Hi everyone please I am writing a proof in beamer with \begin{proof} \end{proof} but the proof is too long so I can't have it in one frame, I wanted to share the proof to many frames but it didn't work. Thanks for helping.
For a presentation with Beamer, I would like to have a slide where a long list of formulas (maybe mixed with text) scrolls automatically within a certain part of the slide. This should look like e.g. closing credits in movies.   I have tried the idea from  based on animateinline from the , but whenever I use environments like \begin{equation} ... \end{equation} or commands like \textcolor{red}{...} I get an error:  TeX capacity exceeded, sorry [main memory size=3000000]. \end{frame}   Ideally, the formulas/the text should be given within the document itself, but if a solution with an external PDF works fine, I'll take it!  Any ideas how that might work?
Displaying and/or searching across thousands of contacts? The Need  Our application has a chat feature that allows our users to chat with other users within their database.   However, we're adding a feature that allows clients to share data between multiple databases as an "Enterprise" configuration, and we've been asked to update the design of the chat feature to allow users to chat with other users within any of those databases.   We've looked at the numbers of possible users in these scenarios, and the number of chat users in these shared database environments can be as many as 5,000.   Users need to be able find people they need to chat with out of these thousands of contacts. However, up til now, we've had a list of contacts the users can browse. Since that is a user expectation, we were hoping to still maintain a browse-able list of contacts. We have ruled out trying to display a list of 5,000 contacts, as this doesn't seem to be anywhere close to useful or usable.   We've come up with three approaches to solve the problem:  Option One  Show a list of contacts from the local database that user's can browse. Have a second tab that allows users to search the "Full Directory", returning results from multiple databases.     Pros:   Allows us to show a browse-able list of relevant contacts  Clearly indicates to users that another source of chat contacts are available   Cons:   Search would be limited to the current tab, meaning users would have to be in the right tab to return the right result Adds a possible source of user confusion: whats the difference between "contacts" and the "full directory"? Adds another tab to what should be a simple chat list     Option Two  Show a list of contacts from the user's local database, but if they search for a contact, include results from other databases as well.     Pros:   Maintains a list of contacts the user can browse Avoids adding adding another tab One search field searches all sources   Cons:   Searching a finite contact list and returning results that weren't in the contact list may defy user's mental model of how the search will work. Not clear that another source of contacts exist. May need to add explanatory text that says "Search to see contacts from the full directory".      Option Three  Add a dropdown that allows a user to select a what to search in: the local database, or the full directory.     Pros:    No additional tab required Indicates to users that another source of chat contacts are available, though not as clearly as option 1   Cons:    Search still isn't universal: the right search category must be selected to return the right result Users may not notice the dropdown or understand intuitively what its function is. May require explanatory text in search results, reminding the user they can expand the search to the full directory     Thoughts?  Right now I'm leaning towards option two, but is there another option I'm not considering here? Is there another alternative worth considering?   Thanks! Normal subgroups of $S_4$ Can anyone tell me how to find all normal subgroups of the symmetric group $S_4$?  In particular are $H=\{e,(1 2)(3 4)\}$ and $K=\{e,(1 2)(3 4), (1 3)(2 4),(1 4)(2 3)\}$ normal subgroups? Compare means of group with a mean of a subgroup Getting file names without extensions I'm trying to prove that if $A\neq 0$ is a commutative ring and there is an injective $A$-module homomorphism $A^m\hookrightarrow A^n$ then $m\leq n$ must necessarily hold.    This is exercise 2.11 from Atiyah and MacDonald's Introduction to Commutative Algebra, but unfortunately all the solutions available online are either very sparse with regard to this question or seem to use (to my surprise) a generalized version of Cramer's rule. Is there perhaps some other "cleaner" approach to solving this problem? Show that Function Compositions Are Associative My intent is to show that a composition of bijections is also a bijection by showing the existence of an inverse.  But my approach requires the associativity of function composition.  Let $f:  X \rightarrow Y, g:  Y \rightarrow Z, h:  Z \rightarrow W$ be functions.  $((f \circ g) \circ h)(x) = h((f \circ g)(x)) = h(g(f(x)))$, and $(f \circ (g \circ h))(x) = (g \circ h)(f(x)) = h(g(f(x)))$.  However, I am having problems in justifying that the two compositions, $(f \circ g) \circ h$ and $f \circ (g \circ h)$, have the same domain and range.  When I consulted ProofWiki, whose link is at the bottom, I got even more confused.  Specifically, for $(f \circ g) \circ h = f \circ (g \circ h)$ to be defined, ProofWiki requires that dom$g =$ codom$f$ and dom$h =$ codom$g$.  First of all, I think that it should be dom$g =$ range$f$ ....  Moreover, as you can see in the example below, you actually have to adjust domains and ranges of $f, g, h$ for the requirement to hold true.  Let $f: \mathbb R \rightarrow \mathbb R$ be $f(x) = 2x$, $g: \mathbb R^+ \rightarrow \mathbb R$ be $g(y) = ln(y)$, $h: \mathbb R \rightarrow \mathbb R$ be $h(z) = z - 10$.   Then $((f \circ g) \circ h)(x) = ln(2x) - 10 = (f \circ (g \circ h))(x)$, with dom$((f \circ g) \circ h) = \mathbb R^+$ = dom$(f \circ (g \circ h))$.  As a result, we need to set dom$f = \mathbb R^+$, range$f = \mathbb R^+$; dom$g$, range$g$, dom$h$, and range$h$ remain the same.  Am I allowed to do that?  This adjustment implies that when we say dom$f = X$, $f$ must be defined for all elements in $X$, but $X$ may not be the entire set of elements for which $f$ is defined. Proving the quotient of a principal ideal domain by a prime ideal is again a principal ideal domain Please help me prove that      the quotient of a principal ideal domain by a prime ideal is again a principal ideal domain.   This was from Abstract Algebra (Google) Title of page not updated, everything else did My website changed significantly this month.  Amongst the changes was the website's title.  Two weeks ago, I finally used Webmaster Tools and submitted the page.  So far, Yahoo and Bing have updated the title in the search results, but not Google.  I've resubmitted via Webmaster Tools the website, and the bots have crawled the page, and have added "content keywords".  But the title still doesn't match.  The "content keywords" lists many of the words in the new title (but contains keywords of the existing title also).  Must say that 4 of the top 5 key words in the "significance" list are the new title's.  Still, no change.  Any remedies for this problem?  Is adding the "alt" tag for the logo significant enough to trigger the change?  There isn't an appropriate place to put the company name between  tags on the front page.  Any suggestions?  Thanks. Do companies have IT "spyware" on corporate iPhones? How is Bohr's model related to electron cloud models via the correspondence principle? Some ships in Star Trek are referred to with definite articles. "The Enterprise", "The Defiant". Other ships are referred to without them. "Voyager", "Enterprise". Is there a pattern to or explanation for this behavior? I'd like to be able to find out which process is currently using a certain port in Linux. Is there any way to do this? Expected number of hamiltonian paths in a tournament The following theorem is from Alon&amp;Spencer's The probabilistic method, in the beginning of chapter 2:  Theorem 2.1.1: There is a tournament $T$ with $n$ vertices and at least $\frac{n!}{2^{n-1}}$ Hamiltonian paths.  Briefly the theorem is proved by looking at $X_\sigma$ the indicator random variable for a permutation $\sigma$ giving a Hamiltonian path, that is, satisfying $(\sigma(i),\sigma(i+1))\in T$ for $1\leq i \lt n$. Then   $\sum \mathbb{E}\left[X_\sigma\right] = \frac{n!}{2^{n-1}}$.  I don't understand where did this value ($\frac{n!}{2^{n-1}}$) came from! could someone explain it please? Geometric series and big theta Consider the following function:  $$S(n)=1+ c + c^2 + ··· + c^n,$$  where c is a positive real number.  (A) This function is the sum of a geometric series. Give a precise closed-form formula for $S(n)$, interms of $c$ and $n$, in the case where $c \neq 1$.  (B) Show that $S(n)$  is:  $\theta(1)$ if $c&lt;1$  $ \theta(n)$ if $c=1$  $\theta(c^n)$ if $c&gt;1$    can someone please explain how i can go about finding the solutions for these problems, i have never encountered these and am stuck, i know that big theta means that the function is bond on both sides but how do i got about finding the answer?  (edit) im still having trouble understanding part B. can someone please explain? (can someone walk me through the second one) so far i have $n \leq c^{n+1} \leq ???$  (edit #2) i tried doing the first one can someone verify if i did it right  $S(n)$ is $\Theta(1)$ if c&lt;1  (1)$S(n)$ is $O(1)$ if c&lt;1  (2)$S(n)$ is $\Omega(1)$ if c&lt;1  (1) $$\frac{C^{n+1}-1}{C-1} \lt (C_1)(1)$$ $$\frac{C^{n+1}-1}{C-1} \lt (C_1)$$ $$C^{n+1} \lt C_1(C-1)$$ $$C^nC-1 \lt C_2C-1$$ $$C^n \lt C_2$$ $$C^n \lt 1$$ since c&lt;1 this means that any number that will be $C^n$ will be less then 1 for all positive n  i did the omega part pretty much the same way and got $$C^n \gt C_2$$ $$C^n \gt 0$$ since c&lt;1, $C^n$ can potentially become very very small so the only thing bounding it is 0 for all positive n What is the definition of $2.5!$? (2.5 factorial) I was messing around with my TI-84 Plus Silver Edition calculator and discovered that it will actually give me values when taking the factorial of any number $n/2$ where $n$ is any integer greater than $-2$. Why does this happen? I thought factorials were only defined for positive integers and $0$, so what is my calculator doing to get the answer $3.32335097$ when I enter $2.5!$? Is there actually a definition of $2.5!$ or is my calculator just being weird? How is the factorial function implemented?  I understand the binary implications of $2.5$, so that could possibly have something to do with it. I get a domain error when trying to take the factorial of $-1, 2.3, e$, and any number that is not of the form $n/2$ where $n$ is any integer greater than $-2$. Plotting discontinuous functions without spurious vertical segments Let K={(1),(12)(34),(13)(24),(14)(23)} show that K is a normal subgroup of the alternating group of degree A4 and find all the members of A4/K and write down the group table for A4/K.   i know that K is the Klein four group and i have already proven it is a normal subgroup but i need a start on approaching the A4/K part of this question no direct answers are necessary,just ideas thank you! :) I have multiple cameras in a scene. What I want to do is to change which one is active. This way I can view the scene from different angles. How would I do this? What is the best way to increase my reputation and privileges? I'm not sure if this is a stupid question, but here goes anyway... I have only recently started learning PHP and Javascript and I'm reasonably proficient. One of the reasons I'm so proficient is because I LOVE Stack Overflow and find most of the solutions to my dilemmas here.   But I've found that I never get the chance to ask a question because it has normally been asked many times before and when I go through the unanswered questions to try and help someone else, they are generally WAY over my head.  So I'm not sure how to get more privileges. I can't even vote up answers that I like or that helped me. I can't comment to thank someone for their solution. It is most tragic and demoralising... How do I change the situation? Symbolizing layer with graduated colors by date/time field? I've got a layer of polygons where each has a date associated with it.  I want to color the polygons with graduated colors, but I want the colors to start with the newest polygons and fade to the oldest polygons.  So far I haven't been able to find a way to do this in ArcMap (&lt;= 9.3.1) other than to create a new field and calculate some integer or float value based on the date.  Is it possible to use a date or time field directly?   The data is natively a date-time value There are far too many records to use any unique-value renderer Converting the date to a numerical field is an answer for a different question.