jackie

applied mathematician in training // AHHHHHpplied math because math makes me want to scream (out of excitement)

[ahhhhhpplied-math]

goodbye, and see you again.

Hello, the person reading this post.

Goodbyes are hard, aren't they. I used to hate saying goodbye. Maybe this goodbye doesn't mean much to you, but it means a lot to me.

When I started this blog, I entered a new chapter of my life. It was the summer of first year. I was full of hope; I believed in myself and others. I was full of love. Lots had happened since then. I don't want to say I stopped believing in myself, nor I stopped believing in people, but there were a lot of doubts, a lot of hurting, and a lot of healing. The silver lining is that I have matured a lot. I am wiser than second-year Jackie, more secured than second-year Jackie, and more established than second-year Jackie. The person writing this post now, is not the same person who started this blog, nor is the one writing the last post. So, I have decided so leave this blog behind, and close this chapter of my life.

Second-year Jackie, you have done a good job. I'm proud of the person I am today, and that is for the decisions you have made along the way. You have met amazing people and made great friends, but you also met not-so-amazing people who hurt you, lied to you, and made you doubt yourself. It is not your fault, and I hope to continue to have empathy and compassion for those people, just like you do. Of course, you also did not-so-amazing things to people. You weren't a good friend, lied to people, and hurt others. But you learned, and you grew. I am proud of you. I now have a great group of friends, who care for me and love me, thanks to you.

So, goodbye second-year Jackie, and goodbye to you, the reader. I hope to cross paths with you again, when you and I both have grown into better people. You might even see my name on a research paper!

Thank you for reading, and I wish you good luck in the future.

Jackie

[ahhhhhpplied-math]

Mathematics in Movies: Good Will Hunting - An Introduction to Graph Theory

I wrote this piece in the beginning of summer, but I never got around to posting it until now. Lots had happened since this post, so look out for another post relatively soon.

Why Am I Watching a 22 year-old Movie?

School has ended for me, even though it doesn’t feel like it. My quarantine routine consists of waking up at noon, working on my research project, and escaping into the wonderful world of Animal Crossing. To be completely honest, being in isolation is taking a toll on my productivity; math recently feels like a chore. I needed to feel motivated again, so of course, like a true mathematician, I watched Good Will Hunting for the 100th time. But this time was different. I could finally understood the “impossible” problems posed in the movie.

If you haven’t watched the movie yet, what are you doing with your life? You uncultured swine. I still remember the first time someone mentioned the movie to me. I was sorting books in alphabetical order as a way to earn my volunteer hours at my high school’s library, when the librarian approached me for a chat. She recommended the movie to me, and I thought it was so lame. A MATH movie. Gross. At that point, I was determined to pursue business school and work in marketing. Over the course of my undergraduate career, Good Will Hunting evolved from an incredible movie about insecurities, love, and uncertainty to a movie that I put on whenever I have imposter syndrome. The movie reminds me of the excitement I feel when I sit down, pick up a pencil, and start writing the solution to a math problem. It reminds me that it is a privilege to do math.

Now, enough rambling. Let’s get to it.

A VERY brief Recap for The Uncultured People (Watch the movie!!) 

Will (Matt Damon), a genius with a dark past, works as a janitor at M.I.T. After a lecture, Professor Lambeau (Stellan Skarsgård) posted an “impossible” math problem on the blackboard in the hallway, which Will solved within days. There are 2 problems posted in total, but in this post I will talk about the first one.

The Math: A Crash Course on Graph Theory

There are many resources on the solutions to these “impossible” problems that you can find online. My favourites are: The Medium and Harvard Mathematics. Many of these concepts are introduced in introductory discrete mathematics courses.

My research project this summer is about graph theory, network theory, and random matrix theory, so you can imagine my excitement when I saw the term “adjacency matrix” in the movie.

The Future

I will be going into my fourth year of undergrad in the fall of 2020. With the pandemic going on, the future seems so uncertain and fragile. I have been stressing out about graduate school application; I updated my CV 5 times already and drafted my letter of intent even though it is only May. So, instead of being afraid of the future, I’m going to celebrate the past. 

Because Will’s talent was discovered because he was doing math on a blackboard, I will share some of my favourite pictures of black( and white ) boards on my phone. 

For the people out there who happened to read this post, send me some of your favourite pictures of black/white boards. (If you don’t have any, bro do you even do math? jkjk)

References

Algebraic Graph Theory, by Chris Godsil and Gordon Royle

Graph Spectra for Complex Networks, by Piet Van Mieghem

[prokopetz]

One of the striking things about math education for me is that most of the common objections to how the material is taught have really simple answers, but I have never in my life heard a math instructor provide those answers.

For example, something you hear a lot is: “why am I losing points for not showing my work when I got the correct answer?”, or even “why are we being told to use this procedure at all when the answers are so obvious?”.

There answer to both of those questions, of course, is: “Because what’s actually being taught is a problem-solving method that works for big and complicated problems as well as small and simple ones. We practice it with the simple ones first so that you can easily compare your intuitive solution with the results of applying the method and know whether you did it right. That way, when we get to the complicated ones where the intuitive approach doesn’t work, you can have confidence that you practised the method correctly.”

Not once in two decades of schooling did I hear that rationale offered – if an instructor deigned to address the objections at all, their response typically boiled down to some variation of “because this is how it’s done”.

Like, what’s difficult about this?

[ahhhhhpplied-math]

Have you ever wondered what do the eigenvalues of a million binary matrices look like on the complex plane?

Me neither.

[ahhhhhpplied-math]

What Do A Soccer Team and Network Theory Have In Common?

Introduction

Firstly, I need to explain the context of this vague question. February this year, two of my friends asked me if I would like to join their team for a math modelling competition, but then proceeded to pulled out last-minute, forcing me to do this competition with 2 strangers. Am I still cheesed? Maybe. But my teammates turned out to be amazing, so I'm happy that it worked out this way. After working non-stop for 3 days straight with little sleep, we were able to come up with a network theory approach to a data analysis problem.

I'm writing this post to organize my thoughts and explain how the connection between a .csv file and network theory was made.

The Problem

Let's say the coach of a soccer team with 30 players hands you a GIANT .csv file. This file contains EVERYTHING (passes, scores, positions of the players, when the passes were made, etc.) that happened in 38 games of the team. If you want to study how closely the players work together, how should you analyse the data given? I expect very few people would say "Network Theory". One of the cons of this approach is the fact that most of the information given must be ingnored, such as positions of the players and the time of the passes. This approach only considers who passed to who. Then, we can ask questions such as:

Who receives the most passes?

Is there a group of players that are getting all of the passes?

Definitions

Give these pages a read if you want a more in-depth explanation!

Adjacency Matrix: Wiki

Graph Theory: Wiki

Eigenvector Centrality: Wiki

Degree Distribution: Wiki

Community Graph: Wiki

A very cool paper: Network Theory and Models

The Process

To begin, I want to mention that this project was done entirely using Mathematica. I have said this once, and I will say it again: Mathematica is beautiful. Mathematica is easy to pick up due to its detailed and comprehensive documentation, while being an incredibly poweful tool. So, thank you @wolframalpha.

Back to our soccer team, I loaded the .csv file into my Mathematica notebook, and extracted the relevant data, i.e all of the passes that happened in all of the games. The next step is to count the number of passes between an abitrary pair of players. We shall order the players (in no particular order), and assign them each a number from 1 to 30. A matrix can be made, where the ij entry (row i, column j, where i,j are in between 1 and 30) is the number of passes between player i and player j. From this, we yield this heat map.

We want to look for "redder" entries, as this means this particular pair works together more often. With this information, we are able to construct an adjacency matrix for the team.

Let's try to understand this weird-looking matrix. The entry ij , tells us if there is a pass between player i and player j or not. If this entry is 0, there is NO pass and if this entry is 1, there exists a pass between these 2 players. You must wonder,

"Why are we not considering how many passes were made between them?"

We are getting to it, don't worry, but the reason why we want the binary version of the matrix is because we can create a graph. What the heck is a graph? In our context, each player is a node in our graph, and where it is placed in the picture does not matter. The adjacency matrix tells us if there is a line (an edge, if you want to be formal) in between an abitrary pair of nodes. The number of passes, acts as a weight for each node. From those assumptions, we yield this graph

This adjacency matrix enables us to turn the team into a network, whose characteristics such as the eigenvector centrality and the degree distribution (the degree of a node is the number of connections it receives) can be studied. Community graphs can also be constructed, which highlight the clusters formed within the team.

To Wrap Up

While I think there are more accurate and efficient ways to analyse the data given, this network theory approach shows us a glimpse of how to visually represent something so vague such as "teamwork". Beside, you get prettier graphs this way.

[the-real-numbers-deactivated202]

[ahhhhhpplied-math]

Original Comic

[ahhhhhpplied-math]

What The Fuck is a Pell’s Equation?

This might be the most useless thing you can learn about, but it’s undeniably interesting. Pell’s equation is an equation of the form x^2 - ny^2 = 1 (I'm pissed that I can’t use Latex). In the example above, we discuss the case where n = 2. 

How Do You Solve One?

Sure, you can read all about this equation here, but this post discusses how fucking cool the proof is. I didn’t come up with this btw, because I’m a moron. 

In How to Solve It by George Pólya, one of my favorite books about Mathematics and problem solving in general, one of the claims is that:

If you can’t solve a problem, there is a simpler problem that you can’t solve. Find it.

The Pell’s equation is hard as fuck to solve. The proof above started out by finding a simple solution.The trivial solution x = 1 or x = -1 and y = 0 is easy to notice, but not interesting (just like me). The solution x= 3 and y = 2 would be the next pair of numbers that works if you are just plugging in positive integers. From the analysis attached, we notice a pattern in the pair of solution. So we grab on to it like Rose to that fucking door in the Titanic and never let go. We boldly make a claim (bde) which applies to ALL solution of the Pell’s Equation and prove it using induction.

This is an example of effective problem-solving. We start out with a guess, analyze the results of that guess, notice a pattern, and prove that the pattern applies to every solution to the problem. 

[ahhhhhpplied-math]

im a happy cookie

im an idiot who doesnt check my tumblr but i just want to announce to the world that @the-real-numbers​ followed me back and im so happy.

[ahhhhhpplied-math]

[ahhhhhpplied-math]

Hi I’m not dead

or am I?

[boomeyer]

Do yourself a favor. Learn to code. Here's how.

I’ve said this to my non-techie friends countless times. It’s no secret that being able to code makes you a better job applicant, and a better entrepreneur. Hell, one techie taught a homeless man to code and now that man is making his first mobile application.

Learning to code elevates your professional life, and makes you more knowledgeable about the massive changes taking place in the technology sector that are poised to have an immense influence on human life.

(note: yes I realize that 3/5 of those links were Google projects)

But most folks are intimidated by coding. And it does seem intimidating at first. But peel away the obscurity and the difficulty, and you start to learn that coding, at least at its basic level, is a very manageable, learnable skill.

There are a lot of resources out there to teach you. I’ve found a couple to be particularly successful. Here’s my list of resources for learning to code, sorted by difficulty:

Novice

Never written a line of code before? No worries. Just visit one of these fine resources and follow their high-level tutorials. You won’t get into the nitty-gritty, but don’t worry about it for now:

Dash - by General Assembly

CodeAcademy

w3 Tutorials (start at HTML on the left sidebar and work your way down)

Intermediate

Now that you’ve gone through a handful of basic tutorials, it’s time to learn the fundamentals of actual, real-life coding problems. I’ve found these resources to be solid:

Khan Academy

CodeAcademy - Ruby, Python, PHP

Difficult

If you’re here, you’re capable of building things. You know the primitives. You know the logic control statements. You’re ready to start making real stuff take shape. Here are some different types of resources to turn you from someone who knows how to code, into a full-fledged programmer.

Programming problems

Sometimes, the challenges in programming aren’t how to make a language do a task, but just how to do the task in general. Like how to find an item in a very large, sorted list, without checking each element. Here are some resources for those types of problems

Talentbuddy

TopCoder

Web Applications

If you learned Python, Django is an amazing platform for creating quick-and-easy web applications. I’d highly suggest the tutorial - it’s one of the best I’ve ever used, and you have a web app up and running in less than an hour.

Django Tutorial

I’ve never used Rails, but it’s a very popular and powerful framework for creating web applications using Ruby. I’d suggest going through their guide to start getting down-and-dirty with Rails development.

Rails Guide

If you know PHP, there’s an ocean of good stuff out there for you to learn how to make a full-fledged web application. Frameworks do a lot of work for you, and provide quick and easy guides to get up and running. I’d suggest the following:

Cake PHP Book

Symfony 2 - Get Started

Yii PHP - The Comprehensive Guide

Conclusion

If there’s one point I wanted to get across, it’s that it is easier than ever to learn to code. There are resources on every corner of the internet for potential programmers, and the benefits of learning even just the basics are monumental.

If you know of any additional, great resources that aren’t listed here, please feel free to tweet them to me @boomeyer.

Best of luck!

[mementoluceat-blog]

I’d also like to add some more specialized resources!

Video games:

Easy game engines (virtually no coding): 

Game Maker Studio (2D; free and paid versions)

GameSalad (2D)

RPG Maker (2D; numerous versions ranging from free to $69.99)

Stencyl (2D; free and paid subscription versions)

Scratch (good for kids and is more general; 2D; free)

More difficult game engines: 

Unity (lots and lots of platforms; C# and JavaScript script; 2D, 3D, VR; free and paid versions)

Unreal (specializes in graphics; C++ and visual script; 2D, 3D, VR; free with a royalty on successful products)

CryEngine (Lua script; 3D; paid subscription and full license versions)

Mobile game development: 

Corona (free and paid subscription versions)

SpriteKit (2D) and SceneKit (3D) which are built into the official compiler to create iOS apps (see iOS apps for more resources)

also all of the above game engines (cross-platform)

Game console development: 

Game Maker Studio (with a paid subscription)

Unity

Unreal

CryEngine

Note that games can also be created on more general platforms like iOS and Android apps, but the resources listed above are specialized for game development.

iOS apps:

In order to develop iOS apps, you’ll need to purchase an iOS developer program membership for $99 a year, which requires an Apple account. Here are some general resources:

Xcode (the official IDE for iOS apps; can be installed on OS X)

Start Developing iOS Apps Today (Objective-C)

Ray Wenderlich iOS tutorials (Objective-C and Swift)

Code School: Try iOS (Objective-C)

Developing iOS 8 Apps (2, 3, 4, 5, 6, 7; Swift)

TutorialsPoint: iOS Tutorial (Objective-C)

How to Make iPhone Apps With No Programming Experience (Swift)

Swift Tutorial: Building an iOS Application (2, 3)

iOS apps are developed in the 2 official languages of Apple: Objective-C and Swift, the latter of which is newer and generally much easier to learn.

Objective-C resources:

the official documentation

TutorialsPoint: Objective-C Tutorial

Ry’s Objective-C Tutorial

Swift resources:

the official documentation

The Swift Programming Language (free official e-book)

Swift: A Quick Reference Guide

Xcode also has SpriteKit, SceneKit, and Metal built in, all of which are incredibly useful for creating apps that require elaborate graphics, particularly games.

SpriteKit resources:

How to Make a Game Like Candy Crush With Swift (2)

Sprite Kit Swift Tutorial

Create Space Invaders with Swift and Sprite Kit

iOS SpriteKit Physics Tutorial in Swift

Build the Game of Life (Swift)

SceneKit resources:

Scene Kit Tutorial: Getting Started (Swift)

An Introduction to SceneKit (2; Swift)

Metal resources:

the official documentation + other resources (Obj-C)

iOS 8 Metal Tutorial with Swift (2, 3)

Getting Started With Metal (Obj-C)

An introduction to 3D graphics with Metal in Swift

Also, in order to publish iOS apps, you’ll have to juggle certificates, app ids, and provisioning profiles. This process can be convoluted at times so here are some resources:

How to Submit Your App to Apple: From No Account to App Store (2)

Beginner Tutorial: iOS Certificates & Provisioning Profiles

Android apps:

In order to develop Android apps, you’ll need to register as a developer for a one-time fee of $25. Here are some general resources:

Android Studio (the official IDE for Android app development; free; can be installed on Windows, OS X, and Linux)

the official documentation

Getting Started

Android Tutorial For Beginners (2, 3)

Learn Android SDK From Scratch

Introduction to Android Development With Android Studio

Android apps are developed in Java and the layout is coded with XML.

Java resources:

W3schools: Java Tutorial series

TutorialsPoint: Java Tutorial

Core Java Tutorial

XML resources:

W3schools: XML Tutorial series

TutorialsPoint: XML Tutorial

For publishing (which is somewhat easier than publishing iOS apps):

How to Publish to the Android Market

3D modeling/animation:

Blender (can also be used to create games; Python script; free and open-source; can be installed on Windows, OS X, and Linux)

Maya (specialized script; free trial, free 3-year student subscription, and paid subscription versions; can be installed on Windows, OS X, and Linux to an extent)

3ds Max (Python script; free trial, free 3-year student subscription, and paid subscription versions; can be installed on Windows and OS X)

RenderMan (specialized script; free for non-commercial/educational use and pay-per-license for commercial use; can be installed on Windows, OS X, and Linux)

Misc. resources:

Stack Overflow is an ask-and-answer community for programmers. It’s amazing and will save your life. Sign up and don’t be afraid to ask for help.

Github offers a student pack (here) if you create an account and prove you’re a student. This gives you free access to a bunch of great programming resources for free for a certain period of time, such as Unreal Engine. Also, Github in general is a site that you can host your code on. Other users can see it, and “fork” it to make a copy of your code and modify it.

Parse is a backend service that allows you to store data in databases it hosts on its own servers. It lets you use push notifications, create users, store and retrieve data, etc. It’s compatible with iOS apps, Android apps, Windows apps, Xamarin, React, Unity, OS X, Windows, JavaScript, PHP, .net, Arduino, and Embedded C. It’s free up to a certain limit that depends on the services you use.

Cloud9, Codebox, and Squad are online IDEs that allow for real-time collaboration and support a variety of languages, so they’re useful for team projects.

And some general advice:

Your program will not work right away, 99% of the time. That’s okay. Do your best to figure out where the error is. Here is some advice on debugging (written for PHP but the methods can be generalized).

If you’re stuck, Google. Google like there’s no tomorrow.

Ask questions on a community like Stack Overflow.

For that matter, browse relevant Stack Overflow questions. You can probably find some solutions there.

Don’t be afraid to copy and paste.

Take breaks sometimes if you’re getting burned out. But don’t stay away from your projects for too long or you’ll lose track of its status.

Backup your code. On the cloud, on a USB drive, wherever. If your IDE has a backing up feature like snapshots, use it whenever you hit a milestone.

If your project is big, split it up into milestones and set goals. Don’t tackle everything at once.

Like the OP said, coding isn’t just for professionals and “geeks” anymore. Anyone can learn it if you really try, and with the rapidly expanding tech industry, learning coding can really broaden your opportunities.

If any of the links are broken, or you have a question or some information/resources to add, you can contact me through the askbox or the OP through his Twitter (as mentioned in his post).

If you’re interested, try some of these out and best of luck!

[boomeyer]

Great work expanding on my humble list to include a much fuller collection of resources for learning how to code! Cheers!

[browningtons-deactivated2018041]

[ahhhhhpplied-math]

STEP by STEP guide to The Mean Value Theorem

No jokes about the theorem being mean pls. 

This is another common theorem presented in first-year calculus courses. The theorem might sound scary, but it’s actually quite simple. 

We observe a function f that is continous on [a, b] and differentiable on (a, b). If I draw a line from one end point to another, f(a) to f(b), find the slope of that line, and call it m, the theorem states that somewhere on the interval (a, b), the function f will have the same slope as m. 

The proof is also quite elementary and simple. In order to prove this theorem, we need to explore Rolle’s theorem. I will prove Rolle’s theorem in another post. You can find the theorem stated in the proof.

STEP by STEP guide

Draw a line connecting the end points of the interval.

Find the equation of the line using y=mx+b

Create a new function g(x) = f(x) - the equation of the secant line y (Assuming f(x) > y). We are looking at the space between our function f and the secant line.

We notice that at the end points, g(x) = 0. This is because f(x) = y at the end points, so the space in between them = 0. Therefore Rolle’s theorem applies.

so there exists a c in (a, b) where g’( c ) = 0. 

WE TAKE THE DERIVATIVE OF g(x) and set it to be equal to 0!!!!

REARRANGE THE DERIVATIVE A LITTLE BIT AND WE GET THE THEOREM!!

congratulate yourself by taking a nap.

bam. Now you know.

[ahhhhhpplied-math]

STEP BY STEP guide to prove the Extreme Value Theorem

Definition of the Extreme Value Theorem: If the function f is continuous on [a,b], then there exists constants c and d in the interval [a,b] so that f( c ) is the absolute maximum of f and f(d) is the absolute minimum of f on [a,b].

Sounds *gross* eh? All of that just means given a continuous function f, we can pick an interval [a,b] and we can always find a local maximum and a local minimum in that interval. This is a common theorem covered in first-year calculus courses.

You might think that it’s obvious. 

But how do we prove that it’s true for all x? There are an infinite amount of x that we can sub into function f. It’s rather silly to test them all out. It will take us an infinite amount of time, and idk about you but I’m too lazy for that. So, we will prove this theorem, mathematically. 

We will prove this theorem by infinitely dividing the interval into halves. I have the written formal proof below, but it’s not important if you just want to know the basic idea of the proof. 

Step by step guide:

We will divide up our closed interval [a,b] into 2 parts.

We then look at the local maxima of each interval. 

We then pick the interval containing the maximum value of the original interval to be our new interval.

Wash, rinse, repeat.

Notice that everytime we split up our interval, the new one gets smaller. And when we repeat this process an infinite amount of time, our interval gets *REALLY, REALLY, REALLY* small. we want to divide it up to the point where the interval is reallyyyyyy closed to completely *disappear*. This is a very abstract thought. 

Remember that while we’re dividing up interval, we’re also choosing the one where the local maximum of the original interval is. So we’re approaching a value where f reaches its maximum. We won’t be arriving exactly where the maximum is ( because then the interval is non-existent ), but we’re getting arbitrarily closed to it. We can call this point c, and the maximum value f( c ). Because the function is continuous on the interval, that point we’re approaching exists and is the maximum. The existence of the minimum can also be proven this way.

Ok so what the fuck did i write in the proof?

The proof I wrote below is a bit more technical, but using the same idea. We divide up the interval n times. Let M be the least upper bound of f(x), meaning f( c ) is smaller or equal to M. Using the epsilon-delta definiton, I proved that f( c ) is larger or equal to M. But for these 2 statements to both be true, f( c ) must be M, the supremum of f(x) on the interval [a, b], and because f( c ) is on f(x), it’s the local maximum. 

coolio. 

[the-real-numbers-deactivated202]

*Coughs Really Loudly in Python* Digital Signals Processing

*Coughs Really Loudly in Python*  Bayes Theorem

*Coughs Really Loudly in Python*  Complexity Science

*Coughs Really Loudly in Java* Data Structures

[the-real-numbers-deactivated202]

*Coughs Really Loudly In C++*

*PDF* *PDF* 

Ahahahah you know what fuck it here’s the original link see what else you can find

[the-real-numbers-deactivated202]

I probably should have mentioned these are all high quality, free, online textbooks. The top three links are textbooks that aim to teach you math things under the assumption you know python. And yes there’s a textbook on python too 👉🏼😎👉🏼

If you’re a Java or C++ programmer, there are also textbooks. There are a few more, even. Have fun.

[the-real-numbers-deactivated202]

hi. so i have a ? about matrix exponentials. you know how if a matrix is diagonal and becomes zero at some power of it then the matrix exponential is the power series for e up to that power. im confused why that is. am i missing something obvious??idk if you can help. sorry if this doesn't make sense or is incorrect terminology btw. have a good day!

*This user specified that the matrix was triangular in a different ask*

Well I’m no expert in matrix exponentials, but if I understand you correctly…

Haha sorry I just felt like LaTeXing it.

[the-real-numbers-deactivated202]

Whoop! I should have specified m is non-negative.