The disjoint cliques problem for complete graphs.
Difficult variation of the committee problem
Rotating hatch labels based on orientation changes in Data Driven Pages Strip maps? I've created a strip map for a linear feature and have it loaded as a data driven page.  I want to show stationing for the line.  The stationing displays properly on the first data driven page but on each subsequent page the stationing labels get messed up as the map orientation changes. I want the stationing labels perpendicular to the linear feature as shown in the first image.  The tick marks seem to rotate and stay perpendicular to the line but the labels won't.  How can I fix this? The wavefunction $\Psi(x,t)$ for a free particle is given by  $$\Psi(x,t) = A e^{i(kx-\frac{\hbar k}{2m}t)}$$  This wavefunction is non-normalisable. Does this mean that free particles do not exist in nature?  Why then do we use free particles $\psi(\vec{x}) = e^{ikz}$, for example, in scattering theory? Problem with "scopes" of variables in try catch blocks in Java The recent user profile tab name change - from "activities" to "all actions" - was not accompanied by a hyperlink update from ...?tab=activities... to ...?tab=allactions...    Click on the "posts edited" link to see the 404-error. Possible Duplicate:          I got two answers for the same question from different users and both of them correct and working perfectly, but I am unable to accept both of them. Why? There may be some questions that can be answered in several ways, but I can't here in Stack Overflow. Why? Is there anyway to sync Outlook's calendar with Google's calendar? I've tried  before but it doesn't work properly. I'm using Windows 7 and Outlook 2010.  The thing is, it's really easy to check my Google Calendar on my phone since I have an Android terminal, but on my computer I prefer just opening Outlook instead of opening a web browser and logging into Gmail. How to declare a variable inside bash -c I can't figure out how to declare a variable inside bash -c command.   bash -c "var=3; echo $var" grub issues message at boot - "error - can't find command hwmatch" This question  is exactly what I'm asking but was closed as non reproducible. Mine is very reproducible.  Ubuntu 18.04 freshly installed on a 64-bit UEFI machine. Default grub boot options work fine, but I wanted to add a brief countdown (my normal config).  After editing /etc/default/grub as normal and running sudo update-grub, the countdown works, but I get the error shown in the question title.  I tried the solution (copying files) in the closed question but it didn't seem to fit my configuration. Frankly I found the answer confusing.  I also tried editing /etc/grub.d/10_linux to comment out the IF-THEN structure that uses hwmatch. After that I did a sudo update-grub but I still got the error.  How can I get my countdown and not see this error?  Edit per comments -  Stock grub as installed (all non-commented-out lines shown)  GRUB_DEFAULT=0 GRUB_TIMEOUT_STYLE=hidden GRUB_TIMEOUT=0 GRUB_DISTRIBUTOR=`lsb_release -i -s 2&gt; /dev/null || echo Debian` GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" GRUB_CMDLINE_LINUX=""   Edited grub to give countdown (all non-commented-out lines shown)  GRUB_DEFAULT=0 GRUB_TIMEOUT_STYLE=countdown GRUB_TIMEOUT=5 GRUB_DISTRIBUTOR=`lsb_release -i -s 2&gt; /dev/null || echo Debian` GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" GRUB_CMDLINE_LINUX="" How to put legend below the chart? I have the following code that produces a line chart:  \begin{figure}[p]     \centering \begin{tikzpicture}     \begin{axis}[     use units,     cycle list name=exotic, %This defines the color. It produces nicer colors         width=15cm, height=8cm,     % size of the image         grid = major,         grid style={dashed, gray!30},         %xmode=log,log basis x=10,         %ymode=log,log basis y=10,         xmin=0,     % start the diagram at this x-coordinate         ymin=0,     % start the diagram at this y-coordinate         %/pgfplots/xtick={0,5,...,60}, % make steps of length 5         %extra x ticks={23},         %extra y ticks={0.507297},         axis background/.style={fill=white},         ylabel=Time,         xlabel=Number of Datapoints (n),         y unit=s,y unit prefix=m,         tick align=outside]        % import the correct data from a CSV file       \addplot table [col sep=comma,trim cells=true,y=Best Case (Ascending)] {datasets/time_mergesort.txt};           \addplot table [col sep=comma,trim cells=true,y=Average Case (Random)] {datasets/time_mergesort.txt};                     \addplot table [col sep=comma,trim cells=true,y=Worst Case (Descending)] {datasets/time_mergesort.txt};                      \legend{Ascending,Random,Descending}     \end{axis}  \end{tikzpicture}     \caption{Mergesort Run Time for Sorting Random Data}     \label{fig:merge_sort_from_csv} \end{figure}   And at the moment the legend appears on the chart on the upper right corner. I need to move that below the chart, or even to leave that on the chart but to move that on the upper left corner.   I have tried the solution mentioned at , but it did not work for me.  How can I move the axis in a different position? Prove that a non-empty closed convex $S$ has an extreme point iff it does not contain any line I am wondering how to prove the following theorem: Let $S \subset \mathbf{R}^{n}$ be a non-empty closed convex set. Then $S$ has at least one extreme point iff $S$ does not contain any line. Extreme point $x \in S$ means that there is no couple of points $y, z \in S$ and $\lambda \in (0;1)$ such that $x = \lambda y + (1-\lambda)z$. I know that every non-empty closed convex set can be represented as an intersection of all closed half-spaces containing the set but I do not know if it is helpful in proving the theorem. I'm struggling with some concepts in my algebra class. I've chosen these two completely unrelated examples that demonstrate my struggling topics (characteristics and function rings) the two are the following:  $1$) Let $X$ be a set, and let $S$ be a ring. Give a proof that the left distributive law holds for $(S^X, +, *)$.  $2$) Suppose $Y^3 = Y$ for each element $Y$ of a unital ring $R$. Show that $R$ has a finite characteristic d that is a divisor of $6$.  Any example solutions for these proofs would be helpful. Thank you. Tags on badge question list I'm trying to get the Weed-Eater hat on Stack Overflow. To do so, I'm looking at the list of questions . The problem is, no tags show up there, and so it's rather difficult to see which ones I have a shot at. Any chance of showing the tags on this page?  I realize that getting this before Winter Bash is over would be difficult, but it would be good to keep such things in mind for the future. Is there an advantage to use a Synchronized Method instead of a Synchronized Block? Product of path connected spaces is path connected Will product of path connected topological spaces be necessarily path connected? Why or why not?  Give me some hints. Thank you. lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $ I´m not sure how to start with this proof, how can I do it? $$ \limsup ( a_n b_n ) \leqslant \limsup a_n \limsup b_n  $$ I also have to prove, if $ \lim a_n $ exists then: $$ \limsup ( a_n b_n  ) = \limsup a_n \limsup b_n  $$ Help please, it´s not a homework I want to learn. I have the need to synchronize the data in certain Salesforce tables (user, account, opportunity, lead, task, contact, etc.) to a local database. For reasons that I won't go into here, I can't use the streaming API. I want to kick off a local process every X minutes that will:   Go through each of the tables on my list Insert any new records Update any existing records that were updated Delete any records locally that were deleted on the server (either deleted directly or indirectly (e.g. through a foreign key relationship, like when an account is deleted its associated tasks are deleted as well).   Can you please offer advice on what API's might be good to use here, and whether one stands out more than the others? Are there any specific design recommendations that might be useful for achieving this goal? Or is there really no practical difference between what REST vs SOAP can do?  It comes down to what is the easiest to maintain/implement? Is there some other route that might be beneficial to use (example: bulk data load to start out followed by outbound messaging tied to all of the objects in question to get updates/inserts/deletes)?   I am using C#/.net. I have a DisplayPort based monitor, and a laptop with a DP out.  I recently got a second laptop with HDMI out, and went looking for a HDMI -> DP converter.  I was very surprised to find that such a thing effectively doesn't seem to exist.   and the  seem to imply that it's just not possible to go HDMI -> DP, without a very expensive converter box (not just an adaptor).  However, I can't find anything that explains why this is the case.  Is DP designed to be backwards compatible but can't interpret HDMI?  Is it a case of HDMI-quality signal not being high enough quality to be convertible?  Some other reason?  Note that this question is not looking for an adaptor - I'm looking for the underlying cause of the lack of them. Assume $x$ doesn't occur free in $\alpha$, show that: $$\vdash (\forall x \beta \to \alpha) \leftrightarrow \exists x (\beta \to \alpha)$$   This is an exercise on page 130, A Mathematical Introduction to Logic, Herbert B. Enderton（2ed).   Here's my attempt: For the '$\to$' direction, since $x$ doesn't occur free in $\alpha$, we have $α→∀x α$ and $\forall x \alpha \to \alpha$ as logical axiom(In the second formula, we have$\alpha_x^x = \alpha$). Thus it suffice to show $\vdash (\forall x \beta \to \forall x \alpha) \rightarrow \exists x (\beta \to \alpha)$. By logical axiom $∀ x(α→β)→( ∀ x α→∀x β)$, it would suffice to show $\vdash \forall x (\beta \to   \alpha) \rightarrow (\exists x (\beta \to \alpha))$. It has been shown in a previous exercise that there's a deduction $∀ x ϕ→∃x ϕ$, so we are done for the "$\to$" part.  The problem for me is how to show the "$\leftarrow$" direction.  Added: Here's logical axioms that we can employ:         Tautologies(in the sense of propositional logic);   $∀ x α →α^x_ t $, where $t$ is substitutable for $x$ in $α$ ($α^x_ t$ is the formula derived from $\alpha$ by replacing $x$ by a term $t$);   $∀ x(α→β)→( ∀ x α→∀x β)$;   $α→∀x α$, where $x$ does not occur free in $α$.      Besides, generalization theorem and deduction principle can be utilized:     If $\Gamma \vdash ϕ$ and $x$ do not occur free in any   formula in$\Gamma$, then$\Gamma \vdash ∀ x ϕ$.      If $\Gamma ; γ \vdash ϕ,$ then $\Gamma \vdash  γ \to ϕ$. I don't understand this. These identities are given in the online notes for MIT's 18.01 calculus class. It's related to taking the sum of two trig functions and transforming them into a single trig function. I can use the formulas to do this, but I am having trouble finding anything on the internet to use a proof for my understanding.  Thanks for any responses!  EDITED (added after the first answer):  Sorry, the identity goes like this:  $A\sin{k(x-c)}=a\sin{kx}+b\cos{kx}$ ,  with the relationships  $a=A\cos{kc}$ , $b=-A\sin{kc}$ , $A=\sqrt{a^2+b^2}$ , and $\tan{kc}=-b/a$ How to evaluate potential advisers on grounds other than their research/publications? While accepting an offer to grad school, one is basically entering into a lasting relationship with one's adviser - most likely, someone whom the applicant has never met before, and the only exposure has been through the potential adviser's website/publications. It is in the interest of both parties to ensure (to the greatest possible extent) that there are no personal/professional traits of either that hamper the formation of a pleasant working relationship - no-one would want to go through the ordeal of having to change advisers midway!  While the faculty has a chance to have a good look at the applicant's profile as well as his motivations (through his grades and SOP), the applicant doesn't have a similar opportunity. So, I'm interested to know what parameters can be used to gauge a potential fit. I've thought of the following:   Past students:  Did anyone ever drop out/change advisers midway, and if so, for what reasons? These would be a bit hard to find though, as I don't expect the faculty concerned would list them on their website. It would be great if anyone could let me know how to find out the list of incoming students to a department for any year. Publication rate, taking into account the venues where they were accepted. Time taken to graduate - though I accept this is more dependent on the student, a median figure should be telling ... What they did post Ph.D. - did anyone get tenure if they went into academia, or is almost everyone unable to break out of being a post-doc?  Tenure status: I'm a bit unsure about this, so wanted the community's opinions about it. Just so that I'm clear, I'm only trying to calibrate the applicant's expectations about the working style of his potential adviser - and hence need to know to what extent are the following "typical" assumptions valid.  Tenured professor  A full professor is more likely to get grant funding, hence less time spent on TAship - but could also mean less time/effort spent on interactions with students (either being busy with other projects/talks, or due to more commitments to family at that age).  Tenure-track faculty (Assistant profs)  More likely to be young and energetic, and could translate to more time spent on one-to-one discussions with grad students - but funding may prove to be an issue, and may have to be on TA for a longer period.   What other factors would be relevant in this matter, and to what extent am I correct/incorrect in either the factors considered, or for undertaking this exercise at all?