OFF ON A TANGENT
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
March 9, 2005 Vol. 3, No. 11
http://www.math.hope.edu/newsletter.html


Research students to give tomorrow's mathematics colloquium

When: Thursday, March 10 at 4:00 p.m.
Where: VWF 104
Tea time: 3:30 p.m. in VWF 222

Two research students, Andrew Craker from Notre Dame and Erin Wicker from Alma College, will give tomorrow's colloquium titled, "Spiraling to our Doom."  These students worked with Prof. Aaron Cinzori last summer on this REU project. 

This colloquium is an extension of the one given a couple of week's ago by Prof. Tom Scofield from Calvin College.  In that talk we were introduce to an algorithm that, given n points in the plane and a parameter t between 0 and 1, produced a spiral.  He showed where the spiral goes, when it has finite length, and gave an approximation for that length.

In this talk, Erin and Andrew will discuss the special case of three initial points.  They will demonstrate a method that produces the exact length of the spiral (as a geometric series) for infinitely many t > 1/4.  They will discuss the problem with extending the method to t < 1/4 and produce an error bound in this case.  We will again be knocked out by the wonders of SVD.

This talk uses ideas from Calculus 2 (infinite series) and Multivariable 1 (matrices).


The 29th Lower Michigan Mathematics Competition will be held soon

The 29th Lower Michigan Mathematics Competition will be held at UM-Flint on Saturday, April 2. Students from colleges and universities in Michigan will gather to challenge themselves on 10 interesting problems, working together in teams of up to three people. The competition runs from 9:30 a.m. to 12:30 p.m. After the problem session in the morning, there will be a break for lunch (provided by LMMC) and a solutions session in the afternoon. A one-year calculus background is assumed, and advanced topic problems will be self-contained, so students in Math 132 and beyond are encouraged to consider participating. Interested students may sign up individually or in teams. Transportation and meals will be provided. The deadline for registering is Thursday, March 17.

Registration deadline: Thursday, March 17
Contest date: Saturday, April 2, 9:30 a.m. to 12:30
How to register: Sign up on the sheet on Dr. Pearson's door (VWF 212), or email him at pearson@hope.edu.

Hope has a history of strong showings at the LMMC, including several championships, and we'd like to regain the title this year and bring the Klein Bottle Trophy back to Hope!

Information about the GRE available

The Hope College Pew Society and the Office of Career Services are sponsoring an information session on the Graduate Record Examination (GRE). Professor Charles Behensky of the Department of Psychology will discuss the mechanics of the GRE, what students might do to prepare for the exam, and answer questions.  The GRE is an exam that is required for admittance in most graduate schools.  The session will be on Monday, March 14 (Pi Day) from  4:00 to 5:00 p.m. in 1118 Science Center.

Information about the GRE is also available on the Career Service’s GRE web page: http://www.hope.edu/student/career/GRE.html. This site provides more information on the GRE, including subject test dates, and announces the availability of some practice test software.


The Mathematics Department web page has been updated

CIT has been updating department web pages this past year and has finally gotten around to the Math Department page.  While there still needs to be some fine tuning done, you can check out the new version at http://www.math.hope.edu/.  There are a number of different pictures that will appear on the main page.  If you hit reload on your computer, you can watch these change.  Perhaps there is a picture of you there!  Check it out.


Problem Solvers of the Fortnight

Making the honor roll in Professor Collectsumup's class are Sommer Amundsen, Benjamin Crumpler, James Daly, Alex Larson, Becky Lathrop, Kevin Vander Bosch and Ryan Weaver, all of whom correctly determined that on average Professor Collectsumup would collect a homework assignment every 2.7 days or so, roughly 36% of the time.  100% of our problem solvers are invited to drop by Dr. Pearson's office to claim their rewards!


Problem of the Fortnight

Last Monday snow began to fall in Holland (again!) sometime before noon and fell at a constant rate until about dinner time.  At noon a snow plow started to plow River Street.  The plow cleared one mile of River Street during the first hour and one-half mile during the second hour.  What time did it start to snow?

(Hint: You may assume that at any instant of time the volume of snow removed is constant; i.e. the snow plow clears snow at a constant rate.  What does this tell you about how the depth of the snow and the linear distance traveled by the plow are related to each other?  There's a neat calculus problem buried in the snow here.  Can you dig it out?)

Write your solution on the back of a paper snowflake and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 on Thursday, March 17 before you head out of town on Spring Break to bask in the sun.


Mathography: Max Zorn (1900 - 1993)

Born in Krefeld, Germany (about 20 km northwest of Dusseldorf), Max Zorn went on to study algebra with Emil Artin at the University of Hamburg and later emigrated to the United States in 1933, forced to leave Germany because of Nazi policies although he was not Jewish.  Zorn is best known for "Zorn's lemma," an important contribution to set theory which he originally conceived when he was a postdoctoral fellow at Yale from 1934-36.  Following his years at Yale, Zorn took a position at UCLA, where I.N. Herstein was one of his doctoral students.  Although Zorn stopped publishing papers in 1947, he remained active in mathematics for the remainder of his life and in his later years became fascinated with the Riemann hypothesis.  Zorn passed away March 9, 1993 in Bloomington, IN.

The lemma which bears his name states that if S is a partially ordered set in which every chain has an upper bound, then S has a maximal element; it is equivalent to the well ordering principle (every nonempty subset of positive integers has a least element) and to the axiom of choice (given any set of mutually exclusive nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets).

In addition to being a foundation of mathematics, Zorn's lemma is also the brunt of quite a few bad mathematical jokes, such as the following:

Q: What's sour, yellow, and equivalent to the axiom of choice? A: Zorn's lemon.



Q: What is brown, furry, runs to the sea, and is equivalent to the axiom of choice? A: Zorn's lemming.




To read more about Zorn, please visit http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Zorn.html. 


Got a Math Question?

Ask Elvis ...

... email him at elvis@hope.edu

Dear Friends,

My e-mail box has been quite busy the past couple weeks.  However, not a single math question.  I had one person that wanted my help to get some money out of his country.  Another wanted me to send my PIN from my credit card to him to “verify my account.”  There were a few that were offering me credit.  Finally, one made promises to me about making increases in a certain aspect of my body that I’m pretty sure is impossible since a trip I had to the vet a couple of years ago.  I don’t think all those that have been writing me lately really knew who they were addressing.  I guess this is the bad type of spam that I have been hearing about. 

There are a couple of special days coming up next week (besides the first day of spring break).  While you are probably aware that next week Thursday is St. Patrick’s Day, you may not be aware that next Monday (3/14) is Pi Day.  I heard that there will be a St. Patrick’s Day parade in Holland, but I haven’t heard anything yet about Pi Day celebration.  Even if there is no parade, day off from classes, or extra kibble in your bits for that day, you can send a loved one a Pi Day electronic greeting card.  Just go online to http://www.123greetings.com/events/pi/ and chose one of their “interesting” (but not entirely mathematically correct) cards.  These can be personalized for your recipient and needn’t start out by saying something like, “Dear Washington Mutual Customer.”

Anyway, while I haven’t received any legitimate letters this past fortnight, I do have one leftover from last time.

Take care and have a safe spring break!




Dear Elvis,
I have a math question about my fish tank that has had me stumped for a while. I took a glass cylinder and filled it with water in the tank and then lifted the cylinder out so that the opening was still underwater. I know the fact that air pressure is what is holding the water up there. However, my question (see my attached drawing) is about the water pressure in the tube as compared to the tank. What is the pressure at the points that I have labeled? Does fish 1 feel the same pressure as fish 2? If not, how is the pressure at the surface of the tank different than the pressure at the top of my cylinder? My fish had no difficulty swimming up and down the cylinder, so it looked like the pressure depends on how much water I have pulled above into the cylinder, but I can't figure it out. kept getting less as he went up, even though the water is above the actual tank surface level. I have a feeling it kind of
-Kyle

Dear Kyle,

A had to get help from a physicist on this one.  She told me that this system you have set up is effectively a barometer, though typical barometers are filled with Mercury.  (Don't fill your fish tank with mercury, Kyle.  Your fish will have a hard time swimming to the bottom of the tank.)

Let's do the easy one first:

In the tank, Marlin (who you lovingly call Fish 1) is feeling the weight of the a meters of water and atmospheric pressure pushing down. He is feeling a pressure from below that is equal to atmospheric pressure and a + hM meters of water above his lower surface.

Now, in a typical barometer, there is a vacuum above the liquid in the tube. The pressure at the top of the tube zero. A distance a from the top would experience a pressure due to the weight of a meters of water, so Nemo (who you cleverly named Fish 2) would feel the pressure of a + hN meters of water above his lower surface. Thus Nemo would feel less pressure at each surface due to the lack of atmospheric pressure. However, in both cases, the net force is zero (the gravitational force on the fish plus the downward force from the pressure on the top must equal the upward force from the pressure on the bottom if the fish is not accelerating up or down).

Now, in the picture drawn, Nemo actually feels something in between what Marlin feels and what he would feel if there was a vacuum between the top surface of the water and the top cylinder surface. The contact between the cylinder and the top of the fluid column will create a contact force downward and increase the pressure in the fluid it the tube.

The downward pressure is proportional to the height of the column (H).  The downward pressure will be patm(1 - h/hmax) where hmax is the highest water column air pressure could support which is patm/(density of air)/(acceleration due to gravity). Once h = hmax, this downward pressure would be zero and the previously described situation would be present.

Since the fluid is water, hmax is quite high, so my guess is that the difference in pressure was negligible, but I'll leave it as an exercise for the reader to actually calculate the difference.

Elvis


If you don't do it this year, you'll be another year older when you do.
Warren Miller