scholarly journals Multiplicative Decomposition of Heterogeneity in Mixtures of Continuous Distributions

Entropy ◽  
2020 ◽  
Vol 22 (8) ◽  
pp. 858
Author(s):  
Abraham Nunes ◽  
Martin Alda ◽  
Thomas Trappenberg

A system’s heterogeneity (diversity) is the effective size of its event space, and can be quantified using the Rényi family of indices (also known as Hill numbers in ecology or Hannah–Kay indices in economics), which are indexed by an elasticity parameter q≥0. Under these indices, the heterogeneity of a composite system (the γ-heterogeneity) is decomposable into heterogeneity arising from variation within and between component subsystems (the α- and β-heterogeneity, respectively). Since the average heterogeneity of a component subsystem should not be greater than that of the pooled system, we require that γ≥α. There exists a multiplicative decomposition for Rényi heterogeneity of composite systems with discrete event spaces, but less attention has been paid to decomposition in the continuous setting. We therefore describe multiplicative decomposition of the Rényi heterogeneity for continuous mixture distributions under parametric and non-parametric pooling assumptions. Under non-parametric pooling, the γ-heterogeneity must often be estimated numerically, but the multiplicative decomposition holds such that γ≥α for q>0. Conversely, under parametric pooling, γ-heterogeneity can be computed efficiently in closed-form, but the γ≥α condition holds reliably only at q=1. Our findings will further contribute to heterogeneity measurement in continuous systems.

1954 ◽  
Vol 58 (526) ◽  
pp. 703-719 ◽  
Author(s):  
R. E. D. Bishop

SummaryComplicated oscillatory systems may be broken down into component “ sub-systems ” for the purpose of vibration analysis. These will generally submit more readily to analytical treatment. After an introduction to the concept of receptance, the principles underlying this form of analysis are reviewed.The dynamical properties of simple systems (in the form of their receptances) may be tabulated. By this means the properties of a complicated system may be found by first analysing it into convenient sub-systems and then extracting the properties of the latter from a suitable table. A catalogue of this sort is given for the particular case of conservative torsional systems with finite freedom.The properties of the composite system which may be readily found in this way are (i) its receptances and (ii) its frequency equation. Tables are given of expressions for these in terms of the receptances of the component sub-systems. All of the tables may easily be extended. The tabulated receptances may also be used for determining relative displacements during free vibration in any principal mode.A method of presenting information on the vibration characteristics of machinery, which is effectively due to Carter, is illustrated by means of an example. More general adoption by manufacturers of this method (which requires no more computational effort than must normally be made) would lead to enormous savings of labour in calculating natural frequencies of composite systems.


2020 ◽  
Vol 994 ◽  
pp. 162-169
Author(s):  
Štěpánka Dvořáčková ◽  
Dora Kroisová

This experimental study deals with the problematics of thermal expansion α [10-6/K] of the composite systems based on recycled carbon fibres reinforced epoxy resin. The epoxy resin CHS – EPOXY 520 (EPOXY 15), cured with the hardener P11 (Districhem, s.r.o.), was chosen as a sample matrix. Recycled carbon fibres with a diameter of 7 μm and a length of 100 μm (Easy Composites Ltd.) were the filler. In the experiment, samples with the fulfilment of 10, 20, 40, 60, 80, 90 and 100 phr were prepared. The samples were being poured into silicone molds, cured at an overpressure of 0.7 MPa and a temperature of 23 ± 2 °C for 24 hours. A thermomechanical analyzer was used to determine the thermal expansion of composite systems. The addition of recycled carbon fibers to epoxy resin can reduce the coefficient of linear thermal expansion at 20°C in half, from the original α = 45 to 55 × 10-6/K for non-filled epoxy resin to α = 25 to 27 × 10-6/K for filled epoxy resin. Optimal filling is at the level of 40 phr, with higher filling there is no further reduction of the linear thermal expansion coefficient.


2005 ◽  
Vol 08 (02n03) ◽  
pp. 193-208
Author(s):  
MAN LIU ◽  
YOU ZHANG ◽  
YUAN-WEI JING ◽  
SI-YING ZHANG

The design of the state observer for a kind of uncertain symmetric circulant composite systems based on regional pole assignment is studied. The "decentralized observer gain matrices" have been obtained by taking use of the special structure of the system. The central observer gain matrix for such a system has the same block symmetric circulant structure as the original system and can be constructed by the "decentralized observer gain matrices." Thus, the problem of designing state observer for such an uncertain system with n·N dimensions based on regional pole assignment can be transformed into the subproblems for N/2+1 or (N + 1)/2 uncertain systems with n dimensions. The central D-stabilizable state observer is also a symmetric circulant composite system. And the design of the observer makes best use of the advantage of interconnections of the system.


2004 ◽  
Vol 449-452 ◽  
pp. 765-768
Author(s):  
Chong Sung Park ◽  
Hyun Seok Hong ◽  
Myung Ho Kim ◽  
Chong Mu Lee

A new approach for the CTE on the basis of Ashelby.s cutting and welding process was made for the analysis of the thermal expansion behaviors of Al-Si alloys and composites. In this theoretical approach, it was considered that relaxation of residual stress could create an elastoplastic zone in the matrix around a particle during cooling. A comparison of the measured CTEs with the calculated ones for the Al-Si-SiCp and Al-Si-Al2O3 composite systems was performed in terms of the volume percent and the size of reinforced phases. The calculated results revealed that the linear CTE of the both composite depends on the size of the reinforce phases, especially at the composite systems with a low volume percent of the reinforce phases. The increase in the volume percentages of Al2O3, SiCp and Si phase lowers the linear CTEs of the systems. The measured CTEs was deviated less than about ten percents from the calculated ones at composites with a high volume percent. The deviations of the CTEs of reinforced phases are about 4 - 6 vol% from real composite systems.


SIMULATION ◽  
1970 ◽  
Vol 14 (2) ◽  
pp. 61-72 ◽  
Author(s):  
David Arthur Fahrland

Surface ◽  
2020 ◽  
Vol 12(27) ◽  
pp. 100-136
Author(s):  
T. V. Krupska ◽  
◽  
V. M. Gun'ko ◽  
I. S. Protsak ◽  
I. I. Gerashchenko ◽  
...  

The formation of a composite system based on equal amounts of hydrophobic, porous polymethylsiloxane and hydrophilic nanosilicon A-300 was studied. It is shown that during the formation of a composite system the specific surface of the material is significantly reduced, which is due to the close contact between hydrophobic and hydrophilic particles. When water is added to the composite system, in the process of homogenization under conditions of dosed mechanical loading, the effect of nanocoagulation is manifested – the formation of nanosized particles of hydrated silica inside the polymethylsiloxane matrix, recorded on TEM microphotographs. When measuring the value of the interfacial energy of PMS and PMS/A-300 composite by low-temperature 1H NMR spectroscopy, it was found that the effect of nanocoagulation is manifested in a decrease (compared to the original PMS) energy of water interaction with the surface of the composite obtained under small mechanical conditions. its growth when using high mechanical loads. In the process, the binding of water in heterogeneous systems containing PMS, pyrogenic nanosilica (A-300), water and surfactants – decamethoxine (DMT) was studied. Composite systems were created using metered mechanical loads. It is shown that when filling the interparticle gaps of PMS by the method of hydrosealing, the interphase energy of water in the interparticle gaps of hydrophobic PMS with the same hydration is twice the interfacial energy of water in hydrophilic silica A-300. This is due to the smaller linear dimensions of the interparticle gaps in PMS compared to A-300. In the composite system, A-300/PMS/DMT/H2O there are non-additive growth of binding energy of water, which is probably due to the formation, under the influence of mechanical stress in the presence of water, microheterogeneous areas consisting mainly of hydrophobic and hydrophilic components (microcoagulation). Thus, with the help of mechanical loads, you can control the adsorption properties of composite systems and create new materials with unique adsorption properties.


2021 ◽  
Author(s):  
Janani S. Gopu

Composite materials help realize high strength to weight ratio requirements of the Aerospace Industry. Composite structures and sandwich composite structures are susceptible to moisture ingress. Moisture ingress causes degradation of thermo-mechanical properties of the composite panels. Water accumulation in sandwich composite structures causes rapid degradation of face to core bondline, damage of cells frozen water and even blow off skins owing to sudden pressure build up in the cells of the honeycomb structure. Mechanisms of moisture ingress can be broadly classified into direct and indirect mechanisms. Direct ingress occurs through pre-existing pathways formed by defects in the composite system. Indirect mechanisms are diffusion, Capillary actions, Wicking actions, and Osmosis. The first form of damage in FRP materials is microcracking. The rate of microcracking increases with moisture ingress. Microcracking fracture toughness is a material property for the susceptibility of a composite system to the formation of microcracks. This work implores the mechanisms and the fracture mechanics dominating the formation of microcracks.


1978 ◽  
Vol 33 (11) ◽  
pp. 1406-1421
Author(s):  
A. Rieckers

The theory of thermodynamic stability is worked out in terms of composite systems starting from basic principles formulated partially in operational language. In this connection a macroscopic ordering concept is introduced, and entropy is characterized as numerical measure for the degree of disorder. Various aspects of thermodynamic stability are investigated in operational and analytical terms laying special emphasis on appropriate neighbourhood sets. Some implications of metastability and neutral stability for the thermodynamic system concept are pursued.


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