How do I type 1A in LaTeX as in the snippet below?
I am trying to put two subscripts at the left and right of a character. For example, something like:  _{t} p_{x}  where p is in the middle. How do you do this?
What does increased caster level affect? Mod shows up with OP background when viewing deleted comments Jointly Complete Sufficient Statistics: Uniform(a, b) Let $\mathbf{X}= (x_1, x_2, \dots x_n)$ be a random sample from the uniform distribution on $(a,b)$, where $a &lt; b$.   Let $Y_1$ and $Y_n$ be the largest and smallest order statistics.  Show that the statistic $(Y_1, Y_n)$ is a jointly complete sufficient statistic for the parameter $\theta  = (a, b)$.     It is no problem for me to show sufficiency using factorization.   Question: How do I show completeness? Preferably I would like a hint.    Attempt: I can show $\mathbb E[g(T(x))] = 0$ implies $g(T(x)) = 0$ for the one parameter uniform distribution, but I am getting stuck on the two parameter uniform distribution.   I tried playing around with $\mathbb E[g(Y_1, Y_n)]$ and using the joint distribution of $Y_1$ and $Y_n$, but I am not sure if I am going in the correct direction, as the calculus is tripping me up. I registered my Stack Overflow account using the Stack Exchange OpenID provider. I later changed my e-mail, but I cannot change the e-mail address associated with my OpenID for some reason. Why is that, and how do I change it? I logged in to Stack Exchange and it lets me change my password, but never my e-mail. I am trying to animate in 2d. I do it with planes that have textures on them. I recently came up with the idea of having a plane with an image sequence of mouths because animating actual lip syncing by hand takes too long and isn't really necessary for what I'm trying to acomplish. However, when playing back the animation in textured viewport shading the animated texture stays at the last frame until the animation stops. I can use the arrow keys and it works fine as well as scrubbing through the timeline, just not when playing back the animation with alt+a. Can anyone help me with this problem? Visa for transit in Seoul This is question 2.9 #9 from Topics in Algebra by Herstein:     If $o(G)$ is $pq$ where $p$ and $q$ are distinct prime numbers and if $G$ has a normal subgroup of order $p$ and a normal subgroup of order $q$, prove that $G$ is cyclic.   Let $N$ denote the normal subgroup of order $p$ and $M$ denote the normal subgroup of order $q$. Here are a couple things I noted while exploring the problem:   $N$ and $M$ are cyclic, since they have prime order. $G = NM$ [1].   Any hints where to look next?    [1] Since $N,M$ are normal, $NM \le G$. Then $o(NM) \mid o(G)$, so $o(NM)$ is either $1$, $p$, $q$, or $pq$. The first three don't work because they force one of $N$ or $M$ to be $(e)$, so we have $o(NM) = pq$. Therefore, $NM = G$. Can I use a variable which has a non-linear relationship to the dependent variable in logistic regression? Let's say I am building a logistic regression model where the dependent variable is binary and can take the values $0$ or $1$. Let the independent variables be $x_1, x_2, ..., x_m$ - there are $m$ independent variables. Let's say for the $k$th independent variable, the bivariate analysis shows a U-shaped trend - i.e., if I group $x_k$ into $20$ bins each containing roughly equal number of observations and calculate the 'bad rate' for each bin - # observations where y = 0 / total observations in each bin - then I get a U shaped curve.  My questions are:   Can I directly use $x_k$ as input while estimating the beta parameters? Are any statistical assumptions violated which might cause significant error in estimating the parameters? Is it necessary to 'linearize' this variable through a transformation (log, square, product with itself, etc.)? Very strange error suddenly appeared running Biber How to add texture to fill colors in ggplot2 Error 404 : While using flag What does "Package  has no installation candidate" mean? The ship flying away from Rey in her flashback of being left on Jakku looks like one of the ships in the group that takes her away when she is captured.  If so, and it is a First Order ship that dropped her off, this is an interesting Easter egg. Spaces in variable assignments in shell scripts Recursively rename files and directories How to find all subgroups of a direct product? I am wondering how do we find all subgroups of a direct product? Is there a method to find it?  For example, how can we find all the subgroups of $\mathbb{Z}_2\times\mathbb{Z}_2$?  There is the answer: $\{(0, 0), (1, 0)\}, \{(0, 0), (0, 1)\}, \{(0, 0), (1, 1)\}$, and the improper and trivial subgroup. But I do not know the method how to find it.   All I can think about is the following: since the order of $\mathbb{Z}_2\times\mathbb{Z}_2$ is $4$, the order of a subgroup of it can be $1,2$ or $4$.   If order is $1$, then it is trivial, and if order is $4$, it is proper. Then, the order of a subgroup is $2$. We know all the elements of   $\mathbb{Z}_2\times\mathbb{Z}_2$, so we try all the pairs.   Is that the method?   But what if we are given a bigger order product, such as $\mathbb{Z}_2\times\mathbb{Z}_8\times\mathbb{Z}_3\times\mathbb{Z}_9$? It looks very long to try all possibilities.   Can anyone help?  Thanks I have a 4GB DDR3 800Mhz recently I installed another 4GB. It turns out it's  DDR3 667MHz. Does it really matter I should own both rams 800MHz?    Slot1 shows recent RAM installation. ALSO I don't know if this is 1600MHz (it shows 800x2 and 667x2 for other) or just 800MHz ram. Inline elements shifting when made bold on hover I created a horizontal menu using a HTML lists and CSS. Everything works as it should except when you hover over the links. You see, I created a bold hover state for the links, and now the menu links shift because of the bold size difference.  I encounter the same problem as . However, the post does not have proper solution. I've searched everywhere for a solution and can't find one. Surely I can't be the only one trying to do this.  Does anyone have any ideas?  P.S: I don't know the width of the text in menu items so I cannot just set the width of the li items.    .nav { margin: 0; padding: 0; } .nav li {      list-style: none;      display: inline;      border-left: #ffffff 1px solid;  } .nav li a:link, .nav li a:visited {      text-decoration: none;      color: #000;      margin-left: 8px;      margin-right: 5px;  } .nav li a:hover{      text-decoration: none;      font-weight: bold;  } .nav li.first { border: none; } &lt;ul class="nav"&gt;     &lt;li class="first"&gt;&lt;a href="#"&gt;item 0&lt;/a&gt;&lt;/li&gt;     &lt;li&gt;&lt;a href="#"&gt;item 1&lt;/a&gt;&lt;/li&gt;     &lt;li&gt;&lt;a href="#"&gt;item 2&lt;/a&gt;&lt;/li&gt;     &lt;li&gt;&lt;a href="#"&gt;item 3&lt;/a&gt;&lt;/li&gt;     &lt;li&gt;&lt;a href="#"&gt;item 4&lt;/a&gt;&lt;/li&gt; &lt;/ul&gt; BeanFactory vs ApplicationContext Prove that $|\log(1 + x^2) - \log(1 + y^2)| \le |x-y|$ I need to show that $ \forall x,y \in \mathbb R, |\log(1 + x^2) - \log(1 + y^2)| \le |x-y|$  I tried using concavity of log function: $\log(1 + x^2) - \log(1 + y^2)=\log(\frac{1 + x^2}{1 + y^2})=\log(\frac{x^2y^2}{(1 + y^2)y^2}+\frac{1}{1 + y^2}) \ge \frac{2(\log(x)-\log(y))}{1+y^2}$  Also the middle value theorem: $0&lt;\frac{\log(1 + x^2) - \log(1 + y^2)}{x^2-y^2} &lt;1$  But both attempts has not led too far.