What would be an example of a convex set in one dimension? How can such a set exist in one dimension? What would this look like geometrically? I recently started learning about convex sets and I don't know how to visualise one in 1-dimension. Any help is appreciated. Thanks.
How can we define convexity in one dimension? Definition of a convex set: A set $S\subset\mathbb R^n$ is said to be convex if for each $x_1, x_2\in S$ the line segment $\lambda x_1 + (1-\lambda)x_2$ for $\lambda\in (0,1)$ belongs to $S$. This says that all points on the line connecting two points in the set are in the set. My question is: how can we define convexity in one dimension?
Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian? Somebody asked me this question in a job interview and I replied that their joint distribution is always Gaussian. I thought that I can always write a bivariate Gaussian with their means and variance and covariances. I am wondering if there can be a case for which the joint probability of two Gaussians is not Gaussian? How to refer Javascript cloud page in my email in content builder? Use XeLaTeX in Texmaker instead of pdfLaTeX Can I make my Ubuntu make sounds when I press Caps Lock etc.? Seeking detailed description of qgs file structure? Is there a detailed description on the .qgs file structure somewhere?   I checked at  and got     The requested URL /qgis.dtd was not found on this server.   and at  which wasn't very detailed. Best way to hide a window from the Alt-Tab program switcher? I've been a .NET developer for several years now and this is still one of those things I don't know how to do properly. It's easy to hide a window from the taskbar via a property in both Windows Forms and WPF, but as far as I can tell, this doesn't guarantee (or necessarily even affect) it being hidden from the Alt+↹Tab dialog. I've seen invisible windows show up in Alt+↹Tab, and I'm just wondering what is the best way to guarantee a window will never appear (visible or not) in the Alt+↹Tab dialog.  Update: Please see my posted solution below. I'm not allowed to mark my own answers as the solution, but so far it's the only one that works.  Update 2: There's now a proper solution by Franci Penov that looks pretty good, but haven't tried it out myself. Involves some Win32, but avoids the lame creation of off-screen windows. Position of a relative clause before/after a verb Pass a PHP string to a JavaScript variable (and escape newlines) What is the easiest way to encode a PHP string for output to a JavaScript variable?  I have a PHP string which includes quotes and newlines.  I need the contents of this string to be put into a JavaScript variable.  Normally, I would just construct my JavaScript in a PHP file, à la:  &lt;script&gt;   var myvar = "&lt;?php echo $myVarValue;?&gt;"; &lt;/script&gt;   However, this doesn't work when $myVarValue contains quotes or newlines. using tar showing, "No Such file or directory" But when I try to run it using  tar zxvf jre-8u151-linux-i586   I get tar (child): jre-8u151-linux-i586: Cannot open: No such file or directory tar (child): Error is not recoverable: exiting now tar: Child returned status 2  What is wrong? The name and place matches prefectly. Is there a way to hide pictures from being shown in Gallery Possible to change where Android Virtual Devices are saved? Ship an application with a database I'm trying to load a field value (machine name: field_theme_image) from a content of node value 3, in "page--front.tpl.php".  &lt;?php   $node=node_load(3);   print $node-&gt;field_theme_image[$node-&gt;language][0]['uri']; ?&gt;   Using the above code I'm not getting full URL value. Please see the image below. I can see the path value in Devel themer. Why the CMB has not been dispersed so far? Imagine you have a box of black body radiation. What happens if you open the box for a long time? It becomes dispersed and no radiation remains in the box.   Now, apply this example to the Cosmic Microwave Background radiation. The CMB has been produced about 380,000 years after the Big Bang. Giving that the space is flat as many observations suggest, that radiation has been produced in a universe with no boundaries.   Now, my question is this: in these conditions, why that radiation has not been dispersed completely so far? It is true that the radiation has been produced everywhere in the space but giving that the space is infinite, why has not it been dispersed do far?   Calculations in the standard textbooks are done in such a way as if the CMB has been within a physical box, however expanding with the universal expansion. But, this is not the actual situation. The CMB has not been and is not enclosed by walls of a box, so it must have completely dispersed so far. What has prevented this to happen? How can I execute a function on pressing the enter key in an  field? I can easily find files with given string in name:  me@comp:/usr/local/hydra/hydra-7.4.2$ find -D stat -name "*hack*" ./hack881663129.txt ./hack881663129_7.txt ./hack881663129_5.txt ./hack881663129_4.txt ./hack881663129_4_7.txt ./hack881663129_6.txt ./hack881663129_1_6.txt ./hack881663129_8.txt   How can I also print file details, like size, date of creation etc.? Proof of sandwich/squeeze theorem for series. Computers on board Starfleet vessels are quite primitive compared to on board computers such as . At their best, they can decipher requests such as "Computer, delete that last sentence" and utter things such as "That crewman is no longer on board this vessel". Even real life computers will be capable of this within a decade or so (by some estimates). Even computers such as HAL 9000 are likely to be available within a few decades.  So, in a genre where intelligent computers are abound, why this strange absence? Did the Federation run into some Battlestar-Galactica-like problem with AI in the past and decide to keep on board computers decentralized?  But if that is the case, how can one explain Data and highly intelligent holographic characters? The technology and programming constructs clearly exist, yet Starfleet vessels continue to be manned by hundreds of people standing in front of consoles. Do muggleborns get their letter on their 11th birthday or the summer before they start Hogwarts? A lot of the fanfiction seems to like to give Hermione her letter in September to coincide with her birthday. That just doesn't seem very logical for the Statute of Secrecy, as you then have 10 out of every 12 10-year-old wizards and witches knowing about magic while finishing the school year (assuming even distribution of birthdays).   It seems to me that this is just asking at least one of them to brag about knowing magic and spill the secret.   Also this would mean that the poor kid born on 30 of August would only have 2 days to adjust to being magical before being shepherded off to a boarding school in the middle of nowhere.  From another point of view, getting such news on your birthday can screw up a lot of celebration plans.  Also, from rereading the memory from Dumbledore where he introduced magic to Tom, there is no mention of snow or even closeness to his birthday from the caretaker.  I believe this view came from the fact that Harry got his letter on his birthday but that piece of fanon seems to forget that letters were being sent up to at least a week or 2 beforehand (leading to the Dursleys hiding on a rock in the sea in the first place). I'm struggling to find the limit at infinity of : $$\lim \limits_{n\rightarrow\infty}\sin^2(\pi\sqrt{n^2 + n}), n\in\Bbb N$$ I know it is $1$ but I don't understand why this is wrong :  $\sin^2(\pi\sqrt{n^2 + n}) = \sin^2(\pi*n\sqrt{1 + \frac{1}{n}})$.  So $\lim \limits_{n\rightarrow\infty}\sin^2(\pi\sqrt{n^2 + n}) = \lim \limits_{n\rightarrow\infty}\sin^2(\pi*n\sqrt{1 + \frac{1}{n}}) =  \lim \limits_{n\rightarrow\infty} \sin^2(\pi*n) = 0$.  Thanks.