VITAL SINES Online
 Vol. 5, Number 2 November 2003
A Web Newsletter of the Plattsburgh State University Mathematics Department

The limits of Brute Force
As a long-time computer nerd, I've grown accustomed to writing quick little programs to take the drudgery out of all kinds of routine computational tasks. In fact, it was just this use of computers that first convinced me to become a computer nerd, way back in the 1980s. More recently, I found an opportunity to write a quick little script to take some of the drudgery out of an Abstract Algebra problem:
 Let p be a prime number. Find the number of generators of the cyclic group, Zp^r where r is an integer ≥ 1.
By the time I turned to the computer, I had already settled on a possible solution. Generating and testing a couple of small groups by hand (24 and 33, for example), I thought pr - pr might be a solution. Testing this solution involved just the sort of tedious, repetitive tasks for which computers are so well suited:
1. Compute pr.
2. Work through each element, from 2 to pr, checking whether it's relatively prime to pr (and thus a generator of the cyclic group), and
3. Count the generators.

Because it's quick and easy, I decided to write a short routine in JavaScript to prompt for values of p and r, and then to do the tedious work of looping through the elements of the cyclic group, testing for and counting the generators. I quickly discovered that my pet formula did not work, and in the process learned a little about the limits of the “brute force” approach.
Because speed is such a valued commodity on the Web, browsers are intolerant of slow-running scripts. Testing relatively high values of pr results in a warning message encouraging the user to abort the script. Of course, there's another limitation of testing solutions in this way. Even a billion positive results can't prove that a theorem works. On the other hand, a little well-written code can make it quicker and easier to find a counter example that proves it doesn't.
See the script on Prof. Morrow's web page.
Jim Owens (Math Major)

 From Math Horizons Mathematicians Socha and Starbird write “Mathematics is everywhere. Once math suffuses your soul, you'll see its shadows in the most surprising settings”. They prove their point in Math Horizons (available for purusal in Hawkins 244) where they show how “Goldilocks and the Three Bears” is just an application of the Mean Value Theorem and “Jack and the Beanstalk” is a tale about exponential growth.

Math club news

Dr. Mohamed Djerdjour (a Mathematician hiding in the department of management and marketing) spoke to the math club about the field of operations research. The name comes from World War II, when generals realized that mathematicians had techniques that would enable them to give optimal solutions to practical problems of combat.
Operations research now plays a crucial role in business. Operations research experts solve problems such as scheduling (think of airlines), and allocation of raw materials and manufactured goods (think of a large company with 5 or 6 production plants, and umpteen clients). As Djerdjour pointed out, mathematicians find not just good solutions to such problems, but solutions they know to be the very best possible.
Here is yet another excellent career for a math major. Math students who are interested in this field might consider taking some related courses in Management; ask Dr. Djerdjour about this.

 What team is now history? Correct for 1000! The answer is … “ the department that lost to us in Jeopardy”. The math dept. team (Diana LeBarron, Amanda McNeil, and professors Kenoyer and Northshield) beat the history department in a white-knuckle contest. It came down to the question “What North American City is named after the mountain it surrounds”. The Historians found out the hard way that it isn't “Denver”. Come see us take on the Communications department at 4:30pm on Dec. 4 in the Warren Ballroom.

 Kudo's to . . . Prof. Rob Keever for working with the developing Learning Communities to ensure that Mathematics is well represented. Prof. Sam Northshield, for giving a talk “On Square Roots of 2x2 Matrices” at the recent MAA Seaway section meeting at RIT in Rochester. Prof. Tom Powell, for representing the Math department at the recent Open House. Our math club for contributing interesting material to this issue of Vital Sines, and to our very competent math club co-presidents, Anna Gadway and Marvel Roberts who are making it all happen.

So what is math, anyway?

Members of the Math club decided to ask mathematics faculty (including a mathematician who hides in C.S.) how they would define mathematics. Here, in no particular order, are the responses they obtained from Professors Bodenrader, D'Aristotile, Hofer, Keever, Keiser, Liu, Morrow, Northshield, Petro, Plaza, Powell, and Solsten.

"The interplay between generality and individuality, deduction and construction, logic and imagination." (Quoting Courant)

It involves discovering patterns and connections, and seeking explanation for why those patterns and connections exist.

It is the discovery of universal laws that govern numbers, shapes and sets. Beyond discovery, it is the formulation of these laws into axioms, theorems and methodologies. On a more poetic level, mathematics is to the universe as current is to a stream.

It is a class of sciences that using specialized notations treats exact relations abstracted or reduced from existing or supposed quantities, shapes, spaces and their interrelationships.

It is a body of knowledge containing generic tools for modeling those aspects of the world which are considered in all sciences.

It is the creation and study of axiomatic systems and morphisms of such systems with particular emphasis on properties preserved under system morphism.

It is the study of infinity.

It is a human endeavor combining the best characteristics of science and of art; the precision of thought and devotion to detail and to 'the real' as in science, the creativity and freedom of expression as in art.

Mathematics is pure and applied and investigates areas such as numbers and structure (Algebra), figures and shape (Topology), surfaces (Geometry), and randomness (Probability). Its enormous development will continue to be driven forward by the passion of its devotees who have so much in common with poets, artists, and musicians. Indeed, Mathematics is the finest of the arts.

“Mathematics” is derived from the Greek, “mathein”, a verb meaning “to learn, understand or process information”. Learning is the root of mathematics.

Mathematics is the logical abstraction and application of quantitative relations. We first identify the relevant quantities and variables, determine their relationships and ignore the unimportant words. By abstracting properties of arithmetic, we obtain, and can then apply the algebra necessary to give a unique solution to the problem. Further abstraction leads to the field of linear algebra.

In mathematics we try to understand and seek relationships in situations in the world around us.

Alumni News

Salim Dhirani (CS '03, Mathematics minor) is working for Crescent Tech. Systems in Orlando, FL. He writes: “I miss my college days already but not the winter that came with it. Here in Orlando the weather is very good, like back home, hence no complaints.”

Jim Pombrio ('91), a Senior Clinical Data Coordinator at Target Research Associates writes: “Lately I've sort of transitioned myself into more of a programmer's role in my job.  I'm technically a data manager, meaning that I do a lot of project management as well as programming.  An aspect of my programming work is what we refer to as "edit checks" where we check logical consistency of data (e.g., males do not get a pregnancy test) and validity of data (e.g., stop dates of medications must be after start dates).  The most interesting part of my programming work is when we work with outside vendors that provide us with electronic laboratory data.  I am able to use my science background a lot!  I use it to categorize lab tests (e.g., blood chemistry versus microbiology).  An awful lot of my math background gets used, although no calculus; more of the sets/functions/relations variety, because I have to pay close attention to logic, as well as concepts of mapping.  The programming language that we use is called SAS (statistical analysis software) which is THE standard in the pharmaceutical business, although it is widely used in insurance/finance as well. News:  I got married and we will be moving into our new house in two weeks.”

 "It is, in fact, nothing short of a miracle that the modern methods of instruction have not yet entirely strangled the holy curiosity of inquiry; for this delicate little plant, aside from stimulation, stands mainly in need of freedom; without this it goes to wrack and ruin without fail. It is a very grave mistake to think that the enjoyment of seeing and searching can be promoted by means of coercion and a sense of duty." ~~ Albert Einstein

Problem

From “Car Talk” on National Public Radio: You, who are blindfolded, are given a deck of 52 playing cards with exactly 13 cards facing up. Find a way to divide the cards into two piles such that each of the two piles has exactly the same number of cards facing up.

Please submit your solution, preferably written on the back of one of those new twenty dollar bills, to Prof. Northshield. The first and/or best solution will permit you to choose a prize from the 'big box'o'prizes' in Northshield's office.

Closing Credits:
Editor: Sam Northshield
Assistant Editor: Margaret Morrow
Web Editor: Don West