Book Review: Research Methods in Education, 7th Edition

This article reviews the book, “Research Methods in Education” by Louis Cohen, Lawrence Manion, Keith Morrison (Authors)

Positioning systems in Minkowski space-time: Bifurcation problem and observational data

In the framework of relativistic positioning systems in Minkowski space-time, the determination of the inertial coordinates of a user involves the {\em bifurcation problem} (which is the indeterminate location of a pair of different events receiving the same emission coordinates). To solve it, in addition to the user emission coordinates and the emitter positions in inertial coordinates, it may happen that the user needs to know {\em independently} the orientation of its emission coordinates. Assuming that the user may observe the relative positions of the four emitters on its celestial sphere, an observational rule to determine this orientation is presented. The bifurcation problem is thus solved by applying this observational rule, and consequently, {\em all} of the parameters in the general expression of the coordinate transformation from emission coordinates to inertial ones may be computed from the data received by the user of the relativistic positioning system.Comment: 10 pages, 7 figures. The version published in PRD contains a misprint in the caption of Figure 3, which is here amende

A physical application of Kerr-Schild groups

The present work deals with the search of useful physical applications of some generalized groups of metric transformations. We put forward different proposals and focus our attention on the implementation of one of them. Particularly, the results show how one can control very efficiently the kind of spacetimes related by a Generalized Kerr-Schild (GKS) Ansatz through Kerr-Schild groups. Finally a preliminar study regarding other generalized groups of metric transformations is undertaken which is aimed at giving some hints in new Ans\"atze to finding useful solutions to Einstein's equations.Comment: 18 page

Contact seaweeds II: type C

This paper is a continuation of earlier work on the construction of contact forms on seaweed algebras. In the prequel to this paper, we show that every index-one seaweed subalgebra of $A_{n-1}=\mathfrak{sl}(n)$ is contact by identifying contact forms that arise from Dougherty's framework. We extend this result to include index-one seaweed subalgebras of $C_{n}=\mathfrak{sp}(2n)$. Our methods are graph-theoretic and combinatorial

Classification of contact seaweeds

A celebrated result of Gromov ensures the existence of a contact structure on any connected, non-compact, odd dimensional Lie group. In general, such structures are not invariant under left translation. The problem of finding which Lie groups admit a left-invariant contact structure resolves to the question of determining when a Lie algebra $\mathfrak{g}$ is contact; that is, admits a one-form $\varphi\in\mathfrak{g}^*$ such that $\varphi\wedge(d\varphi)^k\neq 0.$ In full generality, this remains an open question; however we settle it for the important category of the evocatively named seaweed algebras by showing that an index-one seaweed is contact precisely when it is quasi-reductive. Seaweeds were introduced by Dergachev and Kirillov who initiated the development of their index theory -- since completed by Joseph, Panyushev, Yakimova, and Coll, among others. Recall that a contact Lie algebra has index one -- but not characteristically so. Leveraging recent work of Panyushev, Baur, Moreau, Duflo, Khalgui, Torasso, Yakimova, and Ammari, who collectively classified quasi-reductive seaweeds, our equivalence yields a full classification of contact seaweeds. We remark that since type-A and type-C seaweeds are de facto quasi-reductive (by a result of Panyushev), in these types index one alone suffices to ensure the existence of a contact form