Prove $B(x,\delta)$ is open.   What this question is asking me to prove...? I don't understand nor have a clue to approach the question...
Prove that for any $x_0 \in X$ and any $r&gt;0$, the open ball $B_r(x_o)$ is open.   My attempt: Let $y\in B_r(x_0)$. By definition, $d(y,x_0)&lt;r$. I want to show there exists an $r_1\in\mathbb{R^+}$ s.t. $B_{r_{1}}(y)\subseteq B_r(x_0)$. Let $a\in B_{r_{1}}(y)$. Then, $d(a,y)&lt;r_1$. For $a\in B_{r}(x_0)$, $d(a,x_0)&lt;r$. I want to show $d(a,y)&lt;r_1$ implies $d(a,x_0)&lt;r$. By triangle inequality, $d(a,y)\leq d(a,x_0) + d(y,x_0) \rightarrow$  $d(a,y)&lt;r_1\leq d(a,x_0)+d(y,x_0)&lt;2r...$  I'm a little stuck after this point.
I have recently submitted a paper to arXiv, but the arXiv moderators put my paper in status "on hold", because they have to make a decision on the right classification for the paper.   To those who have experienced a similar situation, how long did it take before they made a decision and published the paper on the arXiv? I just want to have an idea of how long I will have to wait. What are the differences between proofs based on simulation and proofs based on games? It has been a while since I read the Lord of the Rings novels (don't have a copy at the moment) but I seem to recall he went to Hobbiton at least once, as the young hobbits were always excited to see him.   Did he travel there often? download.01.org certificate verification failed - CRLfile missing? So I had intel-Linux-graphics-installer on my computer  but lately when I   sudo apt-get update   I get the following error  Err https://download.01.org trusty/main amd64 Packages                          server certificate verification failed.  CAfile: /etc/ssl/certs/ca-certificates.crt CRLfile: none   Tried reinstalling intel-linux-graphics-installer but nothing happend. And various solutions found here on askubuntu.com on certificates problems but didnt end up anywhere.  Any ideas on how to get that CRLfile? Also is this problem on my side or on the servers side? How to show $\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$? I am able to evaluate the limit $$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$$ for a given $n$ using l'Hôspital's (Bernoulli's) rule.  The problem is I don't quite like the solution, as it depends on such a heavy weaponry. A limit this simple, should easily be evaluable using some clever idea. Here is a list of what I tried:   Substitute $y = x - 1$. This leads nowhere, I think. Find the Taylor polynomial. Makes no sense, it is a polynomial. Divide by major term. Dividing by $x$ got me nowhere. Find the value $f(x)$ at $x = 1$ directly. I cannot as the function is not defined at $x = 1$. Simplify the expression. I do not see how I could. Using l'Hôspital's (Bernoulli's) rule. Works, but I do not quite like it.   If somebody sees a simple way, please do let me know.    Added later: The approach proposed by Sami Ben Romdhane is universal as asmeurer pointed out. Examples of another limits that can be easily solved this way:   $\lim_{x \to 0} \frac{\sqrt[m]{1 + ax} - \sqrt[n]{1 + bx}}{x}$ where $m, n \in \mathbb{N}$ and $a, b \in \mathbb{R}$ are given, or $\lim_{x \to 0} \frac{\arctan(1 + x) - \arctan(1 - x)}{x}$.   It sems that all limits in the form $\lim_{x \to a} \frac{f(x)}{x - a}$ where $a \in \mathbb{R}$, $f(a) = 0$ and for which $\exists f'(a)$, can be evaluated this way, which is as fast as finding $f'$ and calculating $f'(a)$.  This adds a very useful tool into my calculus toolbox: Some limits can be evaluated easily using derivatives if one looks for $f(a) = 0$, without the l'Hôspital's rule. I have not seen this in widespread use; I propose we call this Sami's rule :). LaTeX Error: Option clash for package xcolor I have an option clash between package xcolor and another package, but I don't know which. Here is my MWE  \documentclass[10pt]{beamer} \setbeamertemplate{navigation symbols}{} \usepackage{caption} \usepackage{subcaption} \usepackage{url} \usepackage{color} \usepackage[usenames]{xcolor}  \author{The author} \title{Package conflicts} \date{October 3rd, 2012}  \begin{document}  This is supposed to be \textcolor{olivegreen}{olivegreen} \end{document}   And the error message is :  This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./error_xcolor.tex LaTeX2e &lt;2009/09/24&gt; &lt;blablabla&gt; ! LaTeX Error: Option clash for package xcolor.   I hope this is enough log for you guys. At this time of the day, I'm hoping for America to save my french ass ... again : ) Generating a DEM and DSM from correct LiDAR point classification I have downloaded classified LAS data from    I am trying to generate accurate DEM and DSM for further analysis. I created a LAS dataset with a projected CRS of  NAD_1983_UTM_Zone_19N in meters and the z CRS in meters as well. My question is on which classifications do I choose for the DEM and DSM. Here is the filter options I have in my dataset:     According to unh LiDAR data report class 2 is ground.     What I have done so far:  DEM I chose the class 2 points and used the las dataset to raster.   DSM In the predefined settings I chose first return and used the las dataset to raster tool.   Is this an accurate way of generating these two rasters? Do I not have to take into account the the unassigned class 1, noise class 7, reserved 11, reserved 17, reserved 18?  Additionally (I can ask this as a separate question if it gets requested to), when using the las dataset to raster tool the sampling value is defaulted to 10. I would like to change it to 2 or 3 (would be meters) to match the LiDAR resolution. I also plan on using the generated DEM and DSM for a least cost path analysis and I want them to be higher resolution. Will making the sampling_value 2 or 3 throw off the results or should I leave it at 10? A classic exponential inequality: $x^y+y^x>1$ This is a classic problem, but I haven't seen it on the site before.     Suppose $x$ and $y$ are real numbers that both lie in the interval   $(0,1)$. Prove  $$x^y+y^x&gt;1.$$   The following spoiler box contains a hint that indicates the solution I know. Does anyone know a more natural solution?       Use Bernoulli's inequality to show $$x^y\ge \frac{x}{x+y}.$$   Edit: I recently learned the source is the 1996 French math olympiad. How do I stop my 2013 Macbook Air from putting the computer to sleep? Authenticate to Stack Exchange using GitHub I've seen sign in methods using  as an authentication provider, for example : (AWS Cloud9)        Is it possible to authenticate into Stack Exchange using this method? At the moment there's other methods such as OpenID, Google, Yahoo and Facebook, but it would be interesting to also have the GitHub authentication. How does the size of insert affects the rate of Homologous Recombination in yeast? When performing genetic knockouts in yeast using homologous recombination to replace a target gene sequence via a vector DNA, does the region between the flanking regions in the vector have to be the same length as the site of recombination in the host cell's gene? How can I produce a 'ring (or wheel) chart' like that on page 88 of the PGF manual? I really like the 'ring chart' or 'wheel chart' given as an example on page 88 of the PGF manual. It is like a pie chart, but with a label in the centre. Unfortunately, the manual doesn't give the source for that chart. Does anyone have an example? Thanks. What class of Devil Fruit does the Gomu Gomu no Mi come under? Considering the three types of Devil Fruits that has been explained are 1. Paramecia Type - where the user can get super human abilities and change their body parts into weapons, etc. 2. Zoan type - where the use can turn into a particular animal and inherit its abilities 3. Logia Type - where the user's body inherits a particular element and the body becomes like air which cannot be touched or damaged by non Haki users.  So what exactly is the type of Gomu Gomu no Mi? It cannot be Zoan or Logia. It's very close to a Paramecia but still doesn't have a logical explanation to be of Paramecia type! So is it some special type of Devil Fruit? CMake & CTest : make test doesn't build tests Two trains travel on the same track towards each other, each going at a speed of 50 kph. They start out 100km apart. A fly starts at the front of one train and flies at 75 kph to the front of the other; when it gets there, it turns around and flies back towards the first. It continues flying back and forth till the two trains meet and it gets squashed. How far did the fly travel before it got squashed?  The question is relatively simple - the trains take an hour to collide, and in an hour the fly would travel 75 km. However, the question got me wondering: Lets say you are given a time $t$ in hours(where $t&lt; 1$) How many &quot;rounds&quot; would the fly make in $t$ time? A round  is defined as moving from one train to another and back. How would I model the equation? SharePoint Migration 2007 to 2013 How to remove items from a list while iterating? How can I apply a -p0 patch from any working directory? Did they decide to destroy it simply because they were Wise and knew it was dangerous to wield? Did they create the Fellowship simply because they were heroic; and decided to face Sauron after the Ring was destroyed?  Or did they know that destroying the ring would defeat Sauron, and they wouldn't actually have to do anything once that is taken care of? Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field I'm reading Galois Theory by Steven H. Weintraub (second edition), and finding that I'm at least somewhat short on the prerequisites. However the following proof looks wrong to me - am I misunderstanding something, or is it actually an incorrect proof?  Lemma 2.2.3. Let $F$ be a field and $R$ an integral domain that is a finite-dimensional $F$-vector space. Then $R$ is a field. Proof. We need to show that any nonzero $r \in R$ has an inverse. Consider $\{1, r, r^2, \cdots\}$. This is an infinite set of elements of $R$, and by hypothesis $R$ is finite dimensional as an $F$-vector space, so this set is linearly dependent. Hence $\sum_{i=0}^n{c_i r^i} = 0$ for some $n$ and some $c_i \in F$ not all zero.  It then goes on to show, given the above, that we can derive an inverse for $r$. However, if I consider examples like $r = 2 \in Q[\sqrt{2}]$, $r = \sqrt{2} \in Q[\sqrt{2}]$ or $r = 2 \in Q[X]/{&lt;X^2&gt;}$, the set $\{1, r, r^2, ...\}$ doesn't look linearly dependent to me. I do believe the lemma is true (and might even be able to prove it), but this does not look like a correct proof to me. Am I missing something? [Edit] Well yes, I am. Somehow I had managed to discount the possibility of any $c_i$ being negative, despite repeatedly looking at each fragment of the quoted text in an attempt to find what I might be misunderstanding.