Given a graph $G$ with $n \ge 4$ vertices, where $n$ is even, prove that if every vertex has degree $n/2 + 1$, then $G$ must contain a clique of size $3$.   Recall that a clique of size $3$ is cycle on $3$ vertices. Therefore, you must show that there always exists $3$ vertices $a, b, c$ in $G$ such that $ab$ is an edge, $bc$ is an edge and $ac$ is an edge.
I'm trying to prove : Given a simple graph $G$ with $n$ vertices, where $n$ is even, prove that if every vertex has degree $\dfrac n2 + 1$, then $G$ must contain a (simple) $3$-cycle. A (simple) $3$-cycle is a set of $3$ (distinct) vertices, $a, b, c$ such that $ab$ is an edge, $bc$ is an edge and $ac$ is an edge.  I know there are similar ones already on here, but I can't comprehend any of the answers. This is what I have so far:  Base Case: Since the graph has to have at least $3$ vertices to have a simple $3$-cycle, the lowest possible even number of vertices is $n = 4$.   $\dfrac42 + 1 = 3$ degree (edges attached to node). This means each vertex connects to each of the other $3$ vertices since $G$ is a simple graph. Thus, any combination of $3$ vertices will be a simple $3$-cycle.  Inductive Hypothesis: Assume a simple graph with $k$ vertices, where $k$ is even, contains a simple $3$-cycle if each vertex has degree $\dfrac k2 + 1$.  Inductive Step: Since only concerned about when the graph has an even number of vertices, going to show a simple graph with $k + 2$ vertices, where $k$ is even, contains a simple $3$-cycle if every vertex has degree $\dfrac{k+2}{2} + 1$   This is as far as I get and don't know how to finish the Inductive Step (Also please feel free to let me know if I messed anything up, up to this point)
There is a formal definition for the Big O notation in . Up to now I have come across Big O in Numerical Analysis, Calculus and Algorithms which all are pretty distinct fields. What I am wondering is if that definition is global and is the only one that is used in every field where the Big O is involved or there are other ways that the Big O is defined in other areas  of Mathematics. Why would a spinning space station create a centrifugal force on an astronaut rather than simply spinning around him/her? We often see films with spinning space station that create  by having the astronauts pulled outwards by centrifugal force.  I'd like to know if this would really happen, and if so, why is the following scenario not true:   Take an astronaut in open space. He doesn't move. Put a big open spinning cylinder around him - surely he still doesn't move. Close the cylinder. I still see no reason for him to be pulled outwards. find the number of analytic functions which vanish only on $S$? "Next tag badge" panel score and answers counter discrepancy On the "Next tag badge" panel of  for C# tag, I see:      92/100 score   88/20 answers     But, when I open the "Select your next badge" panel, there's:     97/100 score   90/20 answers       I'm not sure which one is correct. But I believe that both displays should show the same counts.  The discrepancy is not due to a caching, I've noticed it the first time many weeks ago.  For these particular current counts: I have a single C# answer with 3 votes only in the last 5 days, last upvoted 13 hours ago (2015-10-12 19:30:05Z). So it should not cause the 5 votes and 2 answers difference.  I believe that the counters on the "Select your next badge" are correct and the counters on the "Next tag badge" are wrong, as I have:     Asked 1 non-wiki question with a total score of 6. Gave 90 non-wiki answers with a total score of 97.       Today (2015-10-16), the "Next tag badge" got bumped to 97/100. But "Select your next badge" shows 98/100 already. what is the area of the region [ (x,y):0<x , y<1 , 3/4<x+y<3/2 ]? Saving Working Files as PSD or TIFF? I was reading a few articles about how people saved their working files and it got me thinking about my own method. I used to save everything as a PSD, but recently I have started saving my working files as TIFF's. I use Windows, so I don't know if it's different for Apple users but with a PSD, all I can see is that "PSD" icon. But a TIFF file shows up as a normal image (that I can edit), so I found it really great when I have a lot of working files for one project and could just quickly preview it without opening the program.  That being said, I am no expert in which method is better. And I know a lot of people here are pretty seasoned users in all things Adobe, so I was wondering what your two cent on the whole topic is?   Is there really any difference to saving it as a PSD or TIFF? Is there something I should specifically know about saving my files as a TIFF rather than a PSD? What are the basics needed to learn quantum computing? I was very inspired by Michio Kaku's  on the possibilities of quantum computing and also listening to Talia Gershon's  on it.   As I come from a business &amp; analytics background, what materials can I begin exploring to prepare to learn more about the field of quantum computing? Eg. what would I need to study to understand Peter Shor's algorithm,  on physics, etc. Using recordtypes of the same name across multiple SObject types I have one recordtype named 'India' in both Account and Contact Objects.My requirement is whenever i create a new account record of type 'India' a new contact record of type 'India' should be created automatically.  i tried with the following code:  trigger AccRecTypeTrigger on Account (before insert) {     List&lt;RecordType&gt; recType = [select id,name from RecordType where sobjectType = 'Account' and isActive = true];    Map&lt;String,Id&gt; m = new Map&lt;String,Id&gt;();    for(RecordType r : recType){      m.put(r.name,r.id);    }     List&lt;Contact&gt; conlst = new List&lt;Contact&gt;();    for(Account a : trigger.new){        a.recordTypeId = m.get('India');        Contact con = new Contact();        con.accountId = a.id;        con.recordTypeId = a.recordTypeId;        con.recordType.name = a.recordType.name;         conlst.add(con);       }    insert conlst; }   But i am getting following error when i try to save     Apex trigger AccRecTypeTrigger caused an unexpected exception, contact   your administrator: AccRecTypeTrigger: execution of BeforeInsert   caused by: System.NullPointerException: Attempt to de-reference a null   object: Trigger.AccRecTypeTrigger: line 14, column 1   Please help me how to achieve this...... How do you remove the axis multiplier? There is some unwanted extra spaces in between each site in the Your Sites section.  Steps to reproduce:   Touch the Nav Icon in the left top corner Touch the EDIT link in YOUR SITES Scroll the screen; there is unwanted spaces in between the each site   For reference I recorded the same:    Configurations:   App Version: 1.0.85 Device Manufacturer: Motorola Device Model: XT1033 OS Version: 5.1 (2) Appendix in LaTeX How to get the key for a given object from an NSMutableDictionary? Is it correct to say "I write children books" (not possessive case)? Although Children's books is what everybody says, I would like to understand why the genitive case is applied in such case.  If I write books for children, children is an adjective here; not the owners of my book! The word "children" just defines or characterizes the type of books I write. Therefore, it's an adjective.   So, I understand that genitive/possessive case ("I write children's book")  is incorrect grammar.  My question is: is the genitive case here really accepted as right? If I use "I write children books" (following the grammar principle) as as I say "I write pets books" (books about pets, and not possessions of pets) - would I be incorrect? Why? I am a Swedish citizen traveling to Kyrgyzstan via Russia. Do I need a transit visa while waiting for my flight iin 24 hours? This is from  .      Let $\mathcal{A}$ be a locally small category, and let $A,A' \in \mathcal{A}$ with $\operatorname{Hom}(-,A) \cong \operatorname{Hom}(-,A')$. Prove directly that $A \cong A'$.     My thoughts:      Let $\eta$ denote the natural isomoprhism between the two functors.    We have    $$H(A,A) \xrightarrow{ \eta_A} H(A,A')$$   So $\eta_A(id_A): A \rightarrow A'$ is a candidate. Now I want to construct inverse.    $$ H(A',A') \xrightarrow{\eta_{A'}} H(A',A)$$   Then the map $\eta^{-1}_{A'} (id_{A'}):A' \rightarrow A$.    So I'd like to show that the composition is identity. By the naturality condition of $\eta$, from $H_A(A) \rightarrow H_{A'}(A')$ (and similarly on other direction)    $$ \eta_A(id_A) \circ \eta^{-1}_{A'}(id_{A'}) = \eta_{A'}(\eta_{A'}^{-1}(id_{A'}))= id_{A'} $$   We deduce these two maps are inverses. Hence $A' \cong A$.    Is this correct? Or is there a neat way to see this? Are Android OS updates region specific? Well, I have a Galaxy Ace with 2.3.3, build XWKPY and Samsung Kies doesn't show an update to 2.3.4 yet.  I'm thinking of manually updating it through Odin and I need to know if I can just download any Gingerbread 2.3.4 update file or is there anything else that I need to be worried about? Why does a minimal prime ideal consist of zerodivisors? Let $A$ be a commutative ring. Suppose $P \subset A$ is a minimal prime ideal. Then it is a theorem that $P$ consists of zero-divisors.  This can be proved using localization, when $A$ is noetherian: $A_P$ is local artinian, so every element of $PA_P$ is nilpotent.  Hence every element of $P$ is a zero-divisor. (As Matt E has observed, when $A$ is nonnoetherian, one can still use a similar argument: $PA_P$ is the only prime in $A_P$, hence is the radical of $A_P$ by elementary commutative algebra.)  Can this be proved without using localization? My laptop has a function key which turns F keys into keys which change volume or brightness. I would like to swap a number of them so that they perform that function under normal operation and are an Fx key while the function key is being held.  I assumed that the function keys were just normal keys codes with the set of keys being large enough to include all the possible speical keys. I tried to so as this tutorial suggests () and use xev to obtain the key codes and then would use xmodmap to swap their value.  This does not seem to work though. The F12 key gives the xev output:  KeyPress event, serial 41, synthetic NO, window 0x4a00001,     root 0xb1, subw 0x0, time 468005, (23,-15), root:(89,37),     state 0x10, keycode 96 (keysym 0xffc9, F12), same_screen YES,     XLookupString gives 0 bytes:      XmbLookupString gives 0 bytes:      XFilterEvent returns: False   Which I think is to be expected. However, the Increase Volume button give this response:  KeymapNotify event, serial 41, synthetic NO, window 0x0,     keys:  65  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0               0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0    And so does the Decrease Volume key. How can I perform the swapping of the keys that I would like?  Thank you $E|X-m|$ is minimised at $m$=median For a continuous random variable $X$, I want to show that $E|X-m|$ is minimum implies $m$ is the median of the distribution. Assume that the distribution function is $F$ and the density function is $f$.  In this case, we basically need to minimize  $$\int_x |x-m|f(x)\,dx=\int_{x&gt;m}(x-m)f(x)\,dx+\int_{x&lt;m}(m-x)f(x)\,dx.$$  I am finally being left with $2mF(m)-m+E(X)-2\int_{x&lt;m}xf(x)\,dx$. What does it mean to minimise this expression? Is there some way out?  Well, my intuition says that since I have to prove that $F(m)=0.5$, I somehow will need to show that $E|X-m|$ is minimised if $2mF(m)-m=0$ (as then I will get $F(m)=0.5$ I'm  trying to create a FPS game in blender game engine. When I do KeyBoard[Spacebar] sensor and hook it up to a simple motion of z +1 or something like that, the guys pops up really fast as if he didn't have any motion between being on the ground and being in the air. Is there a way to make a more realistic, more slow jump in BGE?