Ken’s Math Journal

Ken’s Math Journal

(Interesting articles I’ve come across in mathematics and science)

2006

Springfield Theory, (or on my laptop), on the Simpsons and math.

One of my favorite programs is reviewed in Science magazine!

Cayley and Sylvester – Daniel Silver reviews two biographies, Arthur Cayley: Mathematician Laureate of the Victorian Age and James Joseph Sylvester: Jewish Mathematician in a Victorian World. Cayley and Sylvester were influential in the development of modern algebra, including linear algebra. (20061115)

Life and Left-handedness – a review by James Franklin of a Martin Gardner book (The New Ambidextrous Universe) includes a discussion on “What is Mathematics?”

I should probably try to get a copy of Hermann Weyl’s Symmetry. (20061204)

On the million dollar math problems from the Dec 06 article of Discover Magazine.

Foolproof by Brian Hayes, from the American Scientist, on the difficulty of mathematical proof.

Einstein’s Wife, on the fad to make Einstein’s wife into a scientist from whom Einstein stole good ideas. Also on my laptop.

The Sudoku Epidemic, from Focus Magazine.

A good introduction to Sudoku.

Underachieving colleges, by Derek Bok. Also see the article in the Boston Globe and this book review.

Francis Collins is on the Colbert Report!

Collins promotes both evolution and Christianity in a brief appearance on the delightful Colbert Report. (20061223)

From the Feb 2005 Monthly

p. 118, a short proof of Cauchy’s interlacing theorem. Photocopy this short article…

p. 141, On the perimeter and area of the unit disk. Deals with Minkowski planes – and more generally Hilbert’s suggestion to find all geometries which “stand next to Euclidean geometry”, that is, are off from Euclidean geometry by a single axiom.

From the March 2005 Monthly

Ivor Grattan-Guinness, The Ecole Polytechnique, 1794-1850, Differences over Educational Purpose and Teaching Practice.

A good paper for math history.

Quadratic Reciprocity in a Finite Group, by Duke, Hopkins.

Some results that are true even in nonabelian groups.

Reviews of linear algebra texts. Includes discussion of the Linear Algebra Curriculum Study Group.

p. 288, the linear combination idea of matrix multiplication (pre-multiplication is a linear combination of rows…)

An angle trisection on p 200, used by the Greeks.

Suppose an acute angle with vertex at the origin has one ray on the x-axis and the other on the line s. Draw a circle K of radius one around the origin. Let P be the point of intersection of the line s and the circle K in the first quadrant. Then there is a point D on the x-axis where the line DP intersects K at a point C in the second quadrant and CD has length 1. Then the angle ODC is one-third of the angle made by s at the origin. This is cute, but obviously the angle ODC can’t be constructed with compass and straightedge, even though it can be easily approximated.

Combinatorial Proofs of Fermat’s, Lucas’s and Wilson’s Theorem, by Anderson, Benjamin and others.

Based on a very useful lemma – if a permutation of prime order p acts on a set then the number of fixed points is congruent mod p, to the size of the set.

From the June-July 2005 Monthly

Counting using determinants.

Theorem 1. Let G be a directed acyclic graph with n designated origin and destination nodes and let A be the n x n matrix whose (i,j) entry is the number of pass from the ith origin to the jth designation. The following statements hold.

(a) The number of n-paths is equal to the permanent of A

(b) If G is nonpermutable, the number of nonintersecting n-paths is equal to the determinant of A.

(c) In general, det(A) = Even(G)-Odd(G) where Even(G) is the number of nonintersecting n-paths corresponding to even permutations, etc. (This theorem shows up in Bressoud’s recent book on signed matrices.)

Are there applications to design theory?

The isoperimetric problem – on the typical calculus “maximize the area” problem, from antiquity.

From the Aug-Sept 2005 Monthly

Dresden, Finding Factors of Factor Rings over the Gaussian Integers, by Greg Dresden and Wayne Dymacek.

What are the factor rings of the Gaussian integers? A little more interesting than I would’ve first thought. Obviously extendable to factor rings of cyclotomic rings which are PIDs – maybe even not needing PIDs – so worth remembering.

Tolstoy’s Integration Metaphor from War and Peace

Tolstoy’s metaphor, of integrating the differential of history, is an interesting one. (Maybe each of us is the “differential”?)

An elementary proof that every singular matrix is a product of idempotent matrices, by Araujo, Mitchell.

The result is believable, after reading the article, so I guess I’m not surprised this is elementary. Maybe save this for a linear algebra class.

A group theoretic approach to a famous partition formula.

The number of subgroups of Z x Z of index n is p(n), the number of partitions of n.

The number of permutation representations of Z x Z on n symbols is n! p(n). Uses Burnside’s orbit counting lemma….

From the October 2005 Monthly

From the 65^th Putman competition, the success rate of a binary distribution is below 80 percent at one point and above 80 percent later. Is there a time at which it is exactly 80 percent? Yes. (And the argument works if we replace .80 by (n-1)/n for any n>1.) Sort of an intermediate value theorem for certain distributions.

Isaacs, others, Generalizations of Fermat’s Little theorem via GroupTheory.

For all a, the sum over d dividing n, of mu(n/d) a^d is equal to zero mod n. For example, if n=15 then a–a^3–a^5+a^15 = 0 mod 15. (mu is the Mobius function.) Orbit counting and other things … worth rereading and working through.

From the November 2005 Monthly

Dan Kalman, Virtual Emperical Investigation…,

Talks about the MathWright library but it seems to charge an annual fee and (more importantly) only for Windows.

The Modular Tree of Pythagorean Triples, by Alperin,

On the set of Pythagorean Triples being an infinite ternary tree. Also in the November 2005 Monthly and available as pdf article. Related to 2x2 matrices with determinant 1. Given Pythagorean triple (s,c,n), set

delta_1 = 2(n+s-c), delta_2 = 2(n-s+c), delta_3 = 2(n+s+c),

Then the new triples are

(-s+delta_1, c+delta_1 , n+delta_1), (s+delta_2, -c+delta_2 , n+delta_2), (-s+delta_3, -c+delta_3 , n+delta_3).

For example, beginning with the root (3,4,5), we have

delta_1=8, delta_2 = 12, delta_3=24.

Therefore the next three Pythagorean triples are

(-3+8,4+8,5+8), (3+12, -4+12, 5+12), (-3+24, -4+24, 5+24)= (5,12,13), (15, 8, 17), (21,20, 29)

(The webpage has some errors here, since only the absolute value is important.)

Surely there are some directions to go here – how do we know we get all of them? What is the distance between two triples in this graph? How does the group theory works? (PTs are related to rational parametrization of circle. To rational line, then to circle via linear fractional transformations.) Can we do this with other diophantine equations?

See also Rationals and the modular group and PSL(2,Z) is a free group, by Alperin. These are also on my laptop: Rationals and the modular group, PSL2Z is a free group.

December 2005 Monthly,

Irreducible quartic polynomials with factorizations modulo p. The Klein-4 group as Galois group leads to some irreducible quartic polynomials which factor for every prime p. Some are reducible modulo every integer n. Quadratic reciprocity plays a part.

On the zeroes of the Nth partial sum of the exponential series, by Zemyan. (They form a “wave” in the shape of a horseshoe, moving away from the origin.) What about other series?

Transpositions and Representability, by Chen and Mullen. Every function from Zn to Zn can be represented by a polynomial only if n is a prime. (Note, every function, not just the permutations.) Of particular interest is the permutation (0,1).

The evolution of nonstandard analysis, by Arkeryd. A nonstandard view of the real line is logically legitimate, so wouldn’t it have been possible for that to be developed first? (A scifi story … have an alien culture that views “nonstandard” analysis as standard.)

Problem 11187: find the closed form for the number of ways to tile a 4xn rectangle with dominoes. Answer: there is a recursion formula: s(n+2)=s(n+1)+2s(n), with initial conditions s(1)=1, s(2)=3 so in general s(n) = (2^n – (-1)^n)/3.

What primes are near our date? 1949, 1951, 1973, 1979, 1987 (the 300^th prime), 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053. (I’ve marked the twin primes.)

Favorite comics from xkcd. (Thanks to David Horstman for pointing me to this comic strip!)

Paths (for Calculus)

Substitute (on velociraptors for calculus)

Move:

100 years of finite group theory by Peter Neumann. (Also on my laptop.)

Applications of group theory to particle physics by Freeman Dyson (Also on my laptop.)