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Densest local sphere-packing diversity. II. Application to three dimensions

Physical Review E 2011年第1期83卷 011304-011304页

作者： Adam B. Hopkins Frank H. Stillinger Salvatore Torquato Department of Chemistry and Department of Physics Princeton Center for Theoretical Science Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

The densest local packings of N three-dimensional identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained for selected values of N up to N=1054. In the predecessor to this paper [A. B. Hopkins, F. H. Stillinger, and S. Torquato, Phys. Rev. E 81, 041305 (2010)], we described our method for finding the putative densest packings of N spheres in d-dimensional Euclidean space Rd and presented those packings in R2 for values of N up to N=348. Here we analyze the properties and characteristics of the densest local packings in R3 and employ knowledge of the Rmin(N), using methods applicable in any d, to construct both a realizability condition for pair correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings. In R3, we find wide variability in the densest local packings, including a multitude of packing symmetries such as perfect tetrahedral and imperfect icosahedral symmetry. We compare the densest local packings of N spheres near a central sphere to minimal-energy configurations of N+1 points interacting with short-range repulsive and long-range attractive pair potentials, e.g., 12–6 Lennard-Jones, and find that they are in general completely different, a result that has possible implications for nucleation theory. We also compare the densest local packings to finite subsets of stacking variants of the densest infinite packings in R3 (the Barlow packings) and find that the densest local packings are almost always most similar as measured by a similarity metric, to the subsets of Barlow packings with the smallest number of coordination shells measured about a single central sphere, e.g., a subset of the fcc Barlow packing. Additionally, we observe that the densest local packings are dominated by the dense arrangement of spheres with centers at distance Rmin(N). In particular, we find two “maracas” packings at N=77 and N=93, each consisting of a few unjammed sphere

关键词： DENSITY functionals SPHERES MATHEMATICAL models EUCLIDEAN metric CORRELATION (Statistics) SYMMETRY

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Comprehensive analysis of compositional interface fluctuations in planar lipid bilayer membranes

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Physical Review E 2011年第5期84卷 051903-051903页

作者： Tao Han Mikko Haataja Department of Mechanical and Aerospace Engineering Princeton Institute for the Science and Technology of Materials (PRISM) and Program in Applied and Computational Mathematics (PACM) Princeton University Princeton New Jersey 08544 USA

In this paper we present a comprehensive analysis of line tension-driven compositional interface fluctuations in planar lipid bilayer membranes. Our starting point is the advective Cahn-Hilliard equation for the local lipid composition in symmetric membranes, which explicitly incorporates both advective and diffusive lipid transport processes, and which is coupled to the continuum hydrodynamic equations governing the flow behavior of the membrane and surrounding solvent with finite subphase thickness. In order to extract the interface dynamics from the continuum phase-field formalism, we first derive the appropriate sharp-interface limit equations. We then carry out a linear perturbation analysis for the relaxational dynamics of small-amplitude sinusoidal interface fluctuations to yield the general dispersion relation ωk as a function of perturbation wave number k. The resulting expression incorporates the effects of diffusive and advective lipid transport processes within the membrane, viscous or viscoelastic membrane properties, coupling between membrane and solvent, and inertial effects within the membrane and solvent. It is shown that previously considered scenarios naturally emerge as limiting cases of the general result. Furthermore, we discuss two additional scenarios amenable to analysis, one in which the inertia of the solvent is relevant, and another one in which the membrane displays significant viscoelastic properties. Finally, we numerically evaluate the general dispersion relation for three representative model membrane systems.

关键词： BILAYER lipid membranes FLUCTUATIONS (Physics) PERTURBATION (mathematics) VISCOELASTICITY HYDRODYNAMICS

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Coarse graining the dynamics of heterogeneous oscillators in networks with spectral gaps

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Physical Review E 2011年第3期84卷 036708-036708页

作者： Karthikeyan Rajendran Ioannis G. Kevrekidis Department of Chemical and Biological Engineering Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics (PACM) Princeton University Princeton New Jersey 08544 USA

We present a computer-assisted approach to coarse graining the evolutionary dynamics of a system of nonidentical oscillators coupled through a (fixed) network structure. The existence of a spectral gap for the coupling network graph Laplacian suggests that the graph dynamics may quickly become low dimensional. Our first choice of coarse variables consists of the components of the oscillator states—their (complex) phase angles—along the leading eigenvectors of this Laplacian. We then use the equation-free framework, circumventing the derivation of explicit coarse-grained equations, to perform computational tasks such as coarse projective integration, coarse fixed-point, and coarse limit-cycle computations. In a second step, we explore an approach to incorporating oscillator heterogeneity in the coarse-graining process. The approach is based on the observation of fast-developing correlations between oscillator state and oscillator intrinsic properties and establishes a connection with tools developed in the context of uncertainty quantification.

关键词： GRAINING COMPUTER-aided design LAPLACIAN operator EIGENVECTORS HARMONIC oscillators

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Direct minimization of the optimized effective problem based on efficient finite differences

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Physical Review B 2011年第16期84卷 165122-165122页

作者： Chen Huang Emily A. Carter Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics and the Andlinger Center for Energy and the Environment Princeton University Princeton New Jersey 08544-5263 USA

To avoid the difficult-to-solve optimized effective potential (OEP) integral equation, we introduce an efficient direct minimization scheme for performing OEP calculations within Kohn–Sham density functional theory (KS-DFT). We reformulated the functional derivative of the total energy with respect to the KS effective potential in terms of efficient finite differences. Our method only uses the orbitals involved in the construction of the KS exchange-correlation functionals. We demonstrate our scheme by performing exact-exchange OEP for sodium clusters, in which only occupied KS orbitals are needed to obtain the OEP. Our efficient direct minimization scheme should aid future development of orbital-dependent density functionals and render OEP to be a practical choice for various applications.

关键词： INTEGRAL equations DENSITY functionals TOTAL energy systems (On-site electric power production) FINITE differences PHYSICS

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Linear optimal estimation problems in systems with actuator faults

Linear optimal estimation problems in systems with actuator ...

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International Conference on Information Fusion

作者： Daniel Sigalov Yaakov Oshman Program for Applied Mathematics Technion-Israel Institute of Technology Haifa Israel Department of AeroSpace Engineering Technion-Israel Institute of Technology Haifa Israel

ISBN: (纸本)9781457702679

We consider estimating the state of a dynamic system subject to actuator faults. The discretely-valued fault mechanism renders the system hybrid, and results in anomalous changes in the dynamics equation that may be interpreted as random accelerations. Two closely related problem formulations are considered. In the first formulation multiple models are used to describe the system's behavior: one model stands for the nominal, fault-free actuator condition, all other models correspond to various actuator fault conditions, and the system can freely assume any model at any time. In the second formulation the abnormal mode is described by a single dynamical model, and the system can switch between the nominal and anomalous conditions a bounded number of times, with the bound assumed known. In both formulations, the minimum mean squared error (MMSE) optimal state estimator requires a polynomially growing number of primitive Kalman filters, and is, thus, computationally infeasible. We derive sequential, linear MMSE-optimal state estimation algorithms for both problem formulations. Depending only on the first two moments of the random quantities of the problem, linear optimal filters are robust with respect to the actual driving noise distributions, in the sense that they achieve the smallest worst-case estimation error of all other (nonlinear) filters. Although derived assuming seemingly different problem formulations, both filters share essentially the same structure, thus exposing a certain duality between the underlying problems. The performance of both estimators is demonstrated in a simulation study, where they are compared to the interacting multiple model filter.

关键词： Actuators Mathematical model Noise Equations Switches Covariance matrix Aerodynamics

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Tracking maneuvering targets with a soft bound on the number of maneuvers

Tracking maneuvering targets with a soft bound on the number...

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International Conference on Information Fusion

ISBN: (纸本)9781457702679

We revisit the problem of tracking the state of a hybrid system capable of performing a bounded number of mode switches. In a previous paper we have addressed a version of the problem where we have assumed the existence of a deterministic, known hard bound on the number of mode transitions. In addition, it was assumed that the system can possess only two modes, e.g., the maneuvering and non-maneuvering regimes of a tracked target. In the present paper we relax both assumptions: we assume a soft, stochastic bound on the number of mode transitions, and altogether remove the restriction on the number of modes of the system (thus, e.g., the target can have multiple different maneuvering modes, in addition to the non-maneuvering one). Similarly to the case where the number of transition was deterministically hard-bounded, the existence of the bound renders the mode switching mechanism non-Markov. Thus, the two formulations address similar, though not identical, problems, that cannot be solved by direct application of standard algorithms for hybrid systems. The novel solution approach is based on transforming the non-Markovian mode switching mechanism to an equivalent Markovian one, at the price of augmenting the mode definition. A standard interacting multiple model (IMM) filter is then applied to the transformed problem in a straightforward manner. The performance of the new method is demonstrated via a simulation study comprising three examples, in which the new method is compared with 1) the filter for hard-bounded mode transitions, and 2) a standard IMM filter directly applied to the original problem. The study shows that even when working outside its operating envelope, the new filter closely approximates the best filter for the scenario.

关键词： Markov processes Switches Heuristic algorithms Estimation Target tracking Adaptation models

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Frame coherence and sparse signal processing

Frame coherence and sparse signal processing

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IEEE International Symposium on Information Theory

作者： Dustin G. Mixon Waheed U. Bajwa Robert Calderbank Program in Applied and Computational Mathematics Princeton University Princeton NJ USA Department of Electrical and Computer Engineering Duke University Durham NC USA

The sparse signal processing literature often uses random sensing matrices to obtain performance guarantees. Unfortunately, in the real world, sensing matrices do not always come from random processes. It is therefore desirable to evaluate whether an arbitrary matrix, or frame, is suitable for sensing sparse signals. To this end, the present paper investigates two parameters that measure the coherence of a frame: worst-case and average coherence. We first provide several examples of frames that have small spectral norm, worst-case coherence, and average coherence. Next, we present a new lower bound on worst-case coherence and compare it to the Welch bound. Later, we propose an algorithm that decreases the average coherence of a frame without changing its spectral norm or worst-case coherence. Finally, we use worst-case and average coherence, as opposed to the Restricted Isometry Property, to garner near-optimal probabilistic guarantees on both sparse signal detection and reconstruction in the presence of noise. This contrasts with recent results that only guarantee noiseless signal recovery from arbitrary frames, and which further assume independence across the nonzero entries of the signal-in a sense, requiring small average coherence replaces the need for such an assumption.

关键词： Coherence Signal processing algorithms Noise Harmonic analysis Noise measurement Geometry

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Particle-scale structure in frozen colloidal suspensions from small-angle x-ray scattering

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Physical Review E 2011年第2期83卷 021402-021402页

作者： Melissa Spannuth S. G. J. Mochrie S. S. L. Peppin J. S. Wettlaufer Department of Geology and Geophysics Yale University New Haven Connecticut 06520 USA Department of Physics Yale University New Haven Connecticut 06520 USA and Program in Applied Mathematics Yale University New Haven Connecticut 06520 USA

During directional solidification of the solvent in a colloidal suspension, the colloidal particles segregate from the growing solid, forming high-particle-density regions with structure on a hierarchy of length scales ranging from that of the particle-scale packing to the large-scale spacing between these regions. Previous work has concentrated mostly on the medium- to large-length scale structure, as it is the most accessible and thought to be more technologically relevant. However, the packing of the colloids at the particle scale is an important component not only in theoretical descriptions of the segregation process, but also to the utility of freeze-cast materials for new applications. Here we present the results of experiments in which we investigated this structure across a wide range of length scales using a combination of small-angle x-ray scattering and direct optical imaging. As expected, during freezing the particles were concentrated into regions between ice dendrites forming a microscopic pattern of high- and low-particle-density regions. X-ray scattering indicates that the particles in the high-density regions were so closely packed as to be touching. However, the arrangement of the particles does not conform to that predicted by standard interparticle pair potentials, suggesting that the particle packing induced by freezing differs from that formed during equilibrium densification processes.

关键词： Small angle X ray scattering Solid state forms Freezing Colloids Frozen Segregation process Directional solidification

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Overlapping batches for the assessment of solution quality in stochastic programs 11

Overlapping batches for the assessment of solution quality i...

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Simulation Winter Conference

作者： David Love Güzin Bayraksan Program in Applied Mathematics University of Arizona Tucson Tucson AZ USA Department of Systems and Industrial Engineering University of Arizona Tucson Tucson AZ USA

We investigate the use of overlapping batches for assessing solution quality in stochastic programs. Motivated by the original use of overlapping batches in simulation, we present a variant of the multiple replications procedure that reuses data via variably overlapping batches to obtain alternative variance estimators. These estimators have lower variances, where the degree of variance reduction depends on the amount of overlap. We provide several asymptotic properties and present computational results to examine small-sample behavior.

关键词： Materials requirements planning Upper bound Approximation methods Computational modeling Stochastic processes Optimization Data models

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Socially induced ovulation synchrony and its effect on seabird population dynamics

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Journal of Biological Dynamics 2011年第5期5卷 495-516页

作者： Henson, Shandelle M. Cushing, J.M. Hayward, James L. Department of Mathematics Andrews University Berrien Springs MI 49104 United States Department of Mathematics Interdisciplinary Program in Applied Mathematics University of Arizona Tucson AZ 85721 United States Department of Biology Andrews University Berrien Springs MI 49104 United States

Spontaneous oscillator synchrony is a form of self-organization in which populations of interacting oscillators ultimately cycle together. This phenomenon occurs in a wide range of physical and biological systems. In rats and humans, oestrous/menstrual cycles synchronize through social stimulation with pheromones acting as synchronizing signals. In previous work, we showed that glaucous-winged gulls (Larus glaucescens) can lay eggs synchronously on an every-other-day schedule, and that synchrony increases with colony density. We posed a discrete-time mathematical model for reproduction during the breeding season based on the hypothesis that pre-ovulatory luteinizing hormone surges synchronize by means of visual, auditory and/or olfactory cues. Here, we extend the seasonal model in order to investigate the effect of ovulation synchrony on population dynamics across reproductive seasons. We show that socially stimulated ovulation synchrony can enhance total population size and allow the population to persist at lower birth rates than would otherwise be possible. © 2011 Taylor & Francis.

关键词： Bifurcation Cannibalism Discrete-time model Gull colony Ovulation synchrony

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