'Is it just me...?': Q methodology and representing the marginal

In this paper Q as a constructionist methodology is considered through its engagement with marginality. Drawing primarily on debates within and examples from the discipline of psychology, we aim to illustrate ways in which issues of marginality become relevant to constructionist concerns around knowledge production. A key focus of constructionism(s) is on multiple versions of social phenomena in situated and local contexts. This position represents a move away from, and a challenge to, totalising forms of knowledge associated with more objectivist epistemologies. Broadly speaking, Q’s ability to tap into a range of perspectives or, what we will refer to here as, narratives – marginal or otherwise – provides a way to explicate constructionist concerns with multiplicity and unsettle mainstream notions of coherent and total knowledges of the social world. To contextualise the ways in which Q works with notions of marginality, this paper begins by delineating how Q itself is (re)produced as an othered methodology in debates around its location within the quantitative ‐qualitative dichotomy. We move on to consider the ways in which Q may offer a distinctive contribution within constructionist‐informed research through its ability to make expressions of marginality manifest

An Upper Bound on the Average Size of Silhouettes

It is a widely observed phenomenon in computer graphics that the size of the silhouette of a polyhedron is much smaller than the size of the whole polyhedron. This paper provides, for the first time, theoretical evidence supporting this for a large class of objects, namely for polyhedra that approximate surfaces in some reasonable way; the surfaces may be non-convex and non-differentiable and they may have boundaries. We prove that such polyhedra have silhouettes of expected size $O(\sqrt{n})$ where the average is taken over all points of view and n is the complexity of the polyhedron

Motion Planning of Legged Robots

We study the problem of computing the free space F of a simple legged robot called the spider robot. The body of this robot is a single point and the legs are attached to the body. The robot is subject to two constraints: each leg has a maximal extension R (accessibility constraint) and the body of the robot must lie above the convex hull of its feet (stability constraint). Moreover, the robot can only put its feet on some regions, called the foothold regions. The free space F is the set of positions of the body of the robot such that there exists a set of accessible footholds for which the robot is stable. We present an efficient algorithm that computes F in O(n2 log n) time using O(n2 alpha(n)) space for n discrete point footholds where alpha(n) is an extremely slowly growing function (alpha(n) <= 3 for any practical value of n). We also present an algorithm for computing F when the foothold regions are pairwise disjoint polygons with n edges in total. This algorithm computes F in O(n2 alpha8(n) log n) time using O(n2 alpha8(n)) space (alpha8(n) is also an extremely slowly growing function). These results are close to optimal since Omega(n2) is a lower bound for the size of F.Comment: 29 pages, 22 figures, prelininar results presented at WAFR94 and IEEE Robotics & Automation 9

If I am woman, who are 'they'? The construction of 'other' feminisms

Characterizations of feminist identities are presented, represented and, arguably, misrepresented within current public debates and popular media. Issues of sameness and difference have come to the fore as both timely and politically relevant. This paper aims to address issues arising from engagement with feminisms, in particular those which we experience as 'other' but which, concurrently, resonate with many of our concerns. Conflicting views revolve around the viability of constructing stable political identities for women who elect to include the term 'feminist' in their selfdescription. These debates become increasingly complex when contextualized within relative power positionings of knowledge production in differing arenas. Drawing on the literature around the legitimization of gender and political identities, the authors reflect in this paper on the possibilities of engaging with these identities, both in our capacity of 'others', but also as individuals whose theoretical positioning resonates with the issues under consideration

On tangents to quadric surfaces

We study the variety of common tangents for up to four quadric surfaces in projective three-space, with particular regard to configurations of four quadrics admitting a continuum of common tangents. We formulate geometrical conditions in the projective space defined by all complex quadric surfaces which express the fact that several quadrics are tangent along a curve to one and the same quadric of rank at least three, and called, for intuitive reasons: a basket. Lines in any ruling of the latter will be common tangents. These considerations are then restricted to spheres in Euclidean three-space, and result in a complete answer to the question over the reals: ``When do four spheres allow infinitely many common tangents?''.Comment: 50 page

Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph

A standard way to approximate the distance between any two vertices $p$ and $q$ on a mesh is to compute, in the associated graph, a shortest path from $p$ to $q$ that goes through one of $k$ sources, which are well-chosen vertices. Precomputing the distance between each of the $k$ sources to all vertices of the graph yields an efficient computation of approximate distances between any two vertices. One standard method for choosing $k$ sources, which has been used extensively and successfully for isometry-invariant surface processing, is the so-called Farthest Point Sampling (FPS), which starts with a random vertex as the first source, and iteratively selects the farthest vertex from the already selected sources. In this paper, we analyze the stretch factor $\mathcal{F}_{FPS}$ of approximate geodesics computed using FPS, which is the maximum, over all pairs of distinct vertices, of their approximated distance over their geodesic distance in the graph. We show that $\mathcal{F}_{FPS}$ can be bounded in terms of the minimal value $\mathcal{F}^*$ of the stretch factor obtained using an optimal placement of $k$ sources as $\mathcal{F}_{FPS}\leq 2 r_e^2 \mathcal{F}^*+ 2 r_e^2 + 8 r_e + 1$, where $r_e$ is the ratio of the lengths of the longest and the shortest edges of the graph. This provides some evidence explaining why farthest point sampling has been used successfully for isometry-invariant shape processing. Furthermore, we show that it is NP-complete to find $k$ sources that minimize the stretch factor.Comment: 13 pages, 4 figure

An expansion of the Jones representation of genus 2 and the Torelli group

We study the algebraic property of the representation of the mapping class group of a closed oriented surface of genus 2 constructed by VFR Jones [Annals of Math. 126 (1987) 335-388]. It arises from the Iwahori-Hecke algebra representations of Artin's braid group of 6 strings, and is defined over integral Laurent polynomials Z[t, t^{-1}]. We substitute the parameter t with -e^{h}, and then expand the powers e^h in their Taylor series. This expansion naturally induces a filtration on the Torelli group which is coarser than its lower central series. We present some results on the structure of the associated graded quotients, which include that the second Johnson homomorphism factors through the representation. As an application, we also discuss the relation with the Casson invariant of homology 3-spheres.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-3.abs.htm