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Probably everyone has at least seen the Mandelbrot set in some form, as it’s a popular object of mathematical artists. Here’s a picture from Wikipedia: The formal definition is as follo…

It seems everyone’s heard of Pascal’s triangle. However, if you haven’t then it is an infinite triangle of integers with 1’s down each side and the inside numbers determined…

Problem: Optimally pack n unit circles into the smallest possible equilateral triangle. Let L(n) denote the length of the side of the smallest equilateral triangle in which n circles have been pack…

A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example,  1 + 2 + 3 = 6 i…

In 1911, Otto Toeplitz asked the following question. Inscribed Square Problem: Does every plane simple closed curve contain all four vertices of some square? This question, also known as the square…

In 1937 Lothar Collatz proposed the 3n+1 conjecture (known by a long list of aliases), is stated as follows. First, we define the function $f$ on the set of positive integers: If the number \$…

Recall the Fibonacci numbers $F_n$ given by 1,1,2,3,5,8,13,21… There is no need to define them. You all know. Now take the Euler numbers (OEIS) $E_n$ 1,1,1,2,5,16,61,272… Th…

Source: Fibonacci times Euler

On Thursday, the first in a series of public discussions on scientific topics was put on by an organisation called Mass Interaction (the name comes from a statement by Richard Feynman that “a…

Source: The nature of infinity, 2

H. C. Chan, $\pi$ in terms of $\phi$: Some Recent Developments, Proceedings of the Twelfth International Conference in Fibonacci Numbers, (2010): 17-25. Read Pi in terms of Phi (Fib Conf 2006).

H. C. Chan, $\pi$ in terms of $\phi$, Fibonacci Quart. 44 (2006): 141–144. Read Pi in terms of phi.

H. C. Chan, More Formulas for $\pi$, Amer. Math. Monthly 113: 452-455. Read More formulas for Pi.

H. C. Chan, Machin-type formulas expressing $\pi$ in terms of $\phi$, Fibonacci Quart. 46/47 (2008/2009): 32–37 Read Pi via Machin.

This information, for undergraduate students, is hosted on the MAA website.

Professional Mathematical Sciences organizations

In addition to the MAA, there are a variety of special professional mathematical sciences organizations:

The American Mathematical Association of Two-Year Colleges (AMATYC) is the only organization exclusively devoted to providing a national forum for the improvement of the instruction of the mathematics in the first two years of college.

The American Mathematical Society (AMS), through its programs and services, aims to promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and to everyday life.

The American Statistical Association (ASA) is a scientific and educational society founded to promote excellence in the application of statistical science.

The Association for Computing Machinery (ACM) is a professional organization whose main goal is advancing the skills of information technology professionals and students worldwide.

The Association for Women in Mathematics (AWM) is dedicated to encouraging women and girls in the mathematical sciences.

The Computing Research Organization (CRA) is an organization of academic departments of computer science and related fields whose mission is to strengthen research and education in computing, to expand opportunities for women and minorities, and to improve public understanding of the importance of computing.

The Institute for Operations Research and the Management Sciences (INFORMS) is a professional organization that helps practitioners apply scientific tools and methods to improve systems and operations and to assist in managerial decision making.

The National Association of Mathematians (NAM) promotes excellence in the mathematical sciences and the mathematical development of underrepresented minorities.

The National Council of Teachers of Mathematics (NCTM) serves the needs of mathematics teachers at all levels from elementary school teachers through college professors.

The Society for Industrial and Applied Mathematics (SIAM) promotes the development of the mathematical methods needed in a variety of applications areas.

The Society of Actuaries (SOA) has as its mission, to advance actuarial knowledge and to enhance the ability of actuaries to provide expert advice and relevant solutions for financial, business, and societal problems involving uncertain future events.

Mathematics, M.S.

The University of West Florida Department of Mathematics and Statistics is proud to offer a synchronous, fully online masters degree in mathematics using Elluminate web conferencing software. Utilizing this free software, students can participate in graduate courses in real-time on their personal computers. Because of this real-time, interactive instruction, each student receives personal attention just like they would if they attended the class in our main campus in Pensacola, Fl. All the courses are offered after 4:00 p.m. Central Time. Course Schedule.

The Master of Science Mathematics program offers students who hold a bachelor’s degree in mathematics, statistics, or related fields an opportunity to broaden their knowledge in several fields of mathematics, statistics, and their applications. The M.S. Math program is designed for students seeking careers in science, business, industry, or government; for students who want to teach in high schools or at the community college level; or for students who plan to pursue doctoral studies. The M.S. Math program offered by the Department of Mathematics and Statistics permits students considerable flexibility in choosing courses. For example, students who want to seeking careers in financial/investment industries, banks, insurance companies, or government may choose more statistics courses that emphasize the use, adoption, and development of statistical methods and state-of-the-art computer technology in the analysis of data from problems in all fields of study. Go to web site for more information.

I just purchased two tickets to see The Dark Knight Rises! Considering my wife and I have not been to a movie theater since our daughter was born 16 months ago, we are both very excited to see this movie on the big screen!

Now for some fun math. Here is what you get when you type bat-insignia in WolframAlpha:

To generate the bat-insignia in Sage use the following code:
 x,y = var('x,y') f1 = ((x/7)^2*sqrt((abs(abs(x)-3))/(abs(x)-3))+(y/3)^2*sqrt((abs(y+3*sqrt(33)/7)/(y+3*sqrt(33)/7)))-1) f2 = abs(x/2)-(3*sqrt(33)-7)/112*x^2-3+sqrt(1-(abs(abs(x)-2)-1)^2)-y f3 = 9*sqrt(abs((abs(x)-1)*(abs(x)-0.75))/((1-abs(x))*(abs(x)-0.75)))-8*abs(x)-y f4 = -y+3*abs(x)+0.75*sqrt(abs((abs(x)-0.75)*(abs(x)-0.5))/(-(abs(x)-0.75)*(abs(x)-0.5))) f5 = 2.25*sqrt(abs((x-0.5)*(x+0.5))/(-(x-0.5)*(x+0.5)))-y f6 = 6*sqrt(10)/7+(1.5-0.5*abs(x))*sqrt(abs(abs(x)-1)/(abs(x)-1))-6*sqrt(10)/14*sqrt(4-(abs(x)-1)^2)-y f=[f1,f2,f3,f4,f5,f6] num = 2000 sum([implicit_plot(g,(x,-8,8),(y,-5,5),plot_points=num)for g in f]) 

Taking free online courses is gaining momentum. There are many options to choose from: Khan Academy, Education – YouTube, and Udacity. Some come with ivy league and other top universities backing such as MIT’s and Harvard’s edX (a Joint venture builds on MITx and Harvard distance learning), and Coursera which offers host courses from Princeton University, Stanford University, University of California, Berkeley, University of Michigan-Ann Arbor, and University of Pennsylvania.

The world is watching and people have welcomed this movement with open arms! For instance, Udacity’s (founded by three roboticists) first class, “Introduction to Artificial Intelligence,” over 160,000 students in more than 190 countries enrolled! They aim to continue this trend by teaching 160,000 plus students statistics with it’s latest course: “Intro to Statistics (ST101) Making Decisions Based on Data.” This class is being taught by leading computer scientist and Udacity co-founder Sebastian Thurn a PhD in computer science and statistics.

Khan Academy is not-for-profit with the goal of changing education for the better by providing a free world-class education to anyone anywhere.

Coursera is a social entrepreneurship company that partners with the top universities in the world to offer courses online for anyone to take, for free.

Udacity is a private educational organization that believes university-level education can be both high quality and low cost (For some courses, you will have an opportunity to take a certified exam, which will have some cost.)

I recently experienced three consecutive days that are major anniversaries for me: June 12 is my baptismal anniversary, June 13 is my birthday, and June 14 is my wedding anniversary (also my daughter turned 15 months old on this day). These events put to the forefront of my mind a famous problem in probability, the birthday problem.

— The Uniform Probability Model for a Finite Sample Space —

There are many different probability models. We’ll start with the simplest of these models, the uniform probability model for a finite sample space.

Let ${S}$ be a finite sample space. The uniform probability model ${P}$ for ${S}$ is defined by

$\displaystyle P(E)=\frac{|E|}{|S|}.$

${P}$ as defined satisfies the axioms for a probability measure. By adopting this model, we are predicting that in a sequence of experiments with sample space ${S}$ the observed relative frequency of events${E}$ in the sequence of experiments will be equal to the relative frequency, ${|E|/|S|}$, of events ${E}$ in the set ${S}$. In particular, for each outcome ${s\in S}$, we have ${P(\{s\})=1/|S|}$ , or as one says, “All outcomes in ${S}$ are equally likely (i.e., equally probable)." Incidentally, for events ${E=\{s\}}$ consisting of a single outcome are sometimes called elementary events.

Problem 1 (The birthday problem) In a room of ${n}$ people, what is the probability ${p_{n}}$ that at least two have the same birthday (i.e., same month and day of the year)? Find the smallest ${n}$ such that ${p_{n}\geq\frac{1}{2}}$. Neglect February 29 and assume a year of ${365}$ days.

Solution: Let ${S}$ be the sample space that consists of ${365^{n}}$ ${n-}$tuples ${(b_{1},b_{2},\ldots,b_{n})}$, where ${b_{i}}$ is birthday of the ${i^{th}}$ person in the room. Note, that the probability is

$\displaystyle \begin{array}{rcl} p_{n} & = & 1-P(\mbox{all n individuals have different birthdays})\\ & = & 1-\frac{365{}^{\underline{n}}}{365^{n}}, \end{array}$

assuming all ${n-}$tuples ${(b_{1},b_{2},\ldots,b_{n})}$ are equally likely. Below are some values of $p_{n}$:

$\begin{array}{|c|c|} p & p_{n} \\ \hline 5 & 0.027 \\ 10 & 0.117 \\ 20 & 0.411 \\ 23 & 0.507 \\ 30 & 0.706 \\ 40 & 0.891 \\ 60 & 0.994 \\ \hline \end{array}$

Remarkably, in a room with just ${23}$ people, the probability is greater than ${1/2}$ that two or more people have the same birthday! $\Box$

— Binomial Random Variables and Poisson Approximations —

In the case of a binomial experiment there are ${n}$ independent trials, with the results of each trial labeled “success" or “failure." On each trial the probability of success is ${p}$ and (hence) the probability of failure is ${1-p}$. The sample space ${S}$ consists of the set of ${2^{n}}$ words in the alphabet ${\{s,f\}}$ and the random variable ${X}$ on ${S}$ is defined by ${X}$ (each word) ${=}$ the number of ${s}$‘s in that word. The probablity density function (pdf) ${f_{X}}$ for ${k=0,1,\ldots,n}$ is given by

$\displaystyle f_{X}(k)=P(X=k)=\binom{n}{k}p^{k}(1-p)^{n-k}.$

We say that such a random variable ${X}$ is a binomial random variable with parameters ${n}$ and ${p}$, abbreviating this with the notation ${X\sim binomial(n,p)}$. We also say that X has a binomial distribution with parameters ${n}$ and ${p}$, and that ${f_{X}}$ is a binomial density function with parameters ${n}$ and ${p}$. Note, in general, if ${X\sim binomial(n,p)}$, then ${P(X\geq1)=\sum_{k=1}^{n}P(X=k)=1-P(X=0)=(1-p)^{n}}$.

If ${X\sim binomial(n,p)}$, where ${n}$ is “large" and ${p}$ is “small," then for ${k=0,\ldots,n}$

$\displaystyle P(X=k)\approx\frac{\lambda^{k}}{k!}e^{-\lambda},$

where ${\lambda=np}$. This is called the Poisson approximation (Denis Poisson, 1781-1840) to the binomial distribution.

Remark 1 It can in fact be shown that the accuracy of this approximation depends largely on the value of ${p}$, and hardly at all on the value of ${n}$. The errors in using this approximation are of the same order of magnitude as ${p}$, roughly speaking.

Problem 2 Given ${400}$ people, estimate the probability that 3 or more will have a birthday on June 13.

Solution: Assuming a year of ${365}$ days, each equally likely to be the birthday of a randomly chosen individual, if ${X}$ denotes the number of people with a birthday on June 13 among ${400}$ randomly chosen individuals, then ${X\sim binomial(400,\frac{1}{365})}$. The exact answer to this question

$\displaystyle \begin{array}{rcl} P(X\geq3)& = & 1-P(X\leq2)\\ & = & 1-\sum_{k=0}^{2}\binom{400}{k}\left(\frac{1}{365}\right)^{k}\left(\frac{364}{365}\right){}^{400-k}\\ & = & 0.09850825486213655 \end{array}$

The Poisson approximation of this quantity is

$\displaystyle \begin{array}{rcl} 1-\sum_{k=0}^{2}\frac{(400/365)^{k}}{k!}e^{-400/365} & = & 1-e^{-1.096}(1+1.096+\frac{(1.096)^{2}}{2})\\ & \approx & 0.099. \end{array}$ $\Box$

In the television show Futurama, episode “The Prisoner of Benda,” Dr. Farnsworth created a machine that swaps the minds of two individuals. Unfortunately, they discover that the machine cannot be used on the same pair of bodies again.

The question is, given a collection of jumbled up minds and bodies, how can you get them all back to normal? Well, they have to come up with some equation to prove that, with enough people switching, eventually everyone will end up in their rightful form.

It was first mentioned in an interview that head writer and executive producer David X. Cohen gave to the American Physical Society:

“In an APS News exclusive, Cohen reveals for the first time that in the 10th episode of the upcoming season, tentatively entitled “The Prisoner of Benda,” a theorem based on group theory was specifically written (and proven!) by staffer/PhD mathematician Ken Keeler to explain a plot twist. Cohen can’t help but chuckle at the irony: his television-writing rule is that entertainment trumps science, but in this special case, a mathematical theorem was penned for the sake of entertainment.”

This theorem is now known as Keeler’s Theorem, or the “Futurama Theorem”.

The Futurama Theorem proves that no matter the size of the collection or how many mind swaps have been made, all minds can be restored to their original bodies using only two extra “clean” bodies (individuals who haven’t swapped minds with anyone) are needed, since a collection of just two people with swapped minds could never get back to normal.

If you want to know more about the details on group theory (group, the symmetric group, permutation diagrams, transpositions, cycle notation, products of disjoint cycles, products of distinct transpositions, etc.) read the talk given by Dana C. Ernst about the Futurama Theorem. He has blogged twice on this talk:
1. Talk about the Futurama Theorem
2. Another talk about the Futurama Theorem

and he has given the talk three times:
1. November 3, 2011 Gordon Talk
2. December 7, 2011 PSU Talk
3. February 28, 2012 UNO Talk

I personally like the first talk. He mentions:

“The slides for the second talk are very similar to the first set, but there are a few differences:

• The second talk is shorter. I trimmed a few things from the first talk that were not completely necessary.
• I’ve improved the wording in a few spots.
• In the second talk, multiplication of permutations is right to left.”

Talks 2 and 3 are almost identical.

If you read the talk, you will understand the proof that appeared in the show,

which uses nothing other than basic facts about group theory and permutations. Furthermore, you will understand applying the algorithm to the arrangement of minds in Futurama requires 13 moves. However, since Fry and Zoiberg only swapped with each other and since there is only one other cycle, we could use Fry and Zoiberg as $x$ and $y$, respectively in the algorithm and thus it only requires 9 moves.

Below are the implementations of the 13 move algorithm in Sage and WolframAlpha. (Note, Sage and WolframAlpha will multiply cycles left to right):
 S_11 = SymmetricGroup(11) c7 = S_11("(1,2,3,4,5,6,7)") c1 = S_11("(8,9)") t1 = S_11("(8,10)") t2 = S_11("(9,11)") t3 = S_11("(9,10)") t4 = S_11("(8,11)") t5 = S_11("(10,1)") t6 = S_11("(11,7)") t7 = S_11("(11,6)") t8 = S_11("(11,5)") t9 = S_11("(11,4)") t10 = S_11("(11,3)") t11 = S_11("(11,2)") t12 = S_11("(11,1)") t13 = S_11("(10,7)")

 

c1*t1*t2*t3*t4*c7*t5*t6*t7*t8*t9*t10*t11*t12*t13 

Click here to view the 13 move algorithm for the Futurama permutation solution in WolframAlpha

Here are some questions posed by Dana C. Ernst:
1. Can we do better (i.e., fewer moves to fix the cycles)?
2. Under what circumstances do we not need to introduce two “clean” people?
3. Would only using one “clean” person ever be useful?
4. Could you use fewer moves if you allowed more than two “clean” people?

We now know the answer to some these questions. Students from the University of California, San Diego have improved the result, giving a better algorithm for finding the minimum number of switches to put everyone’s head back in the right places, give optimal solutions for two particular situations, and give necessary and sufficient conditions for it being possible to represent the identity permutation as $m$ distinct transpositions in $S_n$.

Here is the paper submitted on April 26, 2012.