# All Questions

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### How do you prove that $\{ Ax \mid x \geq 0 \}$ is closed?

Let $A$ be a real $m \times n$ matrix. How do you prove that $\{ Ax \mid x \geq 0, x \in \mathbb R^n \}$ is closed (as in, contains all its limit points)? The inequality $x \geq 0$ is ...
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### seating at two round tables

We have $n$ people sitting in two round tables like the picture. We randomly change the place of the people we decide two of them and change their places. We can only change the place of the ...
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### Boundedness of the Fourier transform of a Battle-Lemarie scaling function

Could anyone please give a short and simple proof of the following proposition: the Fourier transform $\hat\phi$ of Battle-Lemarie scaling function (of arbitrary order) is bounded on $\mathbb{R}$, ...
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### How to determine the existence of all subsets of a set?

Given The definition of subset; The axiom of power set: for any set $S$, there exists a set $\wp$ such that $X \in \wp$ if and only if $X\subseteq S$ we know what a subset is and what a power set ...
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### A question about many-one reducibility of two sets

We want to show that $\big\{x:W_{x}$ is finite }$=Fin \leq _m Cof=\big\{x : W_{x}$ is cofinite}. But I really have not any idea. Would be grateful for your help.
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### An intuitive question about the metric $d(\cdot,\cdot)$ on a complete Riemannian manifold

Let $(M,g)$ be a complete Riemannian manifold. Suppose $\gamma_1,\gamma_2:[0,1] \to M$ are two different segments(i.e.minimizing geodesics) from $p$ to $x=\exp_p(v)$. Suppose further that \$\gamma'_1(1)...