computing hopf bifurcations ii three examples from neurophysiology in guckenheimer myers and sturmfels siam j numer anal 34 1997 we present algorithms for detecting hopf bifurcations in two parameter families of vector fields based on classical algebraic constructions in addition to their utility as augmented systems for use with standard newton type continuation methods they are shown to be particularly well adapted for solution by computer algebra techniques for vector fields of small or moderate dimension the present study examines the performance of these methods on test problems selected from models of current research interest in neurophysiology implementation issues are examined and the numerical properties of the proposed methods are compared with several alternative algorithms for hopf pathfollowing that appear in the literature

rigorous results on the bipartite mean field model we consider a bipartite mean field model in which interaction coefficients and magnetic fields depend only on the groups the particles belong to we rigorously compute the value of the limiting pressure per particle using tail estimation techniques we study the phase space of the model in the symmetric regime without an external field and when the interaction coefficients within the two groups are identical magnetic field perturbations are considered

survey design under the regression superpopulation model abstract the construction of sample designs and estimators under a linear regression superpopulation model is considered the anticipated variance the variance of the predictor computed with respect to the sampling design and the superpopulation model is used as a criterion for evaluating probability designs and model unbiased predictors regression predictors that are model unbiased and design consistent are constructed

a statistical mechanics approach to granovetter theory abstract in this paper we try to bridge breakthroughs in quantitative sociology econometrics pioneered during the last decades by mac fadden brock durlauf granovetter and watts strogatz by introducing a minimal model able to reproduce essentially all the features of social behavior highlighted by these authors our model relies on a pairwise hamiltonian for decision maker interactions which naturally extends the multi populations approaches by shifting and biasing the pattern definitions of a hopfield model of neural networks once introduced the model is investigated through graph theory to recover granovetter and watts strogatz results and statistical mechanics to recover mac fadden and brock durlauf results due to the internal symmetries of our model the latter is obtained as the relaxation of a proper markov process allowing even to study its out of equilibrium properties the method used to solve its equilibrium is an adaptation of the hamilton jacobi technique recently introduced by guerra in the spin glass scenario and the picture obtained is the following shifting the patterns from 1 1 0 1 implies that the larger the amount of similarities among decision makers the stronger their relative influence and this is enough to explain both the different role of strong and weak ties in the social network as well as its small world properties as a result imitative interaction strengths seem essentially a robust request enough to break the gauge symmetry in the couplings furthermore this naturally leads to a discrete choice modelization when dealing with the external influences and to imitative behavior a la curie weiss as the one introduced by brock and durlauf

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

the aggregate path coupling method for the potts model on bipartite graph in this paper we derive the large deviation principle for the potts model on the complete bipartite graph kn n as n increases to infinity next for the potts model on kn n we provide an extension of the method of aggregate path coupling that was originally developed in the work of kovchegov otto and titus j stat phys 144 5 1009 1027 2011 for the mean field blume capel model and in kovchegov and otto j stat phys 161 3 553 576 2015 for a general mean field setting that included the generalized curie weiss potts model analyzed in the work of jahnel et al markov process relat fields 20 601 632 2014 we use the aggregate path coupling method to identify and determine the threshold value s separating the rapid and slow mixing regimes for the glauber dynamics of the potts model on kn n

inference from stratified samples properties of the linearization jackknife and balanced repeated replication methods the asymptotic normality of both linear and nonlinear statistics and the consistency of the variance estimators obtained using the linearization jackknife and balanced repeated replication brr methods in stratified samples are established the results are obtained as l oa within the context of a sequence of finite populations jil with l strata in ji l and are valid for any stratified multistage design in which the primary sampling units psu s are selected with replacement and in which independent subsamples are taken within those psu s selected more than once in addition some exact analytical results on the bias and stability of these alternative variance estimators in the case of ratio estimation are obtained for small l under a general linear regression model

jackknife variance estimation under imputation for estimators using poststratification information abstract poststratified estimators are commonly used in sample surveys to improve the efficiency of estimators and to ensure calibration to known poststrata counts similarly generalized regression estimators are used to handle two or more poststratifiers with known marginal counts in addition weighting adjustment within weighting classes is used to handle unit nonresponse and imputation within imputation classes is used to handle item nonresponse for the full response case asymptotic consistency of the jackknife variance estimator under stratified multistage sampling is established using mild regularity conditions on residuals similar to those of scott and wu for ratio and regression estimation under simple random sampling a jackknife linearization variance estimator obtained by linearizing the jackknife variance estimator is also given for unit nonresponse the general case of poststrata cutting across weighting classes is considered and a jackknife variance estimator and the corresponding

a survey of condition number estimation for triangular matrices we survey and compare a wide variety of techniques for estimating the condition number of a triangular matrix and make recommendations concerning the use of the estimates in applications each of the methods is shown to bound the condition number the bounds can broadly be categorised as upper bounds from matrix theory and lower bounds from heuristic or probabilistic algorithms for each bound we examine by how much at worst it can overestimate or underestimate the condition number numerical experiments are presented in order to illustrate and compare the practical performance of the condition estimators

a linpack style condition estimator for the equation ax xb t c estimation of the condition number of ax xb t c requires an approximation of the smallest l 2 singular value of the linear transformation x rightarrow ax xb t the approximation is obtained from a step of inverse iteration applied to a heuristically chosen starting matrix in the style of linpack the cost of the estimator is kept down by using the same matrix factorizatlons to make the estimate as are used to solve ax xb t c computational experiments verify its reliability

weighted likelihood estimation under two phase sampling we develop asymptotic theory for weighted likelihood esti mators wle under two phase stratied sampling without replacement we also consider several variants of wle s involving estimated weights and calibration a set of empirical process tools are developed including a glivenko cantelli theorem a theorem for rates of convergence of z estimators and a donsker theorem for the inverse probability weighted em pirical processes under two phase sampling and sampling without replace ment at the second phase using these general results we derive asymptotic distributions of the wle of a nite dimensional parameter in a general semiparametric model where an estimator of a nuisance parameter is es timable either at regular or non regular rates we illustrate these results and methods in the cox model with right censoring and interval censor ing we compare the methods via their asymptotic variances under both sampling without replacement and the more usual and easier to analyze assumption of bernoulli sampling at the second phase ams 2000 subject classications primary 62e20 secondary 62g20 62d99 62n01

functional central limit theorems for single stage sampling designs for a joint model based and design based inference we establish functional central limit theorems for the horvitz thompson empirical process and the h ajek empirical process centered by their finite population mean as well as by their super population mean in a survey sampling framework the results apply to single stage unequal probability sampling designs and essentially only require conditions on higher order correlations we apply our main results to a hadamard differentiable statistical functional and illustrate its limit behavior by means of a computer simulation

fluctuation results for general block spin ising models we study a block spin mean field ising model i e a model of spins in which the vertices are divided into a finite number of blocks with each block having a fixed proportion of vertices and where pair interactions are given according to their blocks for the vector of block magnetizations we prove large deviation principles and central limit theorems under general assumptions for the block interaction matrix using the exchangeable pair approach of stein s method we establish a rate of convergence in the central limit theorem for the block magnetization vector in the high temperature regime

numerically robust delta domain solutions to discrete time lyapunov equations a problem of numerical conditioning of a special kind of discrete time lyapunov equations is considered it is assumed that a discretisation procedure equipped with the zero order holder mechanism is utilised that leads to the data matrices that are affinely related to the sampling period and matrices that are independent or linearly related to the squared sampling period it is shown that common forward shift operator techniques for solving these equations become ill conditioned for a sufficiently small sampling period and that numerical robustness and reliability of computations can be significantly improved via utilising the so called delta operator form of the origin equations

generalized schur methods with condition estimators for solving the generalized sylvester equation stable algorithms are presented for solving the generalized sylvester equation they are based on orthogonal equivalence transformations of the original problem perturbation theory and rounding error analysis are included condition estimators dif sup 1 estimators are developed which when substituted into derived error bounds give accuracy estimates of a computed solution results from numerical experiments on well conditioned and ill conditioned problems are reported

on the numerical properties of the schur approach for solving the matrix riccati equation abstract an a posteriori estimate of the relative error in the solution of the algebraic matrix riccati equation found by the schur approach is derived it is shown that in some cases the schur approach is numerically unstable

scaling limits for multi species statistical mechanics mean field models we study the limiting thermodynamic behavior of the normalized sums of spins in multi species curie weiss models we find sufficient conditions for the limiting random vector to be gaussian or to have an exponential distribution of higher order and compute the covariance matrix in terms of model parameters

sensitivity of the stable discrete time lyapunov equation the sensitivity of the stable discrete time lyapunov equation is analyzed through the spectral norm of the inverse lyapunov operator this leads to a directly computable easy to interpret sensitivity measure which involves only the open loop state matrix and also provides insight into the connection between sensitivity stability radius and conditioning of the eigenproblem of the open loop state matrix these results are an extension to the discrete time case of analogous results for the continuous time lyapunov equation

rescaled magnetization for critical bipartite curie weiss models we consider a bipartite generalization of the curie weiss model in a critical regime in order to study the asymptotic behavior of the random vector of the total magnetization we apply the change of variables that diagonalizes the hessian matrix of the pressure functional associated to the model we obtain a new vector that suitably rescaled weakly converges to the product of a gaussian distribution and a distribution proportional to exp xi x 4 where the positive constant xi can be computed from the pressure functional

some numerical considerations and newton s method revisited for solving algebraic riccati equations analyzes some of the numerical aspects of solving the algebraic riccati equation are this analysis applies to both the symmetric and unsymmetric cases the author reconsiders the numerically relevant problems of balancing the are and the conditioning properties of the are and shows how these can be exploited by a solution algorithm he proposes an estimator for the condition number of the sylvester equation ax xb c based on iterative refinement also he interprets newton s method as a sequence of similarity transformations on the underlying system matrix this closes the gap between so called global and iterative methods for solving the are and also suggests an altogether revised implementation of newton s method one of the advantages of this revised implementation is that in the case where newton s method converges to a solution different from the desired solution enough information emerges to allow a switch to the desired solution the author examines the roundoff properties of the new algorithm and provides implementation considerations and numerical examples to highlight pros and cons

spin flip dynamics of the curie weiss model loss of gibbsianness with possibly broken symmetry we study the conditional probabilities of the curie weiss ising model in vanishing external field under a symmetric independent stochastic spin flip dynamics and discuss their set of points of discontinuity bad points we exhibit a complete analysis of the transition between gibbsian and non gibbsian behavior as a function of time extending the results for the corresponding lattice model where only partial answers can be obtained for initial temperature beta 1 geq 1 we prove that the time evolved measure is always gibbsian for frac 2 3 leq beta 1 1 the time evolved measure loses its gibbsian character at a sharp transition time for beta 1 frac 2 3 we observe the new phenomenon of symmetry breaking in the set of points of discontinuity bad points corresponding to non zero spin average appear at a sharp transition time and give rise to biased non gibbsianness of the time evolved measure these bad points become neutral at a later transition time while the measure stays non gibbs in our proof we give a detailed description of the phase diagram of a curie weiss random field ising model with possibly non symmetric random field distribution based on bifurcation analysis

a newton squaring algorithm for computing the negative invariant subspace of a matrix by combining newton s method for the matrix sign function with a squaring procedure a basis for the negative invariant subspace of a matrix can be computed efficiently the algorithm presented is a variant of multiplication rich schemes for computing the matrix sign function such as the well known inversion free schulz method which requires two matrix multiplications per step however by avoiding a complete computation of the matrix sign and instead concentrating only on the negative invariant subspace the final newton steps can be replaced by steps which require only one matrix squaring each this efficiency is attained without sacrificing the quadratic convergence of newton s method

on the two phase framework for joint model and design based inference we establish a mathematical framework that formally validates the two phase super population viewpoint proposed by hartley and sielken biometrics 31 1975 411 422 by defining a product probability space which includes both the design space and the model space the methodology we develop combines finite population sampling theory and the classical theory of infinite population sampling to account for the underlying processes that produce the data under a unified approach our key results are the following first if the sample estimators converge in the design law and the model statistics converge in the model then under certain conditions they are asymptotically independent and they converge jointly in the product space second the sample estimating equation estimator is asymptotically normal around a super population parameter

equilibrium statistical mechanics of bipartite spin systems the aim of this paper is to give an extensive treatment of bipartite mean field spin systems pure and disordered at first bipartite ferromagnets are investigated and an explicit expression for the free energy is achieved through a new minimax variational principle then via the hamilton jacobi technique the same structure of the free energy is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one the same is investigated for bipartite spin glasses by the borel cantelli lemma we obtain the control of the high temperature regime while via the double stochastic stability technique we also obtain the explicit expression of the free energy in the replica symmetric approximation uniquely defined by a minimax variational principle again we also obtain a general result that states that the free energies of these systems are convex linear combinations of their independent one party model counterparts for the sake of completeness we show further that at zero temperature the replica symmetric entropy becomes negative and consequently such a symmetry must be broken the treatment of the fully broken replica symmetry case is deferred to a forthcoming paper as a first step in this direction we start deriving the linear and quadratic constraints to overlap fluctuations

a stability enhancing scaling procedure for schur riccati solvers abstract scaling riccati equations so that the solution has unit norm eliminates instabilities associated with the inversion step of invariant subspace based schur methods without changing the relative condition number of the underlying problem such optimal scaling makes it easy to estimate this condition number and to bound the relative error of the computed solution moreover the determination of whether the first step of newton s method will improve this solution is also simplified