# All Questions

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### 2D Helmholtz equation on rectangular domain

What are some hints to solve (analytically) this equation: $$\frac{\partial^2}{\partial x^2}u(x,y) + \frac{\partial^2}{\partial y^2}u(x,y) =a^2 u(x,y)+bx+cy+d$$ on the ...
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### Compact factors of Lie groups; possibly varying definitions

Let $G$ be a real connected semisimple Lie group. Are the following equivalent?: (1) $G$ has no proper cocompact Normal subgroups. (2) $G$ has no proper cocompact connected Normal subgroups. In ...
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### Inequality for the maximum of Gaussian variables

Let $X=(X_1,\dots,X_n)$ and $Y=(Y_1,\dots,Y_n)$ be centered Gaussian vectors with variance matrix $\Gamma_X$ and $\Gamma_Y$. We assume that the matrix $\Gamma_Y-\Gamma_X$ is positive definite. Is it ...
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### Horn's spectrum problem with random Hermitian matrices

An important problem in matrix analysis, completely solved in the early 2000's by A. Knutson & T. Tao (The honeycomb model of GLn(C) tensor products. I. Proof of the saturation conjecture. J. ...
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### relation between algebraic geometry and complex geometry

As a complex manifold $\mathbb{P}^n$ is locally the euclidean space $\mathbb{C}^n$, as a projective variety it is locally $\mathbb{C}^n$ with the Zariski topology, as a scheme it is locally ...
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### Compute the index of the Dirac operator on $C_0(R^2)$ to obtain Bott element in $K_0$

I am studying the paper of Baum-Connes-Higson to understand the Connes-Kasparov conjecture. In example 4.23, they discuss the case $G=\mathbb{R}^2$. I have constructed the Dirac operator, but I’m ...
### Jensen formula in $\mathbb{C}^n$?
Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function with zero set $X\subset \mathbb{C}$. Jensen's formula reads  \log(|f(0)|)+\int_0^R\frac{|X\cap B_t(0)|}{t}dt = ...