Appendix/Ramblings

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.

1. Files on LSU P&A servers (myPhys)
2. Orthogonal Curvilinear Coordinate Systems
   1. Direction Cosines
   2. (Confocal) Elliptic Cylinder Coordinates plus Concentric Analog (T5 Coordinates)
   3. Concentric Ellipsoidal Coordinates:
      1. T6 Coordinates (pt. 1)
      2. T6 Coordinates (pt. 2)
      3. T6 Coordinates (pt. 3)
      4. Concentric Ellipsoidal (T8) Coordinates
      5. T9 Coordinates
      6. Concentric Ellipsoidal (T12) Coordinates
   4. Daring Attack
3. Relationship between HNM82 models and T1 coordinates
4. Playing with the Spherical Wave Equation
5. Analyzing Azimuthal Distortions
   1. Summary for Hadley & Imamura
   2. Detailed Notes   🎦
   3. Supplementary database generated by the Hadley & Imamura collaboration
   4. Large supplementary dataset accumulated by the Hadley & Imamura collaboration … access to this server may be denied
   5. YouTube videos that supplement simulations of J. W. Woodward, J. E. Tohline, & I. Hachisu (1994)
   6. Stability Analyses of PP Tori (Part 1A)
   7. Stability Analyses of PP Tori (Part 2A)
6. Integrals of Motion
1. Old discussion
2. T3 Coordinates
1. Special (quadratic) case: Joel's Derivation vs. Jay's Derivation
3. Killing Vector Approach; Jay Call's related Talk session
4. Characteristic Vector for T3 Coordinates
5. T4 Coordinates (Abandoned by Joel 7/6/2010 because non-orthogonal)
7. Marcello's Radiation-Hydro Simulations
1. Radiation-Hydrodynamics
2. Determining Code Units
3. Summary of Scalings
4. Initial Temperature Distributions
8. Photosphere of Stably Accreting DWD
9. Binary Polytropes
10. A* Scheme
    1. Hybrid Scheme Preface
    2. A^* Scheme
    3. Initial Effort to Explain Jay Call's Hybrid Scheme in the Context of Zach Byerly's Dissertation
    4. Implications of Hybrid Scheme
11. Other Structural and/or Stability Considerations for Spherically Symmetric Configurations
    1. Exploring the Properties of Radial Oscillations in Pressure-Truncated n = 5 Polytropes
    2. Radial Pulsation Neutral Mode
    3. Other Analytic Models, e.g., Parabolic Density Distribution
    4. Other Analytic Ramblings
    5. R. Van der Borght (1970), Proc. Astronomical Soc. Australia, Vol. 1, Issue 7, pp. 325 - 326):   Adiabatic Oscillations of Stars
12. Instabilities Associated with Equilibrium Sequence Turning Points
13. Derivations Related to Ledoux's Variational Principle
14. More on Zero-Zero Bipolytropes
1. Pt 1: Radial Oscillations of a Zero-Zero-Bipolytrope (Early Flawed Summary)
2. Pt 2: Details
3. Pt 3: Searching for Additional Eigenvectors
4. Pt 4: Good Summary
5. Numerically Determined Eigenvectors
15. Analyzing Five-One Bipolytropes
1. Assessing the Stability of Spherical, BiPolytropic Configurations
2. Searching for Analytic EigenVector for (5,1) Bipolytropes
3. See (below) Discussing Patrick Motl's 2019 BiPolytrope Simulations
4. Continue Search
5. Renormalize Structure
6. Renormalize Structure (Part 2)
7. Renormalize Structure (Part 3)
8. More Carefully Exam Step Function Behavior
9. More Focused Search for Analytic EigenVector if (5,1) Bipolytropes
10. Do Not Confine Search to Analytic Eigenvector
11. Clean, Methodical Examination
12. Rethink Handling of n = 1 Envelope
13. Improved Treatment of Core-Envelope Interface:
    1. Part 1
    2. Part 2
16. Interrelating 51 and 00 Bipolytropes
1. Organization
17. On the Origin of Planetary Nebulae (Investigation Resulting from a July, 2013 Discussion with Kundan Kadam)
18. Looking outward, from Inside a Black Hole
19. Dyson (1893a) Part I:  Some Details
20. Saturn
21. Doctoral students Tohline has advised over the years
22. Discussing Patrick Motl's 2019 BiPolytrope Simulations
23. For Richard H. Durisen
24. For Shangli Ou
25. For Howard Cohl:
    1. Ongoing Collaborative Discussions
    2. Critical Points along the Maclaurin Spheroid Sequence
26. For Paul Fisher
27. For PJ in April 2021
28. Riemann Meets COLLADA and Oculus Rift S: Example (b/a, c/a) = (0.41, 0.385)
    1. Virtual Reality and 3D Printing
    2. Success Importing Animated Scene into Oculus Rift S
    3. Carefully (Re)Build Riemann Type S Ellipsoids Inside Oculus Rift Environment
    4. Other Example S-type Riemann Ellipsoids:
       1. (b/a, c/a) = (0.90, 0.333)
       2. (b/a, c/a) = (0.74, 0.692)
       3. (b/a, c/a) = (0.28, 0.256)
    5. Pointers to a collection of COLLADA-based Models of Riemann S-Type Ellipsoids
29. Challenges Constructing Ellipsoidal-Like Configurations
    1. Riemann Type 1 Ellipsoids
    2. Construction Challenges (Pt. 1)
    3. Construction Challenges (Pt. 2)
    4. Construction Challenges (Pt. 3)
    5. Construction Challenges (Pt. 4)
    6. Construction Challenges (Pt. 5)
    7. Related discussions of models viewed from a rotating reference frame:
       1. PGE/RotatingFrame
       2. NOTE to Eric Hirschmann & David Neilsen... I have moved the earlier contents of this page to a new Wiki location called Compressible Riemann Ellipsoids.
30. Bordeaux University
    1. External Gravitational Potential of Toroids
    2. Spheroid-Ring Sequences
    3. Discussions Following Dissertation Defense
31. Copyright Issues
32. Old VisTrails Title Page
    1. Entire (Old) Vistrails Title Page.     <==   NO! Replaced by Old (Vistrails) Cover Page
    2. Musings Regarding Dark Matter and Dark Energy
    3. Radial Dependence of the Strong Nuclear Force
33. Journal Refereeing
    1. ApJ (2023):  Binary-driven Hypernovae (BdHN)
34. Temp001

Mathematics

1. Roots of Cubic Equation
1. In the context of T2 Coordinates, when $q^2=(a_1/a_3{)}^2=3$.
2. PP Tori — Also includes cube root of a complex number
3. Srivastava's F-Type solution for $n=5$ polytropes.
4. Murphy & Fiedler's Bipolytrope with $(n_c,n_e)=(1,5)$
5. Analytic Eigenfunctions for Bipolytropes with $(n_c,n_e)=(0,0)$ — also involves cube root of a complex number
2. Roots of Quartic Equation
1. Analytic Eigenfunction for Bipolytropes with $(n_c,n_e)=(0,0)$
2. Determine temperature from total pressure
3. Singular Sturm-Liouville (eigenvalue) Problem
1. Oscillations of PP Tori in the slim torus limit
2. Characteristics of unstable eigenvectors in self-gravitating tori
4. Approximate Power-Series Expressions
5. Fourier Series
6. Analytic Expressions for Selected Trigonometric Functions
7. Special Functions & Other Broadly Used Representations
1. Spherical Harmonics and Associated Legendre Functions
2. Multipole Expansions
3. Familiar Expression for the Cylindrical Green's Function Expansion
4. Toroidal Functions
8. Green's Function in terms of Toroidal Functions
1. Compact Cylindrical Green Function
2. Toroidal configurations & related coordinate systems — Includes EUREKA! moment; also uses wikitable overflow (scrolling) box
3. Toroidal Coordinate Integration Limits $\Leftarrow$ 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.
4. Using Toroidal Coordinates to Determine the Gravitational Potential (Initial Presentation)
5. Using Toroidal Coordinates to Determine the Gravitational Potential (Improved Presentation)   $\Leftarrow$    includes series expansions for K(k) and E(k)
6. Relationships between Toroidal Functions $\Leftarrow$ 5 plots of [MF53] data included here
7. Confusion Regarding Whipple Formulae
8. Pulling It All Together $\Leftarrow$ 2 additional plots of [MF53] data included here
9. Scale Factors for Orthogonal Curvilinear Coordinate Systems
10. Euler Angles
11. Step Function and its Derivative
12. Hypergeometric Equation and Its Solutions

Other Pages Worth Visiting

• Old (VisTrails) Cover Page
• Introductory Web Page
• Temporary …
• Key Directories and Files on xxx.phys.lsu.edu
• Experimenting With the a2Hosting Environment
• Riemann (1861)
• Appendix: Equation Templates
• Appendix: Special Functions
• Appendix: Variables Templates
• Appendix: SGF Images
• Appendix: Mathematics/EulerAngles
• Appendix: References
• Appendix: Ramblings
• Appendix: Permissions     <==     Not usable; editing error
• Appendix: Copyright Permissions
• Appendix: Format Recommendations
• Tiled Menu
• Table of Contents
• Total pressure quartic solution
• Spherically Symmetric Configurations (SSC) Index
• Virial Equilibrium of Adiabatic Spheres
• Virial Equilibrium of Adiabatic Spheres (Summary)
• Virial Equilibrium of Isothermal Spheres
• Virial Equilibrium of Uniform-Density Spheres
• Virial of Embedded Polytropes (Outline)
  • First Effort     $\Leftarrow$     Same as "Virial Equilibrium of Adiabatic Spheres"
  • Second Effort     $\Leftarrow$     Same as "Virial Equilibrium of Adiabatic Spheres (Summary)"
  • Third Effort     $\Leftarrow$     FormFactors
  • First Effort, Second Time Around     $\Leftarrow$     We reproduce discussion associated with "First Effort," but correct expressions for ${\mathfrak{𝔣}}_W$ and ${\mathfrak{𝔣}}_A$ as identified in our "Third Effort" and, accordingly reserve various affected expressions that follow.
  • Second Effort, Second Time Around     $\Leftarrow$     We reproduce the discussion associated with our "Second Effort," but revise key sections to incorporate corrected expressions for the structural form factors.
• SSC Synopsis (Using Style Sheet)
• PP Tori: Structure & Stability
  • Menu Tile:  Papaloizou-Pringle Tori (1984) = Apps/PapaloizouPringleTori; this chapter focuses on equilibrium structure.
  • Menu Tile:  (Massless) Papaloizou-Pringle Tori = Apps/PapaloizouPringle84; in mid-July of 2020, Blaes granted permission to include in this chapter some reprinted material from his 1985 paper.
  • Menu Tile:  Analytic Analysis by Blaes (1985) = Apps/ImamuraHadleyCollaboration; compare Blaes85 against Imamura-Hadley Collaboration]; in mid-July of 2020, Blaes granted permission to include in this chapter some reprinted material from his 1985 paper.
  • Oscillations in the Slim Torus limit = Apps/Blaes85SlimLimit
  • Apps/Blaes85SlimLimit
• Earlier version of the PGE/Euler chapter
• Explains Whitworth's chosen scaling
• Details regarding Stahler's (1983) derived mass-radius relation
• Practice use of image map (Image:SchonbergChandra1942.jpg)
• Mass Upper Limits
• Bipolytrope Generalization:
  • Bipolytrope Generalization (original)
  • Bipolytrope Generalization (Version 2)
• Note that …Template: "SSCstructure" points to original "table of contents"
• Stability of Spherically Symmetric Configurations (Eulerian Perspective) <-- Includes scratch work with rarely used html characters, such as ⑧
• Supports Free-Energy PowerPoint Presentation
• Free-Energy Stability Analysis (early version)
• Attempt at Formulating an Analytic Solution to oscillating n = 1 polytrope; see also the Challenge
• ProjectsUnderway/Core_Collapse_Supernovae  $\Leftarrow$   Referenced in our discussion of Goldreich & Weber (1980)
• Power-Law Density Distribution
• Nonlinear PGEs in Cylindrical Coordinates
• Linearized Equations in Cylindrical Coordinates     <==     Referenced, for example, in our discussion of PP84
• Murphy UV Plane regarding $(n_c,n_e)=(1,5)$ bipolytropes

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Computer-Generated Holography

Computer Generated Holgram (Fall 2004) in collaboration with Richard Muffoletto and others from utsouthwestern.edu as cited

0. Lead in …
1. Original Table of Contents
2. Preface
1. Apertures that are Parallel to the Image Screen:
1. One-dimensional Aperture
   1. Initial Ideas
   2. Consolidate Expressions
   3. T. Kreis, P. Aswendt, & R. Höfling (2001), Optical Engineering, vol. 40, no. 6, 926 - 933:   Hologram reconstruction using a digital micromirror device
2. Two-dimensional, Rectangular Aperture
3. Relevance to Holograms
4. Caution and Words of Wisdom
2. Apertures that are Tilted with Respect to the Image Screen:
3. Building Holograms from VRML Files:
4. ZebraImaging and Southwestern Medical Center
5. Embracing COLLADA (2020)
1. Principal Illustration
2. Demonstration Steps
6. Quantum Mechanics
• What is Real?
• Speculation Regarding Quantum Transitions
7. On 4/15/2021, Google brought the following article to my attention:  S. Igarashi, T. Nakamura, K. Matsushima, & M. Yamaguchi (2018), Optics Express, Vol. 26, Issue 8, pp.10773-10786, Efficient tiled calculation of over-10-gigapixel holograms using ray-wavefront conversion. It heavily references [22] the 2007 (Opt. Express, 15(9), 5631-5640, Shifted Fresnel diffraction for computational holography) work that I published in collaboration with R. Muffoletto and John Tyler.

Computer Algorithms

1. Directory …/fortran/FreeEnergy/EFE: README
2. Directory …/numRecipes/EllipticIntegrals/Riemann