The clever way curvature is described in math Mathemaniac 2024-08-02 | Second channel video: youtu.be/b8b5qyLovewHow do mathematicians describe curvature of surfaces? There are two measures: Gaussian and mean curvatures, and both are useful in differential geometry, the study of surfaces and higher-dimensional manifolds (or lower-dimensional curves).I know I have talked about Gaussian curvature before in this video: youtube.com/watch?v=7Ju9f9odKX4, but I want to reintroduce it slightly differently with a fuller explanation of the shape operator. This will allow mean curvature in the picture, and is something that I want to focus on for the two future videos on minimal surfaces.I deliberately didn't say principal curvatures, which are the eigenvalues of the shape operator. The eigenvalues are guaranteed to be real, and the eigenvectors must also be orthogonal, because the shape operator is real and symmetric. However, getting to the point where we can prove the shape operator is real and symmetric is a bit tricky (can be proved rather easily with computations, but I'm not sure how to do it "intuitively"); and getting from real symmetric matrices to real eigenvalues and orthogonal eigenvectors is another thing that I still don't know how to think about intuitively.Files for download:Go to mathemaniac.co.uk/download and enter the following password: shapeoperatorMore on my criteria on choosing videos in #SoMEpi: mathemaniac.co.uk/blog/my-somepi-criteriaSources:- Paternain’s differential geometry notes dpmms.cam.ac.uk/~gpp24/dgnotes/dg.pdf (see pp. 28 - 33)- Visual Differential Geometry and Forms by Tristan NeedhamFor this whole series, I have not consulted this book, but it should be a nice resource anyway for the geometric intuitions.- Soap film images: commons.wikimedia.org/wiki/File:Bulle_cat%C3%A9no%C3%AFde.pngcommons.wikimedia.org/wiki/File:Minimal_surface.jpg Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
The real meaning of trace of matrix | Lie groups, algebras, brackets #5 Mathemaniac 2024-01-07 | Can we visualise this algebraic procedure of adding diagonal entries? What is really happening when we add them together? By visualising it, it is possible to almost immediately see how the different properties of trace comes about.Files for download:Go to mathemaniac.co.uk/download and enter the following password: traceisdivThe concept of the whole video starts from one line the Wikipedia page about trace, and I am surprised this isn't on YouTube: "A related characterization of the trace applies to linear vector fields. Given a matrix A, define a vector field F on R^n by F(x) = Ax. The components of this vector field are linear functions (given by the rows of A). Its divergence div F is a constant function, whose value is equal to tr(A)."Actually, this is one of the last concepts in linear algebra that I really wanted a visualisation for, the other being transpose, but this is already on the channel: youtube.com/watch?v=g4ecBFmvAYUChapters:00:00 Introduction00:48 Matrix as vector field02:24 Divergence04:50 Connection between trace and divergence10:12 Trace = sum of eigenvalues13:32 Determinant and matrix exponentials15:15 Trace is basis-independent18:10 Jacobi's formulaFurther reading:Trace (the origin of the whole video): en.wikipedia.org/wiki/Trace_(linear_algebra)#Derivative_relationshipsDivergence (more qualitative, and subtly different from the video): en.wikipedia.org/wiki/Divergence#Physical_interpretation_of_divergenceJacobi's formula (more formal proof): en.wikipedia.org/wiki/Jacobi%27s_formulaOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
Can we exponentiate d/dx? Vector (fields)? What is exp? | Lie groups, algebras, brackets #4 Mathemaniac 2024-01-02 | Part 5: youtu.be/B2PJh2K-jdUCan we exponentiate vectors? What does e^(d/dx) mean? Does it make sense to exponentiate a whole bunch of vectors? Well yes! While what these exponentials do seem very different at first, they can be recast into the same framework.Files for download:Go to mathemaniac.co.uk/download and enter the following password: expderivativeshiftCHAPTERS:00:00 Introduction01:03 What is exponentiation?04:15 Exponentiating vectors11:23 Exponentiating derivatives24:04 Exponentiating vector fields❗Remark❗1️⃣ I know that many people would be thinking of series expansion of exponentials. I deliberately avoid this because it is not conducive to learning the intuition of the exponential, and more crucially, it does not apply to the exponential of vectors on manifolds. The result is very manifold-dependent, and I will be very impressed if there is a series-like explanation for the exponential map in differential geometry.2️⃣ However, I want to know: is there a generalisation of the translation operator statement in the video to manifolds? For a flat plane, we have exp(a * nabla) f(x) = f(x + a). And in fact,the exponential map on the flat manifold of R^n gives x + a = exp_x (a). Hence, for flat R^n, we have exp(a * nabla) f(x) = f(exp_x(a)). Can this be generalised to general manifolds? Is it true if I interpret nabla as not a normal gradient, but covariant derivative? Please let me know if you have any ideas for it. I want this to be true because it connects different “exponential” ideas.📖 Further reading 📖1️⃣ Exp vectorsExponential map in Riemannian geometry (if you actually want to know how this is just a generalised exponential map in the usual sense, rather than just having the same “philosophy”, then go to the relationship to Lie theory section - when they say translations, they mean multiply on the left/right by g, a group element of the Lie group): en.wikipedia.org/wiki/Exponential_map_(Riemannian_geometry)Why exponential map in differential geometry is useful: en.wikipedia.org/wiki/Normal_coordinates❗You might also need to learn these first before tackling the link above❗Riemannian metric: https://www.ime.usp.br/~gorodski/teaching/mat5771/ch1.pdfConnection (the introduction is the most illuminating part): en.wikipedia.org/wiki/Affine_connectionmath.stackexchange.com/questions/2145255/what-is-the-affine-connection-and-what-is-the-intuition-behind-for-affine-conneLevi-Civita connection (a particular kind of connection that makes the metric invariant): en.wikipedia.org/wiki/Levi-Civita_connection2️⃣ Exp d/dxWhat I have described is kind of solving PDE with the method of characteristics (identifying characteristic curves along which it becomes an ODE): en.wikipedia.org/wiki/Method_of_characteristics#ExampleThe partial differential equation is part of the wave equation: en.wikipedia.org/wiki/Wave_equation#General_solutionTranslation operator in quantum mechanics: en.wikipedia.org/wiki/Translation_operator_(quantum_mechanics)Time-ordering: solving differential equations of the form ∂f/∂t = X(t) f, where X(t) is a time-dependent differential operator, e.g. t^2*∂^2/∂x^2: en.wikipedia.org/wiki/Ordered_exponentialTime-ordering in example in QM: en.wikipedia.org/wiki/Dyson_series3️⃣ Exp vector fieldVector flow: en.wikipedia.org/wiki/Vector_flowThis is more related to the video: en.wikipedia.org/wiki/Flow_(mathematics)Textbook: worldscientific.com/doi/pdf/10.1142/9789813276628_0001I actually wanted to say the following, but I think the video is long enough and didn’t include it into the script, but vector field is actually related to (and most often described by) differential operators, and in that sense both exponential of (1st-order) differential operators and exponential of vector fields yield very similar things: en.wikipedia.org/wiki/Vector_fieldOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
What is Lie theory? Here is the big picture. | Lie groups, algebras, brackets #3 Mathemaniac 2023-08-10 | Part 4: youtu.be/9CBS5CAynBEA bird's eye view on Lie theory, providing motivation for studying Lie algebras and Lie brackets in particular.Basically, Lie groups are groups and manifolds, and thinking about them as manifolds, we know that we want to understand Lie algebras; and thinking about them as groups, we know what additional structure we want on the Lie algebras - the Lie bracket.YouTube, please do not demonetise this video for me saying “Tits group”. This is an actual mathematical object named after a French mathematician Jacques Tits.Files for download:Go to mathemaniac.co.uk/download and enter the following password: so3embeddedin5dimSO(3) embedded in R^5: http://at.yorku.ca/b/ask-an-algebraic-topologist/2020/2618.htmen.wikipedia.org/wiki/Whitney_embedding_theorem (n-dim manifold can be properly embedded in R^(2n): if you only want “the overall picture”, but perhaps distances are distorted)en.wikipedia.org/wiki/Nash_embedding_theorems (n-dim Riemannian manifold can be isometrically embedded in n(3n+11)/2 dim if compact, n(n+1)(3n+11)/2 dim if not compact: if you want everything to remain intact, i.e. distances are preserved)BCH formula (why Lie brackets are useful): en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formulaFinite simple groups as building blocks: en.wikipedia.org/wiki/Composition_seriesClassification of finite simple groups: en.wikipedia.org/wiki/Classification_of_finite_simple_groupsLevi decomposition (the more precise meaning of “building blocks” in Lie algebra): en.wikipedia.org/wiki/Levi_decompositionE8 (the monster group of Lie algebras): aimath.org/E8/e8.htmlen.wikipedia.org/wiki/E8_(mathematics)en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory_of_EverythingVideo chapters:00:00 Introduction01:26 Lie groups - groups05:41 Lie groups - manifolds10:23 Lie algebras14:16 Lie brackets18:03 The "Lie theory picture"Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
How to rotate in higher dimensions? Complex dimensions? | Lie groups, algebras, brackets #2 Mathemaniac 2023-07-29 | Part 3: youtu.be/ZRca3Ggpy_gAround 11:50, can't imagine that this error got in - it should have been SU(n) = {U in U(n), det U = 1}.Orthogonal and unitary groups. Rotational symmetries, real and complex, are particularly useful in the field of Lie theory, because their (complexified) Lie algebras, together with that of the symplectic group Sp(n), are the only infinite families of simple Lie algebras. This video is to familiarise with the SO(n), SU(n) notations, and provides further motivation to study Lie theory.Files for download:Go to mathemaniac.co.uk/download and enter the following password: orthogonalunitaryVideo chapters:00:00 Introduction01:04 Real rotation in n dimensions07:03 Complex rotationOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
Why study Lie theory? | Lie groups, algebras, brackets #1 Mathemaniac 2023-07-23 | Next video: youtu.be/erA0jb9dSm0Lie’s theory of continuous symmetries was originally for differential equations, but turns out to be very useful for physics because symmetries are manifest in many physical systems. This is the start of a series on Lie groups, Lie algebras, and Lie brackets.Files for download:Go to mathemaniac.co.uk/download and enter the following password: lievideoseriesSources:History of group theory: en.wikipedia.org/wiki/History_of_group_theoryLie’s theory for differential equations (actually developed further after Lie died):arxiv.org/pdf/1901.01543.pdfhttps://www.physics.drexel.edu/~bob/LieGroups/LG_16.pdfresearchgate.net/profile/Michael-Frewer/publication/269465435_Lie-Groups_as_a_Tool_for_Solving_Differential_Equations/links/548cbf250cf214269f20e267/Lie-Groups-as-a-Tool-for-Solving-Differential-Equations.pdfen.wikipedia.org/wiki/Lie_point_symmetryLie theory applications to mathematics: math.stackexchange.com/questions/1322206/what-are-applications-of-lie-groups-algebras-in-mathematicsOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
How many terms do we need in the exp series? Mathemaniac 2023-07-05 | The exponential function is expressed in terms of the series, but practically, how many terms do we need to compute it to some desired accuracy? Well, it turns out the answer to that is quite interesting.Oscar's video on how computers actually compute the exponential series: youtu.be/NorMJ5PO3Hc For the files created for this video, please visit mathemaniac.co.uk/download and enter the password: cltoutofnowhereand follow the instructions on the website. If you can't enter the website, watch the latest video! It always changes when a new video is up.Sources:Central Limit Theorem: en.wikipedia.org/wiki/Central_limit_theoremThe problem that inspired this video (last question on the problem sheet): dpmms.cam.ac.uk/study/IA/Probability/2017-2018/ex4.pdfOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
Queuing theory and Poisson process Mathemaniac 2023-06-27 | Queuing theory is indispensable, but here is an introduction to the simplest queuing model - an M/M/1 queue. Also included is the discussion on Poisson process, which is the underlying assumption for the M/M/1 queue.Second channel video on variance of Poisson distribution: youtu.be/xiX1QoX8i6MTo me, this is mainly a "prequel" which serves as a prerequisite for the next video, even though the next video is not as long.For the files created for this video, please visit mathemaniac.co.uk/download and enter the password: queuingmodelM/M/1and follow the instructions on the website. If you can't enter the website, watch the latest video! It always changes when a new video is up.Sources:Different queues:M/M/1 queue: en.wikipedia.org/wiki/M/M/1_queueM/M/c queue: en.wikipedia.org/wiki/M/M/c_queueM/M/∞ queue: en.wikipedia.org/wiki/M/M/%E2%88%9E_queueM/G/1 queue: en.wikipedia.org/wiki/M/G/1_queueM/G/k queue: en.wikipedia.org/wiki/M/G/k_queueG/M/1 queue: en.wikipedia.org/wiki/G/M/1_queueG/G/1 queue: en.wikipedia.org/wiki/G/G/1_queueJackson Network: en.wikipedia.org/wiki/Jackson_networkMore general queues: en.wikipedia.org/wiki/Kendall%27s_notationThe (transient) solution: Computer Networks and Systems. New York, NY: Springer New York. p. 72 (uses moment-generating function and Laplace transforms); for more details, see Gross, D. and Harris, C.M., Fundamentals of Queueing Theory, Wiley, New York, 1974, 1985. (Section 3.11.2)Other related sources:Markov Chains: statslab.cam.ac.uk/~james/MarkovBirth-and-death chain: en.wikipedia.org/wiki/Birth%E2%80%93death_processBessel functions (for the solution to the differential equations): en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions_:_I%CE%B1,_K%CE%B1 Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
Theorema Egregium: why all maps are wrong Mathemaniac 2023-04-26 | Head to squarespace.com/mathemaniac to save 10% off your first purchase of a website or domain using code mathemaniac.Website for files download (remember to use the password shown in the video!): mathemaniac.co.uk/downloadThe Mercator projection is the standard world map, but it famously makes Greenland and Africa the same size, but in reality, Greenland is so much smaller. Gall-Peters projection aims to solve exactly this area mismatch problem, but the shape resulted is horrible, and actually unsuitable for any navigation, unlike Mercator. Can we make a world map that preserves both areas (like Gall-Peters) and angles (like Mercator)? No, and the reason why is Theorema Egregium, the subject of the video.Traditionally, Theorema Egregium was proved with a lot of tedious calculations, and somehow magically, you can compute the curvature with the "first fundamental form", whatever that means. It took until more than a century later than its original discovery that a geometric proof was found, and is presented here.Theorema Egregium, more traditional proof, going through first and second fundamental forms: dpmms.cam.ac.uk/~gpp24/dgnotes/dg.pdfTristan Needham's book on visual differential geometry: amazon.co.uk/Visual-Differential-Geometry-Forms-Mathematical/dp/0691203709Video chapters:00:00 Introduction02:40 Chapter 1: Curvature10:32 Chapter 2: Spherical areas17:34 Chapter 3.1: Gauss map preserves parallel transport22:15 Chapter 3.2: Geodesics preserved27:16 Chapter 3.3: Parallel transport preserved31:46 Chapter 3.4: Area = holonomy on sphere36:43 Chapter 4: Tying everything togetherOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
The biggest misconception about spin 1/2 Mathemaniac 2023-03-23 | Head to squarespace.com/mathemaniac to save 10% off your first purchase of a website or domain using code mathemaniac.My website link: mathemaniac.co.uk/about“If you rotate a spin 1/2 particle by 360 degrees, it doesn’t go back to its original state, rather you need 720 degrees”. This is only technically correct if you interpret the words “rotate” and “state” in the way it’s intended, and here is the video on what they really mean, and what this sentence is saying.----------------------Many people have talked about spin 1/2 before, but I do want to chime in because there are a few things that I myself am not satisfied with all the explanations.(1) I personally don’t like **any** “demonstration” of spin 1/2, aimed at demonstrating in some physical situations you need two full rotations to get back to where you started, like the Dirac belt trick / spinning your hand trick. This is simply because physically, you literally can’t tell the difference before and after rotation - even in principle. You can only tell the difference when you have a superposition. And these demonstrations, to me, give the false perception that physically there is something different. This is, I think, an extremely important point that people miss out when talking about spin 1/2 and how “weird” this is.(2) From U(2) to SU(2): the “usual” explanation for the choice of determinant is that the phase factor does not matter, so in passing from U(2) to SU(2), we sort of “remove the redundancies'' in the description of our transformation. But why did we leave the ±1 factor redundancy untouched? Either you remove **every** redundancy, or **none**. It doesn’t make sense (at least to me) that you don’t also remove that sign redundancy as well.This is actually because the projective representations (obtained using Lie algebra methods) of SO(3) **must** have determinant 1, and this is known prior to constructing such a (projective) representation. This might be briefly explained if I decide to make a video on Lie algebras / groups / representations. I think that using the Lie algebra method means that we are imposing the constraint about analyticity and genuine representation in the neighbourhood of the identity, but I can’t be sure about this.(3) When actually constructing the (projective) representation, the usual “trick” is to do some conjugation - but actually with Bloch sphere, we can visualise the representation extremely visually. That’s almost all the reason why I love the Bloch sphere visualisation method. The conjugation trick is not wrong, just that it seems way more complicated to me. For more information on this old trick, see this: https://indico.cern.ch/event/243629/attachments/415251/576988/L2.pdf---------------------There are alternative perspectives out there. What I have described is **how** SU(2) and SO(3) **acts**. But there are explanations using algebraic topology to describe **what** SU(2) and SO(3) really **are** as manifolds. For more on this, see damtp.cam.ac.uk/user/examples/D18S.pdf---------------------------------In a similar vein, people often think that spin-s particles are just particles that magically goes back to its original state after 2pi / s rotations. Even Wikipedia says this! Except this is not even technically correct - it is just plain wrong. I’ll talk a bit about that, again if I decide to make a video about higher spins / Lie groups / algebras / representations.Hopf fibration / quaternions:en.wikipedia.org/wiki/Hopf_fibration#Geometric_interpretation_using_rotationsVideo chapters:00:00 Introduction00:47 Chapter 1: "State"07:42 Chapter 2: "Rotate"17:46 Chapter 3: The construction25:41 Chapter 4: The "spin-1/2 property"Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
I use PowerPoint to edit all* videos (and hit 100k subs!) Mathemaniac 2023-01-10 | Head to squarespace.com/mathemaniac to save 10% off your first purchase of a website or domain using code mathemaniac.My blog on how to use GeoGebra: mathemaniac.co.uk/blog/how-i-use-geogebra-to-make-animations-in-videos-part-iYes, I use PowerPoint to edit all* my videos... This is actually a much more efficient way of making videos, and yet it can be made to look professional, like most of the videos on the channel. I hope that people don't judge me for using PowerPoint.*except for the George Green's life video, and the Cambridge interview video with Tom Rocks MathsA bit of addendum to my opinion of math education: if you think that making math not compulsory will hinder STEM education, I would like to remind you that those who hate math would most likely not be pursuing STEM-related degrees anyway. The emphasis is that the math being taught is too computational - if we are talking about more basic skills like reading statistics, then I would have no problem with it being compulsory, because most people definitely need to use those in their daily life. But for the kind of computational math in the current curricula, it is very difficult to make the argument that it is anywhere near useful in daily life, hence a lot of memes about how people have gone by a day without knowing Pythagoras theorem, or quadratic formula.LaTeX online equation editor (to export picture files): latex.codecogs.com/eqneditor/editor.phpBooks mentioned:(1) Prime Obsession by John Derbyshire(2) e: the story of a number by Eli Maor(3) Trigonometric Delights by Eli MaorDexter’s notes: dec41.user.srcf.net/notesTranslating videos: mathemaniac.co.uk/faq/#translateMy Quora account: quora.com/profile/Trevor-1366/answersTom Scott’s video series: youtube.com/watch?v=h0AMaW4XRCI&list=PL96C35uN7xGL1V_TCp5OLlkBl0qwjQ1PlThere are many things that I would like to share, but I can’t possibly answer all the questions I received. On the other hand, there are certain things that I love to share, but none of the questions actually ask for those parts.Video chapters:00:00 How do you make the animations?03:34 My math journey08:08 Opinions on math14:55 Miscellaneous questionsOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
The deeper meaning of matrix transpose Mathemaniac 2022-12-06 | 100k Q&A: https://forms.gle/dHnWwszzfHUqFKny7Transpose isn’t just swapping rows and columns - it’s more about changing perspective to get the same measurements. By understanding the general idea of transpose of a linear map, we can use it to visualise transpose much more directly. We will also heavily rely on the concept of covectors, and touch lightly on metric tensors in special/general relativity, and adjoints in quantum mechanics.As far as I know, this way of visualisation of transpose is original. Most people use SVD (singular value decomposition) for such visualisation, but I think it is much less direct than this one, and also SVD is mostly used for numerical methods, so it feels somewhat unnatural to use a numerical method to explain linear transformations (though, of course, SVD is extremely useful). Please let me know if you know that other people have this specific visualisation.The concept I am introducing here is usually called a “pullback” (and actually the original linear transformation would be called “pushforward”), but as said in the video, another way of thinking about transpose is the notion of “adjoint”.Notes:(1) I am calling covectors a “measuring device”, not only because the level set representation of covectors looks like a ruler when you take a strip of the plane, but also because of its connections with quantum mechanics. A “bra” in quantum mechanics is a covector, and can be thought of as a “measurement”, in the sense of “how likely will you measure that state” (sort of).(2) I deliberately don’t use row vectors to describe covectors, not only because this only works in finite-dimensional spaces, but also because it is awkward for the ordering when we say a transpose matrix *acts* on the covector. We usually apply transformations on the *left*, but if you treat the covector as a row vector, you have to act the transpose matrix on the *right*.(3) You can do the sort of “exercise” to verify this visualisation of transpose for all (non-singular) matrices, but I think the algebra is slightly too tedious. This is the reason why I spent a lot of time talking about the big picture of transpose - to make the explanation as natural as possible.Further reading:**GENERAL**(a) Transpose of a linear map (Wikipedia)en.wikipedia.org/wiki/Transpose_of_a_linear_map(b) Vector space not isomorphic to its dual (for infinite-dimensional vector spaces):math.stackexchange.com/questions/35779/what-can-be-said-about-the-dual-space-of-an-infinite-dimensional-real-vector-spamath.stackexchange.com/questions/58548/why-are-vector-spaces-not-isomorphic-to-their-duals/58598#58598**RELATIVITY**(a) Metric / inverse metric as the vector-covector correspondence: en.wikipedia.org/wiki/Raising_and_lowering_indicesen.wikipedia.org/wiki/Minkowski_space#Raising_and_lowering_of_indices**ADJOINT**(a) Inner product (the prerequisite of even defining adjoints, the analog of dot products in Euclidean space): en.wikipedia.org/wiki/Inner_product_space(b) Adjoints (another way of thinking about transposes, but I think this is mostly about the complex analogue of transpose): en.wikipedia.org/wiki/Hermitian_adjoint(c) Reisz representation theorem (more relevant to adjoints, but in regards to the statement that “we choose certain covectors to act on”: here, it is the “continuous” dual, very relevant in QM): en.wikipedia.org/wiki/Riesz_representation_theorem(d) Self-adjoint operators (Hermitian operators in QM, but also useful in Sturm-Liouville theory in ODEs): en.wikipedia.org/wiki/Self-adjoint_operatorVideo chapters:00:00 Introduction00:56 Chapter 1: The big picture04:29 Chapter 2: Visualizing covectors09:32 Chapter 3: Visualizing transpose16:18 Two other examples of transpose19:51 Chapter 4: Subtleties (special relativity?)Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
How hard was my Cambridge interview? (ft. @TomRocksMaths) Mathemaniac 2022-11-04 | To prove once and for all that Cambridge is better than O*ford 😤 (just kidding)Tom's channel: youtube.com/c/TomRocksMaths/featured100k Q and A Google form: https://forms.gle/WtpM3HqhkdPuwsbP9You might recognize Tom from his Navier-Stokes equations on the Numberphile channel, and he is a tutor at Oxford teaching students at St. Edmund Hall college in Oxford. I just thought it was funny to have a Cambridge student to interview a teacher at Oxford.**I HAVE TO EMPHASISE THAT THIS IS ON A MUCH MORE DIFFICULT LEVEL THAN THE OTHER QUESTIONS I HAVE HEARD FROM MY FRIENDS. THIS IS AGREED BY MY FRIENDS, SO IT'S NOT JUST ME SAYING MINE ARE HARDER.**I am a fourth-year Cambridge math student, and so I thought I would share with you the interview questions that I have gone through! I have signed a sort of NDA for not disclosing the exact interview questions, but it has now been 4 years since my last interview, so it is fine!Actually, my interviewer knows about this channel, but I am not sure whether he watches it or not. I hope I don't get into trouble, because in the spirit of the confidentiality agreement I signed, I should be able to disclose the details of the interview questions now. It has been 4 years since my interview, so unless they are so uncreative to come up with new interview questions, I think this should be fine.A bit of notes:(1) Sorry for a lot of the technical glitches - we were very unlucky, because Tom has also used the same setup in his exam series without any problems.(2) There is an organ playing in the background - they are practising for the evening service at the college! If you don’t like it, I’m sorry; if you like it, consider it background music.(3) For Cambridge applicants: the interview format differs from college to college. If you are interviewing for St John’s, there are separate pure and applied interviews; for Trinity (which I strongly discourage any potential applicants to apply to, simply due to the competitive nature), your interview will be based on some questions you’ve attempted in a test prior to the interview. However, because of COVID, I am not sure if all these have changed.(4) For the people who are here for the maths: for the second question, what I meant to say was that because there are 999 consecutive integers, when you shift by one place, then you are either (a) adding an even number and deleting an odd number, or (b) adding an odd number and deleting an even number.In the case of (a), adding an even number does not change the number of primes, but deleting an odd number might or might not decrease the number of primes by 1, so the number of primes in the interval either changes by 0 or -1.In the case of (b), deleting an even number again does not change the number of primes, but adding an odd number might or might not increase the number of primes by 1, so the number of primes in the interval either changes by 0 or +1.Actually, even if it is 1000 integers, a similar argument applies, though I would say this parity argument is slightly more complicated. Anyway, the idea is that initially you have more than 10 primes, and then you have also constructed an interval with exactly 0 prime, so somewhere in between, there must be an interval with exactly 10 primes, because every time you are only either changing the number of primes by +1,0, or -1.Video chapters:00:00 Introduction01:24 First question: graph sketching08:54 Second question (Part I): 0 prime19:59 Second question (Part II): 10 primes29:47 Third question: physics50:04 Final point: SAQ (now called My Cambridge)Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
Random walks in 2D and 3D are fundamentally different (Markov chains approach) Mathemaniac 2022-10-04 | Second channel video: youtu.be/KnWK7xYuy00100k Q&A Google form: https://forms.gle/BCspH33sCRc75RwcA"A drunk man will find his way home, but a drunk bird may get lost forever." What is this sentence about?In 2D, the random walk is "recurrent", i.e. you are guaranteed to go back to where you started; but in 3D, the random walk is "transient", the opposite of "recurrent". In fact, for the 2D case, that also means that you are guaranteed to go to ALL places in the world (the only constraint is, of course, time). [Think about why.]Markov chains are also an important tool in modelling the real world, and so I feel like this is a good excuse for bringing it up.At the end, I also compare this phenomenon to Stein's paradox - in both cases, there is a cutoff between 2 and 3 dimensions, and they have similar intuitive explanation - is that a coincidence?Video chapters:00:00 Introduction00:59 Chapter 1: Markov chains03:20 Chapter 2: Recurrence and transience10:08 Chapter 3: Back to random walksFurther reading:Larry Brown’s paper: http://stat.wharton.upenn.edu/~lbrown/Papers/1971b%20Admissible%20estimators,%20recurrent%20diffusions,%20and%20insoluble%20boundary%20value%20problems.pdfUsing electric circuits to prove recurrence / trasience: https://math.dartmouth.edu/~pw/math100w13/mare.pdfMore complicated, but more general proof: https://sites.math.washington.edu/~morrow/336_19/papers19/Legrand.pdfActual probability for 3D random walk to come back: mathworld.wolfram.com/PolyasRandomWalkConstants.htmlOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
The weirdest paradox in statistics (and machine learning) Mathemaniac 2022-08-30 | 🌏 AD: Get Exclusive NordVPN deal here ➼ nordvpn.com/mathemaniac. It's risk-free with Nord's 30-day money-back guarantee! ✌Second channel video: youtube.com/watch?v=3ne9yghOtw8Stein's paradox is of fundamental importance in modern statistics, introducing concepts of shrinkage to further reduce the mean squared error, especially in higher dimensional statistics that is particularly relevant nowadays, in the world of machine learning, for example. However, this is usually ignored, because it is mostly seen as a toy problem. Precisely because it is such a simple problem that illustrates the problem of maximum likelihood estimation! This paradox is the subject of many blogposts (linked below), but not really here on YouTube, except in some lecture recordings, so I have to bring this up to YouTube.This is not to say that maximum likelihood estimator is not useful - in most situations, especially in lower dimensional statistics, it is still good, but to hold it to such a high place, as statisticians did before 1961? That is not a healthy attitude to this theory.One thing I did not say, but perhaps a lot of people will want me to, is that this is an emprical Bayes estimator, but again, more links below.Video chapters:00:00 Introduction04:38 Chapter 1: The "best" estimator09:48 Chapter 2: Why shrinkage works15:51 Chapter 3: Bias-variance tradeoff18:45 Chapter 4: ApplicationsFurther reading:The “baseball paper”: https://efron.ckirby.su.domains//other/Article1977.pdfWikipedia: en.wikipedia.org/wiki/Stein%27s_exampleDominating the (positive-part) James-Stein estimator: projecteuclid.org/journals/annals-of-statistics/volume-22/issue-3/Improving-on-the-James-Stein-Positive-Part-Estimator/10.1214/aos/1176325640.fullWikipedia (Empirical Bayes): en.wikipedia.org/wiki/Empirical_Bayes_methodOther writeups:http://www.ime.unicamp.br/~veronica/MI677/steinparadox.pdfjoe-antognini.github.io/machine-learning/steins-paradoxjchau.org/2021/01/29/demystifying-stein-s-paradoxnaftaliharris.com/blog/steinvizaustinrochford.com/posts/2013-11-30-steins-paradox-and-empirical-bayes.htmlduphan.wordpress.com/2016/07/10/steins-paradox-or-the-power-of-shrinkagehttp://www.statslab.cam.ac.uk/~rjs57/SteinParadox.pdf(Philosophical implications) http://philsci-archive.pitt.edu/13303/1/Philosophical%20significance%20of%20Stein%20Preprint.pdfOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
Why you cant solve quintic equations (Galois theory approach) #SoME2 Mathemaniac 2022-07-03 | An entry to #SoME2. It is a famous theorem (called Abel-Ruffini theorem) that there is no quintic formula, or quintic equations are not solvable; but very likely you are not told the exact reason why. Here is how traditionally we knew that such a formula cannot exist, using Galois theory.Correction: At 08:09, I forgot to put ellipsis in between. Video chapters:00:00 Introduction00:23 Chapter 1: The setup04:38 Chapter 2: Galois group11:15 Chapter 3: Cyclotomic and Kummer extensions19:43 Chapter 4: Tower of extensions27:25 Chapter 5: Back to solving equations35:23 Chapter 6: The final stretch (intuition)43:25 Chapter 7: What have we done?Notes:I HAVE to simplify and not give every technical detail. This is made with the intent that everyone, regardless of their background in algebra, can take away the core message of the video. This can only be done if I cut out the parts that are not necessary for this purpose. As with my previous video series on “Average distance in a unit disc”, this is made to address the question I always had when I was small - treat this as a kind of a video message to my past self.For the “making everything Galois extension” bit, we will need to show that the only things fixed by ALL automorphisms over Q must be in Q itself. This is intuitive, but difficult to justify rigorously. All proofs I know involve “degree of field extension”, and the very satisfying result called the “tower law”, which I deliberately avoided throughout the video because it turns out not to be necessary for the core part of the video. For instance, this proof: math.stackexchange.com/questions/962898/on-a-proof-that-the-splitting-field-of-a-separable-polynomial-is-galoisThe reason we have this mess is that we defined Galois extension using the splitting field of a (separable, i.e. no repeated roots) polynomial. The usual definition given is exactly as above - only things fixed by ALL automorphisms over Q must be in Q itself. This typical definition will of course solve the problem above, but will now create the problem of why this definition implies the larger field is made by adjoining the roots of some polynomial. These two definitions are equivalent, but I just think that it makes much more sense to define it the way I did in the video, in the context of the video; and also I think this is an easier definition to accept.Resources on Tower law: en.wikipedia.org/wiki/Degree_of_a_field_extension#The_multiplicativity_formula_for_degreesartofproblemsolving.com/wiki/index.php/Tower_lawQuotients of solvable groups are solvable (the elementary proof): math.stackexchange.com/questions/2203250/quotient-of-solvable-groups-is-solvable-whats-wrong-with-this-proof[The question is already the proof - it is a really elementary way to show the result that we want]More resources on proofs that A_n is not solvable:Sign of permutations: en.wikipedia.org/wiki/Parity_of_a_permutationAlternating groups: mathworld.wolfram.com/AlternatingGroup.htmlThe proof that A_n is simple (i.e. no non-trivial normal subgroups): http://ramanujan.math.trinity.edu/rdaileda/teach/s19/m3362/alternating.pdf[You need to only go up to Page 5 towards the end of the proof of Theorem 2, but you definitely need group theory lingo]If you know a bit of group theory (orbit-stabiliser and Cauchy), then you can see that the polynomial x^5 - 6x + 3 has the full S5 Galois group, because it is (i) irreducible [this requires Eisenstein’s criterion, see link below], and (ii) exactly two complex roots [and hence the Galois group contains a transposition, i.e. complex conjugation]. Note that the Galois group is transitive. This again needs quite a bit of justification. For the proof assuming transitivity, see here: math.stackexchange.com/questions/3075225/f-irreducible-polynomial-with-p-2-real-roots-rightarrow-gal-mathbbq-fEisenstein’s criterion: en.wikipedia.org/wiki/Eisenstein%27s_criterionOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukKo-fi: ko-fi.com/mathemaniac [for one-time support]For my contact email, check my About page on a PC.See you next time!
The simplest reason why π/4 = 1 - 1/3 + 1/5 -... Mathemaniac 2022-04-29 | Without differentiation or integration, or number theory, could we still prove this infinite sum? | Visit http://brilliant.org/Mathemaniac to get started learning STEM for free, and the first 200 people will get 20% off their annual premium subscription.π/4 = 1 - 1/3 + 1/5 -..., also called the Leibniz series, is quite famous, but the usual proof involves differentiation or integration. The more visual geometric proof still relies a lot on some advanced theorems from number theory, but given how simple the series is, is it possible to have an even simpler proof? Yes! And this video tries to explain this.Actually, similar to the previous proof, this proof has been at the video ideas list for quite a long time - so it's good to finally see this out!By the way, on second thought, this looks similar to Fourier coefficients, at least the time average bit - though I can't see whether this is the same proof as the one using Fourier series of sgn(x). It feels very connected, but also very different in the sense that we don't need to use the Cesàro sums of the Fourier series. Please let me know if you have any ideas regarding this.Video chapters:00:00 Intro00:45 Chapter 1: The setup03:16 Chapter 2: The main argument14:55 Chapter 3: Making this rigorous17:33 Sponsored segmentThis video was sponsored by Brilliant.Sources (in order of appearance in the video):(1) Pi hiding in prime regularities (3Blue1Brown): youtube.com/watch?v=NaL_Cb42WyY(2) Fermat’s Christmas Theorem (Mathologer): youtube.com/watch?v=00w8gu2aL-w(3) The proof itself: quora.com/How-can-I-show-that-frac-pi-4-1-frac-1-3-frac-1-5-frac-1-7-cdots-perhaps-geometrically/answer/Sridhar-Ramesh(4) Cesàro sums: en.wikipedia.org/wiki/Ces%C3%A0ro_summationA little more reading:(1) Calculating pi by hand (Matt Parker): youtube.com/watch?v=HrRMnzANHHs(2) Regular sum, if defined, equals Cesàro sum: math.stackexchange.com/questions/1134265/cesaro-summable-seriesmath.stackexchange.com/questions/514802/convergence-of-series-implies-convergence-of-cesaro-mean(3) Wikipedia proof (involving integration): en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80(Worth noting that this requires a bit more than just plugging in x = 1 in arctangent series - this requires the justification from Abel’s theorem)(4) If you want to prove the Cesàro sum of the positions to be 0 more rigorously, here is the justification:wolframalpha.com/input?i=sum+from+k%3D1+to+n+%281-k%2F%28n%29%29*cos%28kx%29and wolframalpha.com/input?i=lim+n+to+infty+sum+from+k%3D1+to+n+%281-k%2F%28n%29%29*cos%28kx%29&assumption=%22LimitHead%22+-%3E+%7B%22Discrete%22%7D(and the formula in the first link would only require repeatedly applying product to sum formulas and telescoping series - still no differentiation / integration involved. If you want even further assistance, I made a video on this: youtube.com/watch?v=DzhK3UJj5IY )Music:Asher Fulero - BeseechedAakash Gandhi - Heavenly, Kiss the Sky, Lifting Dreams(From YouTube audio library)Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukFor my contact email, check my About page on a PC.See you next time!
The geometric interpretation of sin x = x - x³/3! + x⁵/5! -... Mathemaniac 2022-03-24 | We first learnt sin x as a geometric object, so can we make geometric sense of the Taylor series of the sine function? For a long time, I thought it was just my dream, but actually, it is not!This proof only uses very elementary methods, and depending on your definition of calculus, doesn't actually use calculus. The most we are using is a limiting process, and definitely no differentiation or integration here.This proof is very beautiful - not only that it unveils the geometric meaning of each term in the series very beautifully, but also understandable by a normal high-school student with a little bit of patience. I am very surprised that it has not appeared on YouTube before, and even if it does exist on the internet, it is far too unpopular, and so I have to bring this up!Obviously this is not my proof. See the sources below.Sources:jstor.org/stable/2974881?seq=1#metadata_info_tab_contentsarxiv.org/pdf/1610.04825.pdf (equations of the involutes)math.stackexchange.com/questions/2758418/deriving-the-power-series-for-cosine-using-basic-geometry?noredirect=1&lq=1 (the StackExchange post)en.wikipedia.org/wiki/Involute (More about involutes)support.google.com/youtube/answer/6162278#zippy=%2Cguide-to-self-certification (Advertiser-friendly guidelines)unicef.org.uk/donate/donate-now-to-protect-children-in-ukraine (UNICEF Ukraine Appeal)Music:Asher Fulero - BeseechedAakash Gandhi - White River, Heavenly, Lifting Dreams, Kiss the SkyVideo Chapters:00:00 Introduction00:50 Preliminaries02:10 Main sketch06:03 Details - Laying the ground work09:42 The iteration process11:11 Finding lengths of involutes14:57 What? Combinatorics?18:44 Final calculation20:45 Fundraiser appealOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukFor my contact email, check my About page on a PC.See you next time!
Complex integration, Cauchy and residue theorems | Essence of Complex Analysis #6 Mathemaniac 2022-01-22 | Unlock new career opportunities and become data fluent today! Use my link bit.ly/MathemaniacDCJan22 and check out the first chapter of any DataCamp course for FREE! I can't pronounce "parametrisation" lolA crash course in complex analysis - basically everything leading up to the Residue theorem. This is a more intuitive explanation of complex integration using Pólya vector field.You might notice that my explanation on parametrisation is a bit similar to the Jacobian, and you will be right! Jacobian is really important in this area (and also understanding complex differentiation and Cauchy-Riemann equations).I have made this slower in comparison with some of my other videos, because when I myself watched some of my other videos that are faster, I couldn't comprehend if I was not paying too much attention on the screen, let alone the audience watching it for the first time. If somehow, miraculously, you think this is way too slow, feel free to speed it up!I said the more general Cauchy integral formula is related, because in my original plan, I did want to say that Laurent coefficients take on exactly the same form, but it just occurred while I was finally editing the video that we don't find Laurent coefficients using integrals, and I don't want to send my Cauchy integral formula bit to waste, so here it is.Throughout this video series, of course I have left out lots of theorems in complex analysis, only talking about the things that I find more "applicable" (read: more audience want to watch). Things like Fundamental theorem of algebra, or maximum modulus principle, or even winding numbers are not presented, but in my defense, they are not really "essence of" anymore, because they use the concepts that we have developed in this series instead - like Cauchy integral formula as seen here.This video was sponsored by DataCamp.📖📖MORE READING📖📖(Simply connected version) en.wikipedia.org/wiki/Cauchy%27s_integral_theorem(Homologous to 0 version)https://www-users.cse.umn.edu/~brubaker/docs/8701-F13/dixon.pdfSurely, You’re Joking, Mr Feynman: amazon.co.uk/Surely-Youre-Joking-Feynman-Adventures/dp/009917331XMath StackExchange thread: math.stackexchange.com/questions/253910/the-integral-that-stumped-feynmanResidue theorem example links:Wikipedia: en.wikipedia.org/wiki/Residue_theorem#An_integral_along_the_real_axis[The example that I nicked from]You will need - estimation lemma (a.k.a. ML lemma): en.wikipedia.org/wiki/Estimation_lemmaIf you want to watch other videos on the exact same integral instead (although I think the Wikipedia page is a more “elementary” way of finding residues), you might want to have a look at:Classifying singularities: https://sites.oxy.edu/ron/math/312/14/ws/24.pdf (page 1)More ways of calculating residues: en.wikipedia.org/wiki/Residue_(complex_analysis)#Calculating_residuesYouTube videos talking about exactly the same integral (though they all assume quite a bit of familiarity of the above):Flammable maths: youtube.com/watch?v=YWBdwYr6PGgqncubed3: youtube.com/watch?v=DvND8J32gOAtahiri math science online teaching: youtube.com/watch?v=oGEo_BxqZ3c🎶🎶Music used🎶🎶Aakash Gandhi - Heavenly / Kiss the Sky / Lifting Dreams / White RiverAsher Fulero - The Closing of SummerVideo chapter:00:00 Complex integration (first try)06:01 Pólya vector field08:18 Complex integration (second try)12:27 Cauchy's theorem18:39 Integrating 1/z22:28 Other powers of z28:26 Cauchy integral formula31:43 Residue theorem36:14 But why?Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukFor my contact email, check my About page on a PC.See you next time!
What if we define 1/0 = ∞? | Möbius transformations visualized Mathemaniac 2021-12-17 | Head to brilliant.org/Mathemaniac to get started for free with Brilliant's interactive lessons. The first 200 listeners will also get 20% off an annual membership.Defining 1/0 = ∞ isn't actually that bad, and actually the natural definition if you are on the Riemann sphere - ∞ is just an ordinary point on the sphere! Here is the exposition on Möbius maps, which will explain why 1/0 = ∞ isn't actually something crazy. And this video will also briefly mention the applications of the Möbius map.There will also be things like circular and spherical inversion, which are really neat tools in Euclidean geometry to help us establish lots of interesting results, this one included.This video was sponsored by Brilliant.Video chapters:00:00 Intro02:38 Chapter 1: The 2D perspective08:43 Chapter 2: More about inversion14:33 Chapter 3: The 3D perspective (1/z)19:38 Chapter 4: The 3D perspective (general)---------------------------------------------------SOURCES:[That 2012 paper] Rigid motion 1-1 Möbius map: https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1218&context=rhumjMöbius transformations revealed: youtube.com/watch?v=0z1fIsUNhO4Accompanying paper: https://www-users.cse.umn.edu/~arnold//papers/moebius.pdfUnitary iff rotation: https://users.math.msu.edu/users/shapiro/pubvit/Downloads/RS_Rotation/rotation.pdfMöbius iff sphere: https://home.iitm.ac.in/jaikrishnan/MA5360/files/mobius.pdfRotation of Riemann sphere: https://people.reed.edu/~jerry/311/rotate.pdfCircle-preserving implies Möbius: onlinelibrary.wiley.com/doi/epdf/10.1002/mana.19670330506Problem of Apollonius video: youtube.com/watch?v=Z6GG8zsMWH8Power of a point: nagwa.com/en/explainers/798164323509--------------------------------------------------------MORE CONNECTIONS OF MÖBIUS MAPS:Sir Roger Penrose lecture on the book with Rindler (Spinors and space-time): youtube.com/watch?v=fzYV6VrsHyQThe book: cambridge.org/core/books/spinors-and-spacetime/B66766D4755F13B98F95D0EB6DF26526Hyperbolic geometry: assets.cambridge.org/97811071/16740/excerpt/9781107116740_excerpt.pdfConformal mapping (fluid mechanics): https://math.berkeley.edu/~iliopoum/Topics_121A/Conformal%20mapping%20in%20fluid%20mechanics.pdf--------------------------------------------------------Music used:Aakash Gandhi - Heavenly / Kiss the Sky / Lifting Dreams / White RiverAsher Fulero - The Closing of Summer--------------------------------------------------------Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukFor my contact email, check my About page on a PC.See you next time!
What do complex functions look like? | Essence of complex analysis #4 Mathemaniac 2021-10-25 | A compilation of plots of different complex functions, like adding and multiplying complex constants, exponentiation, the power function (including nth roots), and logarithm. Issues like branch cuts, branch points, and branches in general will also be discussed as the result of inability to construct the plots. Finally, we will do a 4D rotation (composed of two 4D reflections) to the typical Riemann surfaces pictures, and see that it should be the same as its inverse functions.The video is going to be jam-packed with visuals and animations, so while it may sometimes be too quick, you can pause the video; or you can just simply appreciate the visuals, the plots, and the animations.Some interesting plots are usually the vector plots, like for the power functions, we have different regions of flow. The formula of 2(n+1) when n is positive can be left as an exercise - it is not TOO difficult to see why, but it is not the focus of the video, or not the primary feature that I want to discuss; or for negative powers, we have dipole, quadrupole, and octupole, and in general multipole, which might be familiar to physicists, because in electromagnetism, we use multipole expansions to see the dominant effects of the electric field.Watch the previous video to see what the 5 methods of visualisation I am referring to, and also watch the Problem of Apollonius video for the next video!Video chapters:00:00 Introduction01:01 Adding constant02:51 Multiplying constant06:14 Exponentiation09:47 Power function - integer powers14:11 Power function - complex inversion15:39 Power function - square root branches20:37 Power function - Riemann surfaces22:53 Logarithm26:51 Logarithm - 4D rotationMusic used:Recollections - Asher FuleroStinson - Reed MathisBeseeched - Asher FuleroWhite River - Aakash GandhiOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Merch: mathemaniac.myspreadshop.co.ukFor my contact email, check my About page on a PC.See you next time!
The 5 ways to visualize complex functions | Essence of complex analysis #3 Mathemaniac 2021-09-12 | Complex functions are 4-dimensional: its input and output are complex numbers, and so represented in 2 dimensions each, so how do we visualize complex functions if we are living in a 3D world? There are actually 5 different ways to visualize a complex function, and this video is going to explore a bit about each of them.Some of you commented that you have already studied complex analysis in full, but hopefully there are still some things that you haven't seen before, because a typical university course on complex analysis wouldn't contain as many visuals as seen in this video.I know this might not be recommended by YouTube as much simply because the video is not that long (less than 20 minutes), and it seems like YouTube only puts my videos in recommendations when my video is very long. Originally I wanted to put things that will be covered in the next video into this particular video, but I figured that it doesn't make sense to cram two quite separate things into one video just for the sake of watch time. So please consider sharing this video, liking and commenting so that more people can watch it!LEARN A LITTLE BIT MORE HERE:More about complex visualisation: http://www.nucalc.com/ComplexFunctions.htmlJoukowsky transform: complex-analysis.com/content/joukowsky_airfoil.htmlJoukowsky transform (NASA): grc.nasa.gov/www/k-12/airplane/map.htmlCredits to Yehuda, there is an interactive tool to obtain the domain colouring plot for complex functions here: https://people.math.osu.edu/fowler.291/phase/Music used:Heavenly - Aakash Gandhifrom YouTube audio libraryVideo chapters:00:00 Introduction01:03 Domain colouring03:35 3D plots05:45 Vector fields07:50 z-w planes10:53 Riemann spheresOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)For my contact email, check my About page on a PC.See you next time!
What are complex numbers? | Essence of complex analysis #2 Mathemaniac 2021-08-18 | A complete guide to the basics of complex numbers. Feel free to pause and catch a breath if you feel like it - it's meant to be a crash course!Complex numbers are useful in basically all sorts of applications, because even in the real world, making things complex sometimes, oxymoronically, makes things simpler. This is the basis for a lot of physical applications, like circuit analysis, as well as slightly more far-fetched like Fourier analysis or Laplace transforms. Even for pure mathematicians, this is very useful due to its algebraic closure, and the beautiful world of complex analysis. As said before in the previous video, complex analysis is an extremely powerful tool that everyone should be excited about!This video will cover basically all of the topics you will ever learn in a normal course in complex numbers, or things considered basics in complex analysis, namely, Cartesian and polar ways of representation, Euler's formula and identity, the different operations of complex numbers, including complex exponentiation like i^i, de Moivre's theorem helping us find nth root of any complex numbers and so on. Even if you know a lot about complex numbers, maybe the proofs of Euler's formula is something that you might not have seen, or just maybe use this as a refresher! This video also touches on the argument issue, which could naturally lead to branch cuts, but this will be postponed to the next video.This is a part of a video series on complex analysis. Next video is going to be about visualising complex functions, and of course, the messy stuff about it, like the branch points and branch cuts. Don't forget to subscribe with the bell icon on so that you can be notified of my new uploads in the series!Video chapters:00:00 Sarcastic and serious introductions01:32 1.1 Complex plane - Cartesian way02:47 1.2 Complex plane - Polar way (Intro)03:25 1.3 Arguments about arguments04:57 1.4 Interconversion 07:48 2.1 Euler's formula - classic proof09:22 2.2 Euler's formula - 2nd proof13:03 3.1 Operations - addition/subtraction14:12 3.2 Operations - multiplication16:06 3.3 Operations - conjugation17:45 3.4 Operations - division19:23 3.5 Operations - exponentiation20:11 3.6 Operations - logarithm21:50 3.7 Operations - sine/cosine23:56 4.1 de Moivre's theorem - intro24:49 4.2 de Moivre's theorem - nth roots28:53 4.3 de Moivre's theorem - Euler's formula 3rd proof31:12 OutroMusic used:Asher Fulero - BeseechedAakash Gandhi - Kiss the SkyAsher Fulero - Night SnowAsher Fulero - RenunciationAsher Fulero - UnrequitedAll obtained from YouTube audio library.THINGS MENTIONED AND FURTHER READING/WATCHING:youtube.com/watch?v=v0YEaeIClKY (3Blue1Brown video on the second proof)youtu.be/xowA7Z5aee0 (My video on pi calculation)en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem (Gelfond-Schneider theorem: why e^(-pi/2) is transcendental)youtube.com/watch?v=N-KXStupwsc (Mathologer video on cubic formula)en.wikipedia.org/wiki/Cubic_equation (Cubic equation)brilliant.org/wiki/de-moivres-theorem (More uses of de Moivre’s theorem; not sponsored)youtube.com/watch?v=-dhHrg-KbJ0 (Mathologer video on the final proof of Euler’s formula)en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci_identity (Originally wanted to put this in, but don’t know where to put it, so: a nice application of complex numbers)Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)For my contact email, check my About page on a PC.See you next time!
Why care about complex analysis? | Essence of complex analysis #1 Mathemaniac 2021-08-02 | Complex analysis is an incredibly powerful tool used in many applications, specifically in solving differential equations (Laplace's and others via inverse Fourier / Laplace transforms), and of course, fundamental theorem of algebra, Riemann hypothesis, as well as solving complicated integrals to show off!This is the start of a series on complex analysis, which focuses on the visual insights rather than the more traditional rigorous approach, so it is not very likely that we will touch upon more advanced, yet very remarkable theorems in complex analysis, like Riemann mapping theorem, or the Picard's theorems. This is more likely of more use to the applied mathematicians, physicists and engineers, at least in the first few videos, simply because the interesting theorems are most likely pushed towards the end, and grouped as "applications of complex analysis", like FTA and RH.This is only currently my plan, and things are likely to change, but hopefully this is the backbone of what's coming next. I might not be uploading like weekly, but hopefully as frequently as I hope. Videos mentioned:youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF (Welch Labs complex numbers playlist)youtube.com/watch?v=0z1fIsUNhO4 (Möbius transformations video)youtube.com/watch?v=b8_3PFjiJvY (Zach Star complex derivatives)youtube.com/playlist?list=PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7 (Khan academy playlist on vector calculus)youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa (Dr Trefor Bazett’s playlist on vector calculus)Some (sort of) unexpected applications / connections to complex analysis:en.wikipedia.org/wiki/Joukowsky_transform (historical aerofoil application)http://math.colorado.edu/~kstange/papers/notes-Spin.pdf (Relationship between Möbius map and special relativity)Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)For my contact email, check my About page on a PC.See you next time!
Greens functions: the genius way to solve DEs Mathemaniac 2021-07-20 | Green's functions is a very powerful and clever technique to solve many differential equations, and since differential equations are the language of lots of physics, including both classical mechanics, electrodynamics, and even quantum field theory, so it is important to know how it works. Of course, this includes some explanation and perhaps a pretty different motivation for Dirac delta function, which is pretty weird, but also not really when you think about it a different way. Correction: in 19:11, the Green's function lacks a factor of 1/m.This video simply aims to introduce the Green's functions, what it is supposed to do, how the motivation of it all comes to be, and why it works. If you do need a lot more than introductory knowledge on Green's functions, and you are comfortable in basic differential equation solving, here are some links:brilliant.org/wiki/greens-functions-in-physics (not sponsored)roe.ac.uk/japwww/teaching/fourier/fourier_lectures_part4.pdfMy further write-up on the examples of Green's functions (how to find): drive.google.com/file/d/1D6E857eTvqM1CQgS1vYwcLqhLeGFL-aV/view?usp=sharingFor those who want some answers for the exercise towards the end of Chapter 3, i.e. around 15:47:Essentially, what I intended was that using that momentum change = integral of force over small period of time, you can obtain the first answer (by a similar definition of delta function in 1D), and I am expecting "point impulse / impulse" on Q2.For Q3: It is supposed to be that "applied force can be thought of as a 'continuous sum' of point impulses".For Q4: the Green's function describes the displacement of the oscillator after we apply an impulse. For this reason, Green's function is usually called the "impulse response".For Q5: Exactly copying the "adding different charge distributions (implies) adding up the electric potential", so in this case, "adding different forces (implies) adding up the displacement"For Q6: From the formula that x(t) = int G(t, tau)*F(tau) d(tau), we can interpret that the displacement is a continuous sum of the impulse responses.I stopped saying anything more because (1) the video is already very long, (2) this video assumes only basic knowledge of calculus (it is actually better if you don't know too much of the rigour in real analysis, since this is really hand-wavy - and it has to be! Otherwise this would be a lecture in distribution theory, which I am not quite well-versed in), and (3) this really just aims to provide motivation for Green's functions and doing examples would make this more "textbook-y" than it already is.Of course, the link to the Wikipedia table of Green's functions:en.wikipedia.org/wiki/Green%27s_function#Table_of_Green's_functionsNote: they don't state the boundary / initial conditions explicitly, and they don't even use x and xi, or t and tau, usually just their difference. Usually it is that the Green's functions vanish when the position is far away from the origin, and for those involving time, 0 before time tau, assuming that tau is greater than 0 (the so-called "advanced" Green's function)A little bit of remark after viewing the video once again: in some places, it is a little bit quick, so please treat yourself by pausing if necessary: YouTube allows you to do so! In my defense, different people require different time to pause, and also I don't want too much of dead air, so... that is probably also how a lot of other math videos on YouTube are doing right now.Video chapters:00:00 Introduction01:01 Linear differential operators03:54 Dirac delta "function"09:56 Principle of Green's functions15:50 Sadly, DE is not as easyOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Ko-fi: ko-fi.com/mathemaniacFor my contact email, check my About page on a PC.See you next time!
From miller to entering Cambridge at age 40 - George Green Mathemaniac 2021-06-30 | This is the legend of George Green (1793 - 1841), a mathematician / physicist, "son of a miller". First time going outside to film, and I only have my phone, so I'm really sorry that the video quality is not very good.Green's theorem, identities, and functions are immensely useful in 19th century physics, and even in the 20th century for quantum electrodynamics, but not a lot of people know who this person is, unless you have been to college or uni. When Albert Einstein, Julian Schwinger, Lord Kelvin, Sturm and Liouville all speak very highly of Green, yet not everyone knows who he is, is a little bit sad... This is really an injustice, so I hope to bring this mathematician to more attention so that people recognise his efforts in mathematical physics. This video was made because I remember my lecturer talked about George Green only being a miller while he discovered Green's theorem, so I decided to look more into the story and continued to be amazed. Hopefully you are as well!(Please don't tell the Cambridge University Library about this, because I don't think posting stuff about the library on the internet is allowed. Shhhhh...)This is hopefully a new side of Mathemaniac you haven't seen before, but the next video is going to be a more typical one. There are loads of stuff that I came across in the research, primarily by reading Cannell's biography, that I haven't put into the video, just because I don't feel like this is too relevant to the whole story, but this is just to tell you that there is a lot more history going on behind the scenes, e.g. why did Green use Leibniz notation throughout the paper, while the majority of the UK was still using Newton's notation? If you want to know, refer to the biography!Correction: 9:25 The German journal name should be Klassiker der Exacten Wissenschaf*t*en instead.SOURCES:1) Cannell, D. M., and Society for Industrial Applied Mathematics, George Green, Mathematician & Physicist, 1793-1841 : the Background to His Life and Work, 2nd ed. (Philadelphia: Society for Industrial and Applied Mathematics, 2001)2) "The Green of Green's functions" from the University of Nottingham: nottingham.ac.uk/physics/documents/historical/greenphystoday1203.pdf3) "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism" by George Green, online here: arxiv.org/pdf/0807.0088.pdfWHAT'S USED IN THIS VIDEO:1) Einstein in Nottingham (on YouTube): youtube.com/watch?v=161UNSza_qkNote: This is not really the instance where he said "Green was 20 years ahead of his time", but the only footage I could find that sort of represents Einstein's visit to Nottingham back then.2) Music used: Nine Lives - Unicorn HeadsNight Run Away - An JoneWolf Moon - Unicorn Heads(All from YouTube Audio Library)FURTHER READING:1) George Green: An Enigmatic Mathematician, by D.M. Cannell: jstor.org/stable/2589050?seq=13#metadata_info_tab_contents2) Einstein's Blackboard (Periodic Videos): youtube.com/watch?v=KIwpGEvmgvsOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH63) Math/Maths History Tour of Nottingham - George Green: Miller, Mathematician, and Physicist: youtube.com/watch?v=vK9NQ6e6rng4) Green's Functions (Sixty Symbols): youtube.com/watch?v=ji-i6XCkgC0Note: Yes, I know this is similar to what I did in this video, but hopefully slightly more researched, and a completely different style.If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)For my contact email, check my About page on a PC.See you next time!
Solved simply: the impossible integral Mathemaniac 2021-04-16 | Yes, it can't be done using substitution, by parts or changing variables (and using the Jacobian); but there is a very clever trick to actually compute this integral, which is attributed to Crofton, an English mathematician.This clever trick only requires the law of total expectation and some very simple algebraic manipulations, and is very elegant in solving this very complicated integral, and it is incredibly powerful in the sense that it can be used in much more general situations, not just this integral - when we want an average of some quantities (which needs to be a bounded symmetric function of n points), we can use the Crofton's differential equation already to convert the problem to the average quantity when 1 point is on the boundary of the domain. In this case, the differential equation is easy since we already know that the average distance is proportional to the radius.The problem can be made even more interesting when we think of higher dimensions: what about the average distance in a unit ball, or an n-dimensional ball? The calculations might be a bit tedious, but doable, and it again simply relies on the Crofton's differential equation. The only difficult part would be to figure out the limits of integration and the Jacobian determinant when using higher-dimensional spherical polar coordinates, and you can see that in the sources below.Even if you don't know the Jacobian, or multiple integrals, you can still at least understand the clever trick behind this, which is the more important message of this video.I currently have plenty of video ideas, but none of them really forms fully into a plan yet, so if you do have any video ideas, drop a comment below!SOURCE FOR THIS VIDEO SERIES:tandfonline.com/doi/abs/10.1080/07468342.1997.11973860 [The paper that I'm following, with a bit of biological motivation of the problem]jstor.org/stable/2589434?seq=1#metadata_info_tab_contents (a more technical version for Crofton's differential equation)If you want to know the higher-dimensional analog of spherical polar coordinates, and possibly want to derive the average distance in n-dimensional ball yourself, you can see the exercise 5.19 in Chapter 4 (Multiple integrals) on page 268 of C.H. Edwards, Jr., Advanced Calculus of Several Variables, Academic Press, San Diego, 1973Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)For my contact email, check my About page on a PC.See you next time!
What is Jacobian? | The right way of thinking derivatives and integrals Mathemaniac 2021-03-18 | Jacobian matrix and determinant are very important in multivariable calculus, but to understand them, we first need to rethink what derivatives and integrals mean. We can't think of derivatives as slopes if you want to generalise - there are four dimensions to graph the function! This video hopes to explain what the Jacobian matrix and determinant really mean, and essentially why they are actually very natural for changing variables; and also explaining something that might be glossed over when you use them - for example, we require absolute value, and the changing variables function is injective.In the video, we have only talked about 2D transformations, but the Jacobian can be easily generalised to any number of dimensions you like - you just need to introduce linear maps in higher dimensions! Think about what that means in 3 dimensions for a start!This video simply aims to introduce the intuition of the Jacobian, and so a lot of things said in the video is not going to be very rigorous - for example, what does approximate mean? It has a specific meaning in mathematics, but we are not getting there; and also not all functions have this nice property of looking like a linear map near a point. These belong to the realm of real analysis, which is well beyond the scope of this video. So please don't shout Fubini's theorem when you see flipping the order of integration at about 17:09. Video chapter feature:00:00 Introduction01:20 Chapter 1: Linear maps06:01 Chapter 2: Derivatives in 1D08:08 Chapter 3: Derivatives in 2D13:01 Chapter 4: What is integration?17:26 Chapter 5: Changing variables in integration (1D)19:25 Chapter 6: Changing variables in integration (2D)22:59 Chapter 7: Cartesian to polarIf you are interested in thinking about how the formula for the determinant came about, here is it: https://moodle.tau.ac.il/2018/pluginfile.php/403616/mod_resource/content/1/Nelsen%201993%20Proofs%20without%20Words.pdf (p. 134, 135)Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)Ko-fi: ko-fi.com/mathemaniacFor my contact email, check my About page on a PC.See you next time!
The numerical simulation is NOT as easy as you think! - Average distance #2 Mathemaniac 2021-01-26 | Continuing from part 1 (intro), we conduct a numerical simulation to calculate the average distance between two points in a unit disc. It turns out that the simulation is not as straightforward as you previously thought - it requires a bit of tweaking to sample points in the unit disc correctly.There will be concepts including inverse transform sampling, t-distribution, and t-tests in the video, with inverse transform sampling having a more detailed explanation, because it is a considerably simpler concept which doesn't require too much prior knowledge. There is also a passing mention of the Box-Muller transform, which is used as an example of the pitfall of the inverse transform sampling - even though it works for all distributions, sometimes it isn't computationally efficient.Even though this problem "highlights the unity and utility of the undergraduate mathematics curriculum" (from the paper below), I would assume you know nothing, so don't worry if you are not in university / have a degree in mathematics! If you are in college / university, hopefully the first few videos can be a nice revision and application of the concepts, and possibly a new perspective on the concepts.The paper that I am following (very readable, for an undergraduate at least):tandfonline.com/doi/abs/10.1080/07468342.1997.11973860I do notice that MindYourDecisions made a similar video (youtube.com/watch?v=i4VqXRRXi68 ) a few years ago but for a unit *square* instead. I still make this video series because (1) the unit disc version is much harder to tackle in the sense that we are not even attempting to evaluate the integral, and (2) Presh's video seemed to pull pdf's and Jacobian out of nowhere, which might be confusing to people who have not gone to college to study mathematics, and genuinely quite a different level of difficulty from his other videos, so I am going to actually explain what those are.Thank you so much for the overwhelming support for the video about the Dream Minecraft speedrun cheating drama! Hopefully this channel makes you like mathematics a bit more!**CORRECTION**6:06 I said F^(-1)(Y) less than r, but actually should be x, as said on the screen, because my script has been revised.8:11 I mean *sample size* not the number of samples.**SLIGHT CAVEAT**Technically, we should consider F(x) to be the probability that X smaller than *or equal to* x, because this will be different if the distribution does not have a well-defined probability density. In those cases, the inverse of F is not as straightforward, but we can still define the inverse. See the inverse transform sampling method Wikipedia page below.I might not have stressed enough, but it is shown on the screen, that in the general case, Y is a random number generator from 0 to 1. The range here is important because this will allow us to say that the probability that Y is less than or equal to F(x) is exactly F(x).**RESOURCES FOR MORE DETAILS**Numerical simulation data:docs.google.com/spreadsheets/d/1ltcLYBESXmATluMYh_lUl2YUXctDNNcxI4lknqRhllY/edit?usp=sharingMore about sampling methods described:en.wikipedia.org/wiki/Inverse_transform_samplingen.wikipedia.org/wiki/Box%E2%80%93Muller_transformen.wikipedia.org/wiki/Ziggurat_algorithm (an alternative algorithm for sampling a normally distributed variable)Single-sample and unpaired two-sample t-tests:youtube.com/watch?v=QZ7kgmhdIwA Single-sample t-test (more detailed):youtube.com/watch?v=NQWZefn41VY(or consult any statistics textbook!)Crash Course statistics playlist:youtube.com/watch?v=zouPoc49xbk&list=PL8dPuuaLjXtNM_Y-bUAhblSAdWRnmBUcr&index=1Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)For my contact email, check my About page on a PC.See you next time!
Dream cheating scandal - explaining ALL the math simply Mathemaniac 2020-12-30 | In short, you don't need a PhD to understand the math going on in the entire Dream cheating scandal; you just need patience and this video ;)Recently, a very popular Minecraft player Dream got caught cheating in his speedrunning attempt by the moderation team. The moderation team did a mathematical analysis, and Dream tried to debunk the math used by hiring an astrophysicist, but the math used in astrophysicist's paper was questionable at best. This video will explain ALL the math involved, including from both the moderation team and the astrophysicist, and doesn't offer any value judgement of the character of anyone involved. This hopefully can help you make an informed decision on who to trust / what to believe / what to think.If you watch this video just because you are a fan of this channel (good for you), this is also a very good opportunity of knowing the statistical tools for bias corrections, or just probability and statistics in general - binomial distribution, Fisher method, Bonferroni correction; or even just coding. This is quite possibly the longest video that I will EVER put on my channel.00:00 Disclaimer00:34 Background of the drama03:41 Binomial distribution05:46 Applying binomial distribution08:12 Bias 1: Stopping criterion12:02 Bias 2: Stream selection bias14:08 Bias 3: Runner selection bias15:24 Bias 4: p-hacking17:25 Mod team math summary18:03 Main criticism from astrophysicist19:37 Minor criticisms from astrophysicist21:14 Blatant mistake of astrophysicist25:06 Outro and endcard stuff***CORRECTIONS***:2:40: p-values mean getting AT LEAST as lucky as these success rates, not just as lucky.13:00: The streams aren't exactly independent, so this is not an exact answer, but it works as an overestimate nonetheless because they are positively correlated; if you are not convinced, you can also consider the Bonferroni correction, which is very close to this.13:50: I said 1.19 * 10^(-12), but the figure shown on screen is correct: 1.19*10^(-11).14:51: 1000 is the UPPER BOUND, not an estimation19:30: This is based on the faulty assumption that all speedrun attempts are streamed.24:20: The astrophysicist's 1 in 6300 is somewhat close to the chance of 18 heads in a row, or 19 of heads or tails in a row, so maybe some coding mistakes, or that he is exceptionally lucky in obtaining so many runs of 20 heads in his simulation.Also, the early stopping identified by the astrophysicist is not even true - in Dream's speedruns, he just throws a lot of gold to a lot of piglins in parallel to see what is traded.Further reading:1) Markov Chain solution: bit.ly/3pHcrRo2) Numpy.round/Numpy.around and numpy.random.uniform() function implementationbit.ly/3pGubN8bit.ly/3pDA8dr4) Fisher method of combining p-values: bit.ly/3pxjFaG5) (not reading, but a video) Biases in statistical analysis in science: bit.ly/34Zex7D6) Karl Jobst (another speedrunner) video on this situation, focusing on simulations: bit.ly/35072wX***SOURCES***Mod team original paper: mcspeedrun.com/dream.pdfDream's astrophysicist paper: bit.ly/3nYQa1bMod team rebuttal: mcspeedrun.com/dream/rebuttal.pdfCodes that I used in this video for calculation: bit.ly/2KGFLZNLoot table and Minecraft images:bit.ly/2KOg3Cobit.ly/34Z3Hykbit.ly/2Jz7riBbit.ly/3rGndJDbit.ly/3o4WdBsbit.ly/3hJ2dxvbit.ly/3n3M0Uubit.ly/2X09cZmbit.ly/3pEQMJWbit.ly/34Z3LOAOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)For my contact email, check my About page on a PC.See you next time!
What is the average distance of two points in a disc? (PART 1) Mathemaniac 2020-12-23 | This seemingly simple question actually encompasses very rich connections between different topics in mathematics, including statistics / probability theory, Jacobian / multivariable calculus, and even differential equations.This video is the first part in a video series tackling this question, which tries to make the problem more mathematically precise by breaking down the words "random" and "average", which can have precise mathematical definition, using probability density function, and the concept of mean of a function.The "joke" in the beginning of the video is inspired by (read: "copied from") CGPGrey's "How many countries are there?" youtube.com/watch?v=4AivEQmfPpk,Tom Scott's "How many languages are there?" youtube.com/watch?v=qYlmFfsyLMo,and Tom Scott's "The Never-Used Road Where The BBC Crash Cars" youtube.com/watch?v=nQOroZNr_fwEven though this problem "highlights the unity and utility of the undergraduate mathematics curriculum" (from the paper below), I would assume you know nothing, so don't worry if you are not in university / have a degree in mathematics! If you are in college / university, hopefully the first few videos can be a nice revision and application of the concepts, and possibly a new perspective on the concepts.The paper that I am following (very readable, for an undergraduate at least):tandfonline.com/doi/abs/10.1080/07468342.1997.11973860I do notice that MindYourDecisions made a similar video (youtube.com/watch?v=i4VqXRRXi68) a few years ago but for a unit *square* instead. I still make this video series because (1) the unit disc version is much harder to tackle in the sense that we are not even attempting to evaluate the integral, and (2) Presh's video seemed to pull pdf's and Jacobian out of nowhere, which might be confusing to people who have not gone to college to study mathematics, and genuinely quite a different level of difficulty from his other videos, so I am going to actually explain what those are.Thanks to all my subscribers, because this channel has grown a lot since the beginning of the year, and this cannot happen without your support! Merry Christmas / happy holidays! We shall see next year, which hopefully will be better :)Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytFor my contact email, check my About page on a PC.See you next year!
Icosahedral symmetry - conjugacy classes and simplicity Mathemaniac 2020-12-07 | How do we prove the rotational symmetries of icosahedron form a simple group? But wait, how do we prove *any* group is simple? The key to that involves the concept of conjugacy classes. This video explains intuitively why a normal subgroup has to be a union of conjugacy classes.This video is a continuation of the summary of my previous video series, and it is highly recommended that you watch the entire video series before this video, because there are a lot of intuitions developed throughout the video series, like conjugation is simply viewing symmetries in different perspectives. It might not make sense if you have not heard of this intuition of conjugation before.Essence of Group Theory video series: youtube.com/playlist?list=PLDcSwjT2BF_VuNbn8HiHZKKy59SgnIAeOI haven't had the time to talk about centralisers and centre, which are strongly associated with the concept of conjugacy classes, because these two other concepts are less related to simplicity of the group under consideration. Maybe another video on these two concepts?Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytFor my contact email, check my About page on a PC.See you next time!
Without integration, why is the volume of a paraboloid half of its inscribing cylinder? (DIw/oI #8) Mathemaniac 2020-10-12 | Rather than using integration, can we find the volume of a paraboloid? Yes, if we accept a precursor to calculus - Cavalieri's principle. Usually, integration is needed to find the volume of a paraboloid, for example using shell method, but using Cavalieri's principle, and a sneaky little trick, we can find the volume very easily - half of the volume of the circumscribing cylinder!The idea for this video isn't actually mine, but thanks to Yehuda Simcha Waldman for suggesting the idea of this video! He emailed me about the proof, and I modified it a little bit and adapted it into this video that you are watching here.Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytFor my contact email, check my About page on a PC.See you next time!
Problem of Apollonius - what does it teach us about problem solving? Mathemaniac 2020-09-21 | This video uses the problem of Apollonius as a way to introduce circle inversion and an important problem-solving technique - transforming a hard problem into a simpler one; then solve for the simpler, transformed version of the problem before doing the inverse transformation so that we obtain the solution to the hard problem. This problem-solving technique is the motive behind Fourier transform and other transforms like that (Laplace / Mellin): transforming the forced ODEs to a polynomial function, which is much easier than the original problem.Circle inversion is also an important technique in Euclidean geometry - it grants access to many more advanced geometric problems, like Pappus chain. This is because it really changes the form of the geometric object as opposed to translation, reflection, rotation and dilations; but it still preserves circles or lines (or generalised circles - lines can be thought of as circles with infinite radius), i.e. generalised circles mapped to generalised circles, and its anticonformal property: preserving angles while reversing orientations. This can be used to prove that complex inversion 1/z is conformal; and hence an important tool in complex analysis as well, not just Euclidean geometry.Most people know other solutions to the problem of Apollonius, but I think this is a much easier solution to understand. If you are interested in checking out other solutions, be prepared to hold on to all your knowledge of Euclidean geometry (like power of a point / radical axis and so on) and check out these links:(1) Problem of Apollonius wikipedia page: en.wikipedia.org/wiki/Problem_of_Apollonius(2) Inversive solution of Problem of Apollonius: https://www.ntg.nl/maps/40/03.pdf(3) Other methods of solving: cut-the-knot.org/pythagoras/Apollonius.shtmlhttp://users.math.uoc.gr/~pamfilos/eGallery/problems/ApolloniusProblem.pdfSource of photo of Apollonius: researchgate.net/figure/Apollonius-of-Perga-252-170-BC_fig1_317673086Correction:At around 8:12, I should have said "we already know that *the centre of* the solution circle...", but it shouldn't materially impact the video. It's just a bad writing of script on my part, which I apologise.Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytFor my contact email, check my About page on a PC.See you next time!
Burnsides Lemma (Part 2) - combining math, science and music Mathemaniac 2020-08-27 | Part 1 (previous video): youtu.be/6kfbotHL0fsOrbit-stabilizer theorem: youtu.be/BfgMdi0OkPUBurnside's lemma is an interesting result in group theory that helps us count things with symmetries considered, e.g. in some situations, we don't want to count things that can be transformed into one another by rotation different, like in this case of an isomer - rotating an isomer yields the same isomer; and in the case of music theory, we might not want to count chords obtained by transposition the same.There is usually also something called Pólya Enumeration Theorem, which is a generalization of Burnside's lemma, and can be used to tackle a wider set of problems, but most problems that Pólya Enumeration Theorem can be applied to can also be tackled by Burnside's lemma, so this is usually more important. The theorem simply extracts the core of what we are doing (considering cycles for example) and put it in a nice generating function, which can be useful, but the notation and the computations required are very troublesome, and does not fit well too well with the theme of this channel.Non-mathematical applications like counting the number of isomers of an organic molecule (organic chemistry) and the number of trichords (music theory) are usually tackled by the theorem mentioned above, but this can really be tackled by Burnside's lemma, just with a bit more care. By the way, this lemma is not actually first discovered by Burnside, and the Pólya Enumeration theorem is also not first discovered by Pólya, but this phenomenon is also prevalent throughout mathematics and science, known as Stigler's law of eponymy.This is not a part of the "Essence of Group Theory" video series, because it is not "essence", but an application of the orbit-stabilizer theorem, which is in Chapter 2 of the video series: youtu.be/BfgMdi0OkPU****RESOURCES FOR THE POLYA ENUMERATION THEOREM****All the resources here illustrate the theorem using different examples, and the last one also presents the proof of the theorem. You will need to know cycle notation of permutations (a topic in symmetric groups that I previously said wouldn't be covered on this channel, because it is not very visual and that I couldn't give unique enough insight on YouTube)(1) https://www.whitman.edu/Documents/Academics/Mathematics/Huisinga.pdf(2) cp-algorithms.com/combinatorics/burnside.html(3) http://pi.math.cornell.edu/~apatotski/IHS2015/Lecture%2010.pdf(4) http://www.diva-portal.org/smash/get/diva2:324594/FULLTEXT01.pdf(5) https://www.math.cmu.edu/~af1p/Teaching/Combinatorics/Slides/Polya.pdf****GOOGLE FORM****Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!****SOCIAL MEDIA****Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytFor my contact email, check my About page on a PC.See you next time!#mathemaniac #grouptheory #abstractalgebra #burnside #mathematics
Burnsides Lemma (Part 1) - combining group theory and combinatorics Mathemaniac 2020-08-20 | A result often used in math competitions, Burnside's lemma is an interesting result in group theory that helps us count things with symmetries considered, e.g. in some situations, we don't want to count things that can be transformed into one another by rotation different, like in this case - how many ways are there to paint a cube's faces when we are given three colors, if two coloring patterns are considered the same when they differ just by a rotation?There is usually also something called Pólya Enumeration Theorem, which is a generalization of Burnside's lemma, and can be used to tackle a wider set of problems, but most problems that Pólya Enumeration Theorem can be applied to can also be tackled by Burnside's lemma, so this is usually more important. The theorem simply extracts the core of what we are doing and put it in a nice generating function, which can be useful, but the notation and the computations required are very troublesome, and does not fit well too well with the theme of this channel.Non-mathematical applications like counting the number of isomers of an organic molecule (organic chemistry) and the number of trichords (music theory) are usually tackled by the theorem mentioned above, but this can really be tackled by Burnside's lemma, just with a bit more care. We will explore how this can be applied in those situations in the next video.By the way, this lemma is not actually first discovered by Burnside, and the Pólya Enumeration theorem is also not first discovered by Pólya, but this phenomenon is also prevalent throughout mathematics and science, known as Stigler's law of eponymy.This is not a part of the "Essence of Group Theory" video series, because it is not "essence", but an application of the orbit-stabilizer theorem, which is in Chapter 2 of the video series: youtu.be/BfgMdi0OkPUOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytFor my contact email, check my About page on a PC.See you next time!#mathemaniac #grouptheory #abstractalgebra #burnside #mathematics
Limitations of mathematical models; historical context of BGW process [PART III] Mathemaniac 2020-07-27 | Part 3 of a series on a stochastic process approach to model the spread of coronavirus (COVID-19) as opposed to the compartmental deterministic SIR model. This model is generally known as branching process, but this video only focuses on the simplest type, called Bienaymé-Galton-Watson (BGW) process. This video will especially be on the inevitable limitations on the BGW process model, to illustrate the limitations of any mathematical model in general. There can be problems in applicability and difficulty in interpreting data from the predictions of the model, but we can always strive to make the model more realistic. But the cost would be more complicated mathematics, and higher computing power necessary.The historical context of the Bienaymé-Galton-Watson (BGW) process will be also discussed in this video, and it is actually a bit more interesting than usual. It is not just that the three mathematicians got together and studied this process. I currently don't have a solid plan for my future videos, but feel free to comment below about the topics that I can cover in a future video! REFERENCES / SOURCES:(1) Branching Processes: Their Role in Epidemiology [basis / limitations / improvement of the model]ncbi.nlm.nih.gov/pmc/articles/PMC2872325/pdf/ijerph-07-01186.pdf(2) Branching Processes Since 1873 [historical context of the Bienaymé-Galton-Watson process]londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms/s1-41.1.385(3) The Genealogy of Genealogy Branching Processes before (and after) 1873 [historical context of the Bienaymé-Galton-Watson process]londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms/7.3.225(4) The Educational Times (March 1873) [Galton's original famous Problem 4001; historical context of the Bienaymé-Galton-Watson process]web.archive.org/web/20170123114453/http://ioearc.da.ulcc.ac.uk/9344/1/Educational%20Times%20Vol%2025%20Iss%20143.PDF(5) The Educational Times (August 1873) [Watson's (brief) answer to the Problem 4001; historical context of the Bienaymé-Galton-Watson process]web.archive.org/web/20161201212518/http://ioearc.da.ulcc.ac.uk/9309/1/Educational%20Times%20Vol%2026%20Iss%20148.PDF(6) On the probability of extinction of families [Watson's paper / more detailed albeit partially incorrect answer to the Problem 4001; historical context of the Bienaymé-Galton-Watson process]jstor.org/stable/pdf/2841222.pdf(7) BBC news [breaking lockdown rules complaint]: bbc.com/news/uk-england-humber-53361096(8) Guardian [KFC party breaking recent lockdown in Melbourne]:theguardian.com/australia-news/2020/jul/10/kfc-birthday-party-in-melbourne-costs-26000-in-covid-19-fines-after-police-track-order(9) Metro [18000 people breaking lockdown rules]: metro.co.uk/2020/06/28/18000-people-fined-breaking-lockdown-rules-since-march-12913927(10) Contact tracing map: ourworldindata.org/grapher/covid-contact-tracing?year=2020-07-10(11) What is contact tracing (CDC): cdc.gov/coronavirus/2019-ncov/daily-life-coping/contact-tracing.htmlOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!#mathemaniac #math #coronavirus #COVID_19 #SIRmodel #statistics #probability #epidemic #pandemicStay safe everyone! Please do wear a mask to protect ourselves, and stay at home as much as possible. We will get through this together.Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytFor my contact email, check my About page on a PC.See you next time!
How likely is coronavirus (COVID-19) eradicated? [PART II] Mathemaniac 2020-07-24 | Part 2 of a series of videos on a stochastic process approach to model the spread of coronavirus (COVID-19) as opposed to the compartmental deterministic SIR model. This model is generally known as branching process, but this video only focuses on the simplest type, called Bienaymé-Galton-Watson (BGW) process. This video will explore how we can extract the extinction probability (probability that coronavirus will eventually get eradicated) from the BGW process using the cobwebbing technique. Although it is not a rigorous approach, it is a very nice visual way to see what's happening. We then, not too surprisingly, come up with the concept of basic reproduction number, a concept that is seen in both the BGW model and the SIR model.This video will involve concepts like distribution, independence, expected value, generating function, and the cobwebbing techniques to visualise iteration processes and so on, but a basic understanding of the concept of probabilities will be good, and basic understanding on differentiation (definition as slope of a function, power rule and linearity of the differential operator) will be required.The next video on the limitations / improvement and the historical context of the BGW process should be released in a couple of days, because they are already done.REFERENCES / SOURCES (which I will explain in much more detail in later videos that I promised, but if you are impatient, you can read these):(1) Branching Processes: Their Role in Epidemiology [basis / limitations / improvement of the model]ncbi.nlm.nih.gov/pmc/articles/PMC2872325/pdf/ijerph-07-01186.pdf(2) Branching Processes Since 1873 [historical context of the Bienaymé-Galton-Watson process]londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms/s1-41.1.385(3) The Genealogy of Genealogy Branching Processes before (and after) 1873 [historical context of the Bienaymé-Galton-Watson process]londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms/7.3.225(4) The Educational Times (March 1873) [Galton's original famous Problem 4001; historical context of the Bienaymé-Galton-Watson process]web.archive.org/web/20170123114453/http://ioearc.da.ulcc.ac.uk/9344/1/Educational%20Times%20Vol%2025%20Iss%20143.PDF(5) The Educational Times (August 1873) [Watson's (brief) answer to the Problem 4001; historical context of the Bienaymé-Galton-Watson process]web.archive.org/web/20161201212518/http://ioearc.da.ulcc.ac.uk/9309/1/Educational%20Times%20Vol%2026%20Iss%20148.PDF(6) On the probability of extinction of families [Watson's paper / more detailed albeit partially incorrect answer to the Problem 4001; historical context of the Bienaymé-Galton-Watson process]jstor.org/stable/pdf/2841222.pdfOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!#mathemaniac #math #coronavirus #COVID_19 #SIRmodel #statistics #probability #epidemic #pandemicStay safe everyone! Please do wear a mask to protect ourselves, and stay at home as much as possible. We will get through this together.Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytFor my contact email, check my About page on a PC.See you next time!
Alternative to SIR: Modelling coronavirus (COVID-19) with stochastic process [PART I] Mathemaniac 2020-07-20 | A stochastic process approach to model the spread of coronavirus (COVID-19) as opposed to the compartmental deterministic SIR model. This model is generally known as branching process, but this video only focuses on the simplest type, called Bienaymé-Galton-Watson (BGW) process, because the math involved will get a LOT more complicated if we relax a few constraints that we impose for this simple process.This video will involve concepts like distribution, independence, expected value, generating function and so on, but a basic understanding of the concept of probabilities will suffice, because these will all be introduced in this video. It also demonstrates some problem-solving strategy of mathematicians (generating function: encoding [countably] infinite data into one single thing, and this can be decoded uniquely, which is justified with standard tools in real analysis).By the way, during the "rigour" part of the video where the video displayed a lot of text, the "potential generating function" needs to also be equal to its own Taylor series itself, because this is not necessarily guaranteed for a smooth function.The next videos on the likelihood that coronavirus will be eradicated (extinction probability), limitations / improvement and the historical context of the BGW process should be released in a couple of days, because they are already done.REFERENCES / SOURCES (which I will explain in much more detail in later videos that I promised, but if you are impatient, you can read these):(1) Branching Processes: Their Role in Epidemiology [basis / limitations / improvement of the model]ncbi.nlm.nih.gov/pmc/articles/PMC2872325/pdf/ijerph-07-01186.pdf(2) Branching Processes Since 1873 [historical context of the Bienaymé-Galton-Watson process]londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms/s1-41.1.385(3) The Genealogy of Genealogy Branching Processes before (and after) 1873 [historical context of the Bienaymé-Galton-Watson process]londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms/7.3.225(4) The Educational Times (March 1873) [Galton's original famous Problem 4001; historical context of the Bienaymé-Galton-Watson process]web.archive.org/web/20170123114453/http://ioearc.da.ulcc.ac.uk/9344/1/Educational%20Times%20Vol%2025%20Iss%20143.PDF(5) The Educational Times (August 1873) [Watson's (brief) answer to the Problem 4001; historical context of the Bienaymé-Galton-Watson process]web.archive.org/web/20161201212518/http://ioearc.da.ulcc.ac.uk/9309/1/Educational%20Times%20Vol%2026%20Iss%20148.PDF(6) On the probability of extinction of families [Watson's paper / more detailed albeit partially incorrect answer to the Problem 4001; historical context of the Bienaymé-Galton-Watson process]jstor.org/stable/pdf/2841222.pdfOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!#mathemaniac #math #coronavirus #COVID_19 #SIRmodel #statistics #probability #epidemic #pandemic #maskStay safe everyone! Please do wear a mask to protect ourselves, and stay at home as much as possible. We will get through this together.Social media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytFor my contact email, check my About page on a PC.See you next time!
Summary: an example covering ALL group theory concepts!! | Essence of Group Theory Mathemaniac 2020-07-05 | The summary of the entire video series! After a quick recap on all the important concepts covered in the series, we see a very interesting, yet a bit involved example to see how these concepts can be applied to prove an interesting result.The concepts that we used are:(1) The correspondence between action and homomorphism (where symmetric group comes in)(2) The three statements of isomorphism theorem(3) Lagrange's theoremBut as an aside, the group in the example is actually the group of rotational symmetries of a regular icosahedron (and dodecahedron, because they are dual to each other and has the same symmetry groups), and one can use Orbit-stabiliser theorem to verify that this group has 60 elements, and the intuition of conjugation to see that it is a simple group. I haven't filled up the details in the video, so leave a comment for the proof!I could not promise when the next video will be out, but hopefully it will be out in a few weeks (?), and I don't really want to give a time frame for that. Currently, the plan is to have the next video to be about the current epidemic, but there might be some other videos that get in the way as well.Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!#mathemaniac #math #grouptheory #abstractalgebra #mathematicsSocial media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytFor my contact email, check my About page on a PC.See you next time!
Chapter 7: Group actions, symmetric group and Cayley’s theorem | Essence of Group Theory Mathemaniac 2020-06-28 | Group action can be thought of as a homomorphism to a symmetric group, so apart from orbit-stabiliser theorem, we can also use the isomorphism theorem to analyse any group action. It turns out that this correspondence between group action and homomorphism can be visualised rather easily. This correspondence is very important in group theory, but often neglected.Symmetric group is also briefly mentioned here as a concept to facilitate the introduction of the above correspondence, but a more detailed analysis of the symmetric group would not be in this video series, because I don't think it is as intuitive as concepts discussed in the video series and therefore does not fit the theme of the series too well. However, I will do a video on Burnside's lemma and its interesting application, but it is not "Essence of Group Theory" anymore, because it is an application of the orbit-stabiliser theorem. I haven't mentioned in the video that I will also not do a video on matrix groups because it requires the knowledge of linear algebra, and again, the concepts discussed in the matrix groups will usually be algebraic.The next few videos will be a summary of this video series with one very cool example that really covers all the concepts discussed in this video series; another video on the current epidemic explaining an alternative model to the SIR discussed on this channel before, and will be about a stochastic branching process; then the Burnside's lemma video. There might be some videos in between these, but I will definitely do all of these videos in some time in the future.Cayley's theorem is named in honour of Arthur Cayley, a British (Cambridge) mathematician who is also known for a lot of mathematical results, like Cayley table, Cayley graph, Cayley's theorem and the famous Cayley-Hamilton theorem in linear algebra. However, even though it is a motivation for the study of symmetric groups, you might not see Cayley's theorem too often in more advanced studies of group theory. It mainly acts as a direct application of the correspondence discussed in this video.Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!#mathemaniac #math #grouptheory #groupaction #homomorphism #CayleySocial media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacyt
Two opposite games involving golden ratio (ft. Tom Rocks Maths) Mathemaniac 2020-05-27 | Thanks Tom for the little cameo in the beginning of the video! Dr. Tom Crawford, who got his PhD in Cambridge, is currently at the University of Oxford teaching undergraduates in St. Edmund Hall (nicknamed Teddy Hall, hence the name of the competition), St. Hugh's College and St. John's College. You might recognise him that he appeared on Numberphile talking about the Navier-Stokes equation.Tom Rocks Maths: youtube.com/channel/UCRfo-DAifrP3lzcxUHtGm_ATeddy Rocks Maths competition: tomrocksmaths.com/oxford-university/seh/teddy-rocks-mathsThe essay I submitted to the Teddy Rocks Maths competition: tomrocksmaths.files.wordpress.com/2020/05/trevor-cheung.pdf(The essay was written in kind of a hurry, so there were a lot of typos there, and hopefully this video is a kind of "enhanced" version of the essay.)The golden ratio appears unexpectedly in a game of removing stones, similar in style to NIM. It is a game played in the ancient Chinese, and there is also a variant called the Wythoff's game. The proof of the winning strategy is related to the Beatty's theorem and is shown in detail in this video.There are two points of fascination during this video: the fact that two seemingly opposite games very similar winning strategy, and that the golden ratio is involved in all of this.For people who might complain in the comments, phi should properly be pronounced as "fee" - that's the pronunciation in Greek. I adopted this pronunciation right from the Fibonacci video.Sources:(1) Wythoff's game: en.wikipedia.org/wiki/Wythoff%27s_game(2) Beatty sequences and theorem: en.wikipedia.org/wiki/Beatty_sequenceOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!#mathemaniac #math #goldenratio #tomrocksmaths #gamesSocial media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_Twitter: twitter.com/mathemaniacytFor my contact email, check my About page on a PC. See you next time!
Chapter 6: Homomorphism and (first) isomorphism theorem | Essence of Group Theory Mathemaniac 2020-04-15 | The isomorphism theorem is a very useful theorem when it comes to proving novel relationships in group theory, as well as proving something is a normal subgroup. But not many people can understand it intuitively and remember it just as a kind of algebraic coincidence. This video is about the intuition behind the idea of homomorphism - functions preserving group structures, and the closely related isomorphism theorem.There are second, third and even fourth isomorphism theorem (the fourth one is usually disputed), but all can be derived from the first one, using clever constructions of homomorphisms.Apparently, when I was typing the description (after the video is edited), I knew that the name "homomorphism" is probably mistranslated from German. Originally, it was supposed to mean "similar", not "same".This video series is about understanding the group theory intuitively, complementing how most people learn about it, because it is usually introduced as part of abstract algebra. Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video! SUBSCRIBE and see you in the next video! If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe! #mathemaniac #grouptheory #isomorphism #homomorphismSocial media: Facebook: facebook.com/mathemaniacyt Instagram: instagram.com/_mathemaniac_ Twitter: twitter.com/mathemaniacyt For my contact email, check my About page on a PC. See you next time!
Chapter 5: Quotient groups | Essence of Group Theory Mathemaniac 2020-04-01 | Quotient groups is a very important concept in group theory, because it has paramount importance in group homomorphisms (connection with the isomorphism theorem(s)). With this video series, abstract algebra needs not be abstract - one can easily develop intuitions for group theory!In fact, the concept of quotient groups is one way to define modular arithmetic formally, which allows us to prove a lot of number theory theorems once we draw parallels between group theory and number theory. For example, Fermat's little theorem and Euler's totient theorem are just corollaries of the Lagrange's theorem introduced in Chapter 3 of the video series.Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!SUBSCRIBE and see you in the next video!#mathemanic #grouptheory #abstractalgebra #quotientgroup #intuitionSocial media: Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_ Twitter: twitter.com/mathemaniacyt For my contact email, check my About page on a PC. See you next time!
How will the COVID-19 (coronavirus) epidemic end? Mathemaniac 2020-03-15 | When will the COVID-19 / coronavirus epidemic end? How many people will die from it? How many people will get an infection? How much should you worry about it? This video hopefully can give you a sense of what to expect via a simple mathematical model. It is a standard one with reasonable accuracy, called the SIR model, which illustrates exponential growth / decay depending on the ratio between two constants, called the basic reproduction ratio / number (i.e. R0). The prediction from the model is not that optimistic...We are still in the exponential growth stage, but eventually it has to be a logistic curve, and the video discusses where the plateau will be, and the time scale at which the plateau will be achieved.The video seems to be making a wild claim, but it really isn't. This is what some UK health officials predicted as well: theguardian.com/world/2020/mar/15/uk-coronavirus-crisis-to-last-until-spring-2021-and-could-see-79m-hospitalised?fbclid=IwAR2XWpGi3diiS4rrWwBjtSbOWY8A85THZL7D3w06X17Xlmo9yrF4Z9KW9pkThe measures that can be taken to reduce the number of infected individuals, as in the last part of the video, should give enough context to show how seemingly insignificant local measures can have a huge impact, and ultimately persuading the public to stay on alert and take part in those measures described.Sources:The spreadsheet generating the numbers: docs.google.com/spreadsheets/d/14XWEmLefkh-jRiHMeWc8kvM3sk9Q-Mx4EYHHlOSC_1Q/edit?usp=sharingWorldometer statistics: worldometers.info/coronavirusSIR model: en.wikipedia.org/wiki/Compartmental_models_in_epidemiology#The_SIR_modelOther than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe! Stay safe everyone! Please do wear a mask to protect ourselves, and stay at home as much as possible. We will get through this together. #coronavirus #COVID_19 #epidemic #mathematics #mathemaniac #math #washyourhands #dontgoout #pandemic #flattenthecurveSocial media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_ Twitter: twitter.com/mathemaniacyt For my contact email, check my About page on a PC. See you next time!
Chapter 4: Conjugation, normal subgroups and simple groups | Essence of Group Theory Mathemaniac 2020-03-09 | A VERY important concept of group theory, but often taught without any intuition, so much that it often confuses a lot of people when they first learned it (including me). Conjugation can be visualised easily with a (literal) change of perspective! This video also lays the foundation for quotient groups, which gives rise to some unexpected relationship with number theory.Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!#mathemaniac #math #conjugation #grouptheory #simplegroups #normal #abstractalgebra
Chapter 3: Lagranges theorem, Subgroups and Cosets | Essence of Group Theory Mathemaniac 2020-02-27 | Lagrange's theorem is another very important theorem in group theory, and is very intuitive once you see it the right way, like what is presented here. This video also discusses the idea of subgroups and cosets, which are crucial in the development of the Lagrange's theorem. Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!#mathemaniac #math #grouptheory #groups #abstractalgebra #lagrange
Chapter 2: Orbit-Stabiliser Theorem | Essence of Group Theory Mathemaniac 2020-02-17 | An intuitive explanation of the Orbit-Stabilis(z)er theorem (in the finite case). It emerges very apparently when counting the total number of symmetries in some tricky but easy way. This video series continues to develop your intuition towards some fundamental concepts and results in Group theory.Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!#mathemaniac #math #grouptheory #groups #orbitstabiliser #orbitstabilizer
Chapter 1: Symmetries, Groups and Actions | Essence of Group Theory Mathemaniac 2020-02-06 | Start of a video series on intuitions of group theory. Groups are often introduced as a kind of abstract algebraic object right from the start, which is not good for developing intuitions for first-time learners. This video series hopes to help you develop intuitions, which are useful in understanding group theory.In particular, this video is going to be about thinking groups as symmetries (or isometries to be precise) because they are much more visualisable, and that symmetries of an object do form a group using the abstract definition of the group that is usually given.Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!#mathemaniac #math #grouptheory #groups #intuitionSocial media:Facebook: facebook.com/mathemaniacytInstagram: instagram.com/_mathemaniac_ Twitter: twitter.com/mathemaniacytPatreon: patreon.com/mathemaniac (support if you want to and can afford to!)For my contact email, check my About page on a PC. See you next time!
Power rule of integrals not using integration; definition of e | DIw/oI #7 Mathemaniac 2020-01-27 | You know the power rule for integration, but can we derive the rule without integration? (What does this question even mean?) And can we have a visual definition of e? This video answers them all.This should really be the last video in the series of Doing Integrals without Integration, unless there are other ideas that fall into the umbrella of calculating some (curved) areas without integration.This idea was originally from Fermat, right before the invention of calculus. I cannot find any easy-to-read literature / source on the exact same topic, so if you happen to find one, please let me know in the comments.This whole video is adapted from a book called e: The Story of a Number by Eli Maor, which is a very good recreational mathematics book that I recommend all to read.Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!#mathemaniac #math #integration #powerrule
How do I love math? (Part II - Proof without words) Mathemaniac 2020-01-17 | Proofs without words are the gem of mathematics. Together with animations, if it doesn't make you love math, I don't know what will :)Proofs include:1a and 1b) Two formulas relating pi and inverse tangent functions2) Sum of first n natural numbers3) Sum of first n cubes4) Geometric series formulaSource: Proofs without Words https://moodle.tau.ac.il/2018/pluginfile.php/403616/mod_resource/content/1/Nelsen%201993%20Proofs%20without%20Words.pdfIf you want to treat yourself with some more proofs without words, go to the above link for some more. Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:https://forms.gle/QJ29hocF9uQAyZyH6If you want to know more interesting Mathematics, stay tuned for the next video!SUBSCRIBE and see you in the next video!
