spectral formulation and wkb approximation for rare event statistics in reaction systems we develop a spectral formulation and a stationary wkb approximation for calculating the probabilities of rare events large deviations from the mean in systems of reacting particles with infinite range interaction describable by a master equation we compare the stationary wkb approximation to a recent time dependent semiclassical approximation developed for the same class of problems by elgart and kamenev phys rev e 70 41106 2004 as a benchmark we use an exactly solvable problem of the binary annihilation reaction 2a 0

a numerical ode solver that preserves the fixed points and their stability in this paper we provide a one step predictor corrector method for numerically solving first order differential initial value problems with two fixed points the method preserves the stability behaviour of the fixed points which results in an efficient integrator for this kind of problem some numerical examples are provided to show the good performance of the method

order and stepsize control in extrapolation methods the paper presents a new theory for joint order and stepsize control in extrapolation methods this theory defines a locally optimal order that can be determined along any trajectory to be computed in addition shannon s information theory is applied to derive some ideal convergence model that is expected to describe the behavior of an extrapolation method over a large set of test problems extensive numerical comparisons document a drastic acceleration in stiff integration and a mild acceleration in non stiff integration by the new device moreover a significant increase in reliability robustness and portability of the extrapolation codes is achieved

coefficients for studying one step rational schemes for ivps in odes ii abstract in 1 we derived variable order variable stepsize one step nonlinear methods in which global formulation is facilitated by defining some certain coefficients the analysis has a broader application and facilitates the derivation of new formulae the methods are based on a variant self adjusting rational interpolant of luke et al 2 and otunta and ikhile 3 4 in this paper we motivate an extension of this analysis to obtain a more generalized class of the rational methods in 1 for an arbitrarily desired order

extinction rates for fluctuation induced metastabilities a real space wkb approach the extinction of a single species due to demographic stochasticity is analyzed the discrete nature of the individual agents and the poissonian noise related to the birth death processes result in local extinction of a metastable population as the system hits the absorbing state the fokker planck formulation of that problem fails to capture the statistics of large deviations from the metastable state while approximations appropriate close to the absorbing state become in general invalid as the population becomes large to connect these two regimes a real space wkb method based on the master equation is presented and is shown to yield an excellent approximation for the decay rate and the extreme events statistics all the way down to the absorbing state the details of the underlying microscopic process smeared out in a mean field treatment are shown to be crucial for an exact determination of the extinction exponent this general scheme is shown to reproduce the known results in the field to yield new corollaries and to fit quite precisely the numerical solutions moreover it allows for systematic improvement via a series expansion where the small parameter is the inverse of the number of individuals in the metastable state

a boundary value technique for singular perturbation problems abstract a new way to solve singular perturbation problems is introduced it is designed for the practicing engineer or applied mathematician who needs a practical tool for these problems easy to use modest problem preparation and ready computer implementation as with other methods the original problem is partitioned into inner and outer solution differential equation systems however asymptotic solution techniques are not employed the method is distinguished by the following facts the inner solution problem is solved as a two point boundary problem where the terminal boundary conditions are supplied by the solution of the outer solution problem in turn the outer solution differential equations are solved as an initial value system where the initial conditions are supplied by the inner solution at its terminal point the method is iterative on the terminal point of the inner solution problem three numerical examples are included

demographic stochasticity and extinction in populations with allee effect we study simple stochastic scenarios based on birth and death markovian processes that describe populations with the allee effect to account for the role of demographic stochasticity in the mean field deterministic limit we recover well known deterministic evolution equations widely employed in population ecology the mean time to extinction is in general obtained by the wentzel kramers brillouin wkb approximation for populations with the strong and weak allee effects an exact solution for the mean time to extinction can be found via a recursive equation for special cases of the stochastic dynamics we study the conditions for the validity of the wkb solution and analyze the boundary between the weak and strong allee effect by comparing exact solutions with numerical simulations

a galerkin method with rational basis functions for burgers equation abstract the aim of this paper is to use rational basis functions as an alternative to upwinding techniques in the petrov galerkin finite element method

families of backward differentiation methods based on a new type of mixed interpolation abstract backward differentiation methods based on a new type of mixed interpolation for the first order initial value problems whose solutions are known to be periodic are constructed the angular frequency k is calculated by minimizing the local truncation error within each integration interval the resulting methods depend on a parameter hk h the step length and reduce to classical backward methods if k 0 numerical examples are provided to illustrate the algorithms

stochastic dynamics and logistic population growth the verhulst model is probably the best known macroscopic rate equation in population ecology it depends on two parameters the intrinsic growth rate and the carrying capacity these parameters can be estimated for different populations and are related to the reproductive fitness and the competition for limited resources respectively we investigate analytically and numerically the simplest possible microscopic scenarios that give rise to the logistic equation in the deterministic mean field limit we provide a definition of the two parameters of the verhulst equation in terms of microscopic parameters in addition we derive the conditions for extinction or persistence of the population by employing either the momentum space spectral theory or the real space wentzel kramers brillouin approximation to determine the probability distribution function and the mean time to extinction of the population our analytical results agree well with numerical simulations

two low accuracy methods for stiff systems two low accuracy explicit one step methods for stiff ordinary differential equations are extended directly to solve systems of equations some defects of the component form of these methods are avoided to perform these a new set of vector computations are introduced some numerical experiments are presented to show the superiority of the new methods

a non linear single step explicit scheme for non linear two point singularly perturbed boundary value problems via initial value technique abstract in this paper a method based on initial value technique is proposed for solving non linear two point singularly perturbed boundary value problems for second order ordinary differential equations odes with a boundary layer at one either left or right end the original singularly perturbed boundary value problem is reduced to an initial value problem approximated by its outer solution asymptotic approximation the new initial value problem is solved by proposed non linear single step explicit scheme followed the idea given in f d van niekerk non linear one step methods for initial value problems comput math appl 13 1987 367 371 the proposed scheme has been shown to be of order two to demonstrate the applicability of the proposed scheme several linear and non linear problems have been solved it is observed that the present scheme approximate the exact solution very well

an explicit finite element method for convection diffusion equations using rational basis functions abstract an explicit galerkin method is formulated by using rational basis functions the characteristics of the rational difference scheme are investigated with regard to consistency stability and numerical convergence of the method numerical results are also presented

on the exponential fitting of composite multiderivative linear multistep methods exponentially fitted composite multiderivative linear multistep methods of orders up to 5 are derived a numerical investigation of these formulae indicates that they may be a stable for all choices of the fitting parameters an important aspect of the class of integration formulae considered is that each formula contains a built in local error estimate a facility lacking in most other exponentially fitted formulae some preliminary numerical results are given and these show that for problems for which exponential fitting is appropriate the new formulae provide a large saving in computational effort compared with conventional ones

coefficients for studying one step rational schemes for ivps in odes iii extrapolation methods abstract this paper considers extrapolation of rational methods for the numerical solution of initial value problems ivps in ordinary differential equations odes the extrapolation code difex2 of fatunla 1 which is a modification of the automatic extrapolation code difexi of deuflhard 2 is further modified to accommodate the basic integrators the extrapolation method which we refer to as difex2 for reference purposes is compared with results from difexi and difex2 and gbs extrapolation method

some comments on preference order dynamic programming models abstract a simple deterministic dynamic programming model is used as a general framework for the analysis of stochastic versions of three classical optimization problems knapsack traveling salesperson and assembly line balancing problems it is shown that this model can provide an alternative to the preference order models proposed for these problems counterexample to the optimality of the preference order models are presented

a technique for the numerical solution of initial value problems based on a class of birkhoff type interpolation method in this paper a class of birkhoff type interpolation problems on arbitrary nodal points is studied the explicit representation characterization the uniqueness and the error function are explicitly given furthermore we apply the obtained birkhoff type interpolation method to find i the numerical solution of high order initial value problems ivps and the corresponding error of this approximation ii the approximation of some special functions with their explicit error functions and iii new interpolatory type quadrature formulae of precision degree at least m n 1 and m kn 1 m n email protected n n k 2 numerical examples are included to demonstrate the validity and applicability of the technique proposed in this paper and a comparison is made with the existing results the results reveal that the new method is effective simple and accurate

numerical treatment of singular initial value problems abstract an efficient extrapolation scheme whose basic integrator is the inverse euler scheme cf fatunla 1982 is proposed for nonlinear singular initial value problems y x y y 0 y 0 the automatic polynomial rational extrapolation code difexi deufhard 1983 1985 is modified to accomodate the basic integrator the new algorithm was implemented in variable step variable order mode and compares favourably with the earlier works

double precision rational approximation algorithms for the standard normal first and second order loss functions abstract we present double precision algorithms based upon piecewise rational approximations for the standard normal first and second order loss functions these functions are used frequently in inventory management no direct approximation or closed formulation exists for the standard first and second order loss functions current state of the art algorithms require intermediate computations of the cumulative normal distribution or tabulations and they do not compute to full double precision we deal with both these issues and present direct double precision accurate algorithms which are valid in the full range of double precision floating point numbers

nonlinear multistep methods for initial value problems abstract some k step kth order explicit nonlinear multistep methods nmm are proposed for both stiff and singular initial value problems the algorithms are based on a local representation of the solution by the reciprocal of a polynomial for a k step scheme there is need to solve a linear system of equations for the k coefficients of the inverse polynomial which are apparently global for linear differential equations the resultant integration schemes are stable and convergent and compare favorably with the methods of lambert and shaw 2 luke fair and wimp 7 and lambert 1

on compactly supported spline wavelets and a duality principle let v 1 v 0 v 1 be a multiresolution analysis of l 2 generated by the mth order b spline n m x we exhibit a compactly supported basic wavelet m x that generates the corresponding orthogonal complementary wavelet subspaces w 1 w 0 w 1 consequently the two finite sequences that describe the two scale relations of n m x and m x in terms of n m 2x j j z yield an efficient reconstruction algorithm to give an efficient wavelet decomposition algorithm based on these two finite sequences we derive a duality principle which also happens to yield the dual bases n m x j and 59 m x j relative to n m x j and m x j respectively

linearization techniques for singularly perturbed initial value problems of ordinary differential equations linearization methods for singularly perturbed initial value problems in ordinary differential equations based on taylor series expansions and piecewise analytical solutions are presented if only the linear terms are retained it is shown that exponentially fitted techniques are obtained whereas if quadratic terms are kept then riccati s equations result both linear and quadratic linearization methods are shown to be nonuniformly convergent in the small perturbation parameter and the former are more accurate than the latter in equally spaced grids linearization methods are also used with three different adaptive mesh refinement methods that account for the magnitude of the jacobian the variation of the solution and the resolution of the initial layer and it is shown that mesh refinement methods based on the variation of the solution can provide very accurate solutions although they may yield large errors at the outer edge of the initial layer if the grid spacing increases rapidly it is also shown that for values of the perturbation parameter on the order of unity quadratic linearization methods are more accurate than linearization methods and can account for the presence of internal layers it is also shown that linearization methods on grids whose spacing obeys a geometric progression are not robust and may suffer from a loss in the order of convergence compared with adaptive methods that resolve the boundary layer or adapt the grid according to the numerical solution

contributions to the development of differential systems exactly solved by multistep finite difference schemes abstract the motivation underlying this contribution has been to complete some of the topics concerning exact schemes for numerically solving ordinary differential equations a procedure for obtaining differential systems exactly solved by a given finite difference method is described examples illustrating the application of the procedure for obtaining first order second order and systems of differential equations exactly solved by different numerical methods are given among the numerical methods considered there are the trapezoidal rule the two step adams bashforth method and the numerov method some numerical examples are presented to provide evidence that the procedure works properly

recent progress in extrapolation methods for ordinary differential equations the paper is intended to give a survey on the state of the art of extrapolation methods for initial value problems in ordinary differential equation systems all basic discretizations which are su

a rational basis for second kind abel integral equations abstract a rational basis set is constructed using 1 1 rational interpolation the basis is used in a product integration method for solving second kind abel integral equations criteria are established for determining the amount of bias to be used in the basis set when the integral equation has a nonsmooth solution several test problems are solved to illustrate the performance of the approximation method

stability and accuracy of time discretizations for initial value problems this paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems here the treatment is extended to cover also implicit schemes and by placing the accuracy of the schemes into a more central position in the discussion general method free statements are again obtained more specialized results are additionally given for linear multistep methods and for the taylor series method

colonization of a territory by a stochastic population under a strong allee effect and a low immigration pressure we study the dynamics of colonization of a territory by a stochastic population at low immigration pressure we assume a sufficiently strong allee effect that introduces in deterministic theory a large critical population size for colonization at low immigration rates the average precolonization population size is small thus invalidating the wkb approximation to the master equation we circumvent this difficulty by deriving an exact zero flux solution of the master equation and matching it with an approximate nonzero flux solution of the pertinent fokker planck equation in a small region around the critical population size this procedure provides an accurate evaluation of the quasistationary probability distribution of population sizes in the precolonization state and of the mean time to colonization for a wide range of immigration rates at sufficiently high immigration rates our results agree with wkb results obtained previously at low immigration rates the results can be very different

rare event statistics in reaction diffusion systems we present an efficient method to calculate probabilities of large deviations from the typical behavior rare events in reaction diffusion systems this method is based on a semiclassical treatment of an underlying quantum hamiltonian encoding the system s evolution to this end we formulate the corresponding canonical dynamical system and investigate its phase portrait this method is presented for a number of pedagogical examples

immigration extinction dynamics of stochastic populations how high should be the rate of immigration into a stochastic population in order to significantly reduce the probability of observing the population become extinct is there any relation between the population size distributions with and without immigration under what conditions can one justify the simple patch occupancy models which ignore the population distribution and its dynamics in a patch and treat a patch simply as either occupied or empty we answer these questions by exactly solving a simple stochastic model obtained by adding a steady immigration to a variant of the verhulst model a prototypical model of an isolated stochastic population

a galerkin method using rational basis functions abstract rational basis functions were constructed without any poles these basis functions were applied to the galerkin method in order to solve a wide class of differential equations

stochastic analysis of complex reaction networks using binomial moment equations the stochastic analysis of complex reaction networks is a difficult problem because the number of microscopic states in such systems increases exponentially with the number of reactive species direct integration of the master equation is thus infeasible and is most often replaced by monte carlo simulations while monte carlo simulations are a highly effective tool equation based formulations are more amenable to analytical treatment and may provide deeper insight into the dynamics of the network here we present a highly efficient equation based method for the analysis of stochastic reaction networks the method is based on the recently introduced binomial moment equations barzel and biham phys rev lett 106 150602 2011 the binomial moments are linear combinations of the ordinary moments of the probability distribution function of the population sizes of the interacting species they capture the essential combinatorics of the reaction processes reflecting their stoichiometric structure this leads to a simple and transparent form of the equations and allows a highly efficient and surprisingly simple truncation scheme unlike ordinary moment equations in which the inclusion of high order moments is prohibitively complicated the binomial moment equations can be easily constructed up to any desired order the result is a set of equations that enables the stochastic analysis of complex reaction networks under a broad range of conditions the number of equations is dramatically reduced from the exponential proliferation of the master equation to a polynomial and often quadratic dependence on the number of reactive species in the binomial moment equations the aim of this paper is twofold to present a complete derivation of the binomial moment equations to demonstrate the applicability of the moment equations for a representative set of example networks in which stochastic effects play an important role

extinction in the lotka volterra model birth death processes often exhibit an oscillatory behavior we investigate a particular case where the oscillation cycles are marginally stable on the mean field level an iconic example of such a system is the lotka volterra model of predator prey interaction fluctuation effects due to discreteness of the populations destroy the mean field stability and eventually drive the system toward extinction of one or both species we show that the corresponding extinction time scales as a certain power law of the population sizes this behavior should be contrasted with the extinction of models stable in the mean field approximation in the latter case the extinction time scales exponentially with size

a b wavelet based noise reduction algorithm a wavelet based method is introduced for removing structured noise e g impulsive spikes or unwanted harmonic components from data for this type of noise the time and frequency localization capabilities of wavelets provide better noise detection and less signal distortion than direct filtering of data the procedure is applied to time series data with impulsive noise and transfer function data with multipath interference the authors use a single set of scaling and wavelet bases that can effectively represent a wide range of signals and then they perform noise reducing operations on the scaling and wavelet coefficients multiresolution analysis is introduced in order to describe the noise reduction algorithm

extinction of an infectious disease a large fluctuation in a nonequilibrium system we develop a theory of first passage processes in stochastic nonequilibrium systems of birth death type using two closely related epidemiological models as examples our method employs the probability generating function technique in conjunction with the eikonal approximation in this way the problem is reduced to finding the optimal path to extinction a heteroclinic trajectory of an effective multidimensional classical hamiltonian system we compute this trajectory and mean extinction time of the disease numerically and uncover a nonmonotone spiral path to extinction of a disease we also obtain analytical results close to a bifurcation point where the problem is described by a hamiltonian previously identified in one species population models

an approximate method for solving a class of singular perturbation problems abstract an approximate method for the numerical solution of a class of singularly perturbed two point boundary value problems is presented the given region is divided into inner and outer regions the original second order problem is replaced by an asymptotically equivalent first order problem and solved as an initial value problem in the inner region a terminal boundary condition is then obtained from the solution of the inner region problem in turn an outer region problem is obtained by replacing the second order differential equation by an approximate first order differential equation with a small deviating argument and solved efficiently by employing the trapezoidal formula coupled with a discrete invariant imbedding algorithm the proposed method is iterative on the terminal point of the inner region problem several numerical examples have been solved to demonstrate the applicability of the method

extinction of metastable stochastic populations we investigate the phenomenon of extinction of a long lived self regulating stochastic population caused by intrinsic demographic noise extinction typically occurs via one of two scenarios depending on whether the absorbing state n 0 is a repelling scenario a or attracting scenario b point of the deterministic rate equation in scenario a the metastable stochastic population resides in the vicinity of an attracting fixed point next to the repelling point n 0 in scenario b there is an intermediate repelling point n n 1 between the attracting point n 0 and another attracting point n n 2 in the vicinity of which the metastable population resides the crux of the theory is a dissipative variant of wkb wentzel kramers brillouin approximation which assumes that the typical population size in the metastable state is large starting from the master equation we calculate the quasistationary probability distribution of the population sizes and the exponentially long mean time to extinction for each of the two scenarios when necessary the wkb approximation is complemented i by a recursive solution of the quasistationary master equation at small n and ii by the van kampen system size expansion valid near the fixed points of the deterministic rate equation the theory yields both entropic barriers to extinction and pre exponential factors and holds for a general set of multistep processes when detailed balance is broken the results simplify considerably for single step processes and near the characteristic bifurcations of scenarios a and b

on the convergence of the restrictive pade approximation to the exact solutions of ibvp of parabolic and hyperbolic types some accurate finite difference sophisticated methods for solving initial boundary value problem for partial differential equations gives its exact solution with certain values of the mesh sizes of space and time the restrictive pade approximation is one of these very promising methods enclosed is a model structure of its local truncation error differential forms for problems of parabolic and hyperbolic types

restrictive chebyshev rational approximation and applications to heat conduction problems in this paper we construct a restrictive type of chebyshev rational approximation it yields more accurate results and exact values at some selected points we use this method to approximate the exponential function we apply this approach to define a new implicit finite difference scheme to parabolic partial differential equations this approach will exhibit several advantages features highly accurate fast and the absolute error still very small whatever the exact solution is too large stability conditions are obtained and utilized in numerical computations finally some numerical results to describe the performance of this approach are given

a maximum likelihood digital receiver using coordinate ascent and the discrete wavelet transform in this paper a maximum likelihood ml method is presented for joint estimation of amplitude phase time delay and data symbols in a single user direct sequence spread spectrum communication system since maximization of the likelihood function is analytically intractable a novel coordinate ascent algorithm is used to obtain sequential updates of the data symbols and all unknown nuisance parameters the novelty of the algorithm is due to the use of a multiresolution expansion of the received signal and the use of polynomial rooting in the complex plane in place of a line search over the signal delay parameter the multiresolution structure of the algorithm is exploited to reduce sensitivity to impulsive noise via wavelet thresholding computer simulations of the single user system show that the algorithm has fast convergence and comparison with theoretical lower bounds establishes that the algorithm achieves nearly optimal error performance

an introduction to stochastic modeling stochastic modeling probability review the major discrete distributions rtant continuous distributions some elementary exercises useful functions integrals and sums conditional probability and conditional expectation the discrete case the dice game craps random sums conditioning on a continuous random variable markov chains introduction definitions transition probability matrices of a markov chain some markov chain models first step analysis some special markov chains functionals of random walks and success runs another look at first step analysis the long run behavior of markov chains regular transition probability matrices examples the classification of states the basic limit theorem of markov chains reducible markov chains sequential decisions and markov chains poisson processes the poisson distribution and the poisson processes the law of rare events distributions associated with the poisson process the uniform distribution and poisson processes spatial poisson processes compound and marked poisson processes continuous time markov chains pure birth processes ptire death processes birth and death processes the limiting behavior of birth and death processes birth and death processes with absorbing states finite state continuous time markov chains set valued processes renewal phenomena definition of a renewal process and related concepts some examples of renewal processes the poisson process viewed as a renewal process the asymptotic 3ehavior as renewal process

effect of reaction step size noise on the switching dynamics of stochastic populations in genetic circuits when the messenger rna lifetime is short compared to the cell cycle proteins are produced in geometrically distributed bursts which greatly affects the cellular switching dynamics between different metastable phenotypic states motivated by this scenario we study a general problem of switching or escape in stochastic populations where influx of particles occurs in groups or bursts sampled from an arbitrary distribution the fact that the step size of the influx reaction is a priori unknown and in general may fluctuate in time with a given correlation time and statistics introduces an additional nondemographic reaction step size noise into the system employing the probability generating function technique in conjunction with hamiltonian formulation we are able to map the problem in the leading order onto solving a stationary hamilton jacobi equation we show that compared to the usual case of single step influx bursty influx exponentially decreases the population s mean escape time from its long lived metastable state in particular close to bifurcation we find a simple analytical expression for the mean escape time which solely depends on the mean and variance of the burst size distribution our results are demonstrated on several realistic distributions and compare well with numerical monte carlo simulations

rate of convergence of multistep codes started by variation of order and stepsize variable stepsize variable order multistep codes are among the most popular for the solution of the initial value problem for a system of ordinary differential equations the typical code starts with a formula of order 1 and builds up the stepsize and order rapidly to values appropriate for the problem according to the classical theory the starting errors persist and hence the order of convergence of the whole integration is at most 2 this is not what is observed in practice this paper develops a theory of the start that models the way codes work and establishes the high rate of convergence observed it is even possible to determine the asymptotic behavior of the error which turns out to be remarkably simple

an error corrected euler method for solving stiff problems based on chebyshev collocation in this paper we present error corrected euler methods for solving stiff initial value problems which not only avoid unnecessary iteration process that may be required in most implicit methods but also have such a good stability as all implicit methods possess the proposed methods use a chebyshev collocation technique as well as an asymptotical linear ordinary differential equation of first order derived from the difference between the exact solution and the euler s polygon these methods with or without the jacobian are analyzed in terms of convergence and stability in particular it is proved that the proposed methods have a convergence order up to 4 regardless of the usage of the jacobian numerical tests are given to support the theoretical analysis as evidences

numerical treatment of singularly perturbed two point boundary value problems we propose a method for numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval this is a practical method and can be easily implemented on a computer the original problem is divided into inner and outer region differential equation systems the reduced problem is solved to obtain the terminal boundary condition then a new inner region problem is created and solved as a two point boundary value problem tpbvp in turn the outer region problem is also solved as a tpbvp both these tpbvps are efficiently treated by employing a slightly modified classical finite difference scheme coupled with discrete invariant imbedding algorithm to obtain the numerical solutions the stability of some recurrence relations involved in the algorithm is investigated the proposed method is iterative on the terminal point some numerical examples are included and the computational results are compared with exact solutions it is observed that the accuracy predicted can always be achieved with very little computational effort

rational one step methods for initial value problems abstract first second and third order explicit nonlinear one step methods are proposed for singular and stiff initial value problems ivp s the algorithms are based on the representation of the solution by a finite continued fraction numerical examples are provided to illustrate the algorithms

convergence on error correction methods for solving initial value problems higher order semi explicit one step error correction methods ecm for solving initial value problems are developed ecm provides the excellent convergence o h 2 p 2 one wants to get without any iteration processes required by most implicit type methods this is possible if one constructs a local approximation having a residual error o h p on each time step as a practical example we construct a local quadratic approximation further it is shown that special choices of parameters for the local quadratic polynomial lead to the known explicit second order methods which can be improved into a semi explicit type ecm of the order of accuracy 6 the stability function is also derived and numerical evidences are presented to support theoretical results with several stiff and non stiff problems it should be remarked that the ecm approach developed here does not yield explicit methods but semi implicit methods of the rosenbrock type both ecm and rosenbrock s methods require to solve a few linear systems at each integration step but the ecm approach involves 2p 2 evaluations of the jacobian matrix per integration step whereas the rosenbrock method demands one evaluation only however it is much easier to get high order methods by using the ecm approach

on the commutant algebras corresponding to the permutation representations of the full collineation groups of pg k s and eg k s s pr k 2 abstract in this paper the dimension t and a linear basis of the commutant algebra corresponding to the representation of the full collineation group as matrices permuting the flags incident point line or point hyperplane pairs have been determined for each one of the four geometries pg 2 s eg 2 s pg k s and eg k s s pr k 3 for the four geometries t 6 7 7 and 8 respectively and the corresponding linear bases are i g b t bt tb i g b t bt tb btb i g b t bt tb s and i g b t bt tb btb s i g b t are the relationship matrices of james ann math statist 28 1957 993 1082 and the matrix s was introduced by sysoev and shaikin avtomat i telemekh 5 1976 64 73

double precision rational approximation algorithm for the inverse standard normal second order loss function abstract we present a double precision algorithm based upon rational approximations for the inverse standard normal second order loss function this function is used frequently in inventory management no direct approximation or closed formulation exists for the inverse standard normal second order loss function calculations are currently based on root finding methods and intermediate computations of the cumulative normal distribution or tabulations results then depend on the accuracy and valid range of that underlying function we deal with these issues and present a direct double precision accurate algorithm valid in the full range of double precision floating point numbers

commitment versus persuasion in the three party constrained voter model in the framework of the three party constrained voter model where voters of two radical parties a and b interact with centrists c and c we study the competition between a persuasive majority and a committed minority in this model a s and b s are incompatible voters that can convince centrists or be swayed by them here radical voters are more persuasive than centrists whose sub population comprises susceptible agents c and a fraction of centrist zealots c whereas c s may adopt the opinions a and b with respective rates 1 a and 1 b with a b 0 c s are committed individuals that always remain centrists furthermore a and b voters can become susceptible centrists c with a rate 1 the resulting competition between commitment and persuasion is studied in the mean field limit and for a finite population on a complete graph at mean field level there is a continuous transition from a coexistence phase when b whereas all species coexist when a b when c and the initial density of centrists is low one finds lnn when n 1 our analytical findings are corroborated by stochastic simulations

population extinction in a time modulated environment the extinction time of an isolated population can be exponentially reduced by a periodic modulation of its environment we investigate this effect using as an example a stochastic branching annihilation process with a time dependent branching rate the population extinction is treated in eikonal approximation where it is described as an instanton trajectory of a proper reaction hamiltonian the modulation of the environment perturbs this trajectory and synchronizes it with the modulation phase we calculate the corresponding change in the action along the instanton using perturbation techniques supported by numerical calculations the techniques include a first order theory with respect to the modulation amplitude a second order theory in the spirit of the kapitsa pendulum effect and adiabatic theory valid for low modulation frequencies

an efficient variable step size rational falkner type method for solving the special second order ivp in this paper firstly a rational one parameter family of falkner type explicit methods is presented for directly solving numerically special second order initial value problems in ordinary differential equations the proposed family of methods has second algebraic order of convergence imposing that the principal term of the local truncation error of the proposed family vanishes we get an expression for the free parameter at the grid point xn yn by substituting this value of the free parameter in the family a new rational third order method is obtained further by combining the third order method with any member of the second order family their variable step size formulation as an embedded pair is considered some numerical experiments are given to illustrate the performance and efficiency of the proposed methods

robust l estimation based forms of signal transforms and time frequency representations the l estimation based signal transforms and time frequency tf representations are introduced by considering the corresponding minimization problems in the huber 1981 1998 estimation theory the standard signal transforms follow as the maximum likelihood solutions for the gaussian additive noise environment for signals corrupted by an impulse noise the median based transforms produce robust estimates of the non noisy signal transforms when the input noise is a mixture of gaussian and impulse noise the l estimation based signal transforms can outperform other estimates in quadratic and higher order tf analysis the resulting noise is inherently a mixture of the gaussian input noise and an impulse noise component in this case the l estimation based signal representations can produce the best results these transforms and tf representations give the standard and the median based forms as special cases a procedure for parameter selection in the l estimation is proposed the theory is illustrated and checked numerically

population switching under a time varying environment we study the switching dynamics of a stochastic population subjected to a deterministically time varying environment our approach is demonstrated on a problem of population establishment which is important in ecology at the deterministic level the model we study gives rise to a critical population size beyond which the system experiences establishment notably the latter has been shown to be strongly influenced by the interplay between demographic and environmental variations here we consider two prototypical examples of a time varying environment a temporary change in the environment and a periodically varying environment by employing a semiclassical approximation we compute within exponential accuracy the change in the establishment probability and mean establishment time of the population due to the environmental variability our analytical results are verified by using a modified gillespie algorithm which accounts for explicitly time dependent reaction rates importantly our theoretical approach can also be useful in studying switching dynamics in gene regulatory networks under external variations

non linear one step methods for initial value problems abstract a one step first order non linear method is proposed for initial value problems the algorithms are based on a representation of the solution by the reciprocal of a polynomial

a numerical study of multistep methods based on continued fractions abstract we investigate rational multistep methods of both osculatory and adams type for solving singular initial value problems formulas that permit variable steplength are constructed to avoid possible difficulties arising from the rational interpolation in two test examples it is found that explicit formulas of the adams type perform generally better than those based on the osculatory method the most accurate solutions are obtained from predictor corrector formulas based on the osculatory method

given a one step numerical scheme on which ordinary differential equations is it exact a necessary condition for a non autonomous ordinary differential equation to be exactly solved by a one step finite difference method is that the principal term of its local truncation error be null a procedure to determine some ordinary differential equations exactly solved by a given numerical scheme is developed examples of differential equations exactly solved by the explicit euler implicit euler trapezoidal rule second order taylor third order taylor van niekerk s second order rational and van niekerk s third order rational methods are presented

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

spectral theory of metastability and extinction in a branching annihilation reaction we apply the spectral method recently developed by the authors to calculate the statistics of a reaction limited multistep birth death process or chemical reaction that includes as elementary steps branching a 2a and annihilation 2a 0 the spectral method employs the generating function technique in conjunction with the sturm liouville theory of linear differential operators we focus on the limit when the branching rate is much higher than the annihilation rate and obtain accurate analytical results for the complete probability distribution including large deviations of the metastable long lived state and for the extinction time statistics the analytical results are in very good agreement with numerical calculations furthermore we use this example to settle the issue of the lacking boundary condition in the spectral formulation

a class of inverse runge kutta schemes for the numerical integration of singular problems a class of methods for the numerical solution of initial value problems whose solution has singular point s shall be discussed they are a and l stable and were found to perform efficiently well when faced with singularity problem but perform poorly when applied to non singular problems

a numerical algorithm for singular perturbation problems exhibiting weak boundary layers abstract a class of singularly perturbed two point boundary value problems bvp for second order ordinary differential equations is considered here to avoid the numerical difficulties in the solution to these problems we divide the domain into two subdomains the first bvp is a layer domain problem and the second bvp is a regular domain problem error estimates are derived for the numerical solution numerical examples are provided in support of the proposed method

a non standard explicit integration scheme for initial value problems in this paper we present the construction of a non standard explicit algorithm for initial value problems the method results to be of second order and a stable this new algorithm has been proven to be suitable for solving different kind of initial value problems specifically non singular problems singular problems stiff problems and singularly perturbed problems some numerical experiments are considered in order to check the behaviour of the method when applied to a variety of initial value problems

noise driven unlimited population growth demographic noise causes unlimited population growth in a broad class of models which without noise would predict a stable finite population we study this effect on the example of a stochastic birth death model which includes immigration binary reproduction and death the unlimited population growth proceeds as an exponentially slow decay of a metastable probability distribution mpd of the population we develop a systematic wkb theory complemented by the van kampen system size expansion for the mpd and for the decay time important signatures of the mpd are a power law tail such that all the distribution moments except the zeroth one diverge and the presence in the solution of two different wkb modes

solving first order initial value problems by using an explicit non standard a stable one step method in variable step size formulation this paper presents the construction of a new family of explicit schemes for the numerical solution of initial value problems of ordinary differential equations odes the one parameter family is constructed by considering a suitable rational approximation to the theoretical solution resulting a family with second order convergence and a stable imposing that the principal term in the local truncation error vanishes we obtain an expression for the parameter value in terms of the point xn yn on each step with this approach the resulting method has third order convergence maintaining the characteristic of a stability finally combining this last method with other of order two in order to get an estimation for the local truncation error an implementation in variable step size has been considered the proposed method can be used in a wide range of problems for solving numerically a scalar ode or a system of first order odes several numerical examples are given to illustrate the efficiency and performance of the proposed method in comparison with some existing methods in the literature

numerical solution of singular perturbation problems via deviating arguments we present a new approach to numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the interval the original problem is divided into outer and inner region problems a terminal boundary condition in the implicit form is derived then the outer region problem is solved as a two point boundary value problem tpbvp and an explicit terminal boundary condition is obtained in turn a new inner region problem is obtained and solved as a tpbvp using the explicit terminal boundary condition the proposed method is iterative on the terminal point of the inner region some numerical examples have been solved to demonstrate the applicability of the method

population extinction under bursty reproduction in a time modulated environment in recent years nondemographic variability has been shown to greatly affect dynamics of stochastic populations for example nondemographic noise in the form of a bursty reproduction process with an a priori unknown burst size or environmental variability in the form of time varying reaction rates have been separately found to dramatically impact the extinction risk of isolated populations in this work we investigate the extinction risk of an isolated population under the combined influence of these two types of nondemographic variation using the so called momentum space wentzel kramers brillouin wkb approach and accounting for the explicit time dependence in the reaction rates we arrive at a set of time dependent hamilton equations to this end we evaluate the population s extinction risk by finding the instanton of the time perturbed hamiltonian numerically whereas analytical expressions are presented in particular limits using various perturbation techniques we focus on two classes of time varying environments periodically varying rates corresponding to seasonal effects and a sudden decrease in the birth rate corresponding to a catastrophe all our theoretical results are tested against numerical monte carlo simulations with time dependent rates and also against a numerical solution of the corresponding time dependent hamilton equations

reducing the extinction risk of stochastic populations via nondemographic noise we consider nondemographic noise in the form of uncertainty in the reaction step size and reveal a dramatic effect this noise may have on the stability of self regulating populations employing the reaction scheme ma ka but allowing e g the product number k to be a priori unknown and sampled from a given distribution we show that such nondemographic noise can greatly reduce the population s extinction risk compared to the fixed k case our analysis is tested against numerical simulations and by using empirical data of different species we argue that certain distributions may be more evolutionary beneficial than others

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed