Two nonmixed symmetric end-loadings of an elastic waveguide

This investigation treats the response of the semi-infinite plate with free faces subjected to suddenly applied normal loads on its end. The plate is elastic and in plane strain. The normal loads are symmetric and act in the absence of shear stress, i.e. the plate has nonmixed end conditions. A double Laplace transform technique is used to obtain long-time information for two particular normal loads; the uniform load and the line-load. Near-field and far-field approximations are found. Results in the long-time near-field for the uniform load reduce to elementary forms; for the line-load however, the corresponding forms are quite complex entailing singular terms and some numerically evaluated contributions. The far-field approximations give rise to integral of the Airy function forms for both loads and, if the forces applied under the two loadings are equal, these far-field responses are shown to be identical.     

On the singular behavior at the vertex of a bi-material wedge

Information on the singular behavior at the vertex of a bi-material wedge is the objective of this paper. A summary of the necessary conditions, which depend heavily on the associated eigenvalue equation, for stress singularities of O(r ^- 1n r) as r0 or O(r ^-) as r0 is stated. The eigenvalue equations arising from a wide range of boundary and interface conditions are then provided. Bi-material wedge problems that have been subjected to singularity analyses of some generality in the literature are briefly reviewed.     

A continuous path of singular masas in the hyperfinite II1 factor

Using methods of Tauer, we exhibit an uncountable family ofsingular masas in the hyperfinite II₁ factor R all with Pukánszkyinvariant {1}, no pair of which is conjugate by an automorphismof R. This is done by introducing an invariant (A) for a masaA in a II₁ factor N as the maximal size of a projection eA forwhich A e contains non-trivial centralizing sequences for eNe. The masas produced give rise to a continuous map from theinterval [0, 1] into the singular masas in R equipped with thed_{, 2}-metric. A result is also given showing that the Pukánszky invariantis d_{, 2}-upper semi-continuous. As a consequence, the sets ofmasas with Pukánszky invariant {n} are all closed.     

A Note on the Influence of Cohesive Stress-Separation Laws on Elastic Stress Singularities in Antiplane Shear

Elastic stress singularities occur at the vertices of reentrant corners with traditional, stress-free, boundary conditions. By recognizing cohesive stress action at sharp corners, and also introducing consistent cohesive laws ahead of corners, these singular stresses can be removed. This was earlier shown asymptotically for plane strain states: Here it is shown asymptotically for antiplane shear states.      

On the pressure and stress singularities induced by steady flows of incompressible viscous fluids

Design for structural integrity requires an appreciation of where stress singularities can occur in structural configurations. While there is a rich literature devoted to the identification of such singular behavior in solid mechanics, to date there has been relatively little explicit identification of stress singularities caused by fluid flows. In this study, stress and pressure singularities induced by steady flows of viscous incompressible fluids are asymptotically identified. This is done by taking advantage of an earlier result that the Navier-Stokes equations are locally governed by Stokes flow in angular corners. Findings for power singularities are confirmed by developing and using an analogy with solid mechanics. This analogy also facilitates the identification of flow-induced log singularities. Both types of singularity are further confirmed for two global configurations by applying convergence-divergence checks to numerical results. Even though these flow-induced stress singularities are analogous to singularities in solid mechanics, they nonetheless render a number of structural configurations singular that were not previously appreciated as such from identifications within solid mechanics alone.     

Correlation Functions for <Emphasis Type="Italic">β</Emphasis>=1 Ensembles of Matrices of Odd Size

Using the method of Tracy and Widom we rederive the correlation functions for β=1 Hermitian and real asymmetric ensembles of N×N matrices with N odd. This research was supported in part by the National Science Foundation (DMS-0801243).     

The Dynamic Geometrisation of Computer Programming

The goal of this paper is to explore dynamic geometry environments (DGE) as a type of computer programming language. Using projects created by secondary students in one particular DGE, we analyse the extent to which the various aspects of computational thinking—including both ways of doing things and particular concepts—were evident in their work, drawing specifically on frameworks for computational thinking that are designed for the purpose of mathematics education. We show how many of the practices associated with the use of propositional programming languages also feature in the more spatial and temporal register of the geometric ‘language’ of DGEs.