Does, in a simple graph $G$ with $100$ vertices where $G \cong \bar G$, exist a vertex with a degree of $50$?  So I think that the simple graph $G$ can be isomorphic with $\bar G$ iff $|V(G)| = \frac {C_{100}^2}{2}$, so we can divide $G$ into $2$ parties: the complete graph $K_{50}$ and the edgeless graph, Hence there can't be a vertex with a degree $50$ in $K_{50}$.  Is that right? or I'm wrong? would you help please.
Question Prove that for every integer k≥1, exists a self-complementary graph with 4k vertices half of which are of degree 2k-1 and the other half of degree 2k.  My approach So, I think the easiest way to prove what is asked is by induction. I can prove the base case for k=1, but as far as the inductive step is concerned, I cannot seem to reach a conclusion as to why if the statement holds for n, then it holds for n+1.  Any help would be appreciated.
Fast way to come up with solutions to $x(x-1)(x-2)(x-3)=1$? I can solve this equation $x(x-1)(x-2)(x-3)=1$ using the usual method but I am looking for a fast analytical method to solve this. Any hints ? I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre.  Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know which root was the extra ).  I noticed that $z=a+bi$ and there exists $\theta$ such that the trigonometric representation of $z$ is $\left ( \sqrt{a^2+b^2} \right )\left ( \cos \theta + i \sin \theta \right )$ .  It seems that $-\bar{z} = -\left ( \sqrt{a^2+b^2} \right )\left ( \cos (-\theta) + i \sin (-\theta) \right )$  However, my trig is pretty rusty and I'm not quite sure where to go from here. If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$ If a group $G$ is isomorphic to $H$, prove that ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$.   Can someone provide a step by step solution? Explaining along the way our strategy in proving this. "Each person's car" vs. "each persons' car" Which of the following is correct?        Each person's car has four wheels.   Each persons' car has four wheels. Why do I have the "required reviews completed" status on my submission in Elsevier? I had submitted my paper in an Elsevier journal one year back. Now, status update on the submission tracking says "Required reviews completed." However, for the last two weeks still there is no change in the status.  Should I ask the editor what is going on with my submission? Network reputation graph becomes nervous when clicking quickly, backs away slowly An awful identity We take place on $\mathbb C(x_1,...,x_r,x'_1,...,x'_p,u_0,...,u_r,u'_0,...,u'_p)$ with $r,p\in \mathbb N$     Show that :   $$\displaystyle{\sum_{i=1}^r \left(   \frac{\prod_{j=0}^r (u_j-x_i) \prod_{j=1}^p (x'_j-x_i) }{  \prod_{1\leq j\leq r, j\neq i     }(x_j-x_i) \prod_{j=0}^p (u'_j-x_i)}      \right)}-\displaystyle{\sum_{i=0}^p \left(   \frac{\prod_{j=0}^r (u'_i-u_j) \prod_{j=1}^p (u'_i-x'_j) }{  \prod_{0\leq j\leq p, j\neq i     }(u'_i-u'_j) \prod_{j=1}^r (u'_i-x_j)}      \right)}=\displaystyle{\sum_{i=1}^p x'_i +\sum_{i=0}^r u_i -\sum_{i=1}^r x_i-\sum_{i=0}^p u'_i}$$   I have seen this exercise in an old book at the library. Unfortunately there was no indication how can I solve the problem.   Anyway if someone can give me some ideas it will be greatful.   NB: Sorry for the title, I have not had other ideas Typical software & tools short-list for *manual* QA specialist (tester) working environment For QA workstations preparation (Windows-based PCs) our IT department uses typical batch deployment tools. I'm not at all interested in the details of such process, but IT guys ask me:  What typical set of software you need for everyday manual testing tasks? We could include these in our deployment so that you guys could save more time for testing, not for environment prep &amp; tuning.  Sure thing we do test various range of projects, from web applications to standalone solutions, and each application requires its own set of tools for manual testing, e.g.:  Browsers Virtualization utilities DB clients Screens capturing tools FTP clients XML editors etc.  However, and that's for sure - there's a list of tools every QA MUST have installed on its PC, regardless of project types under testing. My area of experience is mostly standalone / complex web apps, that's why my vision on the task is a bit subjective. So, your suggestions of what are tools / software an average QA specialist can't live without will be highly appreciated. 2 assumptions:  Manual testing is a point of interest. Let's limit the list to not more than 15 (better 10) items.  Thanks in advance! nodes near coords with logarithmic axis: Wrong values I created a horizontal bar plot to compare benchmark times. Since one of the tested components is really slow, I need a logarithmic y axis for the other values to be distinguishable. Combined with "nodes near coords" this results in the calculated values showing, instead of the original ones. Is there any way to fix this?  What it looks like:   What the values(!) should look like:   Clearly, I'm overlooking something terribly obvious. I'm just a beginner at PGFPlots, so please don't kill me. :)  MWE:  \documentclass[fontsize=12pt]{scrartcl}  \usepackage{pgfplots} \pgfplotsset{compat=1.8}  \begin{document}   \begin{tikzpicture}   \begin{axis}[      width=12.5cm,     xbar=0.1cm,     xmin=0,     y=0.6cm,     bar width=0.4cm,     xlabel={Time (ms)},     xmode=log,     major x tick style = transparent,     major y tick style = transparent,     symbolic y coords={       {SpreadsheetGear},       {Aspose.Cells},       {NPOI},       {C1.Excel},       {Spire.XLS},       {GemBox.Spreadsheet},       {SmartXLS},     },     ytick=data,     y dir = reverse,      nodes near coords,     nodes near coords align={horizontal},   ]    \addplot coordinates {       (3078.6,{Aspose.Cells})       (10059.2,{C1.Excel})       (28794.8,{GemBox.Spreadsheet})       (7437079.8,{SmartXLS})       (19975.2,{Spire.XLS})       (573.2,{SpreadsheetGear})       (10025.2,{NPOI})};   \end{axis}   \end{tikzpicture} \end{document} Can I change my itinerary and hotel reservation after getting a Schengen visa? Which annotation should I use: @IdClass or @EmbeddedId The JPA (Java Persistence API) specification has 2 different ways to specify entity composite keys: @IdClass and @EmbeddedId.   I'm using both annotations on my mapped entities, but it turns out to be a big mess to people who aren't very familiar with JPA.   I want to adopt only one way to specify composite keys. Which one is really the best? Why? Where do I place my own .sty or .cls files, to make them available to all my .tex files? In Book 1, Artemis et al escaped from the time stop by drugging themselves to sleep.  In the same time stop, Julius Root "accidentally" plugged Briar Cudgeon with a tranquilizer dart.  Cudgeon fell asleep, but did not disappear from the time stop.  Why not? IBus methods do not show up in text entry settings I installed both Korean and Japanese using the Language Support window, and expected Korean (Hangul) and Japanese (Anthy) to show up in the input sources for Text Entry Settings (having both packages installed). However, neither shows up. How to install software provided in software-center via terminal? Answers in top Network Posts showing incorrect score There is already a question  which is posted around a month ago (which was fixed). My question reports a similar issue of showing inaccurate scores along with another bug.   A couple of answers in the Top Answers in my network profile are showing incorrect score.         As we can see they are showing a score of 10. But as of writing this question, the score of both the answers is 9.  and  both have a score of 9 instead of 10.  Even though the current score is 9, both these answers reached a score of 10 a few months ago and I was awarded  for both.         The vote was removed either when a user was removed or there was an unupvote. As far as I can tell, I don't remember an unupvote notification in the achievement box from these answers. My guess is the votes are removed when user accounts were deleted.   So, I suspect the score of answers in Top Network Posts is not updated either when there is an unupvote or a vote thrown away due to account deletion.   Is this the right reason why score is inaccurate? Can this be fixed? Can I connect copper wire to aluminum wire? How do I move Steam games to a new computer without re-downloading them? I have just bought a new computer and I have everything migrated over except for my Steam games.  I looked in Program Files/Steam/ and I found a folder full of the games I have downloaded, but I do not know if it is safe to just copy these folders from one computer to another (seeing as a lot of other applications will not work if you do this), or is there another recommended way?  I don't really feel like re-downloading all my games. I don't have the time nor bandwidth to download 20GB over a 1.5mbps connection.  Is it possible to contact Valve and get a disk with the games on it? How do you make a deep copy of an object? If someone who is otherwise healthy decided to drink fairly heavily and consistently over a long period of time, what is the healthiest way to do so?  Example: Stay extra hydrated by drinking more water, take a multivitamin, get more sleep, etc.  Obviously the best solution would be to cease drinking, but IF that person is stubborn and won't quit - what's the best way to go about it?