scholarly journals Symmetry group of the equiangular cubed sphere

2021 ◽  
Vol 80 (1) ◽  
pp. 69-86
Author(s):  
Jean-Baptiste Bellet
Keyword(s):  
Symmetry Group ◽  
Geodesic Distance ◽  
Computational Physics ◽  
Rotational Invariance ◽  
Numerical Schemes ◽  
Metric Properties ◽  
Mathematical Properties ◽  
Cubed Sphere

The equiangular cubed sphere is a spherical grid, widely used in computational physics. This paper deals with mathematical properties of this grid. We identify the symmetry group, i.e. the group of the orthogonal transformations that leave the cubed sphere invariant. The main result is that it coincides with the symmetry group of a cube. The proposed proof emphasizes metric properties of the cubed sphere. We study the geodesic distance on the grid, which reveals that the shortest geodesic arcs match with the vertices of a cuboctahedron. The results of this paper lay the foundation for future numerical schemes, based on rotational invariance of the cubed sphere.

scholarly journals Best predictors in logarithmic distance between positive random variables

2019 ◽  
Vol 15 (2) ◽  
pp. 15-28
Author(s):  
H. Gzyl
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Geodesic Distance ◽  
Geometric Mean ◽  
Random Variables ◽  
Random Variable ◽  
Metric Properties ◽  
The Central Limit Theorem ◽  
Logarithmic Distance

Abstract The metric properties of the set in which random variables take their values lead to relevant probabilistic concepts. For example, the mean of a random variable is a best predictor in that it minimizes the L2 distance between a point and a random variable. Similarly, the median is the same concept but when the distance is measured by the L1 norm. Also, a geodesic distance can be defined on the cone of strictly positive vectors in ℝn in such a way that, the minimizer of the distance between a point and a collection of points is their geometric mean. That geodesic distance induces a distance on the class of strictly positive random variables, which in turn leads to an interesting notions of conditional expectation (or best predictors) and their estimators. It also leads to different versions of the Law of Large Numbers and the Central Limit Theorem. For example, the lognormal variables appear as the analogue of the Gaussian variables for version of the Central Limit Theorem in the logarithmic distance.

Interpolating data on the Cubed Sphere with Spherical Harmonics

2021 ◽  
Author(s):  
Jean-Pierre Croisille ◽  
Jean-Baptiste Bellet ◽  
Matthieu Brachet
Keyword(s):  
Numerical Simulation ◽  
Symmetry Group ◽  
Spherical Harmonics ◽  
Recent Progress ◽  
Quadrature Rules ◽  
Geometrical Properties ◽  
Truncation Scheme ◽  
Cubed Sphere

<p>The Cubed Sphere is a grid commonly used in numerical simulation in climatology. In this talk we present recent progress<br>on the algebraic and geometrical properties of this highly symmetrical grid.<br>First, an analysis of the symmetry group of the Cubed Sphere will be presented: this group <br>is identified as the group of the Cube, [1]. Furthermore, we show how to construct a discrete Spherical Harmonics (SH) basis associated to <br>the Cubed Sphere. This basis displays a truncation scheme relating the zonal and longitudinal <br>mode numbers reminiscent of the rhomboidal truncation on the Lon-Lat grid.<br>The new analysis allows to derive new quadrature rules of  interest for applications in any kind of spherical modelling. In addition,<br>we will comment on applications in mathematical climatology and meteorology, [2].</p><p>[1] J.-B. Bellet, Symmetry group of the equiangular Cubed Sphere, preprint, IECL, Univ. Lorraine, 2020, submitted</p><p>[2] J.-B. Bellet, M. Brachet and J.-P. Croisille, Spherical Harmonics on The Cubed Sphere, IECL, Univ. Lorraine, 2021, Preprint.</p>

scholarly journals The Perceived Beauty of Regular Polygon Tessellations

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 984
Author(s):  
Jay Friedenberg
Keyword(s):  
Symmetry Group ◽  
Regular Polygon ◽  
Circular Aperture ◽  
Geometric Shapes ◽  
Single Vertex ◽  
Mathematical Properties ◽  
Circular Arrangements

Beauty judgments for regular polygon tessellations were examined in two experiments. In experiment 1 we tested the three regular and eight semi-regular tilings characterized by a single vertex. In experiment 2 we tested the 20 demi-regular tilings containing two vertices. Observers viewed the tessellations at different random orientations inside a circular aperture and rated them using a numeric 1–7 scale. The data from the first experiment show a peak in preference for tiles with two types of polygons and for five polygons around a vertex. Triangles were liked more than other geometric shapes. The results from the second experiment demonstrate a preference for tessellations with a greater number of different kinds of polygons in the overall pattern and for tiles with the greatest difference in the number of polygons between the two vertices. Ratings were higher for tiles with circular arrangements of elements and lower for those with linear arrangements. Symmetry group p6m was liked the most and groups cmm and pmm were liked the least. Taken as a whole the results suggest a preference for complexity and variety in terms of both vertex qualities and symmetric transformations. Observers were sensitive to both the underlying mathematical properties of the patterns as well as their emergent organization.

EXPONENTIATED HALF-LOGISTIC LOMAX DISTRIBUTION WITH PROPERTIES AND APPLICATION

2019 ◽  
Vol XVI (2) ◽  
pp. 1-11
Author(s):  
Farrukh Jamal ◽  
Hesham Mohammed Reyad ◽  
Soha Othman Ahmed ◽  
Muhammad Akbar Ali Shah ◽  
Emrah Altun
Keyword(s):  
Real Data ◽  
Continuous Model ◽  
Model Parameters ◽  
Data Set ◽  
Lomax Distribution ◽  
Mathematical Properties ◽  
Proposed Model ◽  
Probability Weighted Moment ◽  
Record Statistics ◽  
Maximum Likelihood Criterion

A new three-parameter continuous model called the exponentiated half-logistic Lomax distribution is introduced in this paper. Basic mathematical properties for the proposed model were investigated which include raw and incomplete moments, skewness, kurtosis, generating functions, Rényi entropy, Lorenz, Bonferroni and Zenga curves, probability weighted moment, stress strength model, order statistics, and record statistics. The model parameters were estimated by using the maximum likelihood criterion and the behaviours of these estimates were examined by conducting a simulation study. The applicability of the new model is illustrated by applying it on a real data set.

Novel geodesic distance Fourier shape descriptor

2010 ◽  
Vol 30 (2) ◽  
pp. 362-363
Author(s):  
Sheng CHEN ◽  
Xun LIU
Keyword(s):  
Geodesic Distance ◽  
Shape Descriptor

Rotating Shallow-Water Models with Moist Convection

2018 ◽  
Author(s):  
Vladimir Zeitlin
Keyword(s):  
Baroclinic Instability ◽  
Potential Temperature ◽  
Moist Convection ◽  
Mathematical Properties ◽  
Vertical Fluxes ◽  
Equivalent Potential ◽  
Shallow Water Models ◽  
Wave Emission ◽  
Rotating Shallow Water ◽  
Convective Fluxes

It is shown how the standard RSW can be ’augmented’ to include phase transitions of water. This chapter explains how to incorporate extra (convective) vertical fluxes in the model. By using Lagrangian conservation of equivalent potential temperature condensation of the water vapour, which is otherwise a passive tracer, is included in the model and linked to convective fluxes. Simple relaxational parameterisation of condensation permits the closure of the system, and surface evaporation can be easily included. Physical and mathematical properties of thus obtained model are explained, and illustrated on the example of wave scattering on the moisture front. The model is applied to ’moist’ baroclinic instability of jets and vortices. Condensation is shown to produce a transient increase of the growth rate. Special attention is paid to the moist instabilities of hurricane-like vortices, which are shown to enhance intensification of the hurricane, increase gravity wave emission, and generate convection-coupled waves.

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