Abstracts of Talks at the Mathematics Conference in Honor Professor Tom Osler’s 70^th Birthday

Rowan University, Rowan Hall

Glassboro, NJ 08028

April 16 – 17, 2010

Friday April 16

3:30- 3:40        Opening Remarks by Hieu Nguyen (Chair, Math Dept) and Parivz Ansari (Dean, CLAS) 

3:40 - 4:30       Bruce Brendt, University of Illinois Urbana-Champaign

Ramanujan's Life and Notebooks

Ramanujan was born in southern India in 1887 and died there in 1920 at the age of 32.  He attended college for only one year, but his mathematical discoveries, made mostly in isolation, have made him one of this century's most influential mathematicians.  An account of Ramanujan's life will be presented.  Most of Ramanujan's mathematical discoveries were recorded without proofs in notebooks, and a description and history of these Notebooks will be provided. The lecture will be accompanied by photographs depicting Ramanujan, his home, his school, his notebooks, and those influential in his life, including his mother and wife.

Saturday April 17

9:00- 9:10        Opening Remarks by Ali Houshmand, Provost

9:15- 10:05      George Andrews, Penn State University 

Lessons from Ramanujan's Lost Notebook

In 1976 quite by accident, I stumbled across a collection of about 100 sheets of mathematics in Ramanujan's handwriting; they were stored in a box in the Trinity College Library in Cambridge. I titled this collection "Ramanujan's Lost Notebook" to distinguish it from the famous notebooks that he had prepared earlier in his life.  On and off for the past 34 years, I have studied these wild and confusing pages. Some of the weirder results have yielded entirely new lines of discovery.  Sometimes, if you pay close attention, you can gain some possible insights about the searches that Ramanujan undertook and the questions he must have asked himself.  Even if such speculations may be far from Ramanujan's actual thinking, they are nonetheless valuable exercises to undertake. Some of these flights of fancy will form the topics in this talk.

10:10- 10:30    Walter Jacob, Fidelity National Information Systems

A Modern Look at a Neglected Summation Formula by Euler

In this talk we will look at results from a paper of Leonhard Euler's (E46) and discuss his method for approximating sums of series using integrals, his discussion of approximating integrals using both rectilinear and curvilinear triangles, and his evaluations of the first ten terms of ZETA(2), the first million terms of the harmonic series, and the exact value of the infinite sum of the reciprocals of squares. Also discussed will be the connection between this summation formula and the Euler-Maclaurin summation formula and some results from contemporary mathematics.

 10:30- 10:45 Coffee Break

10:50- 11:40    Richard Askey, University of Wisconsin Madison

The Start of Trigonometry and Related Results

Serious trigonometry started with Ptolemy's theorem on cyclic quadrilaterals. Some refinements of Ptolemy's theorem and related results will be described

11:45- 12:05   TR Chandrupatla, Rowan University                  

Planar Bi-arc Curves

Planar curves comprised of piecewise circular arcs find many engineering applications. These curves are easily produced accurately using numerically controlled milling machines. A geometric view will be given in this presentation to generalize the construction of a curve comprised of two circular arcs to match end slopes on a segment. Two circular arc approximation of a quadrant of an ellipse is an interesting application in engineering drawing.

12:10- 2:00      Lunch Break 

2:00- 2:50 Victor Moll, Tulane University  

A Heuristic Method for the Evaluation of Integrals 

The method of brackets developed by I. Gonzalez consists of a simple set of rules that yields the evaluation of a large class of definite integrals.  This method was created as a new approach to evaluate (complicated) integrals coming from Feynman diagrams.  Some elementary examples will be described.

 2:55- 3:10 Jason Scaramazza (Student) Rowan University

A Musical Conjecture by Euler

A basic premise of musical theory is that consonant tones have frequencies contained in uncomplicated whole number ratios, while dissonant tones do not. Euler is puzzled by the apparent denial of this law by two chords known as the "seventh" chord and the "triad add-6". He observes that though these chords sound pleasant to the ear and are used with quite a bit of regularity, a simple mathematical analysis of the ratios of the frequencies reveals that they should be cast aside as unbearable dissonances. In an attempt to resolve this strange paradox of sense versus mind, Euler makes a simple yet astute observation that a modification of one tone in these chords makes them fit the theory perfectly. He furthermore explains that the ear approximates the actual chord in this way. His support draws from two different types of tuning in which the ear must approximate tonal intervals, or else they would sound terribly dissonant. 

 3:10- 3:25 Coffee Break

3:30 – 4:20       Bruce Brendt, University of Illinois Urbana-Champaign

Ramanujan Reaches His Hand from His Grave to Snatch Your Theorems from You

It has often happened that a mathematician proves a theorem of which she or he is justly proud, only to discover that, alas, his or her beautiful theorem  is not original, but is ensconced somewhere in Ramanujan's notebooks, lost notebook, or unpublished papers.  The purpose of this lecture is to provide several instances of this phenomenon, beginning only three or four years after Ramanujan's death.