# User:Tohline/Appendix/Ramblings

(Difference between revisions)
 Revision as of 17:36, 20 December 2016 (view source)Tohline (Talk | contribs) (→Mathematics: Point to some chapters where there is a need to solve a cubic or quartic equation)← Older edit Current revision as of 11:10, 28 May 2021 (view source)Tohline (Talk | contribs) (→Ramblings) (97 intermediate revisions not shown) Line 1: Line 1: + __FORCETOC__ + {{LSU_HBook_header}} {{LSU_HBook_header}} Line 6: Line 8:
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1. [[User:Tohline/Appendix/Ramblings/ToroidalCoordinates|Toroidal configurations & related coordinate systems]]
2. +
3. Orthogonal Curvilinear Coordinate Systems +
+
1. [[User:Tohline/Appendix/Ramblings/DirectionCosines|Direction Cosines]]
2. +
3. [[User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates|(Confocal) Elliptic Cylinder Coordinates]] plus Concentric Analog ([[User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates|T5 Coordinates]])
4. +
5. [[User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates|Concentric Ellipsoidal (T6) Coordinates]]
6. +
7. [[User:Tohline/Appendix/Ramblings/ConcentricEllipsodalT8Coordinates|Concentric Ellipsoidal (T8) Coordinates]]
8. +
9. [[User:Tohline/Appendix/Ramblings/ConcentricEllipsodalT12Coordinates|Concentric Ellipsoidal (T12) Coordinates]]
10. +
11. [[User:Tohline/Appendix/Ramblings/ConcentricEllipsodalDaringAttack|Daring Attack]]
12. +
+
4. [[User:Tohline/Appendix/Ramblings/T1Coordinates|Relationship between HNM82 models and T1 coordinates]]
5. [[User:Tohline/Appendix/Ramblings/T1Coordinates|Relationship between HNM82 models and T1 coordinates]]
6. -
7. [[User:Tohline/Appendix/Ramblings/DirectionCosines|Orthogonal Curvilinear Coordinates]]
8. [[User:Tohline/Appendix/Ramblings/SphericalWaveEquation|Playing with the Spherical Wave Equation]]
9. [[User:Tohline/Appendix/Ramblings/SphericalWaveEquation|Playing with the Spherical Wave Equation]]
10. Analyzing Azimuthal Distortions
11. Analyzing Azimuthal Distortions Line 39: Line 49:
• [[User:Tohline/Appendix/Ramblings/Photosphere|Photosphere of Stably Accreting DWD]]
• [[User:Tohline/Appendix/Ramblings/Photosphere|Photosphere of Stably Accreting DWD]]
• -
• [[User:Tohline/Appendix/Ramblings/Hybrid_Scheme_old|Initial Effort to Explain Jay Call's ''Hybrid'' Scheme in the Context of Zach Byerly's Dissertation]]
• +
• [[User:Tohline/Appendix/PolytropicBinaries|Binary Polytropes]]
• +
• A* Scheme +
+
1. [[User:Tohline/Appendix/Ramblings/Hybrid_Scheme_old|Initial Effort to Explain Jay Call's ''Hybrid'' Scheme in the Context of Zach Byerly's Dissertation]]
2. +
3. [[User:Tohline/Appendix/Ramblings/Hybrid_Scheme_Implications|Implications of Hybrid Scheme]] +
+
• [[User:Tohline/Appendix/Ramblings/Nonlinar_Oscillation|Exploring the Properties of Radial Oscillations in Pressure-Truncated n = 5 Polytropes]]
• [[User:Tohline/Appendix/Ramblings/Nonlinar_Oscillation|Exploring the Properties of Radial Oscillations in Pressure-Truncated n = 5 Polytropes]]
• [[User:Tohline/Appendix/Ramblings/Turning_Points#Turning_Points|Instabilities Associated with Equilibrium Sequence Turning Points]]
• [[User:Tohline/Appendix/Ramblings/Turning_Points#Turning_Points|Instabilities Associated with Equilibrium Sequence Turning Points]]
• -
• +
• [[User:Tohline/Appendix/Ramblings/LedouxVariationalPrinciple|Derivations Related to Ledoux's Variational Principle]] -
• [[User:Tohline/Appendix/Ramblings/OriginOfPlanetaryNebulae|Investigation Resulting from a July, 2013 Discussion with Kundan Kadam]] +
• More on Zero-Zero Bipolytropes
• - +
+
2. +
4. +
6. +
8. +
9. [[User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors|Numerically Determined Eigenvectors]]
10. +
+
• Analyzing Five-One Bipolytropes
• +
+
1. [[User:Tohline/Appendix/Ramblings/BiPolytropeStability|Assessing the Stability of Spherical, BiPolytropic Configurations]]
2. +
3. [[User:Tohline/Appendix/Ramblings/BiPolytrope51AnalyticStability|Searching for Analytic EigenVector for (5,1) Bipolytropes]]
4. +
5. [[User:Tohline/Appendix/Ramblings/PatrickMotl|Discussing Patrick Motl's 2019 Simulations]]
6. +
7. [[User:Tohline/Appendix/Ramblings/BiPolytrope51ContinueSearch|Continue Search]]
8. +
+
• [[User:Tohline/Appendix/Ramblings/OriginOfPlanetaryNebulae|On the Origin of Planetary Nebulae]] (Investigation Resulting from a July, 2013 Discussion with Kundan Kadam)
• +
• [[User:Tohline/Appendix/Ramblings/InsideOut|Looking outward, from Inside a Black Hole]]
• +
• [[User:Tohline/Appendix/Ramblings/StrongNuclearForce|Radial Dependence of the Strong Nuclear Force]]
• +
• [[User:Tohline/Appendix/Ramblings/Dyson1893Part1|Dyson (1893a) Part I:  Some Details]]
• +
• +
• [[User:Tohline/Appendix/Ramblings/Saturn#Saturn|Saturn]]
• +
• [[User:Tohline/Appendix/Ramblings/MyDoctoralStudents|Doctoral students Tohline has advised]] over the years
• +
• [[User:Tohline/Appendix/Ramblings/ForDurisen|For Richard H. Durisen]]
• +
• [[User:Tohline/Appendix/Ramblings/ForOuShangli|For Shangli Ou]]
• +
• [[User:Tohline/Appendix/Ramblings/ForPaulFisher|For Paul Fisher]]
• +
• [[User:Tohline/Appendix/Ramblings/ForPJ_April2021|For PJ in April 2021]]
• +
+
1. [[User:Tohline/Appendix/Ramblings/VirtualReality#Virtual_Reality_and_3D_Printing|Virtual Reality and 3D Printing]]
2. +
3. [[User:Tohline/Appendix/Ramblings/OculusRift_S|Success Importing Animated Scene into Oculus Rift S]]
4. +
5. [[User:Tohline/Appendix/Ramblings/RiemannMeetsOculus|Carefully (Re)Build Riemann Type S Ellipsoids Inside Oculus Rift Environment]]
6. +
7. Other Example S-type Riemann Ellipsoids: +
+
1. [[User:Tohline/Appendix/Ramblings/RiemannB90C333|(b/a, c/a) = (0.90, 0.333)]]
2. +
3. [[User:Tohline/Appendix/Ramblings/RiemannB74C692|(b/a, c/a) = (0.74, 0.692)]]
4. +
5. [[User:Tohline/Appendix/Ramblings/RiemannB28C256|(b/a, c/a) = (0.28, 0.256)]]
6. +
+
8. +
+
• +
• Challenges Constructing Ellipsoidal-Like Configurations +
+
1. [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids|Riemann Type 1 Ellipsoids]]
2. +
3. [[User:Tohline/ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations|Construction Challenges (Pt. 1)]]
4. +
5. [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2|Construction Challenges (Pt. 2)]]
6. +
7. [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3|Construction Challenges (Pt. 3)]]
8. +
9. [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt4|Construction Challenges (Pt. 4)]]
10. +
11. [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt5|Construction Challenges (Pt. 5)]]
12. +
13. Related discussions of models viewed from a rotating reference frame: +
+
1. [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame|PGE]]
2. +
3. NOTE to Eric Hirschmann & David Neilsen...  I have moved the earlier contents of this page to a new Wiki location called [[User:Tohline/Apps/RiemannEllipsoids_Compressible|Compressible Riemann Ellipsoids]].
4. +
+
14. + +
+
• +
• Bordeaux University +
+
1. [[User:Tohline/Appendix/Ramblings/Bordeaux|External Gravitational Potential of Toroids]]
2. +
3. [[User:Tohline/Appendix/Ramblings/BordeauxSequences|Spheroid-Ring Sequences]]
4. +
5. [[User:Tohline/Appendix/Ramblings/BordeauxPostDefense|Discussions Following Dissertation Defense]]
6. +
+
• +
• + ==Mathematics== ==Mathematics== Line 51: Line 131:
• Roots of Cubic Equation
• Roots of Cubic Equation
1. +
2. In the context of [[User:Tohline/Appendix/Ramblings/T1Coordinates#Second_Special_Case_.28cubic.29|T2 Coordinates]], when [itex]~q^2 = (a_1/a_3)^2=3[/itex].
3. [[User:Tohline/Appendix/Ramblings/PPTori#Cubic_Equation_Solution|PP Tori]] — Also includes [[User:Tohline/Appendix/Ramblings/PPTori#CubeRootImaginary|cube root of a complex number]]
4. [[User:Tohline/Appendix/Ramblings/PPTori#Cubic_Equation_Solution|PP Tori]] — Also includes [[User:Tohline/Appendix/Ramblings/PPTori#CubeRootImaginary|cube root of a complex number]]
5. [[User:Tohline/SSC/Structure/Polytropes#CubicRoot|Srivastava's F-Type solution]] for [itex]~n=5[/itex] polytropes.
6. [[User:Tohline/SSC/Structure/Polytropes#CubicRoot|Srivastava's F-Type solution]] for [itex]~n=5[/itex] polytropes.
7. [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Analytic_Solution_of_Key_Interface_Relation|Murphy & Fiedler's Bipolytrope]] with [itex]~(n_c, n_e) = (1,5)[/itex]
8. [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Analytic_Solution_of_Key_Interface_Relation|Murphy & Fiedler's Bipolytrope]] with [itex]~(n_c, n_e) = (1,5)[/itex]
9. +
10. [[User:Tohline/Appendix/Ramblings/Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes#Solve_Cubic_Equation|Analytic Eigenfunctions for Bipolytropes]] with [itex]~(n_c, n_e) = (0, 0)[/itex] — also involves cube root of a complex number
• Roots of Quartic Equation
• Roots of Quartic Equation
• Line 60: Line 142:
• [[User:Tohline/SR/Ptot_QuarticSolution#Quartic_Equation_Solution|Determine temperature from total pressure]]
• [[User:Tohline/SR/Ptot_QuarticSolution#Quartic_Equation_Solution|Determine temperature from total pressure]] +
• Singular Sturm-Liouville (eigenvalue) Problem
• +
+
1. [[User:Tohline/Apps/Blaes85SlimLimit#Singular_Sturm-Liouville_Problem|Oscillations of PP Tori]] in the slim torus limit
2. +
3. [[User:Tohline/Apps/ImamuraHadleyCollaboration#Singular_Sturm-Liouville_Problem|Characteristics of unstable eigenvectors in self-gravitating tori]]
4. +
+
• [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions|Approximate Power-Series Expressions]]
• +
• [[User:Tohline/Appendix/Ramblings/FourierSeries|Fourier Series]]
• +
• +
+
1. [[User:Tohline/AxisymmetricConfigurations/PoissonEq#Spherical_Harmonics_and_Associated_Legendre_Functions|Spherical Harmonics and Associated Legendre Functions]]
2. +
3. [[User:Tohline/AxisymmetricConfigurations/PoissonEq#Multipole_Expansions|Multipole Expansions]]
4. +
5. [[User:Tohline/AxisymmetricConfigurations/PoissonEq#Familiar_Expression_for_the_Cylindrical_Green.27s_Function_Expansion|Familiar Expression for the Cylindrical Green's Function Expansion]]
6. +
7. [[User:Tohline/AxisymmetricConfigurations/PoissonEq#Toroidal_Functions|Toroidal Functions]]
8. +
+
• Green's Function in terms of Toroidal Functions
• +
+
1. [[User:Tohline/Appendix/Ramblings/CCGF|Compact Cylindrical Green Function]]
2. +
3. [[User:Tohline/Appendix/Ramblings/ToroidalCoordinates#Relating_CCGF_Expansion_to_Toroidal_Coordinates|Toroidal configurations & related coordinate systems]] — Includes EUREKA! moment; also uses [[User:Tohline/Appendix/Ramblings/ToroidalCoordinates#Examples|wikitable overflow]] (scrolling) box
4. +
5. [[User:Tohline/2DStructure/ToroidalCoordinateIntegrationLimits#Evaluation_of_Elliptic_Integrals|Toroidal Coordinate Integration Limits]] [itex]~\Leftarrow ~~[/itex]  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 [[User:Tohline/Apps/DysonPotential#Our_Attempt_to_Replicate|Our Attempt to Replicate]] Dyson's results.
6. +
7. [[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|Using Toroidal Coordinates to Determine the Gravitational Potential]] (Initial Presentation)
8. +
9. [[User:Tohline/2DStructure/ToroidalGreenFunction#Appendix_B:_Elliptic_Integrals|Using Toroidal Coordinates to Determine the Gravitational Potential]] (Improved Presentation)   [itex]~\Leftarrow[/itex]    includes [[User:Tohline/2DStructure/ToroidalGreenFunction#Series_Expansions|series expansions]] for K(k) and E(k)
10. +
11. [[User:Tohline/Appendix/Mathematics/ToroidalFunctions|Relationships between Toroidal Functions]] [itex]~\Leftarrow ~~[/itex] 5 plots of [[[User:Tohline/Appendix/References#MF53|MF53]]] data included here
12. +
13. [[User:Tohline/Appendix/Mathematics/ToroidalConfusion|Confusion Regarding Whipple Formulae]]
14. +
15. [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01|Pulling It All Together]] [itex]~\Leftarrow ~~[/itex] 2 additional plots of [[[User:Tohline/Appendix/References#MF53|MF53]]] data included here
16. +
+
• [[User:Tohline/Appendix/Mathematics/ScaleFactors|Scale Factors for Orthogonal Curvilinear Coordinate Systems]]
• + ==Computer-Generated Holography== + +
+ Computer Generated Holgram (Fall 2004)
in collaboration with [https://digitalcommons.lsu.edu/gradschool_dissertations/2127/ Richard Muffoletto]
and others from [https://www.utsouthwestern.edu utsouthwestern.edu] as cited +
[[File:Hologram2004.JPG|400px|Muffoletto's CGH]]
+ + + + +
+
2. +
+
2. +
3. [[User:Tohline/Appendix/CGH/Preface|Preface]]
4. +
+
3. Apertures that are Parallel to the Image Screen:
4. +
+
1. One-dimensional Aperture +
+
1. + [[User:Tohline/Appendix/CGH/ParallelApertures|Initial Ideas]] +
2. +
3. + [[User:Tohline/Appendix/CGH/ParallelAperturesConsolidate|Consolidate Expressions]] +
4. +
5. + [[User:Tohline/Appendix/CGH/KAH2001|T. Kreis, P. Aswendt, & R. Höfling (2001)]], Optical Engineering, vol. 40, no. 6, 926 - 933:   ''Hologram reconstruction using a digital micromirror device'' +
6. +
+
2. +
3. [[User:Tohline/Appendix/CGH/ParallelApertures2D|Two-dimensional, Rectangular Aperture]]
4. +
5. [[User:Tohline/Appendix/CGH/ParallelAperturesHolograms|Relevance to Holograms]]
6. +
7. [[User:Tohline/Appendix/CGH/ParallelAperturesWisdom|Caution and Words of Wisdom]]
8. +
+
5. Apertures that are Tilted with Respect to the Image Screen:
6. +
7. Building Holograms from VRML Files:
8. +
9. [[User:Tohline/Appendix/CGH/ZebraImaging|ZebraImaging and Southwestern Medical Center]]
10. +
12. +
+
2. +
4. +
+
13. Quantum Mechanics
14. +
+
• [[User:Tohline/Appendix/CGH/WhatIsReal|What is Real?]]
• +
• [[User:Tohline/Appendix/CGH/QuantumTransitions|Speculation Regarding Quantum Transitions]]
• +
+
15. On 4/15/2021, Google brought the following article to my attention:   [https://doi.org/10.1364/OE.26.010773 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== + +
+
2. +
3. Directory [[User:Tohline/Appendix/ComputerAlgorithms/Riemann|…/numRecipes/EllipticIntegrals/Riemann]]
4. +

{{LSU_HBook_footer}} {{LSU_HBook_footer}}

## 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. Orthogonal Curvilinear Coordinate Systems
2. Relationship between HNM82 models and T1 coordinates
3. Playing with the Spherical Wave Equation
4. Analyzing Azimuthal Distortions
5. 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 page
4. Characteristic Vector for T3 Coordinates
5. T4 Coordinates (Abandoned by Joel 7/6/2010 because non-orthogonal)
7. Photosphere of Stably Accreting DWD
8. Binary Polytropes
9. A* Scheme
10. Exploring the Properties of Radial Oscillations in Pressure-Truncated n = 5 Polytropes
11. Instabilities Associated with Equilibrium Sequence Turning Points
12. Derivations Related to Ledoux's Variational Principle
13. More on Zero-Zero Bipolytropes
14. Analyzing Five-One Bipolytropes
15. On the Origin of Planetary Nebulae (Investigation Resulting from a July, 2013 Discussion with Kundan Kadam)
16. Looking outward, from Inside a Black Hole
17. Radial Dependence of the Strong Nuclear Force
18. Dyson (1893a) Part I:  Some Details
20. Saturn
21. Doctoral students Tohline has advised over the years
22. For Richard H. Durisen
23. For Shangli Ou
24. For Paul Fisher
25. For PJ in April 2021
26. 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:
27. 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
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.
28. Bordeaux University

## 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. Special Functions & Other Broadly Used Representations
7. 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
8. Scale Factors for Orthogonal Curvilinear Coordinate Systems

## Computer-Generated Holography

 Computer Generated Holgram (Fall 2004)in collaboration with Richard Muffolettoand others from utsouthwestern.edu as cited
2. 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
3. Apertures that are Tilted with Respect to the Image Screen:
4. Building Holograms from VRML Files:
5. ZebraImaging and Southwestern Medical Center
7. Quantum Mechanics
8. 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.