Flow-induced crystallization of poly(ethylene terephthalate) melts in a capillary die: Formation and properties of the crystalline toroid

An unusual crystalline aggregate, which is toroidal in shape, has been formed at the entrance region to capillary dies during the high-pressure extrusion of poly(ethylene terephthalate) melts in an Instron capillary rheometer. The toroidal remnant was discovered in capillary dies removed from the rheometer after the occurrence of oscillating flow. Detailed observations of the rheological factors involved in the formation of the crystalline toroid were made. Examination of recovered entrance plugs disclosed that the crystalline toroid was a hard, stagnant zone which throttled the bulk flow of polymer melt into the capillary during the extrusion experiments. Hot-stage optical microscopy, differential scanning calorimetry, and wide-angle x-ray techniques were used to initiate a structural study of the crystalline toroid. The toroid contained clusters of fibrous crystals which persisted to high temperatures. The fibrous clusters were randomly oriented in a matrix polymer which melted normally. Based upon the rheological and structural evidence, a mechanism of crystalline toroid formation is proposed.     

A Note on the Two-Dimensional Rectangular Cutting-Stock Problem

The concept of orthogonality in the Two-Dimensional Rectangular Cutting-Stock Problem is introduced. None of the published approaches for solving this problem permit non-orthogonal cutting patterns. An example is given to show that this may prevent the optimal cutting pattern from being obtained. The application of orthogonality to the problem of obtaining the minimum area rectangular enclosure of a number of rectangles is also discussed.     

Geodesic symmetries of homogeneous Kähler Manifolds

D'Atri and Nickerson [6], [7] have given necessary conditions for the geodesic symmetries of a Riemannian manifold to preserve the volume element. We use their results to show that ifG is a compact simple Lie group,T is a maximal torus ofG, andG/T is not symmetric, then anyG-invariant Kähler metric onG/T does not have volume-preserving geodesic symmetries. From the Kähler/de Rham decomposition of a compact homogeneous Kähler manifold [8], our result extends to the invariant Kähler metrics on a quotient of a compact connected Lie group by a maximal torus. In proving these results we compute directly the Ricci tensor of anyG-invariant Kähler metric onG/T forG compact connected andT a maximal torus ofG. The result is an explicit formula giving the value of the Ricci tensor elements in terms of the root structure of the Lie algebra ofG.     

Graphs for cone preserving maps

Let K be a closed, pointed, full cone in a finite dimensional real vector space. We associate with a linear map A for which AK?K four directed graphs. For two of the graphs the vertex set is the collection of all faces of K, and for two the vertices are all the extreme rays of K. We relate the irreducibility and primitivity of A to the strong connectedness of some of these graphs.