Wednesday, March 25, 2015

Of Jocks and Graph Laplacians

I first read the story via Jason Rosenhouse here:

...just a brief post about a journal article (that you may have been awaiting... or, NOT): "A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians" by mathematician John Urschel... who, just happens to play pro lineman for the Baltimore Ravens football team.
Who knew!?...

The story has received some significant coverage, including this audio piece at NPR:

also, from Huffington Post:

and at Bloomberg news:

Urschel's Erdös number, by the way, is now 4 (rather better, I dare say, than Peyton Manning's).

Monday, March 23, 2015

Flippin' Probabilities

I suspect we all get a kick from mathematical problems that lead us merrily down wrong intuitive paths; statistics is often a source for such misdirecting problems. Mike Lawler pointed to a good one over the weekend. The problem arises in an old Peter Donnelly TEDTalk (below), starting at about the 4-minute mark (but his entire 22-minute talk is definitely worth a listen). The problem is simply this:

When flipping a fair coin repeatedly (heads/tails), which 3-part sequence is more likely to appear first:  HTH or HTT?
i.e., In a running sequence, is HTH likely to show up before HTT, or HTT before HTH, or, over many trials, are the probabilities equal?

MANY people jump to the conclusion that the two possibilities are equally likely, no doubt thinking in terms of the true equal probability of each single flip being an 'H' or a 'T.'
BUT NO, the somewhat surprising answer is that the HTT sequence is more likely to occur first, and the explanation Donnelly gives (again starting at ~4-min. point), has to do with the 'clumping' of overlapping 3-part chunks:

It may be worth noting that one could identify 3-part sequences that DO have equal probability of occurrence, as well as additional sequences, which like the above two, have unequal probabilities.

Donnelly doesn't really expound too fully on the explanation for the outcome, so you may wish to follow along Mike Lawler's videos (linked to above), or read up on "Penney's Game" (named after creator Walter Penney) which relates to all this:

The always-entertaining "Scam School" once did a 15-min. episode on Penney's Game here:

Sunday, March 22, 2015

Computers and Math

"Can computers be used to prove mathematical theorems beyond being a computational aid? Doron Zeilberger of Rutgers University in New Jersey claims that the answer is yes. Moreover, he claims, the computer can reveal mathematical facts outside human reach... Computer programs that operate  on symbolic expressions have existed for many years. Zeilberger used these programs to prove important identities in algebra and used a computer to reveal new identities. He valued the computer's contribution so highly that he added the computer as a coauthor of some of his scientific papers. The computer is named Shalosh B. Ekhad... (At the time of writing these lines, there are twenty-three papers listed in Shalosh B. Ekhad's list of publications, and it has cooperated with thirteen authors.) Beyond the healthy humor, I think there is something basic in this approach. Zeilberger claims, and that claim cannot be ignored, that the day will come when computers will reveal mathematical theorems that will be difficult for humans to understand."

-- Zvi Artstein (from "Mathematics and the Real World")

p.s... for a different style quote to start your day check out Futility Closet's lovely offering this morning from Lewis Carroll:

 p.p.s... also, please note a new interview up over at MathTango this morning.

Thursday, March 19, 2015

Puzzle Redux

I don't mind re-running posts or puzzles that I've posted before that are personal or reader favorites, so below is one from almost 3 years ago that still gets hits each month. (It's the sort of puzzle that Rick Kurshen just might love. ;-))
I'd already seen it multiple times on the Web when I first ran it (though I don't know where it originated???), so I realize many of you have seen it already, but if not, give your brain a whack at it, if you dare!

[The general consensus from those who have coded the problem and run it, is that there is one lone correct answer, which I'll eventually insert in comments if no one else does.]


Given the following 12 statements which of the statements below are true?

1.  This is a numbered list of twelve statements.
2.  Exactly 3 of the last 6 statements are true.
3.  Exactly 2 of the even-numbered statements are true.
4.  If statement 5 is true, then statements 6 and 7 are both true.
5.  The 3 preceding statements are all false.
6.  Exactly 4 of the odd-numbered statements are true.
7.  Either statement 2 or 3 is true, but not both.
8.  If statement 7 is true, then 5 and 6 are both true.
9.  Exactly 3 of the first 6 statements are true.
10.  The next two statements are both true.
11.  Exactly 1 of statements 7, 8 and 9 are true.
12.  Exactly 4 of the preceding statements are true.


Tuesday, March 17, 2015

Encryption and Uncertainty...

Two tidbits I encountered in the last 24 hours that I found interesting, and so am passing along (perhaps, in some odd way, there's even a tiny thread of linkage between the two!??):

1)  RSA encryption researchers make "an astonishing find" (essentially that encryption keys aren't nearly as random as one might expect):


2)  Ed Frenkel advocating the importance of human uncertainty in science (I couldn't agree more) -- even scientists who acknowledge uncertainty, nonetheless often fall into the trap of assuming certainty about specific matters. The epistemological nature of uncertainty is difficult to internalize (I think) because we must operate in our daily lives as if we are certain of a great many things... but this ought not detract from the importance of recognizing uncertainty in a scientific perspective... anyway, read Ed and the Facebook discussion that follows (be sure to click on "See More" to expand Ed's full initial response):

p.s.  -- the Lawrence Krauss New Yorker piece that is referenced (and inspired Ed's response) is here: 

Monday, March 16, 2015

Badass Mathematicians... and Blogs

Here's another fun question from the Quora mathematics site -- "Who is the most badass mathematician ever?":

Fun reading... All the 'nominations' are pretty much who you would expect, but still interesting to see how people make a specific case for their individual choices -- little mini-profiles of the greats of mathematics... EXCEPT, to my utter dismay (unless I missed it), haven't seen anyone make the case for Bernhard Riemann yet! C'mon folks, the man founded perhaps THE most important, far-reaching, unsolved, incredible conjecture in all of mathematics, still being pursued after 150+ years, and nary a mention... show a little respect for your elders!! ;-)

Meanwhile "Grey Matters" just celebrated their 10th Blogiversary (a major achievement, and on Pi Day, no less) and, in celebration, posted links to many of their best posts over the years. Check it out:


Sunday, March 15, 2015

God, Is She a Geometer?...

"Our minds may indeed be just swirls of electrons in nerve cells; but those cells are part of the universe, they evolved within it, and they have been molded by Nature's deep love affair with symmetry. The swirls of electrons in our heads are not random, not arbitrary, and not -- even in a godless universe, if that is what it is -- an accident. They are patterns that have survived millions of years of Darwinian selection for congruence with reality....
"Perhaps we have created a geometer God in our own image, but we have done it by exploiting the basic simplicities that nature supplied when our brains were evolving. Only a mathematical universe can develop brains that do mathematics. Only a geometer God can create a mind that has the capacity to delude itself that a geometer God exists.
"In that sense, God is a mathematician; and She's a lot better at it than we are. Every so often, She lets us peek over her shoulder.

-- Ian Stewart in "Letters To a Young Mathematician"

[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know ( If I use one submitted by a reader, I'll cite the contributor.]   

Saturday, March 14, 2015

Bah Humbug...

For we Pi Day curmudgeons, Vi Hart comes through as usual:

(ohh, but, hey,  Happy Birthday to Albert Einstein... now that's something I can celebrate... with, a piece of pie perhaps!)

Friday, March 13, 2015

Mathematicians Drunk on Moonshine... so-to-speak

 “Math is all about building bridges where on one side you see more clearly than on the other. But this bridge was so unexpectedly powerful that before you see the proof it’s kind of crazy.” -- John Duncan (Case Western Reserve)

Linking number theory and physics... Fantastic Erica Klarreich piece for Quanta Magazine on the Monster group, j-function, Monstrous Moonshine, Kac-Moody algebra, string theory, serendipity in math, and more... these are the sort of connections that are almost spine-tingling, they're so spooky (in a positive way):

Wednesday, March 11, 2015

Answer Me This...

Just a couple of interesting queries on Quora recently:

1)  "What is the largest non-integer that naturally occurs in mathematics?"

You can see the top answer ("Ramanujan's constant"), and others, here:

and, even more broadly,  
2) "What is the most intriguing mathematical concept you have ever encountered?"

Of course many good selections for this one, though I was a bit surprised to not see one of my own favorites, "Cantor's dust" or "Cantor Set," mentioned in the responses when I last leafed through them:

Sunday, March 8, 2015

Doing Mathematics (What It's Like)

I originally posted this last year when MAA first uploaded it to YouTube, but will revisit it now as a nice Sunday reflection via James Tanton, talking about the experience of doing mathematics:

Wednesday, March 4, 2015