I'm new in discrete math. Can someone prove simply that a tree with $n$ nodes must have exactly $n - 1$ edges. I have researched the solution but I haven't founded yet. I know of course, a tree with n nodes must have exactly n - 1 edges. But, I can't prove it. Thank you. induction on $n$
How many edges does an undirected tree with $n$ nodes have?
How can I improve framerate of Minecraft client? "This edit would have to be a suggested edit" / edit button grayed out A couple of times recently when I attempted to edit a post while reviewing it from the "First Posts" queue, I received the following message when I tried to submit the form:  This edit would have to be a suggested edit. (x)   This is a rather cryptic message, considering that edits made from a review are suggested edits, subject to the same peer review process. There was no indication of how the edit would have to be submitted differently, much less why. I searched both help and meta for more information about this message, but came up empty.  The first time it happened, after I cancelled out I saw that someone else had just edited the post, so I thought maybe that's why I couldn't submit an edit from the review (and wondered why, if that's the case, the error message didn't just say that instead of telling me something that doesn't make any sense - that the edit I'm trying to suggest can't be submitted because it would have to be a suggested edit). However, the second time I received this message (last night, for this question: ), there was no other edit. So, I clicked the No action needed button and tried to edit the post outside the review process, only to discover that the edit link was grayed out.   I see several explanations of possible causes for graying out the edit link here: . However, there was no tooltip when I hovered over the disabled link, and the edit queue was in the single digits.  A few minutes later, I noticed that the edit link was no longer grayed out, so I edited the post and my edit was accepted.   My best hypothesis based on my experience with the error so far and the explanations I've read here at meta is that someone submitted a suggested edit while I was reviewing the post, which (a) caused me to get the cryptic error message and (b) caused the edit link to be grayed out when I accessed the post the normal way. Then a few minutes later, the suggested edit was rejected, opening up the question for new edit suggestions.  Rather than speculating about it and trying to determine the answer scientifically, it would be helpful if someone could give me the answer (as Seinfeld fans might put it, "Why don't you just tell me why I can't edit the post?"). It would be even more helpful if the error message itself provided the explanation, or if the message were defined somewhere in the help pages. But, at the very least, hopefully there will soon be an answer on meta to enlighten people who scratch their chins over this in the future. :) How to express a polar coordinate for a macro that only accepts space-separated values? \psElectricfield only accepts a coordinate in space-separated values as follows. For example, a charge of -0.1 unit at (-1,0) is written as [-0.1 -1 0]. I have attempted to write in PS notation but it still does not work.  \documentclass{article} \pagestyle{empty}  \usepackage{pst-electricfield,multido} \def\M{2}  \begin{document} \multido{\i=0+30}{12}{% \begin{pspicture*}(-\M,-\M)(\M,\M)     \psElectricfield     [         Q={[-0.1 -1 0][-0.1 1 0][0.1 !\i sin 2 mul \i cos 2 mul]}     ] \end{pspicture*}\newpage} \end{document}   How to express  a polar coordinate for a macro that only accepts space-separated values? Alternative IDE for VB6 and VBA I've been spoiled by Visual studio 2008 and Eclipse and have to do a little maintainence work on a VB6 app.  Does anyone know of an alternative/ updated IDE for VB6?  A rewrite is not an option I'm just fixing a couple of bugs and it's a big codebase. Shortest code to calculate list min/max in .NET I'd like something like  int minIndex = list.FindMin(delegate (MyClass a, MyClass b) {returns a.CompareTo(b);});   Is there a builtin way to do this in .NET? Russian babel and chemformula How do I install .run files? Should the mean be used when data are skewed? Often introductory applied statistics texts distinguish the mean from the median (often in the the context of descriptive statistics and motivating the summarization of central tendency using the mean, median and mode) by explaining that the mean is sensitive to outliers in sample data and/or to skewed population distributions, and this is used as a justification for an assertion that the median is to be preferred when the data are not symmetrical.  For example:     The best measure of central tendency for a given set of data often depends on the way in which the values are distributed.... When data are not symmetric, the median is often the best measure of central tendency. Because the mean is sensitive to extreme observations, it is pulled in the direction of the outlying data values, and as a result might end up excessively inflated or excessively deflated."   —Pagano and Gauvreau, (2000) Principles of Biostatistics, 2nd ed. (P&amp;G were at hand, BTW, not singling them out per se.)   The authors define "central tendency" thus: "The most commonly investigated characteristic of a set of data is its center, or the point about which observations tend to cluster."  This strikes me as a less-than forthright way of saying only use the median, period, because only using the mean when the data/distributions are symmetrical is the same thing as saying only use the mean when it equals the median. Edit: whuber rightly points out that I am conflating robust measures of central tendency with the median. So it is important to keep in mind that I am discussing the specific framing of the arithmetic mean versus the median in introductory applied statistics (where, mode aside, other measures of central tendency are not motivated).  Rather than judging the utility of the mean by how much it departs from the behavior of the median, ought we not simply understand these as two different measures of centrality? In other words being sensitive to skewness is a feature of the mean. One could just as validly argue "well the median is no good because it is largely insensitive to skewness, so only use it when it equals the mean."  (The mode is quite sensibly not getting involved with this question.) I was setting up up android-studio referring to . While installing oracle-java7-installer, this error is displayed:  sudo apt-get install oracle-java7-installer [sudo] password for cyan:  Reading package lists... Done Building dependency tree        Reading state information... Done oracle-java7-installer is already the newest version (7u80+7u60arm-0~webupd8~1). The following package was automatically installed and is no longer required:   snap-confine Use 'sudo apt autoremove' to remove it. 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. 1 not fully installed or removed. After this operation, 0 B of additional disk space will be used. Do you want to continue? [Y/n] y Setting up oracle-java7-installer (7u80+7u60arm-0~webupd8~1) ... Downloading Oracle Java 7... --2017-11-27 08:40:47--  http://download.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz Resolving download.oracle.com (download.oracle.com)... 223.223.158.138, 223.223.158.136 Connecting to download.oracle.com (download.oracle.com)|223.223.158.138|:80... connected. HTTP request sent, awaiting response... 302 Moved Temporarily Location: https://edelivery.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz [following] --2017-11-27 08:40:47--  https://edelivery.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz Resolving edelivery.oracle.com (edelivery.oracle.com)... 23.58.58.93, 2600:140f:7:18e::2d3e, 2600:140f:7:18d::2d3e Connecting to edelivery.oracle.com (edelivery.oracle.com)|23.58.58.93|:443... connected. HTTP request sent, awaiting response... 302 Moved Temporarily Location: http://download.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz?AuthParam=1511752368_824528560c20825821ecbcac8f4e9be9 [following] --2017-11-27 08:40:48--  http://download.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz?AuthParam=1511752368_824528560c20825821ecbcac8f4e9be9 Connecting to download.oracle.com (download.oracle.com)|223.223.158.138|:80... connected. HTTP request sent, awaiting response... 404 Not Found 2017-11-27 08:40:50 ERROR 404: Not Found.  download failed Oracle JDK 7 is NOT installed. dpkg: error processing package oracle-java7-installer (--configure):  subprocess installed post-installation script returned error exit status 1 Errors were encountered while processing:  oracle-java7-installer E: Sub-process /usr/bin/dpkg returned an error code (1)   How do I fix this error? I did a fresh installation of Ubuntu 16.04. How can I detect inside a TeX file which engine (pdftex/xetex/luatex) is used to compile the file? Is there a LaTeX package that provides this functionality? Let $R$ be a commutative ring with 1 then why does $a\in N(R) \Rightarrow 1+a\in U(R)$? Let $R$ be a commutative ring with 1, we define  $$N(R):=\{ a\in R \mid  \exists k\in \mathbb{N}:a^k=0\}$$  and  $$U(R):=\{ a\in R \mid a\mbox{ is invertible} \}.$$  Could anyone help me prove that if $a\in N(R) \Rightarrow 1+a\in U(R)$?  I've been trying to construst a $b$ such that $ab=1$ rather than doing it by contradiction, as I don't see how you could go about doing that. Integrating $\int \sin^n{x} \ dx$ I am working on trying to solve this problem:     Prove: $\int \sin^n{x} \ dx = -\frac{1}{n} \cos{x} \cdot \sin^{n - 1}{x} + \frac{n - 1}{n} \int \sin^{n - 2}{x} \ dx$   Here are the steps that I follow in the example that I am reading:     $u = \sin^{n - 1}{x}$      $du = (n - 1) \cdot \sin^{n - 2}{x} \cdot \cos{x} \ dx$      $v = -\cos{x}$      $dv = \sin{x} \ dx$            $\int \sin^n{x} \ dx =  \sin^{n - 1}{x} \cdot \sin{x} \ dx$      $\int \sin^n{x} \ dx = \underbrace{\sin^{n - 1}{x}}_{u} \cdot \underbrace{-\cos{x}}_{v} - \int \underbrace{-\cos{x}}_{v} \cdot \underbrace{(n - 1) \cdot \sin^{n - 2}{x} \cdot \cos{x} \ dx}_{du}$      $\int \sin^n{x} \ dx = -\cos{x} \cdot \sin^{n - 1}{x} + (n - 1)\int \sin^{n - 2}{x} \cdot \cos^{2}{x} \ dx$      $\int \sin^n{x} \ dx = -\cos{x} \cdot \sin^{n - 1}{x} + (n - 1)\int \sin^{n - 2}{x} \cdot \left(1 - \sin^{2}{x}\right) \ dx$   Here is where I get lost. How did we go from $\int \sin^{n - 2}{x} \cdot \left(1 - \sin^{2}{x}\right) \ dx$ to $\int \sin^{n - 2}{x} \ dx - (n - 1) \int \sin^{n}{x} \ dx$? Even more specifically, where did $\sin^{n}{x}$ come from?     $\int \sin^n{x} \ dx = -\cos{x} \cdot \sin^{n - 1}{x} + (n - 1)\int \sin^{n - 2}{x} \ dx - (n - 1) \int \sin^{n}{x} \ dx$   I get this part.     $n\int \sin^n{x} \ dx = -\cos{x} \cdot \sin^{n - 1}{x} + (n - 1)\int \sin^{n - 2}{x} \ dx$      $\int \sin^n{x} \ dx = -\frac{1}{n} \cos{x} \cdot x \ \sin^{n - 1}{x} + \frac{n - 1}{n} \int \sin^{n - 2}{x} \ dx$   Could someone please explain what I am missing?  Thank you for your time. How can I determine whether a 2D Point is within a Polygon? Can a legendary creature use a legendary resistance to pass a concentration check? As I was looking through some legendary creatures, I had a thought. Most legendary creatures get three Legendary Resistances per day, which they can use to automatically succeed on a saving throw.  A Concentration check is technically a Constitution saving throw, so does that mean that a creature can use a Legendary Resistance to automatically pass a Concentration check, or is it not considered a saving throw? Question involving the Cauchy-Goursat Theorem Show $\int_C \frac{dz}{(z-z_0)^n} = 2\pi i$ when $n = 1$ and 0 when $n \gt 1$   I'm not entirely sure how to approach this one.  Any help would be much appreciated. Rejecting a PhD offer after having accepted it. Advice? I have accepted a PhD offer from a very good university to do a project very close to my interests. I have told the project supervisor I accept it. In the meanwhile I have managed to secure a PhD offer from another 2 universities much better in world rankings and with better research groups and better faculty. Additionally these 2 universities do research exactly on what I want to devote my PhD research.  My only commitment to the first university is a mail confirming I accept. Nothing else. I understand declining the offer after already having accepted it is quite dishonest but it is a huge decision that will affect my future in a very significant level.   I would like your opinion and I would also like to ask for possible consequences of such an action. As far a postdocs concerned, in the first university there is no chance I ever get a postdoc since they do not actually do research on my area (except for the prospective supervisor and even he not as much).  UPDATE I have decided to do as I was afraid to. I have decided to choose a different university than the first one that I have already accepted the offer. My offer acceptance included:    An online "I accept" statement. An e-mail I sent stating "I accept the scholarship and this University is my first option"   I do not have any legal commitments since I have not signed anything. The supervisor of the first university though is really nice to me and continuously was sending me mails. How do I very nicely tell her I will not be joining in the end? Additionally the place I will go in the end will be for research in Mathematics and not Physics. Can I use this as a part of a sensical excuse?  WHAT I HAVE CHOSEN AND WHAT THE CONSEQUENCES WERE  I think that I should let people, especially other beginning grad students know what happened with my case. I have chosen to nicely tell the first university I got accepted that I will not be attending and will not be pursuing a PhD there since I decided I have different research interests (which is true). I had applied there in the first place since I was not sure what I wanted to do and because it was a nice program. Despite that, I got an offer from a World top university to work on 100% my research interests. I explained this to them, the first uni, and they happily let me go and wished me good luck, especially the supervisor who faught a lot secure me funding. I was lucky. Radical change of favourite tags doesn't reflect in questions list Evaluate the following integral:  $$\int \frac{\sqrt{\sin ^4x+\cos ^4x}}{\sin ^3x \cos x }dx$$ where $x \in \big(0,\frac{\pi}{2} \big)$  Could some give me hint as how to approach this question?  I tried to use the fact that $\sin ^4x+\cos ^4x=1-\frac{sin^22x}{2}$ but it didn't help. How should I proceed? I have an ec2 instance on AWS which already installed a few Wordpress-es in it.  I want to set things up to let more people handle their own projects, with me also being able to go into the server using terminal if I still want to.  I have been following tutorials such as  and a few others.  I figured all of them just teach users how to sftp into their own home folder, also the steps keep saying addUser but some users already existed and removing them might cause some issue.  How can I give existing users the permission to use SFTP for Wordpress projects?  Let's say all my Wordpress projects are already under /var/www/ which has already been set up with www-data:www-data.  Really new with setting things like this.  Thanks in advance for any help. Is it (always) true that $$\mathrm{Var}\left(\sum\limits_{i=1}^m{X_i}\right) = \sum\limits_{i=1}^m{\mathrm{Var}(X_i)} \&gt;?$$