Arrangiamenti di sfere: stime sul "kissing number".

La tesi ha lo scopo di iniziare, in quanto non lo esaurisce completamente, a riordinare il percorso fatto nella stima del kissing number.

Minimizing the object dimensions in circle and sphere packing problems

Given a fixed set of identical or different-sized circular items, the problem we deal with consists on finding the smallest object within which the items can be packed. Circular, triangular, squared, rectangular and also strip objects are considered. Moreover, 2D and 3D problems are treated. Twice-differentiable models for all these problems are presented. A strategy to reduce the complexity of evaluating the models is employed and, as a consequence, instances with a large number of items can be considered. Numerical experiments show the flexibility and reliability of the new unified approach. (C) 2007 Elsevier Ltd. All rights reserved.

Advances in random matrix theory, zeta functions, and sphere packing

Over four hundred years ago, Sir Walter Raleigh asked his mathematical assistant to find formulas for the number of cannonballs in regularly stacked piles. These investigations aroused the curiosity of the astronomer Johannes Kepler and led to a problem that has gone centuries without a solution: why is the familiar cannonball stack the most efficient arrangement possible? Here we discuss the solution that Hales found in 1998. Almost every part of the 282-page proof relies on long computer verifications. Random matrix theory was developed by physicists to describe the spectra of complex nuclei. In particular, the statistical fluctuations of the eigenvalues (“the energy levels”) follow certain universal laws based on symmetry types. We describe these and then discuss the remarkable appearance of these laws for zeros of the Riemann zeta function (which is the generating function for prime numbers and is the last special function from the last century that is not understood today.) Explaining this phenomenon is a central problem. These topics are distinct, so we present them separately with their own introductory remarks.

Computer Simulation Of Random Sphere Packing In An Arbitrarily Shaped Container

Most simulations of random sphere packing concern a cubic or cylindric container with periodic boundary, containers of other shapes are rarely studied. In this paper, a new relaxation algorithm with pre-expanding procedure for random sphere packing in an arbitrarily shaped container is presented. Boundaries of the container are simulated by overlapping spheres which covers the boundary surface of the container. We find 0.4 similar to 0.6 of the overlap rate is a proper value for boundary spheres. The algorithm begins with a random distribution of small internal spheres. Then the expansion and relaxation procedures are performed alternately to increase the packing density. The pre-expanding procedure stops when the packing density of internal spheres reaches a preset value. Following the pre-expanding procedure, the relaxation and shrinking iterations are carried out alternately to reduce the overlaps of internal spheres. The pre-expanding procedure avoids the overflow problem and gives a uniform distribution of initial spheres. Efficiency of the algorithm is increased with the cubic cell background system and double link data structure. Examples show the packing results agree well with both computational and experimental results. Packing density about 0.63 is obtained by the algorithm for random sphere packing in containers of various shapes.

Intruder detection over sensor placement in a hexagonal lattice

The problem of intrusion detection and location identification in the presence of clutter is considered for a hexagonal sensor-node geometry. It is noted that in any practical application,for a given fixed intruder or clutter location, only a small number of neighboring sensor nodes will register a significant reading. Thus sensing may be regarded as a local phenomenon and performance is strongly dependent on the local geometry of the sensor nodes. We focus on the case when the sensor nodes form a hexagonal lattice. The optimality of the hexagonal lattice with respect to density of packing and covering and largeness of the kissing number suggest that this is the best possible arrangement from a sensor network viewpoint. The results presented here are clearly relevant when the particular sensing application permits a deterministic placement of sensors. The results also serve as a performance benchmark for the case of a random deployment of sensors. A novel feature of our analysis of the hexagonal sensor grid is a signal-space viewpoint which sheds light on achievable performance.Under this viewpoint, the problem of intruder detection is reduced to one of determining in a distributed manner, the optimal decision boundary that separates the signal spaces SI and SC associated to intruder and clutter respectively. Given the difficulty of implementing the optimal detector, we present a low-complexity distributive algorithm under which the surfaces SI and SC are separated by a wellchosen hyperplane. The algorithm is designed to be efficient in terms of communication cost by minimizing the expected number of bits transmitted by a sensor.

Experiments on thermal response of low aspect ratio packed beds at high Reynolds numbers with varying inflow temperature

An experimental study for transient temperature response of low aspect ratio packed beds at high Reynolds numbers for a free stream with varying inlet temperature is presented. The packed bed is used as a compact heat exchanger along with a solid propellant gas-generator, to generate room temperature gases for use in applications such as control actuation and air bottle pressurization. Packed beds of lengths similar to 200 mm and 300 mm were characterized for packing diameter based Reynolds numbers, Re-d ranging from 0.6 x 10(4) to 8.5 x 10(4). The solid packing used in the bed consisted of circle divide 9.5 mm and circle divide 5 mm steel spheres with suitable arrangements to eliminate flow entrance and exit effects. The ratios of packed bed diameter to packing diameter for 9.5 mm and 5 mm sphere packing were similar to 9.5 and 18 respectively, with the average packed bed porosities around 0.4. Gas temperatures were measured at the entry, exit and at three axial locations along centre-line in the packed beds. The solid packing temperature was measured at three axial locations in the packed bed. An average Nusselt number correlation of the form Nu(d) = 3.91Re(d)(05) for Re-d range of 10(4) is proposed. For engineering applications of packed beds such as pebble bed heaters, thermal storage systems, and compact heat exchangers a simple procedure is suggested for calculating unsteady gas temperature at packed bed exit for packing Biot number Bi-d < 0.1. (C) 2012 Elsevier Inc. All rights reserved.

A family of asymptotically good lattices having a lattice in each dimension

A new constructive family of asymptotically good lattices with respect to sphere packing density is presented. The family has a lattice in every dimension n >= 1. Each lattice is obtained from a conveniently chosen integral ideal in a subfield of the cyclotomic field Q(zeta(q)) where q is the smallest prime congruent to 1 modulo n.

Strong Dynamical Heterogeneity and Universal Scaling in Driven Granular Fluids

Large-scale simulations of two-dimensional bidisperse granular fluids allow us to determine spatial correlations of slow particles via the four-point structure factor S-4 (q, t). Both cases, elastic (epsilon = 1) and inelastic (epsilon < 1) collisions, are studied. As the fluid approaches structural arrest, i.e., for packing fractions in the range 0.6 <= phi <= 0.805, scaling is shown to hold: S-4 (q, t)/chi(4)(t) = s(q xi(t)). Both the dynamic susceptibility chi(4)(tau(alpha)) and the dynamic correlation length xi(tau(alpha)) evaluated at the alpha relaxation time tau(alpha) can be fitted to a power law divergence at a critical packing fraction. The measured xi(tau(alpha)) widely exceeds the largest one previously observed for three-dimensional (3d) hard sphere fluids. The number of particles in a slow cluster and the correlation length are related by a robust power law, chi(4)(tau(alpha)) approximate to xi(d-p) (tau(alpha)), with an exponent d - p approximate to 1.6. This scaling is remarkably independent of epsilon, even though the strength of the dynamical heterogeneity at constant volume fraction depends strongly on epsilon.

直径任意分布球填充的数值模拟

提出球填充数值算法的新分类方法.改进原有的松弛算法,使其能够模拟直径任意分布的球填充问题,采用可变循环周期使不同球数情形下的填充率基本保持不变.算例数据表明,该算法的填充率和配位数均高于原算法.由于采用背景网格搜索和双向链表组数据结构,使得邻接球搜索效率有相当大的提高,算法的时间复杂度为O（N）（N为球数）.在一台AMD Athlon 3200＋PC上,对于10000个等径球的随机密排列,只需217s,填充率即可达到0.64.

Geometric discretization through primal-dual meshes

This thesis introduces new tools for geometric discretization in computer graphics and computational physics. Our work builds upon the duality between weighted triangulations and power diagrams to provide concise, yet expressive discretization of manifolds and differential operators. Our exposition begins with a review of the construction of power diagrams, followed by novel optimization procedures to fully control the local volume and spatial distribution of power cells. Based on this power diagram framework, we develop a new family of discrete differential operators, an effective stippling algorithm, as well as a new fluid solver for Lagrangian particles. We then turn our attention to applications in geometry processing. We show that orthogonal primal-dual meshes augment the notion of local metric in non-flat discrete surfaces. In particular, we introduce a reduced set of coordinates for the construction of orthogonal primal-dual structures of arbitrary topology, and provide alternative metric characterizations through convex optimizations. We finally leverage these novel theoretical contributions to generate well-centered primal-dual meshes, sphere packing on surfaces, and self-supporting triangulations.

Oxygen carrier dispersion in inert packed beds to improve performance in chemical looping combustion

Various packed beds of copper-based oxygen carriers (CuO on Al2O3) were tested over 100 cycles of low temperature (673K) Chemical Looping Combustion (CLC) with H2 as the fuel gas. The oxygen carriers were uniformly mixed with alumina (Al2O3) in order to investigate the level of separation necessary to prevent agglomeration. It was found that a mass ratio of 1:6 oxygen carrier to alumina gave the best performance in terms of stable, repeating hydrogen breakthrough curves over 100 cycles. In order to quantify the average separation achieved in the mixed packed beds, two sphere-packing models were developed. The hexagonal close-packing model assumed a uniform spherical packing structure, and based the separation calculations on a hypergeometric probability distribution. The more computationally intensive full-scale model used discrete element modelling to simulate random packing arrangements governed by gravity and contact dynamics. Both models predicted that average 'nearest neighbour' particle separation drops to near zero for oxygen carrier mass fractions of x≥0.25. For the packed bed systems studied, agglomeration was observed when the mass fraction of oxygen carrier was above this threshold. © 2013 Elsevier B.V.

A derivation of the source-channel error exponent using nonidentical product distributions

This paper studies the random-coding exponent of joint source-channel coding for a scheme where source messages are assigned to disjoint subsets (referred to as classes), and codewords are independently generated according to a distribution that depends on the class index of the source message. For discrete memoryless systems, two optimally chosen classes and product distributions are found to be sufficient to attain the sphere-packing exponent in those cases where it is tight. © 2014 IEEE.

Temperature dependent SANS from ferritin and apoferritin

SANS from deuterated ferritin and apoferritin solutions over the temperature range 5 to 300 K is presented. Above the freezing point the SANS is well described by Percus-Yevick hard sphere packing. On freezing, highly correlated, partially crystallised, clusters of the proteins form and grow with decreasing temperature. The resulting scattering, characterised by a squared Lorentzian structure factor, indicates a spatial extent of 1000 8, for the protein clusters.

An Extension of Craig's Family of Lattices

Let p be a prime, and let zeta(p) be a primitive p-th root of unity. The lattices in Craig's family are (p - 1)-dimensional and are geometrical representations of the integral Z[zeta(p)]-ideals < 1 - zeta(p)>(i), where i is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions p - 1 where 149 <= p <= 3001, Craig's lattices are the densest packings known. Motivated by this, we construct (p - 1)(q - 1)-dimensional lattices from the integral Z[zeta(pq)]-ideals < 1 - zeta(p)>(i) < 1 - zeta(q)>(j), where p and q are distinct primes and i and fare positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties.

The geometry and the number of contacts of monodisperse sphere packs using X-ray tomography

The three-dimensional structure of very large samples of monodisperse bead packs is studied by means of X-Ray Computed Tomography. We retrieve the coordinatesofeach bead inthe pack and wecalculate the average coordination number by using the tomographic images to single out the neighbors in contact. The results are compared with the average coordination number obtained in Aste et al. (2005) by using a deconvolution technique. We show that the coordination number increases with the packing fraction, varying between 6.9 and 8.2 for packing fractions between 0.59 and 0.64. © 2005 Taylor & Francis Group.