LaTex produces the formula I want:    but it returns some warning messages. The is the code of the formula above, I don't know where is the mistake:  \begin{align} \mathbb{D} (k,n) = \frac{2}{n_q} ~\text{cos} \bigg(\frac{(n-1)(k-1)\pi}{n_q /2} \bigg) \cdot  \left\{ \begin{array}{c l l} 1/2 &amp;  \mbox{if} &amp; n=\{1, n_q /2+1 \}\\ 1   &amp; \multicolumn{2}{l}{\mbox{otherwise}} \\ \end{array} \right \end{align}
\begin{eqnarray*} M&amp;=&amp; \max{\left \{a,b,\frac{a+b}{2},\right}\\ &amp;&amp; a+b,a+b/2\Bigg\} \end{eqnarray*}
Let $X$ be simply connected and let $ \tau $ be a cycle in $X$. If $ \tau(0)=\tau(1)=x_0$ and $\gamma $ is a path between $x\in X$ and $x_0$, is $\tau$ homologous to $ \gamma\tau\gamma^{-1} $?  I think it is, but I'm not sure about the proof. Can I use the fact that $ \gamma\tau\gamma^{-1} $ is homologous to $ \gamma+\tau+\gamma^{-1} $, where $+$ is the operation between 1-chains (that is abelian)? Is there a way I can see all the versions that are in the archives that I have configured in sources.list. I can see the last version of each archive withapt-get policy, but how can I see them all?  Is there any way that this can also include PPA and anything in sources.list.d? In the course of creating the content for a site, we've ended up with orphaned content nodes that are not linked to from anywhere. I'd like to find and examine these. Surprisingly, I can't find a module for this.  Tiny hypothetical example: I'm building a brochureware site. 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Show that $\frac 1{1+x+y^{-1}}+\frac1{1+y+z^{-1}}+\frac1{1+z+x^{-1}}=1$ if $xyz=1$ If $x.y.z=1$ show that $\dfrac 1{1+x+y^{-1}}+\dfrac1{1+y+z^{-1}}+\dfrac1{1+z+x^{-1}}=1$  My attempt - L.H.S$=\dfrac 1{1+x+y^{-1}}+\dfrac1{1+y+z^{-1}}+\dfrac1{1+z+x^{-1}}$  $=\dfrac y{y+xy+1}+\dfrac z{z+yz+1}+\dfrac x{x+zx+1}$  $=\dfrac{yz}{yz+xyz+z}+\dfrac z{z+yz+1}+\dfrac x{x+zx+1}$  What to do next? Let $(X,d)$ be a metric space and for $x,y \in X$ define $$d_b(x,y) = \dfrac{d(x,y)}{1 + d(x,y)}$$ a) show that $d_b$ is a metric on $X$  Hint: consider the derivative of $f(t)$ = $\dfrac{t}{1+t}$  b) show that $ d$ and $ d_b $ are equivalent metrics. c) let $(X,d) $ be $(\mathbb{R}, |\cdot|)$ Show that there exists no $ M&gt;0$ such that $ |x-y| $ $\leq$ $Md_b$ $(x,y)$ for all $ x,y$ $\in$ $\mathbb{R}$  I have calculated $f(t)$ and $f'(t)$ and from this I know $f(t)$ is an increasing function as $f'(t)$ is strictly positive. But I don't know where to go from here or how to do parts b) and c)