Posets arising as 1-skeleta of simple polytopes, the nonrevisiting path conjecture, and poset topology

Given any polytope $P$ and any generic linear functional ${\bf c}$, one obtains a directed graph $G(P,{\bf c})$ by taking the 1-skeleton of $P$ and orienting each edge $e(u,v)$ from $u$ to $v$ for ${\bf c} (u) < {\bf c} ( v)$. This paper raises the question of finding sufficient conditions on a polytope $P$ and generic cost vector ${\bf c}$ so that the graph $G(P, {\bf c} )$ will not have any directed paths which revisit any face of $P$ after departing from that face. This is in a sense equivalent to the question of finding conditions on $P$ and ${\bf c}$ under which the simplex method for linear programming will be efficient under all choices of pivot rules. Conditions on $P$ and ${\bf c}$ are given which provably yield a corollary of the desired face nonrevisiting property and which are conjectured to give the desired property itself. This conjecture is proven for 3-polytopes and for spindles having the two distinguished vertices as source and sink; this shows that known counterexamples to the Hirsch Conjecture will not provide counterexamples to this conjecture. A part of the proposed set of conditions is that $G(P, {\bf c} )$ be the Hasse diagram of a partially ordered set, which is equivalent to requiring non revisiting of 1-dimensional faces. This opens the door to the usage of poset-theoretic techniques. This work also leads to a result for simple polytopes in which $G(P, {\bf c})$ is the Hasse diagram of a lattice L that the order complex of each open interval in L is homotopy equivalent to a ball or a sphere of some dimension. Applications are given to the weak Bruhat order, the Tamari lattice, and more generally to the Cambrian lattices, using realizations of the Hasse diagrams of these posets as 1-skeleta of permutahedra, associahedra, and generalized associahedra.Comment: new results for 3-polytopes and spindles added; exposition substantially improved throughou

Professional Ethics and Social Responsibility: Military Work and Peacebuilding

The paper considers a number of important questions related to the involvement of engineers in peacebuilding and military work, including the preference of many countries for high tech weapons based security over peacebuilding, whether and in what circumstances, if any, it is justified for engineers to be involved in military work; and how engineers can persuade their colleagues to apply their skills to support peacebuilding. It is introduced by an overview of what is meant by the term military work and the extent and consequences of the use of military technology worldwide. This is followed by the applications of different approaches and theories of ethics to discuss the questions presented in the introduction. The approaches and theories applied include considerations of micro-and macro-ethics, codes of ethics, virtue ethics, considerations of gender and paradigms and the ethical imperative. Initial insights include the importance of considering the associated context and the need to avoid othering, which can make different treatment of minority groups, including the use of high tech weapons against them, seem acceptable

Shelling Coxeter-like Complexes and Sorting on Trees

In their work on `Coxeter-like complexes', Babson and Reiner introduced a simplicial complex $\Delta_T$ associated to each tree $T$ on $n$ nodes, generalizing chessboard complexes and type A Coxeter complexes. They conjectured that $\Delta_T$ is $(n-b-1)$-connected when the tree has $b$ leaves. We provide a shelling for the $(n-b)$-skeleton of $\Delta_T$, thereby proving this conjecture. In the process, we introduce notions of weak order and inversion functions on the labellings of a tree $T$ which imply shellability of $\Delta_T$, and we construct such inversion functions for a large enough class of trees to deduce the aforementioned conjecture and also recover the shellability of chessboard complexes $M_{m,n}$ with $n \ge 2m-1$. We also prove that the existence or nonexistence of an inversion function for a fixed tree governs which networks with a tree structure admit greedy sorting algorithms by inversion elimination and provide an inversion function for trees where each vertex has capacity at least its degree minus one.Comment: 23 page

Regular cell complexes in total positivity

This paper proves a conjecture of Fomin and Shapiro that their combinatorial model for any Bruhat interval is a regular CW complex which is homeomorphic to a ball. The model consists of a stratified space which may be regarded as the link of an open cell intersected with a larger closed cell, all within the totally nonnegative part of the unipotent radical of an algebraic group. A parametrization due to Lusztig turns out to have all the requisite features to provide the attaching maps. A key ingredient is a new, readily verifiable criterion for which finite CW complexes are regular involving an interplay of topology with combinatorics.Comment: accepted to Inventiones Mathematicae; 60 pages; substantially revised from earlier version

Overcoming barriers and increasing independence: service robots for elderly and disabled people

This paper discusses the potential for service robots to overcome barriers and increase independence of elderly and disabled people. It includes a brief overview of the existing uses of service robots by disabled and elderly people and advances in technology which will make new uses possible and provides suggestions for some of these new applications. The paper also considers the design and other conditions to be met for user acceptance. It also discusses the complementarity of assistive service robots and personal assistance and considers the types of applications and users for which service robots are and are not suitable

Ethical Engineering and Respect for The ‘Other’

Engineers have a very important role and responsibility in shaping modern society. Diversity amongst engineers is important in fulfilling this responsibility and ensuring that the creativity and needs of the whole population are taken account of. However, only a small percentage of engineers are female and very few of them are disabled. The paper discusses the experiences of women and disabled engineers in the context of othering and considers the way in which the existence of binary divides facilitates marginalisation and exclusion. It also discusses the need to involve end-users in design and development and education to encourage this, with a particular focus on disabled end-users

Fatigue of boron-aluminum composites bonds and joints

Study examines effects of boron filament diameter on bonds and joints in boron-aluminum composite. Data include static strength, fatigue, and dynamic moduli of elasticity. Manson-Coffin analyses and metallurgical and fracture surface evaluation were also performed