magicdragon2 ([info]magicdragon2) wrote,
@ 2004-07-01 15:39:00
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I presented 3 papers at CMU, Pittsburgh, PA, this week
This is the longest I've been away from LiveJournal since I launched.

I was presenting these 3 papers at CASOS 2004, the annual conference of Center for Computational Analysis of Social and Organizational Systems:

The Implications of Peter Lynds 'Time and Classical and Quantum Mechanics: Indeterminacy vs Discontinuity' for Mathematical Modeling
[Proceedings, North American Association for Computation in the Social and
Organizational Sciences, 2004]
Author #1 = Professor Philip V. Fellman, Southern New Hampshire University
Author #2 = Maurice Passman
Author #3 = Professor Jonathan Vos Post, Woodbury University
Author #4 = Professor Christine Carmichael, Woodbury University
Author #5 = Andrew Carmichael Post, California State University Los Angeles

The Nash Equilibrium, Polytopes, and Quantum Computing
[Proceedings of the Fifth International Conference on Complexity Science,
17-21 May 2004] [title may have been changed]
Author #1 = Professor Philip V. Fellman, Southern New Hampshire University
Author #2 = Professor Jonathan Vos Post, Woodbury University

'Time and Classical and Quantum Mechanics' and the Arrow of Time
[Proceedings, North American Association for Computation in the Social and
Organizational Sciences, 2004]
Author #1 = Professor Philip V. Fellman, Southern New Hampshire University
Author #2 = Professor Jonathan Vos Post, Woodbury University

You can find more here about Peter Lynds
On these pages you'll find information and links relating mainly to Peter Lynds' work on the subject of time. His particular areas of interest include time and its relation to classical and quantum mechanics, relativity and cosmology, as well as to brain function and consciousness. He also has an interest in the foundations of assertion and truth.

Recent papers by Peter Lynds:

Time and Classical and Quantum Mechanics: Indeterminacy vs. Discontinuity. Foundations of Physics Letters, 16(4), 2003. Lynds, Peter.
http://cdsweb.cern.ch/search.py?recid=622019

Zeno's Paradoxes: A Timely Solution. Lynds, Peter.
http://philsci-archive.pitt.edu/archive/00001197/

Subjective Perception of Time and a Progressive Present Moment: The Neurobiological Key to Unlocking Consciousness. Lynds, Peter.
http://cogprints.ecs.soton.ac.uk/archive/00003125/

(16 comments) - (Post a new comment)

Kathleen M. Carley chaired the conference
[info]magicdragon2
2004-07-01 03:54 pm UTC (link)
The conference was chaired by:

Professor Kathleen M. Carley

She is an amazing person, so see her home page, linked-to above.

(Reply to this)

NAACSOS Conference 2004: More Information
[info]magicdragon2
2004-07-01 03:59 pm UTC (link)
The magicdragon2 thread on "Asimov Number" was typical of the subjects at this conference.

For more on the conference, see:
NAACSOS Conference 2004
June 27 - 29, 2004, Pittsburgh PA
North American Association for Computational Social and Organizational Science

This conference provides an international forum for interdisciplinary research that combines computation, organizations and society. The goal is to advance the state of science in formal reasoning, analysis, and system building drawing on and encouraging advances in areas at the confluence of social networks, artificial intelligence, complexity, machine learning, sociology, business, political science, economics, and operations research. Such research will lead to the development of new theories that explain and predict the behavior of complex adaptive systems, new computational models and technologies that are responsible to society, business, policy, and law, new methods for integrating data, computational models, analysis and visualization techniques.

Of particular interest is recent work in any of the following areas:

* Computational theorizing about complex socio-cognitive-technical systems, including organizations, commerce, markets, societies, institutions, privacy issues and technology enhanced environments.
New computational, especially agent based, multi-agent based, cognitive, or social network based models for studying, reasoning about, or providing policy guidance with respect to socio-cognitive-technological systems, social-psychological, social, organizational, political or technological systems.

* Advances in grounding, tuning, and validating computational models, particularly multi-agent models, in the social and organizational sciences, including new techniques generalizable across many models, explainable artificial intelligence, and new empirical tests of specific models.

* Papers presenting, validating, or applying network models or computational techniques are strongly encouraged. In addition, papers that take any of these foci are encouraged:

* Applications work using computational models.
* Theoretical research using computational models on fundamental principles of social behavior such as coordination, cooperation, evolution, and destabilization are welcome.
* Computational or network modeling related to corporate, military or intelligence issues, particularly papers on counter-terrorism.
* Computational social, organizational, or economic science.
* New algorithms for or dynamic metrics for network or relational data.
* Complex social or organizational systems models.
* Comparing, contrasting and docking computational models - new approaches and/or actual comparisons.
* Teams, organizations, and swarms of intelligent agents.
* Computational statistics for networks.
* Automated organizational design tools.
* Automated data collection tools for use with computational models.
* Ethical use of, privacy issues related to, relational and computational data.
* Infrastructure for large scale multi-agent simulation.
* Coordination, social cognition, or group performance.
* Social science models using grid based computing or super computers.

(Reply to this)

Dr. Mark Newman, Santa Fe Institute, part 1
[info]magicdragon2
2004-07-01 04:06 pm UTC (link)
The keynote speaker was:

Dr. Mark Newman, Santa Fe Institute

Mark Newman's Recent publications (2000-present)

The statistical mechanics of networks, Juyong Park and M. E. J. Newman, submitted to Phys. Rev. E.

Solution of the 2-star model of a network, Juyong Park and M. E. J. Newman, submitted to Phys. Rev. E.

Uniform generation of random graphs with arbitrary degree sequences, R. Milo, N. Kashtan, S. Itzkovitz, M. E. J. Newman, and U. Alon, submitted to Phys. Rev. E.

Network theory and SARS: Predicting outbreak diversity, Lauren Ancel Meyers, Babak Pourbohloul, M. E. J. Newman, Danuta M. Skowronski, and Robert C. Brunham, submitted to Journal of Theoretical Biology.

A measure of betweenness centrality based on random walks, M. E. J. Newman, submitted to Social Networks.

The physical limits of communication, Michael Lachmann, M. E. J. Newman, and Cristopher Moore, Am. J. Phys., in press.

Identifying the role that individual animals play in their social network, David Lusseau and M. E. J. Newman, Biology Letters, in press.

Fast algorithm for detecting community structure in networks, M. E. J. Newman, Phys. Rev. E, in press.

Who is the best connected scientist? A study of scientific coauthorship networks, M. E. J. Newman, to appear in Complex Networks, E. Ben-Naim, H. Frauenfelder, and Z. Toroczkai (eds.), Springer, Berlin.

Detecting community structure in networks, M. E. J. Newman, Eur. Phys. J. B 38, 321-330 (2004).

Diffusion-based method for producing density equalizing maps, Michael T. Gastner and M. E. J. Newman, Proc. Natl. Acad. Sci. USA 101, 7499-7504 (2004).

Technological networks and the spread of computer viruses, Justin Balthrop, Stephanie Forrest, M. E. J. Newman, and Matthew M. Williamson, Science 304, 527-529 (2004).

Coauthorship networks and patterns of scientific collaboration, M. E. J. Newman, Proc. Natl. Acad. Sci. USA 101, 5200-5205 (2004).

Finding and evaluating community structure in networks, M. E. J. Newman and M. Girvan, Phys. Rev. E 69, 026113 (2004).

Mixing patterns and community structure in networks, M. E. J. Newman and M. Girvan, in Statistical Mechanics of Complex Networks, R. Pastor-Satorras, J. Rubi, and A. Diaz-Guilera (eds.), Springer, Berlin (2003).

Why social networks are different from other types of networks, M. E. J. Newman and Juyong Park, Phys. Rev. E 68, 036122 (2003).
Properties of highly clustered networks, M. E. J. Newman, Phys. Rev. E 68, 026121 (2003).

The origin of degree correlations in the Internet and other networks, Juyong Park and M. E. J. Newman, Phys. Rev. E. 68, 026112 (2003).

The structure and function of complex networks, M. E. J. Newman, SIAM Review 45, 167-256 (2003).

Mixing patterns in networks, M. E. J. Newman, Phys. Rev. E 67, 026126 (2003).

Applying network theory to epidemics: Control measures for outbreaks of Mycoplasma pneumoniae, Lauren Ancel Meyers, M. E. J. Newman, Michael Martin, and Stephanie Schrag, Emerging Infectious Diseases 9, 204-210 (2003).

Modelling Extinction, M. E. J. Newman and R. G. Palmer, Oxford University Press (2003).

Ego-centered networks and the ripple effect, M. E. J. Newman, Social Networks 25, 83-95 (2003).

Random graphs as models of networks, M. E. J. Newman, in Handbook of Graphs and Networks, S. Bornholdt and H. G. Schuster (eds.), Wiley-VCH, Berlin (2003).

Assortative mixing in networks, M. E. J. Newman, Phys. Rev. Lett. 89, 208701 (2002).

Email networks and the spread of computer viruses, M. E. J. Newman, Stephanie Forrest, and Justin Balthrop, Phys. Rev. E 66, 035101 (2002).

Convergence of threshold estimates for two-dimensional percolation, R. M. Ziff and M. E. J. Newman, Phys. Rev. E 66, 016129 (2002).

The structure and function of networks, M. E. J. Newman, Computer Physics Communications 147, 40-45 (2002).

The spread of epidemic disease on networks, M. E. J. Newman, Phys. Rev. E 66, 016128 (2002).

(Reply to this)

Dr. Mark Newman, Santa fe Institute, Part 2
[info]magicdragon2
2004-07-01 04:07 pm UTC (link)
Optimal design, robustness, and risk aversion, M. E. J. Newman, Michelle Girvan, and J. Doyne Farmer, Phys. Rev. Lett. 89, 028301 (2002).

Community structure in social and biological networks, M. Girvan and M. E. J. Newman, Proc. Natl. Acad. Sci. USA 99, 7821-7826 (2002).
Identity and search in social networks, D. J. Watts, P. S. Dodds, and M. E. J. Newman, Science 296, 1302-1305 (2002).

A simple model of epidemics with pathogen mutation, Michelle Girvan, Duncan S. Callaway, M. E. J. Newman, and Steven H. Strogatz, Phys. Rev. E 65, 031915 (2002).

Random graph models of social networks, M. E. J. Newman, D. J. Watts, and S. H. Strogatz, Proc. Natl. Acad. Sci. USA 99, 2566-2572 (2002).

Complex systems theory and evolution, Melanie Mitchell and Mark Newman, in the Encyclopedia of Evolution, M. Pagel (ed.), Oxford University Press, New York (2002).

Percolation and epidemics in a two-dimensional small world, M. E. J. Newman, I. Jensen, and R. M. Ziff, Phys. Rev. E 65, 021904 (2002).

Dynamics of a simple evolutionary process, Dietrich Stauffer and M. E. J. Newman, Int. J. Mod. Phys. C 12, 1375-1382 (2001).

The structure of growing social networks, Emily M. Jin, Michelle Girvan, and M. E. J. Newman, Phys. Rev. E 64, 046132 (2001).

Are randomly grown graphs really random? D. S. Callaway, J. E. Hopcroft, J. M. Kleinberg, M. E. J. Newman, and S. H. Strogatz, Phys. Rev. E 64, 041902 (2001).

Random graphs with arbitrary degree distributions and their applications, M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys. Rev. E 64, 026118 (2001).

Clustering and preferential attachment in growing networks, M. E. J. Newman, Phys. Rev. E 64, 025102 (2001).

Fast Monte Carlo algorithm for site or bond percolation, M. E. J. Newman and R. M. Ziff, Phys. Rev. E 64, 016706 (2001).

Scientific collaboration networks: I. Network construction and fundamental results, M. E. J. Newman, Phys. Rev. E 64, 016131 (2001).

Scientific collaboration networks: II. Shortest paths, weighted networks, and centrality, M. E. J. Newman, Phys. Rev. E 64, 016132 (2001).

A new picture of life's history on Earth, Mark Newman, Proc. Natl. Acad. Sci. USA 98, 5955-5956 (2001).

The structure of scientific collaboration networks, M. E. J. Newman, Proc. Natl. Acad. Sci. USA 98, 404-409 (2001).

Patterns of extinction and biodiversity in the fossil record, R. V. Sole and M. E. J. Newman, in the Encyclopedia of Global Environmental Change, T. Munn (ed.), John Wiley, New York (2001).

Network robustness and fragility: Percolation on random graphs, D. S. Callaway, M. E. J. Newman, S. H. Strogatz and D. J. Watts, Phys. Rev. Lett. 85, 5468-5471 (2000).

Models of the small world, M. E. J. Newman, J. Stat. Phys. 101, 819-841 (2000).

Replica-exchange algorithm and results for the three-dimensional random field Ising model, J. Machta, M. E. J. Newman and L. B. Chayes, Phys. Rev. E 62, 8782-8789 (2000).

Glassiness and constrained dynamics of a short-range non-disordered spin model, J. P. Garrahan and M. E. J. Newman, Phys. Rev. E 62, 7670-7678 (2000).

Exact solution of site and bond percolation on small-world networks, Cristopher Moore and M. E. J. Newman, Phys. Rev. E 62, 7059-7064 (2000).

The power of design, Mark Newman, Nature 405, 412-413 (2000).

Simple models of evolution and extinction, M. E. J. Newman, Computing in Science and Engineering 2, 80-86 (2000).

Epidemics and percolation in small-world networks, Cristopher Moore and M. E. J. Newman, Phys. Rev. E 61, 5678-5682 (2000).

Efficient Monte Carlo algorithm and high-precision results for percolation, M. E. J. Newman and R. M. Ziff, Phys. Rev. Lett. 85, 4104-4107 (2000).

Height representation, critical exponents, and ergodicity in the four-state triangular Potts antiferromagnet, Cristopher Moore and M. E. J. Newman, J. Stat. Phys. 99, 629-660 (2000).

Mean-field solution of the small-world network model, M. E. J. Newman, C. Moore and D. J. Watts, Phys. Rev. Lett. 84, 3201-3204 (2000).

Patterns of biodiversity in the fossil record, M. E. J. Newman and G. J. Eble, in the Encyclopedia of Biodiversity, S. Levin (ed.), Academic Press, London (2000).

(Reply to this)

Professor Philip Vos Fellman
[info]magicdragon2
2004-07-01 04:11 pm UTC (link)
My main co-author of the 3 papers at this conference was:

Professor Philip Vos Fellman

You can find out more about him and his publications at:
More About Professor Philip Vos Fellman

(Reply to this)

Professor Christine M. Carmichael
[info]magicdragon2
2004-07-01 04:25 pm UTC (link)
Another co-author of papers I presented at this conference is:

Professor Christine M. Carmichael

A compressed version of her resume:

CHRISTINE CARMICHAEL, Ph.D.

EDUCATION:

UNIVERSITY OF NEW SOUTH WALES, Sydney, Australia,
Ph.D. Physics (1982)

MORAY HOUSE COLLEGE OF EDUCATION, Edinburgh, Scotland,
Teacher Training Certificate

UNIVERSITY OF EDINBURGH, Edinburgh, Scotland
Diploma in Education

UNIVERSITY OF EDINBURGH, Edinburgh, Scotland
B.Sc. (Honors) Physics

EMPLOYMENT:

1/95-Present MAGIC DRAGON MULTIMEDIA, Pasadena, CA
President/Web Master: commercial web domain averaging 200,000 visitors per month.

5/86-Present COMPUTER FUTURES, INC., Altadena, California
Consultant in thin film deposition/semiconductors for Patent Attorneys

4/97-5/98 EARTHLINK NETWORK, Pasadena, California
Technical Support

6/87-4/95 TALANDIC RESEARCH CORP., Irwindale, California
Technical Expert in Spacecraft Contamination and Thermal Control Coatings for Brilliant Eyes.
Project Manager: thin film high temperature superconductors.
Principal Investigator : won 2 government research proposals.

3/85-12/85 SCHOOL OF PHYSICS, University of New South Wales, Australia
Project Research Scientist and Project Manager of a research group with ten members building a new sputtering system for producing thin, epitaxial films of semiconductors .

5/81-12/84 SCHOOL OF METALLURGY, University of New South Wales Research Associate , used X-ray, optical and transmission electron microscopy to examine deformation and recrystallization effects on the texture and microstructure of f.c.c. and c.p.h. metals. Discovered evidence of superplastic flow in hot rolled zinc.

TEACHING:

9/01-Present WOODBURY UNIVERSITY, Burbank, California
Full-Time Physics Professor
Coordinator of All Sciences
Coordinator of Physics

6/00-Present Mt. SAN ANTONIO COLLEGE, Walnut, California
Physics Lecturer, part-time.

3/87-6/87 WHITTIER COLLEGE, Whittier, California
Physics Lecturer , part-time; managed lab

3/77-5/81 UNIVERSITY OF NEW SOUTH WALES, Australia
Teaching Fellow , Ph.D. student, School of Physics
Taught tutorial classes, instructor-in-charge of laboratory classes of 80 students, voluntary coaching of Vietnamese refugees

* Computer Skills: Microsoft Word, WordPerfect, Excel, Windows, Visual Basic, PhotoShop, QuarkXPress, Macintosh, HTML, Internet, Web Master

* Writing and Publication Success in a wide variety of styles and formats, including scientific papers, progress reports, course outlines, lecture course materials, school reports, scientific reports, conference presentations, and science fiction.

* Founding Member of Sydney Women and Technology, a support group for women working in information technology and related areas, set up computer courses designed specifically for women.

* Verbal Communication: Have presented papers at scientific conferences in Australia, Japan, United Kingdom, and USA. Member of Toastmasters in Sydney. Past Secretary of the Dialectic Society at Edinburgh University.

PAPERS/CONFERENCES:

* "Intrinsic Magnetic Aftereffect", Wagga-Wagga, New South Wales, 1978
* "Magnetic Aftereffect in Dy(Co,Ni)2 Alloys", University of Warwick, England, 1979
* "Investigation of the Intrinsic Magnetic Aftereffect in Dy(Co,Ni)2 Alloys", Wagga-Wagga, Australia, 1979
* "Origin of the Recrystallization Texture in Rolled Low Zinc Brass", International Conference on the Strength of Metals and Alloys, Melbourne, Australia, 1982
* "Mechanism of Intrinsic Magnetic Aftereffect", International Conference on Magnetism, Kyoto, Japan, 1982
* "Gallium Arsenide and Superchips", International Computer Fair, University of Washington, Seattle, Washington, 1986
* "Superchips", Information Systems Forum, Olympia, Washington, 1986
* "Deformation Processes in Hot Worked Copper and Alpha Brass", Acta Metallica , Vol.34, No.11, pp.2247-2257, 1986
* "Materials of the Future", Information Systems Forum, Olympia, Washington, 1989
* "Stacked p-FET Dosimeter for the STRV-2 MWIR Detector: A Joint US-UK Project", IEEE Transactions in Nuclear Science , 1996

(Reply to this)

Today's Pyramidal Numbers discoveries
[info]magicdragon2
2004-07-02 01:32 pm UTC (link)
This has nothing to do with the conference from which I've returned.

In one of my background activities, I have been researching Pyramidal Numbers, and found several publishable things. For instance, when a pyramidal number with one base is equal to a pyramidal number of another base.

Today I uncovered 3 nice previously unknown facts:

For 32-Gonal Pyramidal Numbers 32Pyr(n)=n(n+1)(30n-27)/6

32Pyr(13) = 9Pyr(21) = 11011 (palindrome, same upside down base 10)

32Pyr(18) = Square(171) = Square(Triangular(18)) = 29241

For 33-Gonal Pyramidal Numbers 32Pyr(n)=n(n+1)(31n-28)/6

and the 8-dimensional Simplex Numbers as defined by Kim,

33Pyr(34) = 33Pyr(33Pyr(2)) = 8Simplex(14) = 203490

33Pyr(38) = 6Pyr(75) = 284050



(Reply to this)


[info]slyfoot
2004-07-06 01:50 pm UTC (link)
These are intriguing. I wanted you to know that I put the Ultimate Science Fiction guide in Memories under Ubergeek Links here: http://www.livejournal.com/tools/memories.bml?user=ubergeeks

Feel free to post lectures, lecture notes, links, book reviews, and other such things in [info]ubergeeks. It would give the community some depth to have posts from people with science and technical backgrounds.

~Sly

(Reply to this) (Thread)
Ubergeeks, Ultimate Science Fiction Web Guide
[info]magicdragon2
2004-07-06 04:01 pm UTC (link)
Thank you, Slyfoot! I'm delighted to have more Ubergeeks find their way to the Ultimate Science Fiction Web Guide. I shall, now and then, post lectures, lecture notes, links, book reviews, and other such things in Ubergeeks. Sometimes these will mirror the same (or similar) versions on my web domain, or magicdragon2 blog. But sometimes you may get an exclusive!

By the way, I just returned (710 miles of driving) from Westercon 57, a science fiction convention near Phoenix, Arizona. At this Western regional annual convention, I appeared on 3 panel discussions, all involving the Space Program, and all including others professionally in the Space Program, including some from Canada and Private Enterprise missions. I was moderator of the panel "Space Ship 1 and the X-Prize." The founder of the Heinlein Society, David Silver, moderated the panel on "Future Mars Missions." John Hertz, an attorney active in fandom and space activism, moderated "NASA: Answer or problem?"

I that last panel, which had 2 experts on NASA money (albeit at universities), so devil's advocate, I acted as a prosecutor. I was arguing that NASA should be disbanded, and its assets folded into a new organization with Warren Buffet as CEO. Attorney Hertz counted the audience/jury's votes: by 3-to-1 I'd persuaded them to adopt my radical suggestion!

(Reply to this) (Parent)(Thread)
Re: Ubergeeks, Ultimate Science Fiction Web Guide
[info]slyfoot
2004-07-06 04:09 pm UTC (link)
That is amazing! I'm quite pleased to have met you, because I'm sure you'll be a fountain of information.

So I take it you think NASA should be privatized? If so, what are some of your reasons behind it?

And feel free to point out other links of yours that might fit the community zeitgeist.

~Sly

(Reply to this) (Parent)
When Octahedral Numbers are Polygonal Numbers
[info]magicdragon2
2004-07-08 03:15 pm UTC (link)
Here are some results from the Mathematics research that I've been doing yesterday and today. These are "sporadic" solutions. I have a longer paper, several months in progress, that uses the theory of Elliptic Surfaces, to find four infinite sequences of "parametric" solutions to the question of whn an Octahedral Number is a Polygonal Number.

Where Octh(n) in the nth Octahedral Number,
Octh(n) = n (2n^2 + 1) / 3

I have proven (in the main paper):

The only Octahedral Numbers that are also Perfect Squares (i.e. squares of integers) are:

Octh(1) = Square(1) = 1
Octh(12) = Square(34) = 1156

In the work yesterday and today, I've pointed out that,
where P(n) in the nth Pentagonal Number,
P(n) = n (3n-1) / 2

Octh(1) = P(1) = 1

I’ve conjectured that this is the only Octahedral Pentagonal Number.

I've also shown,where H(n) is the nth Hexagonal Number,
H(n) = n (2n-1)

then:

Octh(1) = H(1) = 1

Octh(2) = H(2) = 6

Octh(7) = H(11) = 231

I conjecture that these are the only Octahedral Hexagonal Numbers.

Further, where Hep(n) is the nth Heptagonal Number,
Hep(n) = n (5n-3) / 2

then:

Octh(1) = Hep(1) = 1

Octh(92) = Hep(456) = 519156

I conjecture that these are the only Octahedral Heptagonal Numbers.

When O(n) is the nth Octagonal Number,
O(n) = n (3n-2)

then:

Octh(1) = O(1) = 1

Octh(136) = O(748) = 1677016

I conjecture that these are the only Octahedral Octagonal Numbers.

and, finally, where Non(n) is the nth Nonagonal Number,
Non(n) = n (7n-5) / 2

then:

Octh(1) = Non(1) = 1

Octh(47) = Non(141) = 69231

Octh(124) = Non(603) = 1271124

I conjecture that these three are the only Octahedral Nonagonal Numbers.

Most of the computations I've done, in Python, involve searches for "near misses" where the difference between Octahedral Numbers through about the 30,000th and Polygonal Numbers are as small as possible. Several nice examples, but that's enough for here, now.

(Reply to this)

When Octahedral Numbers are Near Polygonal Numbers
[info]magicdragon2
2004-07-09 11:31 am UTC (link)
When Octahedral Numbers are Near Polygonal Numbers
by
Jonathan Vos Post
8-9 July 2004

Here are some results from the Mathematics research that I've been doing 7-9 July 2004. See the summary of results in table 1, at the end. These are "sporadic" solutions. I have a longer paper, several months in progress, that uses the theory of Elliptic Surfaces, to find four infinite sequences of "parametric" solutions to the question of when an Octahedral Number is a Polygonal Number.

Where Octh(n) in the nth Octahedral Number,
Octh(n) = n (2n^2 + 1) / 3

I have proven (in the main paper):

The only Octahedral Numbers that are also Perfect Squares (i.e. squares of integers) are:

Octh(1) = Square(1) = 1
Octh(12) = Square(34) = 1156

In the work yesterday and today, I've pointed out that,
where P(n) in the nth Pentagonal Number,
P(n) = n (3n-1) / 2

Octh(1) = P(1) = 1

I've conjectured that this is the only Octahedral Pentagonal Number.

I've also shown, where H(n) is the nth Hexagonal Number,
H(n) = n (2n-1)

then:
Octh(1) = H(1) = 1

Octh(2) = H(2) = 6

Octh(7) = H(11) = 231

I conjecture that these are the only Octahedral Hexagonal Numbers.

Further, where Hep(n) is the nth Heptagonal Number,
Hep(n) = n (5n-3) / 2

then:
Octh(1) = Hep(1) = 1

Octh(92) = Hep(456) = 519156

I conjecture that these are the only Octahedral Heptagonal Numbers.

When O(n) is the nth Octagonal Number,
O(n) = n (3n-2)

then:
Octh(1) = O(1) = 1

Octh(136) = O(748) = 1677016

I conjecture that these are the only Octahedral Octagonal Numbers.

and, finally, where Non(n) is the nth Nonagonal Number,
Non(n) = n (7n-5) / 2

then:
Octh(1) = Non(1) = 1

Octh(47) = Non(141) = 69231

Octh(124) = Non(603) = 1271124

I conjecture that these three are the only Octahedral Nonagonal Numbers.

Only the trivial Octahedral 10-gonal, 11-gonal, 12-gonal, and 13-gonal Numbers.

Where 14gon(n) is the nth 14-gonal Number,
14gon(n) = n (6n-5)

then:

Octh(1) = 14gon(1) = 1
Octh(8) = 14gon(8) = 344

I conjecture that these two are the only Octahedral 14-gonal Numbers.

Only the trivial Octahedral 15-gonal, and 16-gonal Numbers.

Where 17gon(n) is the nth 17-gonal Number,
17gon(n) = n (15n-13) / 2

then:

Octh(1) = 17gon(1) = 1
Octh(7) = 17gon(6) = 231
Octh(90) = 17gon(255) = 486030
Octh(1112) = 17gon(11056) = 916691656

I conjecture that these four are the only Octahedral 17-gonal Numbers.

When 18gon(n) is the nth 18-gonal Number,
18gon(n) = n (8n-7)

then:

Octh(1) = 18gon(1) = 1
Octh(11) = 18gon(11) = 891

I conjecture that these two are the only Octahedral 18-gonal Numbers.

Where 19gon(n) is the nth 19-gonal Number,
19gon(n) = n (17n-15) / 2

then:

Octh(1) = 19gon(1) = 1
Octh(3) = 19gon(2) = 19

Only the trivial Octahedral 20-gonal and 21-gonal Number.

Where 22gon(n) is the nth 22-gonal Number,
22gon(n) = n (10n-9)

then:

Octh(1) = 22gon(1) = 1
Octh(14) = 22gon(14) = 1834

I conjecture that these two are the only Octahedral 22-gonal Numbers.

Only the trivial Octahedral 23-gonal and 24-gonal Numbers.

Most of the computations I've done, in Python, involve searches for "near misses" where the difference between Octahedral Numbers (n<20000) and Polygonal Numbers are small. One example is enough for here, now:

Octh(556) = 114586596
19gon(3672) = 114582924
Octh(556) = 19gon(3672) + 3672

We note this unique example, through n<20, where the index (3672) of an n-gonal number when added to the n-gonal number with that index, is an Octahedral Number.

Table 1
----------------------------------------------------------------------

SELECTED* OCTAHEDRAL NUMBERS Octh(N) EQUAL TO M-GONAL NUMBERS M-gonal(K)

M K M-gon(K) N such that M-gon(K)=Octh(N)
-- --- ------------ --------------------------------------
4 34 1156 12
6 2 6 2
6 7 231 11
7 456 519156 92
8 748 1677016 136
9 141 69231 47
9 603 1271124 124
14 8 344 8
17 6 231 7
17 255 486030 90
17 11056 916691656 1112
18 11 891 11
19 2 19 3
22 14 1834 14
----------------------------------------------------------------------
* 1<N<20000 Trivial solutions Octh(1)=M-gonal(1)=1 omitted. Triangular Number solutions omitted.

(Reply to this)

"Experimental Math" + Octahedral/Polygonal [Continued]
[info]magicdragon2
2004-07-10 10:24 am UTC (link)
Since I posted the previous draft paper, with a table that didn't quite format correctly, I've added the data listed after this note.

You may see a pattern of solutions of the form
Octh(4n+2) = Mgonal(4n+2)

I'd discovered that general solution some months ago, and proved it. Beyond that proof, I've been engaged in what's called "Experimental Mathematics." See mathworld.com (type Experimental Mathematics into its search box) for a lovely definition and mini-essay on how computational can lead to insight. It is the OTHER solutions, neither of that form nor of the form
Octh(1)=Mgonal(1)=1, that are "sporadic" -- and generally hard to find for problems such as this.

We note that the formula for the Octahedral Numbers is an Integer-Valued Cubic polynomial and the formula for M-gonal numbers (for any specific M) is a Quadratic polynomial. Equating a cubic to a quadratic is, in general, an Elliptic Curve problem. Letting M vary, we have an infinite set of related Elliptic Curves, thus establishing a problem involving fiber bundles on an Elliptic Surface. This is "real math." These sporadic solutions correspond to "singular fibers" and do not affect the Group Theory structure of the Elliptic Surface. By Siegel's Theorem, for any specific M, there are only a finite number of such sporadics, but it is hard to know that number, or to find them.

----------

Only the trivial Octahedral 23-gonal, 24-gonal, and 25-gonal Numbers.

When 26gon(n) is the nth 26-gonal Number,
26gon(n) = n (12n-11)

then:

Octh(1) = 26gon(1) = 1
Octh(17) = 26gon(17) = 3281

I conjecture that these two are the only Octahedral 26-gonal Numbers.

Only the trivial Octahedral 27-gonal Number.

When 28gon(n) is the nth 28-gonal Number,
28gon(n) = n (13n-12)

then:

Octh(1) = 28gon(1) = 1
Octh(352) = 28gon(1496) = 29076256

I conjecture that these two are the only Octahedral 28-gonal Numbers.

Only the trivial Octahedral 29-gonal Number.

Where 30gon(n) is the nth 30-gonal Number,
30gon(n) = n (14n-13)

then:

Octh(1) = 30gon(1) = 1
Octh(20) = 30gon(20) = 5340

I conjecture that these two are the only Octahedral 30-gonal Numbers.

---------

(Reply to this)

Optimum cloesest possible "near miss" discovered
[info]magicdragon2
2004-07-10 11:00 am UTC (link)
Finally, after 3 days of experimental math, I've found what I was looking for. I had a hunch there must be one such solution...

Where Octh(n) is the nth Octahedral Number,
Octh(n) = n (2n^2 + 1) / 3

and 32gon(n) is the nth 32-gonal Number,
32gon(n) = n (15n-14)

then:

Octh(1) = 32gon(1) = 1

I conjecture that this is the only Octahedral 32-gonal Number. How about near misses? There are three near misses (near meaning < 5000 in this case) including one optimally good example with n<20,000.

There are three close misses relatively speaking (miss < 5000) which are the smallest through n<20000. However, the rest of these are not as close as the closest near-misses for k-gonal numbers where
3 < k < 32.

One of these is the first "optimally near miss" for k-gonal numbers k<33 and n<20000; that is, a near miss by 1.

Octh(1233) = 1249677969
32gon(9128) = 1249677968
Octh(1233) = 32gon(9128) + 1

The two next closest Octahedral Near 32-gonal Numbers may be expressed as:

Octh(85) = 409445
32gon(165) = 406065
Octh(85) = 32gon(165) + 3380

and:

Octh(120) = 1152040
32gon(277) = 1147057
Octh(120) = 32gon(277) + 4983

There may be other close near-misses for Octahedral 32-gonal Numbers, but roundoff error in my sqrt function makes it problematic to search further.

(Reply to this) (Thread)
Re: Optimum cloesest possible "near miss" discovered
[info]epacris
2004-10-16 09:04 am UTC (link)
It's entirely possible you already know about this, but I just couldn't resist the immediate connexion I felt with your usual blog-mode (mostly from the nielsenhayden area) when I saw this. As your email seems a touch full-up, I'm trying this method to send it 'round.

http://3dpancakes.typepad.com/ernie/2004/10/jcdcg_highlight.html

I can do some arithmetic, only very little maths, but I have worked long enough with science & scientists to realise some of the characteristics, and delight in others' delight in their thought processes and ideas.

(Reply to this) (Parent)(Thread)
Re: Optimum cloesest possible "near miss" discovered
[info]magicdragon2
2004-10-16 10:02 am UTC (link)
This is -- and I say this as a Mathematics professor -- EXTREMELY cool! Thank you for sending it this way.

I've been away from LiveJournal for at least a month, working gruesomely long hours (on top of teaching and research) in writing an Appellate Opening Brief, and filing a late tax return (very complicated when you run several small businesses, have erroneous data from a mortgage company on interest paid, and have a son getting tax credits as tiny rebate for college tuition). But now that I'm back, I'll be catching up.

I've also been very busy submitting and having over 100 accepted and posted on Prime Curios! at primes.utm.edu/curios/

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