Omitting/Defining a restrictive clause
Restrictive vs Non-restrictive Relative Clause
Why can templates only be implemented in the header file? Quote from :     The only portable way of using templates at the moment is to implement them in header files by using inline functions.   Why is this?  (Clarification: header files are not the only portable solution. But they are the most convenient portable solution.) Elementary properties of closure Hi everyone I'd like to know if the following is correct. I really appreciate any suggestion. (Honestly the only one that matters me is the second property the others are easy, I think) Thanks.  Definition: Let $X\subset \mathbb{R}$ and let $x'\in \mathbb{R}$, we say that $x'$ is an adherent point of $X$ iff $\,\forall\varepsilon&gt;0\,\exists x\in X \text{ s.t.} \; d(x',x)\le \varepsilon$. We say that $\overline{X}$ is the closure of $X$ iff contain all the adherent points of $X$.  Lemma (Elementary properties of Closures): Let $X$ and $Y$ be arbitrary subset of $\mathbb{R}$. Then $X\subset \overline{X}, \; \overline{X\cup Y}=\overline{X}\cup\overline{Y},\; \overline{X\cap Y}\subset\overline{X}\cap\overline{Y}$. If $X\subset Y$ then $\overline{X}\subset \overline{Y}$.  Proof:  Clearly if $x\in X$,  $x$ is $\varepsilon$-adherent to $x$ (indeed is $0$-close to itself) for every $\varepsilon$, hence $x\in \overline X$.   Let $z'$ be an adherent point of $X\cup Y$, then there exists some $z\in X\cup Y$ such that $d(z',z)$ for a given $\varepsilon&gt;0$. Suppose  $z\in X$, hence $z'$ is an adherent point of $X$, i.e., $z'\in \overline{X}\subset \overline{X}\cup \overline{Y}$. Similarly when  $z\in Y$, $z'\in \overline{Y}\subset \overline{X}\cup \overline{Y}$. Thus $ \overline{X\cup Y}\subset\overline{X}\cup \overline{Y}$. We  know that $X\cup Y \subset\overline{X\cup Y}\subset\overline{X}\cup \overline{Y}$. Then $\overline{\overline{X\cup Y}}=\overline{X}\cup \overline{Y}$. Since $\overline{\overline{X\cup Y}} = \overline{X\cup Y}$ we're done.  Let $z'\in \overline{X\cap Y}$, i.e., $z'$ is an adherent point of $X\cap Y$. So, there is some $z\in X\cap Y \subset X$ such that $d(z',z)\le \varepsilon$ for any given $\varepsilon&gt;0$. It follows that $z'$ is an adherent point of $X$, i.e., $z'\in \overline{X}$. Similarly since $z\in X\cap Y \subset Y$ we have $z'\in \overline{Y}$. Hence $z'\in \overline{X}\cap\overline{Y}$.   Let $x'\in \overline{X}$ we'd like to show that $x'\in \overline{Y}$. Let $\varepsilon&gt;0$ be arbitrary then $\exists x\in X\subset Y$ such that $d(x',x)\le \varepsilon$, so $x'$ is an adherent point of $Y$, i.e., $x'\in \overline{Y}$ as desired. $\Box$     Claim 1: If $X\subset Y \subset \overline{X}$. Then $\overline{Y}=\overline{X}$.    Proof claim: Let $y'$ be an adherent point of $Y$, i.e., $y'\in \overline{Y}$ and let $\varepsilon&gt;0$ be given. Thus, $\exists y \in Y\subset \overline{X}$ s.t. $d(y,y')\le \varepsilon$. So, $y$ is an adherent point of $X$. Thus $\exists x\in X$ s.t. $d(x,y)\le \varepsilon$ and therefore  we must have $d(x,y')\le 2\varepsilon$ which shows that $y'$ is an adherent point of $X$, i.e., $\overline{Y}\subset \overline{X}$. Conversely if $x'$ is an adherent point of $X$. Then $\exists x\in X\subset Y$ s.t. $d(x,x')\le \varepsilon$, and so is adherent to $Y$. $\Box$     Claim 2: $\overline{\overline{X}} = \overline{X}$    Proof: It will suffice to show that if $x''\in \overline{\overline{X}}$, then $x''$ is an adherent point of $X$. Let $\varepsilon &gt;0$ be arbitrary and let $x''$ be an adherent point of $\overline{X}$. Then there is some $x' \in \overline{X}$ such that $d(x'',x')\le \varepsilon$. Since $x' \in \overline{X}$ by definition is an adherent point of $X$, so there is some $x\in X$ for which $d(x,x')\le \varepsilon$. Thus $d(x,x'')\le d(x,x')+d(x',x'')\le 2\varepsilon$ it follows that $x''$ is an adherent point of $X$ as desired. $\Box$ How tall is Supreme Leader Snoke? In Star Wars: The Force Awakens, we see Supreme Leader Snoke a few times dispensing orders to his subordinates.      However, we always see him communicating via hologram, and we've seen in previous Star Wars films that these holograms are often projected at a different size than the original person is.   So how tall is he really?  I don't think we ever got a good shot of him with something to give a sense of scale. Change Gmail account to make in-app purchase? Thus is a picture of both of my hard-drives:    My C Drive and my D drive. I want to put Ubuntu onto my D Drive. Now my question here is if i do install onto my D Drive then how should I do it in order to retain use of the drive. I intend to partition about 60 gigs from it. Now after doing that I should still have about 830 gigs left. I want to still be able to use that in Windows. So how and will I be able to retain use of the drive after Installing? I have installed once before and I believe i did it wrong because not only could Windows not recognize my D drive but I also lost all the data on it.   My computer is a Lenovo Y700 It is a UEFI (I believe i got that right). A question about long exact sequence of finite dim vector spaces Q) Prove that if $0\rightarrow V_1\rightarrow V_2\rightarrow...\rightarrow V_k\rightarrow 0$ is an exact sequence of finite dimensional vector spaces, then $\sum_{i=1}^{k}(-1)^idim (V_i) = 0$.  For $k=3$, I see that the statement is true. For $k&gt;3$, am I correct in understanding that $V_2 = V_1\oplus V_3 (\implies dim(V_2)=dim(V_1)+dim(V_3))$ and $V_3 = V_2\oplus V_4(\implies dim(V_3)=dim(V_2)+dim(V_4))$ and so on?   If it is true, then for $k=4$, $\sum_{i=1}^{4}(-1)^idim (V_i) = -m_1+m_1+m_3-m_2-m_4+m_4$ and am not sure why $m_2=m_3$ for the sum to be $0$? where $m_i = dim(V_i)$. Thanks. Mass Uninstalling Just been working on a new laptop today (Acer) and it was pre-installed with all sorts of crapware. Mostly games and "backup utilities" and McAfee. I had to manually uninstall about 30 apps via Vista's Programs and Features  So it got me wondering is there any way of mass uninstalling or queuing an app to be uninstalled? Through Vista/XP you need to wait for Install Shield or whatever to close before beginning to uninstall the next app. Took way more time than it should have.  Just another one of those Windows annoyances! Any help appreciated as I often need to do this when I come across a slow machine that's just been clogged with a bunch of useless apps Editing default custom object home screen I would like to add a custom button (URL to Flow) to my default home page for my custom object. Currently, I have a default "New" button to create a new record (see attached screen shot). I'm wanting to add my Flow custom button to this page. Is there a way to do this?  Cheers, Sean. I'm just starting to learn LaTeX. I tried typing \begin{equation*}  (math goes here)  \end{equation*}  and I got a whole bunch of errors. It seems LaTeX interpreted the empty line as the equation: when it saw nothing there, I guess it just ended math mode? It then said my other commands needed $'s so I assume that's what happened. When I typed \begin{equation*} (math goes here) \end{equation*}  everything was fine. So I'm confused. If LaTeX sees an empty line in an environment, does it just interpret this as me changing my mind, and deciding not to use the declared environment? ssh master - execute su remotely I have a server S that I need to connect to.  I'm connecting to S as user asdf with control master and after that I perform some stuff:  ssh -N -f -M -o ControlPath=$SSHSOCKET ${USER}@${IP} -p ${PORT} -i id_rsa ssh -to ControlPath=$SSHSOCKET ${USER}@${IP} -p ${PORT} "su -c whoami &amp;&amp; hostname &amp;&amp; exit"   It works as expected, but second line asks me  for password.  I am trying such fix:  echo 'qwerty' | ssh -t -to ControlPath=$SSHSOCKET ${K3_USER}@${IP} -p ${PORT} "su -c whoami &amp;&amp; hostname &amp;&amp; exit"   but it gets even worse:  muxserver_accept_control: tcgetattr: Inappropriate ioctl for device tcgetattr: Inappropriate ioctl for device Password:   For me, it's mandatory to ssh as asdf first. I also can't make any other ssh control master connections. I also can't modify any config on the target machine.  I just need to switch to root via ssh somehow. Any ideas ? If G is a group of order n=35, then it is cyclic I've been asked to prove this.   In class we proved this when $n=15$, but our approached seemed unnecessarily complicated to me. We invoked Sylow's theorems, normalizers, etc. I've looked online and found other examples of this approach.    I wonder if it is actually unnecessary, or if there is something wrong with the following proof:   If $|G|=35=5\cdot7$ , then by Cauchy's theorem, there exist $x,y \in G$ such that $o(x)=5$, $o(y)=7$. The order of the product $xy$ is then $\text{lcm}(5,7)=35$. Since we've found an element of $G$ of order 35, we conclude that $G$ is cyclic.   Thanks. How do I delete Windows.old in Windows? How can I explain to a non-native why the plural of a noun isn't used adjectivally? How can I explain to a non-native speaker that while this is correct:     The felling of trees.    And this is also correct:     tree felling   This is not:     *trees felling   To a non-native it seems very reasonable that since many trees are being felled and given that the felling of trees uses the plural, the inverse form would also use the plural and become trees felling.  To a native speaker such as myself, the latter is very obviously wrong, but I couldn't figure out how to formalize this into a rule that a non-native can understand. Why does as.factor return a character when used inside apply? I need a command that will wait for a process to start accepting requests on a specific port.    Is there something in linux that does that?  while (checkAlive -host localhost -port 13000 == false)   do some waiting  ... How to self study Linear Algebra I have no idea if this question is appropriate for this forum, but I hope you guys can overlook the fact that I asked it on a wrong forum (if I did) and still help me answer it (of course, if this is indeed the wrong forum for the type of question I'm about to ask, do please say so).  I am a 16 year old guy who is passionate about physics and as a result wants to increase his knowledge in mathematics, the language of physics. I've read and heard a lot about Linear Algebra and how crucial it is to physics and I am deeply motivated to self study this intriguing part of mathematics. However, I have limited knowledge of maths. I (for example) know basic algebra, trig, diff/int calc and some analytical geometry, but I wouldn't say I master these subjects past the high school curriculum. Now my question is: Would you kind people say I am able to self study Linear algebra or is it too tough and/or does it require too much of a math background? And are there any good books out there for BEGINNERS in LA? I found this (free) e-book called: Elementary Linear Algebra by Kenneth Kuttler and another one called 'Linear Algebra: Theory and its applications' also by Kuttler? Are these any good? Or would you recommend other books? If you guys have any tips regarding books for LA but also tips in general, please, I'd appreciate them! Let me tell the story from the beginning: I installed Big Sur on an unsupported Mac using guidelines at Medium. In the guide it was stated that I should create a separate partition for Big Sur, and shouldn't install it on my current one. I had my SSD Called &quot;Macintosh SSD&quot; and later I created a partition named &quot;Macintosh SSD Big Sur&quot; of size 40GBs. After installation I wanted to clear the &quot;Macintosh SSD&quot; of size 80GBs(with old MacOS) and merge it with the new &quot;Macintosh SSD Big Sur&quot;. This is what I initially had after formatting partition with the old MacOS  So, I used the command sudo diskutil apfs deleteContainer disk3 after which I had (free space) in /dev/disk1  I want to extend that 39.6GB partition which free space of 80,1GB but all the commands I googled do not work:(( Could you please guide me on what I should do now? Thank you in advance! Using Laptop Screen as a monitor If $X_1, ..., X_n$ are independent identically-distributed random variables, what can be said about the distribution of $\min(X_1, ..., X_n)$ in general? Find the integer which is closest to the value of   $\frac{1}{\sqrt[6]{5^6+1}- \sqrt[6]{5^6-1}}$  I have tried putting them between n and n+1 and tried manipulating  n but I cant seem to find a strategy.