User:Tohline/Appendix/Ramblings
From VisTrailsWiki
(→Mathematics) |
(→Mathematics) |
||
Line 100: | Line 100: | ||
<li>Roots of Cubic Equation</li> | <li>Roots of Cubic Equation</li> | ||
<ol style="list-style-type:lower-latin"> | <ol style="list-style-type:lower-latin"> | ||
- | <li>In the context of [[User:Tohline/Appendix/Ramblings/T1Coordinates#Second_Special_Case_.28cubic.29| | + | <li>In the context of [[User:Tohline/Appendix/Ramblings/T1Coordinates#Second_Special_Case_.28cubic.29|T2 Coordinates]], when <math>~q^2 = (a_1/a_3)^2=3</math>.</li> |
<li>[[User:Tohline/Appendix/Ramblings/PPTori#Cubic_Equation_Solution|PP Tori]] — Also includes [[User:Tohline/Appendix/Ramblings/PPTori#CubeRootImaginary|cube root of a complex number]]</li> | <li>[[User:Tohline/Appendix/Ramblings/PPTori#Cubic_Equation_Solution|PP Tori]] — Also includes [[User:Tohline/Appendix/Ramblings/PPTori#CubeRootImaginary|cube root of a complex number]]</li> | ||
<li>[[User:Tohline/SSC/Structure/Polytropes#CubicRoot|Srivastava's F-Type solution]] for <math>~n=5</math> polytropes.</li> | <li>[[User:Tohline/SSC/Structure/Polytropes#CubicRoot|Srivastava's F-Type solution]] for <math>~n=5</math> polytropes.</li> |
Revision as of 10:11, 14 October 2020
| Tiled Menu | Tables of Content | Banner Video | Tohline Home Page | |
Contents |
Ramblings
Sometimes I explore some ideas to a sufficient depth that it seems worthwhile for me to archive the technical derivations even if the idea itself does not immediately produce a publishable result. This page, which has a simple outline layout, provides links to these various pages of technical notes.
- Relationship between HNM82 models and T1 coordinates
- Orthogonal Curvilinear Coordinates
- Playing with the Spherical Wave Equation
- Analyzing Azimuthal Distortions
- Summary for Hadley & Imamura
- Detailed Notes 🎦
- Supplementary database generated by the Hadley & Imamura collaboration
- Large supplementary dataset accumulated by the Hadley & Imamura collaboration
- YouTube videos that supplement simulations of J. W. Woodward, J. E. Tohline, & I. Hachisu (1994)
- Stability Analyses of PP Tori
- Stability Analyses of PP Tori (Part 2)
- Integrals of Motion
- Old discussion
- T3 Coordinates
- Special (quadratic) case: Joel's Derivation vs. Jay's Derivation
- Killing Vector Approach; Jay Call's related Talk page
- Characteristic Vector for T3 Coordinates
- T4 Coordinates (Abandoned by Joel 7/6/2010 because non-orthogonal)
- Marcello's Radiation-Hydro Simulations
- Photosphere of Stably Accreting DWD
- Binary Polytropes
- A* Scheme
- Exploring the Properties of Radial Oscillations in Pressure-Truncated n = 5 Polytropes
- Instabilities Associated with Equilibrium Sequence Turning Points
- Derivations Related to Ledoux's Variational Principle
- More on Zero-Zero Bipolytropes
- Pt 1: Radial Oscillations of a Zero-Zero-Bipolytrope (Early Flawed Summary)
- Pt 2: Details
- Pt 3: Searching for Additional Eigenvectors
- Pt 4: Good Summary
- Numerically Determined Eigenvectors
- Analyzing Five-One Bipolytropes
- Assessing the Stability of Spherical, BiPolytropic Configurations
- Searching for Analytic EigenVector for (5,1) Bipolytropes
- Discussing Patrick Motl's 2019 Simulations
- Continue Search
- On the Origin of Planetary Nebulae (Investigation Resulting from a July, 2013 Discussion with Kundan Kadam)</lli>
- Looking outward, from Inside a Black Hole
- Radial Dependence of the Strong Nuclear Force
- Dyson (1893a) Part I: Some Details
- Radiation-Hydrodynamics
- Saturn
- Doctoral students Tohline has advised over the years
- For Richard H. Durisen
- Riemann Meets COLLADA and Oculus Rift S: Example (b/a, c/a) = (0.41, 0.385)
- Bordeaux University
- Copyright Issues
Mathematics
- Roots of Cubic Equation
- In the context of T2 Coordinates, when .
- PP Tori — Also includes cube root of a complex number
- Srivastava's F-Type solution for polytropes.
- Murphy & Fiedler's Bipolytrope with
- Analytic Eigenfunctions for Bipolytropes with — also involves cube root of a complex number
- Roots of Quartic Equation
- Singular Sturm-Liouville (eigenvalue) Problem
- Oscillations of PP Tori in the slim torus limit
- Characteristics of unstable eigenvectors in self-gravitating tori
- Approximate Power-Series Expressions
- Fourier Series
- Special Functions & Other Broadly Used Representations
- Spherical Harmonics and Associated Legendre Functions
- Multipole Expansions
- Familiar Expression for the Cylindrical Green's Function Expansion
- Toroidal Functions
- Green's Function in terms of Toroidal Functions
- Compact Cylindrical Green Function
- Toroidal configurations & related coordinate systems — Includes EUREKA! moment; also uses wikitable overflow (scrolling) box
- Toroidal Coordinate Integration Limits Includes Table of Example K(k) and E(k) Function Values; see a separate set of K(k) and E(k) evaluations in the context of Our Attempt to Replicate Dyson's results.
- Using Toroidal Coordinates to Determine the Gravitational Potential (Initial Presentation)
- Using Toroidal Coordinates to Determine the Gravitational Potential (Improved Presentation) includes series expansions for K(k) and E(k)
- Relationships between Toroidal Functions 5 plots of [MF53] data included here
- Confusion Regarding Whipple Formulae
- Pulling It All Together 2 additional plots of [MF53] data included here
Computer-Generated Holography
Computer Generated Holgram (Fall 2004) |
- Lead in …
- Apertures that are Parallel to the Image Screen:
- One-dimensional Aperture
- Initial Ideas
- Consolidate Expressions
- T. Kreis, P. Aswendt, & R. Höfling (2001), Optical Engineering, vol. 40, no. 6, 926 - 933: Hologram reconstruction using a digital micromirror device
- Two-dimensional, Rectangular Aperture
- Relevance to Holograms
- Caution and Words of Wisdom
- Apertures that are Tilted with Respect to the Image Screen:
- Building Holograms from VRML Files:
- ZebraImaging and Southwestern Medical Center
- Embracing COLLADA (2020)
- Quantum Mechanics
Computer Algorithms
- Directory …/fortran/FreeEnergy/EFE: README
- Directory …/numRecipes/EllipticIntegrals/Riemann
© 2014 - 2021 by Joel E. Tohline |