On the Polytopal Generalization of Sperner’s Lemma

We introduce and prove Sperner’s lemma, the well known combinatorial analogue of the Brouwer fixed point theorem, and then attempt to gain a better understanding of the polytopal generalization of Sperner’s lemma conjectured in Atanassov (1996) and proven in De Loera et al. (2002). After explaining the polytopal generalization and providing examples, we present a new, simpler proof of a slightly weaker result that helps us better understand the result and why it is correct. Some ideas for how to generalize this proof to the complete result are discussed. In the last two chapters we provide a brief introduction to the basics of matroid theory before generalizing a matroid generalization of Sperner’s lemma proven in Lovász (1980) to polytopes. At the end we present some partial progress towards proving the polytopal generalization of Sperner’s lemma using this matroid generalization

Dance-the-music : an educational platform for the modeling, recognition and audiovisual monitoring of dance steps using spatiotemporal motion templates

In this article, a computational platform is presented, entitled “Dance-the-Music”, that can be used in a dance educational context to explore and learn the basics of dance steps. By introducing a method based on spatiotemporal motion templates, the platform facilitates to train basic step models from sequentially repeated dance figures performed by a dance teacher. Movements are captured with an optical motion capture system. The teachers’ models can be visualized from a first-person perspective to instruct students how to perform the specific dance steps in the correct manner. Moreover, recognition algorithms-based on a template matching method can determine the quality of a student’s performance in real time by means of multimodal monitoring techniques. The results of an evaluation study suggest that the Dance-the-Music is effective in helping dance students to master the basics of dance figures

The basics of gravitational wave theory

Einstein's special theory of relativity revolutionized physics by teaching us that space and time are not separate entities, but join as ``spacetime''. His general theory of relativity further taught us that spacetime is not just a stage on which dynamics takes place, but is a participant: The field equation of general relativity connects matter dynamics to the curvature of spacetime. Curvature is responsible for gravity, carrying us beyond the Newtonian conception of gravity that had been in place for the previous two and a half centuries. Much research in gravitation since then has explored and clarified the consequences of this revolution; the notion of dynamical spacetime is now firmly established in the toolkit of modern physics. Indeed, this notion is so well established that we may now contemplate using spacetime as a tool for other science. One aspect of dynamical spacetime -- its radiative character, ``gravitational radiation'' -- will inaugurate entirely new techniques for observing violent astrophysical processes. Over the next one hundred years, much of this subject's excitement will come from learning how to exploit spacetime as a tool for astronomy. This article is intended as a tutorial in the basics of gravitational radiation physics.Comment: 49 pages, 3 figures. For special issue of New Journal of Physics, "Spacetime 100 Years Later", edited by Richard Price and Jorge Pullin. This version corrects an important error in Eq. (4.23); an erratum is in pres

Decumulation 101: the basics of drawing down capital in retirement

Decumulation in the retirement income context is the using up of retirement savings by way of drawing out regular income – for example, a fixed amount each month. It’s the converse of accumulating retirement savings while in paid work by regularly putting money aside. And return on investment plays its part in both: in the accumulation phase it enhances the amount saved; in the decumulation phase it enhances the regular amount that can be paid out. The first question is whether decumulation is something that actually needs any policy attention. Since the answer to that question here is a qualified yes, the next part of this article outlines a number of decumulation methods, considering such matters as cost, risk and flexibility. The desirability of any form of decumulation to individuals will naturally vary according to their preferences in respect of those points. The article concludes by positing some public good components in respect of each decumulation method, and setting out some possible government interventions in response. • Geoff Rashbrooke is a Senior Associate of the Institute for Governance and Policy Studies. He is a consultant on insurance and pension schemes and was formerly Government Actuary

How a colloidal paste flows – scaling behaviors in dispersions of aggregated particles under mechanical stress –

We have developed a novel computational scheme that allows direct numerical simulation of the mechanical behavior of sticky granular matter under stress. We present here the general method, with particular emphasis on the particle features at the nanometric scale. It is demonstrated that, although sticky granular material is quite complex and is a good example of a challenging computational problem (it is a dynamical problem, with irreversibility, self-organization and dissipation), its main features may be reproduced on the basis of rather simple numerical model, and a small number of physical parameters. This allows precise analysis of the possible deformation processes in soft materials submitted to mechanical stress. This results in direct relationship between the macroscopic rheology of these pastes and local interactions between the particles

Basics of Generalized Unitarity

We review generalized unitarity as a means for obtaining loop amplitudes from on-shell tree amplitudes. The method is generally applicable to both supersymmetric and non-supersymmetric amplitudes, including non-planar contributions. Here we focus mainly on N=4 Yang-Mills theory, in the context of on-shell superspaces. Given the need for regularization at loop level, we also review a six-dimensional helicity-based superspace formalism and its application to dimensional and massive regularizations. An important feature of the unitarity method is that it offers a means for carrying over any identified tree-level property of on-shell amplitudes to loop level, though sometimes in a modified form. We illustrate this with examples of dual conformal symmetry and a recently discovered duality between color and kinematics.Comment: 37 pages, 10 figures. Invited review for a special issue of Journal of Physics A devoted to "Scattering Amplitudes in Gauge Theories", R. Roiban(ed), M. Spradlin(ed), A. Volovich(ed