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Dense packings of polyhedra: Platonic and Archimedean solids

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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Interactions leading to disordered ground states and unusual low-temperature behavior

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Physical Review E 2009年第3期80卷 031105-031105页

作者： Robert D. Batten Frank H. Stillinger Salvatore Torquato Department of Chemical Engineering Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Princeton Materials Institute Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA School of Natural Sciences Institute for Advanced Study Princeton New Jersey 08544 USA

We have shown that any pair potential function v(r) possessing a Fourier transform V(k) that is positive and has compact support at some finite wave number K yields classical disordered ground states for a broad density range [R. D. Batten, F. H. Stillinger, and S. Torquato, J. Appl. Phys. 104, 033504 (2008)]. By tuning a constraint parameter χ (defined in the text), the ground states can traverse varying degrees of local order from fully disordered to crystalline ground states. Here, we show that in two dimensions, the “k-space overlap potential,” where V(k) is proportional to the intersection area between two disks of diameter K whose centers are separated by k, yields anomalous low-temperature behavior, which we attribute to the topography of the underlying energy landscape. At T=0, for the range of densities considered, we show that there is continuous energy degeneracy among Bravais-lattice configurations. The shear elastic constant of ground-state Bravais-lattice configurations vanishes. In the harmonic regime, a significant fraction of the normal modes for both amorphous and Bravais-lattice ground states have vanishing frequencies, indicating the lack of an internal restoring force. Using molecular-dynamics simulations, we observe negative thermal-expansion behavior at low temperatures, where upon heating at constant pressure, the system goes through a density maximum. For all temperatures, isothermal compression reduces the local structure of the system unlike typical single-component systems.

关键词： Ground state

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Exact constructions of a family of dense periodic packings of tetrahedra

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Physical Review E 2010年第4期81卷 041310-041310页

作者： S. Torquato Y. Jiao Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Science 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

The determination of the densest packings of regular tetrahedra (one of the five Platonic solids) is attracting great attention as evidenced by the rapid pace at which packing records are being broken and the fascinating packing structures that have emerged. Here we provide the most general analytical formulation to date to construct dense periodic packings of tetrahedra with four particles per fundamental cell. This analysis results in six-parameter family of dense tetrahedron packings that includes as special cases recently discovered “dimer” packings of tetrahedra, including the densest known packings with density ϕ=40004671=0.856347…. This study strongly suggests that the latter set of packings are the densest among all packings with a four-particle basis. Whether they are the densest packings of tetrahedra among all packings is an open question, but we offer remarks about this issue. Moreover, we describe a procedure that provides estimates of upper bounds on the maximal density of tetrahedron packings, which could aid in assessing the packing efficiency of candidate dense packings.

关键词： constructions family tetrahedra tetrahedrons family Dense Platonic solid

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Wave properties using displaced phase amplitude

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IOP Conference Series: Earth and Environmental science 2019年第1期364卷

作者： D Fadhiliani M Ramli S Rizal S Munzir M Syafwan Doctor of Mathematics and Applied Science Study Program Universitas Syiah Kuala Banda Aceh Indonesia Department of Mathematics Universitas Syiah Kuala Banda Aceh Indonesia Department of Marine Sciences Faculty of Marine and Fisheries Universitas Syiah Kuala Banda Aceh Indonesia Department of Mathematics Faculty of Mathematics and Natural Sciences Universitas Andalas Padang Indonesia

Displaced phase-amplitude variable in polar form. This variable is used to investigate changes in amplitude in complex fields with phases that depend only on the position in the propagation. Soliton on Finite Background (SFB) which is an exact solution of Nonlinear Schrodinger (NLS) equation has been widely used in investigating wave propagation dynamics so that it is the basic for the proposed displaced phase-amplitude. Using displaced phase-amplitude, the results obtained can be described in Argand Diagrams. Wave equation used as a model is the Benjamin Bona Mahony (BBM) equation where the envelope of this wave evolves following the NLS equation. This wave is unidirectional long wave on the surface and has low amplitude characteristics. Step by step to obtain a SFB solution that contains displaced phase-amplitude described and displayed in an argand diagram. In additional, the envelope graph is given.

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Nonlinear Geometric Effects in Mechanical Bistable Morphing Structures

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Physical Review Letters 2012年第11期109卷 114302-114302页

作者： Zi Chen Qiaohang Guo Carmel Majidi Wenzhe Chen David J. Srolovitz Mikko P. Haataja Department of Mechanical and Aerospace Engineering Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials (PRISM) Princeton University Princeton New Jersey 08544 USA Department of Biomedical Engineering Washington University St. Louis Missouri 63130 USA College of Materials Science and Engineering Fuzhou University Fuzhou Fujian 350108 China Department of Mathematics and Physics Fujian University of Technology Fuzhou Fujian 350108 China Department of Mechanical Engineering Carnegie Mellon University Pittsburgh Pennsylvania 15213 USA Department of Materials Science and Engineering Fujian University of Technology Fuzhou Fujian 350108 China Institute for High Performance Computing 1 Fusionopolis Way Singapore 138632 Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA School of Mathematics Institute for Advanced Study Princeton New Jersey 08540 USA

Bistable structures associated with nonlinear deformation behavior, exemplified by the Venus flytrap and slap bracelet, can switch between different functional shapes upon actuation. Despite numerous efforts in modeling such large deformation behavior of shells, the roles of mechanical and nonlinear geometric effects on bistability remain elusive. We demonstrate, through both theoretical analysis and tabletop experiments, that two dimensionless parameters control bistability. Our work classifies the conditions for bistability, and extends the large deformation theory of plates and shells.

关键词： NONLINEAR theories DEFORMATIONS (Mechanics) PARAMETER estimation OPTICAL bistability STRUCTURAL shells PLATES (Engineering) CRYSTAL structure

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Phase behavior of colloidal superballs: Shape interpolation from spheres to cubes

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Physical Review E 2010年第6期81卷 061105-061105页

作者： Robert D. Batten Frank H. Stillinger Salvatore Torquato Department of Chemical Engineering Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Physics 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 08544 USA

The phase behavior of hard superballs is examined using molecular dynamics within a deformable periodic simulation box. A superball’s interior is defined by the inequality |x|2q+|y|2q+|z|2q≤1, which provides a versatile family of convex particles (q≥0.5) with cubelike and octahedronlike shapes as well as concave particles (q<0.5) with octahedronlike shapes. Here, we consider the convex case with a deformation parameter q between the sphere point (q=1) and the cube (q=∞). We find that the asphericity plays a significant role in the extent of cubatic ordering of both the liquid and crystal phases. Calculation of the first few virial coefficients shows that superballs that are visually similar to cubes can have low-density equations of state closer to spheres than to cubes. Dense liquids of superballs display cubatic orientational order that extends over several particle lengths only for large q. Along the ordered, high-density equation of state, superballs with 1

关键词： Spheres

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Method for obtaining upper bounds on photonic band gaps

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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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Model of virus therapy and chemotherapy for cancer

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IOP Conference Series: Earth and Environmental science 2019年第1期364卷

作者： E Safitri T Tarmizi M Ramli S Wahyuni M Ikhwan Mathematics Graduate Study Program Universitas Syiah Kuala Banda Aceh 23111 Indonesia Department of Mathematics Universitas Syiah Kuala Banda Aceh 23111 Indonesia Graduate School of Mathematics and Applied Science Universitas Syiah Kuala Banda Aceh 23111 Indonesia

This article discusses the mathematical model of changes in the number of cancer cells as a result of the intervention. Here the effect of combination treatment, in the form of virus injection and chemotherapy on tumor cells, is presented in detail. There are two different tumor cells, uninfected tumor cells and infected tumor cells. Model simulations with different parameters were performed for 700 d. The initial values for the population of virus-infected tumor cells are divided into two categories. The number of tumor cells injected with the virus is ½ (half) and all tumor cells that are not infected. Based on the simulation results, it was concluded that the more the number of tumor cells injected with the virus, the faster the decrease in the number of tumor cell populations. It was also concluded that the reduced population of tumor cells reduces the amount of chemotherapy concentration used.

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Forest fires and carbon emission release in semi-arid regions of Indonesia: the evidence from medium resolution of satellite imagery

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IOP Conference Series: Earth and Environmental science 2021年第1期622卷

作者： AN Armanto Perdinan BD Dasanto R Hidayati Applied Climatology Study Program Faculty of Mathematics and Natural Science Graduate School of IPB University Bogor West Java Indonesia 16680 Department of Geophysics and Meteorology IPB University Bogor Indonesia 16680

Forest fires are one of the causes of forest degradation and deforestation in Indonesia. Tropical savanna forests are one of the forest ecosystems found in the Indonesian region with a semi-arid climate type prone to drought and fire. Forest fires contribute to the release of greenhouse gas emissions into the atmosphere. The calculation method to estimate the burnt area is still an obstacle in calculating greenhouse gas emissions due to forest fires. This research aims to assess the burnt area in tropical savanna forest cover type using medium resolution satellite imagery data by utilizing the Normalized Burn Ratio index in four villages in East Sumba Regency and estimating greenhouse gas emissions these activities. The results showed that the NBR method does not produce an estimate of the burned area with a fair degree of accuracy because the characteristic of channel 7 (SWIR) is sensitive to vegetation moisture content. The savanna forest cover type is the type of land cover identified as the most dominant burned area. It produces the highest greenhouse gas emissions compared to other land cover types.

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Fuzzy sentiment analysis using convolutional neural network

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AIP Conference Proceedings 2021年第1期2329卷

作者： Sugiyarto Joko Eliyanto Nursyiva Irsalinda Meita Fitrianawati 1Mathematics Department Faculty of Applied Science and Technology Universitas Ahmad Dahlan Indonesia 2Mathematics Education Department Postgraduate Program Universitas Ahmad Dahlan Indonesia 3Elementary School Teacher Study Program Faculty of Teacher Training and Education Universitas Ahmad Dahlan Indonesia

Sentiment analysis is one part of natural language processing. Sentiment analysis can be done by lexicon based, or machine learning based. Sentiment analysis based on machine learning has advantage of dynamism to meet with new language datasets or new vocabulary. Sentiment analysis seeks to understand the sentiments contained in a sentence. A sentence can be positive, neutral or negative, based on its sentiments. A sentence can have positive, neutral or negative sentiments. However, the fact is each sentence does not always have positive, negative or neutral sentiment clearly. We try to develop a sentiment analysis method that can show the sentiment degree of a sentence. Fuzzy sentiment analysis using convolutional neural network are introduced in this paper to produce more accurate sentiment analysis results. Convolutional neural networks are a popular machine learning method for sentiment analysis. The concept of fuzzy sets is used to express the sentiment degree of a sentence. Euclidean distance analysis to determine the proximity of two vectors is used to show that this method is better than the standard method. The method we propose successfully produces a value that indicates the degree of sentiment of a sentence. Comparison of the euclid distance between the results of the standard sentiment analysis and our method shows that the results of the fuzzy sentiment analysis using convolutional neural network have a distance that is relatively close to the true sentiment value. Fuzzy convolutional neural network analysis sentiment is proven to be able to produce better and smoother sentiment analysis results than standard methods.

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