Cutting a mesh with another mesh
Making holes in a mesh
I am reading Velleman's "How to Prove it", and in one example he proves  $\neg \exists x P(x)  \implies \forall x \neg P(x)$    in the following way:    Suppose $\neg \exists x P(x)$. Let x be arbitrary. Suppose $P(x)$. Since we have a specific $x$ for which P(x) is true, it follows that $\exists x P(x)$, which is a contradiction. Therefore $\neg P(x)$. Since $x$ was arbitrary, we can conclude that $\forall x \neg P(x)$, so $\neg \exists x P(x)  \implies \forall x \neg P(x)$    What makes me confused is the line in bold text, that is "Suppose $P(x)$". The reason is that he writes that he is doing proof by contradiction, but $P(x) \not\iff \neg\forall x \neg P(x)$. He is not treating all possible cases where $\forall x \neg P(x)$ is false by assuming $P(x)$, does he? So how can this be a valid proof by contradiction?    Edit: I deleted my first example since it was clearly not representative for the problem.  Edit 2: So the thing that I am opposing I guess is that $\neg \forall x \neg P(x)$ is not $P(x)$. And from what I have undestood, a proof by contradiction should use the negation of the consequent which I do not think that Velleman does. Where is the itoa function in Linux? itoa() is a really handy function to convert a number to a string. Linux does not seem to have itoa(), is there an equivalent function or do I have to use sprintf(str, "%d", num)? Correct usage of "lest" Can more than one antibody bind the same antigen I'd like to clear something up about antibodies that I'm not sure I've understood in the articles I've read. Looking at concepts such as "affinity maturation", "monovalent antigens" and "polyvalent antigens" it seems as if there are multiple antibodies which can bind to the same antigen. (Although I think I'm right in saying that individual antibodies can only bind to one antigen?) However, when I read sentences like "It has been estimated that humans generate about 10 billion different antibodies, each capable of binding a distinct epitope of an antigen" on the Wikipedia page (for antibody) it does make it sound like it's one antibody per antigen? Is this the case? Or is it one antibody per epitope. Either way, to put this question in a nutshell:  Is it true to say that the same substance could bind with (and, therefore, be recognised by) more than one antibody, or not?  Also, if the answer is no, does that mean that antibodies with different binding affinities are classed as the same antibody or not? Tenure and tenure denial rates by universities Is there a way or website to find the tenure and tenure denial rates by universities and departments? I heard from some faculty that the tenure rate at X university is Y%, so I wonder if there is a way to find the statistics for all universities. What is the difference between varchar and nvarchar? (Posted here as probably applies to all sites).  I was examining my reputation gain yesterday on Stack Overflow: 190. I believe I should have hit the 200 cap. Mainly due to         Respectively, I had 10, 7, 3, and 2 upvotes on those questions alone. There are a couple of other minnows in the list but the above alone should enable me to hit the cap.  There was a user removed sometime yesterday which knocked off 10 points (and possibly accounts for the disparity) but I think (1) the cap should have been recalibrated and (2) I answered (3) after this event before turning in for the night.  I checked all the usual suspects: the reputation graph on your profile and  the achievements tab, but they are all showing 190. Indentation in tcolorbox package I'd like to have the same indentation within a tcolorbox as I have in the rest of the document.  How can I get it?  Thanks.  MWE:  \documentclass[a4paper]{book}  \usepackage{lipsum} \usepackage[most]{tcolorbox}  \begin{document}  \chapter{Example with \texttt{tcolorbox}} \lipsum[1-8]  \begin{tcolorbox}[breakable,enhanced] \lipsum[9-14] \end{tcolorbox}  \end{document}   The first block of text has the typical indentation, but that in the box doesn't. Dollar Sign corrupts question In  my question () contains a "$" in a quoted section of the question.  Originally I had a second "$" in a second quoted  section of the same question.    However when I did this the second $ seemed to trigger a style that made the text italic and lost spacing between words.  I took out the second "$" and instead emphasized it with the word "dollars" which solved the immediate problem.  However, in the listing of the question, (not the question itself) the same thing happens after the first "$".  The summary shows the text without spaces between words and in italic.  Seems like a bug.  Below is a verbatim copy/paste of the original issue (two dollar signs):    I was able to find a small ESC for about $12 off of ebay.  If you were designing a robot, would you see that and think?       $12 bucks for an ESC that connects to simple pulse-wave interface -   sign me up!   Or would you think:     $12 just to control a motor? I could throw together an H-bridge for   $0.50 and be done with it.   My robot in particular actually has two motors and therefor $24 to control the two of them.  But the interface is really easy (plus has the added advantage of being R/C vs computer controlled with a simple change of connectors.  Which way would you go? I found some articles related to this, but they were not sufficient for me. I want to know how the localhost works and why is it the same on many different machines?   How does the localhost block the access to the website when we use it in the host file to block the website? Etymology for “Mc‑” and “O’‑” prefix in surnames There is clearly a prefix in names like McDonald, McChrystal, O’Brian, O’Neal.  What does this Mc- and O- prefix signify? It looks like Donald, Chrystal, Brian, Neal are perfectly fine names on their own, so why is there a prefix before it? Suppress system() output Show a message if directly accessing a deleted answer or comment Currently, if you access a deleted question you will get a short message informing you that the question has been deleted. However, if you click on a direct link to a deleted answer or comment (which you can get via notifications), nothing is displayed.  This is naturally quite confusing as you will get the message about a new answer or comment via notifications, click on the link and simply see nothing.  Therefore, I propose to display some sort of message to the user, informing her/him that the answer or question has been deleted. Note that this is different from always showing a deleted answer/comment, as I specifically request it in case you access the specific URL.  One example for a comment would be:   One example for a deleted answer is:   This question is related to ,  but requests a feature to circumvent the problem altogether. Came across the following exercise in Bartle's Elements of Real Analysis. This is the solution I came up with. Would be grateful if someone could verify it for me and maybe suggest better/alternate solutions.  I also looked up these related questions - , ,  - but was not happy with proofs given there. I seem to need some help understanding these. Any such help is appreciated.      Show that every sequence in $\Bbb R$ either has a monotone increasing sub-sequence or a monotone decreasing sub-sequence.   Let $(x_n)$ be a sequence in $\Bbb R$. Suppose $(x_n)$ is not bounded. Without loss of generality we may assume that $(x_n)$ is not bounded above. Therefore given any real number there is a member of the sequence which is greater. Let $x_{n_1}$ be any member of the sequence.   There is $x_{n_2} \gt \sup\{x_1, x_2, ..., x_{n_1} \}$. For $i \gt 1$ let $x_{n_i} = \{x_1, x_2, ..., x_{n_{i - 1}}\}$ then $(x_{n_k})$ forms a monotone subsequence of $(x_n)$.   Now suppose instead that $(x_n)$ is bounded. By the Bolzano-Weierstrass Theorem there is a subsequence $(y_n)$ of $(x_n)$ which converges to a limit $y$. Without loss of generality there are infinitely many distinct values in $(y_n)$ that are unequal to $y$. Let $y_{k1}$ be the first such element. Let $y_{k2}$ be any element in $\{ y' \in (y_n) \ \ | \ \ |y' - y | \lt |y - y_{k1}|  \}$.   For $i \gt 1$ let $y_{ki} \in \{ y' \in (y_n)  \ \ | \ \ |y' - y | \lt |y - y_{k \ i - 1}|  \}$. Such $y_{ki}$ exists for every $i \in  \Bbb N$ since $ \lim (y_n) = y $. Now let $(y_{kn})$ be the sub-sequence of $(y_n)$ thus formed. At least one of the two following sets must contain infinitely many elements.    $\{ y \in (y_{kn}) \ \ | \ \ y \gt x\}$ $\{ y \in (y_{kn}) \ \ | \ \ y \lt x\}$   The one which does forms a monotone subsequence. Ubuntu kernel not upgrading with upgrade to 15.10 I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form $a_nz^n$ but don't have a clue what to do with these. Any help would be greatly appreciated!  (i) $\displaystyle\sum_{n=0}^\infty \bigg(\dfrac{1}{2n}\bigg)z^{3n}$.  (ii) $\displaystyle\sum_{n=0}^\infty(2^n+i)(z-i)^n$.  (iii) Show that the radius of convergence of $\displaystyle\sum_{n=0}^\infty z^{n!}$ is $1$ and that there are infinitely many $z \in \mathbb{C}$ with $|z|=1$ for which the series diverges. Problem about angle in isosceles triangle How can we compute the following limit:     $$\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$$   Mathematica gives the answer $\sqrt{e}$. However, I do not know to do it. Is there a way of moving unwanted villagers out faster? Is there any way to speed up this process or indirectly 'force a villager to move out'?  I know they move out if it's time for a new one to move in, and I know you can't actually 'force' them persay.  I have a villager who has caused nearly all my other villagers to wear his.. rather hideous 'kiddie shirt'. ((the character's default shirt)) Aside from that his home blocks the perfect spot, very close to my river, for building a convenient bridge...  I've taken to pushing him, hitting him with a net, and I've been trying to find a Toy Hammer... He wanted to leave once and I told him to go, but then he said he wouldn't leave until I, "made an effort to get to know him" or some BS..  Are there any other ways I can get an unwanted, 'pesty', villager to leave town? I received a journal paper for a review. The paper suggests a solution to an important problem, but the solution is very complicated and has mediocre performance. While reading the paper, I thought of a different solution, which is much simpler and has better performance (to give some scale: their solution is 1 page to write and 5 pages to prove; my solution is 5 lines to write and half a page to prove).  What should I do now?  A. Write my solution in the review, hoping the authors will remember to "thank the anonymous reviewer"?  B. Recommend that the paper published as is, then write a paper of my own with my improved solution?  C. Something else?  EDIT: Before I found the superior solution, I thought of recommending "resubmit with major revision", with the reasons that I mentioned above: your solution is too complicated and its performance is too weak.  However, now I am not sure it is the good course of action. A major revision may take a long time. The authors might even decide to delay or stop working on the paper without telling the editor (or me). Meanwhile, I will have to postpone the publishing of the superior solution for an indefinite amount of time. What do you think?