Strong d-collapsibility

We introduce a notion of strong d-collapsibility. Using this notion, we simplify the proof of Matoušek and the author showing that the nerve of a family of sets of size at most d is d-collapsible

Simpson's Paradox and Collapsibility

Simpson's paradox and collapsibility are two closely related concepts in the context of data analysis. While the knowledge about the occurrence of Simpson's paradox helps a statistician to draw correct and meaningful conclusions, the concept of collapsibility deals with dimension-reduction aspects, when Simpson's paradox does not occur. We discuss in this paper in some detail the nature and the genesis of Simpson's paradox with respect to well-known examples and also various concepts of collapsiblity. The main aim is to bring out the close connections between these two phenomena, especially with regard to the analysis of contingency tables, regression models and a certain measure of association or a dependence function. There is a vast literature on these topics and so we focus only on certain aspects, recent developments and some important results in the above-mentioned areas.Comment: 19 page

Effects of variation in water content, clay fraction and sodium carbonate additions on the synthetic moulding properties of Igbokoda clay and silica sand

Loto, C.A. and Adebayo, H., 1990. Effects of variation in water content, clay fraction and sodium carbonate additions on the synthetic moulding properties of Igbokoda clay and silica sand. Appl. Clay Sci., 5: 165-181. In this investigation, the Igbokoda pure silica sand is characterised and used as a base sand for the Igbokoda clay- sand mixture in an attempt to develop an efficient synthetic moulding sand. The mechanical properties of the clay-sand-water mixture were determined. The effect of additions ofNa2C03 to the moulding sand was examined with particular attention to its influence on mechanical properties of the synthetic moulding sand. The fineness number, screen bulk fraction, total coarse fraction and the total fine fraction of Igbokoda silica sand were found to be within the range that could give good mouldable properties. The values obtained for the green compressive strength, the dry compressive strength, the green and dry shear strengths, collapsibility and toughness indicate that Igbokoda clay has good values as a binder for synthetic moulding sand. In general, the addition of Na2C03 gives improved properties to the moulding sand though with a tendency towards impaired collapsibility values

Analysis and Development of lgbokoda Clay as a Binder for Synthetic Moulding Sand

Clay obtained from lgbokoda in the southwestern part of Nigeria has been analysed and developed as a binder for synthetic moulding sand. The analysis was compared with the analytical results obtained for the pre-treated bentonite clay imported from the U.S.A. and used by the Nigerian Foundries Ltd. (NFL). X-ray analysis indicated the presence of kaolin, illite and montmorillonite (parent mineral of bentonite), with kaolin having the highest percentage in the lgbokoda clay and montmorillonite was further confirmed by the flame test which shows the existence of exchangeable cations not present in kaolin. The effects of mixing varying amounts of clay, tempering water and sand on moulding properties were investigated. The green and dry strength and shatter index of the mixtures were determined. The green and the dry strength (both in shear and compression) of the lgbokoda test clay, mixed with the sub angular lagoon silica sand and determined at optimum water content, give good values for synthetic moulding sand. The shatter index test shows a decrease in collapsibility as the water content decreases at constant clay addition. lgbokoda clay is more collapsible but less tough than bentonite as shown by the shatter inde

Collapses, products and LC manifolds

Durhuus and Jonsson (1995) introduced the class of "locally constructible" (LC) triangulated manifolds and showed that all the LC 2- and 3-manifolds are spheres. We show here that for each d>3 some LC d-manifolds are not spheres. We prove this result by studying how to collapse products of manifolds with exactly one facet removed.Comment: 6 pages; added references; minor changes. Accepted for J. Comb. Theory, Series

Graphical Models for Inference Under Outcome-Dependent Sampling

We consider situations where data have been collected such that the sampling depends on the outcome of interest and possibly further covariates, as for instance in case-control studies. Graphical models represent assumptions about the conditional independencies among the variables. By including a node for the sampling indicator, assumptions about sampling processes can be made explicit. We demonstrate how to read off such graphs whether consistent estimation of the association between exposure and outcome is possible. Moreover, we give sufficient graphical conditions for testing and estimating the causal effect of exposure on outcome. The practical use is illustrated with a number of examples.Comment: Published in at http://dx.doi.org/10.1214/10-STS340 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Vertex decompositions of two-dimensional complexes and graphs

We investigate families of two-dimensional simplicial complexes defined in terms of vertex decompositions. They include nonevasive complexes, strongly collapsible complexes of Barmak and Miniam and analogues of 2-trees of Harary and Palmer. We investigate the complexity of recognition problems for those families and some of their combinatorial properties. Certain results follow from analogous decomposition techniques for graphs. For example, we prove that it is NP-complete to decide if a graph can be reduced to a discrete graph by a sequence of removals of vertices of degree 3.Comment: Improved presentation and fixed some bug