I tried this \begin{equation*}      \begin{cases}         \begin{tabular}{c|c}              $r_0=3.0000$  \\              $r_0^2=9.0000$          \end{tabular}      \end{cases}      \rho_0=2.76518\rightarrow r_0^2=8.56814  \end{equation*}
How can I get a brace on the right, not the left, to group cases, using cases? (I know actual cases look better with the brace on the left, but the items I need to group are not literally cases. They are a set of steps in an algorithm which I want to consider as an item.)  The following puts the brace on the left. All I need to do is to flip it to the right.  \begin{align} \begin{cases} a&amp;=bbb\\ c&amp;=ddddddd\\ e&amp;=ffffffffff \end{cases} \end{align}   Thanks for any help with this!
What are the odds of hitting exactly 100 rolling a fair die Let $E$ be a Banach space, $\mathcal B(E)$ the Banach space of linear bounded operators and $\mathcal I$ the set of all invertible linear bounded operators from $E$ to $E$. We know that $\mathcal I$ is an open set, and if $E$ is finite dimensional then $\mathcal I$ is dense in $\mathcal B(E)$. It's not true that $\mathcal I$ is dense if we can find $T\in\mathcal B(E)$ injective, non surjective with $T(E)$ closed in $E$, since such an operator cannot be approximated in the norm on $\mathcal B(E)$ by elements of $\mathcal I$ (in particular $E$ has to be infinite dimensional).  So the question is (maybe a little vague): is there a nice characterization of $\overline{\mathcal I}^{\mathcal B(E)}$ when $E$ is infinite dimensional? Is the case of Hilbert space simpler? Help needed with definite integral for $\lim_{n\rightarrow \infty}\sum_{k=1}^{n} \frac{1}{n+k}$ For $x$ and $\lambda$ complex, if $|x| \gt \frac{1}{2} + \sqrt{\frac{5}{4} + |\lambda|}$ then $|x^2+\lambda| \gt |x| +1$ Define $f(z) = z \ ^ 2 + \lambda$ and $V = \left\{x \in \Bbb C : |x| \gt \frac{1}{2} + \sqrt{\frac{5}{4} + |\lambda|} \right\}$, for some $\lambda \in \Bbb C$.   Then, if $x \in V$ then $|f(x)| \gt |x| +1$.   I tried the following :   Let $x\in V$ then $|f(x)| = |x \  ^ 2 +\lambda | \ge |x |  \ ^ 2 - | \lambda| \gt (\frac{1}{2} + \sqrt{\frac{5}{4} + |\lambda|}) \  ^ 2 - |\lambda| = 1.5 + \sqrt{\frac{5}{4} + |\lambda|} $.  But this doesn't get me anywhere. Quadratic Formula in Complex Variables Showing that $\lnot Q \lor (\lnot Q \land R) = \lnot Q$ without a truth table I've done a truth table after reducing it to this and it seems to be equal to $\neg Q$:  $$\lnot Q \lor (\lnot Q \land R) = \lnot Q$$  But when I try to show it without a truth table (with just transformations), I end up in a loop between that and:  $$\lnot Q \land (\lnot Q \lor R)$$  Is there a way to show this is true without using a truth table? What am I missing?!  Thanks in advance! Git for beginners: The definitive practical guide How can I get all three Kalos starters? How can i make my Java Swing GUI Components [Right To Left] for Arabic language from NetBeans Desktop Application? How is the Gmail password stored in Android - and where? I have looked around and found no information on how Android manages to store passwords on the device. Especially Gmail passwords. I'm looking to learn how Android encrypts and stores passwords ? What key does it use and where is this key stored, and what encryption algorithm it uses. Partitions of $n$ into distinct odd and even parts proof Let $p_\text{odd}(n)$ denote the number of partitions of $n$ into an odd number of parts, and let $p_\text{even}(n)$ denote the number of partitions of $n$ into an even number of parts.  How do I prove that    $|p_\text{even}(n) - p_\text{odd}(n)|$ is equal to the partitions of $n$ into distinct odd parts. Show that the number of partitions of $n$ for which no part appears exactly once is equal to the number of partitions of n for which every part is divisible by 2 or 3. Show that the number of partitions of $n$ for which no part appears more than twice is equal to the number of partitions of $n$ for which no part is divisible by 3. Optimal estimation of a single parameter given data at different scales Let $A,B,C,D$ be independent normal random variables distributed according to $\mathcal{N}(0,\sigma^2)$ and $E = A+B+C+D$.  The goal is to provide an optimal estimate of $\sigma^2$ given samples of $A,B,E$ but not $C,D$.  In this case optimal refers to minimising the root-mean square deviation of the estimate $\hat{\sigma}^2$:  $$RMSD(\hat{\sigma}^2) = \sqrt{\mathbb{E}\left[\left(\hat{\sigma}^2-\sigma^2\right)^2\right]}$$  I can think of a simple approach to estimate $\sigma^2$ given samples of $A,B$ which is to simply compute the sample variance. Also I know that the variance of $E$ is $4\sigma^2$ so I could compute the sample variance of $E$ and divide it by 4 to get an estimate of $\sigma^2$. However it should be possible to combine the information from both $E$ and $A,B$ to improve estimation $\sigma^2$. Could you outline the approach to use both sets of information? Performance of a Trigger vs Stored Procedure in MySQL "It looks straightforward, but actually it isn't" In , I asked about proof of statements which are simple but incorrect.   Here, I ask about statements which seems, at a first glance, straightforward, but if we try to write a proof, we can see it's much harder than it looked. So I expect the answers to contain:   the statement; why it looks easy to prove; why actually it isn't. Get list of all tables in Oracle? How do I put /home in a different partition? Yeah, so I have 2 hard drives, on with 2 partitions(backups and windows), and I have another that also has 2 partitions with only one used. I have Ubuntu on hard drive 2:partition one, but I have come to a point when the /home folder can't hold any more files, but the partition is only about 15% filled up. So I did some researching and figured out that you can put it on a separate partition, but most of the guides were vague on what I needed to do. Could someone give me step-by-step instructions on what I need to do, or even tell me a, easier or different way to fix this /home folder capacity problem? How to get my own IP address and save it to a variable in a shell script? What did Dumbledore say his scar on his left knee was? In Philosopher's Stone, we hear Dumbledore replying to McGonagall's query as to whether he could do something about the scar baby Harry has on his forehead:     “Even if I could, I wouldn’t. Scars can come in handy. I have one   myself above my left knee that is a perfect map of the London Underground." - Chapter 1, The Boy Who Lived.   Now, I have no idea what that last sentence means.  It could be because English is not my native tongue. Is it some kind of a joke, or does it really mean something? How to run Visual Studio post-build events for debug build only How can I limit my post-build events to running only for one type of build?  I'm using the events to copy DLL files to a local IIS virtual directory, but I don't want this happening on the build server in release mode. Any suggestions for a good undergraduate introductory textbook to statistics? I am hoping you can give me some suggestions. I am teaching in a very diverse (made of minority groups) college and the students are mostly Psychology majors. Most students are fresh from high school but some of them are older returning students above 40. Most of the students have motivational problems and aversion to math. But I am still looking for a book that covers the basic curriculum: from descriptive to sampling and testing all the way to ANOVA, and all in the context of experimental methods. The department requires me to use SPSS in class, but I like the idea of building the analysis in a spreadsheet such as excel.  p.s. the other teachers use a book that I don't like because of the extensive reliance on computational formulae. I find using these computational formulas - rather than the more intuitive and computationally intensive formula that is consistent with the rational and basic algorithm- unintuitive, unnecessary and confusing. This is the book I refer to Essentials of Statistics for the Behavioral Sciences, 7th Edition Frederick J Gravetter State University of New York, Brockport Larry B. Wallnau State University of New York, Brockport ISBN-10: 049581220X Thank you for reading!