The applications of eigenvectors and eigenvalues | That thing you heard in Endgame has other uses
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This video covers the applications of eigenvectors and eigenvalues (in and outside of mathematics) that I definitely didn't learn in school.
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The applications of matrices: https://youtu.be/rowWM-MijXU
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The intuition behind Fourier and Laplace transforms I was never taught in school
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This video covers a purely geometric way to understand both Fourier and Laplace transforms (without worrying about imaginary numbers).
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How do complex numbers actually apply to control systems?
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This video mainly covers the underlying math behind the 'nyquist stability criterion' seen in control systems.
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The beauty of complex numbers
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This video covers complex numbers, euler's formula, fractals, and more of the mechanics of complex numbers.
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Proof of Euler's Theorem: https://youtu.be/_h49ilnTmW4
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Engineering professors be like
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What does the Laplace Transform really tell us? A visual explanation (plus applications)
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This video goes through a visual explanation of the Laplace Transform as well as applications and its relationship to the Fourier Transform.
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The Mathematics of Signal Processing | The z-transform, discrete signals, and more
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This video goes through an overview of what you would learn in a discrete time signals (or digital signal processing) course. Sampling, digital filters, the z-transform, and the applications of these are some of the things included. I had to over simplify a lot to barely keep this video under 30 minutes but hopefully it gives you an idea of how this type of math is used in the real world.
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The Applications of Matrices | What I wish my teachers told me way earlier
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This video goes over just a few applications of matrices that may give you some insight into how they can be used in the real world. Linear algebra was never explained well to me in school and I had very little motivation to learn matrices at the time so hopefully this helps you if you're in a similar situation. Also note there are so many applications there's just no way to fit them into one video but here you'll find some of my favorite instances of when they come up.
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►Videos Mentioned
Google PageRank Algorithm: https://youtu.be/qxEkY8OScYY
Mathematics Used to Solve Crime: https://youtu.be/-cXBgHgX5UE
Essence of Linear Algebra: https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab
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The algorithm that started google
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This video goes over the very basics of the PageRank algorithm and how a google search works. The video is oversimplified and doesn't cover everything of course but note this was the was first algorithm used by Google. Now they use much more (that is kept secret for the most part) but this should still give you an idea of how websites are ranked.
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A visibility problem, how many guards are enough?
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The video explains the proof of 'The Art Gallery Problem'. There are different versions of this problem (such as where guards can be placed) but we will focus on the most well known one where guards are forced to be in corners (and can see in all directions at once).
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The second most beautiful equation and its surprising applications
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More to check out
►Euler's Gem: https://amzn.to/319uBzw
►5 Color Theorem Proof (Up and Atom channel): https://www.youtube.com/watch?v=42-ws3bkrKM
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Is it possible? Simple questions, not so simple solutions
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Mathematics for Computer Science (MIT Course)
►Lectures: https://www.youtube.com/watch?v=L3LMbpZIKhQ&list=PLB7540DEDD482705B
►Course Page (assignments and textbook found here): https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/
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This completely changed the way I see numbers | Modular Arithmetic Visually Explained
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What math and science cannot (yet?) explain
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This video only covers a few things that we cannot yet explain including the WOW signal, FRB's, turbulence, P Vs NP, and some unsolved ciphers.
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►Further explanations/details about some of the mentioned stories
Minute Physics Video: https://www.youtube.com/watch?v=oZAc5t2lkvo
Somerton Man (detailed explanation): https://somerandomstuff1.wordpress.com/2018/11/09/the-ultimate-guide-to-the-somerton-man-mystery/
Unsolved Ciphers: http://scienceblogs.de/klausis-krypto-kolumne/the-top-50-unsolved-encrypted-messages/
The WOW Signal (in more detail): https://www.youtube.com/watch?v=Qtb5TRa09yw
P Vs NP (in more detail): https://www.youtube.com/watch?v=YX40hbAHx3s
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If higher dimensions exist, they aren't what you think | Exploring Worlds Beyond Our Own
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Previous video about 4 dimensional spacetime : https://youtu.be/KT5Sk-62-pg
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The Mathematics of our Universe
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Metric Tensor Explanation for Beginners: https://www.youtube.com/watch?v=C76lWSOTqnc
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This is how easy it is to manipulate public perception
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►Sources
MIT Lecture (who has more sexual partners?): https://www.youtube.com/watch?v=h9wxtqoa1jY&list=PLB7540DEDD482705B&index=6
MIT Study (Optimal Predictions in Everyday Cognition): http://web.mit.edu/cocosci/Papers/Griffiths-Tenenbaum-PsychSci06.pdf
Stephen Gould: https://people.umass.edu/biep540w/pdf/Stephen%20Jay%20Gould.pdf
Trends in violence discussed on the news: http://www.penelopeironstone.com/GlassnerFearMongering.pdf
Death Penalty Study (Confirmation Bias): http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.372.1743&rep=rep1&type=pdf
►Other videos you may enjoy
How easy it is to lie with statistics: https://youtu.be/bVG2OQp6jEQ
The Friendship Paradox: https://youtu.be/GEjhO65FYks
When numbers don't tell the full story: https://youtu.be/FUknTs9AzYA
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The Friendship Paradox | This is in your recommended because it relates to the spread of diseases
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This video covers what the friendship paradox is, why it occurs, and what its applications are.
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Friendship Paradox (Technical Explanation): https://www.quora.com/The-%E2%80%9Cfriendship-paradox%E2%80%9D-states-that-most-people-have-fewer-friends-than-their-friends-on-average-have-Is-the-friendship-paradox-fallacious
Friendship Paradox (less technical): https://paradoxicon.quora.com/Friendship-paradox
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So why are these the only (regular) dice you'll ever see?
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This is a follow up to the last video (https://youtu.be/dr2sIoD7eeU) regarding higher dimensional shapes. Here we answer why there can only be 5 shapes in 3 dimensions that meet certain criteria (as listed in the video).
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The things you'll find in higher dimensions
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This video covers a range of what shapes and properties you'd encounter in higher dimensions.
Why there are only 5 platonic solids proof: https://youtu.be/04eVLnPrpEY
Numberphile Video on Klein Bottles: https://www.youtube.com/watch?v=AAsICMPwGPY
Sphere Packing: https://www.quantamagazine.org/sphere-packing-solved-in-higher-dimensions-20160330/
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Math that will make you think. Twice....maybe
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Can You Outsmart an Economist?: https://amzn.to/2EPQAlQ
Airport Puzzle: https://www.theguardian.com/science/2016/jul/18/did-you-solve-it-wheres-the-best-place-to-tie-your-shoe-in-an-airport
Fraction of Girls Explanation: https://mathoverflow.net/questions/17960/google-question-in-a-country-in-which-people-only-want-boys/17963#17963
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So why do the volumes of all even dimensional unit spheres sum to e^π?
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This is a follow up to my video on Euler's number (https://youtu.be/AAir4vcxRPU). Here you'll see why all unit spheres in even dimensions have volumes that sum to e^π
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Proof of Volume Formula: https://www.youtube.com/watch?v=eEJtMFdkzzU
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When numbers don't tell the full story
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e (Euler's Number) is seriously everywhere | The strange times it shows up and why it's so important
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Animations: Brainup Studios (email: mail@brainup.in)
Timestamps/Extra Resources
2:42 - Derangements (https://en.wikipedia.org/wiki/Derangement)
3:11 - Optimal Stopping (https://en.wikipedia.org/wiki/Optimal_stopping)
5:25 - Infinite Tetration (https://en.wikipedia.org/wiki/Tetration)
6:25 - 1958 Putnam exam question (https://mks.mff.cuni.cz/kalva/putnam/putn58.html)
9:33 - Fourier Transform (GIF credit to 3blue1brown, check out his video on the FT here: https://www.youtube.com/watch?v=spUNpyF58BY)
10:56 - Gamma Function (https://en.wikipedia.org/wiki/Gamma_function)
12:53 - Casimir Effect Paper (https://arxiv.org/pdf/hep-th/9901011.pdf)
13:20 - Higher Dimensional Spheres (https://en.wikipedia.org/wiki/Volume_of_an_n-ball)
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The (strange) Mathematics of Game Theory | Are optimal decisions also the most logical?
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