Convex and bounded function is constant
Show bounded and convex function on $\mathbb R$ is constant
The word that always gets me mixed up : JUST I had just arrived at the hotel, checked in and gotten into my room. Just as I was about to unpack, I noticed that one of my bags was missing. At first I thought I had left it back at the airport, but it turned out that I had left it on the taxi.  Here, what's the meaning and role of "just"?  Why do you guys input the word "just" here?  What exactly does the word do grammatically?   Could you help me? the converse of Pythagoras Theorem Is it possible to prove the converse of the Pythagoras theorem with out a geometric proof?.That is from $a^2+b^2=c^2$ to $\Theta=90^\circ$ For distinct prime numbers $p_1,...,p_n$, what is the Galois group of $(x^2-p_1)\cdots(x^2-p_n)$ over $\mathbb{Q}$?     This problem appears to be quite common, however my understanding of Galois theory is quite poor, and I have no idea how to do this problem. Change audio output depending on which one is on I have a PC that is hooked up to my HDTV (via a long hdmi cable) and to a monitor, which is in another room. I have speakers directly plugged to the PC audio out. The PC is next to the monitor and speakers.   I am not sure but I think Windows can play sound on only 1 audio device at a time. And I can only set 1 device as the default output. I can get sound on the TV or speaker depending on which device I set to default. But I would have to do this every time I switch between using my TV or my monitor!   Is there some way to configure such that sound plays through the TV if the TV is on, else it plays in the speakers? If this is not possible, then the next best alternative would be to get the sound to play on both the devices at the same time. Why can't certain programs (like readlink) take input from a pipe? At the end of the movie we are shown      a bowl with berries in it and Seneca Crane looked like he was locked in a room with them and was supposed to eat them. Did he eat them?   Why did that happen? What does $\rightarrow$ mean in $p \rightarrow q$ I was looking at an exercise where it asked the following:  $$\begin{array}{ccc} p&amp;q&amp;p\rightarrow q \\ T&amp;T&amp;T \\ &amp;\ldots \end{array}$$  So, for the third column, I just put $T$ which was correct but I didn't understand what $\rightarrow$ meant. I have seen $\implies$ but I haven't the arrow. Are they the same thing?  Thanks a bunch! My sister, cousins, and other relatives speculate that my mom's grandmother was Jewish (for various reasons). I've heard people say at times things like, for example, "I'm 50% German" or "I'm 25% Swedish." This has to do with their ancestry. But, how exactly is this calculated? Say for example that my mother's paternal grandmother was indeed 100% Jewish, what would that make me and my siblings, and how is it calculated? Parsing DateTime on Localized Systems Hypernym for “import” and “export”? What word could describe an operation of either importing or exporting?  X is to import and export as migration is to immigration and emigration.  Porting isn't the word I'm looking for, I don't think, but that's what I'm getting at. Cheers! Hump, Rump, Lump, Bump I’m referring to the similar definitions of these four nouns – something raised and rounded.  Why do these four rhyming words have similar meanings?  I have not found very specific sources for these words on the  and . Hump is “probably from Dutch homp.” Rump is “from a Scandinavian source,” “from Old Norse rumpr” ultimately from Proto-Germanic “rumpo.” Lump is “perhaps from a Scandinavian source” and finally, Bump is “perhaps from Scandinavian,” “probably of North Germanic origin.”   Could these words come from the same root? Other than the fact that they all might come from northern Europe, the only hint I have in this direction is the following from , tracing the origin of Hump through Dutch, Middle Low German, Old Saxon and Proto-Germanic, to Proto-Indo-European *kumb-, *kumbʰ- (“curved”). Could this PIE root, which is not given in these online sources for any of the other three words, be a common root to all four words? I find myself looking at other –ump words and “seeing” a curved or rounded line: jump (a curved trajectory), grump (a downturned mouth), pump (the action of the pump handle), clump (a rounded clod of earth, not sharp), etc. Am I just seeing this because the words sound similar, or is there something more to my tendency? For example, the  suggests the origin of jump is "perhaps onomatopoeic (cf. bump)" although this view is not echoed on Wiktionary. Is it possible that these words do all share a common root and are simply onomatopoeic variations of each other? Are there other pairs or groups of words like these in English? (Wham! Bam! Ka-blam!) Does     this phenomenon have a name? Trying to remember a short story (most likely from the sixties).  It starts with a soldier at the front line fighting an unknown enemy on the other side of the front line.  He then gets withdrawn and you realise that time is passing slower the further he moves from the front. He then spends years in peace away from the front, before being sent back to the front where only a short time (hours or days perhaps) have passed.      I seem to remember their was an inference that they were actually fighting against themselves, with their attacks coming back at themselves, which was perhaps the entire cause of the war, and an interesting critique on war.  I can't remember if this all this was stated explicitly, or just left for the reader to infer. Maximal order of an element in a symmetric group As I found solutions on making site-to-site VPN between Azure and AWS using a software VPN server, I'm asking myself if it's possible to connect the VPN gateways (Azure VNET / AWS VPC) directly.  Is there a limitation that forbids it ? fake proof of $\forall a. \forall b. a = b \to 1 = 0$ I saw a less formal version of this fake proof that claimed to prove $2=1$ but because it assumed $a=b$ from the start I knew why it was wrong. It does seem however that the proof can be used to prove an certain implication I find very odd however. Clearly $\forall a. \forall b. a = b \to 0 = 1$ is false because I could pick $a$ and $b$ to be $1$ and then I would have derived 0 = 1 with no other conditions. So there must be something wrong with this proof that I am not seeing.  proof of $\forall a. \forall b. a = b \to 0 = 1$:  $a = b \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (assume)\\ a+a=a+b  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (x=y \to f(x) = f(y), f(x)=a+x)\\ 2a=a+b \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  (a+a=2a)\\ 2a-2b=a+b-2b \,\,\, (x=y \to f(x) = f(y), f(x)=x-2b)\\ 2(a-b)=a+b-2b \,\, (distributive)\\ 2(a-b)=a-b \,\,\,\,\,\,\,\,\,\,\,\,\,\, (-b=b-2b)\\ 2=1 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (x=y \to f(x) = f(y), f(x) =x/(a-b))\\ 1=0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (x=y \to f(x) = f(y), f(x) =x-1) $  now that I have "proven" $\forall a. \forall b. a = b \to 0 = 1$ I can now pick a and b to be 1.  $ 1 = 1 \to 1=0 \,\,\,\,\,\, (a=1, b=1, above)\\ 1 = 1 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (\text{reflexivity})\\ 1=0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (\text{modus ponens}) \\ $  So now it seems that I have managed to derive $1=0$. Clearly I messed up given that no one has been able to show $ZFC$ inconsistent and ostensibly this fits well within the most studied aspects of $ZFC$. Either that or a random guy that makes meme's online found the most profound result in mathematics ever.  So where have I missed the mistake? Book identification: technologically-enhanced man (1970s/80s YA) How do I disable the drum beat sound on the login screen? I would have expected System ▸ Administration ▸ Login Screen ▸ Play login sound to control this, but unchecking it does not prevent the drum beat sound from playing. Unable to install Ubuntu on Lenovo Y500 Counting and rasterizing polygon overlaps in ArcGIS Desktop? How do you take a vector layer with overlapping polygons and turn it into a raster where each cell counts the number of polygons containing that cell?    We are looking at polygons of fire burn areas between 1976 to 2000 and want to see how many fires would be contained in each raster cell. I'm sure this is a newbie question, but I can't figure this out. I have tried to follow the advice here:  and here:   But I can not seem to be able to generate any font change. I either get no change or error messages.  Here is the output of one failed attempt to change the font to Times using this command:  \rmfamily \renewcommand{\rmdefault}{ptm}     Apparently I either don't have a correct package installed or...  The output says " Metric (TFM) file not found." but I am not sure where to go from there.