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 ve...
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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
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 analy...
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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.
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 ...
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.
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 e...
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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.
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 impl...
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.
We show that edge stresses introduce intrinsic ripples in freestanding graphene sheets even in the absence of any thermal effects. Compressive edge stresses along zigzag and armchair edges of the sheet cause out-of-pl...
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We show that edge stresses introduce intrinsic ripples in freestanding graphene sheets even in the absence of any thermal effects. Compressive edge stresses along zigzag and armchair edges of the sheet cause out-of-plane warping to attain several degenerate mode shapes. Based on elastic plate theory, we identify scaling laws for the amplitude and penetration depth of edge ripples as a function of wavelength. We also demonstrate that edge stresses can lead to twisting and scrolling of nanoribbons as seen in experiments. Our results underscore the importance of accounting for edge stresses in thermal theories and electronic structure calculations for freestanding graphene sheets.
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 dens...
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.
We consider a basic model for two-hop transmissions of two information flows which interfere with each other. In this model, two sources simultaneously transmit to two relays (in the first hop), which then simultaneou...
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We consider a basic model for two-hop transmissions of two information flows which interfere with each other. In this model, two sources simultaneously transmit to two relays (in the first hop), which then simultaneously transmit to two destinations (in the second hop). While the transmission during the first hop is essentially the transmission over a classical interference channel, the transmission in the second hop enjoys an interesting advantage. Specifically, as a byproduct of the Han-Kobayashi transmission scheme applied to the first hop, each of the relays (in the second hop) has access to some of the data that is intended to the other destination, in addition to its own data. As recently observed by Simeone et al., this opens the door to cooperation between the relays. In this paper, we observe that the cooperation can take the form of distributed MIMO broadcast, thus greatly enhancing its effectiveness at high SNR. However, since each relay is only aware of part of the data beyond its own, full cooperation is not possible. We propose several approaches that combine MIMO broadcast strategies (including ldquodirty paperrdquo) with standard non-cooperative strategies for the interference channel. Numerical results are provided, which indicate that our approaches provide substantial benefits at high SNR.
Almost all studies of the densest particle packings consider convex particles. Here, we provide exact constructions for the densest known two-dimensional packings of superdisks whose shapes are defined by |x1|2p+|x2|2...
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Almost all studies of the densest particle packings consider convex particles. Here, we provide exact constructions for the densest known two-dimensional packings of superdisks whose shapes are defined by |x1|2p+|x2|2p≤1 and thus contain a large family of both convex (p≥0.5) and concave (0
We describe a computational framework linking uncertainty quantification (UQ) methods for continuum problems depending on random parameters with equation-free (EF) methods for performing continuum deterministic numeri...
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