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Arago's spot is a classical (and classic) example of a beautiful mathematical theory anticipating a beautiful physical fact. Briefly, the story goes like this: Back in the 1800's, scientists were ...

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Natural transformations are a &quot;natural&quot; example of this. Mathematicians knew for a long time that certain maps--e.g. the canonical isomorphism between a finite-dimensional vector space and ...

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Exotic spheres are a feature only of dimensions higher than 3. These are topological spaces which are homeomorphic to a sphere, but with different differential structure. Informally, you can ...

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Yes, with the appropriate definition of "circle". Namely, define a circle of radius $R$ centered at $x$ on manifold $M$ to be the set of points which can be reached by a geodesic of length $R$ ...

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1. Definition of a matrix. The question of what a matrix is, precisely, is one I had for a long time as a high school student. It took many tries to get a straight answer, because people tend to ...

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The main question here seems to be &quot;why can we differentiate a function only defined on integers?&quot;. The proper answer, as divined by the OP, is that we can't--there is no unique way to ...

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The granddaddy of ill-behaved functions is $e^{1/x}$. Even in one dimension it is very strange (having what is called an &quot;essential singularity&quot; at the origin), but in 2D or in the complex ...

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Historically, a very important, computationally intensive problem arising from physics was lattice QCD (LQCD). LQCD is a theoretical framework for computing basic quantities like the mass of the ...

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Case work: If 3 or 4 of the rolls come up 1, it's straightforward to see that the desired inequality holds. If 0 or 1 of the rolls come up 1, then we can show that the inequality never holds. (The ...

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Bell's theorem on the foundations of quantum mechanics showed that not all philosophical questions are impervious to experiment, to the extreme surprise of pretty much every physicist on Earth. (It ...

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If somebody speaks only one language, then everybody speaks that language, and we're done. If somebody speaks all four languages, then the desired statement follows by induction after removing that ...

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Fermat's Last Theorem states that there are no positive integer solutions $a,b,c$ to $a^n+b^n=c^n$ for $n&gt;2$. This one irked mathematicians for 350 years until the theory of elliptic curves was ...

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It depends on what you mean by "self-taught" and "mathematician". What is definitely true is that you probably have the chops to learn any particular branch of math that you are interested in. And ...

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Rewrite it as $$\frac{2y'y''}{(y')^2+1} = \frac{2ky'}{y}$$ Integrate to get $$\ln \left((y')^2+1\right) = 2k \ln y + c$$ So $$(y')^2+1 = C_1y^{2k}$$ Rearrange to get $$1=\frac{y'}{\sqrt{C_1y^{2k}-1}}$$...

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According to Wikipedia, the "tangent half-angle substitution" (a.k.a. the Weierstrauss substitution) is the "world's sneakiest substitution". It consists of subbing $x=\arctan(2t)$, which allows you ...

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There are 9 of these numbers having 1 digit, $\binom{9}{2}$ having 2 digits, and in general $\binom{9}{k}$ having $k$ digits. The total number of elements in this set is thus $\binom{9}{1} + \binom{9}... View answer 1 answers 4 votes 2k views 14 votes Here's one solution: Your graph$H_n$is an n-dimensional hypercube. For the induction step, separate the cube into two "faces" by cutting along one dimension. Do parallel Hamiltonian cycles on each ... View answer 14 answers 74 votes 5k views 13 votes In non-standard analysis, to any set$A$there is an associated set$^*\!A$, which consists of the original set$A$plus a bunch of new points infinitesimally close to$A$. There are two natural ... View answer 5 answers 8 votes 1k views 12 votes The easy way: You don't actually need to find the two times when the hour and minute hand are at right angles. Just note that the difference between these two times is how long it takes for the ... View answer 12 answers 58 votes 7k views 12 votes At the risk of sounding flippant, I'm going to propose the universe as one such example. It is a peculiar, but experimentally well verified fact, that the universe has four dimensions (three spatial ... View answer 4 answers 18 votes 1k views 9 votes You requested a proof in three lines, and conveniently the proof actually consists of three computations :) I'll break the proof up along these computations. However, in the interest of clarity, I ... View answer 4 answers 12 votes 2k views 8 votes Choose a basis which puts$AB-BA$in Jordan normal form (both$\det$and$\text{Tr}$are invariant under basis change, so this is allowed). Then since a commutator is traceless, the diagonal must ... View answer 1 answers 7 votes 156 views Accepted answer 7 votes The question seems to be requiring that the subsets be distinct. In this case, certain sums can only be attained in one way, and it is thus impossible to put two pigeons in these pigeonholes, ... View answer 1 answers 9 votes 250 views Accepted answer 7 votes This is not a complete answer, but since you are looking for intuition, it may be helpful. Some background: Historically, the spectrum originated in quantum mechanics, and in particular in the energy ... View answer 7 answers 24 votes 2k views 7 votes The Hadamard Conjecture states that a Hadamard matrix of order$4k$should exist for every positive integer$k$. It has been numerically verified for all orders up to 668. The Circulant Hadamard ... View answer 4 answers 12 votes 811 views 6 votes Note that if we call$a=e^{ix/2}$, then we have$$\frac{\sin(21x/2)}{\sin(x/2)} = \frac{a^{21}-a^{-21}}{a-a^{-1}} = a^{-20}\frac{a^{42}-1}{a^2-1}=a^{-20}\frac{(a^2-1)(a^{40}+a^{38}+a^{36}+\cdots+1)}{a^... View answer 4 answers 15 votes 6k views 6 votes There is a one-parameter infinite family of solutions. Below is an algorithm to find them all. As a spoiler, I'll point out up front that this algorithm works equally well for any convex subset of$\...
The union of the two spaces $P\cup P'$ is not all of $V$, so there is a vector $v_1\notin P\cup P'$. Now proceed inductively on the spaces $P\oplus \text{span}\{v_1\}$ and $P'\oplus \text{span}\{v_1\}... View answer 6 answers 46 votes 6k views 6 votes What you are asking, in some sense, is "Is abstraction useful?" Abstraction is the art of embedding easy-to-understand ideas inside of harder-to-understand ideas. As to the usefulness of this ... View answer 1 answers 6 votes 216 views Accepted answer 6 votes Your answer is right, your book is wrong. Here are two alternative ways to prove it: The coefficient of$x^{11}$is the number of ways to put 11$x\$'s in 3 baskets. This is the "stars and bars" ...