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

Thematic mapper studies band correlation analysis

Spectral data representative of thematic mapper candidate bands 1 and 3 to 7 were obtained by selecting appropriate combinations of bands from the JSC 24 channel multispectral scanner. Of all the bands assigned, only candidate bands 4 (.74 mu to .80 mu) and 5 (.80 mu to .91 mu) showed consistently high intercorrelation from region to region and time to time. This extremely high correlation persisted when looking at the composite data set in a multitemporal, multilocation domain. The GISS investigations lend positive confirmation to the hypothesis, that TM bands 4 and 5 are redundant

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

Hexagonal columnar phase in 1,4-trans-polybutadiene: morphology, chain extension, and isothermal phase reversal

Morphol., lamellar thickening, and the transition between the crystal and the hexagonal columnar phase were studied in 1,4-trans-polybutadiene (I) by optical and electron microscopy, time-resolved synchrotron x-ray diffraction, and DSC. I could form several-thousand-.ANG.-thick lamellas of the extended or nearly extended chain type. These could form either by direct growth from the melt or through annealing of thinner chain-folded lamellas, provided they were in the columnar phase. Lamellar thickness increased by several hundred .ANG. in soln.-crystd. mats immediately upon heating above the crystal-columnar transition, which contrasted with some previous reports on single crystals of I. The low-temp. crystal phase reappeared spontaneously on annealing above the temp. of the initial transition to the high-temp. columnar phase. [on SciFinder (R)

On the Study of Hyperbolic Triangles and Circles by Hyperbolic Barycentric Coordinates in Relativistic Hyperbolic Geometry

Barycentric coordinates are commonly used in Euclidean geometry. Following the adaptation of barycentric coordinates for use in hyperbolic geometry in recently published books on analytic hyperbolic geometry, known and novel results concerning triangles and circles in the hyperbolic geometry of Lobachevsky and Bolyai are discovered. Among the novel results are the hyperbolic counterparts of important theorems in Euclidean geometry. These are: (1) the Inscribed Gyroangle Theorem, (ii) the Gyrotangent-Gyrosecant Theorem, (iii) the Intersecting Gyrosecants Theorem, and (iv) the Intersecting Gyrochord Theorem. Here in gyrolanguage, the language of analytic hyperbolic geometry, we prefix a gyro to any term that describes a concept in Euclidean geometry and in associative algebra to mean the analogous concept in hyperbolic geometry and nonassociative algebra. Outstanding examples are {\it gyrogroups} and {\it gyrovector spaces}, and Einstein addition being both {\it gyrocommutative} and {\it gyroassociative}. The prefix "gyro" stems from "gyration", which is the mathematical abstraction of the special relativistic effect known as "Thomas precession".Comment: 78 pages, 26 figure

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

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

The Hyperbolic Derivative in the Poincaré Ball Model of Hyperbolic Geometry

AbstractThe generic Möbius transformation of the complex open unit disc induces a binary operation in the disc, called the Möbius addition. Following its introduction, the extension of the Möbius addition to the ball of any real inner product space and the scalar multiplication that it admits are presented, as well as the resulting geodesics of the Poincaré ball model of hyperbolic geometry. The Möbius gyrovector spaces that emerge provide the setting for the Poincaré ball model of hyperbolic geometry in the same way that vector spaces provide the setting for Euclidean geometry. Our summary of the presentation of the Möbius ball gyrovector spaces sets the stage for the goal of this article, which is the introduction of the hyperbolic derivative. Subsequently, the hyperbolic derivative and its application to geodesics uncover novel analogies that hyperbolic geometry shares with Euclidean geometry

Minimum Description Length Penalization for Group and Multi-Task Sparse Learning

We propose a framework MIC (Multiple Inclusion Criterion) for learning sparse models based on the information theoretic Minimum Description Length (MDL) principle. MIC provides an elegant way of incorporating arbitrary sparsity patterns in the feature space by using two-part MDL coding schemes. We present MIC based models for the problems of grouped feature selection (MIC-GROUP) and multi-task feature selection (MIC-MULTI). MIC-GROUP assumes that the features are divided into groups and induces two level sparsity, selecting a subset of the feature groups, and also selecting features within each selected group. MIC-MULTI applies when there are multiple related tasks that share the same set of potentially predictive features. It also induces two level sparsity, selecting a subset of the features, and then selecting which of the tasks each feature should be added to. Lastly, we propose a model, TRANSFEAT, that can be used to transfer knowledge from a set of previously learned tasks to a new task that is expected to share similar features. All three methods are designed for selecting a small set of predictive features from a large pool of candidate features. We demonstrate the effectiveness of our approach with experimental results on data from genomics and from word sense disambiguation problems