Tangencies in invariant disc packings for certain planar piecewise isometries are rare

For planar piecewise isometries (PWIs) (two-dimensional maps that restrict to isometries on some partition) there is a natural coding given by the itinerary of a trajectory between the pieces (atoms) of the partition on which it is defined. The set of points with the same coding is referred to as a cell and under certain general conditions the periodically coded cells define an invariant set that is a disjoint union of discs. In this paper properties of this invariant disc packing are investigated. For a one-parameter family of PWI on a torus, it is proved that tangencies between discs in this packing are rare. More precisely it is shown, using algebraic constraints on the geometry of the centres of the discs, that tangencies between any two discs can only occur at a finite number of parameter values; hence all tangencies will occur at a set of parameter values that is (at most) countably infinite. If such packings are dense it can be shown that they are maximal in a sense of measure. Examples are provided to show that the packing may not be dense if there is continuity over boundaries in the partition, and also that the absence of tangencies in the packing does not necessarily imply that the complement of the packing has positive Lebesgue measure.     

The unique ergodic measure of the symmetric piecewise toral isometry of rotation angle θ=kπ/5 is the Hausdorff measure of its singular set

It is known that the symmetric piecewise toral isometry of rotation angle θ = k π /5, k=1, 2, 3, 4 is uniquely ergodic in 'a certain subset' of its singular set (aka exceptional set). The purpose of this paper is to identify the unique ergodic measure explicitly. In fact, we prove that the unique ergodic measure is none other than the normalized Hausdorff measure of the singular set, consequently proving that the unique ergodicity holds in the entire singular set. We use the 'phantom dynamics' given by a number of symmetry identifications as our main tool.     

On the asymptotic properties of piecewise contracting maps

We are interested in the phenomenology of the asymptotic dynamics of piecewise contracting maps. We consider a wide class of such maps and we give sufficient conditions to ensure some general basic properties, such as the periodicity, the total disconnectedness or the zero Lebesgue measure of the attractor. These conditions show in particular that a non-periodic attractor necessarily contains discontinuities of the map. Under this hypothesis, we obtain numerous examples of attractors, ranging from finite to connected and chaotic, contrasting with the (quasi-)periodic asymptotic behaviours observed so far.     

An empirical approach for the optimisation of aggregate combinations for self-compacting concrete

The fresh and hardened properties of self-compacting concrete (SCC) depend on number of factors such as paste composition, paste content, aggregate content, aggregate gradation etc. In the present investigation, the influence of the packing density of aggregates on the properties of SCC was evaluated. Experiments were conducted to measure the packing density for different combinations of aggregates precisely. A ternary packing diagram (TPD) was developed based on the packing density of measured and interpolated data. Considering the limitations in generalising the TPD and the difficulty involved in adopting mathematical models for aggregates, an attempt was made to establish a simple method for the selection of the combination of aggregates resulting in maximum packing density from the particle size distribution of aggregates (represented by the Coefficient of uniformity??C _u). Further, studies were extended to investigate the effect of aggregate packing density on fresh and hardened SCC properties. The results indicate that for a constant paste volume and paste composition, with increase in packing density of aggregates, the fresh properties and the compressive strength of SCC were improved positively. An attempt was also made to identify the influence of 10 different proportions of aggregates having the same packing density on the properties of SCC. The results indicate that at the same aggregate packing density, the fresh concrete properties were influenced significantly by the choice of the aggregate combination, while there was little or no influence on the hardened properties. Furthermore, the experimental data obtained was used for supplementary validation of the existing model (compressible packing model) for predicting the packing density and the fresh behaviour of SCC.     

Wet packing of blended fine and coarse aggregate

All codified methods for measuring the packing density of aggregate are carried out under dry condition. However, these dry packing methods do not account for the effect of water in the concrete mix. In a previous study, a wet packing method for measuring the packing density of fine aggregate under wet condition has been developed and it was found that the packing density of fine aggregate can be substantially higher under wet condition than dry condition. Nevertheless, many researchers still believe that for coarse aggregate, it does not matter much whether the packing density is measured under dry or wet condition. In this study, the wet packing method was extended to measure the packing density of coarse aggregate and blended fine and coarse aggregate. The results revealed that whilst the packing density of coarse aggregate is only slightly higher under wet condition than dry condition, the packing density of blended fine and coarse aggregate is highly dependent on whether the aggregate is dry or wet. Hence, when measuring the packing density of blended aggregate, the wet packing method should always be used.     

Local and global stability indices for a riddled basin attractor of a piecewise linear map

We consider a piecewise expanding linear map with a Milnor attractor whose basin is riddled with the basin of a second attractor. To characterize the local geometry of this riddled basin, we calculate a stability index for points within the attractor as well as introducing a global stability index for the attractor as a set. Our results show that for Lebesgue almost all points in attractor, the index is positive and we characterize a parameter region, where some points have negative index. We show there exists a dense set of points for which the index is not converge. Comparing to recent results of Keller, we show that the stability index for points in the attractor can be expressed in terms of a global stability index for the attractor and Lyapunov exponents for this point.     

Determining Piezoceramic Parameters on Disks

Approximate formulas are obtained for various parameters of a piezoelectric material in the form of a disk, which do not require the calculation of the least (first) root of the frequency equation. To calculate approximately the planar electromechanical coupling coefficient, one can simplify the measurement procedure because it is not necessary to measure the first overtone frequency and determine Poisson's ratio.     

Consistency results on nonparametric Bayesian estimation of level sets using spatial priors

The main purpose of this paper is to propose a Bayesian nonparametric estimation of density’s level sets from n independent identically distributed observations. We prove the strong consistency of posterior distribution under a fairly general class of priors. This entails, in particular, that the Bayes estimate of the level set is consistent in terms of the Lebesgue measure of the symmetric difference. The considered priors put mass on piecewise-constant densities. The pieces on which the densities are constant are Voronoi tiles generated by a spatial point process. Using conditions similar to those considered in the density framework, we also obtain the consistency of the Bayesian level set estimates in the regression model. We present a simulation study using a special prior in the framework of density level set estimation. This allows us to visualise the numerical performances of the Bayesian estimate and to compare them to a plug-in level set estimate.     

G~1保凸分段二次多项式插值参数曲线

给定平面上一列凸数据点,导出了用具有一阶几何连续性的分段二次多项式参数曲线插值各型值点且具有保凸性的充分必要条件.并用一些实例进行验证.结果表明,这种方法是正确和有效的.     

Wet packing of crushed rock fine aggregate

It is well known that the performance of concrete is dependent on the packing density of the aggregate. All existing methods of measuring the packing density of aggregate are carried out under dry condition. However, these dry packing methods are sensitive to the amount of compaction applied and do not account for the effect of water in the concrete mix. In this research, a new method, which measures the packing density of aggregate under wet condition, has been developed. It is called the wet packing method and has the advantages that it is less sensitive to the amount of compaction applied and that it includes the effect of water. The wet packing method was compared to the dry packing method by applying both methods to measure the packing density of crushed rock fine aggregate. It was found that the packing density of crushed rock fine aggregate can be 24% higher under wet condition than under dry condition and that the addition of superplasticizer can have a marginal contributing effect. Furthermore, it was demonstrated that the beneficial effect of blending different size aggregates together is better revealed by the wet packing method than by the dry packing method.     

Statistical properties of generalized Viana maps

We study quadratic skew-products with parameters driven over piecewise expanding and Markov interval maps with countable many inverse branches, a generalization of the class of maps introduced by Viana. In particular, we construct a class of multi-dimensional non-uniformly expanding attractors that exhibit both critical points and discontinuities and prove existence and uniqueness of an SRB (Sinai–Ruelle–Bowen) measure with stretched-exponential decay of correlations, stretched-exponential large deviations and satisfying some limit laws. Moreover, generically such maps admit the coexistence of a dense subset of points with negative central Lyapunov exponent together with a full Lebesgue measure subset of points which have positive Lyapunov exponents in all directions. Finally, we discuss the existence of SRB measures for skew-products associated with hyperbolic parameters by the study of fibred hyperbolic maps.     

Optical radiation flux illuminating a circular disk from an off-axis point source separated by a homogeneous plane-parallel plate

In a number of techniques that measure weak fluxes of optical radiation, it is frequently necessary to keep a detector in a medium different from that of the radiating source by separating it from the source with a planar transparent window. However, sources such as systems of light-emitting diodes, large-fiber illuminators, and microscopic living objects that emit biological luminescence may sometimes be regarded as multiple-point sources. To estimate the fluxes of optical radiation illuminating a surface from a nonuniformly distributed multiple-point source, a method for calculating fluxes from a single off-axis point source is needed. A formula is derived to estimate a flux of temporally incoherent optical radiation incident on a circular disk from a single off-axis point source separated by a plane-parallel plate (PPP). This formula is expressed by a series of single integrals of some superposed elementary functions. These functions depend on the variables that characterize the point-source-plane-parallel-plate-circular-disk geometry and on the optical properties of the media that separate the source from the PPP and the PPP from the disk. The solution was obtained for isotropic media. For illustrative purposes some examples of the use of the formula are presented. The selected results are illustrated by three-dimensional surface plots and compared with the values of the fluxes calculated for radiation incident on the disk from a point source not separated by a PPP.     

脉动注射保压过程中的压力损失

以低密度聚乙烯(LDPE)为原料,在中心浇口圆盘模腔脉动压力诱导注射成型过程中,测定了模具和喷嘴之间的压力,研究了保压过程中的压力损失,并与稳态注射成型过程做了比较.实验结果表明:存在某一频率和振幅,可使振动力场对保压过程中模腔压力损失最小.在所给实验条件下,频率为4 Hz时压力损失最小.与稳态注射相比,在保压阶段,脉动压力诱导注射模腔中的压力损失要小.脉动压力诱导注射可提高喷嘴压力和模腔压力,有利于保压补缩.     

Absolutely continuous invariant measures for the family of maps x → rxe-bx with application to the Belousov-Zhabotinski reaction

For the family of transformations τ(r,b,x)=rxe^-bx which models the Poincaré sections of the Belousov-Zhabotinski reaction we prove that there is an uncountable set A such that for each τξA,τ(r,b,˙) has an absolutely continuous invariant measure. We also approximate the density of this measure and thereby the Lyapunov exponent of τ(r,b,˙) .This is accomplished by proving that τ(r,b,˙) is conjugate, via an absolutely continuous homeomorphism, to a transormation T, where T^x is piecewise expanding, for some integers     

Packing density of cementitious materials: part 2—packing and flow of OPC + PFA + CSF

The wet packing method developed in Part 1 [Wong HHC, Kwan AKH (2007) Packing density of cementitious materials: part 1 measurement using a wet packing method. Mater Struct (Paper No. MAAS3281)] has been successfully applied to measure the packing densities of cementitious materials containing ordinary Portland cement (OPC), pulverised fuel ash (PFA) and condensed silica fume (CSF). The test results for non-blended materials revealed that whilst the addition of a superplasticiser would always increase the packing densities of OPC and PFA, the addition of a polycarboxylate-based superplasticiser could decrease the packing density of CSF. On the other hand, the results for blended materials showed that the packing density could be improved by double blending OPC with either PFA or CSF, and further improved by triple blending OPC, PFA and CSF together in appropriate proportions. A maximum packing density of 0.752 has been achieved and a ternary packing density diagram for determining the mix proportions for maximum packing density has been produced. Furthermore, the positive influence of a higher packing density on cement paste rheology has been demonstrated using the mini-slump cone test. Based on these results, the concept of excess water ratio, which is the major factor governing the rheology of a paste, is introduced.     

Read compensation in flexible disk drives

The peak shift due to packing density and other effects in flexible disk drives is discussed. Some of the effects merely cause loss of high-frequency components and may be corrected by a suitably designed filter. The design requirements are qualitatively summarized. Comparison with and use with write precompensation is discussed.     

PERIODICITY AND FINE STRUCTURE IN TWO-DIMENSIONAL GRAVITATIONAL RANDOM PACKING

Gravitational random packing of equal disks can be viewed as the evolution of a Markov chain in which a state is the shape of the front line formed by joining the centers of touching disks at the upper boundary. The placement of a new disk adds two straight-line segments and deletes one or more old ones. The replaced segments plus the two new ones form a polygon. Triangles, rhombi, and pentagons are most common, but hexagons and heptagons have been observed. The latter appear only in the early stages of packing while the numbers of triangles and pentagons in horizontal strips decay exponentially with height. Thus, the packing evolves to a purely rhombic structure.

The appearance of triangles, pentagons, or higher polygons indicates a choice between mutually exclusive sites. These branch points, which disrupt the rhombic structure, will occur whenever certain constraints are not satisfied. If the rhombic structure persists long enough, a diamond-shaped repeating unit is formed. The limiting packing density is simply the density of this repeating unit. Since units may differ greatly in their density, estimates should be based on the repeating units from many different runs.     

PERIODICITY AND FINE STRUCTURE IN TWO-DIMENSIONAL GRAVITATIONAL RANDOM PACKING

ABSTRACT

Gravitational random packing of equal disks can be viewed as the evolution of a Markov chain in which a state is the shape of the front line formed by joining the centers of touching disks at the upper boundary. The placement of a new disk adds two straight-line segments and deletes one or more old ones. The replaced segments plus the two new ones form a polygon. Triangles, rhombi, and pentagons are most common, but hexagons and heptagons have been observed. The latter appear only in the early stages of packing while the numbers of triangles and pentagons in horizontal strips decay exponentially with height. Thus, the packing evolves to a purely rhombic structure.

The appearance of triangles, pentagons, or higher polygons indicates a choice between mutually exclusive sites. These branch points, which disrupt the rhombic structure, will occur whenever certain constraints are not satisfied. If the rhombic structure persists long enough, a diamond-shaped repeating unit is formed. The limiting packing density is simply the density of this repeating unit. Since units may differ greatly in their density, estimates should be based on the repeating units from many different runs.     

Granular packing: numerical simulation and the characterisation of the effect of particle shape

The packing of granular particles is investigated using a combined finite-discrete element approach. One of the aims of this paper is to present an application of a recently improved numerical simulation technique for deformable granular material with arbitrary shapes. Our study is focused on the influence of the effect of the particle shape on (1) the emergent properties of a granular pack (packing density, coordination number, force distribution), and on (2) the spatial distribution of the stress. A set of simulations that mimick the sedimentation process is carried out, with varying input parameters, such as contact friction and particle shape. It is shown that the eccentricity of the particles not only significantly influences the final density of the pack but also the distribution of the stress and the contact forces. The presence of surface friction increases the amount of disorder within the granular system. Stress heterogeneities and force chain patterns propagate through the particles more efficiently than for the frictionless systems. The results also suggest that for the monodisperse systems investigated the coordination number is one of the factors that controls the distribution of the stress within a granular medium.     

Packing of fine particles in an electrical field

A numerical model based on the Discrete Element Method (DEM) is developed to study the packing of fine particles in an electrical field related to the dust collection in an electrostatic precipitator (ESP). The particles are deposited to form a dust cake mainly under the electrical and van der Waals forces. It is shown that for the packing formed by mono-sized charged particles, increasing either particle size or applied electrical field strength increases packing density until reaching a limit corresponding to the density of random loose packing obtained under gravity. The corresponding structural changes are analyzed in terms of coordination number, radial distribution function and other topological and metric properties generated from the Voronoi tessellation. It is shown that these properties are similar to those for the packing under gravity. Such structural similarities result from the similar changes in the competition of the cohesive forces and the driving force in the packing. In particular, it is shown that by replacing the gravity with the electrical field force, the previous correlation between packing density and the ratio of the cohesive force to the packing-driven force can be applied to the packing of fine particles in ESP.