The notion of the shell of a Hilbert space operator, which is a useful generalization (proposed by Wielandt) of the numerical range, is extended to operators in spaces with an indefinite inner product. For the most part, finite dimensional spaces are considered. Geometric properties of shells (convexity, boundedness, being a subset of a line, etc.) are described, as well as shells of operators ...

This note derives formulae for Gaussian and mean curvatures for tensor-product and triangular rational Bézier patches in terms of the respective control meshes. These formulae provide more geometric intuition than the generic formulae from differential geometry.

We prove some general theorems which unify results on arithmetic-geometric mean and some other related matrix inequalities. As an application we obtain some results involving Hadamard product of matrices. Mathematics subject classification (2010): 47A30 15A60.

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We use the subgraph replacement method to prove a simple product formula for the tilings of an octagonal counterpart of Propp’s quasi-hexagons (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999), called quasi-octagon.

Due to the development of science and technology, the computer has become a useful tool for supporting engineering activities in product design. Many computer aided tools such as CAD/CAM, product data management (PDM), product life cycle assessment (PLA), etc., have been popularly used in industry for reducing product development lead-time and increasing total product quality. However, the nume...

We define a Cartesian product G ×H for finite simple graphs which satisfies the Künneth formula H(G × H) = ⊕i+j=kH(G) ⊗ H(G) and so pG×H(x) = pG(x)pH(y) for the Poincaré polynomial pG(x) = ∑ k=0 dim(H (G))x and χ(G × H) = χ(G)χ(H) for the Euler characteristic χ(G) = pG(−1). The graph G1 = G × K1 is homotopic to G, has a digraph structure and satisfies the inequality dim(G1) ≥ dim(G) and G1. Hod...

A convenient framework is developed to generalize Berry’s investigation of the adiabatic geometrical phase for a classical relativistic charged scalar field in a curved background spacetime which is minimally coupled to electromagnetism and an arbitrary (non-electromagnetic) scalar potential. The general formalism which is applicable for any scalar field equation which involves time derivatives...

This paper describes an approach at Volvo Truck Corporation where geometry users’ requirements are utilised to define configurable product views. The paper is derived from a research framework on geometry management. The objective of the research framework is to improve reuse of geometry by providing relevant geometry-based product information, so-called geometric product views, through applica...

Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E8, have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework, which affords a simple way of performing reflections and rotations whilst exposing more clearly the underlying geometry. The Clifford approach shows that the...