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Why does an unhinged body rotate about its centre of mass? If a force is applied to a body which does not act through its centre of mass, it rotates about its centre of mass and not any other point. Why? Connectedness of the Hausdorff distance. Does anyone know a proof of connectedness of the Hausdorff distance?  I mean a proof of the following:  Theorem  If $(X, \rho )$ is a connected metric space, then $(F(X), d_h )$ is also connected.  Remark:  $F(X) = \{ A \subseteq X | A \text{ is nonempty, closed and bounded } \} $, and $d_h$ stands for the Hausdorff distance. Circle-Rectangle collision detection (intersection) How can I tell whether a circle and a rectangle intersect in 2D Euclidean space? (i.e. classic 2D geometry) Cross-reference packages: which to use, which conflict? Inspired by my earlier question , here is another similar question.  There are many different packages for making cross references. Among them , , , , … The  package defines its own cross-referencing command for theorems…  Which collection of packages provides which kind of functionality, and which packages cause conflicts with each other? Catching exceptions from a constructor's initializer list Create directory if it does not exist If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t Let $a_n$ be the number of those permutation $\sigma $ on $\{1,2,...,n\}$  such that $\sigma $ is a product of exactly two disjoint cycles. Then find $a_4$ and $a_5$.   Calculating $a_4$: Possible cases which can happen are $(12)(34),(13)(24),(14)(23)$, any cycle of the form $(123)$ or $(12)$ i.e. two-cycles and three cycles thus we have in total $3+\frac{1}{3}4P_3+\frac{1}{2}4P_2=3+8+6=17$ but the correct answer is given to be either $11$ or $14$.  Where am I wrong? Please help. What is the difference between DX format and FX format lenses, and which to choose for what purpose? What is the difference between DX format and FX format lenses, and which to choose for what purpose?  What are the pros and cons of both? Is a passport needed when flying within Mexico as a U.S permanent resident? I have an executable program (no source code, just the compiled executable) that was made in windows (.exe extension).  It doesn't use any graphics... it simply reads and writes files.  I want to be able to run it in a linux shell script so that I don't have to switch operating systems to get my output.  Is there a way to use or convert the executable for linux operating systems? Why $\sqrt[3]{3}\not\in \mathbb{Q}(\sqrt[3]{2})$? We all know that $\sqrt[3]{3}\not\in \mathbb{Q}({\sqrt[3]{2}})$ intuitively. We even know that $\sqrt[3]{p}\not\in \mathbb{Q}(\sqrt[3]{q})$ for two distinct primes $p, q$. However, I don't know how to prove these things rigorously. In case of $\sqrt{p}\not\in \mathbb{Q}(\sqrt{q})$, we can just set assume $\sqrt{p}=a+b\sqrt{q}$ for some $a, b\in\mathbb{Q}$ and deduce a contradiction. However, in cubic case, expansion of $(a+b\sqrt[3]{2}+c\sqrt[3]{4})^{3}$ is very complicated and even I expand this I couldn't get any contradiction by this.  To be specific, it gives us 3 diophantine equations  $$a^{3}+2b^{3}+4c^{3}+12abc=3$$ $$a^{2}b+2b^{2}c+2c^{2}a=0$$ $$ab^{2}+2bc^{2}+ca^{2}=0$$  and I don't know how to check solvability of this kind of diophantine equations (over $\mathbb{Q}$). Even if there exists a way to check solvability, I cannot convince that it can be used to prove $\sqrt[3]{p}\not\in \mathbb{Q}(\sqrt[3]{q})$.   So I tried to use Galois theory, but actually I don't know where to go. I assumed $\mathbb{Q}(\sqrt[3]{3})=\mathbb{Q}(\sqrt[3]{2})$ and take a Galois closure, i.e. $K=\mathbb{Q}(\sqrt[3]{3}, w)=\mathbb{Q}(\sqrt[3]{2}, w)$ where $w^{3}=1$ then tried to use field norm maps. We can check that $\beta=\sqrt[3]{3}\cdot\sigma(\sqrt[3]{3})\cdot\sigma^{2}(\sqrt[3]{3})\in \mathbb{Q}$ when $\sigma\in Gal(K/\mathbb{Q}), \sigma(\sqrt[3]{2})=\sqrt[3]{2}, \sigma(w)=w^{2}$. But I cannot get any information from this.   Edit : It would be great if there is a simple algorithm to check whether $\alpha\in K$ or not for a given algebraic number $\alpha$ and a number field $K$. Determining the default Method used in optimization and root-finding algorithms Secure hash and salt for PHP passwords Why does the dock appear on the left when my screen is locked? The dock doesn't work correctly after I enter my password, either. Are there any examples of Humans and Aliens producing offspring in Star Wars? The title is pretty self explanatory, although to expand on the question further I am looking for any examples of inter-species breeding in the Star Wars universe, not just inter-species (ahem) "relations". Python: using a recursive algorithm as a generator The difference between sys.stdout.write and print? Using Replace Geometry to replace selected feature with circle in ArcGIS Desktop? I have a polygon shapefile and I want replace every irregular polygon with a circle.  I think I can do it with the Replace Geometry tool. But once the Replace Geometry tool is selected, draw a polygon mode is immediately entered. If I select the circle tool in the Draw tool window, the Replace Geometry tool becomes inactive.  Is there any way to use the Replace Geometry tool to draw a circle polygon? Prove that the following inequality holds   $$a^a b^b c^c\ge (abc)^{\frac{a+b+c}{3}}$$   if $a,b,c$ are positive.   I'm not sure how to handle these kinds of powers. Are there any "famous" but not so advanced inequalities that involve this kinds of expressions?  I was trying to do something like ordering them without lost of generality because it's symmetric, but I get nowhere and I just make the thing even more complicated.  I'm asking for solution rather than hints since this is my first time encountering these kinds of problems.