Relations, residuals, regular interiors, and relative regular equivalence

Given a relation α (a binary sociogram) and an a priori equivalence relation π, both on the same set of individuals, it is interesting to look for the largest equivalence πo that is contained in and is regular with respect to α. The equivalence relation πo is called the regular interior of π with respect to α. The computation of πo involves the left and right residuals, a concept that generalized group inverses to the algebra of relations. A polynomial-time procedure is presented (Theorem 11) and illustrated with examples. In particular, the regular interior gives meet in the lattice of regular equivalences: the regular meet of regular equivalences is the regular interior of their intersection. Finally, the concept of relative regular equivalence is defined and compared with regular equivalence.

Computing regular equivalence: practical and theoretical issues

Social network analysts have tried to capture the idea of social role explicitly by proposing a framework that precisely gives conditions under which group actors are playing equivalent roles. They term these methods positional analysis techniques. The most general definition is regular equivalence which captures the idea that equivalent actors are related in a similar way to equivalent alters. Regular equivalence gives rise to a whole class of partitions on a network. Given a network we have two different computational problems. The first is how to find a particular regular equivalence. An algorithm exists to find the largest regular partition but there are not efficient algorithms to test whether there is a regular k-partition. That is a partition in k groups that is regular. In addition, when dealing with real data, it is unlikely that any regular partitions exist. To overcome this problem relaxations of regular equivalence have been proposed along with optimisation techniques to find nearly regular partitions. In this paper we review the algorithms that have developed to find particular regular equivalences and look at some of the recent theoretical results which give an insight into the complexity of finding regular partitions.

The effect of reflow process on the contact resistance and reliability of anisotropic conductive film interconnection for flip chip on flex applications

The work presented in this paper focuses on the effect of reflow process on the contact resistance and reliability of anisotropic conductive film (ACF) interconnection. The contact resistance of ACF interconnection increases after reflow process due to the decrease in contact area of the conducting particles between the mating I/O pads. However, the relationship between the contact resistance and bonding parameters of the ACF interconnection with reflow treatment follows the similar trend to that of the as-bonded (i.e. without reflow) ACF interconnection. The contact resistance increases as the peak temperature of reflow profile increases. Nearly 40% of the joints were found to be open after reflow with 260 °C peak temperature. During the reflow process, the entrapped (between the chip and substrate) adhesive matrix tries to expand much more than the tiny conductive particles because of the higher coefficient of thermal expansion, the induced thermal stress will try to lift the bump from the pad and decrease the contact area of the conductive path and eventually, leading to a complete loss of electrical contact. In addition, the environmental effect on contact resistance such as high temperature/humidity aging test was also investigated. Compared with the ACF interconnections with Ni/Au bump, higher thermal stress in the Z-direction is accumulated in the ACF interconnections with Au bump during the reflow process owing to the higher bump height, thus greater loss of contact area between the particles and I/O pads leads to an increase of contact resistance and poorer reliability after reflow.

Predicting toxic gas concentrations resulting from enclosure fires using local equivalence ratio concept linked to fire field models

A practical CFD method is presented in this study to predict the generation of toxic gases in enclosure fires. The model makes use of local combustion conditions to determine the yield of carbon monoxide, carbon dioxide, hydrocarbon, soot and oxygen. The local conditions used in the determination of these species are the local equivalence ratio (LER) and the local temperature. The heat released from combustion is calculated using the volumetric heat source model or the eddy dissipation model (EDM). The model is then used to simulate a range of reduced-scale and full-scale fire experiments. The model predictions for most of the predicted species are then shown to be in good agreement with the test results