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

On algebraic endomorphisms of the Einstein gyrogroup

We describe the structure of all continuous algebraic endomorphisms of the open unit ball $\mathbf{B}$ of $\mathbb{R}^3$ equipped with the Einstein velocity addition. We show that any nonzero such transformation originates from an orthogonal linear transformation on $\mathbb{R}^3$

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 $B_t^n$

The inhibitors and enablers of emerging adult COVID-19 mitigation compliance in a township context.

Young adults are often scapegoated for not complying with COVID-19 mitigation strategies. While studies have investigated what predicts this population’s compliance and non-compliance, they have largely excluded the insights of African young people living in South African townships. Given this, it is unclear what places young adult South African township dwellers at risk for not complying with physical distancing, face masking and handwashing, or what enables resilience to those risks. To remedy this uncertainty, the current article reports a secondary analysis of transcripts (n=119) that document telephonic interviews in June and October 2020 with 24 emerging adults (average age: 20 years) who participated in the Resilient Youth in Stressed Environments (RYSE) study. The secondary analysis, which was inductively thematic, pointed to compliance being threatened by forgetfulness; preventive measures conflicting with personal/collective style; and structural constraints. Resilience to these compliance risks lay in young people’s capacity to regulate their behaviour and in the immediate social ecology’s capacity to co-regulate young people’s health behaviours. These findings discourage health interventions that are focused on the individual. More optimal public health initiatives will be responsive to the risks and resilience-enablers associated with young people and the social, institutional, and physical ecologies to which young people are connected.Significance:• Emerging adult compliance with COVID-19 mitigation strategies is threatened by risks across multiple systems (i.e. young people themselves; the social ecology; the physical ecology).• Emerging adult resilience to compliance challenges is co-facilitated by young people and their social ecologies.• Responding adaptively to COVID-19 contagion threats will require multisystem mobilisation that is collaborative and transformative in its redress of risk and co-championship of resilience-enablers

Dislocation density and graphitization of diamond crystals

Two sets of diamond specimens compressed at 2 GPa at temperatures varying between 1060 K and 1760 K were prepared; one in which graphitization was promoted by the presence of water and another in which graphitization of diamond was practically absent. X-ray diffraction peak profiles of both sets were analyzed for the microstructure by using the modified Williamson-Hall method and by fitting the Fourier coefficients of the measured profiles by theoretical functions for crystallite size and lattice strain. The procedures determined mean size and size distribution of crystallites as well as the density and the character of the dislocations. The same experimental conditions resulted in different microstructures for the two sets of samples. They were explained in terms of hydrostatic conditions present in the graphitized samples

Mother’s Understanding of the Newborn Infant Hearing Screening (NIHS) and the Audiologist’s Role in Healthy Hearing

The Newborn Infant Hearing Screening (NIHS) is a process mandated by the state, which requires that all children born in New York must receive a hearing screening at birth and prior to discharge. The NIHS enables the early identification and subsequent treatment of hearing loss in infants. To properly facilitate identification, the infants who fail the screening are referred for further testing either to confirm normal hearing at the time of the screen or to identify the extent of the hearing loss, if one is found to exist. An additional way, dictated by New York State, in which early identification is ensured, is that the parents of children who pass the screen are required to be given educational materials. These handouts explain the hearing screening, signs of hearing loss in children, and what to do if a parent is concerned about their child’s hearing. The purpose of this survey is to ascertain the information parents receive when a newborn hearing screening takes place, and when results are shared with family members

Lorentz transformation and vector field flows

The parameter changes resulting from a combination of Lorentz transformation are shown to form vector field flows. The exact, finite Thomas rotation angle is determined and interpreted intuitively. Using phase portraits, the parameters evolution can be clearly visualized. In addition to identifying the fixed points, we obtain an analytic invariant, which correlates the evolution of parameters.Comment: 11 pages, 3 figures. Section IV revised and title change