Happy Belated Valentine’s Day!

Departmental Romantic, Ravi Shankar, asked us earlier this week to share this with everyone.   The delay is entirely due to the Blog staff suffering from a Valentine’s induced chocolate coma.

Happy Valentine’s Day, especially mathematics lovers! On the fun side, check out the graph of this implicit function: $x^2 + (y - x^{2/3})^2 = 1$. To see the full graph on your TI-83 or equivalent graphing calculator, sketch the following two graphs simultaneously on the window, $-2 < x < 2$ and $-2 < y < 2$. Let $Y1 = (x^2)^{1/3} + \sqrt{(1-x^2)}$, and let $Y2 = (x^2)^{1/3} - \sqrt{(1-x^2)}$. Note that if you type $x^{2/3}$ instead of $(x^2)^{1/3}$ for the first term in each equation, you will only see half the graph. Enjoy!

Or, if you don’t have your calculator handy, you can always try Wolfram Alpha.

The Impossible Puzzle

Dr. Shankar’s upcoming talk in the Math Club reminded us of the wonderful problem which Martin Gardner called the Impossible Puzzle.  It seems to have been invented by the mathematician Hans Freudenthal in the 1960′s.   There are now several different versions out there.  We give the version by David Sprows (this and other variations can be found at this website) :

David Sprows, Mathematics Magazine, Volume 49, Number 2, March 1976, page 96:

Let x and y be two integers with 1 < x < y and x+y <= 100. Suppose Ms S is given the value of x+y and Mr P is given the value of x*y.

1. Mr P says: “I don’t know the values of x and y.”
2. Ms S replies: “I knew that you didn’t know the values.”
3. Mr P responds: “Oh, then I do know the values of x and y.”
4. Ms S exclaims: “Oh, then so do I.”

What are the values of x and y?

How to Toss a Coin Over the Phone

This

Wednesday, April 18th at 5 pm in PHSC 1105

our own Dr. Ravi Shankar will be talking in the Math Club.  He answer the puzzling problem of telling someone something without telling them and related conundrums:

Your favorite uncle Mo who lived in Alaska has recently gifted you his zamboni. But he left it to you and his other favorite niece, Eva, who lives in Arizona. You give Eva a call and neither of you wants to share the zamboni. You decide to flip a coin for it. So Eva flips and you call, “Heads!” and she says, “Oops, sorry, no. You lose.” To which you reply, “I don’t believe you!” An argument ensues until you realize you went to a math club talk at OU, where you were shown how to flip a coin over the phone and both be reasonably sure that the other person could not have cheated.

In this talk we will explore how to solve this problem and other similar “authentication” and “signature” problems (how do you convince a bank teller on the phone that you know the secret pin number for your account without actually revealing the pin number?)

– Dr. Shankar’s abstract

There's more than one way to find out what somebody knows!

As always, Free Pizza!

P.S.  Come and quiz Dr. Shankar on Weierstrass gap sequences!

Award Fest 2012

The Math department made out like bandits last week.  Not one, not two, but three OU math faculty won big awards.

David Ross Boyd

Andy Miller received a David Ross Boyd Professorship.  This and the George Lynn Cross professorships are the most prestigious awards given to OU faculty.   According to the University:

To qualify for a David Ross Boyd Professorship a faculty member must have consistently demonstrated outstanding teaching, guidance, and leadership for students in an academic discipline or in an interdisciplinary program within the University.

David Ross Boyd was the first president of OU.  He’s famous (he has his own Wikipedia page!) for planting the thousands of trees at OU, and for being fired by the first governor of Oklahoma for being an “aristocrat, not democratic enough.”

Christian Remling received a Regent’s Award for Research and Creative Activity.  At most three of these are given each year (this year only one was given out!) for excellence in research.  Dr. Remling got it for, of course, his recent paper in the Annals (which we talked about here).  One of the world experts in this area of research called the paper “Remling’s earthquake”!

And Jon Kujawa received the Irene Rothbaum Outstanding Assistant Professor award.  It’s given for:

The nominee should demonstrate outstanding teaching as shown by student-teacher evaluations, chair recommendations and additional information such as student or peer letters of support…. The candidate also should show evidence for good progress in scholarship, such as publication in peer-reviewed outlets and fellowship or grant support.

As you know, our own Ravi Shankar won the award in 2006 and Nikola Petrov won it in 2009.  We reported on Dr. Petrov winning here, but Dr. Shankar won BB (Before Blog).

With Dr. Kujawa winning it in 2012, we fully expect someone in Math to win it in 2015 (we’re looking at you Dr. Pitale!)

It’s Always Sunny in San Diego

It’s the season for applying to REU’s for next summer.  Our own Ravi Shankar let us know of this one.  It’s math and beaches.  What’s not to like?

San Deigo State's Campus

San Diego State University is pleased to invite applications to its summer 2012 Mathematics Research Experiences for Undergraduates.  The program dates this year will be June 17 — August 11, and the project will be in number theory, specifically nonunique factorization theory.

Please announce to any interested and qualified undergraduates, as well as to any graduate students interested in helping to lead one of the two project teams.  The program will pay a stipend of $5000 and provide room and partial board for nonlocal participants. The application deadline is March 5. For a detailed program description and application instructions/materials, please see the program website: http://www.sci.sdsu.edu/math-reu/index.html Best wishes, Vadim Ponomarenko Program Director Math Department Awards! We’re a little late to the party, but the OU Math Department had their annual Awards Ceremony on Friday, April 29th. As always, lots of food, jokes, and awards were handed out. Congratulations to everyone! It’s always nice to see hard work rewarded! The Partij van de Toekomst Sometime soon you’ll be able to find the full list of winners here, (Ed. note: Done! – KK) but we thought we’d share a few pics of the event: Megan Fuzzell and Dr. Shankar Rhyker Benavidez and Dr. Shankar Dr. Shankar, Chris Schroeder, and Dr. Cross A giddy Ore Adekoya and Dr. Shankar Dr. Shankar sure loves to be in front of the camera! Dr. Erin Pearse doing his standup routine. Women in Math conference – the inside scoop On January 28-30, 7 OU undergrads and one graduate student attended the 13th annual Nebraska Conference for Undergraduate Women in Math (NCUWM) in Lincoln, NE. This is a conference where undergraduate women get to give talks about their research, attend panel discussions, and meet with other women who do mathematics. About 250 students attend the conference every year. One of the invited guests this year was Dr. Christine Jerritts of Imagine Math!, who visited the Math Club last semester. The OU students who went up to Lincoln were Ore Adekoya, Michelle Basham, Dana Haymon, Marissa Mercado, Elizabeth Park, Katlyn Seagraves, and Edwina Shwewa. Math graduate student Lynn Greenleaf and faculty member Keri Kornelson accompanied them on the trip. Many thanks go out to the colleges and departments that funded this excursion! Some of the students were kind enough to answer a few questions about their experiences for the blog. Below are some excerpts. Blog HQ: Why did you decide to go to NCUWM? Edwina: I decided to go to the conference because I wanted to find out about opportunities after graduating with a Mathematics degree. Katlin: One of my professors my freshman year (Dr. Shankar, we presume) made it sound like a fun trip. Michelle: I wanted to go to the Nebraska Conference because I wanted to meet other undergraduate women math majors, just to know they exist. I wanted to meet with the top women in the field, and I wanted to learn what I can do with my math major. Marissa: […] This is what college is all about: meeting people and gaining exposure and experiences to help me prepare for the adult world! I have always loved math, but have never really seen it applied outside the classroom. I figured this conference would be a way to see the activities and types of research involved if I were to pursue a degree in math. Over 260 women mathematicians! Can you spot the 9 from OU? HQ: What happens at this conference? What did you do there? Marissa: The time flew by so quickly! This three-day conference was packed with keynote speakers, panel discussions, and presentations from fellow undergraduate students. While I did not give a talk or make a poster, I mainly came to take in whatever the speakers had to offer. Many of the students present were junior or senior math majors looking toward graduate school. While some of the topics were very complex, I was happy to at least gain some mathematics exposure. Lynn: […] The talks given by undergraduate women were impressive. These women put a great deal of effort into their presentations. This also gave them valuable experience at public speaking. There were breakout sessions covering many topics of interest. The undergraduate women asked insightful questions and were obviously engaged in thoughts about their futures. Edwina: […] there were specialized breakout sessions to address particular issues which included juggling a career and life, how to prepare for graduate school, how to find a job with a math degree, etc. The conference helped with networking with people in the mathematics field. Michelle: At the conference, the first day we checked in to the hotel, walked around UN Lincoln, and heard a plenary speech from Dr. Linda Petzold. That night, we went to a banquet where we heard from a panel of women in the field and were able to ask questions and gain insight as to what a career in mathematics will actually be like. We were placed at random so that we would have to interact with women outside of our group, which was wonderful because I was placed next to Dr. Fan Chung, one of the most phenomenal women I have ever met, and the second plenary speaker. […] we went into small groups to discuss topics we chose during registration. My first small group was led by Dr. Petzold on “What Is Research in Mathematics?” Her answers to my questions were so enlightening and I cannot be thankful enough that I chose that talk. […] HQ: Give 5 words that describe the NCUWM conference. The responses! Happy Birthday! OU math professor and friend of the blog, Ravi Shankar, had his birthday on Monday (which, by the way, was the summer solstice!). You should email him a belated Happy Birthday! In his honor, and in the spirit of our previous (and future) posts on probability, we thought we should mention the famous Birthday Problem: If you are in a math class of 50 people and the professor offers to bet you$100 that two people have the same day as their birthday, should you take the bet?

Of course, even without doing the math you should be suspicious.  To paraphrase a quote:

Ha ha! You fool! You fell victim to one of the classic blunders – The most famous of which is “never get involved in a land war in Asia” – but only slightly less well-known is this: “Never go against a math professor when money is on the line”! Ha ha ha ha ha ha ha! Ha ha ha ha ha ha ha! Ha ha ha…

The key of course is that the professor is not betting that you and someone else will have the same birthday, but just that two people in the class will have the same birthday.   For the first scenario, the odds are 49/365 or approximately 13.5% (to see this, you see that each of the 49 other people in the class have a 1/365 chance that they’ll have the same birthday as you*).

In the second scenario, it is easier to count the odds that no two people will have the same birthday; that is, that all 50 birthdays are different.  Well, the odds that two people have different birthdays is (365/365)*(364/365) (the first person can have any birthday, the second person can have any birthday except the day of the first person’s birthday).  If there are three people, then the odds are (365/365)(364/365)(363/365) (the first person can have any birthday, the second any birthday except the first person’s birthday, the third person any day except the two birthdays of the first two), etc.

In general, if you have n people, then the odds that no two of them have the same birthday is given by

$\bar{p}(n)=\frac{365*364*363* \cdots * *(365-n+1)}{365^{n}}.$

So the odds that two people out of a group of n people have the same birthday is given by

$1- \bar{p}(n).$

Note that this gives the odds as a fraction of 1.  If you want it as a percentage, multiply by 100.

So back to our question, what is the odds that two people in a class of 100 have the same birthday?

It’s a shocking 97%!

In a group of 100 people, the odds of two having the same birthday are 99.99996%!

The probability that a pair of people in a group will have the same birthday (from Wikipedia)

We’ll end with a pigeonhole principle question:  How many people do you need to have in the room to ensure that there is a 100% chance that two people will have the same birthday?

* = not counting leap years

Awards, Scholarships, and More Awards

A few weeks before the end of the term the OU Math Department had its annual awards dinner.  As usual, fistfuls of money and fame where handed out right and left.  We wanted to share a few of the goodies which were doled out at the event.

But before we do that we should remind you that the OU math department has a number of different scholarships available for people who study math at OU (including for high school students who will be enrolling next year!).  To get a piece of that pie, you can read about the scholarships and the application process here on the math department website.  Put in your application next year!

Back to this year’s ceremony.  This year, no less than 15 undergraduates received awards!

This includes the Samuel Watson Reeves Award for Outstanding Senior Math Major won by Ted Swang:

Dr. Shankar and Ted Swang

And Nicholas Neal as the Nathan A. Court Award for Outstanding Freshman/Sophomore:

Dr. Shankar and Nicholas Neal

Receiving the C. Eugene Springer Math Scholarship, Oreoluwa Adekoya:

And receiving the American Mathematical Society’s Waldemar J. Trjitzinsky Award, Dana Haymon:

Dr. Shankar and Dana Haymon

Of course this is only a sample of the undergrad award recipients.

Lest you think it was all Dr. Shankar + undergraduates, there were a number of graduate student award recipients, including friend of the Math Club, John Paul Cook:

Dr. Brady and John Paul Cook

Two other people who got awards this spring should get their moment in the spotlight:

A dapper Logan Maingi, recipient of a Goldwater Scholarship:

And recipient of an OU Presidential Professorship, a cheerful Dr. Ralf Schmidt:

Dr. Ralf Schmidt

The Mattress Group

Dr. Strogatz

Steven Strogatz is a professor of applied math at Cornell University.  In addition to doing lots of interesting math, he writes a column for the New York Times about mathematics.

His most recent article is about group theory in the bedroom (which is a lot less racy than it sounds!).  As you know, you are supposed to periodically rotate and/or flip your mattress that it doesn’t get squished flat from people sleeping in the same spot each night.  Dr. Strogatz discusses the fact that if you have a standard rectangular mattress, then there are a total of four possible positions that the mattress can be in (either upside-up or upside-down and either your head on one end or your head on the other end).  But, as Dr. Strogatz discusses, what is actually even more interesting is the actual operations which move the mattress from position to position.  The basic ones are to either rotate the mattress or to flip the mattress, and all the rest are some combination of these two.

Since if you do one of these operations and then another one, you get a third such operation, it turns out that the collection of all four of these forms a group.  Just by checking the various possibilities, it turns out that there are two groups with four elements:

$\mathbb{Z}_{4} \text{ and } \mathbb{Z}_2 \times \mathbb{Z}_2$

The first is the integers with addition modulo 4 (you might have learned it in Discrete Math as modular arithmetic) and the second one is the Klein Four Group.  So which one is the Mattress Group?  Well, in $\mathbb{Z}_4$ there is an element (the number $\bar{1}$) which you can do with itself (ie. add to itself) once, twice, thrice, or four times and each time get a different answer.  But it’s not too hard to see that in the Mattress Group, if you do any operation twice, the mattress is back to where you started!

via the NYTimes

So, by process of elimination, the Mattress Group is the Klein Four Group! (P.S.  Technically, it’s isomorphic to the Klein Four Group.)

You should definitely go to the NYTimes and read the whole article and Dr. Strogatz’s other interesting essay!  For example, check out this one on the Pythagorean Theorem which is closely related to the Primitive Integral Triangles (PITs) that Dr. Miller talked about last month in Math Club.

Thanks to Dr. Ravi Shankar for pointing out the article by Dr. Strogatz.  Dr. Shankar also pointed out that if you have a square mattress or a pillow top, then you get other interesting Mattress Groups.  Which makes us think that Dr. Shankar has a nicer bed than we do!