a linearization algorithm for nonsmooth minimization we present a readily implementable algorithm for finding stationary points for locally lipschitzian functions that are not necessarily convex or differentiable the algorithm is an extension to the nonconvex case of the aggregate subgradient method the user can impose an upper bound on storage and work per iteration whereas no such uniform bound exists for the algorithms due to lemarechal and mifflin on which our method is based the algorithm is globally convergent in the sense that all its accumulation points are stationary moreover the algorithm converges when the objective function happens to be convex this seems to be the first such result for descent methods for nonsmooth minimization

stationarity and superlinear convergence of an algorithm for univariate locally lipschitz constrained minimization this paper introduces an algorithm for minimizing a single variable locally lipschitz function subject to a like function being nonpositive the method combines polyhedral and quadratic approximation a new type of penalty technique and a safeguard in such a way as to give convergence to a stationary point the convergence is shown to be superlinear under somewhat stronger assumptions that allow both nonsmooth and nonconvex cases the algorithm can be an effective subroutine for solving line search subproblems called for by multivariable optimization algorithms

survival probability and order statistics of diffusion on disordered media we investigate the first passage time t j n to a given chemical or euclidean distance of the first j of a set of n ensuremath gg 1 independent random walkers all initially placed on a site of a disordered medium to solve this order statistics problem we assume that for short times the survival probability the probability that a single random walker is not absorbed by a hyperspherical surface during some time interval decays for disordered media in the same way as for euclidean and some class of deterministic fractal lattices this conjecture is checked by simulation on the incipient percolation aggregate embedded in two dimensions arbitrary moments of t j n are expressed in terms of an asymptotic series in powers of 1 mathrm ln n which is formally identical to those found for euclidean and some class of deterministic fractal lattices the agreement of the asymptotic expressions with simulation results for the two dimensional percolation aggregate is good when the boundary is defined in terms of the chemical distance the agreement worsens slightly when the euclidean distance is used

log periodic oscillations for diffusion on self similar finitely ramified structures under certain circumstances the time behavior of a random walk is modulated by logarithmic periodic oscillations using heuristic arguments we give a simple explanation of the origin of this modulation for diffusion on a substrate with two properties self similarity and finite ramification order on these media the time dependence of the mean square displacement shows log periodic modulations around a leading power law which can be understood on the basis of a hierarchical set of diffusion constants both the random walk exponent and the period of oscillations are analytically obtained for a pair of examples one is fractal and the other is nonfractal and confirmed by monte carlo simulations the last example shows that the anomalous diffusion can arise from substrates without holes of all sizes

necessary and sufficient optimality conditions for a class of nonsmooth minimization problems the purpose of this paper is to derive in a unified way second order necessary and sufficient optimality criteria for four types of nonsmooth minimization problems thediscrete minimax problem thediscrete l1 approximation the minimization of theexact penalty function and the minimization of theclassical exterior penalty function our results correct and supplement conditions obtained by various authors in recent papers

second order algorithms for generalized finite and semi infinite min max problems we present two second order algorithms one for solving a class of finite generalized min max problems and one for solving semi infinite generalized min max problems our algorithms make use of optimality functions based on second order approximations to the cost function and of corresponding search direction functions under reasonable assumptions we prove that both of these algorithms converge q superlinearly with rate at least 3 2 this paper is a continuation of e polak l qi and d sun comput optim appl 13 1999 pp 137 161

bayesian prediction for flowgraph models with covariates an application to bladder carcinoma statistical flowgraph models are an efficient tool to model multi state stochastic processes they support both frequentist and bayesian approaches inclusion of covariates is also available in this paper we propose an easy way to perform a bayesian approach with covariates results are presented with an application to bladder carcinoma data

the stability and hopf bifurcation analysis of a gene expression model abstract in this paper we investigate a model for gene expression unlike the models mathematically analyzed previously we have both transcriptional and translational time delays the stability and hopf bifurcation of the equilibrium point are investigated different to previous papers a multiple time scale mts technique is employed to calculate the normal form on the center manifold of system of delay differential equations which is much easier to implement in practice than the conventional method center manifold reduction our results show that when time delay is small the equilibrium is stable when it is at its critical value hopf bifurcation happens and while for very large value of time delay the oscillation sustains which has been confirmed by the published data and proved mathematically by using the global continuity of the hopf bifurcation in this paper

study of iterative methods through the cayley quadratic test many iterative methods for solving nonlinear equations have been developed recently the main advantage claimed by their authors is the improvement of the order of convergence in this work we compare their dynamical behavior on quadratic polynomials with the one of newton s scheme this comparison is defined in what we call cayley quadratic test cqt which can be used as a first test to check the efficiency of such methods moreover we make a brief insight in cubic polynomials

number of common sites visited by n random walkers we compute analytically the mean number of common sites w n t visited by n independent random walkers each of length t and all starting at the origin at t 0 in d dimensions we show that in the n d plane there are three distinct regimes for the asymptotic large t growth of w n t these three regimes are separated by two critical lines d 2 and d d c n 2n n 1 in the n d plane for d d c n w n t const as t exactly at the critical dimensions there are logarithmic corrections for d 2 we get w n t t ln t n while for d d c n w n t ln t for large t our analytical predictions are verified in numerical simulations

multiparticle trapping problem in the half line abstract a variation of rosenstock s trapping model in which n independent random walkers are all initially placed upon a site of a one dimensional lattice in the presence of a one sided random distribution with probability c of absorbing traps is investigated the probability survival probability n t that no random walker is trapped by time t for n 1 is calculated by using the extended rosenstock approximation this requires the evaluation of the moments of the number sn t of distinct sites visited in a given direction up to time t by n independent random walkers the rosenstock approximation improves when n increases working well in the range dt ln 2 1 c ln n d being the diffusion constant the moments of the time lifetime before any trapping event occurs are calculated asymptotically too the agreement with numerical results is excellent

order statistics of rosenstock s trapping problem in disordered media the distribution of times t j n elapsed until the first j independent random walkers from a set of n ensuremath gg 1 all starting from the same site are trapped by a quenched configuration of traps randomly placed on a disordered lattice is investigated in doing so the cumulants of the distribution of the territory explored by n independent random walkers s n t and the probability ensuremath phi n t that no particle of an initial set of n is trapped by time t are considered simulation results for the two dimensional incipient percolation aggregate show that the ratio between the nth cumulant and the nth moment of s n t is for large n i very large in comparison with the same ratio in euclidean media and ii almost constant the first property implies that in contrast with euclidean media approximations of the order higher than the standard zeroth order rosenstock approximation are required to provide a reasonable description of the trapping order statistics fortunately the second property which has a geometric origin can be exploited to build these higher order rosenstock approximations simulation results for the two dimensional incipient percolation aggregate confirm the predictions of our approach

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

self avoiding walks on self similar structures finite versus infinite ramification self avoiding walks saws generated by exact enumeration techniques are studied on the sierpinski carpet d 2 and on the sierpinski sponge d 3 also called sierpinski square lattices a detailed comparison of the results for saws on these infinitely ramified fractals to saws on finitely ramified sierpinski gaskets sierpinski triangular lattices on regular lattices and on the incipient percolation cluster is done providing insight into the behaviour of saws on ordered and disordered structures the saws on sierpinski square lattices are found to display a kind of intermediate behaviour sharing aspects of both saws on ordered and on fractal structures as a consequence a des cloizeaux relation does not seem to hold for this structure as opposed to its validity for saws on regular lattices on sierpinski triangular lattices and on the incipient percolation cluster

stability and bifurcation in a two variable delay model for circadian rhythm of neurospora crassa abstract a two variable delay model for circadian rhythm of neurospora crassa is considered in this paper conditions for the global attractivity of the unique positive equilibrium are given moreover hopf bifurcation and the global continuation of the hopf bifurcation branches are addressed through a global hopf bifurcation result

bifurcation analysis of a mathematical model for genetic regulatory network with time delays in this paper we aim to investigate the dynamics of a gene regulatory network which is a time delayed version of the model proposed by elowitz and leibler nature 403 2000 335 338 based on the normal form theory and center manifold reduction hopf bifurcations including the bifurcation direction and stability of the bifurcated periodic orbits are investigated we also discuss effects of transcriptional rate and time delay on the amplitude and period of the oscillation of the network it shows that variations of time delay or transcriptional rate can change the period and amplitude of the oscillation more precisely i the amplitude increases with small time delay while the change of amplitude is not sensitive to relatively large time delay however the robustness of amplitudes is not true any more for the case of using the transcriptional rate as parameter where amplitude always increases quickly and linearly with the transcriptional rate ii the period of oscillation increases as the time delay increases but it grows up initially as the transcriptional rate increases and then keeps unchanged to certain constant value which implies that the robustness of period to the transcriptional rate variations occurs our numerical simulations also support the theoretical conclusions namely both suggest that time delay and transcriptional rate can be used as control parameters in genetic regulatory networks

global and superlinear convergence of an algorithm for one dimensional minimization of convex functions this paper studies an algorithm for minimizing a convex function based upon a combination of polyhedral and quadratic approximation the method was given earlier but without a good specification for updating the algorithm s curvature matrix here for the case of onedimensional minimization we provide a specification that insures convergence even in cases where the curvature scalar tends to zero or infinity under mild additional assumptions we show that the convergence is superlinear

territory covered by n random walkers on fractal media the sierpinski gasket and the percolation aggregate we address the problem of evaluating the number s n t of distinct sites visited up to time t by n noninteracting random walkers all starting from the same origin in fractal media for a wide class of fractals of which the percolation cluster at criticality and the sierpinski gasket are typical examples we propose for large n and after the short time compact regime an asymptotic series for s n t analogous to that found for euclidean media s n t approximately s n t 1 delta here s n t is the number of sites volume inside a hypersphere of radius l ln n c 1 v where l is the root mean square chemical displacement of a single random walker and v and c determine how fast 1 gamma t l the probability that a given site at chemical distance l from the origin is visited by a single random walker by time t decays for large values of l l 1 gamma t l approximately exp c l l v for the fractals considered in this paper v d l w d l w 1 d l w being the chemical diffusion exponent the corrective term delta is expressed as a series in ln n n ln m ln n with n or 1 and 0 or m or n which is given explicitly up to n 2 this corrective term contributes substantially to the final value of s n t even for relatively large values of n

stability and hopf bifurcation in a model of gene expression with distributed time delays in this paper we consider the effect of distributed time delays on dynamics of a mathematical model of gene expression both the weak and strong delay kernels are discussed sufficient conditions for the local stability of the unique equilibrium are obtained taking the average delay as a bifurcation parameter we investigate the direction of the hopf bifurcations and the stability of the bifurcating periodic solutions by using the method of multiple time scales finally numerical simulation is carried out to illustrate our theoretical results it shows both subcritical and supercritical hopf bifurcations can happen

mathematical analysis of a biochemical oscillator with delay in this work we analyze a nonlinear delay differential equation dde model with negative feedback and constant time delay the model is constructed from a previously studied biochemical reaction network of gene transcription and protein synthesis linear analysis of the associated dde model gives a critical time delay beyond which a periodic motion is born in a hopf bifurcation the method of multiple scales is then used to analyze the nonlinear system to obtain expressions for the amplitude and frequency of oscillation as a function of the system parameters we use our closed form analytical expressions to study the importance of a well balanced ratio between synthesis and degradation rates in the existence of periodic solutions we show that our theoretical results are in agreement with numerical simulations and with experimental evidence found in the biological literature

a quadratic approximation method for minimizing a class of quasidifferentiable functions an unconstrained nonlinear programming problem with nondifferentiabilities is considered the nondifferentiabilities arise from terms of the form max f 1 x f n x which may enter nonlinearly in the objective function local convex polyhedral upper approximations to the objective function are introduced these approximations are used in an iterative method for solving the problem the algorithm proceeds by solving quadratic programming subproblems to generate search directions approximate line searches ensure global convergence of the method to stationary points the algorithm is conceptually simple and easy to implement it generalizes efficient variable metric methods for minimax calculations

a steffensen type method of two steps in banach spaces with applications this paper is devoted to the analysis of a steffensen type of two steps with order of convergence at least three the main advantage of this method is that it does not need to evaluate any frechet derivative or any bilinear operator the method includes extra parameters in the divided difference in order to ensure a good approximation to the first derivative in each iteration we prove using recurrence relations a semilocal convergence result in banach spaces and do a detailed study of the domain of parameters associated to this result finally some numerical results including differentiable and nondifferentiable operators are presented special attention is paid in the approximation of solutions of boundary problems using the multiple shooting method and in the approximation of a nonlinear model related with image processing

first passage properties of mortal random walks ballistic behavior effective reduction of dimensionality and scaling functions for hierarchical graphs we consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites the configuration comprising the graph the starting point of the walk and the locations of the trap sites is taken to be exactly self similar as one goes from one generation of the family to the next under these circumstances the total probability that the walker hits a trap is determined exactly as a function of the single step survival probability q of the mortal walker on the nth generation graph of the family this probability is shown to be given by the nth iterate of a certain scaling function or map q f q the properties of the map then determine in each case the behavior of the trapping probability the mean time to trapping the temporal scaling factor governing the random walk dimension on the graph and other related properties the formalism is illustrated for the cases of a linear hierarchical lattice and the sierpinski graphs in two and three euclidean dimensions we find an effective reduction of the random walk dimensionality due to the ballistic behavior of the surviving particles induced by the mortality constraint the relevance of this finding for experiments involving travel times of particles in diffusion decay systems is discussed