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Viazovska proved that the [Formula: see text] lattice sphere packing is the densest sphere packing in [Formula: see text] dimensions. Her proof relies on two inequalities between functions defined in terms of modular and quasimodular forms. We give a direct proof of these inequalities that does not rely on computer calculations.

The relationship between a company's cost, volume, and profit is important for strategic planning and widely used in business analysis and industrial management. Traditional Cost-Volume-Profit (CVP) analysis is used when a company is trying to determine what single level of sales, prices, and costs is necessary to reach a specific amount of profit. So far, a little attention has been paid to the extension of existing models of CVP analysis to illustrate a set of break even points, profitable sales, prices, and costs. For this purpose, for the first time, we propose a new approach to break even and profitability analysis called sphere packing based on a notion of set of profitability conditions with respect to CVP parameters. This approach uses sphere packing theory [9,10], linear programming and allows industries to handle break even and profitability analysis for multi-product case finding a set of required sales, prices, and costs to ensure profitability of a company. The sphere packing approach also provides practical suggestions and recommendations for managers to choose a set of optimal CVP parameters. The proposed approach is illustrated on some examples providing numerical results. [ABSTRACT FROM AUTHOR]

We have previously explored the hysteresis and reversibility of transitions between ordered packings of soft spheres of diameter d in cylindrical channels of diameter D [A. Mughal, J. Winkelmann, D. Weaire, S. Hutzler, Phys. Rev. E 98, 043303 (2018)]. Here we extend these initial results to include transitions between all columnar structures without inner spheres (i.e. packings in which all of the spheres are in contact with the cylindrical boundary). These results can be represented by a directed network showing permissible transitions between structures. From the hard sphere limit we deduce that there are two different types of transitions, reversible and irreversible. We explore the nature of these transitions for soft spheres as a function of pressure and due to changes in the ratio D/d. These results are illustrated by the use of schematic diagrams, indicating the topological features of each transition. Specific cases are tabulated and can be understood by reference to the appropriate schematic diagram. [ABSTRACT FROM AUTHOR]

The second part of our study is devoted to a detailed convergence analysis of two extensions of the well-known DCA method for solving DC (Difference of Convex functions) optimization problems to the case of general cone constrained DC optimization problems. We study the global convergence of the DCA for cone constrained problems and present a comprehensive analysis of a version of the DCA utilizing exact penalty functions. In particular, we study the exactness property of the penalized convex subproblems and provide two types of sufficient conditions for the convergence of the exact penalty method to a feasible and critical point of a cone constrained DC optimization problem from an infeasible starting point. In the numerical section of this work, the exact penalty DCA is applied to the problem of computing compressed modes for variational problems and the sphere packing problem on Grassmannian. [ABSTRACT FROM AUTHOR]

This paper considers the optimized packing of different spheres into a given spherical container under non-standard placement conditions. A sphere is considered placed in the container if at least a certain part of the sphere is in the container. Spheres are allowed to overlap with each other according to predefined parameters. Ratio conditions are introduced to establish correspondence between the number of packed spheres of different radii. The packing aims to maximize the total number of packed spheres subject to ratio, partial overlapping and quasi-containment conditions. A nonlinear mixed-integer optimization model is proposed for this ratio quasi-packing problem. A heuristic algorithm is developed that reduces the original problem to a sequence of continuous open dimension problems for quasi-packing scaled spheres. Computational results for finding global solutions for small instances and good feasible solutions for large instances are provided. [ABSTRACT FROM AUTHOR]

This note presents simple linear algebraic proofs of theorems due to Šiňajová, Rankin and Kuperberg concerning spherical point configurations. The common ingredient in these proofs is the use of spherical Euclidean distance matrices (EDMs) and the Perron-Frobenius theorem. [ABSTRACT FROM AUTHOR]

We give a short solution of the kissing number problem in dimension three. [ABSTRACT FROM AUTHOR]

This letter investigates the capacity for the multiple-input multiple-output (MIMO) optical intensity channels in the high signal-to-noise ratio (SNR) regime from a sphere-packing (SP) perspective. In such a channel, the inputs represent optical intensities, hence are nonnegative. Considering the peak- and average-power constraints for the inputs, the high-SNR capacity can be expressed in terms of the volume of an image signal space through an SP argument. When the number of transmit antennas $n_{\mathrm{T}} $ and receive antennas $n_{\mathrm{R}} $ satisfies $n_{\mathrm{T}} \leq n_{\mathrm{R}} $ , the image-space volume is derived in terms of singular values of the channel matrix. While for $n_{\mathrm{T}} > n_{\mathrm{R}} $ , the image-space volume is derived by decomposing the image space into multiple non-overlapping subpolytopes, whose volumes can be conveniently calculated. [ABSTRACT FROM AUTHOR]

This paper considers the optimized packing of different spheres into a given spherical container under non-standard placement conditions. A sphere is considered placed in the container if at least a certain part of the sphere is in the container. Spheres are allowed to overlap with each other according to predefined parameters. Ratio conditions are introduced to establish correspondence between the number of packed spheres of different radii. The packing aims to maximize the total number of packed spheres subject to ratio, partial overlapping and quasi-containment conditions. A nonlinear mixed-integer optimization model is proposed for this ratio quasi-packing problem. A heuristic algorithm is developed that reduces the original problem to a sequence of continuous open dimension problems for quasi-packing scaled spheres. Computational results for finding global solutions for small instances and good feasible solutions for large instances are provided.

One of the criteria to a strong principle in natural sciences is simplicity. The conventional view holds that the world is provided with natural laws that must be simple. This common-sense approach is a modern rewording of the medieval philosophical/theological concept of the Multiple arising from (and generated by) the One. Humans need to pursue unifying frameworks, classificatory criteria and theories of everything. Still, the fact that our cognitive abilities tend towards simplification and groupings does not necessarily entail that this is the way the world works. Here we ask: what if singularity does not pave the way to multiplicity? How will we be sure if the Ockham's razor holds in real life? We will show in the sequel that the propensity to reduce to simplicity the relationships among the events leads to misleading interpretations of scientific issues. We are not going to take a full sceptic turn: we will engage in active outreach, suggesting examples from biology and physics to demonstrate how a novel methodological antiunitary approach might help to improve our scientific attitude towards world affairs. We will provide examples from aggregation of SARS-Cov-2 particles, unclassified extinct creatures, pathological brain stiffness. Further, we will describe how antiunitary strategies, plagiarising medieval concepts from William od Ockham and Gregory of Rimini, help to explain novel relational approaches to quantum mechanics and the epistemological role of our mind in building the real world. [ABSTRACT FROM AUTHOR]

In an earlier work, we proposed a generalization for the Apollonian packing in arbitrary dimensions and showed that the resulting object in four, five, and six dimensions have properties consistent with the Apollonian circle and sphere packings in two and three dimensions. In this work, we investigate the generalization in seven and eight dimensions and show that they too have many of the properties of those in lower dimensions. In particular, the hyperspheres are tangent or do not intersect; they fill the hyperspace; the object includes a maximal cluster of mutually tangent hyperspheres; and there exists a perspective where all hyperspheres in the object have integer curvatures. [ABSTRACT FROM AUTHOR]

[Display omitted] We developed a geometrical model to determine the theoretical maximum number of proteins that can pack as a monolayer surrounding a spherical nanoparticle. We applied our new model to study the adsorption of receptor binding domain (RBD) of the SARS-CoV-2 spike protein to silica nanoparticles. Due to its abundance and extensive use in manufacturing, silica represents a reservoir where the virus can accumulate. It is therefore important to study the adsorption and the persistence of viral components on inanimate surfaces. We used previously published datasets of nanoparticle-adsorbed proteins to validate the new model. We then used integrated experimental methods and Molecular Dynamics (MD) simulations to characterise binding of the RBD to silica nanoparticles and the effect of such binding on RBD structure. The new model showed excellent fit with existing datasets and, combined to new RBD-silica nanoparticles binding data, revealed a surface occupancy of 32% with respect to the maximum RBD packing theoretically achievable. Up to 25% of RBD's secondary structures undergo conformational changes as a consequence of adsorption onto silica nanoparticles. Our findings will help developing a better understanding of the principles governing interaction of proteins with surfaces and can contribute to control the spread of SARS-CoV-2 through contaminated objects. [ABSTRACT FROM AUTHOR]

Nearly orthogonal lattices were formally defined in [4] , where their applications to image compression were also discussed. The idea of "near orthogonality" in 2-dimensions goes back to the work of Gauss. In this paper, we focus on well-rounded nearly orthogonal lattices in R n and investigate their geometric and optimization properties. Specifically, we prove that the sphere packing density function on the space of well-rounded lattices in dimension n ≥ 3 does not have any local maxima on the nearly orthogonal set and has only one local minimum there: at the integer lattice Z n. Further, we show that the nearly orthogonal set cannot contain any perfect lattices for n ≥ 3 , although it contains multiple eutactic (and even strongly eutactic) lattices in every dimension. This implies that eutactic lattices, while always critical points of the packing density function, are not necessarily local maxima or minima even among the well-rounded lattices. We also prove that a (weakly) nearly orthogonal lattice in R n contains no more than 4 n − 2 minimal vectors (with any smaller even number possible) and establish some bounds on coherence of these lattices. [ABSTRACT FROM AUTHOR]

• We propose a fast and effective 3D ball packing method for surface meshes. • We present an efficient and effective adaptive surface mesh remeshing method for surface meshes. • Without complicated operations, our method can be implemented easily without relying on any third-party libraries. Triangular mesh has been a prevalent form of 3D model representation in various areas ranging from modeling to finite element analysis due to their simplicity and flexibility. In the paper, we present a triangular mesh remeshing method based on a sphere packing method and a node insertion/deletion method for surface meshes. First, a new set of nodes are generated on the surface mesh via a sphere packing method and added to the original surface mesh. Then, original nodes are deleted through some basic operations. Finally, the mesh is optimized by edge flipping. To regenerate an adaptive mesh, we consider some geometric features to calculate a size field and record and smooth it with an octree background grid. The proposed method remeshes the surface mesh without projection of local areas, the intersection of fronts, Lloyd relaxation, and other complicated calculations, and the proposed method can generate a high-quality mesh without dependence on the quality of the original mesh, which make the method efficient and effective. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of optimally packing spheres of various dimensions into containers of arbitrary geometrical shapes. According to the international classification, this problem belongs to Sphere Packing Problems (SPPs). The problem is to pack a set of spheres (circles, hyperspheres) with given radii into a container with given metric characteristics. The aim of this work is to create an integrated methodology for solving SPPs. The basic formulations of the problem are presented: in the form of the knapsack problem (KP), open dimension problem (ODP), and their corresponding mathematical models. The solution strategy selection is influenced by the form of problem statement, dimension of the space where the spheres are to be packed, metric peculiarities of the spheres (equal or unequal), number of the spheres to be packed, geometric shape of the container, presence of technological restraints, and count time limit. The structural elements of the methodology are mathematical models, methods for constructing initial packings, and methods of local and global optimization. In developing the solution method, we construct the initial feasible packings by using both the random and lattice methods, using a greedy algorithm and solving an auxiliary nonlinear programming problem. As local optimization methods, we consider the modifications of the feasible direction method, interior point method, Lagrange multiplier method, and method of optimization in groups of variables. For global optimization, we use the method of enumerating the subsets of spheres of a given set and method of enumerating the extreme points of the feasible region, which are implemented by using the branch and bound algorithm, the modifications of the decremental neighborhood search method, method of smooth transition from one local minimum to another by increasing problem dimensionality and introducing additional variable metric characteristics, solution method implemented as a sequence of non-linear programming problems of increasing dimensionality, and a multi-start method. Strategies for solving different SPP statements are proposed.

This paper presents a method to calculate the exact average block error probability of some random code ensembles under maximum-likelihood decoding. The proposed method is applicable to various channels and ensembles. The focus is on both spherical and Gaussian random codes on the additive white Gaussian noise channel as well as binary random codes on both the binary symmetric channel and the binary erasure channel. While for the uniform spherical ensemble Shannon, in 1959, argued with solid angles in N-dimensional space, the presented approach projects the problem into two dimensions and applies standard trigonometry. This simplifies the derivation and also allows for the analysis of the independent identically distributed (i.i.d.) Gaussian ensemble which turns out to perform better for short blocklengths and high rates. Moreover, a new lower bound on the average block error probability of the uniform spherical ensemble is found. For codes with more than three codewords, it is tighter than the sphere packing bound, but requires exactly the same computing effort. Furthermore, tight approximations are proposed to simplify the computation of both the exact average error probability and the two bounds. For the binary symmetric channel and the binary erasure channel, bounds on the average block error probability for i.i.d. random coding are derived and compared to the exact calculations. [ABSTRACT FROM AUTHOR]

The concept of jamming has attracted great research interest due to its broad relevance in soft-matter, such as liquids, glasses, colloids, foams, and granular materials, and its deep connection to sphere packing and optimization problems. Here, we show that the domain of amorphous jammed states of frictionless spheres can be significantly extended, from the well-known jammingpoint at a fixed density, to a jamming-plane that spans the density and shear strain axes. We explore the jamming-plane, via athermal and thermal simulations of compression and shear jamming, with initial equilibrium configurations prepared by an efficient swap algorithm. The jamming-plane can be divided into reversible-jamming and irreversible-jamming regimes, based on the reversibility of the route from the initial configuration to jamming. Our results suggest that the irreversible-jamming behavior reflects an escape from the metastable glass basin to which the initial configuration belongs to or the absence of such basins. All jammed states, either compression- or shear-jammed, are isostatic and exhibit jamming criticality of the same universality class. However, the anisotropy of contact networks nontrivially depends on the jamming density and strain. Among all state points on the jamming-plane, the jamming-point is a unique one with the minimum jamming density and the maximum randomness. For crystalline packings, the jamming-plane shrinks into a single shear jamming-line that is independent of initial configurations. Our study paves the way for solving the long-standing random close-packing problem and provides a more complete framework to understand jamming. [ABSTRACT FROM AUTHOR]

It has been a longstanding question how stars are packed or distributed in the universe. Here we show that the spatial distribution of stars follows some geometric rules although there may be local inhomogeneities. The topological and metric properties of their Voronoi tessellations exhibit a lognormal distribution. Moreover, these distributions can be connected with those established for the packing of spheres, corresponding to the loosest packing state. The findings provide a new angle to understand the packing structure of stars in the universe. Unlabelled Image • The packing of stars is studied in terms of the topological and metric properties of the Voronoi tessellation. • The spatial distribution of stars is shown to follow some geometric rules, quantified by the lognormal distribution. • The packing of stars can be linked to the packing of spheres, corresponding to the loosest packing state. [ABSTRACT FROM AUTHOR]

Relationships between microstructure characteristics and effective transport properties of granular materials are crucial for many real-world applications. In the present paper microstructure-property relationships of sphere packings are investigated by means of modeling and simulation. Virtual microstructures are generated with the random close packing algorithm. This algorithm provides initial systems of randomly distributed, non-overlapping and densely-packed spheres of a given class of polydisperse size distributions. Next, the initial sphere packing is further densified until a certain criterion is reached, namely a predefined mean contact angle. In this way, we obtain a large database of slightly overlapping sphere systems. Subsequently, effective transport properties of the sphere systems (solid) and their complementary sets (pores) are determined using the computationally efficient resistor network method. Finally, the generated virtual microstructures are used to establish formulas expressing effective transport properties of the considered sphere packings in terms of the mean contact angle and the standard deviation of the particle radii. Unlabelled Image • RN for effective transport properties in the solid and pore phase of sphere packings • Virtual random sphere packings in a wide range of polydispersity and porosity • Prediction formulas for effective transport parameters in the solid and pore phase [ABSTRACT FROM AUTHOR]

Diabetic retinopathy is one of the most frequent complications of diabetes, which leads to severe consequences, including rapid and irreversible vision loss. The laser coagulation procedure to treat diabetic retinopathy consists in applying a series of microburns to the fundus to deal with macular edema. The existing hardware/software packages are primarily based on the use of a predetermined pattern for coagulate arrangement. However, due to the complex form of edema and vascular system, this approach leads to an uneven arrangement. To solve the problem, we propose a new approach based on the application of sphere packing algorithms (circle packing in two-dimensional images) in the specified area of interest. Since one of the main requirements for the laser coagulation procedure is that it should have the minimum duration, a problem of the computational complexity of the developed algorithms arises. This requirement is completely ignored by the existing approaches, therefore the development of new high-performance coagulate arrangement algorithms is highly relevant. In the paper, we propose seven new algorithms for coagulate arrangement and provide a detailed analysis of key characteristics of the algorithms. The characteristics considered have made it possible to extract information relating to the algorithm properties. Regularity is determined by the median, asymmetry and the kurtosis; determinism is determined by the variance and the mean.

We generalize the recent work of Viazovska by constructing infinite families of Schwartz functions, suitable for Cohn–Elkies style linear programming bounds, using quasi-modular and modular forms. In particular, for dimensions d ≡ 0 (mod 8) , we give new constructions for obtaining sphere packing upper bounds via modular forms. In dimension 8 and 24, these exactly match the functions constructed by Viazovska and Cohn, Kumar, Miller, Radchenko, and Viazovska which resolved the sphere packing problem in those dimensions. [ABSTRACT FROM AUTHOR]

The sphere packing problem is an old puzzle. We consider packings with m spheres in the unit cell (m-periodic packings). For the case m = 1 (lattice packings), Voronoi proved there are finitely many inequivalent local optima and presented an algorithm to enumerate them, and this computation has been implemented in up to d = 8 dimensions. We generalize Voronoi's method to m > 1 and present a procedure to enumerate all locally optimal 2-periodic sphere packings in any dimension, provided there are finitely many. We implement this computation in d = 3 , 4 , and 5 and show that no 2-periodic packing surpasses the density of the optimal lattices in these dimensions. A partial enumeration is performed in d = 6 . [ABSTRACT FROM AUTHOR]

Room segmentation is a basic task for the semantic enrichment of point clouds. Recent studies have mainly projected single-floor point clouds to binary images to realize two-dimensional room segmentation. However, these methods have difficulty solving semantic segmentation problems in complex 3D indoor environments, including cross-floor spaces and rooms inside rooms; this is the bottleneck of indoor 3D modeling for non-Manhattan worlds. To make full use of the abundant geometric and spatial structure information in 3D space, a novel 3D room segmentation method that realizes room segmentation directly in 3D space is proposed in this study. The method utilizes volumetric representation based on a VDB data structure and packs an indoor space with a set of compact spheres to form rooms as separated connected components. Experimental results on different types of indoor point cloud datasets demonstrate the efficiency of the proposed method.

The introduction of 5G with excessively high speeds and ever-advancing cellular device capabilities has increased the demand for high data rate wireless multimedia communication. Data compression, transmission robustness and error resilience are introduced to meet the increased demands of high data rates of today. An innovative approach is to come up with a unique setup of source bit codes (SBCs) that ensure the convergence and joint source-channel coding (JSCC) correspondingly results in lower bit error ratio (BER). The soft-bit assisted source and channel codes are optimized jointly for optimum convergence. Source bit codes assisted by iterative detection are used with a rate-1 precoder for performance evaluation of the above mentioned scheme of transmitting sata-partitioned (DP) H.264/AVC frames from source through a narrowband correlated Rayleigh fading channel. A novel approach of using sphere packing (SP) modulation aided differential space time spreading (DSTS) in combination with SBC is designed for the video transmission to cope with channel fading. Furthermore, the effects of SBC with different hamming distances d(H,min) but similar coding rates is explored on objective video quality such as peak signal to noise ratio (PSNR) and also the overall bit error ratio (BER). EXtrinsic Information Transfer Charts (EXIT) are used for analysis of the convergence behavior of SBC and its iterative scheme. Specifically, the experiments exhibit that the proposed scheme of error protection of SBC d(H,min) = 6 outperforms the SBCs having same code rate, but with d(H,min) = 3 by 3 dB with PSNR degradation of 1 dB. Furthermore, simulation results show that a gain of 27 dB Eb/N0 is achieved with SBC having code rate 1/3 compared to the benchmark Rate-1 SBC codes.

The subject matter of the paper is the problem of optimal packing of spheres of different dimension into a container of arbitrary geometric shape. The goal is to construct a mathematical model which associates different statements of the problem. Sphere packing problems (SPP) are combinatorial optimization problems known as cutting and packing problems. SPP consists in placement of a given set of spheres with given radii into a container of regular or irregular geometric shape. The task to be solved are: to investigate mathematical models of the two formulations according to the classification of cutting and packing problems: knapsack problems (KP) and open dimension problems (ODP); to construct a mathematical model which allow solve KP as ODP. The methods used are: the phi-function technique, increasing the problem dimension, homothetic transformations. KP is formulated as mixed discrete-continuous programming problem. A new approach which reduces solving KP to solving ODP for packing unequal and equal spheres into a container with the variable coefficient of homothety and allows adopt the jump algorithm for KP is suggested. To this end, for a given set of spheres KP is stated as a nonlinear programming problem in which the coefficient of homothety is an independent variable bounded below. The unit value of the coefficient corresponds to the original size of the container. A graphical illustration of the optimization process is presented. Conclusions. The approach suggested is a methodological basis for solving SPP. The generality of the approach lies in the fact that solving SPP does not depend on its formulation (KP or ODP). The approach is suitable for packing unequal and equal spheres into containers of arbitrary spatial shapes for which phi-functions can be constructed.

In this paper we show that for some problem classes it is possible to design Global Optimization algorithms which mimic existing procedures obtaining the same quality at a fraction of their computational cost. We achieved this applying clustering methods to identify regions of attraction of local minima. If we could identify the shape of regions of attraction, a local search starting from each of them would lead to the global minimum. This idea had been a winning one in the 1980s, and later abandoned when large dimensional global optimization problems were used to test global optimization algorithms. In this paper we show that by using the idea of clustering in a feature space of much smaller dimension than the original one, very significant speed ups can be obtained. We apply this idea to two of the most widely studied families of hard, large scale, global optimization problems: the optimization of the potential energy of atomic clusters, and the problem of packing identical spheres of largest radius in the unit hypercube. We could even improve some existing putative optima, thus proving that the proposed method is not only very efficient but also effective in exploring the feasible space. [ABSTRACT FROM AUTHOR]

In this paper we consider the Online Bin Packing Problem in three variants: Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes. The two first ones receive an online sequence of circles (items) of different radii while the third one receives an online sequence of spheres (items) of different radii, and they want to pack the items into the minimum number of unit squares, isosceles right triangles of leg length one, and unit cubes, respectively. For Online Circle Packing in Squares, we improve the previous best-known competitive ratio for the bounded space version, when at most a constant number of bins can be open at any given time, from 2.439 to 2.3536. For Online Circle Packing in Isosceles Right Triangles and Online Sphere Packing in Cubes we show bounded space algorithms of asymptotic competitive ratios 2.5490 and 3.5316, respectively, as well as lower bounds of 2.1193 and 2.7707 on the competitive ratio of any online bounded space algorithm for these two problems. We also considered the online unbounded space right variant of these three problems which admits a small reorganization of the items inside some of the bins after their packing, and we present algorithms of competitive ratios 2.3105, 2.5094, and 3.5146 for Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes, respectively. Throughout the text, we also discuss how our algorithms can be extended to other problems. [ABSTRACT FROM AUTHOR]

This paper discusses various tricks used to reduce an infinitary formulation of a problem to a finitary one, and vice versa. The presented examples are famous problems from arithmetic combinatorics and the Euclidean geometry, while the presented techniques come from mathematical analysis and mathematical logic. [ABSTRACT FROM AUTHOR]