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Pan geometry trigonometry unification

This is a very important and interesting question, the answer to which is perhaps known only partially. Reference to Robert Bonola " Non-Euclidean geometry ", "Pan- geometry" . Circular functions ...
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What is the canonical form of real symmetric 2n\times 2n matrix under unitary congruence?

If $M$ is a $2n\times 2n$ real symmetric matrix, I would like to ask what could be its canonical form under unitary congruence. We view a unitary $n\times n$ matrix $U$ as a real $2n\times 2n$ matrix, ...
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Finding the peak of a bivariate Gaussian distribution via a guess-and-check protocol with noise

I have a bivariate Gaussian distribution $f(x,y)$ with mean $\mu$, covariance matrix $M$, and square roots of the eigenvalues of the covariance matrix $(\sigma_1,\sigma_2)$. We now play a kind of ...
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Hochschild homology of upper triangular matrix algebra?

Let $K$ be a field and $A$ the associative unital $K$-algebra of all $n\times n$ upper triangular matrices with entries in $K$. What is $\dim_K$ of its hochschild homology $HH_k(A;A)$? Is there any ...
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Classical and Quantum Chern-Simons Theory

Please excuse a sloppy question from an old user who hasn't been here in a long time. I think the expertise here is such that it can be answered anyway. Let $\Sigma$ be a two-manifold and $M$ a ...
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Units in residue classes

Let $K$ be a CM-field of degree $2n$. (Quadratic extension of totally real number field) Let $\mathcal{O}_K$ be the ring of integers in $K$, and $m\geq 1$ an integer. Let $U_K$ be the group of units ...