I am writing a mathematical essay and would like to focus on the concept of infinity. I am not sure of any real life applications of infinity to write about or some way to narrow down the topics. Does anyone have any ideas on ways to focus the essay on a section of the topic infinity.
I know what you're thinking: "of course it has, for example, it can be used to tell you how many times you can go around a circle". But that isn't really true, now is it? You'd be dead or the world would go under long before an infinite amount of loops had been reached.  Are there any practical applications for the concept of infinity? Is it a useful concept in maths at all?  I know that Donald E. Knuth has argued that for all practical purposes, a very, very large number has the same effect as infinity, in his book "Things a Computer Scientist Rarely Talks About" (can't remember the exact quote, nor find it online, unfortunately).  Examples are appreciated.
Sometimes (old) comments break the parser, so it would be nice to view their code to check what was (actually) supposed to be there. Is there any way to view a comment's source just like the source link in a post's revision?  edit Another application for this would be the ability to better quote comments in an answer without having to re-markdown them. How to write an RSS feed with Java? I'm using Java, and need to generate a simple, standards-compliant RSS feed. How can I go about this? If $\sin \theta+\cos\theta+\tan\theta+\cot\theta+\sec\theta+\csc\theta=7$, then $\sin 2\theta$ is a root of $x^2 -44x +36=0$ My own bonafide attempt. $$ 0&lt;\theta&lt;\pi/2$$   and $$\sin\theta+\cos\theta+\tan\theta+\cot\theta+\sec\theta+\csc\theta=7$$   then show that $\sin 2\theta$ is a root of the equation $$x^2 -44x +36=0$$   I tried to use the above given equation of all the trigonometric ratios, though I ended up with an expression of $\sin\theta\cos\theta$.   But this too is in the form of $\sin\theta$ and $\cos\theta$ which was $\sin\theta\cos\theta=\frac{1}{6-\sin\theta-\cos\theta}$.  When I put that in the quadratic equation, it would again transform into the form of $\sin\theta$ and $\cos\theta$, and hence at last I couldn't prove the thing.  Even hints would work, as I would like to solve the question myself. How to Straighten Curved UVs into Straight Belt Showing the actual 3D model of a curving ramp.  Shows the UVs of that model that blender did automatically. Shows the UV image that I need.   I am going put a road texture on it., but round faces are not helping me at all. All I need is straight belt kind of UV. Keep box-shadow direction consistent while rotating The following is the residuals vs predicted scatter plot for a regression model with two IVs.   Initially, I thought it was evidence of heteroskedasticity. But, I reasoned that although there is a visible pattern in the plot, the variance across different levels of predicted values is same.  To clear my doubts, I saved the standardized residual and predicted values and ran a bi-variate correlation test. The correlation is zero, as expected. I am, nevertheless, intrigued by this observation. Any idea why I might have obtained this pattern?  PS:- My dependent variable is a sum of two likert-type items (Response scale: 1-5). So, it's theoretical range is 2-10 and it has no absolute zero value. Should operator<< be implemented as a friend or as a member function? That's basically the question, is there a "right" way to implement operator&lt;&lt; ? Reading  I can see that something like:  friend bool operator&lt;&lt;(obj const&amp; lhs, obj const&amp; rhs);   is preferred to something like   ostream&amp; operator&lt;&lt;(obj const&amp; rhs);   But I can't quite see why should I use one or the other.  My personal case is:  friend ostream &amp; operator&lt;&lt;(ostream &amp;os, const Paragraph&amp; p) {     return os &lt;&lt; p.to_str(); }   But I could probably do:  ostream &amp; operator&lt;&lt;(ostream &amp;os) {     return os &lt;&lt; paragraph; }   What rationale should I base this decision on?  Note:   Paragraph::to_str = (return paragraph)    where paragraph's a string. Well, I have come to understand persons with the following traits will be usually be Sorted to Slytherin   Resourcefulness  Cunning Ambition Leadership Qualities Self-Preservation Determination  Cleverness Fraternity  Power   On which trait(s) was Snape Sorted into Slytherin?  One more point: Lily Evans was Sorted before Snape, and when she was placed in Gryffindor, obviously Snape wanted to be in Gryffindor too, because Lily is his only friend.   So why is he Sorted into Slytherin? Is there a stronger reason for the Sorting Hat to override his preference and Sort him into Slytherin? Currently using Ubuntu 18.04, and my wireless driver is not being detected.  I've included a .  $ dmesg | grep iwl [    6.212724] Loading modules backported from iwlwifi [    6.212724] iwlwifi-stack-public:master:6988:0d8892f2 [    6.224583] iwlwifi 0000:00:14.3: enabling device (0000 -&gt; 0002) [    6.229811] iwlwifi 0000:00:14.3: Direct firmware load for iwl-dbg-cfg.ini failed with error -2 [    6.230540] iwlwifi 0000:00:14.3: Direct firmware load for iwlwifi-9000-pu-b0-jf-b0-39.ucode failed with error -2 [    6.230547] iwlwifi 0000:00:14.3: Direct firmware load for iwlwifi-9000-pu-b0-jf-b0-38.ucode failed with error -2 [    6.230552] iwlwifi 0000:00:14.3: Direct firmware load for iwlwifi-9000-pu-b0-jf-b0-37.ucode failed with error -2 [    6.230557] iwlwifi 0000:00:14.3: Direct firmware load for iwlwifi-9000-pu-b0-jf-b0-36.ucode failed with error -2 [    6.230562] iwlwifi 0000:00:14.3: Direct firmware load for iwlwifi-9000-pu-b0-jf-b0-35.ucode failed with error -2 [    6.236670] iwlwifi 0000:00:14.3: loaded firmware version 34.0.0 op_mode iwlmvm [    6.248314] iwlwifi 0000:00:14.3: Detected Intel(R) Dual Band Wireless AC 9560, REV=0x318 [    7.264081] iwlwifi 0000:00:14.3: SecBoot CPU1 Status: 0x3, CPU2 Status: 0x2477 [    7.264084] iwlwifi 0000:00:14.3: Failed to start INIT ucode: -110 [    7.276165] iwlwifi 0000:00:14.3: Failed to run INIT ucode: -110   sudo modprobe iwlwifi returns no output. Prove that any continuous map on $\mathbb{R}$ which is the identity on $\mathbb{Q}$ is the identity map, hence Aut$(\mathbb{R}/\mathbb{Q})= 1.$  proof: Suppose $\sigma$ is continuous on $\mathbb{R}$ which is an identity on $\mathbb{Q}$. let $a \in R $. Then since $\sigma$ is continuous, then by definition there is $\delta_1 &gt; 0$ such that $|a - b| &lt; \delta_1$.  And assume $|\sigma(a) -\sigma(b)| &lt; \epsilon/2$ .  Let $\delta $= min{$\epsilon/2, \delta_1$}.  And now let $b \in \mathbb{Q}$ such that $|a - b| &lt; \epsilon$. Then  $|\sigma(a) - a| = |\sigma(a)  - b + b - a| \leq |\sigma(a) - \sigma(b) | + |b - a| &lt; \epsilon/2 + |b-a|$.  Can someone please help me? I am not sure . Can I say $|b - a| &lt; \epsilon/2 $? In Java is there a way to check the condition:  "Does this single character appear at all in string x"  without using a loop? As the title suggests, I’m simply looking for a method to compute the number of Eulerian Paths in the complete graph $K_5$, where all paths start from the same node. Since there are 5 nodes, all Eulerian paths must be Eulerian circuits as well.  Thanks. Do I need quote marks when I quote myself? Hello language lovers,   English isn't my first language and I would like to know if I need to use quotation marks when I quote something I said in the past. A non-blocking read on a subprocess.PIPE in Python I've noticed the following happen a few times:   A user posts a question. The question is closed as a duplicate and the usual "Possible Duplicate" links are added at the top. The user promptly removes the duplicate links, even though the question is still closed as a duplicate. (excessive use of list items =P)   Now, I could understand this behavior if there was some dispute about whether or not the question was a duplicate, and the OP were to remove the links and post reasons why they disagree with the closing. However, I've noticed this happening without any attempt to defend the question (such as ).  In my opinion, no one should be allowed to remove the "Possible Duplicate" links as long as the question is still closed as a duplicate. The links should remain for the benefit of the community, so they can navigate to the duplicate questions and either post answers there or read them over and judge if the closing was actually fair. Once/if the question is reopened the links could be edited out if one so chooses.  I guess this is part discussion, part feature request:  Could the "Possible Duplicate" links remain un-editable for as long as the question is still closed? "The antipodal map of $S^2$ is the isometry $\bar m$ sending each point $P$ to its opposite point $-P$" (from textbook)  Show that antipodal points remain antipodal under any isometry of $S^2$. In other words, for each isometry f of $S^2$, if $P$ and $P'$ are antipodal, then so are $f(P)$ and $f(P')$  I am unsure how I can define the $P$ and $P'$ so that they are opposites of each other in the antipodal map, and am I supposed to apply the traditional isometries (translation, reflection and rotation) to the points? I don't understand this problem. How do you set a default value for a MySQL Datetime column?  In SQL Server it's getdate(). What is the equivalant for MySQL? I'm using MySQL 5.x if that is a factor. I'd like to use epigrafica specifically for chapters and sections, but like to use times font for main texts.  \usepackage{epigrafica}          % for chapter/section \usepackage{newtxtext,newtxmath} % for main text   Is there any way to do this? I can only use pdftex. Correcting for multiple kruskal wallis tests I am running a kruskal wallis test a number of times (37) and wanted to know if I have to adjust the p-values of the kruskal tests before I look any further into the post hocs? A finite summation of double binomial coefficients I find the following identity and have checked on Mathematica, while I have no idea how to prove it:  $$\sum_{j=0}^n(-1)^{n-j}\binom{p+j}{j}\binom{n+\beta}{n-j}=\binom{p-\beta}{n}, \quad \beta&gt;-1, \quad p&gt;\beta-1.$$  It seems that proof by induction does not work and because of the gamma function, the transform technique does not work either.