Videos uploaded by user “CHALK”

Conformal Mappings are a gem of Complex analysis that play a big role in both the theory behind the analysis of functions of a complex variable as well as studying fluid dynamics and electrostatics in physics, along with general relativity. In this video, a brief introduction to these maps is given in preparation for some even cooler geometric mappings we will talk about next time.
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#CHALK #ConformalMappings #Analysis
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Views: 5133
CHALK

Dimension is a very intuitive idea in the real world, but what is it actually, in order to get a handle on it we look at Hausdorff dimension and the famous Sierpinski's triangle (or inception triforce :D ) along with scaling properties of sets to find something that lies in between our discrete notion of dimension.
P.S. Super glad I was still able to get a video up today. I know it is late, but I'm giving myself a pass on that since I previously said that I may not do one at all.
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#CHALK #Hausdorff #DimensionTheory
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Views: 1396
CHALK

How do Birds Breathe? Well they inhale and exhale and inhale and exhale... but you can find out how that works in the video!
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#CHALK #Birds #AnimalPhysiology
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Views: 4684
CHALK

The Axiom of Choice is often stated in an equivalent form; the Cartesian product of a collection of non-empty sets is non-empty, however, what is the original statement, and what does it have to do with functions? Also, what is a choice function? In this discussion of the axiom of choice, we'll also run into the statements of some paradoxes too!
CORRECTIONS: 0:42: The B is not the Codomain, only the things hit by the function f are in the Codomain and this was not clear from how I was talking about the terminal set.
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#CHALK #AoC #SetTheory
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Views: 3012
CHALK

Zorn's Lemma and The Well-ordering Theorem are seemingly straightforward statements, but they give incredibly mind-bending results. Orderings, Hasse Diagrams, and the Ordinals will come up in this video as tools to get a better view of where the proof of Zorn's lemma comes from.
***Corrections: Near the end of the video, an open interval is mentioned, but a half open half closed interval is drawn. The open interval to which I refer is the one drawn in white chalk.
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#CHALK #ZornsLemma #SetTheory
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Views: 7154
CHALK

What is CHALK? What is chalk? Wait, I can't hear capslock. Anyhow, I'll explain both, or at least what chalkboard chalk is; Gypsum and it's some pretty interesting stuff. From uses in education, construction, and many other areas, gypsum is something we often take for granted and so it's good to think about the places where it is used.
If you catch it in the card there's a poll! I would be super interested in knowing what topics you would like to see in the future.
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Views: 1595
CHALK

What type of math goes into memory and data storage? Well, as it turns out, Symbolic Dynamics, a subfield of Dynamical Systems theory is one such area of mathematics. In this video, we introduce the types of objects studied in symbolic dynamics, along with substitution maps and some recent research regarding shift spaces generated by substitution maps. I also show some of the research I have done in that I just presented at the JMM (Joint Mathematics Meetings) in San Deigo this past week.
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Topological Entropy for Symbolic Dynamics in more generality and rigor: http://www.scholarpedia.org/article/Symbolic_dynamics#Invariants_of_conjugacy_and_variants_of_the_conjugacy_problem
Rust and Spindeler Paper: https://www.math.uni-bielefeld.de/~drust/papers/sss.pdf
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standupmaths: https://www.youtube.com/channel/UCSju5G2aFaWMqn-_0YBtq5A
Numberphile: https://www.youtube.com/user/numberphile
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#CHALK #SymbolicDynamics #DynamicalSystems
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Views: 485
CHALK

Last time we talked about Conformal Mappings, but we didn't really give any specific examples. This episode is dedicated to producing a few of them that fall into the category of Schwarz-Christoffel Mappings. Being a bit handwavy, we'll look at the general form of a Schwarz-Christoffel mapping from the upper-half plane, and other mappings onto polygons. The coolest of which involve the Koch Snowflake, a fractal, which extends the idea of these mappings to "divergent polygons" ones with infinitely many sides and vertices.
This video was inspired by the paper by Gonzol Riera, Hernán Carrasco, and Rubén Preiss (2008) linked here: http://dx.doi.org/10.1155/2008/350326
Last Video: http://bit.ly/2nr4vWb
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Find the CHALKboard on Facebook: http://bit.ly/CHALKboard
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#CHALK #KochSnowflake #Analysis
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Views: 1237
CHALK

There are a lot of things that are pretty intimidating about Hungarian (or Magyar) especially if it is your first encounter with an agglutinative, non-Indo-European language. In this short video, I go over some of the fundamental differences between the two languages.
Also if you want to checkout some of the Hungarian music that I referenced with the hat in the video you can check out some of Wellhello's stuff here.
https://www.youtube.com/channel/UC9yLH0K0gmzfxJH9HjQOe9g
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Views: 233
CHALK

What percentage of people know how to do Mathematics? Sure computations are a good start, but mathematics and computation are different. So what is math, what is computation and how do they work together?
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#CHALK #Computation #Mathematics
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Views: 778
CHALK

Mappings and functions that work on topological spaces other than the real line can be counterintuitive, and in cases like with the sphere, we have things like the Borsuk-Ulam Theorem which can assert the truth of things that would appear, without proof, to be outright lies. In this video, we go through some ideas, like homotopy, antipode preserving mappings, and covering maps/ covering spaces which are at the base of the Borsuk-Ulam theorem and use them to give a sketch of a proof of the theorem.
CORRECTIONS!!! The denominator of the function g given near the end should be in absolute value. (I only wrote half of the absolute value whoops!)
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CHECK OUT THE NOTES THAT GO ALONG WITH THIS VIDEO!!!
http://bit.ly/BorsukUlam
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#CHALK #BorsukUlamTheorem #Topology
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Views: 353
CHALK

Hedgehog spaces and, more generally, metrics/distance functions illustrate what I meant last week when I said a definition of a circle based on distance is ambiguous. Using these ideas, I thought I would go ahead and show you why, and then leave you with a circle and see if you could figure it out.
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#CHALK #Circles #MetricSpaces
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Views: 112
CHALK

How are Math and Stats different? The relationship between the axiomatic logic of mathematics and the experimental nature of statistics and statistical methods makes the distinction clear, but stats is often lumped in with mathematics. So when should math be used and when is math not enough and stats would help out more? In this video, I go through some of these differences in order to explain the nature of statistics, a subject that has always thrown me off guard in my mathematical pursuits.
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#CHALK #Statistics #Math

Views: 209
CHALK

The Dollar Auction. A great example of a game in which escalation of commitment seems like a great idea. A paradox in some sense, this sunk cost fallacy scenario is pretty interesting and lies at the intersection of math (particularly game theory), economics, and psychology.
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#CHALK #DollarAuction #GameTheory
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Views: 1582
CHALK

Cardinality is one of the mathematical designations of a set's 'size'. In this introduction to the idea, I'll answer the questions of how cardinality is equated between two sets, how the natural numbers relate in cardinality to other sets, as well as go over the power set construction and how it is used to construct larger infinites too!
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#CHALK #Cardinality #SetTheory
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Views: 143
CHALK

You've read the title, it seems like a simple question, but an answer requires the p-adic norm, Sperner's lemma, and some more mathematical machinery. In this video, we give a proof of Sperner's lemma for the 2-dimensional case and introduce the p-adic norm in order to provide a proof for Monsky's Theorem a gem that connects the worlds of number theory, combinatorics, graph theory and geometry all to give a counter-intuitive answer to this geometric problem.
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Notes for this video: COMING SOON....
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#CHALK #pAdic #Geometry
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Views: 139
CHALK

When you think of Game Design, you probably think of the coding and art that goes into the game, but have you ever thought about the choice of space, or topology, to play in? In this video, we look at games like Tic-Tac-Toe and Chess to see how playing on different boards in 3 (and even 4) dimensions can change the game, and bring up some examples of video games that utilize these different "boards".
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#CHALK #TicTacToe #Topology
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Views: 212
CHALK

Area, Riemann Sums, Integration formulas, Riemann integrability, and Lebesgue/Measure Theoretic Integrability are the same mathematical beast through different lenses at different levels of mathematical understanding and rigor. Usually you'd spend several years learning about these topics, and because of that, I decided to talk about them all in one video to try and connect/motivate each level based on the others.
Check out the Collaboration with Chimera!!: https://chimeraeditor.com/app/CHALK_levels_of_integration?fbclid=IwAR2LqfNrQZfjGtZ_DlOqEhYfgSfvJMJ0qoMmPLa1Kbm50MDdp4YTMwi_7nk
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#CHALK #Analysis #Integration

Views: 176
CHALK

Categories and Functors can be pretty mindboggling mathematical objects to wrap your head around if you're not used to abstract math, but they come up as useful tools to study different structures in mathematics and beyond. So... What is a Category? Here I introduce the definition of a Category and Functor and give a few examples of each.
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More on Topology and Topological Spaces:
Borsuk-Ulam Theorem: http://bit.ly/BorsukUlam
Topology and Gluing Diagrams: http://bit.ly/2Bdj49i
More on Groups and Examples:
Brief Introduction: http://bit.ly/2tIFw77
Elliptic Curves Example: http://bit.ly/2L0LsQl
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Last video: http://bit.ly/2Lw1vGU
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#CHALK #CategoryTheory #Functor
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Views: 171
CHALK

The Cantor-Schroeder-Bernstein Theorem (or CSB), is a tool used to determine if two sets have the same cardinality. It is particularly useful when equating the cardinality of sets of infinitely many elements. Here, we go over a proof of the theorem and try to illustrate the construction at the heart of the proof as well. This video will probably be easier to parse if you've seen at least the first half of the previous video listed here: https://youtu.be/jh3Bar62swc
Also, I wanted to thank STUDY TIME for the topic suggestion that resulted in this video and the one from last time. If you have any particular areas or topics of math you'd like me to cover, you can leave them in the comments below. Maybe they'll turn into a video too!!!
A statement of the theorem (not in pictures :D ):
CSB: Given two sets, A and B and subsets A' and B' of A and B respectively, if |A|=|B'| and |B|=|A'| then |A|=|B|.
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#CHALK #CSB #SetTheory
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Views: 680
CHALK

Fixed Points and Periodic points are two mathematical objects that come up all over the place in Dynamical systems, Differential equations, and surprisingly in Topology as well. In these videos, I introduce the concepts of fixed points and periodic points and gradually build to a proof of Sharkovskii's Theorem in the n=3 case, while going through some more fixed point magic in the other route.
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#CHALK #Mathematics #DynamicalSystems

Views: 42
CHALK

Boolean Algebra is one of the most fundamental concepts in theoretical mathematics, logic and philosophy. In this video I go through some of the basic operators, what a material implication is, and try to explain what vacuous truth is :D.
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#CHALK #BooleanAlgebra #Logic
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Views: 57
CHALK

What is a Second? How long is a Second? How do we define a second? How precise can our measurements be? All of these questions seem uninteresting at face value, but once you begin to ask and investigate these questions, an impressive amount of complexity arises. In the case of a second, we get to encounter things like atomic and optical lattice clocks that have fascinating abilities.
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Views: 131
CHALK

Search engines like Google are pretty mind-boggling. How do they sort through billions of web pages to present what you want to see? Page rank and the Page rank algorithm are some of the most famous concepts in the mathematics behind many search engines. They use Graph Theory, Linear Algebra, and Probability, three easily intertwined fields of mathematics, to try and fix some fundamental problems with naive searching methods like text frequency searching. Here we go through some of the basics behind the algorithm.
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Source for this video: http://pi.math.cornell.edu/~mec/Winter2009/RalucaRemus/index.html
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#CHALK #Google #InternetMath
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Views: 184
CHALK

Today's video is a bit shorter than others. There is a lot going on, so I wanted to give a bit of intuition for a topic I find interesting; Curvature. In this video, we will go through a few formulas of curvature and compute curvature for a couple of examples, and I'll briefly mention different topics you might want to look into if you're looking for more advanced material at the end of the video too.
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CORRECTIONS:
1. I drew a nonsensical graph ('muahahaha') at the beginning of the video, did you catch it?
2. I will say 't' and write 'x's instead once whoops D:
3. I most likely pronounce Ricci wrong
4. At the end, it should say "notions" on the board not just "notion"
If you find any other errors go ahead and let me know in the comments!!!
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Notes for this episode: http://bit.ly/2sJKH2X
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#CHALK #Curvature #DifferentialGeometry
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Views: 56
CHALK

Lights and Billiards actually have a lot more in common than you might expect. At least, mathematically that is the case. In this video, I briefly go over an illumination problem I was thinking about and its relationship with mathematical billiards too.
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#CHALK #Billiards #DynamicalSystems
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Views: 45
CHALK

Happy Labor Day!, This week, I decided to make a video even though you're supposed to take the day off. It's about being "small and unsuccessful" and trying to put some of that into context. Whether that is by defining "success' or just how you see "growth".
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Views: 53
CHALK

A bit of Algebra is the quickest way to see that .9 repeating equals 1, but there is another approach from the lens of fractal geometry using iterated functions systems (IFSs). These mathematical devices are used to created different fractals, but they can also be used to fundamentally describe what is meant by a decimal expansion, and they provide an avenue for other types of expansions of numbers on the unit interval. In this video we give a brief introduction to iterated function systems and explain how they relate to decimal expansions and the equality of 1 and .9 repeating.
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#CHALK #FractalGeometry #IFS
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Views: 181
CHALK

Getting in front of the camera for once. I often hear that in mathematics you're either a math person or not, but I don't believe that's the case. Just like anything else, doing math is a skill, and all skills require time and energy. So where should you put your time and energy when trying to get better at math? In this video I talk about 4 different methods/activities that you can do when studying mathematics to help you get better accustomed to all of the logic and information density that is involved in the pursuit.
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#CHALK #Mathematics #LearningSkills
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Background Music:
Music by Chillhop: http://youtube.com/chillhopdotcom
Globuldub - Foreign Exchange: https://soundcloud.com/globuldub
Chillhop's 24/7 Live Stream: http://live.chillhop.com
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Views: 122
CHALK

A while back I was asked to do a video on Parameterizations. There were way too many different directions to go, so I just went with how I think about parameterizations. So, here we go on this dive into my brain.
CORRECTIONS:
-One of the little stick guys holds a sign saying “d(x,y)=r for every x,y” should say “d(x,y)=r for some y and every x” 0:39
-There is a blue "F" That should just be a "f" at around 1:45
-I write the set containing the empty set when I just mean the empty set at 4:15
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#CHALK #Parameterization #DifferentialGeometry
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Views: 89
CHALK

Here is the answer to the strange circle question posed in part 1. I take it step by step to get through how you get to this circle that is a square. We'll go through how we're thinking about points, what metric were using and finally end with where the center is and what the radius is of the circle in question.
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#CHALK #Circles #MetricSpaces
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Views: 85
CHALK

Normal Distributions come up all over the place in statistics. Here I go through what the equation for a Normal Curve looks like, as well as how you could quickly generate a normal distribution experimentally and the intution for standardization of a Normal Distribution.
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#CHALK #Statistics #NormalDistribution

Views: 57
CHALK

In mathematics, it is usually pretty easy to take a simple idea and formulate some pretty interesting problems and questions by imposing different constraints or thinking about different scenarios. I've always had a hard time motivating Number Theory, and so, I wanted to illustrate this ability to develop interesting ideas, problems, and questions by starting with the natural numbers and branching out down different paths using number theory to see what I got. After spending some time on it, I made these choose-your-own-adventure style videos. If I were you I would take the multiplicative path, but the additive path, although it is not as fleshed out as some good starter information too for investigating beyond this branch of videos.
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Last video: https://youtu.be/udLr7KM8oH0
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#CHALK #Mathematics #ChooseYourOwnAdventure
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Background Music:
Music by Chillhop: http://youtube.com/chillhopdotcom
Globuldub - Foreign Exchange: https://soundcloud.com/globuldub
Chillhop's 24/7 Live Stream: http://live.chillhop.com
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Views: 65
CHALK

What kinds of things do algebraists study? Well, a group is an example of such a mathematical object that is both important to algebraists and one that pops up in all different fields of mathematics. Here I give an intro to the basics of Group Theory. Namely, the group axioms, Abelian groups, dihedral groups amongst other things.
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#CHALK #Groups #Algebra
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Views: 71
CHALK

The Intermediate Value Theorem is something that I said I covered, but apparently I hadn't yet. So here I tackle a proof of the IVT with the help of Continuity and the Axiom of Completeness so that when it does come up again, I can reference it and actually have something to point at :D
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#CHALK #IVT #RealAnalysis

Views: 31
CHALK

This channel just passed 100 subscribers this past week, and in order to celebrate, I wanted to talk about some places that 100 pops up in mathematics. So, this video contains a bit of combinatorics and number theory that I don't always find time to talk about (but am looking forward to talking more about!). We'll see 100 chilling with some polygons as well as with Nicomachus and self-descriptiveness too!
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#CHALK #ThankYou #Numbers
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Views: 40
CHALK

Limits come up in various forms in many fields of mathematics, in order to understand other ideas of limits it helps to understand limits of sequences in real analysis. In this video, the definition of sequences and limits of sequences is presented along with an example of an epsilon proof argument; how you might go about proving that a limit of a certain sequence is actually the limit from a real analysis point of view.
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#CHALK #Limits #Analysis
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Views: 71
CHALK

I'm trying a new thing where I highlight videos from the new format that are completely proof based on Thursdays. This "Prove it Thursday" I wanted to highlight the video about the fundamental theorem of arithmetic. Here we go through the proof and ask a few questions about what else can happen when things aren't as nice as the Integers.
Lost in the mathematics? This video comes from the "The Natural Numbers, Problems and Questions" video tree here on CHALK. It was made public 6/27/2019. You can start over the original sequence by heading over here: https://youtu.be/VsrPnTs-chg
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#CHALK #ProveItThursday #PrimeFactorization
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Views: 41
CHALK

What could you possibly fear at birth? What fears are you "born with"? Well, in this video, I'll quickly go over a view different things that the study of psychology has brought to light as likely candidates for such fears.
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Views: 34
CHALK

Induction is a proof method that can feel incomplete when first coming across it even though all of the logic sits right in the conditions required of the prinicipal of mathematical induction (PMI) and the principal of strong induction (PSI). In this video, I give an introduction to the former of the logically equivalent twinsees of the induction framework with an example proof and some similar exercises at the end if you want to take a stab at proving somethings yourself!
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#CHALK #Induction #ProofMethods
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Views: 75
CHALK

So you want to make gold nanowire. You have the nanoparticles and some other chemicals, but where do you start? This is where math helps out. The Hairy Ball Theorem remedies some of the difficulties with working with these super small particles. So how does the HBT work and how exactly does it remedy the fact these particles are so small and we can't really pin them together by hand.
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A complete (albeit short) proof of the HBT by Peter McGrath:
https://www.math.brown.edu/~peter_mcgrath/research/hairy-ball.pdf
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***CORRECTIONS***: There are a few times when I say 'p' instead of 'x'. Hopefully it does not cause much confusion...
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#CHALK #HairyBallTheorem #Topology
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Views: 50
CHALK

Hi! Welcome to CHALK. Here, I make mathematics videos on a variety of topics with, well, chalk.Whether you're a fan of math or not, stick around; you might find something interesting if you do!
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#CHALK #Trailer #Math
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Views: 315
CHALK

Series can often be intuitively misleading. When we are taking sums of even very small terms the series we are working with may still grow arbitrarily large, so how slowly can you sum to infinity, and what is the test to see if a series is going to converge or diverge? Here we talk about some of the intuition behind this phenomenon and give a proof of the divergence of the harmonic series as well as a proof of the Cauchy Condensation Test.
****CORRECTIONS: Often I will say "n" but will write "k" *****
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CHECK OUT THE NOTES THAT GO ALONG WITH THIS VIDEO!!!
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#CHALK #Infinity #Analysis
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Views: 62
CHALK

© 2019 Out of the money option call

By mid-month, the idea that Telegram might raise its fundraising round even higher was reported by Bloomberg. Early February. They come up with a lockup period that releases tokens after four waiting periods, the longest one last 18 months. Late February. Finally, Telegram has apparently offered investors some kind of refund provision if it fails to deliver the TON platform by the end of October 2019, Business Insider reported. The leader in blockchain news, CoinDesk is a media outlet that strives for the highest journalistic standards and abides by a strict set of editorial policies. CoinDesk is an independent operating subsidiary of Digital Currency Group, which invests in cryptocurrencies and blockchain startups. The filing names Ton Issuer Inc. and Telegram Group Inc. along with the two individuals, Pavel Durov and Nikolai Durov, as related persons. Apart from building on the extensive userbase Telegram has amassed, and serving as a medium of exchange with a native cryptocurrency called GRAM, the TON platform also aims to include smart contracts and decentralized services such as TON Storage and TON Proxy. Leverage and Margin Explained.