Must mathematical entities necessarily exist?
What is mathematical existence?
New tags page mangles tag wiki excerpts Wikipedia says that the name of concept comes from physics, but I cannot find any similarity between these two concepts. I've been modeling a hermit crab and I don't know why but some areas of the mesh are really dark on some spots, like the legs and the claws.   I hope you can help me fix it. On the meta home page I noticed the  is listed as doing something, but cannot figure out what it is.  There is nothing on the  or its , but the recent activity is listed with the same timestamp. I would like to know the best way to write test methods for controller extension classes that are used by Visualforce.  What is the minimum set code that I need to cover the basic operation of the controller without taking into account any custom functionality that I might add.  Here is my example class with the test method ready to be completed.  public with sharing class myExtension{      private final Account acct;      public myExtension(ApexPages.StandardController stdController){         this.acct = (Account)stdController.getRecord();     }      public static testMethod void testmyExtension(){      } } How to make git ignore changes in case? I'm not too sure what is going on here, but sometimes a particular file in my repository will change the case of its name. e.g.,: before: File.h after: file.h I don't really care why this is happening, but this causes git to think it is a new file, and then I have to go and change the file name back. Can you just make git ignore case changes? [edit] I suspect it is Visual Studio doing something weird with that particular file, because it seems to happen most often when I open and save it after changes. I don't have any way to fix bugs in VS however, but git should be a bit more capable I hope. How to compute the integral $\int_{0}^{\infty} \frac{x^{1/3}}{1+x^{2}} \ dx$ I want to compute this integral $$\displaystyle\int_{0}^{\infty} \frac{x^{1/3}}{1+x^{2}} \ dx$$  What I did was the following. I substituted $x=t^{6}$, so that my $dx= 6t^{5} \ dt$ and so the integral changes to $$\int_{0}^{\infty} \frac{t^2}{1+t^{12}} \cdot 6 t^{5} \ dt =6 \cdot \int_{0}^{\infty} \frac{t^7}{1+t^{12}} \: dt$$  Now If I substitute $t^{4}=v$ then what I will be having is the following integral $$\frac{6}{4}\cdot \int_{0}^{\infty} \frac{v}{1+v^{3}} \ dv$$  Now I can write $1+v^{3} = (1+v) \cdot (1-v+v^{2})$ and so I have   \begin{align*} \int_{0}^{\infty} \frac{v}{1+v^{3}}\: dv &amp;= \int_{0}^{\infty}\biggl[\frac{1}{3}\cdot \frac{v+1}{1-v+v^{2}} - \frac{1}{3} \cdot \frac{1}{1+v}\biggr]\: dv \end{align*}  Now the point is that the integral of $1/(1+v) \to \infty$, so I am not sure if this is the right way to do. Can anyone suggest anything? After Google Play Service update to version 13 I got an error Reactjs convert html string to jsx In Harry Potter and the Deathly Hallows, when Voldemort tries to kill Harry Potter and Harry Potter travel to the Deathly hallows, that is, Kings Cross Station and he later comes back. There it has been mentioned that Voldemort had also fallen down and Bellatrix tried to support him to stand up as if she has some affection towards him.  “My Lord . . . my Lord . . . ” It was Bellatrix’s voice, and she spoke as if to a lover.  Is it true that Bellatrix Lestrange had some affection towards Voldemort? Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$ Let $E \subset F \subset X$ and $f:E\rightarrow Y$. We say that the function $g:F\rightarrow Y$ is an extension of $f$ if $g(x) = f(x)$ for all $x \in E$.   Let $f: (a, b) \rightarrow \mathbb{R}$. Show that $f$ has a continuous extension to $[a, b]$ if and only if $f$ is uniformly continuous on $(a, b)$.   $(X,d)$ is a metric space.  Hint: To prove $\Leftarrow$ start by showing that $f$ maps Cauchy sequences to Cauchy sequences. Then how should you define $g(a)$ and $g(b)$?  The $\Rightarrow$ part of the proof was quite simple, but even with the hint, the other way seems a bit difficult. Doesn't the first part of the "hint" follow from the way we defined $f$? If so, how do I prove this? Help/Hints appreciated! I gave Google my phone number to verify it, not knowing that it would change it, and now my phone number is completely different? What have I done, and how do I unverify it and change my phone number back? How to make `sudo` preserve $PATH? Fastest way to convert string to integer in PHP Using PHP, what's the fastest way to convert a string like this: "123" to an integer?  Why is that particular method the fastest? What happens if it gets unexpected input, such as "hello" or an array? Getting polygon boundaries of City in JSON from Google Maps API? When searching Google Maps for a city, the city's administrative boundary is displayed.    Is there a way to get the polygon boundaries of City in a JSON format using google maps api? Propose Newbie-Overflow site or section Prove that $f(x)\in \mathbb{Z}[x]$ such that $f(0)$ and $f(1)$ are odd has no integer roots Invertibility of the Product of Two Matrices I have shown that if two $n\times n$ matrices $A$ and $B$ are invertible, then $AB$ is also invertible. I am having problems though with showing that $(AB)^{-1}=B^{-1} A^{-1}$. I thought of using their left-multiplication transformations but it seems that this is only associative and not commutative so I can reverse $L_A L_B$ Why did the limit of power change from wizard/witch to wand? So in the first 5 or 6 books, it was very clear that the wand chooses the wizard and the power/level of the spells the wizard produces is based on the knowledge and skill of the wizard.  This is referenced by people constantly being telling Harry that he wasn't skilled enough to perform certain spells, or Bellatrix telling Harry that he needed to mean the Cruciatus curse for it to be effective.  However, in the 7th book, this entire idea is changed to the wand being the thing that's powerful and regardless of the wizard's/witch's skill or knowledge, he/she could still perform a very advanced spell if it's something the wand could handle.  Goyle accidentally creating a fire serpent is something that instantly comes to mind from the movies(I don't remember if that happened in the book).  So, my question is, what actually holds the power, the wizard/witch, or the wand? On convergence of Bertrand series $\sum\limits_{n=2}^{\infty} \frac{1}{n^{\alpha}\ln^{\beta}(n)}$ where $\alpha, \beta \in \mathbb{R}$ Study the convergence of $$\sum_{n=2}^{\infty}  \frac{1}{n^{\alpha}\ln^{\beta}(n)}$$ where $\alpha, \beta \in  \mathbb{R}$   I have proved that:   This series diverges when $\alpha \leq 0$. This series converges when $\alpha &gt; 1, \beta &gt; 0$ This series diverges when $0 &lt; \alpha &lt; 1, \beta &gt; 0$ This series converges when $\alpha = 1, \beta &gt; 1$      Question: What happens when $\alpha &gt; 0$ and $ \beta &lt; 0$?   There are other questions on MSE which ask about this series, but this question is distinct because   I would like an argument which does not rely on the integral test for series convergence, and this question considers all real $\alpha$ and $\beta$, while other questions ask only about $\alpha, \beta &gt; 0$, where we can apply Cauchy condensation criterion