More indecomposable polyhedra

We apply combinatorial methods to a geometric problem: the classification of polytopes, in terms of Minkowski decomposability. Various properties of skeletons of polytopes are exhibited, each sufficient to guarantee indecomposability of a significant class of polytopes. We illustrate further the power of these techniques, compared with the traditional method of examining triangular faces, with several applications. In any dimension $d\neq 2$, we show that of all the polytopes with $d^2+\frac{d}{2}$ or fewer edges, only one is decomposable. In 3 dimensions, we complete the classification, in terms of decomposability, of the 260 combinatorial types of polyhedra with 15 or fewer edges.Comment: PDFLaTeX, 21 pages, 6 figure

NATO's Deterrence Challenges: Report on a Workshop in Vilnius, Lithuania, 10-12 May 2009; Strategic Insights, v. 8, issue 4 (September 2009)

This article appeared in Strategic Insights, v.8, issue 4(September 2009)Approved for public release; distribution is unlimited

Lower bound theorems for general polytopes

For a $d$-dimensional polytope with $v$ vertices, $d+1\le v\le2d$, we calculate precisely the minimum possible number of $m$-dimensional faces, when $m=1$ or $m\ge0.62d$. This confirms a conjecture of Gr\"unbaum, for these values of $m$. For $v=2d+1$, we solve the same problem when $m=1$ or $d-2$; the solution was already known for $m= d-1$. In all these cases, we give a characterisation of the minimising polytopes. We also show that there are many gaps in the possible number of $m$-faces: for example, there is no polytope with 80 edges in dimension 10, and a polytope with 407 edges can have dimension at most 23.Comment: 26 pages, 3 figure

Strictly convex banach algebras

We discuss two facets of the interaction between geometry and algebra in Banach algebras. In the class of unital Banach algebras, there is essentially one known example which is also strictly convex as a Banach space. We recall this example, which is finite-dimensional, and consider the open question of generalising it to infinite dimensions. In

David Yost + ICSC : : Observing Everything Interactions : A Personal Journey

This project is about taking a personal journey to photograph thoughtless acts (human behaviors) or observations of how and why people use products and services in their natural settings. This project contains photographs and personal narrative interpretations of each photo provided by the author. The observations and personalized interpretations of them are intended to get people to inform, inspire, become more curious, aware, and observant of how we as a society design products and services and solve problems efficiently and effectively. These observations and interpretations are also intended to illustrate patterns that point to more a universal need. The final product developed is a draft photo book

Nunavut

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