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John Cook
Works at Singular Value Consulting
Attended University of Texas at Austin
Lives in Houston, TX
28,506 followers|5,661,828 views
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John Cook

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The New Horizons probe is nearly to Pluto, but the images are still fairly low resolution. I suppose that's because Pluto is so small and New Horizons isn't that close, just close relative to where it started.

The probe launched in January 2006 and will arrive in 41 days, so in that sense it's almost there. But it's approaching Pluto very quickly, 14 km/s, so it's still pretty far away in absolute distance, about 50 million km.
What a difference 20 million miles makes! Images of Pluto from NASA’s New Horizons spacecraft are growing in scale as the spacecraft approaches.
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John Cook

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Inside our sego palm.
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It looks very much like a cycad rather than a true palm to me.  There are many species of cycads in Australia, but all have very toxic fruit, dammit.  They look delicious.
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Forgotten airplanes and composers
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Writer Flannery O'Connor will be honored with a Forever postage stamp, the U.S. Postal Service announced Tuesday. Flannery O'Connor to appear on new USPS "Forever" stamp http://t.co/Epx4OOgzGY pic.twitter.com/RYNLrgQDeQ — Los Angeles Times (@latimes) May 27, 2015 The stamp for 3-ounce packages will debut on June 5 and feature peacock feathers, the Los Angeles Times reports, a nod to the fact that O'Connor raised peacocks on her family's farm in...
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John Cook

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With the help of Google Maps, fans of the science fiction franchise "Star Trek" have boldly gone to China to find a new discovery: the USS Enterprise as a work of architecture.
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Nice of them to include the tennis courts for a sense of scale. 
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John Cook

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"Portability is the hobgoblin of little minds. ... It prevents progress. It requires that things are always done using 'lowest common denominator' techniques."

From "The Linux Command Line" by William Shotts, Jr.
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I do agree. Not that portability isn't desirable, but like everything it has costs and benefits. You have to weigh the trade-offs in some context. 

There are people who see portability as an absolute requirement, not subject to trade-offs, and they're necessarily inconsistent. Nothing can run absolutely everywhere, so they implicitly accept some limitations on portability. 
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Ways to create a command line experience on Windows similar to using bash on Linux or Mac OS.
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very useful
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Osmonds.

I'm more country when country wasn't cool. 
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The distinction between code and data is blurry and causes some interesting problems for regulation.
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“You think, I daresay, that our chief job is inventing new words. But not a bit of it! We’re destroying words— scores of them, hundreds of them, every day. We’re cutting the language right down to the bone. Don’t you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it.” — George Orwell, 1984
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Beautiful partial double rainbow tonight.
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wow,beautiful is right,it looks like a dome....
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Richard Green originally shared to Mathematics:
 
The Gamma Function and Fractal Factorials!

This fractal image by Thomas Oléron Evans was created by using iterations of the Gamma function, which is a continuous version of the factorial function.

If n is a positive integer, the factorial of n, n!, is defined to be the product of all the integers from 1 up to n; for example, 4!=1x2x3x4=24. It is clear from the definition that (n+1)! is the product of n+1 and n!, but it is not immediately clear what the “right” way is to extend the factorial function to non-integer values.

If t is a complex number with a positive real part, the Gamma function Γ(t) is defined by integrating the function x^{t–1}e^{–x} from x=0 to infinity. It is a straightforward exercise using integration by parts and mathematical induction to prove that if n is a positive integer, then Γ(n) is equal to (n–1)!, the factorial of (n–1). Since Γ(1)=1, this gives a justification (there are many others) that the factorial of zero is 1.

Using a technique called analytic continuation, the Gamma function can then be extended to all complex numbers except negative integers and zero. The resulting function, Γ(t), is infinitely differentiable, except at the nonpositive integers, where it has simple poles; the latter are the same kind of singularity that the function f(x)=1/x has at x=0. A particularly nice property of the Gamma function is that it satisfies Γ(t+1)=tΓ(t), which extends the recursive property n!=n(n–1)! satisfied by factorials. It is therefore natural to define the factorial of a complex number z by z!=Γ(z+1).

At first, it may not seem very likely that iterating the complex factorial could produce interesting fractals. If n is an integer that is at least 3, then taking repeated factorials of n will produce a sequence that tends to infinity very quickly. However, if one starts with certain complex numbers, such as 1–i, repeated applications of the complex factorial behave very differently. It turns out that (1–i)! is approximately 0.653–0.343i, and taking factorials five times, we find that (1–i)!!!!! is approximately 0.991–0.003i. This suggests that iterated factorials of 1–i  may produce a sequence that converges to 1.

It turns out that if one takes repeated factorials of almost any complex number, we either obtain a sequence that converges to 1 (as in the case of 1–i) or a sequence that diverges to infinity (as in the case of 3). However, it is not possible to take factorials of negative integers, and there are some rare numbers, like z=2, that are solutions of z!=z and do not exhibit either type of behaviour.

By plotting the points that diverge to infinity in one colour, and the points that converge to 1 in a different colour, fractal patterns emerge. The image shown here uses an ad hoc method of colouring points to indicate the rate of convergence or divergence. The points that converge to 1 are coloured from red (fast convergence) to yellow (slow convergence), and the points that diverge to infinity are coloured from green (slow divergence) to blue (fast divergence)

Relevant links

Thomas Oléron Evans discusses these fractals in detail in a blog post (http://www.mathistopheles.co.uk/2015/05/14/fractal-factorials/) which contains this image and many others. He (and I) would be interested in knowing if these fractals have been studied before.

The applications of the Gamma function in mathematics are extensive. Wikipedia has much more information about the function here: http://en.wikipedia.org/wiki/Gamma_function

This post appears in my Mathematics collection at https://plus.google.com/collection/8zrhX

#mathematics #sciencesunday  

Various recent posts by me
Camellia flower: https://goo.gl/8WNrlu
Horse chestnut tree: https://goo.gl/FPCGI3
A Curious Property of 82000: https://goo.gl/1rVg8y
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Nice! My mind automatically starts to preform calculations and configure algorithms when I look at this. Really nice post.
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Work
Occupation
Applied mathematician
Skills
Mathematical modeling, Bayesian statistics, software design, writing clear prose
Employment
  • Singular Value Consulting
    Owner, 2013 - present
    I've consulted for some of the world's largest software and biotech companies, as well as for law firms and smaller companies.
  • M. D. Anderson Cancer Center
    Research Statistician, 2000 - 2013
    Bayesian statistics, clinical trial design, simulation, software project management
  • NanoSoft
    Senior consultant
    Desktop, web, and mobile software development
  • Western Atlas
    Software developer
    Desktop software development, digital signal processing
  • Vanderbilt University
    Assistant professor
    Research in nonlinear partial differential equations and modelling
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Houston, TX
Previously
Texarkana, TX - Nashville, TN - Austin, TX - Washington, DC
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(832) 422-8646
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Introduction
I solve problems by making connections.

I work as a consultant, bringing together the experience I gained from my previous careers as a math professor, programmer, manager, and statistician. I enjoy combining ideas to solve problems and see the solutions carried out.

You can find out more about my work here.
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  • University of Texas at Austin
    Ph. D. in Mathematics
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