General way to find orders of reduced residue classes?
Algorithms for computing multiplicative order modulo $n$
Prove the map has a fixed point Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$ such that $\rho (Ax,Ay) &lt; \rho(x,y)$ for $x\neq y$. Prove A have a unique fixed point in $K$.  The uniqueness is easy. My problem is to show that there a exist fixed point. $K$ is compact, so every sequence has convergent subsequence. Construct a sequence ${x_n}$ by $x_{n+1}=Ax_{n}$,$\{x_n\}$ has a convergent subsequence $\{ x_{n_k}\}$, but how to show there is a fixed point using $\rho (Ax,Ay) &lt; \rho(x,y)$? From Stewart 7e pg 614 # 9  "One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard th eremor and the fraction who have not hear the rumor. a) Write a differential equation that is satisfied by y. b) Solve the differential equation c) A small town has 1000 inhabitants. At 8 am 80 people have heard a rumor. By noon half the town has heard it. At what will 90 percent of the population have heard the rumor? "  The wording of this is very ambiguous to me and I can't really make sense of it.  They mention a product, so I know that something is being multiplied and that y is a fraction which belongs to the population who have seen it so I think that "have not" heard is a constant, and that y is a fraction taht represents who have. I tried to set this up and it is the wrong answer. I am not sure what they want from that, the English usage is too ambiguous to make sense of it. The complete lack of punctuation is what really does it. Rounded corner shape with rounded sides I am trying to make a histogram with the fitted curve plotted on it. Here's the mwe:   \documentclass[a4paper]{scrartcl}     \usepackage{tikz}     \usepackage{pgfplots,pgfplotstable}     \usepackage{filecontents}     \pgfplotsset{compat=1.10}     \usepackage[active,tightpage]{preview}     \setlength\PreviewBorder{2pt}     \newcommand{\plots}{0.611201}     \newcommand{\plotm}{2.19882}     \begin{filecontents}{data.dat}     2 0.0629921259843     3 0.0236220472441     4 0.0314960629921     5 0.125984251969     6 0.0629921259843     7 0.102362204724     8 0.110236220472     9 0.0551181102362     10 0.0629921259843     11 0.0314960629921     12 0.0236220472441     13 0.0314960629921     14 0.0629921259843     15 0.0551181102362     16 0.0393700787402     17 0.0472440944882     18 0.00787401574803     19 0.0393700787402     24 0.00787401574803     27 0.00787401574803     33 0.00787401574803     \end{filecontents}     \begin{document}     \begin{preview}      \begin{tikzpicture}     \begin{axis}[     yticklabel style={/pgf/number format/fixed},     scaled y ticks = false,     minor y tick num={1},     xtick pos=left,     legend cell align = left,     legend style={draw=none},     xlabel = {group size},     ylabel = {ratio}     ]     \addplot[blue,ybar,fill, fill opacity=0.3, bar width = 0.8,] table {data.dat};     \addplot[red, line width = 1,domain=1:40,samples=100] {1/(x*sqrt(2*pi)*\plots)*exp(-(ln(x)-\plotm)^2/(2*\plots^2))};     \legend{empirical,lognormal fit}     \end{axis}     \end{tikzpicture}     \end{preview}     \end{document}     As you can see, the legend is not really good. How can I change the legend so it looks more like this? How can I edit this trigger to only fire once? The trigger creates a related record when a checkbox is updated. The problem is it is creating a new related record everytime the record is edited. I want to edit the trigger to only create the related record once. (When the criteria is first met). The code is below  Trigger copyEquipmentonsite on Unit_Placements_Sales__c(after insert, after update) {      List&lt;Services__c&gt; sub=new List&lt;Services__c&gt;();      for(Unit_Placements_Sales__c u : Trigger.new) {           if(u.Service_Month_is_Next_Month__c == TRUE) {                 Services__c s = new Services__c();                 s.Equipment_Onsite__c=u.ID;                 s.Equipment_Type__c=u.Equipment__c;                 s.Machine_Specific_Notes__c=u.Machine_Specific_Notes__c;                    s.Product__c=u.The_Product__c;                   s.Dell_Tag_Number__c=u.Dell_Tag_Number__c;                 s.Serial_Number__c=u.Serial_Number__c;                  s.XD_Number__c=u.XD_Number__c;                 sub.add(s);           }           if(sub.size()&gt;0)           insert sub;      } } Can I downgrade to an earlier version of Android after installing an OTA update? I installed the KitKat upgrade on my Nexus 7, but I am unhappy with it. Is it possible to roll back to an earlier version of Android (Jelly Bean)? How can I make a gameObject "stick" to another gameObject after it has collided with it? How can I make a gameObject "stick" to another gameObject after it has collided with it? Right now the gameObject just passes through! [C#] Long boot delay on Ubuntu loading/splash screen following regular dist-upgrade on clean SSD install (18.04) Escaping WP_Query tax_query when term has special character(s) Here's my issue. I have a custom post type that I'm attaching a custom taxonomy to. This taxonomy will allow the admin to provide search suggestions that will ensure the post is found when a user searches. This taxonomy is also used on the front-end to autocomplete the search field with these suggestions.  The system works fine, except when the term contains a forward slash, a space or other "special" characters.  &lt;?php $keyword = 'Ski-in%2FSki-out';//Submitted via $_GET $keyword = urldecode($keyword); // (string)'Ski-in/Ski-out'  $taxQuery = new WP_Query(array(     'post_type' =&gt; 'ml_properties',     'tax_query' =&gt; array(         'relation' =&gt; 'OR',         array(             'taxonomy' =&gt; 'mc_tax_lifestyle',             'field' =&gt; 'name',             'terms' =&gt; array($keyword),             'operator' =&gt; 'IN'         )     ) )); ?&gt;   I've looked into using the escaping features provided by WordPress, but haven't had any luck there. I thought this would be a simple problem, but either my Google-fu is not strong, or I'm just not having any luck.  I've also verified that that Ski-in/Ski-out is the exact string stored in MySQL. I've looked into using esc_attr(), $wpdb-&gt;esc_like() and esc_sql() but none have an effect.  Perhaps I'm missing something obvious? $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$ I've been doing some more exercises in , in particular the following:  Show that $H_0(X,A) = 0$ iff $A$ meets each path-component of $X$.  "$\Leftarrow$": Let $x_i \in A \cap X_i \neq \emptyset \forall $ path-components $X_i$. Then for any $x \in X_i$ there is a path $\gamma$ from $x_i$ to $x$. They therefore differ by a boundary: $\partial \gamma = x - x_i$ and therefore $\{&#160;x \} = \{ x_i \}$ in $H_0(X,A) $. Therefore $H_0(X,A) = 0$.  "$\Rightarrow$" Now for this direction I'm not so sure and I'd be glad if you could give me a hint. Let $H_0(X,A) = 0$. By definition, the relative homology group is calculated from the following sequence of chain groups:  $$ 0 \rightarrow C_0(A) \rightarrow C_0(X) \xrightarrow{\partial_1} C_0(X) / C_0(A) \xrightarrow{\partial_0} 0$$  Then $H_0(X, A) = H_0(C_0(X) / C_0(A)) = ker \partial_0 / im \partial_0 = (C_0(X) / C_0(A) ) / im \partial_1 = 0$.  There are two cases where this equality holds:  (i) $C_0(X) / C_0(A) = 0$  (ii) $im \partial_1 = C_0(X) / C_0(A)$  In case (i), $A \cap X_i \neq \emptyset$ for all $X_i$.  In case (ii) I'm not sure how to proceed. Can you give me a hint? Many thanks for your help!  Edit Or maybe I could do the second direction like this:  $$0 = H_0(X,A) = \oplus_i H_0(X_i, A) \implies  H_0(X_i, A) = 0 \forall i$$  $$ \implies A \cap X_i \neq \emptyset \forall i$$? Possible Duplicate:          The FAQ specifies that:     Use your downvotes whenever you encounter an egregiously sloppy, no-effort-expended post, or an answer that is clearly and perhaps dangerously incorrect.   But if I find a correct question, closed (or not yet) as duplicate, is downvote a correct behavior for Stack&nbsp;Overflow policy? How did the English language come to this?       The play was awful.     Is the complete opposite of     The play was awesome.   But if you break it down to awe followed by ful or some, it doesn't make sense at all. Can someone shed a light on this?  Is there a rule to this, or is it one of those things you need to just memorize? Suppose that $X$ and $Y$ are chosen randomly and independently according to the uniform distribution from $(0,1)$. Define $$ Z=\frac{\max(X,Y)}{\min(X,Y)}.$$ Compute the probability distribution function of $Z$. Can anyone give me some hints on how to proceed? I can only note that $\mathbb{P}[Z\geq 1]=1$ and $$F_Z(t)= \mathbb{P}[Z \leq t]=\mathbb{P}[X\leq Y, Y\leq tX]+\mathbb{P}[Y \leq X, x\leq tY]$$ How to use GROUP BY to concatenate strings in SQL Server? I'm looking for a way to insert a &lt;style&gt; tag into an HTML page with JavaScript.  The best way I found so far:  var divNode = document.createElement("div"); divNode.innerHTML = "&lt;br&gt;&lt;style&gt;h1 { background: red; }&lt;/style&gt;"; document.body.appendChild(divNode);   This works in Firefox, Opera and Internet Explorer but not in Google Chrome. Also it's a bit ugly with the &lt;br&gt; in front for IE.  Does anyone know of a way to create a &lt;style&gt; tag that   Is nicer Works with Chrome?   Or maybe   This is a non-standard thing I should avoid Three working browsers are great and who uses Chrome anyway? Unupvote immediately followed by an upvote gives me a net -10 reputation points Rotating clears selected tags Changing the orientation of your device within the tag selection view while asking a question removes all selected tags and clears the textbox.     Steps to reproduce:     Ask a question (does not happen when editing) Edit tags Enter some text in the box  Select a tag from the suggestion list Repeat the previous two steps as many times as you care to Rotate the device (source / target orientation doesn't matter, just need to change it) All selected tags and text typed in the textbox are gone and must be re-entered   I've reproduced this using version 1.0.63 of the SE app on two devices:     HTC One E8 m2ss (the Hong Kong version) rooted and running stock 5.0.2 ATT Galaxy S3 i747, rooted and sim unlocked, running Slimkat 4.4.2.build.4.1.official ROM on Android 4.4.2 How do I install Guest Additions in a VirtualBox VM? How can I delete an email address from Outlook's email history? I have heard many (rather most) people, especially in the USA, saying:      I don't know nothing about it.   Is that correct? I always get a weird feeling hearing this and feel the correct one would be saying:     I don't know anything about it.