# Tagged Questions

If you already have a proof for some result, but want to ask for a different proof (using different methods).

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### Proof of Sophomore's Dream using Contour Integration

Sophomore's dream is a relatively common identity, that states $$\int _0^1 x^{-x} dx = \sum_{n = 1}^\infty n^{-n}$$ The common proof is found using the series expansion for $e^{- x \log x}$ and ...
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### Monomorphisms of monoids are stable under coproducts

Let $M,N,K$ be three monoids (or even groups, if you like) and let $N \to K$ be an injective homomorphism. Then, the induced morphism $M \sqcup N \to M \sqcup K$ is also injective. This is easy to ...
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### A real symmetric matrix $A$ positive definite if all its eigenvalues are positive

Let $A\in \mathbb R^{n \times n},\ A^T=A$ and the eigenvalues $\lambda_i>0$. Then $v^TAv>0$ for every nonzero vector $v$. I know how to prove the above statement by using the fact that if $A$ ...
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### Symmetric proof for the probability of real roots of a quadratic with exponentially distributed parameters

What is the probability that the polynomial has real roots? asked for the probability that the quadratic polynomial $ax^2+bx+c$ has real roots if the parameters $a,b,c$ are exponentially distributed ...
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### Why matrix representation of convolution cannot explain the convolution theorem?

A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
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### Equivalence of Brouwers fixed point theorem and Sperner's lemma

I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. All proof I've found just prove that there must be at least one completely colored n-simplex, not that there must be ...
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### Proof that 10 lines pass through the centroid of a triangle

Let $A$, $B$, $C$, $D$, and $E$ be points on a circle. For any three points, we draw the line going through the centroid of the triangle formed by these three points that is perpendicular to the line ...