Discrete Math - If two natural numbers divide each other, then they are equal I’m not sure how I would go about proving this. Any help would be appreciated.
Prove if $a\mid b$ and $b\mid a$, then $|a|=|b|$ , $a, b$ are integers. Form the assumption, we can say $b=ak$ ,$k$ integer, $a=bm$, $m$ integer.  Intuitively, this conjecture makes sense. But I can't make further step.
Where can I find computer modern fonts for use in other programs, e.g. Adobe Illustrator? For some complicated diagrams, I need to use something like Adobe Illustrator or Inkscape, and would like to include nicely typeset labels. (I'm aware of post-processing solutions, such as pinlabel.) One way to do this is to use a closely-matching font, and then convert fonts to outlines. Better, however, would be to use Computer Modern directly in Illustrator.  Where can I find a Computer Modern font compatible with Adobe Illustrator? CSS Pseudo-classes with inline styles I'm rotating an object on two axes, so why does it keep twisting around the third axis? I see questions come up quite often that have this underlying issue, but they're all caught up in the particulars of a given feature or tool. Here's an attempt to create a canonical answer we can refer users to when this comes up - with lots of animated examples! :)    Let's say we're making a first-person camera. The basic idea is it should yaw to look left &amp; right, and pitch to look up &amp; down. So we write a bit of code like this (using Unity as an example):  void Update() {     float speed = lookSpeed * Time.deltaTime;      // Yaw around the y axis using the player's horizontal input.             transform.Rotate(0f, Input.GetAxis("Horizontal") * speed, 0f);      // Pitch around the x axis using the player's vertical input.     transform.Rotate(-Input.GetAxis("Vertical") * speed,  0f, 0f); }   or maybe  // Construct a quaternion or a matrix representing incremental camera rotation. Quaternion rotation = Quaternion.Euler(                         -Input.GetAxis("Vertical") * speed,                          Input.GetAxis("Horizontal") * speed,                          0);  // Fold this change into the camera's current rotation. transform.rotation *= rotation;   And it mostly works, but over time the view starts to get crooked. The camera seems to be turning on its roll axis (z) even though we only told it to rotate on the x and y!    This can also happen if we're trying to manipulate an object in front of the camera - say it's a globe we want to turn to look around:    The same problem - after a while the North pole starts to wander away to the left or right. We're giving input on two axes but we're getting this confusing rotation on a third. And it happens whether we apply all our rotations around the object's local axes or the world's global axes.  In many engines you'll also see this in the inspector - rotate the object in the world, and suddenly numbers change on an axis we didn't even touch!    So, is this an engine bug? How do we tell the program we don't want it adding extra rotation?  Does it have something to do with Euler angles? Should I use Quaternions or Rotation Matrices or Basis Vectors instead? Can the ship's captain really perform your wedding when in international waters? On , Mark Mayo posted a comment that any marriage has to be on dry land or close to it.  But I'm sure all of us have heard of ships' captains marrying couples at sea.  From the movie Casablanca:     Say, why don't we get married in Marseilles?   ...   Well, let's see. What about the engineer? Why can't he marry us on the train?   ...   Well, why not? The captain on a ship can. It doesn't seem fair that...    So is this fiction after all? If not how does it fit with Mark's comment? What special rules and such should we know?  If I'm travelling at sea with a significant other and wish to tie the knot, can we do so with any captain on any ship on any sea? Must it be international waters? (p → q) ∧ ¬p ⇒ ¬q  I have tried to use the rules of inference in many ways to prove this argument to be valid. However, I cannot seem to find a way to validate the argument, but I am afraid I have missed something.  Using a truth table I find that when p and q is 0, then p → q is 1, ¬p is 1 and ¬q is 1. Would that not make the argument valid?  I apologize in advance if I have misunderstood the whole concept. Any answer would be much appreciated. Thank you multi-layer board copper pour I have a 6-layer board where the internal 4 planes are +15,GND,VCC,-15.  I was wondering if there was any advantage to doing a copper pour on the top and bottom layers?  I would probably leave them floating as I don't want to use micro-vias to say tie it to GND?    Is this actually a bad idea?  I.e. floating copper=antenna.  Would it be just as acceptable to have 4 layer board with the top layer having copper pour to VCC and the bottom having a pour to GND and keep the two internals as +-15?  Note that this is for a fairly low speed circuit which has some analog and digital parts. Best WPF open source projects I have programmed with WPF for more than a year. When I started there weren't so many examples and guidelines to follow, now the situation is much more better.  I would like to ask you some help in listing the best WPF (open) source code examples or projects that you suggest to study in order to learn more about project structure, best practices, etc.  Thank you, Marco Is it appropriate to use short form of "have" ('ve) when it means possession? I feel uncomfortable saying sentences like the following:    "I've a car" instead of "I have a car" "They've a great time" instead of "They have a great time" "He's a pen" instead of "He has a pen" etc   I ask this because I read this sort of thing in a book.  Are they correct? And what is the rule? Can you use such forms in a formal setting? Distributions other than the normal where mean and variance are independent I was wondering if there are any distributions besides the normal where the mean and variance are independent of each other (or in other words, where the variance is not a function of the mean). 8.4 - Use entity browser in a module block from in same way as I would in a custom block The Next tag badge block overlaps the badge score block. Suppose that $f:\Bbb R\to \Bbb R$ is continuous and that $f(x)\to 0$ as $x\to \pm\infty.$Prove that $f$ is uniformly continuous. Suppose that $f:\Bbb R\to \Bbb R$ is continuous and that $f(x)\to 0$ as $x\to \pm\infty$. Prove that $f$ is uniformly continuous.   Since $f(x)\to 0$ as $x\to \pm\infty$ we have $|f(x)|&lt;\epsilon $ whenever $x&lt;-K$ and $x&gt;K$ where $K&gt;0$.  Also $f$ is uniformly continuous on $[-K,K]$ as the domain is compact.  Hence $f$ is uniformly continuous in $(-\infty,-K)\cup [-K,K]\cup (K,\infty)$.  But how to conclude that $f$ is uniformly continuous on $\Bbb R$ from above as I may chose $x\in (-\infty,-K);y\in [-K,K]$ ,how to conclude that $|f(x)-f(y)|&lt;\epsilon $ from above for a given $\epsilon$?  Please suggest the edits required. Workflow for UV orientation How to I deal with unwrapped UVs orientation?   Delete the default Cube. Add a Cylinder Edit mode and edge select the Top and the Bottom and a single connecting edge. Make them into seams UV Unwrap (standard Unwrap mode)   The unwrapped UV now shows two circles above a row of quads. I can save this and open it in an image editor and paint over those areas and get the texture to align on the cylinder back in blender.  But, the pattern(just happens to be text) that was inserted in place of the row of quads is now upside down on the model. Any attempt to flip or align just makes things worse, from upside down to mirrored or the Top/Bottom not aligning.  So my real question is how could I have known that the quads represented an upside down view of the world?  Do other blender users export UVs and mark them with arrows and colours just to see how they align on a model before actually painting for real?  I tried selecting the edges in step 3 in different orders before making them into seams and it made no difference.  I also looked at the Normals but they appear to be only Inside/Outside not Up/Down.  Blender 2.70 There has been talk of how to answer a negative question without ambiguity, most often with a qualifying phrase needed for clarification. (For example, "yes, I do"/"no, I don't.)  I've noticed that other languages have a single word to address a negative question on the contrary. In German, ja means yes and nein means no. Alternatively, doch is used to respond in the positive to a negative question, in essence meaning "No, I intend to do the opposite of what your question presumes."  Take for example the question, "You don't want to go to the movie then?" German speakers could simply reply, "nein" if they don't want to go or they would say "doch" if they did want to go; an unambiguously positive response to a negatively phrased question. No need for, "no, I actually would like to go" or "yes, I'll come actually". In French, the respective equivalents of ja, nein, and doch are oui, non, and si. I imagine many other languages contain this set as well.  I am wondering whether there was at one time an equivalent word in the English language that has since (unfortunately) disappeared. Any ideas? How do you properly use namespaces in C++? What's the correct way to do division using a backslash? If I write $a/bc + d$, will that be equal to $(a/(bc))+d$?   Basically, if I place a slash, will I divide by what's directly behind the slash $(b)$, the term that's behind the slash $(bc)$, or everything after the slash $(bc+d)$? Let $I = \langle 2\rangle$. Prove $I[x]$ is not a maximal ideal of $\mathbb Z[x]$ even though $I$ is a maximal ideal of $\mathbb Z$.   My professor mentioned that I should try adding something to it to prove that $I[x]$ is not the maximal ideal. I want to assume he meant to add $ \langle 2 \rangle$ to $I[x]$? What's a way to implement a flexible buff/debuff system? Is it ever a good idea to close credit cards? I have a bunch of credit cards with no balance, while others have fairly high balances. I'd like to close the 0 balance cards, but I know that will reduce my overall balance and hurt my credit rating, since my ratios of used to unused credit will rise.  Is it ever a good idea to close credit cards? How to get/suppress french spacing before punctuation in `\href` Attention I have already a possible answer and will add it soon. I thought it would be useful to have a solution about it on the site. (Other answers are naturally welcome too).  Question How to activate or suppress the french spacing before some punctuations like a colon in the text argument of \href locally (for one specific \href) and globally? Switching the language is not an answer!  Minimal examples for tests (the argument of \href should normally be a valid url, so it is a bit unfair to add an example without a colon in the link):  pdflatex  With pdflatex it depends on the link how the argument of \href is printed:  \documentclass{article} \usepackage[french]{babel}  \usepackage{hyperref}  \begin{document} With pdflatex no french spacing in links with colons:   \href{link}{a:b}  \href{link:colon}{a:b}  \end{document}     lualatex  With lualatex the spaces are always there:  \documentclass{article} \usepackage[french]{babel}  \usepackage{hyperref}  \begin{document} With lualatex always spaces:  \href{link}{a:b}  \href{link:colon}{a:b}  \end{document}