All Questions

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Beauville's Integrable System with singular spectral curves

Let us consider Beauville's Integrable System. So, we live on $\mathbb{P}^1$. There is the moduli space of matrices $M_r(d)/\mathrm{PGL}(r)$ with polynomial entries of degree less than or equal $d$. ...
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Normal subgroupoid? [on hold]

Is there a definition of normal groupoid? For normal sub-quasi-group I found two: The first one: a sub-quasi-group $H$ is called normal if there exists a normal congruence $\theta$ such that $H$ ...
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K nearest neighbors estimation with a kernel

If I have a bunch of data points $x_1,\dots,x_n$, I can build a density function $f(x)$ based on these data points by defining $f(x) = c/d_k(x)$ for an appropriate constant $c$, where $d_k(x)$ is the ...
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What does b^(3x+1)×b^(2x−5) equal? [on hold]

I am taking a grade 12 math course and this question is really confusing me b^(3x+1)×b^(2x−5). The answer is b^(6x^2−13x−5). However since both the bases ("b") are the same, and they are being ...
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Functors similar to $H^i(\cdot)$

Suppose $T$ is a contravariant functor from the category of pointed topological spaces to the category of abelian groups, then we have homomorphisms $\alpha\colon T(X)\times T(Y)\to T(X\times Y)$ and ...
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group completion theorem by using homology fibrations

In the paper Homology fibrations and group completion theorem, McDuff-Segal (www.maths.ed.ac.uk/~aar/papers/mcdsegal.pdf), page 281: Let $M$ be a topological monoid such that $\pi_0M$ is generated by ...
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binomial/factorial identity mod p

In trying to determine the spectrum of a well-known ergodic transformation, I came up with the following useful (for me) result. Let $p$ be a prime and $a$ a positive integer. Then for $M$ a positive ...
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Weighted Perturbation Bound for Polar Decomposition

Setup: Let $X\in\mathbb{R}^{n\times r}$ be a matrix with orthogonal columns, with $\Sigma = X^TX$, and assume that $\Sigma$ is invertible (note, $\Sigma$ is not necessarily the identity). Suppose we ...
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Number of prime numbers in a range

Denote by $A_n$ the number of prime numbers between $n$ and $n + \log n$. Is it true that $A_n < const$? UPD: Is it true that $A_n > \log \log n$ (or something another) for infinite number ...
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How to determine an unitary operator involved in an unitary transformation?

Let two real matrices $A$ and $B$ be unitarily equivalent. How to determine (computationally or theoretically) the unitary operator $U$ s.t. $A = UBU^\dagger$? Is it possible for some special class of ...
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Do more generalizations of Schur's inequality exist?

I meet this following problem If $$n\ge 3,\sum_{i=1}^{n}\left(\prod_{j\neq i}(a_{i}-a_{j})\right)\ge 0$$ where $a_{i}$ are real numbers. when $n=3$, it is Schur's inequality so which $n$ ...
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Given $x$ in a path-connected open set $S$ on the plane, are there non-crossing paths from $x$ to every point in $\partial S$?

A have a (non-simply) bounded path-connected open set $S\subset\mathbb{R}^2$. Given $x\in S$, there are paths in $S$ from $x$ to any point in the boundary $\partial S$. However, can all these paths ...
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What are a couple of examples of finite sized but interesting categories?

I'm studying category theory and, given that I don't have a background in topology, I'm struggling to think of some finite categories that interesting. The main one I know of is finite preorders -- I ...
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closed range bounded linear operators [migrated]

Let $CL(X,Y)$ be the set of all closed range bounded linear operators from Banach space $X$ to Banach space $Y$. Is $CL(X,Y)$ an open set of $B(X,Y)$?
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residue formula for connections on curves

Let $X$ be a smooth, projective curve over a field $k$ (characteristic zero is enough for me) and $E$ a line bundle on $X$. Assume that $E$ is equipped with an integrable logarithmic connection ...
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Koch snowflake construction in many dimensions

I'm looking for some references to deal with the Koch snowflake construction. The construction basically says we can find a sequence $E_k$ such that $\sup_k|E_k|<\infty$ but \$P(E_k) \rightarrow ...