Mathematics that Swings: The Math Behind Golf

From the YouTube description:

Mathematics is everywhere, and the golf course is no exception. Many aspects of the game of golf can be illuminated or improved through mathematical modeling and analysis. We will discuss a few examples, employing mathematics ranging from simple high school algebra to computational techniques at the frontiers of contemporary research.

Preparation for Industrial Careers in the Mathematical Sciences: Building a Better Filter

The Mathematical Association of America recently published a number of promotional videos showing various mathematics can be used in “the real world.” Here’s the third pair of videos describing how mathematics is used for certain problems in materials science. From the YouTube descriptions:

Dr. Sumanth Swaminathan of W. L. Gore & Associates talks about his career path and the research questions about filtration that he considers. He works to understand the different waste capture mechanisms of filtration devices and to mathematically optimize the microstructure to create better filters.

Prof. Louis Rossi of the Department of Mathematical Sciences of the University of Delaware presents two introductory mathematical models that one can use to understand and characterize filters and the filtration processes.

Mathematics of Juggling

Quite accidentally, I recently stumbled on the following video. The speaker is Allen Knutson, who is a Professor of Mathematics at Cornell. Hope you enjoy it.

Useless Numerology for 2016: Part 1

The following entertaining (but useless) facts about the number 2,016 appeared in a recent Facebook post (and subsequent comments) by the American Mathematical Monthly.

$2016 = 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3$

$2016 = 2^{10} + 2^9 + 2^8 + 2^7 + 2^6 + 2^5$

$2016 = 1+2+3 + \dots + 62 + 63$

$2016 = \displaystyle \sum_{n=0}^{63} (-1)^{n+1} n^2$

$2016 = (1+2+...+8+9)^2 - (1+2)^2$

$(2 + 0 + 1)! = 6$

$2016 = 2^{11} - 2^5$

$2016 = 2016 \times 1$

The last tongue-in-check equation is my favorite.

In this series, I’ll explain why these different expressions for $2016$ have to be equal to each other. I’ll begin with tomorrow’s post.

The Wrong Door, Or Why Math Gets a Bad Rap

I really enjoyed this news article on how to motivate young students to enjoy mathematics. Surprise, surprise: it isn’t by doing a whole bunch of rote arithmetic or algebra problems.

http://www.maa.org/news/the-wrong-door-or-why-math-gets-a-bad-rap

Predicate Logic and Popular Culture (Part 31): The Godfather

Let $p$ be the proposition “I took the gun,” and let $q$ be the proposition “I took the cannoli.” Translate the logical statement

$\lnot p \land q$ .

Obviously, this is an allusion to one of the great lines in The Godfather.

Even though this is a simple example, it actually serves a pedagogical purpose (when I first introduce students to propositional logic) by illustrating two important points.

First, there is an order of precedence with $\lnot$ and $\land$ . Specifically, $\lnot p \land q$ means $(\lnot p) \land q$ (“I did not take the gun, and I took the cannoli”) and not $\lnot (p \land q)$ (“It is false that I took both the gun and the cannoli”).

Second, the actual line from The Godfather is not a proposition because both “Leave the gun” and “Take the cannoli” are commands. By contrast, a proposition must be a declarative sentence that is either true or false. That’s why I had to slightly modify the words to “I took the cannoli” instead of “Take the cannoli.”

Context: This semester, I taught discrete mathematics for the first time. Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

I’m afraid that I found plenty more examples from popular culture to illustrate predicate logic, but a month of posts on this topic is probably enough for now. I’ll return to this topic again at some point in the future.

Predicate Logic and Popular Culture (Part 30): The Platters

Let $R(x)$ be the proposition “ $x$ can make all this the world seem right,” and let $B(x)$ “ $x$ can make the darkness bright.” Translate the logical statement

$R(\hbox{you}) \land B(\hbox{you}) \land \forall x(x \ne \hbox{you} \Longrightarrow \lnot (R(x) \lor B(x)))$ ,

where the domain is all people.

The clunky way of translating this into English is, “You can make all this world seem right, you can make the darkness bright, and everyone else can neither make all this world seem right nor make the darkness bright.” Of course, this is the sentiment expressed by the first two lines of this classic by the Platters.

Predicate Logic and Popular Culture (Part 29): Grease

Let $W(x)$ be the proposition “I want $x$ .” Translate the logical statement

$W(\hbox{you}) \land \forall x(x \ne \hbox{you} \Longrightarrow \lnot W(x))$ ,

where the domain is all people.

The clunky way of translating this into English is, “I want you, and I don’t want anyone who isn’t you.” But it sounds a lot better when John Travolta and Olivia Newton-John sing it.

For professional mathematicians (as opposed to students first learning predicate logic), the more compact way of writing this would be

$W(\hbox{you}) \land \exists! x W(x)$ .

Predicate Logic and Popular Culture (Part 28): High School Musical

Let $L(x)$ be the proposition “ $x$ is a star in heaven” and let $R(x)$ be the proposition “We can reach $x$ ”

$\lnot \exists x(\lnot R(x))$ ,

where the domain for $x$ is the stars in heaven.

The clunky way of translating this into English is, “There is not a star in heaven that we cannot reach,” and this double negative appears in the song Breaking Free from High School Musical.

This example gives students a simple practice problem for using De Morgan’s laws to eliminate the double negative:

$\lnot \exists x(\lnot R(x)) \equiv \forall x(\lnot(\lnot R(x))) \equiv \forall x R(x)$ ,

or “We can reach every star in heaven.”

Predicate Logic and Popular Culture (Part 27): Les Miserables

Let $K(y)$ be the proposition “I know place $y$ ,” let $L(x,y)$ be the proposition “ $x$ is lost at place $y$ ,” and let $C(x,y)$ be the proposition “ $x$ cries at place $y$ .” Translate the logical statement

$\exists y(K(y) \land \lnot \exists x(L(x,y) \lor C(x,y)))$ ,

where the domain for $x$ is all people and the domain for $y$ is all places.

The clunky way of translating this into English is, “There exists a place that I know so that it is false that there is a person at this place who is lost or who cries.” This is the innocent childish dream of Cosette in Les Miserables as she suffers under the Thenardiers.