When did it become incorrect to use apostrophes with possessive pronouns?
Origins of possessive pronouns
Finding Radii of Convergence for $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ Setting: Let $\sum a_n z^n$ have radius of convergence $R$.  We have that  $$ R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} $$  via Hadamard's formula for the radius of convergence.  Question: What are the radii of convergence for (i) $\sum a_n z^{2n}$ and (ii) $\sum a_n^2 z^n$?  Attempt for $\sum a_n^2 z^n$:   For ease of notation, let $R_1$ and $R_2$ denote the radii of convergence for power series $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ respectively. From Hadamard's formula, we then have  $$ R_2 = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n^2 \right|^{1/n}} = \left( \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} \right)^2 = R^2 $$   But what about $R_1$? Movie/TV series with a scene of spiky, black goo growing out of a lab container? There was a scene from a sci-fi horror video, not animated, don't know if it's a movie or TV series, where there was this black goo in a glass container and some scientists were looking at it under a microscope.   The pitch black goo grew and grew so fast, eventually breaking out of the inner box and the scientists that were inside the "inner lab" were begging to get out but the scientists outside that inner lab refused to open the door. The black goo was spiky (had spikes).  It's not any of the "Alien" movies, "The Blob", "Life", or "The Thing". It's not vintage. It was made after 1990, I'm pretty sure. How can I check the existence of an element in jQuery?  The current code that I have is this:  if ($(selector).length &gt; 0) {     // Do something }   Is there a more elegant way to approach this? Perhaps a plugin or a function? How do I protect Python code from being read by users? Performance of a Trigger vs Stored Procedure in MySQL Restart returned to session indicator menu after upgrading to 13.10 This is the first time I'm having problems during Ubuntu upgrade. I had previously enabled the new Unity logout/shutdown dialogs and the restart option had gone. But now it's back again, though utterly useless, because each one leads to the same new Unity dialog: restart or shutdown. I wonder how I can get rid of this duplicate menu option? How do I install Chromium from the command line? I have already tried using the Ubuntu Software Center, but it won't install the Chromium browser. How can I install it using the terminal? How to burn or mount an ISO file? I have downloaded an ISO file recently.  How do I burn it to a CD or DVD or mount it? What is a good introductory book on quantum mechanics? I'm really interested in quantum theory and would like to learn all that I can about it. I've followed a few tutorials and read a few books but none satisfied me completely. I'm looking for introductions for beginners which do not depend heavily on linear algebra or calculus, or which provide a soft introduction for the requisite mathematics as they go along.  What are good introductory guides to QM along these lines? Necessary and Sufficient conditions for $(i \, j)$ and $(1 \, 2 \, \dotsc \, n)$ generate $S_n$. I have a homework question that asks     Find necessary and sufficient conditions on $1 \leq i &lt; j \leq n$ so that $(i \, j)$ and $(1 \, 2 \, \dotsc \, n)$ generate $S_n$.   Here is what I have done so far.  Call $c = (1 \, 2 \, \dotsc \, n)$.  I made the observation that $$ \overbrace{c c \dotsb c}^{n - j + 1} (i \, j) \overbrace{c^{-1} c^{-1} \dotsb c^{-1}}^{n - j + 1} = (1 + i - j + n \, 1). $$ Thus, $\langle (i \, j), c \rangle = \langle (1 \, i - j + n + 1), c \rangle$, so it suffices to look at $\langle (1 \, k), c \rangle$ for positive integers $2 \leq k \leq n$.  I've checked that, for the value $k=2$, $\langle (1 \, 2), c \rangle = S_n$.  This immediately implies $\langle (1 \, n), c \rangle = S_n$.  I suspect that no other values of $k$ will work, but am not sure how to prove it. How is pressure an intensive coordinate? Drop Table and recreate Permissions So we followed an best practice by Andreas de Ruiter ()  The Idea behind it is, that you write SQL Views which are easy to maintain and deploy. You use stored procedures to use these views and create tables from it. He suggests dropping or truncating the tables to maintain consistency over the data. I think it's the easiest approach without writing any complex update logic.  On these tables we have the need to set specific user permissions. By dropping the table we would lose them. So for us the question is, how could we drop the table and recreate the existing permissions on the newly created table? I am trying to automate various tasks in ArcGIS Desktop (using ArcMap generally) with Python, and I keep needing a way to add a shapefile to the current map. (And then do stuff to it, but that's another story).  The best I can do so far is to add a layer file to the current map, using the following ("addLayer" is a layer file object):  def AddLayerFromLayerFile(addLayer):  import arcpy  mxd = arcpy.mapping.MapDocument("CURRENT")  df = arcpy.mapping.ListDataFrames(mxd, "Layers")[0]  arcpy.mapping.AddLayer(df, addLayer, "AUTO_ARRANGE")  arcpy.RefreshActiveView()  arcpy.RefreshTOC()  del mxd, df, addLayer   However, my raw data is always going be shapefiles, so I need to be able to open them. (Equivalently: convert a shapefile to a layer file without opening it, but I'd prefer not to do that). Prove that , any primitive root $r$ of $p^n$ is also a primitive root of $p$ For an odd prime $p$, prove that any primitive root $r$ of $p^n$ is also a primitive root of  $p$   So I have assumed $r$ have order $k$ modulo $p$ , So $k|p-1$.Then if I am able to show that $p-1|k$ then I am done .But I haven't been able to show that.Can anybody help me this method?Any other type of prove is also welcomed. Theorem style with name as argument Is there some way to get a theorem type which will make a theorem with the name passed as a parameter? In other words, is there a way to define "named" such that   \begin{named}[NAME] text \end{named}   will print   "NAME: text"?  I tried using the definition \newtheorem*{named}{}, but that results in   "(NAME): text", which is quite different from what I am trying to do.  I know that I could also create a new instance of \newtheorem for each named theorem I want to use, but that is clunky and inconvenient. Is there a better way? In Einstein's General Relativity, do the space-time dimensions curve? In Einstein's General Relativity, do the space-time dimensions curve according to the positions of stars, planets, and masses? I have a spawn area I am hoping to make a safe haven, so I have a command block on a clock that kills each hostile mob type. This works to keep the place free of any hostile mobs, however every night I see rotten flesh, bones, arrows, and string everywhere.  I would like to add something to my command blocks so that they are killed without dropping loot. How can I accomplish this?  Note that I want mobs to drop loot generally elsewhere in the world, so /gamerule doMobLoot false will not do the trick for me. I'm hoping for a command that only applies to the mobs killed by the command block, and not anything else. Is quantum entanglement affected by time dilation? Is quantum entanglement affected by time dilation? Let's say one of the entangled pair is accelerated to very high speed. When both the entangled particles are observed at the same time, will they have the opposite spin? Port error while changing chrome extension from manifest v1 to v2 Spyder - touchpad scroll does not work I use the scroll function with my mouse and it works fine. But either in script, or in terminal sections of Spyder, I can't use the touchpad for scrolling. I do not experience any problems with my touchpad in other programs like Word, Chrome etc. Is there any option for this in Spyder, specific for touchpad?  I am using Lenovo Legion Y520, Windows 10.