I just thought of a problem earlier today, but wanted to know if there was an easier way of acquiring the answer.  Say I have a standard parabola $y=x^2$ with 3 points on it $P,Q,R$ and another point $S$ such that $PQRS$ is a square.  Now I can clearly see that if I have the origin $(1,1), (-1,1)$ and $(0,2)$, then I form a square with area 2. However, this may not necessarily be the smallest square.  My idea for finding the square with the smallest area is to parameterise the parabola with 3 points $(t,t^2)$ where $t=p,q,r$, then to find the area of that triangle (perhaps using the determinant) and then minimise that quantity with the constraint that the interval $PQ=PR$ and that $PQ \perp PR$.  However, this seems too tedious. Is there a better way perhaps?
I was given this question by a friend and after working tirelessly on it I have not come up with anything substantial. I was hoping someone in the community could provide a pointer or possibly a solution. The person who gave me the question says he has the answer to it, but I am more curious about the working.  Here is the question:     Three points $(B, C, D)$ lie on parabola  $y = x^2$ and a fourth point $A$ is placed such that $ABCD$ forms a square. Find the minimal area of $ABCD$.   So far my working out has involved a collection of distance/mid-point formula's and I haven't even reached the point where I differentiate to minimize the area.  Any pointers would be welcome.   Edit  Apparently the correct answer is $\sqrt{2}$ units squared.
Image viewer without large window borders I am looking for an image viewer software without very large window borders. I don't need any type of toolbar, just a simple image viewer.   It would also be fine if there is no top bar to close, minimize or maximize window. What is the term for replacing a name or brand with a funny pejorative / sardonicism? What is the term for when one replaces a well-known name [(Proper) noun, company, brand, etc.] with one that tries to be funny yet pejoratively descriptive, different but similar enough that one's audience can likely understand the replacement?  For example: Whole Foods ===> Whole Paycheck; Comcast ==> Scumcast; Starbucks ==> Charbucks; Kinko's == Stinko's; eBay ==> Evil Bay  I tend to use sobriquet don't believe this explicitly implies funny as well as insulting. Moniker again is close but not specific to the above description.  There is a similar question which proposes tortonym and malapropsim, but it does not seem to settle the argument definitively:   Another answer here proposes dysphemism, but to me it isn't a full fit:  because it doesn't cover the humor aspect. Singular or plural verb form where subject includes a "parenthetical" element My question arises from , where OP asks whether he should use the singular or plural verb form after "the title, as well as the tone,"  As luck would have it, when I searched Google Books for , I found...        ...and...        At the risk of having this question closed as just peeving, I feel that the first one should have been plural "err", because without the commas it seems to me both "the title" and "the tone" have more or less equal status as "joint" subjects of the verb, making it plural.  On the other hand, I feel the commas in the second example effectively demote "the tone" to "parenthetical" status (the clause is almost incidental). Since this encourages us think of "the title" as the primary subject, it should have been singular "derives".  I'm no great fan of "grammatical rules", but it seems weird to me that the only two relevant examples for this specific wording should both (in opposite ways!) contradict my inclinations.  What's going on? Why did the Jedi want to bring balance to the Force? During the Prequel trilogy, the Jedi wanted to bring balance to the Force, in accordance with a prophecy.  There seemed to be a general agreement that this was a 'good thing', but a disagreement with the assertion that Anakin was the one to do it.  Why did the Jedi Council believe it was good to bring balance to the Force?  The Republic was mighty (in theory), the Council was established as THE source of Force users, the Sith were thought extinct, and the number of Dark Jedi was apparently very, very low.  It seemed, for all intents and purposes, as if the Force was strongly imbalanced in favor of the Light side...so why did the Council want to 'balance' it? Is there a difference between them? If so, how and when are they used?  For example:     I fixed a little/small typo. What are the benefits of nickel-plated chains? Bicycle chains with higher model numbers (and slightly higher prices) are often nickel-plated or half-nickel-plated. What are the benefits of nickel-plated chains over non-nickel-plated chains?  How durable is the plating--does it wear off, or do its benefits last for the life of the chain? How can I echo a newline in a batch file? How can you you insert a newline from your batch file output?  I want to do something like:  echo hello\nworld   Which would output:  hello world Closing a window XGBoost Loss function Approximation With Taylor Expansion How can I make my ad hoc iPhone application's icon show up in iTunes? I've got an iPhone app with icon file Icon.png.  This icon shows up properly when the app is on the phone itself, but it doesn't show up in the applications pane in iTunes.  What do I need to do to get it to show up properly? When using processing tool in pyqgis and run, it always creates a progress bar. Is there a way to disable it?  processing.runalg('qgis:fixeddistancebuffer', _layer, distance_tolerance, 10, False, None)   I want to disable it because I get this error when I run my script in travis:     wrapped C/C++ object of type QProgressBar has been deleted See log for   more details       RuntimeError: wrapped C/C++ object of type QProgressBar has been   deleted A good approximation of $(1+x)^n$ is $1+xn$ when $|x|n &lt;&lt; 1$.  Does this approximation have a name?  Any leads on estimating the error of the approximation? The Canon L lenses are much more expensive and presumably much better quality than non-L lenses. What makes an "L" lens? How much better are they over a similar length/speed "ordinary" canon lenses? Absolute Value Symbols What is the "best LaTeX practices" for writing absolute value symbols?  Are there any packages which provide good methods?  Some options include |x| and \mid x \mid, but I'm not sure which is best... Say you take a dice and you roll it twice so that you have a pair (X,Y) where X represents the first roll, Y represents the second.   When you have a distribution like max(X,Y), what type of distribution is that? It's not uniform, obviously, because the outcomes of max(X,Y) are clearly more favorable towards the higher numbers. I'm being asked to "describe" the distribution. I know that we can look at each distinct outcome, like:  max(1,1) = 1  max(1,2) = 2  max(1,3) = 3  max(1,4) = 4  max(1,5) = 5  max(1,6) = 6  max(2,1) = 2  max(2,2) = 2  max(2,3) = 3  max(2,4) = 4  max(2,5) = 5  max(2,6) = 6  max(3,1) = 3  max(3,2) = 3  max(3,3) = 3  max(3,4) = 4  max(3,5) = 5  max(3,6) = 6  max(4,1) = 4  max(4,2) = 4  max(4,3) = 4  max(4,4) = 4  max(4,5) = 5  max(4,6) = 6  max(5,1) = 5  max(5,2) = 5  max(5,3) = 5  max(5,4) = 5  max(5,5) = 5  max(5,6) = 6  max(6,1) = 6  max(6,2) = 6  max(6,3) = 6  max(6,4) = 6  max(6,5) = 6  max(6,6) = 6  But is that the most effective way to look at a distribution like this? Is there a faster and more concise way to describe this distribution other than saying the probability of each outcome?  i.e.,  P(max = 1) = 1/36  P(max = 2) = 3/36 = 1/12  P(max = 3) = 5/36  P(max = 4) = 7/36  P(max = 5) = 9/36 = 1/4  P(max = 6) = 11/36  So, obviously, I understand how to LOOK at a discrete distribution and see what the probabilities are, but not until I look at each possible outcome. But what does it mean to "describe" a distribution? This isn't uniform or any sort of discrete distribution we're taught. Can someone provide clarity? SVG for images in browsers with PNG fallback Where do the X-Men get all their money/funding? Charles Xavier's father was a "wealthy nuclear scientist". In the beginning, Charles personally bankrolled the Xavier Institute/X-Men using his family's money.   This included, but was not limited to things like upgrading the mansion to house the training levels, providing a helicopter (the predecessor to the Blackbird jet), providing and replacing uniforms, providing health care for injured students, and rebuilding/repairing the mansion and parts of the Danger Room  repeatedly.  These have all been recurring costs. The mansion has been damaged or outright destroyed a ridiculous number of times. X-Men can't get into a fight without someone's uniform getting damaged.   In one of the more recent stories (I believe it was an early issue of Wolverine &amp; The X-Men), a comment is made about how the X-Men are pretty much bankrupt.  Where has all this money come from over all these years?   Xavier may have had a large war chest to start with, but no other income has ever been mentioned to my knowledge. Student's tuition could account for some of it, but there's no way that's going to fund everything. I have a WordPress website that I recently discovered had this error on the login page:     ERROR: Cookies are blocked due to unexpected output. For help, please   see this documentation or try the support forums.   I can only access the login page from wp-login.php, since /wp-admin/ displays a blank white page.  I've searched the web and tried many things with no success:   Removing ?&gt; and space at end of the functions.php Ensuring both functions.php and wp-config.php had no spaces at beginning or end of file Opening wp-config.php and functions.php in Notepad++ and converting to UTF-8 (people recommend 'without BOM' but that is only an option in the newest version of Notepad++ - I'm guessing UTF-8 does that now) I've set WP_DEBUG to true and have received no errors in debug.log,  except for 2 that relate to when I hadn't fully uploaded some files. That particular error mentioned nav-menu.php; this confirmed that it is reporting errors, just not related to my login issues I renamed each plugin folder with _tmp at the end to try and rule out any plugins causing the problem I also renamed the plugins folder to rule out all plugins at the same time   Given these things I've tried, and that I'm not receiving any PHP errors, what else could I try? Excel VBA code to turn a row with a variable number of columns into fixed-length rows I would like to change this:  Filer   ShareName       IPAddress1    IPpaddres2    IPAddress2    ... ... .... ServerA /share1/tools   192.168.1.52  192.168.1.65  192.168.1.158 ServerA /share/library  192.168.1.65  192.168.1.61  192.168.1.155 ServerB /share/tools    192.168.1.158 192.168.1.159 ServerD /share/misc     192.168.1.7 .. ... ..... .......   To this, in a new worksheet:  Filer   ShareName       IPAddress ServerA /share1/tools   192.168.1.52 ServerA /share1/tools   192.168.1.65 ServerA /share1/tools   192.168.1.158 ServerA /share/library  192.168.1.65  ServerA /share/library  192.168.1.61  ServerA /share/library  192.168.1.155 ServerB ... ... ServerD ... ... .. ... ..... ....... Showing a tensor is not a simple tensor Let $(e_1,e_2)$ be the standard basis of $\mathbb{R}^2$. Why can the element $e_1\otimes e_2 + e_2\otimes e_1$ in $ \mathbb{R}^2\otimes_{\mathbb{R}} \mathbb{R}^2 $ not be written as a simple tensor $v\otimes w$ for $v,w\in \mathbb{R}^2$?  If we suppose $e_1\otimes e_2 + e_2\otimes e_1=v\otimes w$, then can we conclude that $e_1=\lambda e_2$ for some $\lambda \in \mathbb{R}$? This would give a contradiction since $e_1$ and $e_2$ are linearly independent. However, I am not sure if the argument is valid.