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1d

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Examples of weird 'modular like mathematical space' that behaves like as if it is infinite until a threshold value is reached?
Your question is disjointed and unclear. It doesn't look like researchlevel mathematics. You should make sure you understand things like hyperreals and Hausdorff measure, and that you may have questions about these that are far from the frontiers of research, and which are not at all about an historical overview. Also, some parts of your question do not seem to make sense. I suggest doing some research and then asking a short question about a specific selfcontained idea on math.stackexchange.com. 
1d

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The view from inside of a mirrored tetrahedron
It helps to build intuition to ask what it looks like one dimension lower in a triangle with angles that are not $\pi/n$. You see only part of a universal cover of a sphere with $3$ punctures, and while angles of $\pi/n$ (or sometimes $2\pi/n$) makes some of the pieces on either side of a puncture fit together, these pieces don't fit together in general. Imagine a telephone pole in the middle of a photo where the scene on the right behind the pole doesn't fit the scene on the left. Analogues happen with the reflecting tetrahedron because the dihedral angles are irrational multiples of $\pi$. 
Nov 22 
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Robotics, Cryptography, and Genetics applications of Grothendieck's work?
@Daniel Litt: The configuration Bob Connelly showed me was made of $32$ regular triangles. Start with a stellated cube, and pull out one square pyramid, placing a square antiprism between the cube and the square pyramid. The apparent motion is an arc. You can do the same on the opposite side to get a two dimensional infinitesimal motion. There are simpler versions if you can fix some points. Connect two unit segments with endpoints fixed at $(0,1)$ and $(0,1)$. Their connection must be at $(0,0)$ but the condition is $y=0, x^2=0$, giving an infinitesimal motion in the $x$ direction. 
Nov 21 
revised 
Robotics, Cryptography, and Genetics applications of Grothendieck's work?
Emphasized question. Deleted sentences about matrices. Added algebraic geometry tag even though some of Grothendieck's work was outside. Made trivial changes. 
Nov 19 
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What is the VapnikChervonenkis dimension of sigmoidal functions?
Whatever threshold you use gives halfspaces. It doesn't matter how complicated the sigmoid function is because it is monotone. This is an exercise. It is not research level. 
Nov 19 
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Robotics, Cryptography, and Genetics applications of Grothendieck's work?
I used to carry around Polydrons to show the following example, which I learned from Bob Connelly: You can build a nonconvex polyhedron from triangles that is provably rigid, so the configuration space mod rigid motions of space is locally just a point. However, the model flexes in your hand. This is because the constraints give a condition like $x^4=0$ instead of $x=0$, and so even small tolerances for error produce macroscopic motions. I think this is natural to describe using schemes. 
Nov 19 
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Robotics, Cryptography, and Genetics applications of Grothendieck's work?
@Dylan Yott: That is not a mathematical scheme, of course, but it is natural to consider a scheme structure on some configuration spaces. 
Nov 19 
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Robotics, Cryptography, and Genetics applications of Grothendieck's work?
Please do not hastily close this question. The use of matrices in robotics is not relevant (as far as I know), but people do study configuration spaces of linkages and other models of robots. I'm not qualified to talk about them, but some people consider a Grothendieck ring of configuration spaces. See Topological Robotics, and this section: books.google.com/… 
Nov 18 
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Is there a bounded sequence of points in the plane with pairwise distances at least $1/\sqrt{ij}$?
@Yaakov Baruch: I looked at that, too. A modification I haven't checked, but which seems to work is to use ternary digits, and use a map $i_6 i_5 i_4 i_3 i_2 i_1 i_0 \mapsto (0.(i_0)(i_2)(i_4)(i_6), 0.(i_1)(i_3)(i_5))$. That is, choose a fixed encoding of the digits so that $.02222$ is not next to $.10000$. 
Nov 17 
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Nash Equilibrium of simple betting game
One complication is that the hands are not linearly ordered. The 90th percentile hand for calling against a wide range is not necessarily the same as the 90th percentile hand for calling against a tight range, and the pushing and calling hands are different. However, a common shortcut for these sorts of Nash equilibrium calculators is to assume a particular ordering of the hands, in which case there is just one parameter needed to specify a range. This isn't necessary for headsup calculations but it is normal for multiway calculations, some of which are computationally hard. 
Nov 17 
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Nash Equilibrium of simple betting game
@domotorp: Yes, it's not something you can calculate directly. It depends on the particular game being played, Texas Hold'em, since that affects the equity when your 90th percentile hand runs into a 95th percentile hand, for example, and the blocking effect that holding a hand like AT means there are fewer combinations of AA, TT, and AK hands. An iterative method is used that is quite fast when you have the cached results of each pair of hands. 
Nov 14 
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Does the boundary of a convex body contain a regular planar pentagon?
@Yoav Kallus: No nonsquare rhombus can be inscribed in a circle. Perhaps cyclic quadrilaterals have the property you need. 
Nov 13 
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Websites for Math Shopping
I designed the math Tshirts for thenerdiestshirts.com, including one on the CalkinWilf tree, a proof of the Fundamental Theorem of Algebra, a truncated icosahedron, and others. Perhaps most mathematicians wouldn't call these high brow, but they are supposed to be a big step above the usual math shirts. 
Nov 7 
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Number of paths through infinite trees with given “growth rates”
There is an assumption in your first theorem that is not present in the question. Your assumption that there are at most $2$ children of each node makes the question more interesting. Without this, it might be that only the first node has multiple children, in which case the number of branches is countable. 
Nov 6 
awarded  Good Answer 
Oct 30 
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Higher dimensional generalization of: Any quadrilateral tiles the plane?
@Matthias Wendt: AFAIK, the construction is mine, but not terribly surprising to some. I mentioned it in a few places but didn't publish it. Start with a "horobrick," something like a fundamental domain for a BaumslagSolitar group, or higher dimensional generalizations, between two horospheres. For example, take a pentagon with $3$ vertices on one horocircle, and $2$ vertices on another concentric horocircle. You can add a bump on one side, and take away a copy of that bump from the other two sides to get polygons of arbitrary areas which tile. There are $2$reptiles in higher dimensions. 
Oct 21 
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Coding for channels with concentrated error
Normally, there is some model for the noise, such as independent chances for flipping bits. Which model are you assuming that would produce that behavior? 
Oct 11 
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Noise reduction in capacity0 channels
The statement "Obviously, this can't carry any information" is obviously false, and I would vote to close this question except that you can't vote to close questions with open bounties. This question is built on a basic error, and although the speculations about reducing the errors in channels with no information sound interesting, I don't see any meaning because of that error. 
Oct 10 
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Zero's in the decimal representation of powers of 3
@James Cranch: The $5\times 10^{n2}$ values hit mod $10^n$ are arbitrary initial sequences with one of the $500$ allowed final $4$ digits. Since $3^8 = 6561$, we can find a power of $3$ that ends in $1111...1116561$. For example, $3^{195508}$ ends in $...1116561.$ 
Oct 10 
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Zero's in the decimal representation of powers of 3
@James Cranch: The multiplicative order of $3$ mod $10^n$ is $5\times 10^{n2}$ for $n \ge 4$. I think this means it would have to work by $n=4$ but it doesn't. 