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 $B_t^n$

Anterior Dental Microwear Texture Analysis of the Krapina Neandertals

Some Neandertal anterior teeth show unusual and excessive gross wear, commonly explained by non-dietary anterior tooth use, or using the anterior dentition as a tool, clamp, or third hand. This alternate use is inferred from aboriginal arctic populations, who used their front teeth in this manner. Here we examine anterior dental microwear textures of the Krapina Neandertals to test this hypothesis and further analyze tooth use in these hominins. Microwear textures from 17 Krapina Dental People were collected by white-light confocal profilometry using a 100x objective lens. Four adjacent scans were generated, totaling an area of 204x276 μm, and were analyzed using Toothfrax and SFrax SSFA software packages. The Neandertals were compared to six bioarchaeological/ethnographic samples with reported variation in diet, abrasive load, and non-dietary anterior tooth use. Results indicate that Krapina anterior teeth lack extreme microwear textures expected of hominins exposed to heavy abrasives or those that regularly generated high stresses associated with intense use of the front teeth as tools. Krapina hominins have microwear attributes in common with Coast Tsimshian, Aleut, and Puye Pueblo samples. Collectively, this suggests that the Krapina Neandertals faced moderate abrasive loads and only periodically used their anterior teeth as tools for non-diet related behaviors

Evaluation of nonmetallic thermal protection materials for the manned space shuttle. Volume 1, task 1: Assessment of technical risks associated with utilization of nonmetallic thermal protection system

Technical problems of design and flight qualification of the proposed classes of surface insulation materials and leading edge materials were reviewed. A screening test plan, a preliminary design data test plan and a design data test plan were outlined. This program defined the apparent critical differences between the surface insulators and the leading edge materials, structuring specialized screening test plans for each of these two classes of materials. Unique testing techniques were shown to be important in evaluating the structural interaction aspects of the surface insulators and a separate task was defined to validate the test plan. In addition, a compilation was made of available information on proposed material (including metallic TPS), previous shuttle programs, pertinent test procedures, and other national programs of merit. This material was collected and summarized in an informally structured workbook

Geometric observation for the Bures fidelity between two states of a qubit

In this Brief Report, we present a geometric observation for the Bures fidelity between two states of a qubit.Comment: 4 pages, 1 figure, RevTex, Accepted by Phys. Rev.

Degree of entanglement for two qubits

In this paper, we present a measure to quantify the degree of entanglement for two qubits in a pure state.Comment: 5 page

Spacecraft Crew Cabin Condensation Control

A report discusses a new technique to prevent condensation on the cabin walls of manned spacecraft exposed to the cold environment of space, as such condensation could lead to free water in the cabin. This could facilitate the growth of mold and bacteria, and could lead to oxidation and weakening of the cabin wall. This condensation control technique employs a passive method that uses spacecraft waste heat as the primary wallheating mechanism. A network of heat pipes is bonded to the crew cabin pressure vessel, as well as the pipes to each other, in order to provide for efficient heat transfer to the cabin walls and from one heat pipe to another. When properly sized, the heat-pipe network can maintain the crew cabin walls at a nearly uniform temperature. It can also accept and distribute spacecraft waste heat to maintain the pressure vessel above dew point