Align two equation environment blocks
Breaking an equation cluster that does not fit on the rest of page across two pages
How to create a GUID / UUID I'm trying to create globally-unique identifiers in JavaScript.  I'm not sure what routines are available on all browsers, how &quot;random&quot; and seeded the built-in random number generator is, etc. The GUID / UUID should be at least 32 characters and should stay in the ASCII range to avoid trouble when passing them around. I always know that the natural logarithm $\log_e$ is the inverse of the exponential function $e$, but to my surprise when reading elementary functions in complex analysis, I discover that that is not always true. $$\log(e^z)=z+2k\pi i$$  My question here is that, why is the above expression true? `--no-install-recommends` only specified packages I want to install xfce4-goodies but I don't want Midori Browser. With --no-install-recommends I don't know how to specify only Midori. I want the rest of packages. How to do that? Thanks! How to view the user's status? Where are my Minecraft saves located? I want to copy/modify/delete a Minecraft world. Where can I find my saves folder? How are POST and GET variables handled in Python? In PHP you can just use $_POST for POST and $_GET for GET (Query string) variables. What's the equivalent in Python? Man/Tree Symbiosis It was an old book... I'm pretty sure it was a book, and pre-1990's I think. The basic plot was this man somehow interacted with this tree, and as a result he became a carrier of the tree's 'seed'. They were in a symbiotic relationship: the man got a much longer life, more strength, etc.; but when he did die the tree would grow up from his body. Benefits of Toad houses I have a few red Toads and a Toad is just hanging out in town. I see an option to buy a red Toad house, ostensibly for him. Are there any added benefits of purchasing this or are the colored Toad houses just cosmetic? While browsing Stack Overflow, I noticed that I lost a lot of reputation:        Is this serial downvoting? Will it be corrected by a rep-recalculation? How often are those done?    Here's a second instance: How to give name to topics created using LDA? UTF-8 encoding a servlet form submission with Tomcat What purpose does downvoting questions serve? What are the biggest changes between UA Ranger and PHB Ranger? Why does the 'b' in absorb change to a 'p' in absorption? What's the best way to communicate between view controllers? According to Theorem 8.8 in Time Series A.W. van der Vaart an ARMA process $$\phi (L)X_t=\theta(L)\epsilon_t$$ has a  unique stationary solution $X_t=\psi(L)\epsilon_t$ with $\psi=\theta/\phi$ if $\phi$ has no roots on the complex unit circle. This would imply that the explosive process, with $\rho&gt;1$, is a stationary process $$X_t=\rho X_{t-1}+\epsilon_t$$ with stationary solution $X_t=\sum_{i=1}^\infty \rho^{-i}\epsilon_{t+i}$.  Now indeed $\sum_{i=1}^{\infty} \rho^{-i} &lt; \infty$ so that weak stationarity can be proved by using this representation.   However, here on stackexchange I see a lot of question/answers that suggest that the process above is not stationary (see for example , ). In particular, the accepted answer of the latter question claims that the process is non-stationary by simulating a series and showing it displays explosive trending behaviour.  I think the only way to reconcile the theorem I mention above and the plots in the accepted answer of () is the following: the explosive process is indeed stationary but non-ergodic, that is, we cannot find the statistical properties of $X_t$ such as $\mathbb{E}(X_t)=\mu$ by observing a single infinitely long sample path of the explosive process, mathematically: $$\lim_{t \to \infty}\frac{1}{t}\sum_{t=1}X_t \neq\mathbb{E}X_t$$  Is this reading correct? siunitx tabular numbers with separated uncertainty I would like to format the space reserved for a number shown in a table with separated uncertainty. It seems that when I use the table-format option the brackets around the mantissa disappear. This is illustrated in the example below:  \documentclass{article}     \usepackage{siunitx}  \begin{document}     \begin{table}   \begin{tabular}{                   S[separate-uncertainty = true]                   S[table-format=1.1(1)e1,                   separate-uncertainty = true]                   }     {Header} &amp; {Header}\\     1.0(2)e2 &amp; 1.0(2)e2\\   \end{tabular} \end{table}     \end{document}     I have also tried to use the bracket-numbers and multi-part-units = brackets without solving this issue. Is there any workaround to center the number and keep the brackets like in the first column of the example? How can I group numerical data into naturally forming "brackets"? (e.g. income) The following describes what I'm trying to accomplish, but it's possible an alternative problem statement can describe my goal:  I want to    divide the following numbers into groups where the variances of the numbers within each group are not too large, and the differences between the averages of groups are not too small  compare the distribution obtained in the end with the "perfect" ones and see how "different" it is from being perfect.     Layman's explanation of goal  , and determine the "income brackets" each population is in.  The income bracket is supposed to be self-adjusting based on the input data.  My goal is to ultimately measure or calculate the difference between the income brackets.  I assume there will be many brackets, and want to see how far "apart" each tier is.  Here is a sample of hourly income for a sample set of a population of 20, and a total income of 3587:  Population= 10                   pop=2   population=5              population =3 10, 11,13,14,14,14,14,14,15,20,  40,50  ,90,91,92,93,94      999,999,900    How can I use mathematical concepts to group, sort and analyze data that acts like income distribution over a given population?  At the end of the calculation, I want to determine tiered income distribution, where a perfect distribution would look (something) like this   (each person makes $10 more per hour than the previous; total is 3587) 89, 99, 109, 119, 129, 139, 149, 159, 169, 179, 189, 199, 209, 219, 229, 239, 249, 259, 269, 279   or this:   (evenly distributed groups of people make the same per hour)  (gaps between income groups is consistent and not "too far") (income total is 3587) 99 99 99   129 129 129   159 159 159   199 199 199  229 229 229  269 269 269    Question  How should I analyze the population groups, and measure the gap in a way that will tell me how much is needed to make it more like the last two model sets listed above? I have to give a combinatorial proof of  $$\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}.$$  I find it difficult to solve such problems. I'm not a brilliant person and never will be so I need to have algorithms to solve most problems. Usually I won't just "see" solutions unless I have a thorough enough understanding of a theory. This is also the case here. I know what the symbols mean, but I don't see what I should count and how to obtain the identity.  I think the right-hand side counts $n$-element subsets of a $(2n-1)$-element set with one element chosen to be "special". I think I should find a partition of the set of such subsets into $n$ parts so that the left-hand side is the sum of the cardinalities of the parts. Is there $n$ of anything on the right-hand side? Well, there are $n$ choices for the "special" elements, but only after I've chosen the $n$-element subset. Otherwise I can choose the "special element" in $2n-1$ ways.  This is a homework problem so I probably shouldn't ask for a full solution. But I would like to have some guidelines for approaching such problems, perhaps based on this example. When I try to print a Unicode string in a Windows console, I get an error . UnicodeEncodeError: 'charmap' codec can't encode character .... I assume this is because the Windows console does not accept Unicode-only characters. What's the best way around this? Is there any way I can make Python automatically print a ? instead of failing in this situation? Edit:  I'm using Python 2.5.  Note: @LasseV.Karlsen answer with the checkmark is sort of outdated (from 2008). Please use the solutions/answers/suggestions below with care!!  is more relevant as of today (6 Jan 2016).