All possible values of $k$ such that $a\le k \le b$ and $k$ is a linear combination of a and b
How to find solutions of linear Diophantine ax + by = c?
Why is "ask" sometimes pronounced "aks"? We've recently moved from New Zealand to New York City, and have noticed that many people (most of whom have good English) pronounce "ask" as "aks". For example:     Could you please go aks her tomorrow?   Sure, I'll ax her! :-)  What's the origin/etymology of "aks"? It seems to be more common among African Americans, but it's definitely used by others too. Negative pressure Footnotes in tables? I would like to have footnotes in tables, but the \footnote command does not work inside a table environment.  Googling one finds that this is , at least in the UK.  That link contains a list of solutions to the problem, but since the author of the FAQ does not believe that this is good style, no comparison is made between the different methods.  I am sure that there are good typographical reasons why in most situations it is undesirable to have footnotes in tables (although I have never come across them), however I have recently felt the need to do just this in a paper I am currently writing.  Therefore I'd like to ask two questions:   What are the arguments against footnotes in tables? If one really must go against the advice in 1., which method (whether considered in the  or not) would you recommend?   Thanks in advance.  Added  In response to ShreevatsaR's comment: I do not much care for where the text of the footnote appears.  I do however insist on the footnote marks being attached to the relevant entries in the table, so that I can avoid having to reference "the second entry in the eighth line of the table", etc... No custom off topic message when flagging on SO I have less than 3000 rep on StackOverflow. Now that the new closing changes are live, I cannot supply a custom off topic message:    I can enter a custom off topic flag message without issue here on meta, though it only has three canned off topic reasons, rather than five.  I'm not sure if this is the same for users with enough rep to vote to close directly, but the lack of a custom message eliminates the ability to use the correct flag reason in some cases.  For example, I wanted to flag , which is simply off topic on SO. However, I can't use a custom message, and none of the canned ones are appropriate.    appears to be the most up to date official information on custom reasons, and indicates that they are remaining in play for now.  supports that.  If this is by design, how should I flag such questions? This is not a case worth bothering the mods over, so I don't want to use a custom diamond attention flag. Bounded in Probability and smaller order in probability Evaluate $\lim_{n\to\infty} [{1\over kn}+{1\over k(n+1)}+{1\over k(n+2)}+\cdots+{1\over k(n+p-k)}]$ where $k<p$ There are four options. Which of them is correct?- (a)${p\over k}$ (b)${k\over p}$ (c)$\log({p\over k})$ (d)${\log p\over k}$ I somehow want to use the rule for evaluating this kind of sum using integration, i.e. $\int_{0}^{1} f(x) \ dx=\lim_{n\to\infty} {1\over n}\sum_{r=1}^{n}f({r\over n})$  But the given expression $\lim_{n\to\infty} [{1\over kn}+{1\over k(n+1)}+{1\over k(n+2)}+\cdots+{1\over k(n+p-k)}]={1\over k}\lim_{n\to\infty}{1\over n}\sum_{r=0}^{p-k}\frac{1}{1+{r\over n}}$, sum running from $0$ to $p-k$ instead of $n$. So, how to evaluate this limit? Thanks for assistance in advance. Can you cast Mirror Image twice on successive turns and have 6 duplicates of yourself instead of 3? (PHB, pg. 260) is a spell that has a casting time of 1 action, and a duration of 1 minute. My question is, can you cast Mirror image twice on successive turns, gaining 6 images by turn 2?  Sample Turns:  Turn 1: Mirror Image (3 images)  Turn 2: Mirror Image again (6 images, with 3 of them lasting 1 turn less)  Since Mirror Image doesn't require concentration, there should be no interference between the 2 casts in terms of concentration. I know that this could be up to DM Fiat, but I was wondering if there was a proper interpretation of the printed rules for this situation. Is a URL allowed to contain a space? Is a URI (specifically an HTTP URL) allowed to contain one or more space characters? If a URL must be encoded, is + just a commonly followed convention, or a legitimate alternative?    In particular, can someone point to an RFC that indicates that a URL with a space must be encoded?  Motivation for question: While beta-testing a web site, I noted that some URLs were constructed with spaces in them. Firefox seemed to do the right thing, which surprised me! But I wanted to be able to point the developers to an RFC so that they would feel the need to fix those URLs. A switch that is only ON if the reference voltage is matched -+ 0.1 volts Im a beginnerish will electronic and im looking for a switch that is only ON if the reference voltage is matched -+ 0.25 volts. So if i have a reference of 2.7v to match and i feed the switch a 2.75v the switch would be ON. But if i feed it  2.5v or 2.85v the switch would is OFF.   This might be a simple google search, but i am lacking the correct terminology. I have an area which is divided into several zones the borders of which I have defined as separate layers.  The layers are defined by sql queries on a postgis table of line segments joined with a table that says which segments are part of which zone boundaries.  What I want to do is produce a set of maps where each map is a zoom to layer extent for each zone and which has standard header, legend and scale (which will preferably vary with the zone but could be fixed across all maps).  My first attempt was to get a new composer for each map -- that worked but was rather tedious.  Next I tried producing one map and printing it and then deleting the map frame and going into qgis to zoom to the next layer extent and make the boundary visible then hopping back into the composer to load it in.  This worked more or less.  Map was fine, legend was OK because it was the same for all the maps and the zone name was easy to edit but the scale needed to be deleted and set up again for each map.  Next I looked at Atlas Generation.  The manual seems to assume that you know how it works and all you need is a reminder about the details (like so many software manuals ;)  It seems to assume that you have something that you are iterating over but I can't see how to make it work -- there is nothing to iterate over to get the layers.  Any suggestions welcome!!  :)  Russell !! Do not use the future tense after unless   [✓]  I won't go unless you go   [✘] [I won't go] unless you will go. [✘]   Why not? Even if the above is prescriptive, what may be some reasons? In my separate question , I quote: unless = if not. If not can precede future tense; so why can't unless?  Footnote:  motivated this. Convert list of ints to one number? Find all $n \in \mathbb{N}$ such that $\phi(n) = 4294967296=2^{32}$.  I managed to find that $16106127360$, $8589934592$, $10737418240$ and $12884901888$ are solutions for the equation, but I do not know how to find all of them. Precision of Coulomb's law Up to which precision has the coulomb law proven to be true? I.e.  if you have two electrons in a vacuum chamber, 5 meters appart, have the third order terms been ruled out? Are there any theoretical limits to measure the precision ( Planck's constant?). Obviously there are practical limitations ( imperfect vacuum, cosmic rays, vacuum fluctuation). Still, does anyone know what was the smallest amount ever correctly predicted by that law?    Edit : Summary  On the high end of the energy spectrum a precision of 10^-16 has been shown ( 42 years ago )  For electron point charges at large distances the law might brake down due to practical reasons.  For moving particles QED gives a correction to the law: Is it possible to have global color swatches or color styles in Photoshop? I'm looking for a way to use one color in multiple different layers, and then be able to change that color in all the layers in one single place.  Lets say that I have four layers using #ffffff. Now I want to know what my design looks like with #ffff00 instead. How can I do this without manually changing all four layers?  I can add that adjustment layer on hue/saturation is not really what I'm looking for. What I need is something like "change all #ffffff to #ffff00." There are various articles on Hearthstone opening strategy which suggest it's a bad idea to use early-game spells and summons to damage your opponent directly, and that you should instead save them for minion removal.  An example: your starting hand contains an Elven Archer, a 1-cost 1/1 minion that does a point of damage when summoned. I've seen it advised that if going first, with that card in your hand, you should not play it, but save it for minion removal.  Another example:  Going second against a Rogue with a 2-cost 2/1 Loot Hoarder, Deathrattle: draw a card in your hand. The advice was not to play it and save the coin, because on her second turn, the Rogue can generate a dagger and kill the Hoarder.   I understand why, in most circumstances, it's better to remove a minion than damage the enemy. What's bugging me about these particular opening examples is why they're a better idea than doing nothing and ending the turn.  In both cases, you've wasted a point of mana, and the chance to take the initiatve. By having a minion on the board first, you can use the following turn to attack with that minion, possibly removing an enemy summon and leaving you clear to summon onto an empty board.   The Loot Hoarder one I find particularly puzzling. If the Rogue gets a dagger and kills it, you still have the initative, you've forced the Rogue to make what's possibly a suboptimal turn-2 play, you've done 2 damage to the Rogue and you got a card into the bargain. Sounds like a pretty good use of the coin to me.  What am I missing here? idempotents acting as local identities Let $R$ be a ring with unity (not necessarily commutative) and $I$ an ideal of $R$.  Suppose that for every element $a \in I$ there exists an element $c\in I$ such that $ac=a$. Note that $c$ is related to $a$. Now we have the following question:  Can we say that  for every element  $a\in I$ there exists an idempotent element $c\in I$ such that $ac=a$?    Of course we have many examples such that the answer is true for them but in general  we don't know. How do you compile OpenSSL for x64? After following the instructions in INSTALL.W64 I have two problems:   The code is still written to the "out32" folder. I need to be able to link to both 32-bit and 64-bit versions of the library on my workstation, so I don't want the 64-bit versions to clobber the 32-bit libs. The output is still 32-bit! This means that I get "unresolved external symbol" errors when trying to link to the libraries from an x64 app. How does ServerName and ServerAlias work? It's the following part of a virtual host config that I need further clarification on:  &lt;VirtualHost *:80&gt;   # Admin email, Server Name (domain name), and any aliases   ServerAdmin example@example.com   ServerName  141.29.495.999   ServerAlias example.com ...   This is and example config, similar to what I currently have (I don't have a domain name at the moment).  &lt;VirtualHost *:80&gt; - Allow the following settings for all HTTP requests made on port 80 to IPs that this server can be contacted on. For instance, if the server could be accessed on more than one IP, you could restrict this directive to just one instead of both.  ServerName - If the host part of the HTTP request matches this name, then allow the request. Normally this would be a domain name that maps to an IP, but in this case the HTTP request host must match this IP.  ServerAlias - Alternate names accepted by the server.  The confusing part for me is, in the above scenario, if I set ServerAlias mytestname.com and then made an HTTP request to mytestname.com, there would have to be a DNS record pointing to the server's IP for this to work? In which case, is ServerAlias just basically EXTRA ServerName entries?  Say I had a DNS entry such that foobar.com = 141.29.495.999 but then I had ServerName = 141.29.495.999 and ServerAlias was empty, would that mean that although foobar.com gets resolved to the right IP, because there is no reference to accept foobar.com in ServerName or ServerAlias?  Or something. Man I'm confused. What is the best way(s) to fake function overloading in Javascript?   I know it is not possible to overload functions in Javascript as in other languages.  If I needed a function with two uses foo(x) and foo(x,y,z) which is the best / preferred way:   Using different names in the first place Using optional arguments like y = y || 'default' Using number of arguments Checking types of arguments Or how?