Synthesis of Planar Stiffness

In this work the problem of designing systems of springs to achieve a desired stiffness matrix is considered. Only planar configurations are studied. After a brief section outlining the theory of the stiffness of planar systems the planar stiffness matrix of three typical design elements are found, simple springs, beams and pairs of stretched springs. A final section shows how arbitrary stiffness matrices can be achieved using three simple springs or two stretched spring pairs

On the use of Klein quadric for geometric incidence problems in two dimensions

We discuss a unified approach to a class of geometric combinatorics incidence problems in $2D$, of the Erd\"os distance type. The goal is obtaining the second moment estimate, that is given a finite point set $S$ and a function $f$ on $S\times S$, an upper bound on the number of solutions of $$f(p,p') = f(q,q')\neq 0,\qquad (p,p',q,q')\in S\times S\times S\times S. \qquad(*)$$ E.g., $f$ is the Euclidean distance in the plane, sphere, or a sheet of the two-sheeted hyperboloid. Our tool is the Guth-Katz incidence theorem for lines in $\mathbb{RP}^3$, but we focus on how the original $2D$ problem is made amenable to it. This procedure was initiated by Elekes and Sharir, based on symmetry considerations. However, symmetry considerations can be bypassed or made implicit. The classical Pl\"ucker-Klein formalism for line geometry enables one to directly interpret a solution of $(*)$ as intersection of two lines in $\mathbb{RP}^3$. This allows for a very brief argument extending the Euclidean plane distance argument to the spherical and hyperbolic distances. We also find instances of the question $(*)$ without underlying symmetry group. The space of lines in the three-space, the Klein quadric $\mathcal K$, is four-dimensional. We start out with an injective map $\mathfrak F:\,S\times S\to\mathcal K$, from a pair of points in $2D$ to a line in $3D$ and seek a combinatorial problem in the form $(*)$, which can be solved by applying the Guth-Katz theorem to the set of lines in question. We identify a few new such problems and generalise the existing ones.Comment: Theorem 5', implicit in the earlier verisons has been stated explicitly in this ArXiv version, giving a family of applications of the Guth-Katz theorem to sum-product type quantities, with no underlying symmetry grou

Parallel Robots with Homokinetic Joints:The Zero-Torsion Case

A two degree-of-freedom (DOF) homokinetic joint provides the freedom of spatially pointing directions without spinning (zero torsion). In this paper, we investigate structural synthesis of several classes of zero-torsion parallel robots using homokinetic joints

Spectroscopic analysis of hot, massive stars in large spectroscopic surveys with de-idealised models

Upcoming large-scale spectroscopic surveys with e.g. WEAVE and 4MOST will provide thousands of spectra of massive stars, which need to be analysed in an efficient and homogeneous way. Usually, studies of massive stars are limited to samples of a few hundred objects which pushes current spectroscopic analysis tools to their limits because visual inspection is necessary to verify the spectroscopic fit. Often uncertainties are only estimated rather than derived and prior information cannot be incorporated without a Bayesian approach. In addition, uncertainties of stellar atmospheres and radiative transfer codes are not considered as a result of simplified, inaccurate or incomplete/missing physics or, in short, idealised physical models. Here, we address the question of "How to compare an idealised model of complex objects to real data?" with an empirical Bayesian approach and maximum a {\it posterior} approximations. We focus on application to large scale optical spectroscopic studies of complex astrophysical objects like stars. More specifically, we test and verify our methodology on samples of OB stars in 30 Doradus region of the Large Magellanic Clouds using a grid of FASTWIND model atmospheres. Our spectroscopic model de-idealisation analysis pipeline takes advantage of the statistics that large samples provide by determining the model error to account for the idealised stellar atmosphere models, which are included into the error budget. The pipeline performs well over a wide parameter space and derives robust stellar parameters with representative uncertainties.Comment: Submitted to MNRAS, 21 pages, 9 figure

The development of an accreditation scheme for accredited exercise physiologists

Background: Accredited Exercise Physiologists provide exercise services for people living with chronic disease, disability or injury and are recognised in Australia as Accredited Exercise Physiologists (AEP) under a national certification system administered by Exercise and Sport Science Australia (ESSA). A major breakthrough occurred for the AEP in 2006 when the Australian Department of Health and Ageing approved the AEP to deliver clinical exercise services for people with chronic medical conditions under the taxpayer-funded national health scheme, Medicare Australia. Aims: In light of these developments, the authors recognised the need for new accreditation criteria, and our report summarises the work that we did on behalf of the profession and ESSA in restructuring the accreditation system. Methods and Outcomes: We first performed a background study that defined the scope of practice of the AEP and benchmarked the AEP against other allied health professions in Australia and Clinical Exercise Physiologists internationally. We then constructed a new set of accreditation criteria comprising sets of pathologyspecific knowledge and experiences, together with a set of generic standards including communication, professional behaviour and risk management. All participating Australian universities (18 out of 27 responded) and 29 practitioner experts were then invited to provide comment and input into the draft guidelines. There was strong support for the new system that was implemented nationally on 1 January 2008 and is now administered by ESSA. Conclusions: This work has stimulated an unprecedented level of activity in the Australian university sector in developing new curricula in clinical exercise science and practice, and is intended to lead to improved standards of clinical exercise practice.<br /

Mobile Icosapods

Pods are mechanical devices constituted of two rigid bodies, the base and the platform, connected by a number of other rigid bodies, called legs, that are anchored via spherical joints. It is possible to prove that the maximal number of legs of a mobile pod, when finite, is 20. In 1904, Borel designed a technique to construct examples of such 20-pods, but could not constrain the legs to have base and platform points with real coordinates. We show that Borel’s construction yields all mobile 20-pods, and that it is possible to construct examples where all coordinates are real

On the Exponentials of Some Structured Matrices

In this note explicit algorithms for calculating the exponentials of important structured 4 x 4 matrices are provided. These lead to closed form formulae for these exponentials. The techniques rely on one particular Clifford Algebra isomorphism and basic Lie theory. When used in conjunction with structure preserving similarities, such as Givens rotations, these techniques extend to dimensions bigger than four.Comment: 19 page