Physical Sciences and Mathematics Commons^™

Supplementary Files For: "Interactive Modeling Of Bear Lake Elevations In A Future Climate", Benjamin D. Shaw, Scout Jarman, Brennan Bean, Kevin R. Moon, Wei Zhang, Nathan Butler, Tommy Bolton, April Knight, Emeline Haroldsen, Abby Funk, Rebecca Higbee

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The water level, or elevation, of Bear Lake has a significant impact on agriculture, power, infrastructure, and recreation for communities around the lake. Climatological variables, such as precipitation, temperature, and snowfall, all have an impact on the elevation of Bear Lake. As the climate changes due to greenhouse gas emissions, the typical behaviors of these climate variables change, leading to new behaviors in Bear Lake elevation. Because of the importance of Bear Lake, it is vital to be able to model and understand how Bear Lake's elevation may change in the face of different climate scenarios and to gain further …

Calculations From On The Existence Of Periodic Traveling-Wave Solutions To Certain Systems Of Nonlinear, Dispersive Wave Equations, Jacob Daniels

Mathematics and Statistics Student Research and Class Projects

In the field of nonlinear waves, particular interest is given to periodic traveling-wave solutions of nonlinear, dispersive wave equations. This thesis aims to determine the existence of periodic traveling-wave solutions for several systems of water wave equations. These systems are the Schr¨odinger KdV-KdV, Schr¨odinger BBM-BBM, Schr¨odinger KdV-BBM, and Schr¨odinger BBM-KdV systems, and the abcd-system. In particular, it is shown that periodic traveling-wave solutions exist and are explicitly given in terms of cnoidal, the Jacobi elliptic function. Certain solitary-wave solutions are also established as a limiting case of the periodic traveling-wave solutions, that is, as the elliptic modulus approaches one.

Supplementary Files For "Adaptive Mapping Of Design Ground Snow Loads In The Conterminous United States", Jadon Wagstaff, Jesse Wheeler, Brennan Bean, Marc Maguire, Yan Sun

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Recent amendments to design ground snow load requirements in ASCE 7-22 have reduced the size of case study regions by 91% from what they were in ASCE 7-16, primarily in western states. This reduction is made possible through the development of highly accurate regional generalized additive regression models (RGAMs), stitched together with a novel smoothing scheme implemented in the R software package remap, to produce the continental- scale maps of reliability-targeted design ground snow loads available in ASCE 7-22. This approach allows for better characterizations of the changing relationship between temperature, elevation, and ground snow loads across the Conterminous United …

A New Non-Inheriting Homogeneous Solution Of The Einstein-Maxwell Equations With Cosmological Term, Charles G. Torre

Research Vignettes

No abstract provided.

When Is A Linear Connection A Metric Connection?, Ian M. Anderson, Charles G. Torre

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In this worksheet we use the DG software to answer the following question: When is there a metric tensor on M whose Christoffel symbols coincide with the components of a given linear connection?

The Differentialgeometry Package, Ian M. Anderson, Charles G. Torre

Downloads

This is the entire DifferentialGeometry package, a zip file (DifferentialGeometry.zip) containing (1) a Maple Library file, DifferentialGeometryUSU.mla, (2) a Maple help file DifferentialGeometry.help, (3) a Maple Library file, DGApplicatons.mla. This is the latest version of the DifferentialGeometry software; it supersedes what is released with Maple.

Installation instructions

What's New In Differentialgeometry Release Dg2022, Ian M. Anderson, Charles G. Torre

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This Maple worksheet demonstrates the salient new features and functionalities of the 2022 release of the DifferentialGeometry software package.

The De Rham Decomposition Theorem, Ian M. Anderson, Charles G. Torre

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In this worksheet we show how the DG software provides for a local implementation of the de Rham decomposition theorem for Riemannian manifolds.

Spacetime Groups, Ian M. Anderson, Charles G. Torre

Publications

A spacetime group is a connected 4-dimensional Lie group G endowed with a left invariant Lorentz metric h and such that the connected component of the isometry group of h is G itself. The Newman-Penrose formalism is used to give an algebraic classification of spacetime groups, that is, we determine a complete list of inequivalent spacetime Lie algebras, which are pairs (g,η), with g being a 4-dimensional Lie algebra and η being a Lorentzian inner product on g. A full analysis of the equivalence problem for spacetime Lie algebras is given which leads to a completely algorithmic solution to the …

How To Make Tetrads, Charles G. Torre

How to... in 10 minutes or less

This is a worksheet which demonstrates tools for creating orthonormal and null tetrads for a given spacetime.

Symmetric Criticality In General Relativity, Charles G. Torre

Research Vignettes

In this worksheet I explore the local Lagrangian version of the Principle of Symmetric Criticality (PSC) due to Anderson, Fels, and Torre], which asserts the commutativity of the processes (i) of symmetry reduction (for finding group-invariant fields) and (ii) forming Euler-Lagrange equations. There are two obstructions to PSC, which I will call the Lie algebra obstruction and the isotropy obstruction. In this worksheet I will illustrate these obstructions in the General Theory of Relativity.

Examples Of The Birkhoff Theorem And Its Generalizations, Charles G. Torre

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In this worksheet I demonstrate three versions of Birkhoff's theorem, which is a characterization of spherically symmetric solutions of the Einstein equations. The three versions considered here correspond to taking the "Einstein equations" to be: (1) the vacuum Einstein equations; (2) the Einstein equations with a cosmological constant (3) the Einstein-Maxwell equations. I will restrict my attention to 4-dimensional spacetimes.

Introduction To The Usu Library Of Solutions To The Einstein Field Equations, Ian M. Anderson, Charles G. Torre

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This is a Maple worksheet providing an introduction to the USU Library of Solutions to the Einstein Field Equations. The library is part of the DifferentialGeometry software project and is a collection of symbolic data and metadata describing solutions to the Einstein equations.

Perihelion Precession In The General Theory Of Relativity, Charles G. Torre

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This is a relatively quick and informal sketch of a demonstration that general relativistic corrections to the bound Kepler orbits introduce a perihelion precession. Any decent textbook on the general theory of relativity will derive this result. My analysis aligns with that found in the good old text "Introduction to General Relativity", by Adler, Bazin and Schiffer.

The Kretschmann Scalar, Charles G. Torre

How to... in 10 minutes or less

On a pseudo-Riemannian manifold with metric g, the "Kretschmann scalar" is a quadratic scalar invariant of the Riemann R tensor of g, defined by contracting all indices with g. In this worksheet we show how to calculate the Kretschmann scalar from a metric.

On The Construction Of Simply Connected Solvable Lie Groups, Mark E. Fels

Research Vignettes

This worksheet contains the implementation of Theorems 4.2, 5.4 and 5.7 in the paper On the Construction of Solvable Lie Groups. All the examples in the paper are demonstrated here, along with one in Section 6 that was too long to include in the article.

Differentialgeometry In Brno, Ian M. Anderson

Presentations

This page will provide files supporting Ian Anderson's presentations in Brno, December 2015. The files can be found and downloaded from "Additional Files", below.

The files include:

(1) DifferentialGeometryUSU.mla: This is the Maple Library Archive file which provides all the DifferentialGeometry functionality. Here are Installation Instructions.

(2) DifferentialGeometry.help : this is the latest version of the DifferentialGeometry documentation. Copy this file to the same directory used for DifferentialGeometryUSU.mla (from step (1)).

Non-Isomorphic Real Simple Lie Algebras Of The Same Complex Type And Character, Ian M. Anderson

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Complex simple Lie algebras are classified by their root types. The type of a real simple Lie algebra is the root type of the associated complex algebra. The character of a real simple Lie algebra is the signature of its Killing form.

For many root types, the character is sufficient to uniquely classify the corresponding real Lie algebras. However, one should not take this statement to be literally true – there are a few cases where the character does not suffice to distinguish all possible real forms.

In this worksheet we will show that the 2 real non-isomorphic Lie algebras …

Cartan Involutions And Cartan Decompositions Of A Semi-Simple Lie Algebra, Ian M. Anderson

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In this worksheet we shall review the basic definitions and properties of Cartan involutions and Cartan decompositions and illustrate these using the DifferentialGeometry software package for Lie algebras.

A Rank 7 Pfaffian System On A 15-Dimensional Manifold With F4 Symmetry Algebra, Ian M. Anderson

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Let I be a differential system on a manifold M. The infinitesimal symmetry algebra of I is the set of all vectors fields X on M such that preserve I. In this worksheet we present an example, due to E. Cartan of a rank 7 Pfaffian system on a 15-dimensional manifold whose infinitesimal symmetry algebra is the split real form of the exceptional Lie algebra f4 .

Cartan Subalgebras, Compact Roots And The Satake Diagram For Su(2, 2), Ian M. Anderson

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In this worksheet we use the 15-dimensional real Lie algebra su(2, 2) to illustrate some important points regarding the general structure theory and classification of real semi-simple Lie algebras.

1. Recall that a real semi-simple Lie algebra g is called a compact Lie algebra if the Killing form is negative definite. The Lie algebra g is compact if and only if all the root vectors for any Cartan subalgebra are purely imaginary. However, if the root vectors are purely imaginary for some choice of Cartan subalgebra it is not necessarily true that the Lie algebra is compact.

2. A real …

Jordan Algebras And The Exceptional Lie Algebra F4, Ian M. Anderson

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This worksheet analyzes the structure of the Jordan algebra J(3, O) and its split and exceptional versions. The algebra of derivations is related to the exceptional Lie algebra f4.

Rainich-Type Conditions For Null Electrovacuum Spacetimes Ii, Charles G. Torre

Research Vignettes

In this second of two worksheets I continue describing local Rainich-type conditions which are necessary and sufficient for the metric to define a null electrovacuum. In other words, these conditions, which I will call the null electrovacuum conditions, guarantee the existence of a null electromagnetic field such that the metric and electromagnetic field satisfy the Einstein-Maxwell equations. When it exists, the electromagnetic field is easily constructed from the metric. In this worksheet I consider the null electrovacuum conditions which apply when a certain null geodesic congruence determined by the metric is twisting. I shall illustrate the these conditions using a …

Gr 20 Workshop, Warsaw, July 2013, Ian M. Anderson, Charles G. Torre

Presentations

These are the Maple worksheets used at the Differential Geometry in Maple Workshop, which was held at the 20th International Conference on General Relativity and Gravitation, in Warsaw, July 2013.

There are 6 worksheets which can be downloaded from the list of files below.

How To Find Killing Vectors, Charles G. Torre

How to... in 10 minutes or less

We show how to compute the Lie algebra of Killing vector fields of a metric in Maple using the commands KillingVectors and LieAlgebraData. A Maple worksheet and a PDF version can be found below.

How To Find A Levi Decomposition Of A Lie Algebra, Ian M. Anderson

How to... in 10 minutes or less

We show how to compute the Levi decomposition of a Lie algebra in Maple using the command LeviDecomposition. A worksheet and corresponding PDF can be found below.

How To Create A Jordan Algebra, Ian M. Anderson, Thomas J. Apedaile

How to... in 10 minutes or less

We show how to create a Jordan algebra in Maple using the commands AlgebraLibraryData and AlgebraData.

How To Create The Quaternion & Octonion Algebras, Ian M. Anderson, Thomas J. Apedaile

How to... in 10 minutes or less

We show how to create the quaternion and octonion algebras with the DifferentialGeometry software. For each algebra, there is a split-form also available.

The Octonions And The Exceptional Lie Algebra G2, Ian M. Anderson

Research Vignettes

The octonions O are an 8-dimensional non-commutative, non-associative normed real algebra. The set of all derivations of O form a real Lie algebra. It is remarkable fact, first proved by E. Cartan in 1908, that the the derivation algebra of O is the compact form of the exceptional Lie algebra G2. In this worksheet we shall verify this result of Cartan and also show that the derivation algebra of the split octonions is the split real form of G2.

PDF and Maple worksheets can be downloaded from the links below.

A Homogeneous Solution Of The Einstein-Maxwell Equations, Charles G. Torre

Research Vignettes

We exhibit and analyze a homogeneous spacetime whose source is a pure radiation electromagnetic field [1]. It was previously believed that this spacetime is the sole example of a homogeneous pure radiation solution of the Einstein equations which admits no electromagnetic field (see [2] and references therein). Here we correct this error in the literature by explicitly displaying the electromagnetic source. This result implies that all homogeneous pure radiation spacetimes satisfy the Einstein-Maxwell equations.

PDF and Maple worksheets can be downloaded from the links below.