spectral averaging techniques for jacobi matrices with matrix entries a jacobi matrix with matrix entries is a self adjoint block tridiagonal matrix with invertible blocks on the off diagonals averaging over boundary condi tions leads to explicit formulae for the averaged spectral measure which can potentially be useful for spectral analysis furthermore another variant of spectral averaging over coupling constants for these operators is presented

sturm liouville operators in the half axis with local perturbations we give conditions which imply equivalence of the lebesgue measure with respect to a measure generated as an average of spectral measures corresponding to sturm liouville operators in the half axis we apply this to prove that some spectral properties of these operators hold for large sets of boundary conditions if and only if they hold for large sets of positive local perturbations

adjustments to the mantel haenszel chi square statistic and odds ratio variance estimator when the data are clustered dependence between observations on a dichotomous variable renders invalid the usual chi square tests of independence and inflates the variances of parameter estimates such a situation occurs for example when subjects consist of members of the same family or with repeated observations on the same person in this paper we describe methods of measuring the extent of correlation or clustering we also develop methods of adjusting the mantel haenszel chi square test statistic and the variance of the mantel haenszel estimate of a common odds ratio in sets of 2 2 contingency tables

global dynamics of a stochastic neuronal oscillator nonlinear oscillators have been used to model neurons that fire periodically in the absence of input these oscillators which are called neuronal oscillators share some common response structures with other biological oscillations such as cardiac cells in this study we analyze the dependence of the global dynamics of an impulse driven stochastic neuronal oscillator on the relaxation rate to the limit cycle the strength of the intrinsic noise and the impulsive input parameters to do this we use a markov operator that both reflects the density evolution of the oscillator and is an extension of the phase transition curve which describes the phase shift due to a single isolated impulse previously we derived the markov operator for the finite relaxation rate that describes the dynamics of the entire phase plane here we construct a markov operator for the infinite relaxation rate that describes the stochastic dynamics restricted to the limit cycle in both cases the response of the stochastic neuronal oscillator to time varying impulses is described by a product of markov operators furthermore we calculate the number of spikes between two consecutive impulses to relate the dynamics of the oscillator to the number of spikes per unit time and the interspike interval density specifically we analyze the dynamics of the number of spikes per unit time based on the properties of the markov operators each markov operator can be decomposed into stationary and transient components based on the properties of the eigenvalues and eigenfunctions this allows us to evaluate the difference in the number of spikes per unit time between the stationary and transient responses of the oscillator which we show to be based on the dependence of the oscillator on past activity our analysis shows how the duration of the past neuronal activity depends on the relaxation rate the noise strength and the impulsive input parameters

map dependence of the fractal dimension deduced from iterations of circle maps every orientation preserving circle mapg with inflection points including the maps proposed to describe the transition to chaos in phase locking systems gives occasion for a canonical fractal dimensiond namely that of the associated set of for whichf g has irrational rotation number we discuss how this dimension depends on the orderr of the inflection points in particular in the smooth case we find numerically thatd r d r 1 r 1 8

complexity in biological systems and hamiltonian dynamics it is well known that in biology and the life sciences in general there are numerous instances of complex behaviour arising from apparently deterministic processes one is familiar with the fractal nature of the shapes of leaves and snowflakes such shapes may be described as the outcome of simple deterministic evolutionary dynamics in another area the stochastic or chaotic response of stimulated cardiac nerve cells is well known and can be modelled by nonlinear deterministic systems of ordinary differential or difference equations in most examples describing complex behaviour in biological systems the underlying models are either ordinary differential or difference equations leading to an analysis of temporal behaviour and its dependence on certain parameter values such as growth rates generation times or reaction constants in this paper we explore both the temporal and spatial nature of complex phenomena in biological systems the examples involve partial dif ferential or partial difference equations and are drawn from models of population genetics mollusc shell patterning and excitable systems in each situation we demonstrate aspects of complexity and show that non integrable hamiltonian dynamical systems play a crucial role this brings into the realms of biology such concepts as structural stability and kolmogorov arnold moser kam theory which lie at the heart of current developments in the theory of dynamical systems

power law subordinacy and singular spectra i half line operators we study hausdorff dimensional spectral properties of certain whole line quasiperiodic discrete schrodinger operators by using the extension of the gilbert pearson subordinacy theory that we previously developed in 19

the analysis of categorical data from complex sample surveys chi squared tests for goodness of fit and independence in two way tables abstract the effect of stratification and clustering on the asymptotic distributions of standard pearson chi squared test statistics for goodness of fit simple hypothesis and independence in a two way contingency table denoted as x 2 and xi 2 respectively is investigated it is shown that both x 2 and xi 2 are asymptotically distributed as weighted sums of independent 1 2 random variables the weights are then related to the familiar design effects deffs used by survey samplers a simple correction to x 2 which requires only the knowledge of variance estimates or deffs for individual cells in the goodness of fit problem is proposed and empirical results on the performance of corrected x 2 provided empirical work on xi 2 indicated that the distortion of nominal significance level is substantially smaller with xi 2 than with x 2 some results under simple models for clustering are also given

observations of order and chaos in nonlinear systems experiments on nonlinear electrical oscillators the belousov zhabotinskii reaction rayleigh benard convection and couette taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom experimental results are presented illustrating the following transition sequences period doubling and the u sequence intermittency the periodic quasiperiodic chaotic sequence frequency locking and an alternating periodic chaotic sequence

odds ratio inference with dependent data summary when data can be presented as a series of k 2 x 2 tables with cell counts xi ni xi yi mi yi it is often assumed that xi and yi are binomially distributed this paper deals with inference for the common odds ratio i l when the binomial assumption is invalid when k increases the consistency and asymptotic normality of the mantel haenszel 1959 estimator is derived the conditional maximum likelihood estimator is shown to be inconsistent and the asymptotic bias is computed when either the first order markov chain or beta binomial model is assumed the mantel haenszel test for testing f 1 is also shown to be inappropriate through some simulation studies two consistent test statistics are proposed and shown to be comparable to each other in terms of size and efficiency some possible further work is described

delocalization in random polymer models a random polymer model is a one dimensional jacobi matrix randomly composed of two finite building blocks if the two associated transfer matrices commute the corresponding energy is called critical such critical energies appear in physical models an example being the widely studied random dimer model it is proven that the lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport showing that it is almost surely overdiffusive even though the models are known to have pure point spectrum with exponentially localized eigenstates for almost every configuration of the polymers furthermore the level spacing is shown to be regular at the critical energy

forced excitations and excitable chaos in the photosensitive oregonator under periodic sinusoidal perturbations abstract the responses of an excitable system to external periodic perturbations are studied by using the photosensitive oregonator that is a three variable model of the photosensitive belousov zhabotinsky reaction the system is set at the vicinity of the subcritical hopf bifurcation point and sinusoidal perturbation is given by light 0 a 0 sin 2 the dynamic threshold for firing appears as a u shaped line in plane under high frequency perturbations the system shows a subthreshold period doubling cascade subthreshold chaotic responses excitable chaos and reaches firing chaos as a result of expansion of orbits around the fixed point

bifurcations of circle maps arnol d tongues bistability and rotation intervals we study the bifurcations of two parameter families of circle maps that are similar tofb w x x w b 2 sin 2 x mod1 the bifurcation diagram is constructed in terms of setstr wheretr is the set of parameter values b w for whichfb w has an orbit with rotation numberr we show that the known structure whenb 1 wherefb w is no longer injective and can have an interval of rotation numbers specifically the tongues overlap in a uniform monotonic manner and forr irrational tr opens into a tongue our other theorems give information about the dynamics offb w e g bistability or aperiodicity for b w in various subsets of parameter space

spectral averaging techniques for jacobi matrices spectral averaging techniques for one dimensional discrete schrodinger operators are revisited and extended in particular simultaneous averaging over several parameters is discussed special focus is put on proving lower bounds on the density of the averaged spectral measures these wegner type estimates are used to analyze stability properties for the spectral types of jacobi matrices under local perturbations

efficient analysis of case control studies with sample weights analysis of population based case control studies with complex sampling designs is challenging because the sample selection probabilities and therefore the sample weights depend on the response variable and covariates commonly the design consistent weighted estimators of the parameters of the population regression model are obtained by solving sample weighted estimating equations weighted estimators however are known to be inefficient when the weights are highly variable as is typical for case control designs in this paper we propose two alternative estimators that have higher efficiency and smaller finite sample bias compared with the weighted estimator both methods incorporate the information included in the sample weights by modeling the sample expectation of the weights conditional on design variables we discuss benefits and limitations of each of the two proposed estimators emphasizing efficiency and robustness we compare the finite sample properties of the two new estimators and traditionally used weighted estimators with the use of simulated data under various sampling scenarios we apply the methods to the u s kidney cancer case control study to identify risk factors published 2012 this article is a us government work and is in the public domain in the usa

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

fitting logistic models under case control or choice based sampling on compare deux approches dans le cas important de l ajustement des modeles de regression logistique

homoclinic solutions for coupled systems of differential equations on utilise des idees topologiques basees sur la notion de flots et d ensembles de wazewski pour etablir l existence d orbites homoclines pour une classe de systemes hamiltoniens

standard errors for attributable risk for simple and complex sample designs adjusted attributable risk ar is the proportion of diseased individuals in a population that is due to an exposure we consider estimates of adjusted ar based on odds ratios from logistic regression to adjust for confounding influence function methods used in survey sampling are applied to obtain simple and easily programmable expressions for estimating the variance of ar these variance estimators can be applied to data from case control cross sectional and cohort studies with or without frequency or individual matching and for sample designs with subject samples that range from simple random samples to sample weighted multistage stratified cluster samples like those used in national household surveys the variance estimation of ar is illustrated with i a weighted stratified multistage clustered cross sectional study of childhood asthma from the third national health and examination survey nhanes iii and ii a frequency matched case control study of melanoma skin cancer

rhythm synchronization and chaotic modulation of coupled van der pol oscillators in a model for the heartbeat we investigate a phenomenological model for the heartbeat consisting of two coupled van der pol oscillators the coupling between these nodes can be both unidirectional or bidirectional and an external driving produced by a pacemaker is also included in this model in order to warrant a robust operation it is desirable that both units oscillate in a synchronized way even though in the presence of external in0uences or parameter mismatches which are unavoidable in a physiological setting we study the synchronization properties of such an association with respect to the nature andintensity of coupling we analyze in particular the generalized synchronization of rhythms characterizedby a chaotic mod ulation of the oscillator frequencies we also investigate the shadowing breakdown of numerically generated chaotic trajectories of the coupled oscillator system via unstable dimension variability in its chaotic invariant set c

bifurcation and chaos in a periodically stimulated cardiac oscillator abstract periodic stimulation of an aggregate of spontaneously beating cultured cardiac cells displays phase locking period doubling bifurcations and aperiodic chaotic dynamics at different values of the stimulation parameters this behavior is analyzed by considering an experimentally determined one dimensional poincare or first return map a simplified version of the experimentally determined poincare map is proposed and several features of the bifurcations of this map are described

proof of non integrability for the henon heiles hamiltonian near an exceptional integrable case abstract in this paper we describe an analytical method for establishing non integrability in hamiltonian systems which are small perturbations of integrable ones we illustrate the method with an application to a system of henon heiles type

transition to topological chaos for circle maps abstract many biperiodic flows can be modelled by maps of a circle to itself for such maps the transition from zero to positive topological entropy can be achieved in several ways we describe all the possible routes for smooth circle maps and discuss the relevance of our results to the transition to chaos for two frequency systems

effect of noise on the critical golden mean quasiperiodic dynamics in the circle map abstract scaling regularities are examined associated with effect of additive noise on a critical circle map at the golden mean rotation number we present an improved numerical estimate for the scaling constant of hamm and graham phys rev a46 1992 6323 responsible for the effect of noise 2 3061852653 decrease of the noise amplitude by this number ensures possibility to distinguish one more level of fractal like structure associated with increase of characteristic time scale by the golden mean factor 5 1 2 numeric results demonstrating evidence of the expected scaling are presented e g portraits of the noisy attractors devil s staircase plots lyapunov charts on the parameter plane in different scales

an analytical and numerical study of the bifurcations in a system of linearly coupled oscillators abstract we study a two parameter family of ordinary differential equations in r 4 that governs the dynamics of two coupled planar oscillators each oscillator has a unique periodic solution that is attracting and the uncoupled product system has a unique invariant torus that is attracting the torus persists for weak coupling and contains two periodic solutions when the coupling is linear and conservative one of these in which the oscillators are synchronized persists and is stable for all coupling strengths the other in which the oscillators are radiant out of phase disappears either in a hopf bifurcation or when fixed points appear on the orbit at a critical ratio of the coupling strength to the frequency the out of phase oscillation is unstable except on an open set in the frequency coupling strength plane which contains moderate values of both parameters furthermore there are tori bifurcating from the out of phase solution which means according to the arnol d theory for hopf bifurcations in maps that there may be periodic solutions of arbitrarily large period and chaotic solutions as well numerous other bifurcations occur and there are a number of higher codimension singularities in a large region of the frequency coupling parameter plane stable steady states coexist with stable periodic solutions

the w sequence for circle maps and misbehaved itineraries the w sequence an ordering of maximal itineraries for circle maps is constructed by means of the farey transformation and the gluing operation the similarity in the dynamical behaviour of the maps is explicitly described in terms of itineraries a class of badly ordered itineraries termed misbehaved is defined with the language of symbolic sequences a misbehaved itinerary is shown to imply a rotation interval

finite sample properties of some old and some new estimators of a common odds ratio from multiple 2 2 tables abstract a simulation study was used to determine the finite sample properties of various estimators of a common odds ratio from multiple 2 2 tables the maximum likelihood mantel haenszel and woolf estimators plus four variants of each were considered the original woolf estimator and one of its variants are dismissed on the grounds that they often cannot be calculated in terms of minimum bias maximum likelihood is the recommended procedure after adding 25 to each cell

effects of cluster sampling on epidemiologic analysis in population based case control studies we consider population based case control designs in which controls are selected by one of three cluster sampling plans from the entire population at risk the effects of cluster sampling on classical epidemiologic procedures are investigated and appropriately modified procedures are developed in particular modified procedures for testing the homogeneity of odds ratios across strata and for estimating and testing a common odds ratio are presented simulations that use the data from the 1970 health interview survey as a population suggest that classical procedures may be fairly robust in the presence of cluster sampling a more extreme example based on a mixed multinomial model clearly demonstrates that the classical mantel haenszel 1959 journal of the national cancer institute 22 719 748 and woolf haldane tests of no exposure effect may have sizes exceeding nominal levels and confidence intervals with less than nominal coverage under an alternative hypothesis classical estimates of odds ratios may also be biased with non self weighting cluster samples the modified procedures we propose remedy these defects

value distribution and spectral analysis of differential operators the theory of value distribution is applied through a study of boundary value distribution for the weyl titchmarsh m function to the spectral analysis of differential operators on the half line particular consequences are the theoretical underpinning of a numerical approach to spectral analysis of differential operators exhibiting absolutely continuous or singular spectra and new identities for the spectral measure in terms of solutions of the associated differential equation

bifurcations from an invariant circle for two parameter families of maps of the plane a computer assisted study we consider a two parameter family of maps of the plane to itself each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point for certain parameter values there is a hopf bifurcation to an invariant circle which is smooth for parameter values in a neighborhood of the bifurcation point however computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point this paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type

adjustments to the mantel haenszel test for data from stratified multistage surveys the usual form of the mantel haenszel test statistic assumes independent observations this is inappropriate for data from a stratified multistage survey two alternative adjustments to the test statistic are developed to deal with this a a modification of the effective sample size for each row of each table using the design effects extending a method proposed by donald and donner for familial aggregation studies b a taylor series approximation to the variance of the square root of the numerator of the mantel haenszel statistic both methods are evaluated by application to both simulated and real data the two methods perform equally well and offer a considerable improvement over the unadjusted test statistic when observations from the same cluster are highly correlated a simplified adjustment is also considered as is the need for correction to the variance of the odds ratio estimator

stability of singular spectral types under decaying perturbations we look at invariance of a e boundary condition spectral behavior under perturbations w of half line continuum or discrete schrodinger operators we extend the results of del rio simon stolz from compactly supported w s to suitable short range w we also discuss invariance of the local hausdorff dimension of spectral measures under such perturbations

two parameter spectral averaging and localization for non monotonic random schrodinger operators we prove exponential localization at all energies for two types of one dimensional random schr6dinger operators the poisson model and the random displacement model as opposed to anderson type models these operators are not monotonic in the random parameters therefore the classical one parameter version of spectral averaging as used in localization proofs for anderson models breaks down we use the new method of two parameter spectral averaging and apply it to the poisson as well as the displacement case in addition we apply results from inverse spectral theory which show that two parameter spectral averaging works for sufficiently many energies all but a discrete set to conclude localization at all energies