patterns in eigenvalues the 70th josiah willard gibbs lecture typical large unitary matrices show remarkable patterns in their eigenvalue distribution these same patterns appear in telephone encryption the zeros of riemann s zeta function a variety of physics problems and in the study of toeplitz operators this paper surveys these applications and what is currently known about the patterns

sticky spheres and related systems some properties of a system of hard core particles with attractive wells in the baxter sticky sphere limit and a related limit are considered as is the approach to these limits a demonstration of the result of stell and williams that sticky spheres of equal diameter in the baxter limit are not thermodynamically stable is given and the way in which size polydispersity can be expected to restore thermodynamic stability is discussed the implications of these results for the py sticky sphere approximation and recent sticky sphere computer simulations are then examined it is concluded that the baxter py sticky sphere approximation for a monodisperse system may well be a reasonable one for a slightly polydisperse system of sticky spheres and that existing simulation results may also be relevant to such a system how polydisperse a system must be in quantitative terms in order for the py approximation to be useful remains to be seen however the question of whether the py sticky sphere approximation may prove to be useful and appropriate in describing monodisperse systems with pair potentials for which the attractive wells arenot extremely narrow is also considered it is noted that firm evidence concerning this question also appears to be lacking implications for systems near but not in the limit of zero attractive well width are also considered especially in terms of the relative size of the well width and the degree of size polydispersity in the repulsive cores the possible pertinence of such considerations to colloidal systems is observed the importance of taking into consideration the extremely long equilibration times that can be expected for systems with very narrow attractive wells is also pointed out in connection both with real colloidal systems and in computer simulations it is further observed that in the baxter limit sticky spheres described quantum mechanically are indistinguishable from hard spheres so described near the zero well width limit the quantal behavior hinges on the number of bound states and thus the well depth as well as the relative size of the de broglie thermal wavelength and the well width related results and investigations relevant to the issues described above are cited

eigenvectors of the multi component generalization of the six vertex model abstract the most general multi component solution of the star triangle equations under an ice type restriction is considered the transfer matrix of this model is diagonalized and the eigenvectors are written down explicitly using the quantum inverse scattering formalism the eigenvectors are shown to have simple symmetry properties under an exchange of arguments

langevin dynamics with constraints and computation of free energy differences in this paper we consider langevin processes with mechanical constraints the latter are a fundamental tool in molecular dynamics simulation for sampling purposes and for the computation of free energy differences the results of this paper can be divided into three parts i we propose a simple discretization of the constrained langevin process based on a standard splitting strategy we show how to correct the scheme so that it samples em exactly the canonical measure restricted on a submanifold using a metropolis rule in the spirit of the generalized hybrid monte carlo ghmc algorithm moreover we obtain in some limiting regime a consistent discretization of the overdamped langevin brownian dynamics on a submanifold also sampling exactly the correct canonical measure with constraints the corresponding numerical methods can be used to sample without any bias a probability measure supported by a submanifold ii for free energy computation using thermodynamic integration we rigorously prove that the longtime average of the lagrange multipliers of the constrained langevin dynamics yields the gradient of a rigid version of the free energy associated with the constraints a second order time discretization using the lagrange multipliers is proposed iii the jarzynski crooks fluctuation relation is proved for langevin processes with mechanical constraints evolving in time an original numerical discretization without time step error is proposed numerical illustrations are provided for ii and iii

a geometrical approach to computing free energy landscapes from short ranged potentials particles interacting with short ranged potentials have attracted increasing interest partly for their ability to model mesoscale systems such as colloids interacting via dna or depletion we consider the free energy landscape of such systems as the range of the potential goes to zero in this limit the landscape is entirely defined by geometrical manifolds plus a single control parameter these manifolds are fundamental objects that do not depend on the details of the interaction potential and provide the starting point from which any quantity characterizing the system equilibrium or nonequilibrium can be computed for arbitrary potentials to consider dynamical quantities we compute the asymptotic limit of the fokker planck equation and show that it becomes restricted to the low dimensional manifolds connected by sticky boundary conditions to illustrate our theory we compute the low dimensional manifolds for identical particles providing a complete description of the lowest energy parts of the landscape including floppy modes with up to 2 internal degrees of freedom the results can be directly tested on colloidal clusters this limit is a unique approach for understanding energy landscapes and our hope is that it can also provide insight into finite range potentials

a note on the de la garza phenomenon for locally optimal designs the celebrated de la garza phenomenon states that for a polynomial regression model of degree p 1 any optimal design can be based on at most p design points in a remarkable paper yang ann statist 38 2010 2499 2524 showed that this phenomenon exists in many locally optimal design problems for nonlinear models in the present note we present a different view point on these findings using results about moment theory and chebyshev systems in particular we show that this phenomenon occurs in an even larger class of models than considered so far

the quantum symmetric xxz chain at delta 1 2 alternating sign matrices and plane partitions we consider the ground state wavefunction of the quantum symmetric antiferromagnetic xxz chain with open and twisted boundary conditions at along with the ground state wavefunction of the corresponding o n loop model at n 1 based on exact results for finite size systems sums involving the wavefunction components and in some cases the largest component itself are conjectured to be directly related to the total number of alternating sign matrices and plane partitions in certain symmetry classes

bayesian estimation and experimental design in linear regression models estimation and design as a bayesian decision problem choice of a prior distribution conjugate prior distributions bayes estimation of the regression parameter optimality and robustness of the bayes estimator bayesian interpretation of estimators using non bayesian prior knowledge bayes estimation in case of prior ignorance further problems the design problem for the linear bayes estimator characterization of optimal designs construction of optimal continuous designs construction of exact optimal designs

effective and efficient global optimization for conceptual rainfall runoff models the successful application of a conceptual rainfall runoff crr model depends on how well it is calibrated despite the popularity of crr models reports in the literature indicate that it is typically difficult if not impossible to obtain unique optimal values for their parameters using automatic calibration methods unless the best set of parameters associated with a given calibration data set can be found it is difficult to determine how sensitive the parameter estimates and hence the model forecasts are to factors such as input and output data error model error quantity and quality of data objective function used and so on results are presented that establish clearly the nature of the multiple optima problem for the research crr model sixpar these results suggest that the crr model optimization problem is more difficult than had been previously thought and that currently used local search procedures have a very low probability of successfully finding the optimal parameter sets next the performance of three existing global search procedures are evaluated on the model sixpar finally a powerful new global optimization procedure is presented entitled the shuffled complex evolution sce ua method which was able to consistently locate the global optimum of the sixpar model and appears to be capable of efficiently and effectively solving the crr model optimization problem

computational experience with confidence regions and confidence intervals for nonlinear least squares we present the results of a monte carlo study of the leading methods for constructing approximate confidence regions and confidence intervals for parameters estimated by nonlinear least squares we examine three variants of the linearization method the likelihood method and the lack of fit method the linearization method is computationally inexpensive produces easily understandable results and is widely used in practice the likelihood and lack of fit methods are much more expensive and more difficult to report in our tests both the likelihood and lack of fit methods perform very reliably all three variants of the linearization method however often grossly underestimate confidence regions and sometimes significantly underestimate confidence intervals the linearization method variant based solely on the jacobian matrix appears preferable to the two variants that use the full hessian matrix because it is less expensive more numerically stable and at least as accurate the bates and watts curvat

the finite size su 3 perk schultz model with deformation parameter q exp 2iπ 3 from extensive numeric diagonalizations of the su 3 perk schultz hamiltonian with a special value of the anisotropy and different boundary conditions we have observed simple regularities for a significant part of its eigenspectrum in particular the ground state energy and nearby excitations belong to this part of the spectrum our simple formulae describing these regularities recall apart from some selection rules the eigenspectrum of the free fermion model based on the numerical observations we formulate several conjectures using explicit solutions of the associated nested bethe ansatz equations guessed from an analysis of the functional equations of the model we provide evidence for some of them

finding the optimal nets for self folding kirigami three dimensional shells can be synthesized from the spontaneous self folding of two dimensional templates of interconnected panels called nets however some nets are more likely to self fold into the desired shell under random movements the optimal nets are the ones that maximize the number of vertex connections i e vertices that have only two of its faces cut away from each other in the net previous methods for finding such nets are based on random search and thus they do not guarantee the optimal solution here we propose a deterministic procedure we map the connectivity of the shell into a shell graph where the nodes and links of the graph represent the vertices and edges of the shell respectively identifying the nets that maximize the number of vertex connections corresponds to finding the set of maximum leaf spanning trees of the shell graph this method allows us not only to design the self assembly of much larger shell structures but also to apply additional design criteria as a complete catalog of the maximum leaf spanning trees is obtained

the importance of being odd in this letter i mainly consider a finite xxz spin chain with periodic boundary conditions and an odd number of sites this system is described by the hamiltonian hxxz j 1n jx j 1x jy j 1y jz j 1z as it turns out the ground state energy is proportional to the number of sites e 3n 2 for a special value of the asymmetry parameter 1 2 the trigonometric polynomial q u the zeros of which are parameters of the ground state bethe eigenvector is explicitly constructed this polynomial of degree n n 1 2 satisfies the baxter t q equation using the second independent solution of this equation that corresponds to the same eigenvalue of the transfer matrix it is possible to find a derivative of the ground state energy w r t the asymmetry parameter this derivative is closely connected with the correlation function jz j 1z 1 2 3 2n2 this correlation function is related to the average number of spin strings for the ground state nstring n 1 n i would like to stress that all the above simple formulae are not applicable to the case of an even number of sites which is usually considered

approximate confidence limits for a parameter function in nonlinear regression on propose une methode a deux etapes la premiere utilisant une transformation des parametres simples pour definir de nouveaux parametres la seconde necessitant l utilisation d une region de confiance ellipsoidale pour obtenir un intervalle de confiance

on hermite interpolation by cauchy vandermonde systems the lagrange formula the adjoint and the inverse of a cauchy vandermonde matrix for a given cauchy vandermonde system and for given multiple nodes a lagrange type formula for the interpolant is derived interpolating a given function in the sense of hermite we give explicit analytic representations of the basic functions in terms of the nodes and prescribed poles they are used to derive formulas for the entries of the adjoint of the confluent cauchy vandermonde matrix corresponding to the interpolation problem thus providing an explicit representation of its inverse

the xxz spin chain at δ 1 2 bethe roots symmetric functions and determinants a number of conjectures have been given recently concerning the connection between the antiferromagnetic xxz spin chain at 1 2 and various symmetry classes of alternating sign matrices here we use the integrability of the xxz chain to gain further insight into these developments in doing so we obtain a number of new results using baxter s q function for the xxz chain for periodic twisted and open boundary conditions these include expressions for the elementary symmetric functions evaluated at the ground state solution of the bethe roots in this approach schur functions play a central role and enable us to derive determinant expressions which appear in certain natural double products over the bethe roots when evaluated these give rise to the numbers counting different symmetry classes of alternating sign matrices

optimal designs for estimating individual coefficients in fourier regression models in the common trigonometric regression model we investigate the optimal design problem for the estimation of the individual coefficients where the explanatory variable varies in the interval a a 0 a le pi it is demonstrated that the structure of the optimal design depends sensitively on the size of the design space for many important cases optimal designs can be found explicitly where the complexity of the solution depends on the value of the parameter a and the order of the term for which the corresponding coefficient has to be estimated the main tool of our approach is the reduction of the problem for the trigonometric regression model to a design problem for a polynomial regression in particular we determine the optimal designs for estimating the parameters corresponding to the cosine terms explicitly if the design space is sufficiently small and prove that under this condition all optimal designs for estimating the parameters corresponding to the sine terms are supported at the same points

hybrid monte carlo methods for sampling probability measures on submanifolds probability measures supported on submanifolds can be sampled by adding an extra momentum variable to the state of the system and discretizing the associated hamiltonian dynamics with some stochastic perturbation in the extra variable in order to avoid biases in the invariant probability measures sampled by discretizations of these stochastically perturbed hamiltonian dynamics a metropolis rejection procedure can be considered the so obtained scheme belongs to the class of generalized hybrid monte carlo ghmc algorithms we show here how to generalize to ghmc a procedure suggested by goodman holmes cerfon and zappa for metropolis random walks on submanifolds where a reverse projection check is performed to enforce the reversibility of the algorithm for any timesteps and hence avoid biases in the invariant measure we also provide a full mathematical analysis of such procedures as well as numerical experiments demonstrating the importance of the reverse projection check on simple toy examples

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

analysis of regression confidence intervals and bayesian credible intervals for uncertainty quantification 1 confidence intervals based on classical regression theories augmented to include prior information and credible intervals based on bayesian theories are conceptually different ways to quantify parametric and predictive uncertainties because both confidence and credible intervals are used in environmental modeling we seek to understand their differences and similarities this is of interest in part because calculating confidence intervals typically requires tens to thousands of model runs while bayesian credible intervals typically require tens of thousands to millions of model runs given multi gaussian distributed observation errors our theoretical analysis shows that for linear or linearized nonlinear models confidence and credible intervals are always numerically identical when consistent prior information is used for nonlinear models nonlinear confidence and credible intervals can be numerically identical if parameter confidence regions defined using the approximate likelihood method and parameter credible regions estimated using markov chain monte carlo realizations are numerically identical and predictions are a smooth monotonic function of the parameters both occur if intrinsic model nonlinearity is small while the conditions of gaussian errors and small intrinsic model nonlinearity are violated by many environmental models heuristic tests using analytical and numerical models suggest that linear and nonlinear confidence intervals can be useful approximations of uncertainty even under significantly nonideal conditions in the context of epistemic model error for a complex synthetic nonlinear groundwater problem the linear and nonlinear confidence and credible intervals for individual models performed similarly enough to indicate that the computationally frugal confidence intervals can be useful in many circumstances experiences with these groundwater models are expected to be broadly applicable to many environmental models we suggest that for environmental problems with lengthy execution times that make credible intervals inconvenient or prohibitive confidence intervals can provide important insight during model development when frequent calculation of uncertainty intervals is important to understanding the consequences of various model construction alternatives and data collection strategies strategic use of both confidence and credible intervals can be critical

exact scheffe type confidence intervals for output from groundwater flow models 2 combined use of hydrogeologic information and calibration data calibration data observed values corresponding to model computed values of dependent variables are incorporated into a general method of computing exact scheffe type confidence intervals analogous to the confidence intervals developed in part 1 cooley this issue for a function of parameters derived from a groundwater flow model parameter uncertainty is specified by a distribution of parameters conditioned on the calibration data this distribution was obtained as a posterior distribution by applying bayes theorem to the hydrogeologically derived prior distribution of parameters from part 1 and a distribution of differences between the calibration data and corresponding model computed dependent variables tests show that the new confidence intervals can be much smaller than the intervals of part 1 because the prior parameter variance covariance structure is altered so that combinations of parameters that give poor model fit to the data are unlikely the confidence intervals of part 1 and the new confidence intervals can be effectively employed in a sequential method of model construction whereby new information is used to reduce confidence interval widths at each stage

physiological pharmacokinetic analysis using population modeling and informative prior distributions abstract we describe a general approach using bayesian analysis for the estimation of parameters in physiological pharmacokinetic models the chief statistical difficulty in estimation with these models is that any physiological model that is even approximately realistic will have a large number of parameters often comparable to the number of observations in a typical pharmacokinetic experiment e g 28 measurements and 15 parameters for each subject in addition the parameters are generally poorly identified akin to the well known ill conditioned problem of estimating a mixture of declining exponentials our modeling includes a hierarchical population modeling which allows partial pooling of information among different experimental subjects b a pharmacokinetic model including compartments for well perfused tissues poorly perfused tissues fat and the liver and c informative prior distributions for population parameters which is possible because the parameters represent real physiological

practical scheffe type credibility intervals for variables of a groundwater model simultaneous scheffe type credibility intervals the bayesian version of confidence intervals for variables of a groundwater flow model calibrated using a bayesian maximum a posteriori procedure were derived by cooley 1993b it was assumed that variances reflecting the expected differences between observed and model computed quantities used to calibrate the model are known whereas they would often be unknown for an actual model in this study the variances are regarded as unknown and variance variability from observation to observation is approximated by grouping the data so that each group is characterized by a uniform variance the credibility intervals are calculated from the posterior distribution which was developed by considering each group variance to be a random variable about which nothing is known a priori then eliminating it by integration numerical experiments using two test problems illustrate some characteristics of the credibility intervals nonlinearity of the statistical model greatly affected some of the credibility intervals indicating that credibility intervals computed using the standard linear model approximation may often be inadequate to characterize uncertainty for actual field problems the parameter characterizing the probability level for the credibility intervals was however accurately computed using a linear model approximation as compared with values calculated using second order and fully nonlinear formulations this allows the credibility intervals to be computed very efficiently

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

spin chains and combinatorics in this paper we continue the investigation of finite xxz spin chains with periodic boundary conditions and odd number of sites initiated in our previous article stroganov yu g 2001 j phys a math gen 34 l179 85 as it turned out for a special value of the asymmetry parameter 1 2 the hamiltonian of the system has an eigenvalue which is exactly proportional to the number of sites e 3n 2 using mathematica we have found explicitly the corresponding eigenvectors for n 17 the obtained results support the conjecture our paper that this special eigenvalue corresponds to the ground state vector we make a lot of conjectures concerning the correlations of the model many remarkable relations between the wavefunction components are noted it turns out for example that the ratio of the largest component to the least one is equal to the number of the alternating sign matrices

models and estimation methods for clinical hiv 1 data clinical hiv 1 data include many individual factors such as compliance to treatment pharmacokinetics variability in respect to viral dynamics race sex income etc which might directly influence or be associated with clinical outcome these factors need to be taken into account to achieve a better understanding of clinical outcome and mathematical models can provide a unifying framework to do so the first objective of this paper is to demonstrate the development of comprehensive hiv 1 dynamics models that describe viral dynamics and also incorporate different factors influencing such dynamics the second objective of this paper is to describe alternative estimation methods that can be applied to the analysis of data with such models in particular we consider i simple but effective two stage estimation methods in which data from each patient are analyzed separately and summary statistics derived from the results ii more complex nonlinear mixed effect models used to pool all the patient data in a single analysis bayesian estimation methods are also considered in particular iii maximum posterior approximations map and iv markov chain monte carlo mcmc bayesian methods incorporate prior knowledge into the models thus avoiding some of the model simplifications introduced when the data are analyzed using two stage methods or a nonlinear mixed effect framework we demonstrate the development of the models and the different estimation methods using real aids clinical trial data involving patients receiving multiple drugs regimens

bayesian design for dose response curves with penalized risk this paper considers some bayesian design problems in quantal response analysis an experimenter must choose a set of dose levels and number of independent observations to take at these levels subject to a total sample size in order to estimate some characteristic phi e g ed50 of a tolerance distribution f theta where theta is the vector of unknown parameters it is shown for the logistic model that the sampling variability of the posterior variance of phi is such that the predicted posterior variance alone is an undesirable criterion for design selection a family of penalty functions is introduced that penalizes any excess in the posterior variance over the expected or predicted variance and protects against unexpected outcomes the goal is to find a design that avoids experimental results with little information at the expense of a small sacrifice in the bayes risk numerical results indicate that the chance of an extreme posterior variance can be reduced by sacrificing a small amount of posterior risk

an exact algorithm for the maximum leaf spanning tree problem given an undirected graph with n vertices the maximum leaf spanning tree problem is to find a spanning tree with as many leaves as possible when parameterized in the number of leaves k this problem can be solved in time o 4kpoly n using a simple branching algorithm introduced by a subset of the authors kneis et al 2008 16 daligault et al 2010 6 improved the branching and obtained a running time of o 3 72kpoly n in this paper we study the problem from an exponential time viewpoint where it is equivalent to the connected dominating set problem here fomin grandoni and kratsch showed how to break the 2n barrier and proposed an o 1 9407n time algorithm fomin et al 2008 11 based on some useful properties of kneis et al 2008 16 and daligault et al 2010 6 we present a branching algorithm whose running time of o 1 8966n has been analyzed using the measure and conquer technique finally we provide a lower bound of 1 4422n for the worst case running time of our algorithm

free fermion branches in some quantum spin models extensive numerical analysis of the eigenspectra of the suq n invariant perk schultz hamiltonian shows some simple regularities for a significant part of the eigenspectrum inspired by those results we have found two sets of solutions of the associated nested bethe ansatz equations the first set is obtained at a special value of the anisotropy q exp i n 1 n and describes in particular the ground state and nearby excitations as a sum of free fermion quasienergies the second set of solutions provides the energies in the sectors whose number ni of particles of distinct species i 0 n 1 are less than or equal to unity except for one of the species for this last set we obtain the eigenspectra of a free fermion model for arbitrary values of the anisotropy

improved methodology for parameter inference in nonlinear hydrologic regression models a new method is developed for the construction of reliable marginal confidence intervals and joint confidence regions for the parameters of nonlinear hydrologic regression models a parameter power transformation is combined with measures of the asymptotic bias and asymptotic skewness of maximum likelihood estimators to determine the transformation constants which cause the bias or skewness to vanish these optimized constants are used to construct confidence intervals and regions for the transformed model parameters using linear regression theory the resulting confidence intervals and regions can be easily mapped into the original parameter space to give close approximations to likelihood method confidence intervals and regions for the model parameters unlike many other approaches to parameter transformation the procedure does not use a grid search to find the optimal transformation constants an example involving the fitting of the michaelis menten model to velocity discharge data from an australian gauging station is used to illustrate the usefulness of the methodology

new conjecture for the suq n perk schultz models we present a new conjecture for the suq n perk schultz models this conjecture extends a conjecture presented in our article alcaraz f c and stroganov yu g j phys a math gen 35 6767 87

designing experiments with respect to standardized optimality criteria we introduce a new class of standardized optimality criteria which depend on standardized covariances of the least squares estimators and provide an alternative to the commonly used criteria in design theory besides a nice statistical interpretation the new criteria satisfy an extremely useful invariance property which allows an easy calculation of optimal designs on many linearly transformed design spaces

emptiness formation probability of the xxz spin heisenberg chain at δ using a multiple integral representation for the correlation functions we compute the emptiness formation probability of the xxz spin heisenberg chain at anisotropy we prove that it is expressed in terms of the number of alternating sign matrices

optimal designs for a class of nonlinear regression models for a broad class of nonlinear regression models we investigate the local e and c optimal design problem it is demonstrated that in many cases the optimal designs with respect to these optimality criteria are supported at the chebyshev points which are the local extrema of the equi oscillating best approximation of the function f 0 0 by a normalized linear combination of the regression functions in the corresponding linearized model the class of models includes rational logistic and exponential models and for the rational regression models the e and c optimal design problem is solved explicitly in many cases

algorithmic design of self folding polyhedra self assembly has emerged as a paradigm for highly parallel fabrication of complex three dimensional structures however there are few principles that guide a priori design yield and defect tolerance of self assembling structures we examine with experiment and theory the geometric principles that underlie self folding of submillimeter scale higher polyhedra from two dimensional nets in particular we computationally search for nets within a large set of possibilities and then test these nets experimentally our main findings are that i compactness is a simple and effective design principle for maximizing the yield of self folding polyhedra and ii shortest paths from 2d nets to 3d polyhedra in the configuration space are important for rationalizing experimentally observed folding pathways our work provides a model problem amenable to experimental and theoretical analysis of design principles and pathways in self assembly

structure of finite sphere packings via exact enumeration implications for colloidal crystal nucleation we analyze the geometric structure and mechanical stability of a complete set of isostatic and hyperstatic sphere packings obtained via exact enumeration the number of nonisomorphic isostatic packings grows exponentially with the number of spheres n and their diversity of structure and symmetry increases with increasing n and decreases with increasing hyperstaticity h n c n iso where n c is the number of pair contacts and n iso 3n 6 maximally contacting packings are in general neither the densest nor the most symmetric analyses of local structure show that the fraction f of nuclei with order compatible with the bulk rhcp crystal decreases sharply with increasing n due to a high propensity for stacking faults five and near fivefold symmetric structures and other motifs that preclude rhcp order while f increases with increasing h a significant fraction of hyperstatic nuclei for n as small as 11 retain non rhcp structure classical theories of nucleation that consider only spherical nuclei or only nuclei with the same ordering as the bulk crystal cannot capture such effects our results provide an explanation for the failure of classical nucleation theory for hard sphere systems of n 10 particles we argue that in this size regime it is essential to consider nuclei of unconstrained geometry our results are also applicable to understanding kinetic arrest and jamming in systems that interact via hard core like repulsive and short ranged attractive interactions

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

free energy of singular sticky sphere clusters networks of particles connected by springs model many condensed matter systems from colloids interacting with a short range potential and complex fluids near jamming to self assembled lattices and various metamaterials under small thermal fluctuations the vibrational entropy of a ground state is given by the harmonic approximation if it has no zero frequency vibrational modes yet such singular modes are at the epicenter of many interesting behaviors in the systems above we consider a system of n spherical particles and directly account for the singularities that arise in the sticky limit where the pairwise interaction is strong and short ranged although the contribution to the partition function from singular clusters diverges in the limit its asymptotic value can be calculated and depends on only two parameters characterizing the depth and range of the potential the result holds for systems that are second order rigid a geometric characterization that describes all known ground state rigid sticky clusters to illustrate the applications of our theory we address the question of emergence how does crystalline order arise in large systems when it is strongly disfavored in small ones we calculate the partition functions of all known rigid clusters up to n 21 and show the cluster landscape is dominated by hyperstatic clusters those with more than 3n 6 contacts singular and isostatic clusters are far less frequent despite their extra vibrational and configurational entropies since the most hyperstatic clusters are close to fragments of a close packed lattice this underlies the emergence of order in sticky sphere systems even those as small as n 10

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

simultaneous confidence and prediction intervals for nonlinear regression models with application to a groundwater flow model methods are presented for computing three types of simultaneous confidence and prediction intervals exact likelihood ratio and linearized on output from nonlinear regression models with normally distributed residuals the confidence intervals can be placed on individual regression parameters or on the true regression function at any number of points in the domain of the independent variables and the prediction intervals can be placed on any number of future observations the confidence intervals are analogous to simultaneous scheffe intervals for linear models and the prediction intervals are analogous to the prediction intervals of hahn 1972 all three types of intervals can be computed efficiently by using the same straightforward lagrangian optimization scheme the prediction intervals can be treated in the same computational framework as the confidence intervals by including the random errors as pseudoparameters in the lagrangian scheme the methods are applied to a hypothetical groundwater model for flow to a well penetrating a leaky aquifer three different data sets are used to demonstrate the effect of sampling strategies on the intervals for all three data sets the linearized confidence intervals are inferior to the exact and likelihood ratio intervals with the latter two being very similar however all three types of prediction intervals yielded similar results the third data set time drawdown data at only a single observation well points out many of the problems that can arise from extreme nonlinear behavior of the regression model

newton raphson and em algorithms for linear mixed effects models for repeated measures data abstract we develop an efficient and effective implementation of the newton raphson nr algorithm for estimating the parameters in mixed effects models for repeated measures data we formulate the derivatives for both maximum likelihood and restricted maximum likelihood estimation and propose improvements to the algorithm discussed by jennrich and schluchter 1986 to speed convergence and ensure a positive definite covariance matrix for the random effects at each iteration we use matrix decompositions to develop efficient and computationally stable implementations of both the nr algorithm and an em algorithm laird and ware 1982 for this model we compare the two methods em vs nr in terms of computational order and performance on two sample data sets and conclude that in most situations a well implemented nr algorithm is preferable to the em algorithm or em algorithm with aitken s acceleration the term repeated measures refers to experimental designs where there are several individuals and several

projection of diffusions on submanifolds application to mean force computation we consider the problem of sampling a boltzmann gibbs probability distribution when this distribution is restricted in some suitable sense on a manifold of r n implicitly defined by n constraints q1 x qn x 0 this problem arises for example in systems subject to hard constraints or in the context of free energy calculations we prove that the constrained stochastic di erential equations i e di usions proposed in w e and e vanden eijnden in multiscale modelling and simulation eds s attinger and p koumoutsakos lncse 39 springer 2004 and g ciccotti r kapral and e vanden eijnden chemphyschem 6 1809 2005 are ergodic with respect to this restricted distribution we also construct numerical schemes for the integration of the constrained di usions finally we show how these schemes can be used to compute the gradient of the free energy associated with the constraints

exact scheffe type confidence intervals for output from groundwater flow models 1 use of hydrogeologic information a new method is developed to efficiently compute exact scheffe type confidence intervals for output or other function of parameters g beta derived from a groundwater flow model the method is general in that parameter uncertainty can be specified by any statistical distribution having a log probability density function log pdf that can be expanded in a taylor series however for this study parameter uncertainty is specified by a statistical multivariate beta distribution that incorporates hydrogeologic information in the form of the investigator s best estimates of parameters and a grouping of random variables representing possible parameter values so that each group is defined by maximum and minimum bounds and an ordering according to increasing value the new method forms the confidence intervals from maximum and minimum limits of g beta on a contour of a linear combination of 1 the quadratic form for the parameters used by cooley and vecchia 1987 and 2 the log pdf for the multivariate beta distribution three example problems are used to compare characteristics of the confidence intervals for hydraulic head obtained using different weights for the linear combination different weights generally produced similar confidence intervals whereas the method of cooley and vecchia 1987 often produced much largermore confidence intervals less

bayesian approach to the analysis of nonlinear models implementation and evaluation a traditional approach to parameter estimation and hypothesis testing in nonlinear models is based on least squares procedures error analysis depends on large sample theory bayesian analysis is an alternative approach that avoids substantial errors which could result frorn this dependence this communication is concerned with the implementation of the bayesian approach as an alternative to least squares nonlinear regression special attention is given to the numerical evaluation of multiple integrals and to the behavior of the parameter estimators and their estimated covariances the bayesian approach is evaluated in the light of practical as well as theoretical considerations 1 inttoduction the traditional approach to the statistical analysis of nonlinear models is first to use some numerical method to minimize the sum of squares objective function in order to obtain least squares estimators of the parameters draper and smith 1966 pp 267 275 nelder and mead 1965 and then to apply linear regression theory to the linear part of the taylor series approximation of the model expanded about these estimators in order to obtain the asymptotic covariance matrix bard 1974 pp 176 179 the distributions of the estimators obtained in this manner are known only in the limit as the sample size approaches infinity jennrich 1969 hence analyses based on these statistics may be inappropriate for small sample problems such as those arising in pharmacokinetics wagner 1975 an alternative approach to the statistical analysis of nonlinear models is to utilize methods based on bayes s theorem box and tiao 1973 pp 1 73 parameters are regarded as random variables rather than as unknown constants if a nonlinear model with known error distribution is assumed a correct probability analysis follows and asymptotic theory is not involved in this communication a bayesian approach to nonlinear regression is implemented and evaluated particular attention is given to numerical integration methods and the calculation of confidence regions using the posterior distributions

e optimal designs for rational models in this paper experimental designs for a rational model y p x q x e are investigated where p x o 1 x p x p and q x 1 p 1 x p q x q are polynomials and e is a random error two approaches bayesian d optimal and bayesian optimal design for extrapolation are examined the first criterion maximizes the expected increase in shannon information provided by the experiment asymptotically and the second minimizes the asymptotic variance of the maximum likelihood estimator mle of the mean response at an extrapolation point x e corresponding locally optimal designs are also discussed conditions are derived under which a p q 1 point design is a locally d optimal design the bayesian d optimal design is shown to be independent of the parameters in p x and to be equally weighted at each support point if the number of support points is the same as the number of parameters in the model the existence and uniqueness of the locally optimal design for extrapolation are proven the number of support points for the locally optimal design for extrapolation is exactly p q 1 these p q 1 design points are proved to be independent of the extrapolation point x e and the parameters in p x the corresponding weights are also independent of the parameters in p x but depend on x e and are not equal

the wavefunctions for the free fermion part of the spectrum of the suq n quantum spin models we conjecture that the free fermion part of the eigenspectrum observed recently for the suq n perk schultz spin chain hamiltonian in a finite lattice with q exp i n 1 n is a consequence of the existence of a special simple eigenvalue for the transfer matrix of the auxiliary inhomogeneous suq n 1 vertex model which appears in the nested bethe ansatz approach we prove that this conjecture is valid for the case of the suq 3 spin chain with periodic boundary condition in this case we obtain a formula for the components of the eigenvector of the auxiliary inhomogeneous 6 vertex model q exp 2i 3 which permits us to find one by one all components of this eigenvector and consequently to find the eigenvectors of the free fermion part of the eigenspectrum of the suq 3 spin chain similarly as in the known case of the suq 2 case at q exp i2 3 our numerical and analytical studies induce some conjectures for special rates of correlation functions