Sphere Packing Theory Research Articles

A sphere packing approach to break even and profitability analysis

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.

Application of Sphere Packing Theory in Government Budget

Application of Sphere Packing Theory in Government Budget

Application of sphere packing theory in financial management

<p style='text-indent:20px;'>The paper illustrates an application of sphere packing theory based on the capital structure of financial management. The entire process is modelled as a new optimization problem formulated as the sphere packing problem to define the capital structure. Numerical results are provided on an example of APU JSC of Mongolia.</p>

Novel method for the facile control of molecular weight cut-off (MWCO) of ceramic membranes

This study demonstrates a simple and novel preparation method to prepare ceramic nanofiltration membranes with a precise and tunable molecular weight cut-off (MWCO) by packing variously sized nanoparticles into existing membrane pores. As a result, ceramic membranes with a MWCO from 1000 Da to 10,000 Da were successfully prepared with the narrow distribution of the pore size after the filtration-coating process. In addition, the effective porosity of the ceramic membranes was calculated from the results of the membrane properties by the Hagen-Poiseuille equation which fit within the range of the sphere packing theory from 17.3% to 41.8%. Furthermore, the results of nonlinear curve fitting between the MWCO and the nanoparticle size show a high accuracy, which implies that the MWCO of the ceramic membranes can be predicted using the curve fitting model with variously sized nanoparticles in the filtration-coating process. In conclusion, the novel filtration-coating method enables precise pore control and provides a tunable MWCO to ceramic membranes by preparing various sizes of nanoparticles.

The smallest convex $k$-gon containing $n$ congruent disks

Consider the problem of fnding the smallest area convex $k$-gon containing $n\in\mathbb{N}$ congruent disks without an overlap. By using Wegner inequality in sphere packing theory we give a lower bound for the area of such polygons. For several cases where this bound is tight we construct corresponding optimal polygons. We also discuss its solution for some cases where this bound is not tight, e.g. $n = 2$ and $k$ is odd, and $n = 3$; $k = 4$. On the way to prove our results we prove a result on geometric invariants between two polygons whose sides are pairwise parallel, and give a new characterisation for the trisectrix of Maclaurin.

Optimization of theoretical maximal quantity of cells to immobilize on solid supports in the rational design of immobilized derivatives strategy.

Current worldwide challenges are to increase the food production and decrease the environmental contamination by industrial emissions. For this, bacteria can produce plant growth promoter phytohormones and mediate the bioremediation of sewage by heavy metals removal. We developed a Rational Design of Immobilized Derivatives (RDID) strategy, applicable for protein, spore and cell immobilization and implemented in the RDID1.0 software. In this work, we propose new algorithms to optimize the theoretical maximal quantity of cells to immobilize (tMQCell) on solid supports, implemented in the RDIDCell software. The main modifications to the preexisting algorithms are related to the sphere packing theory and exclusive immobilization on the support surface. We experimentally validated the new tMQCell parameter by electrostatic immobilization of ten microbial strains on AMBERJET® 4200 Cl- porous solid support. All predicted tMQCell match the practical maximal quantity of cells to immobilize with a 10% confidence. The values predicted by the RDIDCell software are more accurate than the values predicted by the RDID1.0 software. 3-indolacetic acid (IAA) production by one bacterial immobilized derivative was higher (~ 2.6 μg IAA-like indoles/108 cells) than that of the cell suspension (1.5 μg IAA-like indoles/108 cells), and higher than the tryptophan amount added as indole precursor. Another bacterial immobilized derivative was more active (22 μg Cr(III)/108 cells) than the resuspended cells (14.5 μg Cr(III)/108 cells) in bioconversion of Cr(VI) to Cr(III). Optimized RDID strategy can be used to synthesize bacterial immobilized derivatives with useful biotechnological applications.

Honeycomb Scaffolds Fabricated Using Extrusion Molding and the Sphere-Packing Theory for Bone Regeneration

Osteoconductive scaffolds can be used to treat large bone defects that arise due to accidents, tumors, congenital defects, and age-related ailments. Carbonate apatite is a promising material for th...

Manipulation of star-like polymer flooding systems based on their comprehensive solution properties and flow behavior in porous media

In this study, novel star-like polymers (SLPs) with four-gradient molecular weight (Mw), consisting of nano-silica as the core and a layer of hydrophilic chains as the shell, were fabricated via mild water free-radical polymerization. The comprehensive properties of SLP flooding systems prepared with brine were investigated to elucidate their underlying applications by combining static measurements, filterable tests, two-serial cores displacement experiments, and polymer flooding tests for enhanced oil recovery (EOR). The specific injectivity index was proposed, for the first time, to demonstrate the flow behavior of SLP solutions in conglomerate rocks.The results indicated that the rigid core together with the three-dimensional microstructure bestowed SLPs with the desirable viscosification and long-term stability in harsh conditions, outperforming ultrahigh-Mw partially hydrolyzed polyacrylamide. The dynamic hydrodynamic size (Hd) of SLPs was a power function over concentration and significantly depended on Mw. Based on the limitation of specific injectivity index, the resistance factor and residual resistance factor of SLP solutions as a function of permeability were mathematically correlated. SLPs had sustainable mobility control capability with no face plugging or internal filter cake when the average pore size over Hd of SLP solutions was higher than 5.2. This is contradictory to the conventional rigid sphere packing theory. Considering permeability of conglomerate rocks, the SLP flooding systems were designed using polymer concentration and Mw. The incremental oil recovery factor (RF) achieved by the designed SLP flooding systems was 7.0% more than that achieved by the SLP flooding systems that were not well designed but had an equivalent viscosity.This study indicates that the comprehensive performance of polymer solutions associated with the compatibility between Mw and rock permeability is significant to formulate polymer flooding systems with high EOR efficiency.

In-situ measurement of magnetic nanoparticle quantity in a microfluidic device

In this article we demonstrate a method for the accurate in situ determination of the quantity of the entrapped magnetic nanoparticles in the reaction chamber of a microfluidic device. The measurement is based on the resonance frequency shift of a passive electrical resonant circuit where a flat inductor coil integrated in a silicone elastomer film acts as a sensor. The particle amount inside the chamber affects the inductance therefore the resonance frequency is changed. The method also enables the on-line monitoring of the actual particle quantity in the chamber. The effect of particle size and arrangement on the catalytic activity of the reaction was investigated on the basis of the results gathered by the method. It was shown that the arrangement of the particles in the chamber obeys the sphere-packing theory therefore the volumetric ratio of the particles and the chamber is independent on the particle size. In the experiments phenylalanine ammonia-lyase enzyme was immobilized as a biocatalyst on the surface of magnetic nanoparticles. It was demonstrated that decreasing the particle size resulted in increasing catalytic activity due to the increased area to volume ratio. Further improvement could be achieved by using a binary mixture of particles with two different particle sizes. In this case a 40 % increment of the catalytic activity and 17 % increment of the entrapped particle mass could be observed.

New horizons in sphere-packing theory, part II: lattice-based derivative-free optimization via global surrogates

Derivative-free algorithms are frequently required for the optimization of nonsmooth scalar functions in n dimensions resulting, for example, from physical experiments or from the statistical averaging of numerical simulations of chaotic systems such as turbulent flows. The core idea of all efficient algorithms for problems of this type is to keep function evaluations far apart until convergence is approached. Generalized pattern search (GPS) algorithms, a modern class of methods particularly well suited to such problems, accomplish this by coordinating the search with an underlying grid which is refined, and coarsened, as appropriate. One of the most efficient subclasses of GPS algorithms, known as the surrogate management framework (SMF; see Booker et al. in Struct Multidiscip Optim 17:1---13, 1999), alternates between an exploratory search over an interpolating function which summarizes the trends exhibited by existing function evaluations, and an exhaustive poll which checks the function on neighboring points to confirm or confute the local optimality of any given candidate minimum point (CMP) on the underlying grid. The original SMF algorithm implemented a GPS step on an underlying Cartesian grid, augmented with a Kriging-based surrogate search. Rather than using the n-dimensional Cartesian grid (the typical choice), the present work introduces for this purpose the use of lattices derived from n-dimensional sphere packings. As reviewed and analyzed extensively in Part I of this series (see Belitz, PhD dissertation, University of California, San Diego, 2011, Chap. 2), such lattices are significantly more uniform and have many more nearest neighbors than their Cartesian counterparts. Both of these facts make them far better suited for coordinating GPS algorithms, as demonstrated here in a variety of numerical tests.

Some 2/spl times/2 unitary space-time codes from sphere packing theory with optimal diversity product of code size 6

In this correspondence, we propose some new designs of 2/spl times/2 unitary space-time codes of sizes 6, 32, 48, 64 with best-known diversity products (or product distances) by partially using sphere packing theory. In particular, we present an optimal 2/spl times/2 unitary space-time code of size 6 in the sense that it reaches the maximal possible diversity product for 2/spl times/2 unitary space-time codes of size 6. The construction and the optimality of the code of size 6 provide the precise value of the maximal diversity product of a 2/spl times/2 unitary space-time code of size 6.

Optimal ofdm design for time-frequency dispersive channels

Transmission over wireless channels is subject to time dispersion due to multipath propagation and to frequency dispersion due to the Doppler effect. Standard orthogonal frequency-division multiplexing (OFDM) systems, using a guard-time interval or cyclic prefix, combat intersymbol interference (ISI), but provide no protection against interchannel interference (ICI). This drawback has led to the introduction of pulse-shaping OFDM systems. We first present a general framework for pulse shape design. Our analysis shows that certain pulse shapes proposed in the literature are, in fact, optimal in a well-defined sense. Furthermore, our approach provides a simple way to adapt the pulse shape to varying channel conditions. We then show that (pulse-shaping) OFDM systems based on rectangular time-frequency lattices are not optimal for time- and frequency-dispersive wireless channels. This motivates the introduction of lattice-OFDM (LOFDM) systems which are based on general time-frequency lattices. Using results from sphere packing theory, we show how to design LOFDM systems (lattice and pulse shape) optimally for timeand frequency-dispersive channels in order to minimize the joint ISI/ICI. Our theoretical analysis is confirmed by numerical simulations, showing that LOFDM systems outperform traditional pulse-shaping OFDM systems with respect to robustness against ISI/ICI.

Grassmannian frames with applications to coding and communication

For a given class F of unit norm frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation |〈 f k , f l 〉| among all frames {f k} k∈ I ∈ F . We first analyze finite-dimensional Grassmannian frames. Using links to packings in Grassmannian spaces and antipodal spherical codes we derive bounds on the minimal achievable correlation for Grassmannian frames. These bounds yield a simple condition under which Grassmannian frames coincide with unit norm tight frames. We exploit connections to graph theory, equiangular line sets, and coding theory in order to derive explicit constructions of Grassmannian frames. Our findings extend recent results on unit norm tight frames. We then introduce infinite-dimensional Grassmannian frames and analyze their connection to unit norm tight frames for frames which are generated by group-like unitary systems. We derive an example of a Grassmannian Gabor frame by using connections to sphere packing theory. Finally we discuss the application of Grassmannian frames to wireless communication and to multiple description coding.

Computer-based technique for cell aggregation analysis and cell aggregation in in vitro chondrogenesis.

No quantitative methods are currently available to measure different aggregation parameters in cell cultures. In this paper we describe a computer-based technique for the automatic and reliable analysis of cellular aggregates, starting from optical microscopy images of living cells grown in suspension. The method allows determination, on the same sample at different time intervals, of quantitative parameters, including aggregation percentage, average number of cells in aggregates, and aggregate size statistical distribution. To determine the number of cells in an aggregate starting from its two-dimensional microscopic profile, a model has been proposed and verified, using sphere packing theory. Algorithms have been tested on chondrocyte suspension cultures, where cell aggregation is a very early and critical event leading to cell differentiation. Using this technique for the analysis of chick embryo chondrocyte cultures, we observed that aggregate size and development kinetics depend on the culture conditions used. The method, with minor adaptations, is of potential use also in other cell systems to evaluate aggregation indexes or to study aggregation kinetics.

Phonon theory of the entropy of liquid sodium

The Percus-Yevick phonon theory of liquids (1958) is reappraised and extended in the light of recent developments, and calculations are performed for the entropy of liquid sodium. The harmonic contribution is assumed to arise entirely from density fluctuations. The anharmonic contribution arises mainly from the interaction of phonon pairs with large wavenumber difference. The natural description of these effects is through the static structure factor, a(k). A connection is, thereby, in some measure made with the alternative effective hard sphere packing theory of entropy in which the emphasis is on the region around the principal peak of a(k).