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1. Packing Geometric Objects with Optimal Worst-Case Density (Multimedia Exposition)

   https://doi.org/10.4230/LIPIcs.SoCG.2019.63  

   Becker, Aaron T.; Fekete, Sándor P.; Keldenich, Phillip; Morr, Sebastian; Scheffer, Christian (January 2019, 35th International Symposium on Computational Geometry (SoCG 2019))

   We motivate and visualize problems and methods for packing a set of objects into a given container, in particular a set of {different-size} circles or squares into a square or circular container. Questions of this type have attracted a considerable amount of attention and are known to be notoriously hard. We focus on a particularly simple criterion for deciding whether a set can be packed: comparing the total area A of all objects to the area C of the container. The critical packing density delta^* is the largest value A/C for which any set of area A can be packed into a container of area C. We describe algorithms that establish the critical density of squares in a square (delta^*=0.5), of circles in a square (delta^*=0.5390 ...), regular octagons in a square (delta^*=0.5685 ...), and circles in a circle (delta^*=0.5). 

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2. Inside and out: Surface thermodynamics from positive to negative curvature

   https://doi.org/10.1063/5.0099295  

   Martin, Seth C.; Hansen-Goos, Hendrik; Roth, Roland; Laird, Brian B. (August 2022, The Journal of Chemical Physics)

   To explore the curvature dependence of solid–fluid interfacial thermodynamics, we calculate, using Grand Canonical Monte Carlo simulation, the surface free energy for a 2d hard-disk fluid confined in a circular hard container of radius R as a function of the bulk packing fraction η and wall curvature C̄=−1/R. (The curvature is negative because the surface is concave.) Combining this with our previous data [Martin et al., J. Phys. Chem. B 124, 7938–7947 (2020)] for the positive curvature case (a hard-disk fluid at a circular wall, C̄=+1/R), we obtain a complete picture of surface thermodynamics in this system over the full range of positive and negative wall curvatures. Our results show that γ is linear in C̄ with a slope that is the same for both positive and negative wall curvatures, with deviations seen only at high negative curvatures (strong confinement) and high density. This observation indicates that the surface thermodynamics of this system is consistent with the predictions of so-called morphometric thermodynamics at both positive and negative curvatures. In addition, we show that classical density functional theory and a generalized scaled particle theory can be constructed that give excellent agreement with the simulation data over most of the range of curvatures and densities. For extremely high curvatures, where only one or two disks can occupy the container at maximum packing, it is possible to calculate γ exactly. In this limit, the simulations and density functional theory calculations are in remarkable agreement with the exact results. 

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3. Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation

   Aggarwal, Amol and (July 2022, Computational complexity)

   Polynomial approximations for e−x and ex have applications to the design of algorithms for many problems, and our degree bounds show both the power and limitations of these algorithms. We focus in particular on the Batch Gaussian Kernel Density Estimation problem for n sample points in Θ(logn) dimensions with error δ=n−Θ(1). We show that the running time one can achieve depends on the square of the diameter of the point set, B, with a transition at B=Θ(logn) mirroring the corresponding transition in dB;δ(e−x): - When B=o(logn), we give the first algorithm running in time n1+o(1). - When B=κlogn for a small constant κ>0, we give an algorithm running in time n1+O(loglogκ−1/logκ−1). The loglogκ−1/logκ−1 term in the exponent comes from analyzing the behavior of the leading constant in our computation of dB;δ(e−x). - When B=ω(logn), we show that time n2−o(1) is necessary assuming SETH. 

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4. Effect of Wrapping Force on the Effective Elastic Behavior of Packed Cylinders

   https://doi.org/10.1115/1.4056212  

   Yin, Huiming; Cui, Junhe; Teka, Linda G.; Zadshir, Mehdi (March 2023, Journal of Applied Mechanics)

   Abstract When cylinders are packed and wrapped by the bands around the surface, the effective elastic behavior in the cross section of the assembly, which is of significance to its stability and integrity, can be controlled by the wrapping force in the band. The wrapping force is transferred to the cylinders through the Hertz contact between each pair of neighboring cylinders, which is validated by the experiments. The Singum model is introduced to study the mechanical behaviors of the packed cylinders with two-dimensional (2D) packing lattices, in which an inner cylinder is simulated by a continuum particle of Singum and the inter-cylinder force is governed by the Hertz contact model so as to derive the effective stress-strain relationship. The wrapping force will produce configurational forces given a displacement variation, which significantly changes the effective stiffness of the packed cylinders. The hexagonal packing exhibits isotropic elasticity whereas the square packing is anisotropic. The efficacy of our model is demonstrated by comparing the closed form elasticity against the numerical simulation and the previous models. The explicit form of elasticity can be used for packing design and quality control of cable construction and installation. 

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5. The role of crystal packing on the optical response of trialkyltetrelethynyl acenes

   https://doi.org/10.1063/5.0097421  

   Huang, Ling-Yi; Ai, Qianxiang; Risko, Chad (August 2022, The Journal of Chemical Physics)

   The electronic and optical responses of an organic semiconductor (OSC) are dictated by the chemistries of the molecular or polymer building blocks and how these chromophores pack in the solid state. Understanding the physicochemical nature of these responses is not only critical for determining the OSC performance for a particular application, but the UV/visible optical response may also be of potential use to determine aspects of the molecular-scale solid-state packing for crystal polymorphs or thin-film morphologies that are difficult to determine otherwise. To probe these relationships, we report the quantum-chemical investigation of a series of trialkyltetrelethynyl acenes (tetrel = silicon or germanium) that adopt the brickwork, slip-stack, or herringbone (HB) packing configurations; the π-conjugated backbones considered here are pentacene and anthradithiophene. For comparison, HB-packed (unsubstituted) pentacene is also included. Density functional theory and G 0 W 0 (single-shot Green’s function G and/or screened Coulomb function W) electronic band structures, G 0 W 0 -Bethe–Salpeter equation-derived optical spectra, polarized ϵ 2 spectra, and distributions of both singlet and triplet exciton wave functions are reported. Configurational disorder is also considered. Furthermore, we evaluate the probability of singlet fission in these materials through energy conservation relationships. 

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