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Information theory explains a lot of things.

You should watch Wolframs New Kind Of Science series he has up on YT. He makes the connection between Turing universality and the behavior of CA’s under this notion of computational equivalence: that all rules are equivalent (to each other) and to a Turing machine.

I don't really think that Cellular Automata are an alternative to computer programing. Automata theory which CA is a part of is the basis of comp scince.

There is a lateral jump-rope movement pattern called the "dragon roll" where the rope's midline follows the first Lissajous curve described @1:55 in this video: x(t)=sin(t) and y(t)=sin(2t). The rope travels laterally from left to right both in front of and behind the body. It's actually a 3D pattern, because the rope also moves up and down. The rope criss-crosses over the head. The entire 3D curve looks like Viviani's Curve (best shown on the Wolfram MathWorld website). That description of Viv's curve shows the parametric equations for x, y, and z of the motion. The "dragon roll" movement is shown live at th-cam.com/video/J25_41nPFLo/w-d-xo.html -- a demonstration by its inventor. Interestingly, our CNS is able to figure out the movement far faster than my cognitive brain was able to dissect its mathematics. Our nervous system knows how to copy/imitate quite well, but that doesn't explain how Weck was able to imagine/invent the movement in the first place. Brilliant! There's a device called a Harmonograph to plot a Lissajous curve with two decoupled pendulums moving pen or dispensing sand. Beautiful harmonograph-created curves are shown in the book "Harmonograph: A Visual Guide to the Mathematics of Music" (2003). Apparently, this was a popular device in the 1800s for after-dinner entertainment well before the days of TH-cam. Actually, the harmonograph pre-dated essentially all electronics (including vacuum tubes).

Hidden gem this channel

Very nice!

Cantor was a genius who discovered uncountable real numbers cannot be one to one with the natural numbers, so he created two dimensional array of numbers, now the problem isn't that of counting but one of accommodation, in which even a new room can be created out of infinite rooms.

I found this while looking for mvc iron man infinite

What a BRILLIANT video and critical research! Well done! AMAZING and Hats Off!!!

Why does diagonal argument not work on computable real numbers?

15:41 why is d bar defined in this way? it should be the reverse...

That square is not a perfect square so is a rectangle, count the little triangles.

10:34 It is instructive to look at the contraposition as well. If there is a surjective function f: A -> Y^A for some A, then all α: Y -> Y have a fixpoint. These are classically but not constructively equivalent but the proof of this version is very nice and constructs the fixpoint explicitly. To prove this, let α: Y - > Y be any function. Consider now g: A -> Y, a -> α(f(a)(a)). Let a‘ in A be a preimage of g under the surjective function f. Then α(f(a‘)(a‘)) = g(a‘) = f(a‘)(a‘), so f(a‘)(a‘) is a fixpoint of α.

Great talk, nice job!

The intro is very cool, the video as well of course. :)

Great video, especially with the visualizations, e.g. with the hyperplane cut by a line. Interesting extension of this idea: If you consider that *multiple-input* Boolean logical gates/functions are non-linear, then you can recover the full range of possible logical gates/functions by including the non-linear terms as additional inputs to the perceptron. For example, for binary Boolean gates, you need not only an input for x1 and for x2, but *also* another input for the 'product' x1x2, where the 'product' (think like multiplication) is exactly equivalent to Boolean AND. So, if your perceptron now takes a third parameter, let's call it x12 (meaning the AND value of x1 ^ x2), then you can have a third weight w12 on this input, and with these *three* inputs, plus the bias input, you can recreate XOR like so: y(x1, x2, x12) = b + w1x1 + w2x2 + w12x12 With b=-1, w1=2, w2=3, w12=-6. (x1, x2, x12, forumula, y) 0 0 0 -1 0 0 1 0 2 1 1 0 0 3 1 1 1 1 -2 0 I only used those values of weights to show that perceptron can have a variety of weights that generate the same XOR output. But, more clean and simple values for weights would be b=0, w1=1, w2=1, w12=-2. This gives exact output for XOR, no thresholding needed. So, the point is that binary Boolean gates/functions are not fully described by simple linear operation on two Boolean variable inputs. To make them behave *like* a linear operation, also need to supply the product (or conjunction, or join) of the two Boolean variables as a third Boolean input. This is interesting and perhaps useful, but it quickly runs into a problem. In order to handle Boolean functions of *more* variables, say 3 or 4 or more Boolean variables, then you need not only the conjunctions of x1 and x2, but all *pairs* of basic variables, and *also* all triplets of variables, and all *quadruplets* for 4 or more variables. For 2 'basic variables', you need a total of 4 weights (bias, x1, x2, x12). For 3 basic variables, you need 8 weights (bias, x1, x2, x3, x12, x13, x23, x123). For 4 variables, you need 16 weights, etc. For n basic variables, you need 2^n inputs! In some cases, with few Boolean variables, you might be able to handle this. E.g. with 8 variables, you could give each perceptron 2^8 = 256 inputs and weights, but for 16 inputs, you would need 65,536 inputs and weights per perceptron! That's a bit crazy, and it just gets worse the more basic variables you want a single perceptron to handle. However, one beauty of perceptrons is that they themselves are non-linear! And, in particular, a single perceptron can take in two inputs, x1, x2, and produce the conjunction (product / join) of those two, i.e. what we earlier called x12. And so, if you start with two basic variables, then the *first* layer of perceptrons cannot recreate XOR, but if you have a *second* layer of perceptrons, one which produces x1, one produces x2, and one produces x12, then you can combine those three to create XOR in the final output layer! There are multiple ways to do this, and you can use more than one internal/hidden layer, but a simple way is to have: 2 inputs: (x1), (x2) 2 hidden perceptrons: (x1 . not x2), (not x1 . x2); call these as h1 and h2 as inputs for next layer) 1 output perceptron: (h1 + h2) = XOR! (This version does not require any perceptron to take in more than 2 inputs. Of course, can achieve the same thing with a three-input perceptron at the end with inputs x1, x2, x12.) So, my point is that a single layer of perceptron would require more than two inputs, and they would have to be non-linear inputs (e.g. additional conjunctive inputs such as x12), in order to make XOR (or EQU). But you can also get the same non-linear effect by having multiple layers of perceptrons. (In the latter case, however, the extra layers of perceptrons would have to learn to generate the correct non-linear functions such as x1.x2 or whatever. So, there may still be good cases to supply the extra conjunctive inputs directly to the first layer.) Anyway, it's an interesting exploration!

Great work, hoping for more videos

Very good, I like how personal the style is. I had encountered all of the single topics on their own but the way you brought them together is really intriguing. Well done!

It appears to be number factorization and then triangulation, with the hidden layer similar to what may be compared to the mathematics referred to as the vacuum space. Gates are Maxwell mathematically.

That's a good video :D I especially liked the conclusion at the end, as it made it 'click' for me on why NeuronalNetworks are not performing well on certain tasks. That said, I've heard quite a bit that it's not the new hotness to have NeuronalNetworks that do not just work on a given input, but also "concider" previously processed input(s); that for example beeing extremly usefull in object tracking on video and similar applications where a sequence of data points are (more or less) steadily connected and thus their existing connection can be used in processing them instead of relying on extracting all information just from a single point/frame/picture. In essence what I just described above should be some kind of im plementing "a clock" to a NeuronalNetwork so it can also use the information of the previous input / timestep / cycle. How exactly that is done is the next intersting question :D Follow up video? ;)

Very Interesting Great video!

What I found interesting by myself is that we can imagine both a cylinder which rotates left to right (or right to left) *and* a cylinder which rotates top to bottom (or otherwise)! For some frequency ratios it’s easier than for others. And that’s because we can have both cylinders with a wave drawn on them as different projections of one wave drawn on a single 4d torus {(x, y, z, w) : x^2 + y^2 = z^2 + w^2 = 1} where (x, y) = (cos(ω1 t + ϕ1), sin(ω1 t + ϕ1)) trace the first circular motion with angular velocity ω1 (and initial phase ϕ1), and (z, w) likewise trace the second one with ω2, ϕ2. Then we can discard either w or y coordinate to get differently oriented cylinders in x-z-y or x-z-w space, then discard the remaining y or w to get a Lissajous curve in the x-z plane.

This was an amazingly produced video! Really well structured and animated, please make some more! You deserve far more attention..

Beautiful video

Absolutely amazing video!

You should look at the equation (arg(z)+pi)^(i*abs(z))

You make really high quality content ! You should have participated in the Summer of Mathematical Exposition challenge. It would surely have given your channel more visibility.

Fascinating video. Really enjoy maths applied to music. This video should have more views!

Meta

Man, this was so good!

Please continue this series

i've seen chaotic systems as described as pool tables with some form of curved obstruction somewhere on the board... and now i'm watching this and wondering if some wave (or maybe noise, in fact i fear that it would have to be noise 😨) can define a curve that best reflects the chaotic motion of such systems...

You should speak louder

Nice stuff, thanks for breaking this down! I now have a greater appreciation for these curves :)

good video!