the duffing oscillator in the low friction limit theory and analog simulation by a joint use of theory and analog simulation the low friction regime of the duffing oscillator is explored in the weak temperature case it is shown that the low friction regime in turn must be divided in two well distinct subregimes in the former one characterized by the friction ranging from 2 0 0 is the harmonic frequency up to a lower bound t the method of statistical linearization applies and a continued fraction procedure cfp generated by the zwanzig mori projection techniques is shown to provide correct information for both the renormalized frequency of the oscillator and the corresponding line shape the latter subregime characterized by the friction ranging from 0 to t is fraught with the complete breakdown of the statistical linearization method the cfp is shown to provide an incorrect picture of the line shape while suggesting a novel mean field approximation which is then proven analytically via an alternative method of calculation this method consists of expressing the system in a form reminiscent of the model of kubo s stochastic oscillator by using this alternative approach we are in a position to account for the residual linewidth which is shown by the analog experiment to survive for 0

numerical estimation of the path integral and the diffusion in an anharmonic oscillator abstract the solution of the fokker planck equation given in the form of the path integral is estimated by the contribution from paths around the most probable path a numerical procedure for this estimation is given and applied to the regression of an anharmonic oscillator under a random force from unstable positions to a stable position

solution of fokker planck equation using trotter s formula solution of fokker planck equation using trotter s formula is discussed the method is illustrated on the linear fokker planck equation and the ornstein uhlenbeck solution is obtained for the case of a general nonlinear fokker planck equation the method yields an integral representation amenable to approximations in the lowest order approximation suzuki s scaling result emerges physical interpretation and limitations of the approximations are also discussed

dynamics of generalist predator in a stochastic environment modified ratio dependent holling tanner model with prey refuge is considered delays are taken in logistic growth of prey as feedback mechanism and gestation period of predator model system exhibits hopf bifurcation when the delay parameters cross their critical values stochastic analysis of the model system is carried out by incorporating gaussian white noise using fourier transform technique fluctuation and stability of the stochastic model is studied in this paper an attempt has been made to understand the dynamics of a prey predator system with multiple time delays where the predator population is regarded as a generalist type in this regard we consider a modified holling tanner prey predator system where a constant time delay is incorporated in the logistic growth of the prey to represent a delayed density dependent feedback mechanism and the second time delay is considered to account for the length of the gestation period of the predator predator s interference in prey predator relationship provides better descriptions of predator s feeding over a range of prey predator abundances so the predator s functional response is considered to be type ii ratio dependent and foraging efficiency of predator largely varies with the refuge strategy of prey population in accordance with previous studies it is observed that delay destabilizes the system in general and stability loss occurs via hopf bifurcation in particular we show that there exists critical values of the delay parameters below which the coexistence equilibrium is stable and above which it is unstable hopf bifurcation occurs when the delay parameters cross their critical values also environmental stochasticity in the form of gaussian white noise plays a significant role to describe the system and its values numerical computation is also performed to validate and visualize different theoretical results presented the analysis and results in this work are interesting both in mathematical and biological point of views

spectral analysis of a nonlinear oscillator driven by random and periodic forces i linearized theory we have combined the techniques of statistical and harmonic linearization to develop a linearized approximation theory for the calculation of the second order statistics i e autocorrelation functions and spectral densities of nonlinear systems driven by both random and periodic forces for the special case of a duffing oscillator a damped anharmonic oscillator with a cubic nonlinearity driven by gaussian white noise and by a sinusoidal force explicit expressions for the renormalized linearized frequency the autocorrelation function and the spectral density of the oscillator displacement in terms of all the system parameters have been derived we have determined the region of the parameter space in which the applied periodic force has a significant influence on the second order statistics of the oscillator

an approximate master equation for systems driven by linear ornstein uhlenbeck noise abstract by use of an operator method we construct a novel approximate evolution equation for a one dimensional probability distribution of a single degree of freedom system driven by ornstein uhlenbeck noise this equation is an integro differential equation of the time convolutionless type and its steady state solution is presented a class of non linear equations with multiplicative noise is considered

beyond the markovian assumption response excitation probabilistic solution to random nonlinear differential equations in the long time uncertainty quantification for dynamical systems under non white excitation is a difficult problem encountered across many scientific and engineering disciplines difficulties originate from the la