What type of pronoun shoould I use before a gerund?
When is a gerund supposed to be preceded by a possessive adjective/determiner?
Prove a graph Containing $2k$ odd vertices contains $k$ distinct trails I'm reading the book Graphs and Their Uses which contains the following theorem and proof:     THEOREM 2.3. A connected graph with 2k odd vertices contains a family   of k distinct trails which, together, traverse all edges of the graph   exactly once.       PROOF. Let the odd vertices in the graph be denoted by   in some order.       $a_1,a_2,\dots,a_k$ and $b_1,b_2,\dots,b_k$      When we add the $k$ edges $a_lb_l, a_2b_2 ,\dots, a_kb_k$ to the graph, all   vertices become even and there is an Eulerian trail T. When these   edges are dropped out again, T falls into k separate trails covering   the edges in the original graph.   However this doesn't seem to make sense since in the graph whose vertices have degrees 3, 1, 1, 1 there is no way to add 2 edges in such a way that the degree of all odd vertices becomes even.  What am I missing here? I recently received this rather amusing message while reviewing a user's first post:  Congratulations! This was only a test, designed to make sure you were paying attention. This post has already been removed, but thanks for taking time to leave feedback for the author.   What is the purpose of this test? How does the audit work? What happens if I fail? Why was the system expecting me to negatively review a good post, or positively review a bad post? A review is obviously an audit. Should I report it? Do review audits count toward the Reviewer badge? Do robo-reviewers dream of electric sheep? proving that elements of a sequence are pairwise coprime For the sequence $a_n = 6^{2^n} + 1$:   How can I prove that the elements of this sequence are pairwise coprime, i.e. prove that if $m$ is not equal to $n$ then $\gcd(a_m, a_n) = 1$. (begin by proving that $a_n∣ (a_{n+1} - 2)$) Ubuntu kernel not upgrading with upgrade to 15.10 I have detached a process from my terminal, like this:  $ process &amp;   That terminal is now long closed, but process is still running, and I want to send some commands to that process's stdin. Is that possible? Headless linux install How do you do a headless install of linux? No monitor, no keyboard. Machine has floopy and cd drive. Can I configure a live cd to run sshd with a preconfigured pass or something similar and manage it via that?  Instructions for fedora 8+ would be ace but anything else also welcomed.  Cheers. I posted this question but for some reason I can't get back into my account:    Most things I see with 220V have both hot legs going through the switch or relay, current "returns" through the switch.  Is it against code to wire one leg directly to the equipment and the other leg is switched as opposed to having both legs switched.  It seems useless and a waste of wire to switch both legs.  Basically  vs  (minus the GFCI as I'm not allowed to have that)  As a bonus question.  I assume I need a "motor rated switch" as standard switches can weld due to the high startup current.  Is  one legal\safe? What is the distance between the centers? Two unit circles overlap in such a way that exactly half the area of each circle is within the overlap. How far apart are the centers of the two circles? Since I installed TeXLive 2016 I cannot build documents with the standalone document class any more using LuaLaTeX (PDFLaTeX works fine). For example:  \documentclass{standalone} \begin{document} hi \end{document}   Gives me:  ! Undefined control sequence. \sa@placebox -&gt;\newpage \global \pdfpagewidth                                                =\wd \sa@box \global \pdfpageh...  l.4 \end{document}   How to fix that? Say I have this file:  hello world hello world   This program  #!/bin/bash  for i in $(cat $1); do     echo "tester: $i" done   outputs  tester: hello tester: world tester: hello tester: world   I'd like to have the for iterate over each line individually ignoring whitespaces though, i.e. the last two lines should be replaced by  tester: hello world   Using quotes for i in "$(cat $1)"; results in i being assigned the whole file at once. What should I change? How to make tr aware of non-ascii(unicode) characters? I'm trying to remove some characters from file(UTF-8). I'm using tr for this purpose:   tr -cs '[[:alpha:][:space:]]' ' ' &lt;testdata.dat    File contains some foreign characters (like "Латвийская" or "àé"). tr doesn't seem to understand them: it treats them as non-alpha and removes too.  I've tried changing some of my locale settings:  LC_CTYPE=C LC_COLLATE=C tr -cs '[[:alpha:][:space:]]' ' ' &lt;testdata.dat LC_CTYPE=ru_RU.UTF-8 LC_COLLATE=C tr -cs '[[:alpha:][:space:]]' ' ' &lt;testdata.dat LC_CTYPE=ru_RU.UTF-8 LC_COLLATE=ru_RU.UTF-8 tr -cs '[[:alpha:][:space:]]' ' ' &lt;testdata.dat   Unfortunately, none of these worked.  How can I make tr understand Unicode? What happens when you put a Portable Hole inside a Bag of Holding while on the Astral plane? What would happen if someone were to place a Portable Hole within a Bag of Holding while being within the Astral Plane? Would it be the reverse effect? The best stack-up possible with a four-layer PCB? I'm designing a 4 layer PCB and I know that the standard stack-up is  Signals GND VCC Singals  (GND and VCC can be switched depending on the layer with more signals) The problem is, I don't really want to connect all ground pins through vias, there are just too many of them ! maybe because I'm not used to 4 layer PCBs, anyway, I've read a  by Henry W. Ott about a different stack-up  GND Signals Signals GND  (Where the power is being routed with wide traces on the signal planes) According to him, this is the best stack-up possible with a four-layer PCB, for the following reasons: 1.Signal layers are adjacent to ground planes. 2.Signal layers are tightly coupled (close) to their adjacent planes. 3.The ground planes can act as shields for the inner signal layers. (I think this requires stitching ??) 4.Multiple ground planes lower the ground (reference plane) impedance of the board and reduce the common-mode radiation. (don't really understand this one) One problem is cross-talk, but I really don't have any signals in the third layer, so I don't think that corss-talk will be an issue with this stack-up,am I correct in my assumption ? Note: The highest frequency is 48MHz, there's a wifi module on the board too. If someone else is booking international ticket for me using his credit or debit card, does he need to be present physically at the time of check-in?   Suppose I am coming back from Singapore to my home country and my ticket is booked by my friend for me, who lives in my home country. Do airlines in Singapore ask for the person to be present physically to check in? When is $\sin(x)$ rational? Does set difference distribute over set intersection? I am asked to prove that, if $A, B$ and $C$ are sets, then $$A-(B\cap C)=(A-B)\cap(A-C).$$  However, I think that either I have made an error in my working, or the wording of the problem contains a typographical error.    My working so far is as follows: $$\begin{align}x\in(A-(B\cap C)) &amp;\iff (x\in A)\wedge(x\not\in(B\cap C))\\ &amp;\iff(x\in A)\wedge((x\not\in B)\vee(x\not\in C))\\ &amp;\iff((x\in A)\wedge(x\not\in B))\vee((x\in A)\wedge(x\not\in C))\\ &amp;\iff(x\in(A-B))\vee(x\in(A-C))\\ &amp;\iff x\in((A-B)\cup(A-C)) \end{align}$$  First line to second line: by De Morgan's Law. Second line to third line: by distributivity of set intersection over set union. Third line to fourth line: by definitions of intersection and set difference.   If my working is correct, then I have shown that $$A-(B\cap C)=(A-B)\cup(A-C).$$ Am I correct? A game called  is making rounds on social media. I am trying to determine the maximum score attainable for this game.  Let's assume WLOG that only 2s are returned (if 4s are possible the max score is doubled).  First, I'll relax some of the rules to create a game, $\mathcal{G}$, that can achieve a score at least as high as the original, 2048 game.  A run $R$  of $\mathcal{G}_n$ consists of a sequence of moves $m_1,\ldots,m_k$ which manipulate a multiset, $\mathcal{S}$ of maximum size n.  Each $m_i$ is either:   a put $p$, which inserts a 2 into $\mathcal{S}$, or a merge, $\alpha_i$, which replaces two $2^i$ elements from $\mathcal{S}$ with a single $2^{i+1}$ elt.   I allow sequential moves of the same type because 2048 allows multiple merges or no-merge turns.  The score of a position is the sum of $\mathcal{S}$.  What is the maximum score possible on a game of order n?  Note: The final position must have only unique values, else a merge and put would be possible in increase score.  I claim it is $\sum_{i=1}^{n} 2^i$, but couldn't come up with any good induction arguments.  It's pretty easy to prove that an element $2^k$ is producible in a game $\mathcal{G}_k$, but proving an upper bound for production got a bit hairy, and also doesn't cover the full space. Calculate the limit of two interrelated sequences? I'm given two sequences:  $$a_{n+1}=\frac{1+a_n+a_nb_n}{b_n},b_{n+1}=\frac{1+b_n+a_nb_n}{a_n}$$  as well as an initial condition $a_1=1$, $b_1=2$, and am told to find: $\displaystyle \lim_{n\to\infty}{a_n}$.  Given that I'm not even sure how to approach this problem, I tried anyway. I substituted $b_{n-1}$ for $b_n$ to begin the search for a pattern. This eventually reduced to:  $$a_{n+1}=\frac{a_{n-1}(a_n+1)+a_n(1+b_{n-1}+a_{n-1}b_{n-1})}{1+b_{n-1}+a_{n-1}b_{n-1}}$$  Seeing no pattern, I did the same once more:  $$a_{n+1}=\frac{a_{n-2}a_{n-1}(a_n+1)+a_n\left(a_{n-2}+(a_{n-1}+1)(1+b_{n-2}+a_{n-2}b_{n-2})\right)}{a_{n-2}+(a_{n-1}+1)(1+b_{n-2}+a_{n-2}b_{n-2})}$$  While this equation is atrocious, it actually reveals somewhat of a pattern. I can sort of see one emerging - though I'm unsure how I would actually express that. My goal here is generally to find a closed form for the $a_n$ equation, then take the limit of it.  How should I approach this problem? I'm totally lost as is. Any pointers would be very much appreciated!  Edit:  While there is a way to prove that $\displaystyle\lim_{n\to\infty}{a_n}=5$ using $\displaystyle f(x)=\frac{1}{x-1}$, I'm still looking for a way to find the absolute form of the limit, $\displaystyle\frac{1+2a+ab}{b-a}$. Is it okay to publish my personal notes from a lecture? Can I still use a usb flash drive after spilling water on it? So a few months ago I accidentally spilled a little water on a few of my usb flash drives. I really want to test and see if they work, but before I do, I just want to make sure that there's no risk of damaging my computer. I'm pretty sure it's fine, but I just got a new laptop and would hate to ruin it over something stupid.