saari s homographic conjecture of the three body problem saari s homographic conjecture which extends a classical statement proposed by donald saari in 1970 claims that solutions of the newtonian n body problem with constant configurational measure are homographic in other words if the mutual distances satisfy a certain relationship the configuration of the particle system may change size and position but not shape we prove this conjecture for large sets of initial conditions in three body problems given by homogeneous potentials including the newtonian one some of our results are true for n 3

a stochastic approximation algorithm with markov chain monte carlo method for incomplete data estimation problems we propose a general procedure for solving incomplete data estimation problems the procedure can be used to find the maximum likelihood estimate or to solve estimating equations in difficult cases such as estimation with the censored or truncated regression model the nonlinear structural measurement error model and the random effects model the procedure is based on the general principle of stochastic approximation and the markov chain monte carlo method applying the theory on adaptive algorithms we derive conditions under which the proposed procedure converges simulation studies also indicate that the proposed procedure consistently converges to the maximum likelihood estimate for the structural measurement error logistic regression model

time delay effects on coupled limit cycle oscillators at hopf bifurcation abstract we present a detailed study of the effect of time delay on the collective dynamics of coupled limit cycle oscillators at hopf bifurcation for a simple model consisting of just two oscillators with a time delayed coupling the bifurcation diagram obtained by numerical and analytical solutions shows significant changes in the stability boundaries of the amplitude death phase locked and incoherent regions a novel result is the occurrence of amplitude death even in the absence of a frequency mismatch between the two oscillators similar results are obtained for an array of n oscillators with a delayed mean field coupling and the regions of such amplitude death in the parameter space of the coupling strength and time delay are quantified some general analytic results for the n thermodynamic limit are also obtained and the implications of the time delay effects for physical applications are discussed

bi instability and the global role of unstable resonant orbits in a driven laser driven class b lasers are devices which possess quadratic nonlinearities and are known to exhibit chaotic behavior we describe the onset of global heteroclinic connections which give rise to chaotic saddles these form the precursor topology which creates both localized homoclinic chaos as well as global mixed mode heteroclinic chaos to locate the relevant periodic orbits creating the precursor topology approximate maps are derived using matched asymptotic expansions and subharmonic melnikov theory locating the relevant unstable fixed points of the maps provides an organizing framework to understand the global dynamics and chaos exhibited by the laser 2000 published by elsevier science b v

a remarkable periodic solution of the three body problem in the case of equal masses using a variational method we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane the orbit has zero angular momentum and a very rich symmetry pattern its most surprising feature is that the three bodies chase each other around a flxed eight shaped curve setting aside collinear motions the only other known motion along a flxed curve in the inertial plane is the lagrange relative equilibrium in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle our orbit visits in turns every euler conflguration in which one of the bodies sits at the midpoint of the segment deflned by the other two figure 1 numerical computations

amplitude death in the absence of time delays in identical coupled oscillators we study the dynamics of oscillators that are mutually coupled via dissimilar or conjugate variables and find that this form of coupling leads to a regime of amplitude death analytic estimates are obtained for coupled landau stuart oscillators and this is supplemented by numerics for this system as well as for coupled lorenz oscillators time delay does not appear to be necessary to cause amplitude death when conjugate variables are employed in coupling identical systems coupled chaotic oscillators also show multistability prior to amplitude death and the basins of the coexisting attractors appear to be riddled this behavior is quantified an appropriately defined uncertainty exponent in the coupled lorenz system is shown to be zero

hopf bifurcation subject to a large delay in a laser system hopf bifurcation theory for an oscillator subject to a weak feedback but a large delay is investigated for a specific laser system the problem is motivated by semiconductor laser instabilities which are initiated by undesirable optical feedbacks most of these instabilities are starting from a single hopf bifurcation because of the large delay a delayed amplitude appears in the slow time bifurcation equation which generates new bifurcations to periodic and quasi periodic states we determine analytical expressions for all branches of periodic solutions and show the emergence of secondary bifurcation points from double hopf bifurcation points we study numerically different cases of bistability between steady periodic and quasi periodic regimes finally the validity of the hopf bifurcation approximation is investigated numerically by comparing the bifurcation diagrams of the original laser equations and the slow time amplitude equation

stochastic approximation em for large scale exploratory irt factor analysis a stochastic approximation em algorithm saem is described for exploratory factor analysis of dichotomous or ordinal variables the factor structure is obtained from sufficient statistics that are updated during iterations with the robbins monro procedure two large scale simulations are reported that compare accuracy and cpu time of the proposed saem algorithm to the metropolis hasting robbins monro procedure and to a generalized least squares analysis of the polychoric correlation matrix a smaller scale application to real data is also reported including a method for obtaining standard errors of rotated factor loadings a simulation study based on the real data analysis is conducted to study bias and error estimates the saem factor algorithm requires minimal lines of code no derivatives and no large matrix inversion it is programmed entirely in r

stochastic approximation boosting for incomplete data problems boosting is a powerful approach to fitting regression models this article describes a boosting algorithm for likelihood based estimation with incomplete data the algorithm combines boosting with a variant of stochastic approximation that uses markov chain monte carlo to deal with the missing data applications to fitting generalized linear and additive models with missing covariates are given the method is applied to the pima indians diabetes data where over half of the cases contain missing values

what are textons textons refer to fundamental micro structures in natural images and videos and are considered as the atoms of pre attentive human visual perception julesz 1981 unfortunately the word texton remains a vague concept in the literature for lack of a good mathematical model in this article we first present a three level generative image model for learning textons from texture images in this model an image is a superposition of a number of image bases selected from an over complete dictionary including various gabor and laplacian of gaussian functions at various locations scales and orientations these image bases are in turn generated by a smaller number of texton elements selected from a dictionary of textons by analogy to the waveform phoneme word hierarchy in speech the pixel base texton hierarchy presents an increasingly abstract visual description and leads to dimension reduction and variable decoupling by fitting the generative model to observed images we can learn the texton dictionary as parameters of the generative model then the paper proceeds to study the geometric dynamic and photometric structures of the texton representation by further extending the generative model to account for motion and illumination variations 1 for the geometric structures a texton consists of a number of image bases with deformable spatial configurations the geometric structures are learned from static texture images 2 for the dynamic structures the motion of a texton is characterized by a markov chain model in time which sometimes can switch geometric configurations during the movement we call the moving textons as motons the dynamic models are learned using the trajectories of the textons inferred from video sequence 3 for photometric structures a texton represents the set of images of a 3d surface element under varying illuminations and is called a lighton in this paper we adopt an illumination cone representation where a lighton is a texton triplet for a given light source a lighton image is generated as a linear sum of the three texton bases we present a sequence of experiments for learning the geometric dynamic and photometric structures from images and videos and we also present some comparison studies with k mean clustering sparse coding independent component analysis and transformed component analysis we shall discuss how general textons can be learned from generic natural images

negative coupling resonances in pump coupled lasers we consider coupled lasers where the intensity deviations from the steady state modulate the pump of the other lasers most of our results are for two lasers where the coupling constants are of opposite sign this leads to a hopf bifurcation to periodic output for weak coupling as the magnitude of the coupling constants is increased negatively we observe novel amplitude effects such as a weak coupling resonance peak and strong coupling subharmonic resonances and chaos in the weak coupling regime the output is predicted by a set of slow evolution amplitude equations pulsating solutions in the strong coupling limit are described by a discrete map derived from the original model

amplitude response of coupled oscillators abstract we investigate the interaction of a pair of weakly nonlinear oscillators eg each near a hopf bifurcation when the coupling strength is comparable to the attraction of the limit cycles changes in amplitude cannot then be ignored and there are new phenomena we show that even for simple forms of these equations there are parameter regimes in which the interaction causes the system to stop oscillating and the rest state at zero stabilized by the interaction is the only stable solution when the uncoupled oscillators have a local frequency that is independent of amplitude zero shear and the coupling is scalar we give a complete description of the behavior we then analyze more complicated equations to show the effects of amplitude dependent frequency in the uncoupled equations nonzero shear and nonscalar coupling for example when there is shear there can be bistability between a phase locked solution and an unlocked drift solution in which the oscillators go at different average frequencies there can also be bistability between a phase locked solution and an equilibrium nonoscillatory solution the equations for the zero shear case have a pair of degenerate bifurcation points and the new phenomena in the nonzero shear case arise from a partial unfolding of these singularities when the coupling is nonscalar there are a variety of new behaviors including bistability of an in phase and an anti phase solution for two identical oscillators parameters for which only the anti phase solution is stable and phase trapping i e nonlocked solutions in which the oscillators still have the same average frequency it is shown that whether the coupling is diffusive or direct has many effects on the behavior of the system

maximum likelihood estimation for spatial models by markov chain monte carlo stochastic approximation we propose a two stage algorithm for computing maximum likelihood estimates for a class of spatial models the algorithm combines markov chain monte carlo methods such as the metropolis hastings green algorithm and the gibbs sampler and stochastic approximation methods such as the off line average and adaptive search direction a new criterion is built into the algorithm so stopping is automatic once the desired precision has been set simulation studies and applications to some real data sets have been conducted with three spatial models we compared the algorithm proposed with a direct application of the classical robbins monro algorithm using wiebe s wheat data and found that our procedure is at least 15 times faster

improving samc using smoothing methods theory and applications to bayesian model selection problems stochastic approximation monte carlo samc has recently been proposed by liang liu and carroll j amer statist assoc 102 2007 305 320 as a general simulation and optimization algorithm in this paper we propose to improve its convergence using smoothing methods and discuss the application of the new algorithm to bayesian model selection problems the new algorithm is tested through a change point identification example the numerical results indicate that the new algorithm can outperform samc and reversible jump mcmc significantly for the model selection problems the new algorithm represents a general form of the stochastic approximation markov chain monte carlo algorithm it allows multiple samples to be generated at each iteration and a bias term to be included in the parameter updating step a rigorous proof for the convergence of the general algorithm is established under verifiable conditions this paper also provides a framework on how to improve efficiency of monte carlo simulations by incorporating some nonparametric techniques

improving estimates of genetic maps a maximum likelihood approach as a result of previous large multipoint linkage studies there is a substantial amount of existing marker data due to the increased sample size genetic maps estimated from these data could be more accurate than publicly available maps however current methods for map estimation are restricted to data sets containing pedigrees with a small number of individuals or cannot make full use of marker data that are observed at several loci on members of large extended pedigrees in this article a maximum likelihood ml method for map estimation that can make full use of the marker data in a large multipoint linkage study is described the method is applied to replicate sets of simulated marker data involving seven linked loci and pedigree structures based on the real multipoint linkage study of abkevich et al 2003 american journal of human genetics 73 1271 1281 the variance of the ml estimate is accurately estimated and tests of both simple and composite null hypotheses are performed an efficient procedure for combining map estimates over data sets is also suggested

stochastic approximation in monte carlo computation the wang landau wl algorithm is an adaptive markov chain monte carlo algorithm used to calculate the spectral density for a physical system a remarkable feature of the wl algorithm is that it is not trapped by local energy minima which is very important for systems with rugged energy landscapes this feature has led to many successful applications of the algorithm in statistical physics and biophysics however there does not exist rigorous theory to support its convergence and the estimates produced by the algorithm can reach only a limited statistical accuracy in this article we propose the stochastic approximation monte carlo samc algorithm which overcomes the shortcomings of the wl algorithm we establish a theorem concerning its convergence the estimates produced by samc can be improved continuously as the simulation proceeds samc also extends applications of the wl algorithm to continuum systems the potential uses of samc in statistics are discussed through two classes of applications i

stability and rupture of bifurcation bridges in semiconductor lasers subject to optical feedback the bifurcation diagram of a single mode semiconductor laser subject to a delayed optical feedback is examined by using numerical continuation methods for this we show how to cope with the special symmetry properties of the equations as the feedback strength is increased branches of modes and antimodes appear and we have found that pairs of modes and antimodes are connected by closed branches of periodic solutions bifurcation bridges such connections seem generically present as new pairs of modes and antimodes appear we subsequently investigate the behavior of the first connection as a function of the linewidth enhancement factor and the feedback phase our results extend and confirm existing results and hypotheses reported in the literature for large values of the linewidth enhancement factor alpha 5 6 bridges break through homoclinic orbits changing the feedback phase unfolds the bifurcation diagram of the modes and antimodes allowing different types of connections between modes

on the birth of isolas isolas are isolated closed curves of solution branches of nonlinear problems they have been observed to occur in the buckling of elastic shells the equilibrium states of chemical reactors and other problems in this paper we present a theory to describe analytically the structure of a class of isolas specifically we consider isolas that shrink to a point as a parameter of the problem approaches a critical value 0 the point is referred to as an isola center equations that characterize the isola centers are given then solutions are constructed in a neighborhood of the isola centers by perturbation expansions in a small parameter e that is proportional to o with a appropriately determined the theory is applied to a chemical reactor problem

bifurcation analysis of delay induced periodic oscillations in this paper we consider a generic differential equation with a cubic nonlinearity and delay this system in the absence of delay is known to undergo an oscillatory instability the addition of the delay is shown to result in the creation of a number of periodic solutions with constant amplitude and a constant frequency the number of solutions increases with the size of the delay indeed for many physical applications in which oscillatory instabilities are induced by a delayed response or feedback mechanism the system under consideration forms the underlying backbone for a mathematical model our study showcases the effectiveness of performing a numerical bifurcation analysis alongside the use of analytical and geometrical arguments in investigating systems with delay we identify curves of codimension one bifurcations of periodic solutions we show how these curves interact via codimension two bifurcation points double singularities which organise the bifurcations and dynamics in their local vicinity

high dimensional exploratory item factor analysis by a metropolis hastings robbins monro algorithm a metropolis hastings robbins monro mh rm algorithm for high dimensional maximum marginal likelihood exploratory item factor analysis is proposed the sequence of estimates from the mh rm algorithm converges with probability one to the maximum likelihood solution details on the computer implementation of this algorithm are provided the accuracy of the proposed algorithm is demonstrated with simulations as an illustration the proposed algorithm is applied to explore the factor structure underlying a new quality of life scale for children it is shown that when the dimensionality is high mh rm has advantages over existing methods such as numerical quadrature based em algorithm extensions of the algorithm to other modeling frameworks are discussed

coupling induced resonance in two mutually and asymmetrically coupled oscillators we consider two mutually coupled oscillators where we have independent control over the magnitude sign and delay of the coupling signal for appropriate tuning of the coupling constants there is a coupling induced resonance where the amplitude becomes large we investigate the role of nonlinear dissipation and amplitude dependent frequency correction on the coupling resonance with delayed coupling we track the deformation of the resonant bifurcation equation through imperfect bifurcations and the generation of isolas which generate intervals of multistability between oscillations of different amplitudes the resonance should be observable in coupled oscillator systems where the amplitude remains an important dynamical variable but not in limit cycle oscillators modeled by phase only descriptions

bifurcation diagram of a complex delay differential equation with cubic nonlinearity we reduce the lang kobayashi equations for a semiconductor laser with external optical feedback to a single complex delay differential equation in the long delay time limit the reduced equation has a time delayed linear term and a cubic instantaneous nonlinearity there are only two parameters the real linewidth enhancement factor and the complex feedback strength the equation displays a very rich dynamics and can sustain steady periodic quasiperiodic and chaotic regimes we study the steady solutions analytically and analyze the periodic solutions by using a numerical continuation method this leads to a bifurcation diagram of the steady and periodic solutions stable and unstable we illustrate the chaotic regimes by a direct numerical integration and show that low frequency fluctuations still occur

smoothing spline anova frailty model for recurrent event data summary gap time hazard estimation is of particular interest in recurrent event data this article proposes a fully nonparametric approach for estimating the gap time hazard smoothing spline analysis of variance anova decompositions are used to model the log gap time hazard as a joint function of gap time and covariates and general frailty is introduced to account for between subject heterogeneity and within subject correlation we estimate the nonparametric gap time hazard function and parameters in the frailty distribution using a combination of the newton raphson procedure the stochastic approximation algorithm saa and the markov chain monte carlo mcmc method the convergence of the algorithm is guaranteed by decreasing the step size of parameter update and or increasing the mcmc sample size along iterations model selection procedure is also developed to identify negligible components in a functional anova decomposition of the log gap time hazard we evaluate the proposed methods with simulation studies and illustrate its use through the analysis of bladder tumor data

choreographic three bodies on the lemniscate we show that choreographic three bodies x t x t t 3 x t t 3 of period t on the lemniscate x t cn t sn t 1 cn2 t parametrized by the jacobian elliptic functions sn and cn with modulus k2 2 3 4 conserve the centre of mass and the angular momentum where and are the orthogonal unit vectors defining the plane of the motion they also conserve the moment of inertia the kinetic energy the sum of squares of the curvature the product of distances and the sum of squares of distances between bodies we find that they satisfy the equation of motion under the potential energy i j 1 2 ln rij 3 24 rij2 or i j 1 2 ln rij i 3 8 ri2 where rij is the distance between bodies i and j and ri the distance from the origin the first term of the potential energies is the universal gravitation in two dimensions but the second term is a mutual repulsive force or a repulsive force from the origin respectively then geometric construction methods for the positions of the choreographic three bodies are given