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How do I insert the total continuation count in the allowframbreaks frame title? I would like to change the numbering of allowframbreaks frame, for instance, from I II III to (1/3) (2/3) (3/3)， but fail to find out a command to get the total continuation count. Here is the MWE.  \documentclass[compress]{beamer} \usetheme{Warsaw}   %allowframebreaks numbering in the title \setbeamertemplate{frametitle continuation}{(\insertcontinuationcount/)} %\setbeamertemplate{frametitle continuation}{(\insertcontinuationcount/\inserttotalcontinuationcount)}  \title{title} \author{author}  %%%%% \begin{document}  \begin{frame}[allowframebreaks]{allowframebreaks title} frame 1 \framebreak  frame 2 \framebreak  frame 3 \framebreak \end{frame}  \end{document} My stance is that these websites are a great boon to the research community and they were certainly extremely helpful for my research. I think they should be applauded rather than persecuted, and while they may be illegal now, I think a way to make them legal should be found (in the same way I can download articles legally through my university's subscription to all the major publishers and journals).  Can I thank them in the acknowledgements section of my thesis? A field extension of prime degree Suppose that $E$ is an extension of $F$ of prime degree. Show that $~~\forall~ a \in E : ~ F(a)=F$ or $F(a)=E$   Attempt: Suppose that $E$ is an extension of a field $F$ of prime degree, $p$. Therefore $p = [E :F] = [E : F(a)][F(a) : F]$. Since $p$ is a prime number, we see that either $[E : F(a)] = 1$ or $[F(a) : F] = 1$.   Now, $[E : F(a)] = 1 \implies $ there is only one element $x \in E$ which forms a basis and every element in $E$ is generated by $x$ i.e. $E = \{x~c~|~ c \in F(a)\}$  $[F(a) : F] = 1 \implies $ there is only one element $y \in F(a)$ which forms a basis and every element in $F(a)$ is generated by $y$ i.e. $F(a) = \{y~d~|~ d \in F\}$  Have I inferred it correctly? How do I move ahead?  Thank you for your help. Do there Exist Proper Classes that aren't "Too Big" Some proper classes are "too big" to be a set in the sense that they have a subclass that can be put in bijection with $\alpha$ for every cardinal $\alpha$. It is implied in  post that every proper class is "too big" to be a set in this sense, however I have been unable to prove it. It's true if every two proper classes are in bijection, but it's consistent with ZFC for there to be a pair of non-bijective classes.  So, is the following true in ZFC:     For all proper classes, $C$, and $\alpha\in\mathbf{Card}$, $\exists S\subset C$ such that $|S|=\alpha$?   If not, is there something reasonable similar that preserves the intuition about classes that are "too big to be sets"? If $\sum_na_n=\infty$ and $a_n\downarrow 0$ then $\sum\limits_n\min(a_{n},\frac{1}{n})=\infty$ If $\sum_na_n=\infty$ and $a_n\downarrow 0$ then $\sum\limits_n\min(a_{n},\frac{1}{n})=\infty$     My  idea:  $$b_{n}=\min(a_{n},\tfrac{1}{n})\Longrightarrow b_{n}\le a_{n},b_{n}\le\tfrac{1}{n}$$ then I don't know how to go on. IBM Slack links broken Show that $(2,0,4) , (4,1,-1) , (6,7,7)$ form a right triangle What I tried:  Let $A(2,0,4)$, $B(4,1,-1)$, $C(6,7,7)$ then  $$\vec{AB}=(2,1,-5), \vec{AC}=(4,7,3), \vec{BC}=(2,6,8)$$  Then I calculated the angle between vectors:  $$\begin{aligned} \alpha_1 &amp;= \cos^{-1}\left(\frac{(2,1,-5)(4,7,3)}{\sqrt{2^2+1^2+(-5)^2}\sqrt{4^2+7^2+3^2}}\right) \\ &amp;= \cos^{-1}(0)=90° \\ \alpha_2 &amp;= \cos^{-1}\left(\frac{(4,7,3)(2,6,8)}{\sqrt{4^2+7^2+3^2}\sqrt{2^2+6^2+8^2}}\right) \\ &amp;= \cos^{-1}\left(\frac{74}{\sqrt{74}\sqrt{104}}\right)=32.49\\ \alpha_3 &amp;= \cos^{-1}\left(\frac{(2,6,8)(2,1,-5)}{\sqrt{2^2+6^2+8^2}\sqrt{2^2+1^2+(-5)^2}}\right) \\ &amp;= \cos^{-1}\left(\frac{-30}{\sqrt{104}\sqrt{30}}\right)=122.5° \end {aligned}$$  As you can see, these angles don't even form a triangle, what am I doing wrong, any thoughts? Would anyone can help me how to show that a finitely generated projective module over a local ring and PID are free?    What I know about a finitely generated projective module $M$ over a PID $R$ is isomorphic to $R^k\oplus R/(a_1)\oplus\dots\oplus R/(a_n)$, and for the local ring case I don't know how to start. Evaluate $\mathop {\lim }\limits_{x \to 0} \left( {{1 \over {{{\sin }^2}x}} - {1 \over {{x^2}}}} \right)$ I tried l'Hospital but that will require a lot (and I mean A LOT!!!) of differentiating  Is there a shortcut? $$\mathop {\lim }\limits_{x \to 0} \left( {{1 \over {{{\sin }^2}x}} - {1 \over {{x^2}}}} \right)$$  Thanks in advance Apex:OutputText to Currency Format from String Wrapper is there a way to pass a string to the &lt;apex:outputText&gt; so that it can be formatted as a currency or percent value?  Here's what I have inside a column of my visual force table:  &lt;apex:outputText value="{0, number, currecny}"&gt; &lt;apex:param value="{!row.values[1]}" /&gt; &lt;/apex:outputText   this gives me an error at runtime (when the visual force page loads), which I think may be some kind of type mismatch, where &lt;apex:outputText&gt; is expecting a Decimal or number but I am passing a string.  This is the error at runtime:     The value attribute on  is not in a valid format. It must be a positive number, and of type Number, Date, Time, or Choice.   Thanks Can my friend and I play minecraft together with different mods? I'm wondering if I could play Minecraft with my friend. A modded Minecraft, only myself with mods and him without mods.  Is it possible? Validity and Equivalence of two definitions of the real exponential function The Problem :  We state the following two definitions of the real exponential function from the . We're interested in showing that the two definitions are valid $($i.e. the defining sequence/series does converge to a unique real number$)$ and that the two definitions are equivalent.   I'm stuck at a couple of points $($which are described in highlighted lines$)$. Any help would be much appreciated. Thank you!     Definition $1$. The exponential function can be defined as the following limit of a sequence $$\exp x := \lim_{n \to \infty} \left({1 + \frac x n}\right)^n$$      Definition $2$. The exponential function can be defined as a  $$\exp x := \sum_{n = 0}^\infty \frac {x^n} {n!}$$   My Progress and two places where I'm stuck : Essentially the solution consists of three parts,  namely validity of definition $1$, validity of definition $2$ and equivalence of the two definitions.     Validity of Definition $1$. I'm stuck here! Can we show that the sequence $(a_n)$ given by $a_n = \left(1+\frac{x}{n}\right)^n$ converges for every $x \in \mathbb{R} ??$ Is it eventually monotone and bounded $??$   Validity of Definition $2$. The  of the power series is $$r=\lim_{n \to \infty}\left| \frac{\frac{1}{n!}}{\frac{1}{(n+1)!}} \right|=\lim_{n \to \infty}\left| \frac{(n+1)!}{n!} \right|=\lim_{n \to \infty}(n+1)=+\infty$$ Thus the infinite series in the right-hand side of Definition $2$ converges to a unique real number for all $x \in \mathbb{R}$. Hence the definition is well-defined.  Equivalence of Definition $1$ and Definition $2$.  There's this  of Definition $1$ $\implies$ Definition $2$ in the Pr$\infty$fWiki page, but it's kind of under construction and I'm not really convinced by it. So I decided to try to take my own shot at it.  For all $n \in \mathbb{N} \cup \{0\},$ let  $$T_n=\left(1+\frac{x}{n} \right)^n, ~S_n=\sum_{k=0}^n \frac{x^k}{k!}$$ We have to show that $\lim_{n \to \infty} T_n = \lim_{n \to \infty} S_n$  Now, \begin{align} T_n &amp;= \left(1+\frac{x}{n}\right)^n\\ &amp;= 1+n\cdot\frac{x}{n}+\frac{n(n-1)}{2!}\cdot\frac{x^2}{n^2}+\cdots +\frac{n(n-1)\cdots 1}{n!}\cdot\frac{x^n}{n^n}\\ &amp;= 1+x+\left(1-\frac{1}{n}\right)\cdot \frac{x^2}{2!}+\cdots +\left(1-\frac{1}{n}\right)\cdots \left(1-\frac{n-1}{n}\right)\cdot \frac{x^n}{n!} \end{align}  Clearly, $$S_n-T_n=\left\{1-\left(1-\frac{1}{n}\right)\right\}\frac{x^2}{2!}+\cdots +\left\{1-\left(1-\frac{1}{n}\right)\cdots \left(1-\frac{n-1}{n}\right)\right\}\cdot \frac{x^n}{n!}\geq 0$$     I'm stuck at this point. Can we show that $S_n-T_n \leq B_n$ such that $B_n \to 0$ as $n \to \infty ??$ "log in to" or "log into" or "login to" When writing an instruction about connecting to a computer using ssh, telnet, etc., I'm not sure what spacing to use in this familiar spoken phrase:   "Log in to host.com" "Log into host.com" "Login to host.com"   Maybe this is entirely subjective or the realm of industry jargon, but I couldn't think of anywhere else to ask.  Any insight? Strange artifacts along window borders after waking computer from sleep mode What should I do if I want to be able to run a given program regardless of my current directory? Should I create a symbolic link to the program in the /bin folder? Drawing a constant randomly distorted circle I am trying to draw a special and contradictive plot : A circle with random distortions which have to remain constant regardless of how many times I build the document.  An example of what I am trying to achieve can be seen below    This figure was produced with the following code  \documentclass{standalone} \usepackage{tikz}    \usetikzlibrary{decorations.pathmorphing}   \begin{document}     \begin{tikzpicture}         % axes         \draw[ultra thin] (-2.5, 0) -- (2.5, 0);         \draw[ultra thin] (0, -2.5) -- (0, 2.5);         % undistorted sphere         \draw[ultra thin, dashed] (0, 0) circle (2.25cm);         \draw[ultra thin, dashed, -&gt;] (0:0) -- (125:2.25);         \node at (-1, 1) {$R$};         % distrorted sphere         \draw[very thick] plot[domain=0:350,smooth cycle ] (\x:2+rnd*0.5);         \draw[very thick, -&gt;] (0:0) -- (45:2.1);         \draw[ultra thin, -&gt;] (1, 0) arc (0:45:1);         \node at (1.2, 0.4) {$\theta$};         \node at (0.7, 1.25) {$R(\theta)$};     \end{tikzpicture} \end{document}   The problem with the code is that whenever I make a change to it and rebuild the document, the "distorted" circle will have a different shape, since I am using the rnd function. The reason I would like to have a constant-random shape is because I would like to denote its radius at an angle θ, therefore it has to touch the outline of the circle.  Any idea on how to achieve such a thing? Let's say I have a class that has a member called data which is a list.    I want to be able to initialize the class with, for example, a filename (which contains data to initialize the list) or with an actual list.  What's your technique for doing this?  Do you just check the type by looking at __class__?  Is there some trick I might be missing?  I'm used to C++ where overloading by argument type is easy. Where does Drupal store its content? I have been currently learning Drupal while trying to duplicate a Drupal application into CakePHP.  I have been looking at the  database tables to know where data is being stored so I can safely migrate any data or study how it is working.  I have researched and have come across questions like:     So far, I have only seen the title in my Drupal (7) installation.  What I have seen in my node revisions table for a recipe content type, which includes fields like, field_ingredients, field_instructions, field_description, field_preptime, field_cookingtime, just the title of the recipe. This is a sample record on my node_revision table.  nid   |  vid  |  title         |  log   | timestamp  | status | comment | promote  | sticky ------------------------------------------------------------------------------------------- 1     |  1    |  Awesome Sauce |        | 1369998600 | 1      | 2       | 1        | 0   As you can see I can only see the title. Where are the other field contents? Anthology for children; contains story about friendly Martian on postapocalyptic Earth I'm looking for the title of an anthology of children's science fiction stories from the 1950s. The best of the bunch involved a Martian who visited earth, and realized we'd blown ourselves up in a nuclear war and had fallen into savagery. The Martian is a vegetarian who subsists on biscuits made by a machine he brought with him, and communicates by using one of his tentacles to feed messages direction to/from the brain of the person he's talking to.  Inadvertently left behind by his expedition, he initially befriended some children, and eventually managed to rebuild civilization. Presently he dies, but when another expedition comes a generation later, they find the civilization he rebuilt.  there were other stories, but that was the best one.   If I can't find the name of the book it was in, I'd be happy to settle for the name of the author and the title of the story. Cancelled UK visa on entry