Social network analysis is the study of how links between a set of actors are formed. Typically, it is believed that links are formed in a structured manner, which may be due to, for example, political or material incentives, and which often may not be directly observable. The stochastic blockmodel represents this structure using latent groups which exhibit different connective properties, so that conditional on the group membership of two actors, the probability of a link being formed between them is represented by a connectivity matrix. The mixed membership stochastic blockmodel extends this model to allow actors membership to different groups, depending on the interaction in question, providing further flexibility.

Attribute information can also play an important role in explaining network formation. Network models which do not explicitly incorporate covariate information require the analyst to compare fitted network models to additional attributes in a post-hoc manner. We introduce the mixed membership of experts stochastic blockmodel, an extension to the mixed membership stochastic blockmodel which incorporates covariate actor information into the existing model. The method is illustrated with application to the Lazega Lawyers dataset. Model and variable selection methods are also discussed.

To define the pathology in cases of non-Alzheimer primary degenerative dementia (non-AD PDD), we have studied autopsies from four medical centres accessioned in consecutive years since 1976. Neurochemical studies of the basal forebrain-cortical (BF-C) cholinergic system have been conducted in cases from which frozen tissue was available. Twenty-two cases (mean age 70 years, range 47-86) in which the history was consistent with PDD, but which did not meet anatomic criteria for AD, were selected. Approximately 70 cases of PDD, which were accessioned in the same years and met the anatomic criteria for AD, were excluded. The pathologic findings permitted a classification into six groups: Lewy body disease (LBD), 4 cases; Pick's disease, 6 cases; cortical degeneration with motor neuron disease (CDmnd), 2 cases; hippocampal and temporal lobe sclerosis, 3 cases; few or nonspecific abnormalities, 5 cases; other disorders, 2 cases. Our findings suggest that LBD and Pick's disease account for a large proportion of cases of non-AD PDD in the presenile age group, but that a large number of other disorders occasionally present as PDD. Careful examination of the motor systems, as well as cerebral structures relate' to cognitive function, is important in the neuropathologic evaluation. Lesions of the BF-C cholinergic system have been most consistent and severe in LBD, and have not been identified in CDmnd.

We introduce five probability models for random topological graph theory. For two of these models (I and II), the sample space consists of all labeled orientable 2-cell imbeddings of a fixed connected graph, and the interest centers upon the genus random variable. Exact results are presented for the expected value of this random variable for small-order complete graphs, for closed-end ladders, and for cobblestone paths. The expected genus of the complete graph is asymptotic to the maximum genus. For Model III, the sample space consists of all labeled 2-cell imbeddings (possibly nonorientable) of a fixed connected graph, and for Model IV the sample space consists of all such imbeddings with a rotation scheme also fixed. The event of interest is that the ambient surface is orientable. In both these models the complete graph is almost never orientably imbedded. The probability distribution in Models I and III is uniform; in Models II and IV it depends on a parameter p and is uniform precisely when p = 1/2. Model V combines the features of Models II and IV.

The early composers of change ringing music for English church bells were not mathematicians, yet they developed intricate algebraic ideas more than a century before mathematicians independently discovered them. We introduce the mathematical concepts of permutation group and symmetry group by means of elementary change ringing compositions of the 17th and 18th centuries.

This fracture study evaluates the role of a fiber/matrix interfacial glass on the toughening of two different carbon/carbon (C/C) composites. Both composites incorporate a two-dimensional layup of 8-harness satin weave continuous fiber fabric, but differ in several aspects, the most significant of which is the presence of an oxidation inhibitor in one of these. The fracture behavior of both materials was determined in three-point flexure at 20 through 1650 °C. Microstructural studies indicate that the nonhomogeneous distribution of the oxidation inhibitor within the fiber bundles controls the fracture behavior. Electron microprobe results indicate a high concentration of the glass oxidation inhibitor associated with the inter-bundle matrix, while the intra-bundle matrix is composed primarily of carbon. Accordingly, debonding along the inter-bundle interfaces characterizes the oxidation inhibited composite, whereas the nonoxidation inhibited samples debond by individual fibers. Both materials exhibit strongly rising R-curves throughout the test temperature range. At the higher test temperatures the oxidation inhibited C/C shows the greatest cumulative toughening component, although at a lower value of the fracture toughness. This is consistent with the observed increase in the percentage of fibers that experience individual pullout at the higher temperatures.

Hamiltonian circuits with associated word an n-cycle, in Schreier right coset graphs for symmetric groups Sn mod cyclic groups Zn, correspond to change ringing principles on n bells for which the plain course is the extent; that is, neither bobs nor singles are required. This connection is made explicit for the general case, and then specialized to the cases n = 4 (minimus) and n= 5 (doubles). In particular, all 102 no-call doubles principles on three generators are found and catalogued.

The ancient and continuing art of change ringing, or campanology (how the British ring church bells), is studied from a mathematical viewpoint. An extent on n bells is regarded as a hamiltonian cycle in a Cayley colour graph for the symmetric group Sn, embedded on an appropriate surface. Two methods for variable n (Plain Bob for all n and Grandsire for n =; 3 (mod 4)) are discussed, and a new method for n odd is introduced. All minimus methods (n = 4) and five doubles methods (n = 5) are depicted, one of these being the new No Call Doubles.