Monday, January 13, 2014

In the appendix of the paper I reference below is a summary derivation of the relationship  between entropy and fractals (power laws). The appendix is also posted in another blog, Plate Frames. The introduction to the post reads...
In a paper published a couple of years ago (Pilger, 2012), I describe the application of a simple principle, transformed into a distinctive abstract object, to an optimization problem (within the plate tectonics paradigm): simultaneous reconstruction of lithospheric plates for a range of ages from marine geophysical data . It is rare that the relation of the principle, maximum entropy, with a particular transformation, power-series fractals, is recognized, since Pastor-Satorras and Wagensberg derived it. I'm unaware of any other application of fractal forms to optimization problems analogous to the paper. The following derivation is taken from the 2012 paper, with slight modification, in hopes that it might prove useful in other fields, not merely the earth sciences, but beyond. I'm investigating  applications in a variety of other areas, from plate tectonics, to petroleum geology, and, oddly enough, the arts.
Pilger, R. H., Jr. (2012) Fractal Plate Reconstructions with Spreading Asymmetry, Marine Geophysical  Research, Volume 33, 149-168. dx.doi.org/10.1007/s11001-012-9152-6. (rexpilger (at) gmail (dot) com.)

Wednesday, December 22, 2010

Fractals and Plate Tectonics

Can fractal criteria be used in deriving plate reconstructions of asymmetrically spreading ridges? See: link.


Wednesday, November 17, 2010

Peer Review

An article in Physics World describes an "experiment" in peer review and its effect on the quality of published scientific research.
Just a small number of bad referees can significantly undermine the ability of the peer-review system to select the best scientific papers. That is according to a pair of complex systems researchers in Austria who have modelled an academic publishing system and showed that human foibles can have a dramatic effect on the quality of published science.

Monday, November 08, 2010

Jane and Will

Fractal calculations of Jane Austen's six novels imply an unsurprisingly common vocabulary pool. But, what about comparisons with other English literature? Hamlet:

And adding to the Jane Austen plot:

(Click to enlarge; click again to zoom; backspace to return to this post.)

Hamlet and company occupied a smaller "area" in their dramatic fractal space, but note that the slope of the main part of the fractal plot is essentially the same as Jane's novels.

What does it all mean?



Jane Austen

Word frequency usage often shows a logarithmic pattern (e.g., Zipf's distribution). What about fractals? (Click to enlarge graphics; backspace to return to this blog post.)














I suppose one might assume that Jane Austen would draw on the same vocabulary in each novel. How similar are these relationships among the six?

 
The similarity in slopes of the six curves suggests that common vocabulary.

Citation Fractals

Following a suggestion by Murray (2002), I've looked at the indexes of a number of recent scientific monographs and popular scientific accounts and calculated their fractal measures.


Murray combined a large number of text references and normalized them. Applying fractal binning to his results produces:

E. T. Jaynes (2002) Probability Theory:




W. Isaacson (2008) Einstein: His Life and Universe:




L. Gilder (2009) The Age of Entanglement:



S. Pinker (2003) The Blank Slate:




W. Grandy (2008) Entropy and the Time Evolution of Macroscopic Systems:

Sunday, November 07, 2010

Looking for fractals in all the wrong or right places - I

Here is the first of several attempts at documenting fractal structures from science to art.


First, however, a little bit about technique: The magnitude of a particular data set, whatever its source, is ranked from greatest to least. Then, fractal binning is applied over a range of dimensions. The maximum dimension is D(max) = 2*n, in which the value of n produces the smallest value larger than the maximum magnitude of the data set. Each dimension is then determined by D(I) = D(maximum)/M, in which M = 2*N/M, M =1, M*. Each bin I for dimension D(I) is occupied if there is one or more values such that bin I = integer (magnitude/D(i)). Then for each dimension, the number of occupied bins is totaled.
First example: Number of performances of Broadway musicals, ranked from most to least, for the top 100 (not counting some which are still running).
The longest running musicals (Andrew Lloyd Webber's Cats is the current leader among closed shows; however his Phantom of the Opera is still running).

Note that a true fractal would display a linear trend on a log-log plot. However a simple linear trend is apparent for only the top two bins, not one extended over a longer range of scales.

Entropy

Where were you when you first heard about entropy?


Was it in high school chemistry or physics (the most appropriate venue)? Perhaps, metaphorically, it was in college English or political science, or even an op-ed in last year’s Times (NY, LA, or London). What, you say you never heard of entropy before? It’s time to climb onto the bandwagon – there’s a new paradigm in town, recasting a nearly two-hundred year old idea. Some of the “newer” ideas about entropy are more than half a century old, introduced shortly before “paradigms” became paradigmatic, while some of the newest arrived just before the turn of the millennium.

Tectonic Similarity

Spin a globe, tilt it, and center the South Atlantic Ocean, with the coast of South America to the west (and left), that of Africa on the east (and right): see how the South American coastline to the west seem somehow similar to that of Africa on the east. Tilt north and rotate slightly west, to the center of the North Atlantic Ocean: the North American coastline to the west is, with slight imagination, similarly similar to that of the North African Atlantic coast. One more time: east and south, to the center of the Indian Ocean: The facing coasts of East Africa, India, Antarctica, and Australia; rotated back and forth a bit and imagine the coast lines as edges of spherical puzzle pieces, with Madagascar a gap-filling fragment. Might all of these continents, and Eurasia too, have once formed a collective megacontinent? This question has been around in one form or another for not merely decades but two or even three centuries. But, it wasn’t until the mid-1960s that the solid earth scientific community reached near consensus: The answer: a resounding “Yes”.
However, to reach the point at which geologists (stratigraphers, petrologists, volcanologists…), geophysicists (seismologists, paleomagnetists…), and paleontologists (specializing in fossils of plants and both vertebrate and invertebrate animals) could all agree, multidisciplinary results from each field had to be shown as mutually consistent and integrated. 

Misquotation: How difficult is it?

• “Play it again, Sam.” -- Ingrid Bergman
• “…blood, sweat and tears.” – Winston Churchill
• “History is bunk.” – Henry Ford
• “My name is Ishmael.” – Herman Melville
• “Math is hard.” -- Barbie
These famous quotes have something profoundly in common. What is it?
There could be a footnote, upside down, at the bottom of this page, or an end note, somewhere deeper into the blog, with the answer. But, you, dear reader, know the answer, don’t you? Each quote is similar to each of the others. Isn’t it? Aren’t they all similar?

Woody Allen even made a movie with that title, a classic 70’s rock band had that name, who believes history anyway(?), and Ahab was an Arab, wasn’t he? And, mathematics can be difficult,

Mathematical notation can be obscure, Sam really didn’t want to play it, the Prime Minister was playing with a short deck, Ford manufactured the “T” before the “A”, and Melville’s story is a whale of a tale.

Psst…. Don’t tell anyone; don’t include “warning: spoilers” in your review. But you do know, don’t you, that none of the quotes above are original with their auteur? That’s because none of the authors ever said or wrote any of them.

Here are the “real” quotes:
• “Play it Sam. Play ‘As time goes by’” – (Ilsa) Ingrid Bergman in Casablanca
• “…blood, toil, sweat and tears.” – Winston Churchill (and before him Garibaldi and T. Roosevelt)
• “History as it is taught in the schools is bunk.” – Henry Ford (In fairness, there appears to be some disagreement about what the innovative industrialist really said.)
• “Call me Ishmael.” – Herman Melville
• “Math class is tough.” -- Barbie
It is ironic, is it not, that the most quoted line from the most quotable movie of all time is commonly misquoted (even before Woody Allen’s movie). A search for "Play it again Sam" produces 230,000 hits, while a search for “Play it Sam. Play ‘As time goes by’” produces only 69,000 more. Further, even Churchill’s well-known line may have been appropriated from (or at best independently enunciated after) Theodore Roosevelt. Henry Ford came after Karl Marx, so the familiar assertion by the great capitalist could be viewed as a denunciation of the patron saint of communism and his “theory” of history. There is a slight, even significant discrepancy between “My name is…” and “Call me…” is there not?

In any case, whether for miniature mannequins or fully grown adults, math can be really hard. Even Einstein, physics genius, needed help with his math at times.

Oops, I forgot one:
  • "Judy, Judy, Judy..." Cary Grant
Can anyone find anything close to "Judy..." by C. Grant anywhere in his oeuvre? Not even Tony Curtis in either Some Like It Hot or Operation Petticoat came close to it.