A Southwest Turn

Hurricanes are well-known for how unpredictable their paths can be. As wild as they can get, we can usually count on two things for storms that live primarily in the ocean in the Northern hemisphere: their general hook shape, and sharp bends.

source: https://en.wikipedia.org/wiki/File:2018_Atlantic_hurricane_season_summary_map.png

2018 Atlantic Hurricane Season

As has been frequently reported in the news, Florence took a particularly strange turn when it headed southwest. To see how that came about, we can look at the Cauchy momentum equation, one of the Navier Stokes equations that are central to fluid dynamics:

The Cauchy momentum equation

The equation itself can take on several forms, depending on the usefulness for the particular application. For our purposes, however, the above is sufficient since we have the relevant terms. In determining the path for Florence, two out of three of the terms on the right side of the equal sign are important.

The first is the Coriolis force, usually lumped in with other forces. This is the force most famous for causing the spiral pattern of the storms, but, at the largest scale, is also why we have the trade winds and westerlies. This gives us the hook.

The second term is the change in pressure over space. The negative sign simply means that air and liquid prefer to move from areas of high pressure to areas of low pressure. If the pressure is high enough, as it was over New England and the Maritimes during Florence’s landfall, the path can acquire a sharp bend.

Put these two competing terms together, and we get Florence’s odd path.

map source: https://www.flickr.com/photos/internetarchivebookimages/19794489843/

Pressure field during Florence’s landfall.

Peer edited by Gabby Budziewski.

Follow us on social media and never miss an article:

Girls Talk Math – Engaging Girls through Math Media

Girls Talk Math is a non-traditional math camp in that students not only learn challenging Mathematics usually not encountered until college, but also research the life of female mathematicians who have worked on related topics. Campers share what they learned during the two-week day camp by writing a blog post about their Math topic and writing and recording a podcast about the mathematician they researched. Media created by the campers can be found on our website at www.girlstalkmath.web.unc.edu.

Girls Talk Math was founded in 2016 by Francesca Bernardi and Katrina Morgan, then Ph.D. candidates in Mathematics at UNC Chapel Hill. It was born of a desire to create a space for high-school students identifying as female or from an underrepresented gender who are interested in Mathematics. This summer, a sister camp at the University of Maryland at College Park had its first run thanks to Sarah Burnett and Cara Peters, Ph.D. candidates in Mathematics at UMD (www.gtm.math.umd.edu).

During two weeks of July 2018, 39 high schoolers came to the UNC Mathematics Department to participate in the 3rd year of Girls Talk Math. They were divided into groups of 4-5 campers, and each group completed a problem set focused on a different Math topic:

-Number Systems
-Network Science
-RSA Encryption Cryptography
-Elliptic Curve Cryptography
-Mathematical Epidemiology
-Quantum Mechanics
-Knot Theory
-Classification of Surfaces

Each group then wrote a blog post to share what they learned about their topic. Below are excerpts from each post written by the campers, and you can read the full blog posts here. 

 

Number Systems
Miranda Copenhaver, Nancy Hindman, Efiotu Jagun, and Gloria Su

The Number Systems problem set focuses on learning about number bases (in particular, base 2 and 16) to understand how data is stored in computers and how to translate information into a language readable by machines. This problem set included coding in Python.

“[…]  We count in the decimal – or base 10 – system. This means that we count using 10 basic numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In base 10, each place value represents a power of 10.”

“[…]  The two most important bases in coding are binary (base two) and hexadecimal (base sixteen). Binary is quite simple to think about because it only has two numbers that you could possibly use: 1 and 0.”

“[…]  If we had the number 1101010, we would start by labeling each place value with what power of two it represents. Next, we would multiply each digit by its power of two and simplify:                           

                       1101010 = 1*26 + 1*25 + 0*24 + 1*23 + 0*22 + 1*21 + 0*20

                                       = 26 + 25 + 23 + 21

                                       = 64 + 32 + 8 + 2

                                       = 106

We see that the binary number 1101010 is the decimal number 106.”

 

Network Science
Myla James, Shania Johnson, Maya Mukerjee, and Savitha Saminathan

The Network Science problem set focuses on graph theory and how it is utilized for large data sets. Students learned about data storage in networks and how to analyze and study different data sets. This problem set included coding in Python.

“[…]  We were given a map of the city of Königsberg, Prussia that helped us learn about paths and circuits. Euler Paths and Circuits were named after Leonhard Euler, who asked the question: “Is there some route in this city wherein one would cross each of the seven bridges once and only once?”

“An Euler path must include two or less odd degree vertices. […]  In simplest form, an Euler path is a set of edges that is connected, and an Euler circuit is a set of edges that is connected and begins and ends at the same node. An analogy would be an electrical circuit. Electricity can flow in a closed circuit, but not an open path.”

 

RSA Encryption Cryptography
Camille Clark, Layke Jones, Isabella Lane, Aza McFadden, and Lizbeth Otero

The RSA Cryptography problem set introduces the field of Number Theory through modular arithmetic, prime numbers, and prime factorization. RSA cryptography is one of the most widespread methods to transmit codified information and has several applications in everyday technology.

“[…]  A common divisor is an integer that all the numbers in a given set can be divided into without a remainder. To calculate the of 2 numbers, you need to write out the prime factorization. (Camp directors’ note: the greatest common divisor (gcd) of two or more integers, not all zero, is the largest positive integer that divides each of the integers without a remainder.)

For example, let’s consider 8 and 12. The prime factorization of 8 is 23, while the prime factorization of 12 is 3*(22). Then, take the largest factor that overlaps in the two factorizations. Here, 22 is the largest factor in common between the prime factorizations of 8 and 12; then, 4 = gcd(8,12).

We say that two integers a and b are if gcd(a,b) = 1, where a and b don’t need to be prime themselves. For example, if a = 35 and b = 8, then gcd(a,b) = 1, but neither is a prime.”

 

Elliptic Curve Cryptography
Mukta Dharmapurikar, Anagha Jandhyala, Savanna Jones, and Ciara Renaud

In the Elliptic Curve Cryptography (ECC) problem set students learn how to apply this fascinating method of encoding, transmitting, and deciphering information. Elliptic Curve Cryptography is an interesting application of very theoretical concepts from Algebraic Geometry and Abstract Algebra.

“[…]  While the road to understanding Elliptic Curve cryptography was interesting and exciting, there were many twists and turns along the way. Our greatest challenge was that ECC is extremely hard to conceptualize as most of the math differed from our previous understandings and was often very theoretical or abstract.

However, we thoroughly enjoyed learning about topics in math, typically not discussed in school. For example, on the first day, we were learning about modular arithmetic. It was a difficult concept to grasp because it was fundamentally different than what we had learned before. Over time, just by working through the problem set, we became more and more comfortable with the topic, even going as far as being able to explain how it works to other people.

This goes to show, that even when faced with a very difficult problem set, if you keep persevering, eventually you will understand the math. Girls Talk Math has really taught us to never give up, and increased our confidence in learning higher level math.”

 

Mathematical Epidemiology
Camilla Fratta, Ananya Jain, Sydney Mason, Gabby Matejowsky, and Nevaeh Pinkney

The Mathematical Epidemiology problem set introduces the concept of modeling as a whole and in particular focuses on modeling disease spreading in populations. In this problem set campers have used an applet in Python.

“[…]  A mathematical model is an equation used to predict or model the most likely results to occur in a real-world situation. We used these types of equations to model the spread of a disease in a population, tracking the flow of populations from susceptible to infected to recovered. In real life scenarios, there are too many variables to fully account for, so we only were able to place a few in our equations. This made the models less accurate, but at the same time very useful to us in our problem set. They gave us a good idea of how things worked in an actual epidemic and helped us to understand what mathematical modeling really is.”

 

Quantum Mechanics
Izzy Cox, Divya Iyer, Wgoud Mansour, Ashleigh Sico, and Elizabeth Whetzel

The Quantum Mechanics problem set starts by explaining why classical mechanics does not describe properly the behavior of subatomic particles. It then introduces the main concepts of quantum mechanics, in particular focusing on the wave-particle duality, i.e. the fact that mass can be described as both a particle and a wave. As part of their problem set, campers ran a physical experiment to measure Planck’s Constant.

“[…] Quantum Mechanics is the physics of molecular and microscopic particles. However, it has applications in everyday life as well. If someone asked you if a human was a particle or a wave, what would you think? What about a ball? What about light? Not so easy now, is it? It turns out that all of those things, and in fact, everything around us, can be expressed in physics as both a particle and a wave.”

“[…] The realm of physics gets much stranger when it gets smaller! […]  [Quantum mechanics] is arguably one of the most complicated fields of physics, where all traditional rules are wrong. There is much still being added, and so much more to be discovered.”

 

Knot Theory
Jillian Byrnes, Monique Dacanay, Kaycee DeArmey, Alana Drumgold, Ariyana Smith, and Wisdom Talley

The Knot Theory problem set discusses the fascinating field of Abstract Geometry that deals with knots. Maybe surprisingly, there is a Mathematical theory behind tying and untying knots which can be described formally with algebraic symbols. This problem set is a Mathematical approach to knots and how to study and classify them.

“[…]  The Reidemeister moves are the three possible manipulations of knots that are used to find out if two diagrams represent equivalent knots. None of them physically change the knot, because they don’t make any cuts or make the knot intersect itself, so if the two diagrams are equivalent, they are related through a sequence of these three moves:”

Three examples of Reidemeister moves: I, II, III

 

Classification of Surfaces
Ayanna Blake, Lisa Oommen, Myla Marve, Tamarr Moore, Caylah Vickers, and Lily Zeng

The Classification of Surfaces problem set deals with questions of shape, size, and the properties of space. Starting from a Mathematical definition of surfaces, students learn about aspects of a number of shapes, some of which they are already familiar with and some that do not exist in 3-dimensional space, with the aim of classifying them.

“[…]  Before we start off with explaining the basics, we give the definition of a surface, which is an example of a two-dimensional object. When talking about dimensions, basically it’s a way of classifying how many directions of travel an object has.

For example, a line on a piece of paper would be one dimensional because you can only go up or down on that line. A sheet of paper would be two dimensional because you can draw up or down and side to side. A room would be three dimensional because if you imagine throwing a ball in the air, it can move up or down, side to side, and forwards or backwards.”

 

* Girls Talk Math has been funded through the Mathematical Association of America Tensor Women and Mathematics grant which has supported the camp for the past three years. *

 

Peer edited by Rachel Cherney.

Follow us on social media and never miss an article:

Can’t Decide? Use Math!

   

Please help me decide, the hanger approaches.

Too many choices!

I want to present a situation I occasionally find myself in: while visiting a city and looking for dinner, I try my best to find an outstanding place to eat. To help make this decision, I usually turn to my phone and read restaurant reviews and menus online for 30 minutes hoping to find the perfect restaurant. As my hunger continues to grow, I think about all the restaurants I’ve looked at and settle on a restaurant that seems good enough. On my way to the restaurant, I second guess my choice, all the while fighting the hunger nipping at my stomach. After experiencing this restaurant conundrum too many times, I was left wishing there were a more efficient way to handle these choices.

Luckily, within mathematics there exists a theory, named optimal stopping, which has a technique that can be elegantly applied to this problem. Optimal stopping theory presents various methods to choose the best option out of a large number of choices or given a limited amount of time. Applying a technique from optimal stopping theory to my restaurant problem can guide me to the restaurant of my dreams.

Here’s how to handle this restaurant struggle using optimal stopping theory. Consider the same 30 minutes that I usually spend looking for the best restaurant. I take 37% of that time, about 11 minutes, and look at reviews and menus just as I have done many times before. This time, I spend these first 11 minutes only looking at restaurant choices and noting the most appealing choice. Once time is up, I continue scrolling through menus and reviews, remembering my previous tasty sounding choice, until I find a restaurant that sounds even better. This new, better restaurant is where I choose to go. I can now enjoy a meal, confident that I made a good choice and spared myself time before hanger sets in.

Where does this 37% number come from and what makes this technique so optimal? According to optimal stopping theory, when given a large number of options sampling the initial 37% gives sufficient perspective to find the best overall option. When using optimal stopping, as the number of choices increases, the probability picking the absolute best option closes in on 36.8%.

Finding the best option 36.8% of the time doesn’t sound that spectacular, but consider interviewing thousands of applicants for a job position that must be turned down or accepted on the spot. Using optimal stopping, the absolute best choice can be found 36.8% of the time regardless of the choices numbering in the twenties or twenty-thousands!

Optimal stopping is not limited to picking a place to eat; it can be applied to many everyday situations. What if you want to spend as little effort as possible online dating, but still want to find your soulmate? Optimal stopping theory to the rescue! Facing a staggering number of porta potties and want to touch as few handles while finding the cleanest toilet possible, optimal stopping theory will save the day. Trying to get the best deal selling your brother’s beanie babies on Craigslist? Optimal stopping can help you out.

So how can you find the king porta pottie, or the match of your dreams in the most efficient way? When presented a set of choices take 37% of the total number of choices, and skip past this first 37%. Importantly, you must keep in mind the best option from these choices. As you look over the rest of the choices, as soon as you find something better than what was in that first 37% that is your best choice.

This article was directly inspired by Numberphile and the podcast Note to Self, both of which are fantastic sources for how math can be applied to everyday problems.

Peer edited by Christina Parker. 

Follow us on social media and never miss an article: