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Method for obtaining upper bounds on photonic band gaps

Physical Review B 2009年第15期80卷 155126-155126页

作者： Mikael C. Rechtsman Salvatore Torquato Courant Institute of Mathematical Sciences 251 Mercer Street New York New York 10012 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Physics Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton Institute for the Science and Technology of Materials Princeton New Jersey 08544 USA School of Natural Sciences Institute for Advanced Study Princeton New Jersey 08544 USA

We present a method to calculate upper bounds on the photonic band gaps of two-component photonic crystals. The method involves calculating both upper and lower bounds on the frequency bands for a given structure, and then maximizing over all possible two-component structures. We apply this method to a number of examples, including a one-dimensional photonic crystal (or “Bragg grating”) and two-dimensional photonic crystals (in both the TM and TE polarizations) with both four and sixfold rotational symmetries. We compare the bounds to band gaps of numerically optimized structures and find that the bounds are extremely tight. We prove that the bounds are “sharp” in the limit of low dielectric contrast ratio between the two components. This method and the bounds derived here have important implications in the search for optimal photonic band-gap structures.

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Dense packings of polyhedra: Platonic and Archimedean solids

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Physical Review E 2009年第4期80卷 041104-041104页

作者： S. Torquato Y. Jiao Department of Chemistry Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA School of Natural Sciences Institute for Advanced Study Princeton New Jersey 08540 USA Department of Mechanical and Aerospace Engineering Princeton University Princeton New Jersey 08544 USA

Understanding the nature of dense particle packings is a subject of intense research in the physical, mathematical, and biological sciences. The preponderance of previous work has focused on spherical particles and very little is known about dense polyhedral packings. We formulate the problem of generating dense packings of nonoverlapping, nontiling polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem, which we call the adaptive shrinking cell (ASC) scheme. This optimization problem is solved here (using a variety of multiparticle initial configurations) to find the dense packings of each of the Platonic solids in three-dimensional Euclidean space R3, except for the cube, which is the only Platonic solid that tiles space. We find the densest known packings of tetrahedra, icosahedra, dodecahedra, and octahedra with densities 0.823…, 0.836…, 0.904…, and 0.947…, respectively. It is noteworthy that the densest tetrahedral packing possesses no long-range order. Unlike the densest tetrahedral packing, which must not be a Bravais lattice packing, the densest packings of the other nontiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. We also derive a simple upper bound on the maximal density of packings of congruent nonspherical particles and apply it to Platonic solids, Archimedean solids, superballs, and ellipsoids. Provided that what we term the “asphericity” (ratio of the circumradius to inradius) is sufficiently small, the upper bounds are relatively tight and thus close to the corresponding densities of the optimal lattice packings of the centrally symmetric Platonic and Archimedean solids. Our simulation results, rigorous upper bounds, and other theoretical arguments lead us to the conjecture that the densest packings of Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. This can be regarded to be

关键词： Optimization

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Diffusion maps - A probabilistic interpretation for spectral embedding and clustering algorithms

Diffusion maps - A probabilistic interpretation for spectral...

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作者： Nadler, Boaz Lafon, Stephane Coifman, Ronald Kevrekidis, Ioannis G. Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot 76100 Israel Google Inc. United States Department of Mathematics Yale University New Haven CT 06520-8283 United States Department of Chemical Engineering Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States

ISBN: (纸本)9783540737490

Spectral embedding and spectral clustering are common methods for non-linear dimensionality reduction and clustering of complex high dimensional datasets. In this paper we provide a diffusion based probabilistic analysis of algorithms that use the normalized graph Laplacian. Given the pairwise adjacency matrix of all points in a dataset, we define a random walk on the graph of points and a diffusion distance between any two points. We show that the diffusion distance is equal to the Euclidean distance in the embedded space with all eigenvectors of the normalized graph Laplacian. This identity shows that characteristic relaxation times and processes of the random walk on the graph are the key concept that governs the properties of these spectral clustering and spectral embedding algorithms. Specifically, for spectral clustering to succeed, a necessary condition is that the mean exit times from each cluster need to be significantly larger than the largest (slowest) of all relaxation times inside all of the individual clusters. For complex, multiscale data, this condition may not hold and multiscale methods need to be developed to handle such situations.

关键词： Relaxation time

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Locality and low-dimensions in the prediction of natural experience from fMRI

Locality and low-dimensions in the prediction of natural exp...

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21st Annual Conference on Neural Information Processing Systems, NIPS 2007

作者： Meyer, François G. Stephens, Greg J. Center for the Study of Brain Mind and Behavior Program in Applied and Computational Mathematics Princeton University United States Center for the Study of Brain Mind and Behavior Department of Physics Princeton University United States

ISBN: (纸本)160560352X

Functional Magnetic Resonance Imaging (fMRI) provides dynamical access into the complex functioning of the human brain, detailing the hemodynamic activity of thousands of voxels during hundreds of sequential time points. One approach towards illuminating the connection between fMRI and cognitive function is through decoding;how do the time series of voxel activities combine to provide information about internal and external experience? Here we seek models of fMRI decoding which are balanced between the simplicity of their interpretation and the effectiveness of their prediction. We use signals from a subject immersed in virtual reality to compare global and local methods of prediction applying both linear and nonlinear techniques of dimensionality reduction. We find that the prediction of complex stimuli is remarkably low-dimensional, saturating with less than 100 features. In particular, we build effective models based on the decorrelated components of cognitive activity in the classically-defined Brodmann areas. For some of the stimuli, the top predictive areas were surprisingly transparent, including Wernicke's area for verbal instructions, visual cortex for facial and body features, and visual-temporal regions for velocity. Direct sensory experience resulted in the most robust predictions, with the highest correlation (c ∼ 0:8) between the predicted and experienced time series of verbal instructions. Techniques based on non-linear dimensionality reduction (Laplacian eigenmaps) performed similarly. The interpretability and relative simplicity of our approach provides a conceptual basis upon which to build more sophisticated techniques for fMRI decoding and offers a window into cognitive function during dynamic, natural experience.

关键词： Forecasting

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Domain Formation in the Plasma Membrane: Roles of Nonequilibrium Lipid Transport and Membrane Proteins

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Physical Review Letters 2008年第17期100卷 178102-178102页

作者： Jun Fan Maria Sammalkorpi 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

Compositional lipid domains (“lipid rafts”) in plasma membranes are believed to be important components of many cellular processes. The mechanisms by which cells regulate the sizes and lifetimes of these spatially extended domains are poorly understood at the moment. Here we show that the competition between phase separation in an immiscible lipid system and active cellular lipid transport processes naturally leads to the formation of such domains. Furthermore, we demonstrate that local interactions with immobile membrane proteins can spatially localize the rafts and lead to further clustering.

关键词： Lipids

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Bounds on the MMSE of “bad” LDPC codes at rates above capacity

Bounds on the MMSE of “bad” LDPC codes at rates above capa...

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Annual Allerton Conference on Communication, Control, and Computing

作者： Amir Bennatan A. Robert Calderbank Shlomo Shamai Program in Applied and Computational Mathematics (PACM) Princeton University USA Technion Israel Institute of Technology Haifa Haifa IL

We present bounds on the minimum mean square error (MMSE) of LDPC codes at rates above capacity. One potential application for MMSE estimation involves cooperative communication. A relay following a compress-and-forward (CF) strategy could first compute an estimate of the transmitted codeword, to reduce the level of noise in the retransmitted signal. Our first bound is based on an analysis of the LDPC belief-propagation decoder. A second bound relies on the relationship between the mutual information and the MMSE, which was discovered by Guo et al. . We compute our bounds for ldquobadrdquo LDPC codes (requiring SNRs that are far above the Shannon limit, for reliable communications to be possible) and show that such codes substantially outperform ldquogoodrdquo codes. This advantage of ldquobadrdquo codes implies an interesting degree of freedom in the design of codes for cooperative communications.

关键词： Parity check codes Relays Decoding Mutual information Context Mean square error methods Noise reduction Noise level Channel capacity mathematics

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Analytic form for a nonlocal kinetic energy functional with a density-dependent kernel for orbital-free density functional theory under periodic and Dirichlet boundary conditions

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Physical Review B 2008年第4期78卷 045105-045105页

作者： Gregory S. Ho Vincent L. Lignères Emily A. Carter Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544-5263 USA

We derive an analytic form of the Wang-Govind-Carter (WGC) [Wang et al., Phys. Rev. B 60, 16350 (1999)] kinetic energy density functional (KEDF) with the density-dependent response kernel. A real-space aperiodic implementation of the WGC KEDF is then described and used in linear scaling orbital-free density functional theory (OF-DFT) calculations.

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First-principles characterization of Ni diffusion kinetics in β-NiAl

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Physical Review B 2008年第18期78卷 184105-184105页

作者： Kristen A. Marino Emily A. Carter Department of Chemical Engineering Princeton University Princeton New Jersey 08544-5263 USA Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544-5263 USA

First-principles density functional theory calculations are performed to examine five postulated diffusion mechanisms for Ni in NiAl: next-nearest-neighbor (NNN) jumps, the triple defect mechanism, and three variants of the six-jump cycle. In contrast to most previous theoretical work, which employed empirical interatomic potentials, we provide a more accurate nonempirical description of the mechanisms. For each pathway, we calculate the activation energy and the pre-exponential factor for the diffusion constant. Although our quantum mechanics calculations are performed at 0 K, we show that it is critical to include the effect of temperature on the pre-exponential factor. We predict that the triple defect mechanism and [110] six-jump cycle both are likely contributors to Ni diffusion in NiAl since their activation energies and pre-exponential factors are in very good agreement with experimental data. Although the activation energy and pre-exponential factor of NNN jumps agree well with experiment, experimental evidence suggests that this is not a dominant contributor to Ni diffusion. Lastly, the activation energies of the [100] bent and straight six-jump cycles are 1 eV higher than the experimental value, allowing us to exclude both [100] cycle mechanisms.

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Riccati conditioning and sensitivity for a MinMax controlled cable-mass system

Riccati conditioning and sensitivity for a MinMax controlled...

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IEEE Conference on Decision and Control

作者： Lizette Zietsman Katie A. Evans J. Teye Brown Richard A. Idowu Faculty of the Interdisciplinary Center for Applied Mathematics Virginia Technology Blacksburg VA USA Faculty of Mathematics and Statistics Louisiana Tech University Ruston LA USA Computational Analysis and Modeling Program Louisiana Tech University Ruston LA USA

ISBN: (纸本)9781424431236

In this paper we present a numerical study that investigates the relationship between the parameter q, used in the design of the MinMax controller, and the conditioning of the approximate algebraic Riccati equations, the sensitivity of the eigenvalues of I-¿ 2 P¿ to ¿ as well as the effect of q on the stability radia and the stability margin of the system. In order to guarantee accurate numerical solutions to the approximate Riccati equations, the Riccati equations must remain well-conditioned for the values of ¿ that are considered. This condition number reflects the combined sensitivity of the Riccati equations to the system inputs A, B, R, C and ¿. In addition, we also consider the sensitivity of the eigenvalues of I-¿ 2 P¿ to ¿. We study the possibility of these sensitivities serving as an indication of the largest value of ¿ for which I-¿ 2 P¿ remains positive definite. This sensitivity could also serve as an indication of the accuracy of the computation of I-¿ 2 P¿. Lastly, in order to design efficient low order controllers, it is important to ensure the robustness of the design. Stability radius and stability margin serve as measures of the robustness of the controller. A one-dimensional nonlinear cable mass system is considered to illustrate these ideas and numerical results are presented.

关键词： Riccati equations Minimax techniques Control systems Eigenvalues and eigenfunctions Robust control Robust stability Optimal control Contracts mathematics Military computing

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Quantum simulation of materials at micron scales and beyond

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Physical Review B 2008年第5期78卷 054118-054118页

作者： Qing Peng Xu Zhang Linda Hung Emily A. Carter Gang Lu Department of Physics and Astronomy California State University Northridge Northridge California USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey USA Department of Mechanical and Aerospace Engineering Princeton University Princeton New Jersey USA

We present a multiscale modeling approach that can simulate multimillion atoms effectively via density-functional theory. The method is based on the framework of the quasicontinuum (QC) approach with orbital-free density-functional theory (OFDFT) as its sole energetics formulation. The local QC part is formulated by the Cauchy-Born hypothesis with OFDFT calculations for strain energy and stress. The nonlocal QC part is treated by an OFDFT-based embedding approach, which couples OFDFT nonlocal atoms to local region atoms. The method—QCDFT—is applied to a nanoindentation study of an Al thin film, and the results are compared to a conventional QC approach. The results suggest that QCDFT represents a new direction for the quantum simulation of materials at length scales that are relevant to experiments.

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