Given a graph $G$ with $n$ vertices and $k$ connected components, show that the maximum number of edges is ${n - k + 1 \choose 2} $ I am trying to prove the question I wrote in the title. I realized that the maximum number of edges is the maximal value of $ {a_1\choose 2} + ... + {a_k \choose 2}$ such that $a_i \geq 1$ for all $i$ and $a_1+...+a_k = n$.  But now I don't know how to prove that the value of this is the needed one. I know it is reached when all of the $a_i$'s are $1$ except one which is $n-k+1$ but I don't know how to prove this is the maximal case. I have tried induction but it just got to messy.
Graph with $k$ components. Let $G$ be a simple graph on $n$ vertices. If $G$ has $k$ components and every component is complete prove that: $$|E| \le \frac{(n-k)(n-k+1)}{2}$$ I'm asking for advice.
Is a business card necessary for a graduate student? I know some PhD students have their own business cards, but I don't know if it is a standard practice for PhD students to have their own business cards. I think a business card can be useful for networking in conference or collaborating with some industrial partners. So, is are business cards necessary for a PhD student or other graduate students attending conferences?   PS: in case it is field-specific, I would like to know the different practices in different fields I have an area which is divided into several zones the borders of which I have defined as separate layers.  The layers are defined by sql queries on a postgis table of line segments joined with a table that says which segments are part of which zone boundaries.  What I want to do is produce a set of maps where each map is a zoom to layer extent for each zone and which has standard header, legend and scale (which will preferably vary with the zone but could be fixed across all maps).  My first attempt was to get a new composer for each map -- that worked but was rather tedious.  Next I tried producing one map and printing it and then deleting the map frame and going into qgis to zoom to the next layer extent and make the boundary visible then hopping back into the composer to load it in.  This worked more or less.  Map was fine, legend was OK because it was the same for all the maps and the zone name was easy to edit but the scale needed to be deleted and set up again for each map.  Next I looked at Atlas Generation.  The manual seems to assume that you know how it works and all you need is a reminder about the details (like so many software manuals ;)  It seems to assume that you have something that you are iterating over but I can't see how to make it work -- there is nothing to iterate over to get the layers.  Any suggestions welcome!!  :)  Russell I'm code reviewing a change one of my co-workers just did, and he added a bunch of calls to Date.toMonth(), Date.toYear() and other deprecated Date methods.  All these methods were deprecated in JDK 1.1, but he insists that it's ok to use them because they haven't gone away yet (we're using JDK 1.5) and I'm saying they might go away any day now and he should use Calendar methods.  Has Sun/Oracle actually said when these things are going away, or does @deprecated just mean you lose style points? Calculate $E(X^{2n})$ where $X$ is normal (0,1) I need help proving the following:  Let $X$ be normally distributed with parameters $\sigma=0$ and $\mu=1$. Let $n$ be a positive integer. Show that:  $$E(X^{2n})=\frac{(2n)!}{2^nn!}=:(2n-1)!!$$  I've tried the change of variables $Y=X^2$ and then calculating the integral, but got nowhere. Any hints?  Thanks in advance! Bounding lines around tax nodes I am trying to draw lines around a number of nodes. I very much like the definition of \convexpath definition given , but it is not working well on my setup (MacTex 2014 with all available updates).   I tried to compile , but what I get is the following, which is clearly not what I should be getting. Any ideas would be much appreciated. Why do "df" and "du" commands show different disk usage? I cannot access ubuntu system on my pc, the error message: "the system is running in low graphics mode" and I tried some commands I searched from internet.   I found out a problems seems there is no available disk space. I used "df" and "du" commands to check, the results are as follow:  du -j --max-depth=1 23G  ./home 3.3G ./usr  ......  28G   and  df -Th  filesystem   Type   size    used   available   use%    mounted on  /dev/sda5    ext4   68G     68G    0           100%    /   how would I clean up the system for more disk space? Lightning force:createRecord capture redirect after save Is there a way to use force:createRecord but capture the redirect to the newly created record page that occurs after saving the record? Finally, prevent the redirect from completing?  createRecord : function (component) {     var createRecordEvent = $A.get("e.force:createRecord");     createRecordEvent.setParams({         "entityApiName": "Event__c"     });     createRecordEvent.fire(); } $$\tan(\theta_1+\theta_2+\cdots+\theta_n)=\frac{S_1-S_3+S_5+\cdots}{1-S_2+S_4+\cdots}$$  where  $S_i$ denotes the sum of product of tangent of angles taken $i$ at a time.     For example,$$\tan(\theta_1+\theta_2)=\frac{S_1}{1-S_2}=\frac{\tan\theta_1+\tan\theta_2} {1-\tan\theta_1\tan\theta_2}$$   (This formula is given  in my textbook with no derivation or background)  How to derive this? In how many ways can we choose k items from n distinct items put in a circle.   Then further extension of the question, if I want to find out the number of ways to choose k items from n distinct items in a circle such that no 2 are together.  Thanks Examples of provably${}^n$ unprovable statements Given any statement $A$ and a classical theory $T$ which we assume is at least as strong as Peano Arithmetic ($\sf PA$), we have that $T\vdash A$ implies $T\vdash T\vdash A$ (that is, if a statement is provable, then it is provably provable). Let $T_1=T$ and $T_{n+1}=T_n+{\sf Con}(T_n)$. (We need stronger theories if we want to prove unprovability.) Given this, simply by considering the truth values of the relevant statements, we get the following classification of statements by unprovability:   $T\vdash A$ ($A$ is provable) $T_2\vdash T\not\vdash A$ ($A$ is provably unprovable) $T_3\vdash T_2\not\vdash T\not\vdash A$ ($A$ is provably unprovably unprovable)   &nbsp;&nbsp;&nbsp;&nbsp;...  &nbsp;&nbsp;&nbsp;&nbsp;$\omega$. For all $n$, $T_n\not\vdash\dots \not\vdash T_2\not\vdash T\not\vdash A$ (we can't say anything about $A$'s provability)  We can refine this hierarchy by also considering the status of $\lnot A$. Let's call a class $m,n$ statement one such that $A$ is at level $m$ above and $\lnot A$ is level $n$ in the list. Since $T\vdash\lnot A$ implies $T_2\vdash T\not\vdash A$, the only nonempty classes at level $1$ are the class $1,2$ statements (provable statements) and the class $2,1$ statements (refutable statements). Independent statements are in class $2,2$. The class $1,1$ is nonempty iff the theory is inconsistent.  My question is: Are there any (preferably "natural") statements that fall further into this hierarchy? I know many examples of independent statements, like the continuum hypothesis or the axiom of choice relative to $\sf ZF$, or the Paris–Harrington theorem relative to $\sf PA$. But I know no examples of even any class $2,3$ statements. I expect there may also be natural examples of $\omega,\omega$ statements. My guess is that all the classes $m,n$ with $m,n\ne1$ are nonempty, but examples elude me. When are attributive nouns plural? Sorry for the title, it is not very evident and intuitive but I really do not how to tell it better...  Well, you know, several times, or better, many times, we use this form:  If I want to say:      "development of special weapons was   the first point in Hitler's   program..."   I will say this (a better form):      "special weapon development was..."   OK...  is it      "special weapon development"   or      "special weapons development" (note   the plural...)   what's the grammar rule in order to understand how to use this very used form? LWC Navigation: "this[f.NavigationMixin.Navigate] is not a function" When attempting to use the Lightning navigation service in my component's base class (.js), I get the following error when functionally testing in the UI:  this[f.NavigationMixin.Navigate] is not a function  I am calling the NavigationMixin function provided by the lightning/navigation module. I have imported the module: import { NavigationMixin } from 'lightning/navigation';  And am calling the function as below: this[NavigationMixin.Navigate]({     type: 'standard__recordPage',     attributes: {         recordId: this.org.Id,         objectApiName: 'Account',         actionName: 'view'     } }); How to use a C++ string in a structure when malloc()-ing the same structure? Firefox browser opens but URL is not launched Fermat primes relation to $2^n+1$ If $2^n+1$ is prime how to show that  $n$ must be a power of $2$  Being at elementary level I am at a loss what to begin with? I found this very amusing comic on the internet the other day:   The last frame seems to claim that the harmonic series converges if you throw out all the terms with a $9$ in the denominator. Is this true? Where could I find a proof? Is it only true for terms with a $9$ or is it true for any number? If it's only true for $9$, why is that? GLR Test for 2 samples from exponential distributions Is it possible to transfer Minecraft PE worlds to Minecraft PC? How do I automatically remap buttons on my mouse at startup? Finding first and last index of some value in a list in Python