# All Questions

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### $H$ self-adjoint with mass gap, $P≥0,Ω∈D(P),H+λP$ self-adjoint $⟹$ for $λ$ small, $H+λP$ has gap?

Suppose $H$ is a self-adjoint operator on a Hilbert space having a simple isolated least eigenvalue $0$ with gap $1$ ( $H\Omega = 0$, $\Vert \Omega\Vert = 1$ ), $P$ is a non-negative symmetric ...
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### Variant Karamata inequality [on hold]

We observe function $f(x)=x^2$ in the figure as following We can see that: $\frac{f(x_1)+f(x_2)}{2}-f(\frac{x_1+x_2}{2}) \ge \frac{f(y_1)+f(y_2)}{2}-f(\frac{y_1+y_2}{2}) \\$ when $AB \ge CD$ when ...
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### What are the rank 3 boolean intervals [H,G], with G simple group?

The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$. The lattice $B_{3}$ is the following: Question: What are the rank $3$ boolean intervals of the form $[H,G]$, ...
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### Generalization of de Rham cohomology, or cohomology for non-smooth case

Let $\Omega\subseteq \mathbb{R}^{3}$ be a simply connected domain and $f:\Omega\to \mathbb{R}^{3}$ be a vector field on $f$. If $f$ is smooth vector field and $\nabla\cdot f=0$, then $f=\nabla\times g$...
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### What is a universal tree?

I came across some slides talking about the Hrushovski construction. One of the examples was the construction of a "universal tree". I was curious because the collection of finite trees does not ...
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Let $\mathcal{O}_K$ be a complete discrete valuation ring with quotient field $K = \text{Quot}(A)$. Let $L | K$ be an arbitrary finite field extension. Because $K$ is henselian, the integral closure $\... 0answers 76 views ### When is a given polynomial a square of another polynomial? I meet a problem in which I hope to show a special polynomial is not a square of another polynomial. More precisely, let's consider the polynomial$f(x):= 1-x+2bx^n-2bx^{n+1}-b^2x^{2n-1}+2b^2x^{2n}-b^...
This is my first overflow question, so let me apologize in advance if this belongs on http://math.stackexchange.com. Let $G$ be a discrete group. Let $\lambda:G\to B(\ell^{2}(G))$ be the left ...