Which topics are left for Stackoverflow?
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An elementary question regarding a multiplicative character over finite fields Reading Chapter 2 of Koblitz's Introduction to Elliptic Curves and Modular Forms, I got stuck on the following question. I would like to proceed my reading, so I would appreciate any hint to this. I have no background in Algebraic Number Theory, but if someone explains in that language, I would like to see it and learn it as much as I can at this moment. References will also be appreciated!  Let $q$ be a power of prime $p$ such that $q \equiv 1 \mod 4$ and $p \nmid 2n$.  Question. Let $\chi_{4} : \mathbb{F}_{q}^{*} \rightarrow S^{1}$ be the character of order $4$, show that $\chi_{4}(-4) = 1$.  The hint in the book says we may show that $\chi_{4}(4) = \chi_{4}(-1) = 1$ for $q \equiv 1 \mod 8$ and $\chi_{4}(4) = \chi_{4}(-1) = -1$ for $q \equiv 5 \mod 8$.  If $q \equiv 1 \mod 8$, then $\chi_{4}(4) = \chi_{4}(2)^{2} = \chi_{2}(2) = \pm 1$, where $\chi_{j}$ denotes the character of order $j$. So I think $2$ must be quadratic residue in this case, but I cannot figure out why. How do I calculate my skill modifier? I'm trying to fill out the skills section on my first character, but when I look at the character sheets in the Starter Set as examples of how to make a character, the skills that are circled have higher modifiers than their Dexterity or Strength or whatever should make them.  I've filled out all skills but those. What do I add to the ability modifier to get those numbers?   My abilities are Strength 13, Dexterity 14, Constitution 11, Intelligence 15, Wisdom 8, and Charisma 8. Proficiencies are Arcana, Athletics, History, Investigation and Stealth. My saving throws are Strength and Dexterity.   I just started D&amp;D and have the Player's Handbook, DM's Guide, Monster Manual, and the Starter Set. How to add electrical fixtures to (custom) furniture? Having electrical outlets, USB charging ports, night lights, etc would be nice to have on some pieces of furniture, like side tables. Several receptacle manufacturers make combination fixtures with USB and power, for instance, or power plus a night light.   I've thought about nailing an electrical box to a (hidden) portion of the table and putting one of those receptacles in it, but what things do I need to consider doing this? Is it as simple as it sounds,  and then just fashioning some kind of plug out of 12/2 or 14/2 wire? Does the answer change if the box is concealed (though still accessible) vs it being completely visible? Getting always extra button on display .. 1 button dont have any ID I am having problem while looping.  I am always getting an extra button. Please help.   {this.state.items.map(myitems1 =&gt; {     return (        &lt;div key={myitems1.Id}&gt;            &lt;span&gt;{myitems1.Item_x0020_Name}&lt;/span&gt;            &lt;button onClick={e =&gt; this.dosometing(myitems1.Id)}&gt;Offline&lt;/button&gt;        &lt;/div&gt;     ); })}   I can see the collection has only three (3) values, But I am getting 4 buttons on screen.   Getting always extra Prove $\sum\limits_{i=2}^{n}\frac{1}{(i-1)i}$ = $\frac{(n-1)}{n}$ using induction. I need to prove $\sum\limits_{i=2}^{n}\frac{1}{(i-1)i}$ = $\frac{(n-1)}{n}$ using induction. I am getting stuck midway through the inductive step.   Here is what I have:  $\forall n\geq 2$, where $n\in\mathbb{N}, P(n)$ is the statement "$\sum\limits_{i=2}^{n}\frac{1}{(i-1)i}$ = $\frac{(n-1)}{n}$".  Base case:  $\sum\limits_{i=2}^{2}\frac{1}{(i-1)i}$ = $\frac{1}{2}$ = $\frac{(2-1)}{2}$ so $P(2)$ is true.  Inductive assumption: Assume $P(n)$ is true. Need to show $\sum\limits_{i=2}^{n+1}\frac{1}{(i-1)i}$ = $\frac{((n+1)-1)}{(n+1)}$.  $\sum\limits_{i=2}^{n}\frac{1}{(i-1)i}$ + $\frac{1}{n^{2}}$ = $\frac{(n-1)}{n}$ + $\frac{1}{n^{2}}$  = $\frac{n(n-1) + 1}{n^{2}}$ = $\frac{n^{2}-n+1}{n^{2}}$  I got stuck here and tried working backward from $\frac{((n+1)-1)}{(n+1)}$, which gives me:  $\frac{((n+1)-1)}{(n+1)}$ = $\frac{n}{(n+1)}$   Any suggestions on what to do next or where I may be going wrong? How to show that $\sum\limits_{k=0}^n (-1)^k\tfrac{{ {n}\choose{k}}}{{ {x+k}\choose{k}}} = \frac{x}{x+n}$ I'm trying to show that      $$\sum_{k=0}^n (-1)^k\frac{{ {n}\choose{k}}}{{ {x+k}\choose{k}}} = \frac{x}{x+n}$$   I've tried expanding this using definition of binomial coefficient, then binomial expansion, now I'm using induction to simplify $\sum_{k=0}^m (-1)^k\frac{{ {n}\choose{k}}}{{ {x+k}\choose{k}}}$, but it looks very inelegant, is there a better way? Is there a public API for Stack Overflow? Blank white desktop icons on new Windows 7 install I just did a fresh install of Windows 7 professional on my father's Toshiba Satellite laptop and everything went smoothly except for the fact that certain desktop shortcuts appear as a blank white page.    I've tried:    going in and changing the icon to many different things reloading the icon cache by deleting the IconCache.db file uninstalling his Free AVG as he heard the virus scanner might be preventing it (it wasn't and he put it back.)   The one thing these icons have in common is that they are shortcuts to some very old DOS executables.  One of them is Word Star 2000 to give you an idea.  Does anybody have any other suggestions besides what we've tried? What is the maximum RAM supported by the desktop and server editions? I have just downloaded "ubuntu-14.04.1-desktop-amd64.iso" and "ubuntu-14.04.1-server-amd64.iso" through "".   Please can someone confirm the maximum RAM that is supported.  According to many sites, Maximum RAM supported by the OS  ubuntu-14.04.1-desktop : " 2^64 bytes " -- Kindly confirm  ubuntu-14.04.1-server : " 2^64 bytes " -- Kindly confirm Bring Debian to second Position in the GRUB Menu at startup Continuity $\iff$ Preimage is closed whenever set is closed I'm trying to do the following:     Prove that a function $f : S\rightarrow T$ between two topological spaces is continuous iff $f^{-1}(C)$ is closed whenever $C\subset T$ is closed.   To show that the closed-set relationship implies continuity, notice that $T-C$ is open. The preimage of this open set is $f^{-1}\left(T-C\right)=S-f^{-1}(C)$, which is open by assumption ($f^{-1}(C)$ is closed). Of course, any open set $G\subset T$ can be written in the form $T-C$ by letting $C=T-G$. Thus, for every open set $G\subset T$, the preimage is open.  To show the implication in the other way, consider a closed set $C\subset T$. $T-C$ is open, so $f^{-1}(T-C)=S-f^{-1}(C)$ is open as well ($f$ is continuous), which means that $f^{-1}(C)$ is closed. How do I calculate the distance of an object in a photo? (Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems)  I'm aware of Sigma notation, but is there a function/name for e.g.  $$ 4 + 3 + 2 + 1 \longrightarrow 10 ,$$  similar to $$4! = 4 \cdot 3 \cdot 2 \cdot 1 ,$$ which uses multiplication?   Edit: I found what I was looking for, but is there a name for this type of summation? Cleaning ‘staining’ off bottom of MacBook screen A couple months ago, I started to see some very small stain spots on the bottom of my MacBook Pro early 2015 screen. It's very annoying to see while I'm using it. When the screen is turned off, I see it as in the picture below. But when it's turned on, I see it as very small water-ish drops. I tried to clean it using water, screen cleaner, and even alcohol, but it didn't go away.  Also that area doesn't feel as smooth as the unaffected area. So, I think it's on the outside of my screen not inside.  Have you ever seen this before? Any suggestions on how to remove it without the need of replacing the whole screen? How should I structure my URLs for both SEO and localization? Getting a domain name off of the blacklist I have a friend who purchased a domain name recently that he found out (after buying it) is apparently blocked by many corporate firewalls or proxies b/c it's been tagged as porn by the blacklist db they're using.    Are there any common agencies he can appeal to in order to get the domain name off the black lists?  It's not just one corporation, either, he's looking at marketing a subscription-based service to a certain industry, so any contacting a particular firewall/proxy admin and have them manually update the blacklist won't scale well.  Thanks in advance. Paul Bash copy all files that don't match the given extensions When I import pictures from my camera in Shotwell, it also imports the video clips. This is somewhat annoying, as I would like to store my videos in another folder. I've tried to write a bash command to do this, but have not had success.  I need a command that meets the following requirements:   Locate all files in a directory structure that do not have an extension of .jpg, .png, .gif, or .xcf (case insensitive). Move all of these files into a target directory, regardless of whether the file names or directory paths contain spaces or special characters.   Any help would be appreciated!  EDIT: I'm using the default shell in Ubuntu, meaning that some commands are aliased, etc.  EDIT 2: I've attempted this myself (not the copy part, just the listing of files part). I turned on extglob and ran the following command:  $ ls -R /path | awk '   /:$/&amp;&amp;f{s=$0;f=0}   /:$/&amp;&amp;!f{sub(/:$/,"");s=$0;f=1;next}   NF&amp;&amp;f{ print s"/"$0 }'   This lists everything. I tried using grep on the end of it, but haven't the foggiest idea of how to get it to not match a pattern I give it. The extglob switch didn't help much with grep, even though it does help with other commands. Tracking Object Position - Firing on a Trajectory How old is Chewbacca? How old is Chewbacca, anyway? Do Wookiees get gray fur when they are old? When should I not kill -9 a process?