Possibility of the encrypted text being the same as the supplied plain text!
Block cipher fixed points (plaintext equal to ciphertext)
I run into this infinite product today. I don't have that much experience evaluating products therefore I don't have any idea of how to tackle this.   Here is the question. Evaluate (if possible) in a closed form the product:  $$\Pi = \prod_{n=1}^{\infty} \frac{1}{1+\pi^{{\large \frac{1}{2^n}}}}$$  The numerical value seems to be $\Pi= 0.534523$ which is very close to $\frac{\pi}{6}$ taking into account that $\frac{\pi}{6} \approx 0.523598$. In the mean time W|A evaluates it to $0$. I'm lost.   Can the community help?  Edit: Based on the answer we have two products. The product I asked tends to zero and the bonus product provided by @you're in my eye (thanks for that) is $\frac{\ln \pi}{\pi-1}$.   Thanks for the quick response. Algebraic extension of perfect field in which every polynomial has a root is algebraically closed Let $F$ be a perfect field, i.e. every irreducible polynomial over $F$ has distinct roots in the algebraic closure of $F$. Suppose that $K$ is an algebraic extension of $F$ with the property that every non-constant $p(X) \in F[X]$ has a root in $K$.   I want to show that $K$ is algebraically closed, i.e. that $K = \overline{F}$, the algebraic closure of $F$.  If $K$ were a normal extension of $F$, this would follow immediately: Let $\alpha \in \overline{F}$ and $m(X)\in F[X]$ its minimal polynomial. By our assumption, $m(X)$ has at least one zero in $K$. Since $K$ is a normal extension of $F$, it thus contains all other roots of $m(X)$ including $\alpha$.  I don't have an idea as to how to show that $K$ is a normal extension. Index entries duplicated for captions In document class book, and package caption, creating index with makeidx. Index entries for the main text and for footnotes collate properly; but those for captions create their own entries.  Example: for the tag \index{New York!Custom House!\textit{Cartouche}}, text and footnotes collate properly; but used in a caption, a new entry is created at the third level, and is also out-of-alphabetic-sequence. Note: the top level, and first sub are correct; only the second sub is duplicated.  How to get the caption index entries under the same entry? $\chi(G)+\chi(G')\leq n+1$ How to prove, that the sum of chromatic numbers of graph and it's complement is smaller then the number of vertices incremented by one?  $\chi(G)+\chi(G')\leq n+1$  The notes from my classes say to color the first graph using sequential algorithm with vertices ordered by their degrees (the ones with lowest appear first). We would get $k_1$ colors, then we color the graph $G'$ in the backwards order and get $k_2$ colors, and the sum $k_1+k_2$ in my notes is magically smaller than $n+1$, but I just don't see why... Prove directly from the definition of the Ito's integral I am trying to solve the exercises from the book Stochastic differential equations -An Introduction with applications by Bernt Oksendal and I am stuck on 1 question.  Prove directly from the definition of Ito's integral that   $$\int_0^t B_s^2 dB_s=\frac{1}{3}B_t^3-\int_0^t B_s ds$$  I have tried for a couple of hours but I simply cannot prove it . I would be grateful if somebody could help me out , even a hint would be very helpful If $f$ is continuous on $[a,b]$, $f(x)≥0$ on $[a,b]$ and $$\int_{a}^{b} f(x) =0$$   then prove that $f(x)=0$ for all $x \in [a,b]$.  I tried with Riemann's definite integral definition but couldn't proceed How to install Ubuntu on VirtualBox? Should I capitalize the word "that" in a title? Since the version of the word "that" might matter to the correct answer, I'll provide the actual title in question:  Title:  Existing Solutions that Didn't Work  If someone could explain the why of the answer, I would be appreciative.  EDIT:  It has been suggested that this is a duplicate, and that may be, in a sense.  However, the after reading the suggested prior question, I'm no closer to my answer.  I think it may have to do with my use of the word "that."  Perhaps it is incorrect?  Using these definitions of the word "that:" I'm having trouble figuring out which definition of that matches my title.  Am I even using it correctly?  Perhaps I should use which instead:  Possible Title:  Existing Solutions which Didn't Work  If my original usage is incorrect, and I should use which, then the answer is simple since which is always a pronoun or adjective, and those are both capitalized in a title.  However, I'm having difficulty understanding which part of speech that is in my original title. In the recent post , Werner helped me to redefine a proof environment which can be hidden by an optional boolean argument (showProof). Without an argument, somehow the line/paragraph breaking is not correct. How can one fix this?  \documentclass{scrartcl}  \usepackage{ifthen} \usepackage{environ} \usepackage{amsthm}  \newboolean{showProof}% for showing proofs \setboolean{showProof}{true}% setting the switch \newif\ifstarttheorem  % proof environment (with hide feature) \makeatletter \providecommand{\env@proof@save@env}{} \providecommand{\env@proof@process}{} \RenewEnviron{proof}[2][showProof]{   \gdef\@tmp{#1}%   \ifthenelse{\boolean{#1}}{\par\pushQED{\qed}%     \normalfont\topsep2\p@\@plus2\p@\relax     \trivlist\item[\hskip\labelsep\sffamily\bfseries Proof~#2]\gdef\mycurrenvir{proof}\global\starttheoremtrue\mbox{}\hfill\\*\ignorespaces     \BODY}{\ignorespaces} }[\ifthenelse{\boolean{\@tmp}}{%   \gdef\mycurrenvir{\relax}   \popQED\endtrivlist\@endpefalse }{\ignorespacesafterend}] \makeatother  \begin{document} Just some text \begin{proof}   Just some text in a standard proof. As we can see, there is a problem. \end{proof} \begin{proof}[false]   Proof with optional argument set to \texttt{false}. This proof correctly does   not appear. \end{proof} \begin{proof}[showProof]{(Heading)}   Proof with optional argument (\texttt{true}) and second argument ``(Heading)'' \end{proof} \end{document} I have created an FME workspace which inputs a pipeline as well as other linear features which cross it (waterways, roads, railways, fences, other pipelines, etc).  The output is a point file which contains the crossing type.  What I would like to do is calculate the distance along the pipe from the start of the pipeline for each intersection.  So, as you travel along the pipeline, the distance attribute will increase.   I'm thinking that I'll need to use some of the linear referencing tools like MeasureGenerator, MeasureExtractor, or something like that, but I'm not sure how to do so.  Does anyone have an idea of how to do this? Best practice of when to load data into a ListBox How would I go about identifying a port, card or connector on my system which I am unfamiliar with?  Where would I find information on what these parts are? What would be the steps I would take in identifying a unknown port or cable?   This question comes up frequently, and the suggested solutions are usually the same. This community wiki is an attempt to serve as the definitive, most comprehensive answer possible. Feel free to add your contributions via edits. How do you unpin and move the form in the Visual Studio designer? Improve file-info visibility in Finder's preview pane I find the file information in the preview pane in column view is far more useful than the thumbnail.  I wish I did not have to continually resize the Finder window or scroll the preview pane to see the info I'm looking for.  As you can see below, the useful info is hidden in default  size  of Finder's new window under the preview options like rotate etc.        Is there a way to do any of these to achieve the said goal (listed in order of decreasing likelihood!):   reduce the size of the preview "thumbnail" covered by black here,  reduce the amount of whitespace(grey here) around the preview "thumbnail" (only significant for landscape thumbnails), remove the preview "thumbnail" altogether,  reorder the content of the preview column(bring file info up and send thumbnail down)?    includes the options:   resizing the Finder in every new window (which is labor intensive). Using View menu> Show Preview Options to turn off Quick Actions (which must be done for each file-type, and only reveals a small amount of file info). Zero dimensional points do not take up space, so then wouldn't everything in the universe be literally empty? Or is there something that I'm missing? Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$. I'm stuck with the proof of the following:  Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$.   I've tried starting with $s_n \leq t_n \: \forall : n$ and the definitions of each limit (i.e. $|s_n - s| \leq \epsilon \: \forall \: n &gt; N_1$), but I'm not really getting very far. Any help is appreciated! How do I disable the guest session? How do I disable the guest session in Ubuntu 11.10 or higher? I don't want people to be able to use my computer without using a password to log in! I was recently told (  ):     You cannot delete a class in production directly. You will need delete   the class from your sandbox and then deploy the deletions to your   production org.   When you deploy from sandbox to production, the   missing classes will come up in red and you can choose to deploy these   deletions to Production.   However, I'd like some clarification because finding information on how to do this is like pulling teeth, unfortunately.  Am I supposed to:   Delete the Trigger in Sandbox Make a deployment package in Sandbox Add the Trigger which is going to be listed in RED now Deploy (and that deletes it from Production)?   I am not seeing deleted Triggers in Red in Sandbox.  Are they NOT listed under "Triggers" for deployment, but elsewhere?   Frustrated = true.   Does this ONLY apply to Classes??  UPDATE:  I just tried again, with a CLASS, and it still doesn't show up at all as a Deployable package (red or not) after I delete it in the Sandbox.  UPDATE 2:  I was just able to delete a class, that I know was written via IDE from a third party, in both Sandbox and Production.  So now I'm wondering if it has to do with where it was originally deployed from (if from IDE, IDE can delete it - it from Sandbox, you have to jump through hoops).  :-( How to find the square root of a permutation Observe that  $$\begin{pmatrix} 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\  2 &amp; 4 &amp; 1 &amp; 5 &amp; 3 \end{pmatrix} \begin{pmatrix} 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\  2 &amp; 4 &amp; 1 &amp; 5 &amp; 3 \end{pmatrix} =  \begin{pmatrix} 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\  4 &amp; 5 &amp; 2 &amp; 3 &amp; 1 \end{pmatrix},$$  so $\begin{pmatrix} 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\  4 &amp; 5 &amp; 2 &amp; 3 &amp; 1 \end{pmatrix}$ is the square of a permutation. More generally, how do I check if a permutation is square and how can I find its root? Build a basic Python iterator