"that is" or "which is"
When to use “that” and when to use “which”, especially in relative clauses
Finding the limit of $(1-\cos x)/x^2$ $$\lim _{x \to 0}{1-\cos x\over x^2}={2\sin^2\left(\frac{x}{2}\right)\over x^2}={\frac{2}{x^2}\cdot {\sin^2\left(\frac{x}{2}\right)\over \left(\frac{x}{2}\right)^2}}\cdot\left(\frac{x}{2}\right)^2$$  now $$\lim_{x \to 0}{\sin^2\left(\frac{x}{2}\right)\over \left(\frac{x}{2}\right)^2}=\left({\sin\left(\frac{x}{2}\right)\over \left(\frac{x}{2}\right)}\right)^2=1^2$$  So we have $$\frac{2}{x^2}\cdot \left(\frac{x}{2}\right)^2=\frac{2}{x^2}\cdot \left(\frac{x^2}{4}\right)=\frac{1}{2}$$  Are the moves right? How to install driver for TP-Link TL-WN722N(EU)V2 on Ubuntu 17.04 Kylin I can not install the TP-LINK WN722N V2, which includes a 8188 EUS chip. Then I tried to install the backports v4.4.2,but   do not want to recognize either.  The TP-Link driver installation errors thrown out:  man@kman-livve:~$ cd '/home/kman/Documents/Tplink 722 N V2/rtl8188EUS_linux_v4.3.0.8_13968.20150417' kman@kman-livve:~/Documents/Tplink ...$ sudo make  "******************************************" "NO SKRC,we will use default KSRC" "******************************************" make ARCH=x86_64 CROSS_COMPILE= -C /lib/modules/4.10.0-15-generic/build M=/home/kman/Documents/Tplink 722 N V2/rtl8188EUS_linux_v4.3.0.8_13968.20150417  modules make[1]: Entering directory '/usr/src/linux-headers-4.10.0-15-generic' arch/x86/Makefile:140: CONFIG_X86_X32 enabled but no binutils support make[1]: *** No rule to make target '722'.  Stop. make[1]: Leaving directory '/usr/src/linux-headers-4.10.0-15-generic' Makefile:1367: recipe for target 'modules' failed   A method to know him, I do not know about. Thanks  Model:TL-WN722N(EU)_V2_161112_Linux.zip Chipset:rtl8188EUS_linux_v4.3.0.8_13 Why isn't $\mathbb Z [\sqrt{pq}]$ a factorial domain This is exercise 8 on page 147 of Jacobson's Basic Algebra:     Let $p$ be a prime of the form $4n+1$ and let $q$ be a prime such that $ \left( \frac{q}{p} \right) =-1$. Show that $\mathbb Z[\sqrt{pq}]$ is not factorial.   Here, $ \left( \frac{q}{p} \right) =-1$ means that $\overline{q}$ is not a quadratic number in the field $\mathbb Z / (p)$.  In the special case when $p=5$ and $q=2$, $9 = 3 \cdot 3 = (\sqrt{10} +1 )(\sqrt{10 }-1)$, with $3$ and $\sqrt{10} +1$ or $\sqrt{10} - 1$ nonassociates. But I have no idea what can I do with the general case.  I know that primes of the form $4n+1$ can be written in the form $m^2 + n^2$ with $m,n \in \mathbb Z$. But when $q$ is odd and not of the form $4n+1$, $pq$ is not the sum of two quadratic numbers. How can I use the fact $\left( \frac{q}{p} \right) =-1$?  Thanks~ Irreducible conics An algebraic set (not necessairly a variety) $X \subseteq \mathbb{A}^2$ defined by a polynomial of degree $2$ is called a conic.  The problem is:     Show that any irreducible conic is isomorphic either to $Z(y-x^2)$ or to $Z(xy-1)$ after an affine change of coordinates in $\mathbb{A}^2$.   However, with this definition of conic, I only can conclude that if the polynomial is irreducible, then the conic is irreducible (because $f$ irreducible $\Rightarrow Z(f)=X$ is irreducible)     What can I do to prove that if a conic is irreducible, then the polynomial that defines the conic is irreducible?   And for the main problem, how can I do the affine change of coordinates?  Thanks When playing with the complete elliptic integral of the first kind and its Fourier-Legendre expansion, I discovered that a consequence of $\sum_{n\geq 0}\binom{2n}{n}^2\frac{1}{16^n(4n+1)}=\frac{1}{16\pi^2}\,\Gamma\left(\frac{1}{4}\right)^4 $ is:     $$\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \tfrac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2\tag{A}$$   which might be regarded as a sort of Ahmed's integral under steroids.  I already have a proof of this statement (through Fourier-Legendre expansions), but I would be happy to see a more direct and elementary proof of it, also because it might have some consequences about the moments of $K(x)$ of the form $\int_{0}^{1}K(x)\,x^{m\pm 1/4}\,dx$, which are associated with peculiar hypergeometric functions. zero matrix to the power of 0 Why $0^0=I$?   I'd tried prove that considering $N^0$ where N is a Nilpotent matrix and then using the Cayley -Hamilton theorem  Thanks in advance. Windows 10 Icon Gone Missing TL;DR: The Windows 10 upgrade icon went missing. How do I get it back in order to update my system? I am running Windows 7 SP1.   I woke up this morning... er, noon all excited about Windows 10. I had reserved my copy using the clever little icon in the taskbar a few weeks back, so I thought I would be set to go. I booted my computer and checked the icon to see if it would download the update. It still behaved the same as if Windows 10 wasn't launched yet. I supposed this was normal behaviour. After all, Microsoft can't roll out the update to everyone at once.   Windows Update did want me to install several updates though. I installed KB2952664, KB3064209 and KB3048761. Everything went fine, and I was prompted to restart my system.   After restarting, the icon wasn't there anymore. I tried a lot of solutions including the answers provided by our awesome community on , and the answer provided to  posted on the Microsoft Windows support forums.   How do I get the icon back in order to update my system?  UPDATES:   Installing from the Media Creation Tool does not working with an error "Something Happened" with as description "Something Happened". Both are translated from Dutch.  I tried to install it from the Media Creation Tool using PowerISO to mount the created ISO. It will not accept my product key, even though I am able to verify my Windows installation is already registered and the product key is correct. I looked into the Event Log, and GWX-Ins has a bunch of error messages saying: Error message: GWX Main: App failed to start hresult. After that it says what is roughly translated to English as "Cannot find the element". Stop jQuery .load response from being cached I have the following code making a GET request on a URL:  $('#searchButton').click(function() {     $('#inquiry').load('/portal/?f=searchBilling&amp;pid=' + $('#query').val());             });   But the returned result is not always reflected. For example, I made a change in the response that spit out a stack trace but the stack trace did not appear when I clicked on the search button. I looked at the underlying PHP code that controls the ajax response and it had the correct code and visiting the page directly showed the correct result but the output returned by .load was old.  If I close the browser and reopen it it works once and then starts to return the stale information. Can I control this by jQuery or do I need to have my PHP script output headers to control caching? to Global Variables inside functions How can I check if /bin/x86_64/bin/ls is a directory in a bash shell script  Here is what I tried:  #!/bin/bash  if [ $# -eq 2 ]; then     if [[ "$1" = /* ]]     then     cd ./bin/x86_64/bin/ls         if [ -d "$1" ]; then             echo "ok"             i="$1"             echo $i         else             echo "error2"             exit         fi     else         echo "error"         exit     fi fi Reputation has been changed but not reflected on tooltip. What medium should be used for long term, high volume, data storage (archival)? How to make a cycles light emisson object invisible to the camera? In my scene it is required to position the lights close to the main object like this:    How I can make the light source invisible to the camera? 301 redirect blogspot to an existing domain? Is it possible to redirect a blogspot site to an existing URL?  Note that I don't want to buy a new domain and tell blogspot to use that, eg as per this question: .    Instead I am trying to 301 redirect to an existing website in order to combine the website and the blog in one place.  So it needs to be:  301 example.blogspot.com/post to example.com/blog/post Use LaTeX to simulate old typewriter written texts This question led to a new package:      I'm giving my students an old exam and barely remembered to change the date at the top. A colleague and I joked about how long we could reuse the same exam, and it made me wonder if I could use LaTeX to create an exam that looked like it was written in 1963 on a typewriter.  Here's what I'd like to see:   Typewriter font even in math mode. (Several ways to accomplish this; is there a "best" way?) Slight inconsistencies in the printing of each letter. I attempt this below adapting code taken from . Each word has a random vertical shift, though I'd like to see it reduced to each letter having its own vertical shift. Other typewriter-like irregularities would be great. It would be great if math-mode delimeters \left(, \right}, etc., can be created in a piece-meal fashion as they did in days of old. Fraction bars, tops of square-root symbols -- any horizontal line, really -- is made up of a series of en-dashes that don't quite match up.   Here is what I've come up with so far:  \documentclass[10pt]{article}% This is a document class providing more font size options \usepackage{graphicx} \usepackage{ifthen} \usepackage[textwidth=7.5in,textheight=9.5in]{geometry} \pagestyle{empty}  % thanks to Bruno Le Floch: https://tex.stackexchange.com/q/9331/4012 % and in his comments to https://tex.stackexchange.com/a/29458/4012 \usepackage[first=-1,last=1]{lcg}% you can play around with these values \makeatletter \newcommand{\globalrand}{\rand\global\cr@nd\cr@nd} \makeatother  \newcommand{\randomvshift}[1]{\globalrand\raisebox{\value{rand}pt}{#1}}  %%% thanks to Martin Scharrer: https://tex.stackexchange.com/q/11598/4012 \makeatletter \def\typewriter#1{%     \@typewriter#1 \@empty } \def\@typewriter#1 #2{%    \randomvshift{#1}\space    \ifx #2\@empty\else     \expandafter\@typewriter    \fi    #2% } \makeatother \begin{document}\tt  \typewriter{Math 101 Fall 1963}  \begin{enumerate} \item   \typewriter{$\frac{\texttt{d}}{\texttt{dx}}\texttt{(x)}^2$}  \item \typewriter{State the Fundamental Theorem of Calculus.}  \end{enumerate}  \end{document} Why Lasso or ElasticNet perform better than Ridge when the features are correlated I have a set of 150 features, and many of them are highly correlated with each other. My goal is to predict the value of a discrete variable, whose range is 1-8. My sample size is 550, and I am using 10-fold cross-validation.   AFAIK, among the regularization methods (Lasso, ElasticNet, and Ridge), Ridge is more rigorous to correlation among the features.  That is why I expected that with Ridge, I should obtain a more accurate prediction.  However, my results show that the mean absolute error of Lasso or Elastic is around 0.61 whereas this score is 0.97 for the ridge regression. I wonder what would be an explanation for this.  Is this because I have many features, and Lasso performs better because it makes a sort of feature selection, getting rid of the redundant features? When can I say that $\cup H_i$ is a group? Let $G$ be a group and $\{H_i\}_{i\in I}$ a family of subgroups. I would like to find a condition that will imply that $\cup_{i\in I} H_i$ is a subgroup. I know that it's not true in general, I need help to find this condition.  Any help is welcome.  Thanks a lot. How do you handle aspect ratio differences with Unity 2D? I've gotten a lot of answers to this question, but they are all generic and generally not very useful.  None of the tutorials talk about aspect ratio and dealing with mobile devices and there are a zillion ways to do it, all seem to have gotcha's and flaws.  I really would love to know what successful games have used to handle different aspect ratios on iOS and Android without making a zillion different sized assets.  I am strictly speaking mobile, not desktop, specifically with Unity and I don't care about the UI, I only care about the gameplay canvas.  Issues I have in mind is when there are key things that have to be in certain places and cannot fall off the screen.  Using black bars on top or bottom is unacceptable these days. reports:  Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers [...]. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them. Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9.  Do these insights make breaking keys based on primes more mathematically plausible in a shorter amount of time? Can anyone explain how Windows effectively limits the CPU?  Last time I tried some tricks to let my laptop not overheat, so I tried limiting the CPU to 70%.   As for my suprise it stopped my laptop from overheating - even when launching CPU intensive applications which caused my laptop to overheat and shut down, didn't overheat my laptop anymore!  I looked at task manager but the "CPU intensive" application still uses ~95% of CPU and the windows task manager shows a total of 100% CPU usage.  I have limited the CPU usage here ("plugged in" option was on 70% [both min and max] when testing the CPU intensive application):    How does Windows do this magic?   I see 100% CPU usage yet my CPU does not overheat. And I do not see any performance differences.  Does Windows limit the CPU speed? Like, if I have a 2.4 GHz processor, on 70% it will only have 1.68 GHz when limited? How is this possible (changing the CPU speed)? I thought you can only underclock when in BIOS?  If not, how does Windows accomplish this if not by changing the speed?    Side question: If it's about the speed, is there a way to set the CPU max state to 200% ?