Diffrent size of a fraction
Show inline math as if it were display math (and vice versa)
Upon answering the telephone, the person calling asks if Joan is available. If Joan is the person who answered the phone, should she say "This is her" or "This is she"? What permissions should my website files/folders have on a Linux webserver? Heisenberg Uncertainty Principle scientific proof states that:    $$\sigma(x)\sigma(  p_x )\ge \frac {\hbar}{2}.$$  What is the scientific proof of this principle? Consider the factorization into irreducibles of $6$ in $\mathbb{Z}[\sqrt{-5}]$. We have $6=2 \times 3$ and  $6=(1+\sqrt{-5}) \times (1-\sqrt{-5})$, i.e. $2$ distinct factorizations. And,  $$6^2=3 \times 3\times2\times2$$ $$=(1+\sqrt{-5}) \times (1-\sqrt{-5}) \times (1+\sqrt{-5}) \times (1-\sqrt{-5})$$ $$=(1+\sqrt{-5}) \times (1-\sqrt{-5})\times3\times2.$$ More generally, $6^n$ will have $n+1$ distinct factorizations into irreducibles in $\mathbb{Z}[\sqrt{-5}]$ by a simple combinatorial argument. But, can we construct a ring in which there exists an element that has an infinite number of distinct factorizations into irreducibles? To make life harder, can we construct an extension of $\mathbb{Z}$ in which this happens? I have been thinking about this for a while and have managed to find no foothold.. Any help is appreciated. Could we have the ability to mark a change as minor in questions or answers? It would be nice to allow users to mark a change in an answer (or question) as minor (e.g. for a simple spelling or grammar correction). A change marked as minor wouldn't push the question on top of the list of questions on the home page or in the feeds. It wouldn't trigger any notification. This could be implemented with a "Minor change" checkbox when editing an entry (a la Confluence). What do you think? Past tense: learned/learnt, dreamed/dreamt? In my very first question, Carlo made an edit stating that learnt is a rare past tense of learn. I am accustomed to using learnt for past tense and learned as an adjective (as in He is a learned scholar.) I checked into a couple of dictionaries and they say both are valid past tenses. Do I stand a risk of not being understood if I use learnt?  In a similar vein, should dreamed be used as a past tense for dream instead of dreamt? I can't think of other such conflicts, but I guess there must be more. how to shuffle characters of strings in bash? i have this script where it generates random password but it needs to be shuffled. I couldnt find a way to do it. Please Help me.  num=("0" "1" "2" "3" "4" "5" "7" "8" "9") special=("@" "#" "$" "%" "*" "-" "+") upper=(A B C D E F G H I J K L M N O P Q R S T U V W X Y Z) lower=(a b c d e f g h i j k l m n o p q r s t u v w x y z) lower2=${#lower[*]}   #${lower[$((RANDOM%lower2))]} upper2=${#upper[*]}   #${upper[$((RANDOM%upper2))]} num2=${#num[*]}     #${num[$((RANDOM%num2))]} special2=${#special[*]} #${special[$((RANDOM%special2))]} echo "${special[$((RANDOM%special2))]}${num[$((RANDOM%num2))]}${lower[$((RANDOM%lower2))]}${upper[$((RANDOM%upper2))]}${lower[$((RANDOM%lower2))]}${upper[$((RANDOM%upper2))]}${lower[$((RANDOM%lower2))]}${upper[$((RANDOM%upper2))]}"   this is the output: @7nOyIaJ  how do I randomize this? Changing URLs in Google's Index I recently submitted a sitemap.xml in Google's Webmaster Tools and some of the pages already got indexed. The problem now is, that I cleaned my URLs with MOD_REWRITE in the .htaccess file afterwards and I would like the new ones to be indexed. The old ones should be removed from the Google Index.  Indexed Page:  www.example.com/article.php?=article=this%is%an%example   New URL:  www.example.com/article-this-is-an-example   Do I even have to remove them manually from the index in any way or do they refresh if Google crawls my new sitemap? I really wan't to prevent duplicate content.  Thanks in advance. Would the One Ring even work for anyone but Sauron? The temptation of the One Ring, we quickly learn, is always tied up with power: each character's fantasies of possessing it seem to be about what they would do with great power. Even Sam, as I recall, imagines the power to create great huge sweeping gardens. Boromir wants the power to save Gondor, etc.  But does the One Ring actually convey power to anyone but Sauron? It actually seems to diminish its bearers: Bilbo feels "thin" and "stretched", Sméagol becomes the wretched Gollum, Frodo is never quite the same even after it is destroyed. None of them seem more "powerful," even in the abstract way that magic-users in Tolkien operate. No mention is made, that I can recall, of a Ringbearer having greater stature or authority, or of people naturally following them or obeying their commands, while they possess the Ring.  The accepted answer to  implies that a being as powerful as Gandalf could dominate the Ring and break Sauron's hold over it, but even then Tolkien says that "It would have been the master in the end", and his descriptions of a Ringbearing Gandalf sound pretty much like Gandalf is using his own powers, not terribly augmented by the Ring:     [Gandalf] would have remained 'righteous', but self-righteous. He would have continued to rule and order things for 'good', and the benefit of his subjects according to his wisdom (which was and would have remained great).   Now that sounds out of character for a being as humble as Gandalf, but it doesn't actually sound like something he would have been unable to do before. It's a twisted motive, but not really an enhanced power.  Pretty much everyone in Middle-earth agrees that the One Ring will corrupt you, but they all seem to agree that it DOES convey great power even so. Why? Do we ever see any actual, "positive" effects from possessing the One Ring? I have seen that the new  shows off H.265/HEVC playback, the successor of H.264. I wonder how I may get this support in Linux. Is there a need to install a codec specific for this format? How? I won a Samsung Galaxy S6 Edge, signed into it with my Google account, played around with it, wiped it, and sold it.  Now the person can't get past setup because it's saying he must login with an account previously synced to the phone.   Is there anything I can do from my side to remove this lock? Is there a drupal.org API for retrieving the list of available modules? What does a type followed by _t (underscore-t) represent? This seems like a simple question, but I can't find it with the Stack Overflow search or Google. What does a type followed by a _t mean? Such as  int_t anInt;   I see it a lot in C code meant to deal closely with hardware—I can't help but think that they're related. Is it possible to programmatically stop and start ArcGIS Server map services using Python? In Star Wars: "The Force Awakens," how could this lightsaber have turned up where it did? In The Force Awakens:      Maz Kanata tells Rey that the lightsaber she just picked up and had visions from belonged to Anakin and Luke and now it's hers.   The lightsaber that Luke got from Obi-Wan Kenobi in A New Hope was lost/destroyed when Luke fought Darth Vader in Star Wars: The Empire Strikes Back.  In Return of the Jedi Darth Vader compliments Luke on building a new lightsaber and tells Luke that his skills are now complete.   How could that possibly be the lightsaber that Anakin used originally, and that was then passed on to Luke, when that lightsaber no longer exists?  Writers notes, director's notes, interviews, cast interviews, etc. are welcome. Thanks. What is the point of Sauron making the One Ring? Why would Sauron make himself a ring that he invested most of his power in? Wouldn't it be better to keep the power in his physical form and not risk leaving his power in a physical form that could be stolen (which did happen when Isildur took it). Because of its being in the form of a ring, its power could be abused by someone else to attack him. Showing a function on $[0,1]$ is the zero function Projecting a point onto a convex set decreases its distance from each point of that set For a convex set $K$, $x\in \mathbb{R}^d$ and $y=\Pi_K(x) $ (the projection of $x$ onto $K$), we want to show that for any $z\in K$ it holds that: $\|x-z\| \ge \|y-z\|$.  $$ $$  I'm not sure if the following proof is valid:  $$ y=\Pi_K(x) = \arg \min_{y\in K} ||y-x||  \Rightarrow \forall z\in K \langle y-x,y-z\rangle =0 $$ $$ ||y-x||^2_2+||y-z||^2_2=||x-z||^2_2  \quad (Pythagoras)$$  $$ \Rightarrow ||y-z||^2_2 \le ||x-z||^2_2 $$ $$ \Rightarrow||y-z|| \le ||x-z|| $$ Let $L$ denote the set of all lines in the Euclidean plane $\Bbb R^2$. We say that two lines $L$ and $M$ are parallel if either $L = M$ or $L$ and $M$ have no points in common.  Using Euclid’s Parallel Postulate prove that parallelism is an equivalence relation on the set $L$.  I need some help determining if I have the right idea about this.  $L=L$ which means that $L$ is parallel to itself. So this is reflexive.  Suppose we have $L || M$. Then Either $L=M$ or $L$ and $M$ share no points. So Then $M=L$ or $M$ and $L$ share no points. Then $M||L$. So this is symmetric.  Suppose we have $L || M$ and $M || N$. So then either $L=M$ or $L$ and $M$ share no points and either $M=N$ or $M$ and $N$ share no points. So we have that either $L=N$ or $L$ and $N$ share no points. So $L || N$. So then this is transitive.  So from all of this, parallelism is an equivalence relation. I'm not sure if this is correct or not. Any help would be appreciated. Is Java's default value for Boolean 'true'?