Friday, December 28, 2012

Sunday, December 2, 2012

B-1106 – Fibonacci Quarterly (ONU-SOLVE)

Solution to Problem B-1106 – Fibonacci Quarterly 50, Number 2 (May 2012)
Provided by the ONU-SOLVE Problem Group Ohio Northern University

Monday, November 12, 2012

Beyond High School Science Fairs: The Senior Capstone Project

On October 31, 2012, I was invited to speak in the University of Findlay (OH) Mathematics Colloquium. The title of my talk was "Beyond High School Science Fairs: The Senior Capstone Project". The talk was an itinerary through a number of themes investigated in my undergraduate research projects (with an emphasis on quadratic residues and their applications to combinatorics, greater prime factor sequences and the relevant open problems in the area, etc). A difficult problem that I tried to address: what exactly should be considered as a sign of authenticity for a genuine undergraduate research experience, what would make it more than a science fair project (where the emphasis lies - granted with notable exceptions - on contingent data manipulation, with mathematics playing a secondary, albeit cool, "supporting role").  I suppose the answer has something to do with a serious dose of mathematical maturity reflected in a significant focus of action/intentionality on abstract objects, in an increased ability of “bracketing out” the contingent world, and in the sense of freedom implicit in this act.

Monday, November 5, 2012

A nice picture

A nice "Paley digraph" picture - with outgoing edges from x to x+1, x+2, x+3, x+4, x+6, x+8, x+9, x+12, x+13, x+16 and x+18 (modulo 23) for x = 0,1,...,22.


Friday, October 19, 2012

The Role of Research at Undergraduate Institutions

An excellent article by Robert Gavin, in "Academic Excellence - The role of research in the physical sciences at undergraduate institutions" (Michael P. Doyle, Editor - published in the year 2000 by Research Corporation - a foundation for the advancement of science). Even if the paper, which emphasizes the role of publishing in a research-based education, refers to physical sciences, the ideas in there are even better suitable for mathematical sciences, where there is not an excessive need for laboratories and equipment.
Straight to the point: "Publishing research articles, especially those done in collaboration with undergraduate students, should be expected, encouraged and supported both before and after the tenure decision".

Wednesday, August 29, 2012

RIP William P. Thurston (October 30, 1946 – August 21, 2012)


AMS Obituary
“The inner force that drives mathematicians isn’t to look for applications; it is to understand the structure and inner beauty of mathematics.” (source: New York Times
Photo source - wikipedia

Tuesday, July 24, 2012

Between the greenness of the grass and the grip of the big data

Drowning in the Data Deluge - by George E. Andrews (written version of the author's 2012 AMS Retiring Presidential Address). Excellent analysis, definitely a must read. My favorite passage (possibly because it might be read in a phenomenological key):
We are no longer able to assert merely that "Grass in green!" Instead we must add something like:
A team of Harvard scientists has studied 9328 blades of grass from 37 randomly selected countries. They measured the wave length of light emanating from each blade when placed in the noonday sun in Harvard Yard. 98.32% produced light of wave length between 520 and 570 nanometers which is the accepted standard measure for green as certi ed by the International Bureau of Standards.

Thursday, June 28, 2012

ONU-SOLVE, solution to Problem 1873 (Mathematics Magazine)

Proposed problem 1873
Mathematics Magazine, Vol. 84, No. 3, June 2011
Solution provided by the ONU-SOLVE problem group


Wednesday, June 27, 2012

ONU-SOLVE, solution to Problem 1872 (Mathematics Magazine)

Proposed problem 1872 - Mathematics Magazine, Vol. 84, No. 3, June 2011
Solution provided by the ONU-SOLVE problem group.

Tuesday, June 5, 2012

Venus Transit 2012



Camera shot of a telescope projection (courtesy of Ohio Northern University Observatory - public viewing, June 5, 2012, about 7:30 pm)

Thursday, April 12, 2012

Experimental math notes: a curious prime sequence of length 22

The sequence is:
5, 3, 5, 3, 3, 11, 13, 13, 37, 17, 11, 17, 7, 5, 5, 17, 7, 17, 3, 17, 5, 31    (1)
An interesting property of this sequence: if P is the greatest prime factor function, then its infinite periodic extension (X(n)) satisfies the recursion
            X(n)=P[X(n-1)+3X(n-2)+2X(n-3)]    (2)
The intriguing part: for virtually all random choices of the initial conditions that I made - that is, other than the trivial ones (leading to 1-cycles, such as those with X(0)=X(1)=X(2)=p), the prime sequences satisfying (2) ultimately enter the same limit cycle (1). These are strange phenomena, of a similar nature to the GPF-tribonacci sequences (where we discovered that there are at least four distinct GPF-tribonacci limit cycles, of lengths 100, 212, 28 and 6). My last MAPLE experiment with (2) involved a seed consisting of three 80 digits primes,

X(0) = 67525204474446805798439049565857966823399463795779492274451732592404979647691781
X(1) = 43280708928363322606959208124229795876921456250899031972019611375562107511756173
X(2) = 50558523494317177773504742464927157574684415344640904005922668959550436919047547

which eventually led to the cyclic shift of (1) given by 7, 17, 3, 17, 5, 31, 5, 3, 5, 3, 3, 11, 13, 13, 37, 17, 11, 17, 7, 5, 5, 17, with 7=X(50), 17=X(51), 3=X(52), ...

  • Added on April 17, 2012 
Continued the search on possible other nontrivial limit cycles for the recursion X(n)=P[X(n-1)+3X(n-2)+2X(n-3)]. So far (1) is the only one found. Today - improved the process of random selection of initial conditions. A sample in the last series of searches - still leading to (1) (log plot of the graph included below)
X(0)=38790912184195861716665094754005872283807697220069306337918513393335402427949730921338400505241
X(1)=34380504590591201002245563
X(2)=239197136656882070249024044251770704653065407229190712881313724790604984709544507179663

Thursday, February 16, 2012

Thursday, January 19, 2012

Pendulum Waves


http://youtu.be/yVkdfJ9PkRQ
"Fifteen uncoupled simple pendulums of monotonically increasing lengths dance together to produce visual traveling waves, standing waves, beating, and (seemingly) random motion"

Wednesday, January 11, 2012

Interview with George Andrews, AMS Retiring President


http://youtu.be/4TT43NWua38
Former President of the AMS, George Andrews gave JMM TV an exclusive interview to talk about his address to the Joint Mathematics Meeting in Boston looking at "Our Challenges" for the future.

NOTE
George Andrews, the world's leading expert in the theory of integer partitions, discovered Ramanujan's Lost Notebook in 1976. In an interesting news story (Asia Times Online - India celebrates the man who 'knew' infinity), India has declared 2012 as "National Year of Mathematics" as a tribute to Srinivasa Ramanujan.