a direct approach to the exit problem this paper considers the problem of exit for a dynamical system driven by small white noise from the domain of attraction of a stable state a direct singular perturbation analysis of the forward equation is presented based on kramers approach in which the solution to the stationary fokker planck equation is constructed for a process with absorption at the boundary and a source at the attractor in this formulation the boundary and matching conditions fully determine the uniform expansion of the solution without resorting to external selection criteria for the expansion coefficients such as variational principles or the lagrange identity as in our previous theory the exit density and the mean first passage time to the boundary are calculated from the solution of the stationary fokker planck equation as the probability current density and as the inverse of the total flux on the boundary respectively as an application a uniform expansion is constructed for the escape rate in kramers problem o

efficiencies of experimental designs for an illness death model disease development is described by the progressive markov illness death model with one disease the efficiencies of four experimental designs are discussed the simple survival experiment serial sacrifice periodic diagnosis and complete observation the discussion is based on the semiparametric assumption of piecewise constant intensities simple survival experiments are shown to have very low efficiency serial sacrifice experiments to be moderately efficient and periodic diagnosis to be almost as efficient as complete observation an application to the analysis of the development of leukaemia in laboratory mice is given

fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

an index of tumorigenic potency a new index of tumorigenic potency is derived using a survival adjusted quantal response estimator given by bailer and portier 1988 biometrics 44 417 431 the small sample bias and mean squared error properties of this estimator are explored using computer simulations of animal carcinogenicity experiments and are compared to the characteristics of the td50 an estimator proposed by sawyer et al 1984 biometrics 40 27 40 it is observed that this new index tends to have smaller bias and mean squared error relative to the td50 over varying shapes of the tumor onset distribution levels of tumor lethality and levels of treatment lethality

surmounting fluctuating barriers a simple model in discrete time the escape rate is calculated for a piecewise linear map in the presence of two additive weak noises gaussian white noise thermal noise and ornstein uhlenbeck noise barrier fluctuations a transition state theory yields the exact escape rate in the weak noise limit by including the dominant finite noise corrections we find very good agreement with numerical simulations of particular interest is the behavior of the escape rate k ensuremath tau as a function of the correlation time ensuremath tau of the ornstein uhlenbeck noise we show that the qualitative behavior of k ensuremath tau at any fixed thermal noise intensity is the same as in the absence of thermal noise whether k ensuremath tau exhibits a local maximum resonant activation is determined by the detailed ensuremath tau dependence of the colored noise intensity e g resonant activation is found at a correlation time of order 1 if the integral autocorrelation of the colored noise is kept ensuremath tau independent but not if the variance is kept ensuremath tau independent in the continuous time limit neither of these cases shows resonant activation

a stochastic nonlinear schrodinger equation with multiplicative noise we study a conservative stochastic nonlinear schrodinger equation with a multiplicative noise we show the global existence and uniqueness of square integrable solutions for subcritical nonlinearities the critical exponent being the same in dimension 1 or 2 as the critical exponent of the deterministic equation

smoluchowski kramers approximation and large deviations for infinite dimensional nongradient systems with applications to the exit problem in this paper we study the quasi potential for a general class of damped semilinear stochastic wave equations we show that as the density of the mass converges to zero the infimum of the quasi potential with respect to all possible velocities converges to the quasi potential of the corresponding stochastic heat equation that one obtains from the zero mass limit this shows in particular that the smoluchowski kramers approximation is not only valid for small time but in the zero noise limit regime can be used to approximate long time behaviors such as exit time and exit place from a basin of attraction

noise induced escape from bifurcating attractors symplectic approach in the weak noise limit the effect of noise is studied in one dimensional maps undergoing transcritical tangent and pitchfork bifurcations the attractors of the noiseless map become metastable states in the presence of noise in the weak noise limit a symplectic two dimensional map is associated with the original one dimensional map the consequences of their noninvertibility on the phase space structures are discussed heteroclinic orbits are identified which play a key role in the determination of the escape rates from the metastable states near bifurcations the critical slowing down justifies the use of a continuous time approximation replacing maps by flows which allows the analytic calculation of the escape rates this method provides the universal scaling behavior of the escape rates at the bifurcations

surmounting oscillating barriers path integral approach for weak noise we consider the thermally activated escape of an overdamped brownian particle over a potential barrier in the presence of periodic driving a time dependent path integral formalism is developed which allows us to derive asymptotically exact weak noise expressions for both the instantaneous and the time averaged escape rate our results comprise a conceptually different systematic treatment of the rate prefactor multiplying the exponentially leading arrhenius factor moreover an estimate for the deviations at finite noise strengths is provided and a supersymmetry type property of the time averaged escape rate is verified for piecewise parabolic potentials the rate expression can be evaluated in closed analytical form while in more general cases as exemplified by a cubic potential an action integral remains to be minimized numerically our comparison with very accurate numerical results demonstrates an excellent agreement with the theoretical predictions over a wide range of driving strengths and driving frequencies

strong enhancement of noise induced escape by nonadiabatic periodic driving due to transient chaos we have found a mechanism by which a moderately weak nonadiabatic periodic driving may significantly facilitate noise induced interwell transitions in an underdamped multiwell system the mechanism is associated with the onset of a homoclinic tangle in the noise free system if the ratio of the driving amplitude a to the damping ensuremath gamma exceeds a critical value ensuremath sim 1 then the basins of attraction of the linear responses related to different wells are mixed in a complex manner in some layer associated with the separatrix of the undriven nondissipative system and the minimal energy in such layer is lower than the top of the barrier thus the energy to which the system needs to be activated by the noise to be able to make a transition is lower than the top of the barrier

estimating the tumour onset distribution in animal carcinogenesis experiments summary in most survival sacrifice experiments for the detection and quantification of risks from chronic exposure to chemical agents the onset of the condition of interest is not clinically observable cancer is typically such an event in this paper an approximate maximum likelihood method is proposed for parametric estimation of the distribution of unobservable tumour onset times in the presence of competing risks when cause ofdeath information is unavailable

nonparametric estimation of lifetime and disease onset distributions from incomplete observations in this paper we derive and investigate nonparametric estimators of the distributions of lifetime and time until onset associated with an irreversible disease that is detectable only at death the nonparametric maximum likelihood solution requires an iterative algorithm an alternative though closely related pair of estimators for the lifetime and onset distributions exists in closed form these estimators are the familiar kaplan meier estimator and an isotonic regression estimator respectively first order approximations provide variance estimators the proposed methods generalize and shed additional light on the constrained estimators presented by kodell shaw and johnson 1982 biometrics 38 43 58 data from an animal experiment illustrate the techniques

solution of the boundary value problem for optimal escape in continuous stochastic systems and maps topologies of invariant manifolds and optimal trajectories are investigated in stochastic continuous systems and maps a topological method is introduced that simplifies the solution of boundary value problems the activation energy is calculated as a function of a set of parameters characterizing the initial conditions of the escape path the method is applied explicitly to compute the optimal escape path and the activation energy for a variety of dynamical systems and maps

nonparametric methods for survival sacrifice experiments in many carcinogenicity studies the time to disease occurrence is not clinically observable a survival sacrifice experiment is considered for nonparametric inference about the rate of disease occurrence a multistate model for disease development and death is considered and an algorithm of the em type for maximum likelihood estimation is obtained questions of identifiability and estimability are addressed under the model interval hazards for disease occurrence are identifiable for intervals defined by the sacrifice times a score test is developed appropriate for the comparison of two groups with respect to disease development without need of any assumption concerning lethality of the disease concerned

large deviations for stationary measures of stochastic nonlinear wave equation with smooth white noise we prove the freidlin wentzell type large deviations principle for the family of stationary measures of stochastic nonlinear wave nlw equation with white noise we do not assume that the limiting equation possesses a unique equilibrium and do not impose roughness on the noise this allows to provide the first such result in the pde setting

efficiency robust score tests for rodent tumourigenicity experiments summary the standard statistical test for assessing adverse effects in rodent tumourigenicity experiments is based on a time adjusted analysis of lethal and incidental tumours this test is shown to arise as the sum of two score statistics for fatal and incidental lesions respectively the former optimal against extreme value hazard alternatives and the latter optimal against logistic prevalence alternatives in terms of asymptotic relative efficiency in the pitman sense an efficiency robust test is developed which retains near full efficiency in the extreme value logistic situation as well as in other situations where the standard test may be only 70 efficient rodent tumourigenicity experiments are used to determine whether or not a particular agent results in an increase in tumour occurrence in animals exposed to this substance throughout their lifetimes as compared to unexposed controls the tumour status of an individual animal is normally determined at the time of death since many animals will die either from natural mortality or as a result of tumour development prior to the end of the study period information on tumourigenicity will accrue throughout the course of the experiment typically lasting about two years animals surviving to the end of the experiment are sacrificed and subjected to the same pathological evaluation as those dying prior to the termination time peto et al 1980 have proposed life table analysis of the lethal or fatal tumours and a prevalence analysis of the nonlethal or incidental tumours in the present paper we show that when the substance under study affects the occurrence of fatal and incidental lesions in a similar fashion their test can be formally derived as the sum of two score statistics for fatal and incidental tumours respectively the former statistic is the score test under extreme value hazard alternatives whereas the latter statistic is optimal against logistic prevalence alternatives because the true form of the underlying hazard and prevalence functions is unknown we develop in 3 an efficiency robust version of this test this is done by maximizing the minimum asymptotic relative efficiency within a finite class of hazard and prevalence functions applications of the maximin efficiency robust test are discussed in 4

large deviations for a reaction diffusion equation with non gaussian perturbations in this paper we establish a large deviations principle for the non gaussian stochastic reaction diffusion equation srde partial t nu varepsilon mathscr l nu varepsilon f x nu varepsilon varepsilon sigma x nu varepsilon ddot w tx as a random perturbation of the deterministic rde partial t nu 0 mathscr l nu 0 f x nu 0 here the space variable takes values on the unit circle s 1 and mathscr l is a strongly elliptic second order operator with constant coefficients the functions f and sigma are sufficiently regular so that there is a unique solution to the above srde for any continuous initial condition we also assume that there are positive constants m and m such that m leq sigma x y leq m for all x in s 1 and all y in mathbb r the perturbation ddot w tx is the formal derivative of a brownian sheet it is known that if the initial condition is continuous then the solution will also be continuous and moreover if the initial condition is assumed to be holder continuous of exponent kappa for some 0 kappa frac 1 2 then the solution will be holder continuous of exponent kappa 2 as a function of t x in this paper we establish the large deviations principle for nu varepsilon in the holder norm of exponent kappa 2 when the initial condition is holder continuous of exponent kappa for any 0 kappa frac 1 2 and when the initial condition is assumed only to be continuous we establish the large deviations principle for nu varepsilon in the supremum norm moreover we prove that these large deviations principles are uniform with respect to the initial condition

computing discrete logarithms in high genus hyperelliptic jacobians in provably subexponential time we provide a subexponential algorithm for solving the discrete logarithm problem in jacobians of high genus hyperelliptic curves over finite fields its expected running time for instances with genus g and underlying finite field fq satisfying g log q for a positive constant is given by o e f 1 3 2 3 2 o 1 g log q log g log q the algorithm works over any finite field and its running time does not rely on any unproven assumptions

on the effect of a noise on the solutions of the focusing supercritical nonlinear schrodinger equation we investigate the influence of a random perturbation of white noise type on the finite time blow up of solutions of a focusing supercritical nonlinear schrodinger equation we prove that contrary to the deterministic case any initial data gives birth to a solution which develops singularities moreover the singularities appear immediately we use a stochastic generalization of the variance identity and a control argument

approximation of quasi potentials and exit problems for multidimensional rde s with noise we deal with a class of reaction diffusion equations in space dimension d 1 perturbed by a gaussian noise w t which is white in time and colored in space we assume that the noise has a small correlation radius so that it converges to the white noise w t as 0 by using arguments of convergence we prove that under suitable assumptions the quasi potential v converges to the quasi potential v corresponding to space time white noise in spite of the fact that the equation perturbed by space time white noise has no solution we apply these results to the asymptotic estimate of the mean of the exit time of the solution of the stochastic problem from a basin of attraction of an asymptotically stable point for the unperturbed problem

a dose response test via closed form solutions for constrained mles in survival sacrifice experiments in most survival sacrifice experiments in animal carcinogenicity studies the onset of the tumour of interest is not clinically observable due to the complexity of constraints for a biological justification recently developed methods for estimating the tumour onset function and tumour specific survival function employ computer intensive numerical solutions in this paper closed form solutions for nonparametric maximum likelihood estimators are derived under explicit and implicit inequality constraints obtained from the monotonicity of the survival functions our methods do not require cause of death information the proposed methods can be used to estimate the tumour onset function and the survival function of the tumour of interest we use the proposed estimators for the development of our new dose response trend test a modification of the poly k test is made by replacing the time at risk weight to a function of the tumour onset survival function the weighted least square regression method is applied to the estimated survival functions in order to construct a dose response trend test a simulation study is conducted to evaluate the performance of the proposed test and compare it with existing trend tests a real example is used to illustrate the methods

singularities of fluctuational paths for an overdamped two well system driven by white noise abstract noise induced escape from a metastable state in an overdamped double well system is investigated with the wkb approximation employed the pattern of fluctuational paths is given by using the action plot which gives rise to mpeps the pattern is divided into various regions where fluctuational paths inside show qualitatively distinguished characteristics the global quasi potential is calculated and its value at the saddle on the boundary agrees with the one obtained by action plot we show a contour and 3d plot of the quasi potential to illustrate the relative stabilities of metastable states the bifurcations of singularities as the system parameter varies are analyzed by the second order term of the wkb expansion in the vicinity of x axis its asymptotic behavior determines the positions of cusps and furthermore its evolution turns out to delineate the twisting structure of the lagrangian manifold

fluctuational transitions across different kinds of fractal basin boundaries we study fluctuational transitions in discrete and continuous dynamical systems that have two coexisting attractors in phase space separated by a fractal basin boundary which may be either locally disconnected or locally connected theoretical and numerical evidence is given to show that in each case the transition occurs via a unique accessible point on the boundary both in discrete systems and in flows the complicated structure of the escape paths inside the locally disconnected fractal basin boundary is determined by a hierarchy of homoclinic points the interrelation between the mechanism of transitions and the hierarchy is illustrated by consideration of fluctuational transitions in dynamical systems demonstrating fractal fractal basin boundary metamorphosis at some value of a control parameter the most probable escape path from an attractor which can be either regular or chaotic is found for each type of boundary using both statistical analysis of fluctuational trajectories and the hamiltonian theory of fluctuations

effects of treatment induced mortality and tumor induced mortality on tests for carcinogenicity in small samples statistical tests of carcinogenicity are shown to have varying degrees of robustness to the effects of mortality mortality induced by two different mechanisms is studied mortality due to the tumor of interest and mortality due to treatment independent of the tumor the two most commonly used tests the life table test and the cochran armitage linear trend test are seen to be highly sensitive to increases in treatment lethality using small sample simulations increases in tumor lethality are seen to affect the performance of commonly used prevalence tests such as logistic regression a simple survival adjusted quantal response test appears to be the most robust of all the procedures considered

crossing the quasi threshold manifold of a noise driven excitable system we consider the noise induced escapes in an excitable system possessing a quasi threshold manifold along which there exists a certain point of minimal quasi potential in the weak noise limit the optimal escaping path turns out to approach this particular point asymptotically making it analogous to an ordinary saddle numerical simulations are performed and an elaboration on the effect of small but finite noise is given which shows that the ridges where the prehistory probability distribution peaks are located mainly within the region where the quasi potential increases gently the cases allowing anisotropic noise are discussed and we found that varying the noise term in the slow variable would dramatically raise the whole level of quasi potentials leading to significant changes in both patterns of optimal paths and exit locations

computing discrete logarithms in real quadratic congruence function fields of large genus the discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems recently real quadratic congruence function fields were used to construct a public key distribution system the security of this public key system is based on the difficulty of a discrete logarithm problem in these fields in this paper we present a probabilistic algorithm with subexponential running time that computes such discrete logarithms in real quadratic congruence function fields of sufficiently large genus this algorithm is a generalization of similar algorithms for real quadratic number fields

topological features of large fluctuations to the interior of a limit cycle received 29 october 1996 we investigate the pattern of optimal paths along which a dynamical system driven by weak noise moves with overwhelming probability when it fluctuates far away from a stable state our emphasis is on systems that perform self sustained periodic vibrations and have an unstable focus inside a stable limit cycle we show that in the vicinity of the unstable focus the flow field of optimal paths generically displays a pattern of singularities in particular it contains a switching line that separates areas to which the system arrives along optimal paths of topologically different types the switching line spirals into the focus and has a self similar structure depending on the behavior of the system near the focus it may be smooth or have finite length branches our results are based on an analysis of the topology of the lagrangian manifold for an auxiliary purely dynamical problem that determines the optimal paths we illustrate our theory by studying both theoretically and numerically a van der pol oscillator driven by weak white noise s1063 651x 97 02703 7 many nonlinear dissipative systems display stable periodic oscillations i e their state spaces contain stable limit cycles 1 examples include lasers 2 radiofrequency generators 1 a chemical 3 a and biological 3 b systems in many cases the vibrations are nonsinusoidal as in a multimode laser with strong mode coupling 4 passive optically bistable elements 5 and various sorts of engineering structure 6 fluctuations in periodically oscillating systems play an extremely important role they eventually destroy periodicity of the self sustained oscillations so that the oscillation phase becomes random and a stationary probability distribution in the state space of the system is formed 7 8 this distribution reaches a maximum at the cycle does not vary much along the cycle and falls off rapidly in directions transverse to the cycle the shape of the maximum depends on the character of small transverse fluctuations away from the cycle if the fluctuations are due to the system being perturbed by weak noise of intensity d the transverse falloff will usually be approximately gaussian 7 9 with standard deviation proportional to d 1 2 as d 0

stochastic nonlinear schrodinger equation on an upper right quarter plane with dirichlet random boundary in this paper we study the nonhomogeneous stochastic initial boundary value problem for the nonlinear schrodinger equation on an upper right quarter plane with random dirichlet boundary conditions the main novelty of this work is a convenient framework for the analysis of such equations excited by the wiener additive noise on the boundary our approach allows us to show the local existence and uniqueness of solutions in the space h2 the basic properties of the solutions such as the continuity and the boundary layer behavior are also studied using the ito calculus despite several technical difficulties we believe that the approach developed in this paper can be applied to the case of a large class of noise including fractional wiener space time white noise homogeneous noise and levy noise

calculation of carcinogenic potency from long term animal carcinogenesis experiments an index of carcinogenic potency for chemicals tested in chronic animal experiments is described by analogy with the well known lethal dose 50 ld50 of quantal bioassay a tumorigenic dose 50 td50 may be defined in the absence both of tumors in the control group and of intercurrent deaths as that daily dose of chemical which gives 50 of the test animals tumors by some fixed age tumors in the control zero dose group are handled exactly as for the ld50 and intercurrent deaths are handled by life table methods nonparametric procedures are developed for estimating the td50 and for constructing confidence intervals these are based on likelihoods which assume that the tumor hazard is linear in dose

noise induced escape in an excitable system we consider the stochastic dynamics of escape in an excitable system the fitzhugh nagumo fhn neuronal model for different classes of excitability we discuss first the threshold structure of the fhn model as an example of a system without a saddle state we then develop a nonlinear nonlocal stability approach based on the theory of large fluctuations including a finite noise correction to describe noise induced escape in the excitable regime we show that the threshold structure is revealed via patterns of most probable optimal fluctuational paths the approach allows us to estimate the escape rate and the exit location distribution we compare the responses of a monostable resonator and monostable integrator to stochastic input signals and to a mixture of periodic and stochastic stimuli unlike the commonly used local analysis of the stable state our nonlocal approach based on optimal paths yields results that are in good agreement with direct numerical simulations of the langevin equation

smooth ideals in hyperelliptic function fields recently several algorithms have been suggested for solving the discrete logarithm problem in the jacobians of high genus hyperelliptic curves over finite fields some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense we explicitly show how these density results can be derived all proofs are purely combinatorial and do not exploit analytic properties of generating functions

elliptic curve cryptosystems we discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field these elliptic curve cryptosystems may be more secure because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem especially over gf 2 we discuss the question of primitive points on an elliptic curve modulo p and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point

tumour incidence rate alternatives and the cause of death test for carcinogenicity summary lagakos louis 1988 and burnett krewski bleuer 1989 show that the cause ofdeath test of peto et al 1980 for carcinogenicity studies can be viewed as the approximate score test of a one dimensional joint hypothesis about tumour prevalences and the rates of death from tumour we relate this joint prevalence death rate hypothesis to hypotheses about differences in tumour incidence rates for two sample tests we show that when the control group tumour incidence is low when tumours are highly lethal or when the alternative is local the prevalence death rate family of alternatives is approximately equivalent to a proportional tumour incidence rate family more generally the prevalencedeath rate null hypothesis is always equivalent to a hypothesis of equal tumour incidence rates and each member of the family of prevalence death rate alternatives requires the tumour incidence rates not to cross using computer simulation we show that when the model does not hold the test of peto et al can be anti conservative and have poor power we discuss the model s implications for the interpretation of tests of regression hypotheses

quasipotentials for simple noisy maps with complicated dynamics the theory of nonequilibrium potentials or quasipotentials is a physically motivated approach to small random perturbations of dynamical systems leading to exponential estimates of invariant probabilities and mean first exit times in the present article we develop the mathematical foundation of this theory for discrete time systems following and extending the work of freidlin and wentzell and kifer we discuss strategies for calculating and estimating quasipotentials and show their application to one dimensionals unimodal maps the method proves to be especially suited for describing the noise scaling behavior of invariant probabilities e g for the map occurring as the limit of the feigenbaum period doubling sequence we show that the method allows statements about the scaling behavior in the case of localized noise too which does not originally lie within the scope of the quasipotential formalism

noise induced escape through a fractal basin boundary we study noise induced escape within a discrete dynamical system that has two co existing chaotic attractors in phase space separated by a locally disconnected fractal basin boundary it is shown that escape occurs via a unique accessible point on the fractal boundary the structure of escape paths is determined by the original saddles forming the homoclinic structure of the system and by their hierarchical interrelations

noisy precursors of nonlinear instabilities this paper studies the effect of external noise on systems displaying nonlinear instabilities of periodic orbits each class of instability is found to have its own characteristic signature as displayed by the power spectrum results are derived for each of the codimension one instabilities familiar from bifurcation theory

enhancement of noise induced escape through the existence of a chaotic saddle we study the noise induced escape process in a prototype dissipative nonequilibrium system the ikeda map in the presence of a chaotic saddle embedded in the basin of attraction of the metastable state we find the novel phenomenon of a strong enhancement of noise induced escape this result is established by employing the theory of quasipotentials our finding is of general validity and should be experimentally observable

quasipotential approach to critical scaling in noise induced chaos when a dynamical system exhibits transient chaos and a nonchaotic attractor as in a periodic window noise can induce a chaotic attractor in particular when the noise amplitude exceeds a critical value the largest lyapunov exponent of the attractor of the system starts to increase from zero while a scaling law for the variation of the lyapunov exponent with noise was uncovered previously it is mostly based on numerical evidence and a heuristic analysis this paper presents a more general approach to the scaling law one based on the concept of quasipotentials besides providing deeper insights into the problem of noise induced chaos the quasipotential approach enables previously unresolved issues to be addressed the fractal properties of noise induced chaotic attractors and applications to biological systems are also discussed

a class of rank test procedures for censored survival data summary a class of linear rank statistics is proposed for the k sample problem with rightcensored survival data the class contains as special cases the log rank test mantel 1966 cox 1972 and a test essentially equivalent to peto peto s 1972 generalization of the wilcoxon test martingale theory is used to establish asymptotic normality of test statistics under the null hypotheses considered and to derive expressions for asymptotic relative efficiencies under contiguous sequences of alternative hypotheses a class of distributions is presented which corresponds to the class of rank statistics in the sense that for each distribution there is a statistic with some optimal properties for detecting location alternatives from that distribution some monte carlo results are displayed which present small sample behaviour

explict bounds and heuristics on class numbers in hyperelliptic function fields in this paper we provide tight estimates for the divisor class number of hyperelliptic function fields we extend the existing methods to any hyperelliptic function field and improve the previous bounds by a factor proportional to g with the help of new results we thus obtain a faster method of computing regulators and class numbers furthermore we provide experimental data and heuristics on the distribution of the class number within the bounds on the class number these heuristics are based on recent results by katz and sarnak our numerical results and the heuristics imply that our approximation is in general far better than the bounds suggest

tests for differences in tumor incidence based on animal carcinogenesis experiments abstract in animal carcinogenesis studies statistical methods that compare the production of occult tumors should compare tumor incidence rates in general the incidence rate cannot be identified even approximately without both survival and sacrifice information if information from frequent sacrifices is available however a function that approximates the incidence rate can be identified without assumptions about the relationship between tumor incidence and death some earlier proposals that based comparisons on tumor prevalence rates and other functions also compared tumor incidence rates under special conditions but these tests can be biased when the conditions are violated tests that compare functions approximating tumor incidence rates can be developed using survival sacrifice information but these tests may be seriously inefficient in special cases in which sacrifices are not necessary to identify the tumor incidence rate

a penalized likelihood approach for a progressive three state model with censored and truncated data application to aids summary we consider the estimation of the intensity and survival functions for a continuous time progressive three state semi markov model with intermittently observed data the estimator of the intensity function is defined nonparametrically as the maximum of a penalized likelihood we thus obtain smooth estimates of the intensity and survival functions this approach can accommodate complex observation schemes such as truncation and interval censoring the method is illustrated with a study of hemophiliacs infected by hiv the intensity functions and the cumulative distribution functions for the time to infection and for the time to aids are estimated covariates can easily be incorporated into the model

noise induced failures of chaos stabilization large fluctuations and their control noise induced failures in the stabilization of an unstable orbit in the one dimensional logistic map are considered as large fluctuations from a stable state the properties of the large fluctuations are examined by determination and analysis of the optimal path and the optimal fluctuational force corresponding to the stabilization failure the problem of controlling noise induced large fluctuations is discussed and methods of control have been developed

nonparametric estimation of the distribution of time to onset for specific diseases in survival sacrifice experiments this paper concerns the analysis of an animal survival sacrifice experiment designed to investigate the incidence of a particular disease of interest the disease is assumed to be irreversible and detectable only at death for example by a necropsy each observation can be of one of three types i death caused by the disease ii death from a competing cause such as sacrifice with the disease present or iii death with the disease absent a two dimensional em algorithm is proposed for the nonparametric maximum likelihood estimation of the distributions of the time to onset and of the time to death from the disease these can be compared with nonparametric estimators recently proposed by kodell shaw and johnson 1982 biometrics 38 43 58 and by dinse and lagakos 1982 biometrics 38 921 932 a slight modification of the algorithm permits the construction of likelihood based interval estimates of quantiles of the distributions some extensions and generalizations are indicated

a measure of tumorigenic potency incorporating dose response shape many researchers have considered the problem of ranking chemical agents based on their carcinogenic potency sawyer et al 1984 biometrics 40 27 40 proposed a carcinogenic potency estimate that incorporates both intercurrent mortality and background tumor rates since then many authors have either generalized the method outlined by sawyer et al or developed their own method based on a slightly different adjustment for treatment related changes in survival none of these methods however has estimated the shape of the dose response curve and incorporated such an estimate in potency estimation in this manuscript a measure of tumorigenic potency is proposed that utilizes the estimated shape of the dose response relationship in addition to estimated dose effects in order to rank chemicals on the basis of carcinogenic risk comparison of this new measure to that of sawyer et al is done using a large database of animal carcinogenicity experiments from the national cancer institute and the national toxicology program

nonparametric joint estimators for disease resistance and survival functions in survival sacrifice experiments the nonparametric estimators proposed in this paper simultaneously estimate the disease resistance function s t for the distribution of time to occurrence of disease and the survival function p t for the distribution of time to death caused by the disease the estimators are applicable to animal survival data when the disease of interest is irreversible and the cause of death of each animal can be determined diseases need not be classified as universally lethal or universally nonlethal and time to occurrence of the disease of interest need not be observable directly it is shown that the proposed estimators are maximum likelihood estimators under the condition that s t p t is monotonically decreasing application of the derived estimators is illustrated with experimental survival sacrifice data and results of a simulation study of the estimators performance are reported

a probabilistic factorization algorithm with quadratic forms of negative discriminant we propose a probabilistic algorithm for factorization of an integer n with run time expvlog n log log n v54 1 asymptotically our algorithm will be as fast as the wellknown factorization algorithm of morrison and brillhart the latter algorithm will fail in several cases and heuristic assumptions are needed for its run time analysis our new algorithm will be analyzed under the assumption of the extended riemann hypothesis and it will be of las vegas type on input n the new algorithm will factor n with probability 12 in case of prime n the algorithm will prove the primality of n with probability 2 introduction until the last decade the centuries old problem of factoring integers was mainly a problem for specialists worldwide interest in factoring integers increased dramatically in 1978 when rivest shamir and adleman 32 published their public key cryptosystem whose security relies on the fact that some large integers are hard to factor gauss 9 already discovered a close connection between the factorization of a natural number n and the theory of quadratic forms of discriminant 4n now quadratic forms are one of the most important tools for factoring integers examples for efficient factorization algorithms are among others the algorithms of morrison and brillhart 26 and lenstra and schnorr 22 the former works with the continued fraction expansion of fn which is closely related to the theory of quadratic forms of discriminant 4n see 20 while the latter works with quadratic forms of discriminant 4n at present the most efficient factorization algorithm is the quadratic sieve algorithm see 29 which can also be expressed in terms of quadratic forms for an overview of modem factorization algorithms we refer to the papers of guy 10 monier 25 and pomerance 29 a deeper understanding of the theory of quadratic forms is of great importance for the analysis of modem factorization algorithms in this paper we shall only deal with the theory of quadratic forms of negative discriminant which is considerably more simple than the theory of forms of positive discriminants using this theory we obtain a probabilistic factorization algorithm with run time exp vlog n log log n v4 in this paper we denote by log x the natural logarithm of x we have 5 4 1 118 received may 17 1985 revised june 12 1986 1980 mathematics subject classification primary 10a30 68c25 1987 american mathematical society 0025 5718 87 1 00 25 per page

a scaling theory of bifurcations in the symmetric weak noise escape problem we consider two dimensional overdamped double well systems perturbed by white noise in the weak noise limit the most probable fluctuational path leading from either point attractor to the separatrix the most probable escape path or mpep must terminate on the saddle between the two wells however as the parameters of a symmetric double well system are varied a unique mpep may bifurcate into two equally likely mpeps at the bifurcation point in parameter space the activation kinetics of the system become non arrhenius we quantify the non arrhenius behavior of a system at the bifurcation point by using the maslov wkb method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix the approximation is a formal asymptotic solution of the smoluchowski equation our construction relies on a new scaling theory which yields critical exponents describing weak noise behavior at the bifurcation point near the saddle

noise induced escape from attractors the author deals with several aspects of kautz 1987 1988 minimum principle for the noise induced escape from attractors he presents a slightly simpler derivation points out a connection with lyapunov exponents and invariant manifolds and generalises to chaotic maps

real and imaginary quadratic representations of hyperelliptic function fields a hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field if at least one of the rational prime divisors is rational over the field of constants then it also can be represented as an imaginary quadratic extension of the rational function field the arithmetic in the divisor class group can be realized in the second case by cantor s algorithm we show that in the first case one can compute in the divisor class group of the function field using reduced ideals and distances of ideals in the orders involved furthermore we show how the two representations are connected and compare the computational complexity

random perturbations of reaction diffusion equations the quasi deterministic approximation on considere des champs aleatoires u e t x u 1 e t x u n e t x definis comme les solutions d un systeme d equations aux derivees partielles u k e t l k u k e f k x u 1 e u n e e k t x les l k sont des operateurs elliptiques lineaires d ordre 2 k sont des champs gaussiens bruit blanc

estimating tumor incidence rates in animal carcinogenicity experiments tumor incidence is the primary measure of carcinogenesis this article focuses on estimating time dependent incidence rates in animal experiments with few sacrifices when the context of observation is known for none or all of the animals dying with the tumor of interest previous results are obtained under relaxed assumptions the link with existing semiparametric and nonparametric procedures based on latent failure times is exploited by using these methods to compute maximum likelihood estimates of the incidence rates without introducing latent random variables nonparametric estimators that are appropriate when all contexts of observation are known are generalized to the case in which the contexts of observation are unknown for a subset of the tumor bearing animals

exit from a basin of attraction for stochastic weakly damped nonlinear schrodinger equations we consider weakly damped nonlinear schrodinger equations perturbed by a noise of small amplitude the small noise is either complex and of additive type or real and of multiplicative type it is white in time and colored in space zero is an asymptotically stable equilibrium point of the deterministic equations we study the exit from a neighborhood of zero invariant under the flow of the deterministic equation in l 2 or in h 1 due to noise large fluctuations from zero occur thus on a sufficiently large time scale exit from these domains of attraction occur a formal characterization of the small noise asymptotic of both the first exit times and the exit points is given

limiting exit location distributions in the stochastic exit problem consider a two dimensional continuous time dynamical system with an attracting fixed point s if the deterministic dynamics are perturbed by white noise random perturbations of strength epsilon the system state will eventually leave the domain of attraction omega of s we analyze the case when as epsilon to0 the exit location on the boundary partial omega is increasingly concentrated near a saddle point h of the deterministic dynamics we show using formal methods that the asymptotic form of the exit location distribution on partial omega is generically non gaussian and asymmetric and classify the possible limiting distributions a key role is played by a parameter mu equal to the ratio lambda s h lambda u h of the stable and unstable eigenvalues of the linearized deterministic flow at h if mu 1 then the exit location distribution is generically asymptotic as epsilon to0 to a weibull distribution with shape parameter 2 mu on the cal o epsilon mu 2

basin boundary metamorphoses changes in accessible boundary orbits abstract basin boundaries sometimes undergo sudden metamophoses these metamorphoses can lead to the conversion of a smooth basin boundary to one which is fractal or else can cause a fractal or else can basin boundary to suddenly jump in size and change its character although remaining fractal for an invertible map in the plane there may be an infinite number of saddle periodic orbits in a basin boundary that is fractal nonetheless we have found that typically only one of them can be reached or accessed directly from a given basin the other periodic orbits are buried beneath infinitely many layers of the fractal structure of the boundary the boundary metamorphoses which we investigate are characterized by a sudden replacement of the basin boundary s accessible orbit

hyperbolic periodic orbits in nongradient systems and small noise induced metastable transitions abstract small noise can induce rare transitions between metastable states which can be characterized by maximum likelihood paths mlps nongradient systems contrast gradient systems in that mlp does not have to cross the separatrix at a saddle point but instead possibly at a point on a hyperbolic periodic orbit a numerical approach for identifying such unstable periodic orbits is proposed based on string method in a special class of nongradient systems orthogonal type there are provably local mlps that cross such saddle point or hyperbolic periodic orbit and the separatrix crossing location determines the associated local maximum of transition rate in general cases however the separatrix crossing may not determine a unique local maximum of the rate as we numerically observed a counter example in a sheared 2d space allen cahn spde it is a reasonable conjecture that there are always local mlps associated with each attractor on the separatrix such as saddle point or hyperbolic periodic orbit our numerical experiments did not disprove so

decay of metastable states with discrete dynamics we consider the escape from invariant sets of one dimensional piecewise linear maps which are additively disturbed by weak gaussian white noise the escape rates from point attractors and from strange invariant sets in the vicinity of the crisis at fully developed chaos are analytically determined and compared with results from numerical simulations both situations are combined resulting in a model with a point attractor which has a strange invariant set as basin boundary numerically a nonexponential decay of the attractor is found that can be described by a markovian three state model with transition rates known from the previous analysis

use of historical controls in time adjusted trend tests for carcinogenicity we develop a method for incorporating historical control information into time adjusted tests for dose effects in carcinogenicity studies after discretizing the time scale we use a multinomial distribution to model the number of animals dying with tumor in each interval data from past studies are used to estimate the parameters characterizing the prior a score test derived from the resulting dirichlet multinomial generalizes the test of tarone 1982 biometrics 38 215 220 and reduces in the limit to the log rank test in the case of a diffuse prior the methodology is illustrated with data from a study of the fire retardant 2 2 bis bromomethyl 1 3 propanediol