An Introduction to Hyperbolic Barycentric Coordinates and their Applications

Barycentric coordinates are commonly used in Euclidean geometry. The adaptation of barycentric coordinates for use in hyperbolic geometry gives rise to hyperbolic barycentric coordinates, known as gyrobarycentric coordinates. The aim of this article is to present the road from Einstein's velocity addition law of relativistically admissible velocities to hyperbolic barycentric coordinates along with applications.Comment: 66 pages, 3 figure

Gyrations: The Missing Link Between Classical Mechanics with its Underlying Euclidean Geometry and Relativistic Mechanics with its Underlying Hyperbolic Geometry

Being neither commutative nor associative, Einstein velocity addition of relativistically admissible velocities gives rise to gyrations. Gyrations, in turn, measure the extent to which Einstein addition deviates from commutativity and from associativity. Gyrations are geometric automorphisms abstracted from the relativistic mechanical effect known as Thomas precession

Möbius gyrogroups: A Clifford algebra approach

AbstractUsing the Clifford algebra formalism we study the Möbius gyrogroup of the ball of radius t of the paravector space R⊕V, where V is a finite-dimensional real vector space. We characterize all the gyro-subgroups of the Möbius gyrogroup and we construct left and right factorizations with respect to an arbitrary gyro-subgroup for the paravector ball. The geometric and algebraic properties of the equivalence classes are investigated. We show that the equivalence classes locate in a k-dimensional sphere, where k is the dimension of the gyro-subgroup, and the resulting quotient spaces are again Möbius gyrogroups. With the algebraic structure of the factorizations we study the sections of Möbius fiber bundles inherited by the Möbius projectors

Optimised Fabry-Perot (AlGa)As quantum well lasers tunable over 105 nm

Uncoated, Fabry-Perot (AlGa)As semiconductor lasers are tuned over 105nm in a grating-coupled external cavity. Broadband tunability is achieved by optimising the resonator loss so as to invoke lasing from both the first and second quantised states of the single quantum well active region

Harmonic analysis on the Möbius gyrogroup

In this paper we propose to develop harmonic analysis on the Poincaré ball $B_t^n$, a model of the n-dimensional real hyperbolic space. The Poincaré ball $B_t^n$ is the open ball of the Euclidean n-space $R^n$ with radius $t>0$, centered at the origin of $R^n$ and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in $\mathbb{R}^n$. For any $t>0$ and an arbitrary parameter $\sigma \in R$ we study the $(\sigma,t)$-translation, the $( \sigma,t)$-convolution, the eigenfunctions of the $(\sigma,t)$-Laplace-Beltrami operator, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and the associated Plancherel's Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when $t \rightarrow +\infty$ the resulting hyperbolic harmonic analysis on $B_t^n$ tends to the standard Euclidean harmonic analysis on $R^n$, thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on $B_t^n$

A Risk Comparison of Ordinary Least Squares vs Ridge Regression

We compare the risk of ridge regression to a simple variant of ordinary least squares, in which one simply projects the data onto a finite dimensional subspace (as specified by a Principal Component Analysis) and then performs an ordinary (un-regularized) least squares regression in this subspace. This note shows that the risk of this ordinary least squares method is within a constant factor (namely 4) of the risk of ridge regression.Comment: Appearing in JMLR 14, June 201

“Oh, this is What It Feels Like”: a role for the body in learning an evidence-based practice

This paper will present research that explored the experiences of couple and family therapists learning about and using an evidence-based practice (EBP). Using a phenomenological approach called Interpretative Phenomenological Analysis, three themes emerged from the participants’ experiences: the supports and challenges while learning an EBP, the experience of shame while learning, and the embodiment of a therapy practice. This paper will focus on the theme of embodiment. Research participants’ experiences will be reviewed and further explored using Merleau-Ponty’s notion of embodiment and Gendlin’s (1978) more internally focused understanding of how awareness of a felt sense is experienced as a move “inside of a person”. As researchers, educators, administrators, policy makers, and counsellors struggle with what works best with which populations and when, how best to allocate resources, how best to educate and support counsellors, and the complexity of doing research in real-life settings, this research has the potential to contribute to those varied dialogues