does the word PERSONS exist? Should I only use it for singular? PERSON
Person, Persons, People, Peoples
Proving that if $V$ is an inner product space and $U_1, U_2$ are subsets, then $U_1 \subset U_2$ if and only if $U^{\perp}_2 \subset U^{\perp}_1$ I encountered a problem today to prove that $(X_n)$ with $X_n = \cos(n!)$ does not have a limit (when $n$ approaches infinity). I have no idea how to do it formally. Could someone help? The simpler the proof (by that I mean less complex theorems are used) the better. Thanks Is it possible to get rid of the bullets in the miniframes outer theme If I pass PHP variables with . in their names via $_GET PHP auto-replaces them with _ characters. For example:  &lt;?php echo "url is ".$_SERVER['REQUEST_URI']."&lt;p&gt;"; echo "x.y is ".$_GET['x.y'].".&lt;p&gt;"; echo "x_y is ".$_GET['x_y'].".&lt;p&gt;";   ... outputs the following:  url is /SpShipTool/php/testGetUrl.php?x.y=a.b x.y is . x_y is a.b.   ... my question is this: is there any way I can get this to stop? Cannot for the life of me figure out what I've done to deserve this  PHP version I'm running with is 5.2.4-2ubuntu5.3. I can intuitively understand that there are infinitely many irreducible polynomials in $\Bbb{F}_p[X]$, where $p$ is a prime number, but I'm having a hard time actually proving it. What proof strategy should I take for this? Using awk to sum the values of a column, based on the values of another column PHP regular expressions: No ending delimiter '^' found in What happens when anti-matter falls into a black hole? Let's say a black hole of mass $M$ and a very compact lump of anti-matter () also of mass $M$ are traveling toward each other. What does an outside observer see when they meet?  Will they blow themselves apart in a  matter/anti-matter reaction? Or will their masses combine, never quite meeting in the infinite time dilation at the event horizon? How do I use a particular font for a small section of text in my document? The standard font packages make global changes to the fonts in my document. What if I want to use a particular font for just a portion of my text?   How do I find the right name of the font? How do I select the font? How do I restrict the scope of the font change?     There are separate answers for different flavors of TeX (quick links): How to check type of files without extensions in python? How to do line optical illusion art effects? I was wondering how I can do this type of optical illusion. Is it done manually or is there a quicker way to do it. Thanks ahead of time. If $A,B,C$ are matrices I am thinking how to show that $$ A(B + C) = AB + AC$$  Is possible to show without sums like $\sum_i a_i, ..., \sum_j b_j$? It seems if I do the proof with many indexes then is tedious and I don't learn much from it. Change all file type associations from one program to another Is it possible to easily change all file type associations that are currently associated with one program to another program in one batch?  In my case, I am trying to change all files types currently associated with Notepad to instead use Notepad++ in Windows 7.  It's a pain to go through Default Programs &gt; Set Associations because after each type the entire list re-populates, taking several seconds.  Is there a way I can do a search and replace in the registry, or something even easier? I have several covariates in my calculation for a model, and not all of them are statistically significant.  Should I remove those that are not?  This question discusses the phenomenon, but does not answer my question:   There is nothing in the answer to that question that suggests that non-significant covariates be taken out, though, so right now I am inclined to believe that they should stay in.  Before even reading that answer, I was thinking the same since a covariate can still explain some of the variance (and thus help the model) without necessarily explaining an amount beyond some threshold (the significance threshold, which I see as not applicable to covariates).  There is another question somewhere on CV for which the answer seems to imply that covariates should be kept in regardless of significance, but it is not clear on that.  (I want to link to that question, but I was not able to track it down again just now.)  So...   Should covariates that do not show as statistically significant be kept in the calculation for the model?  (I have edited this question to clarify that covariates are never in the model output by the calculation anyway.)  To add complication, what if the covariates are statistically significant for some subsets of the data (subsets which have to be processed separately).  I would default to keeping such a covariate, otherwise either different models would have to be used or you would have a statistically significant covariate missing in one of the cases.  If you also have an answer for this split case, though, please mention it. When punctuating the dimensions of something, if both dimensions share the same units, does the punctuation occur after each measurement or just after the last one. For example, if I'm looking at a sheet of paper, which is correct: {8 1/2 x 11"} or is {8 1/2" x 11"}? Interior of closure of open set in topological space If $U$ is an open set in a topological space, is it true that $U$ is the interior of the closure of itself?   If $U$ is open, it must be the interior of itself. But is it the interior of the closure of itself? In the closure, we include all points $x$ such that any open set containing $x$ also contains a point in $U$. In the interior of the closure, we take the union of all open sets contained in the closure. Sarrus Rule with LaTeX I am currently working on a project to create a (not prepaid) postage label.  In java, I would go about doing this by creating a bufferedimage and adding various other images to it to create what I want.  What I want is to put text, converted to an image, onto a template letter, to make a proper postage label.    For example, imagine I was given a document with blank spaces such as:  "Name: _________"  This document is in the form of an image, and I want to use apex and visualforce (the page is already constructed, I am just stuck at the image manipulation part) to add whatever name the user types in onto the image and pack it and email the new image.  The question I have here is: where do I even start?  Java is a little nicer when it comes to image manipulation, as far as I can see.  I've looked into using a web service and sending a soapxml request to return a proper encoded image, but I haven't found a web service that can do this.  Any pointers would be nice. Thanks. Is changing your passport a good strategy for removing a negative immigration history in Schengen? When certain nationalities of people want to travel in the Schengen zone, they are required to apply for a Schengen visa. This is a straight-forward process for most, but when an applicant has had problems with removals or refusals they may be reluctant to apply because they want to avoid the complexities of serial visa refusals. This reluctance may lead some into wondering how their history can be expunged, and this line of thought inevitably leads to the idea of getting their passport replaced with a new one.  This sounds like a great formula for people who have had prior refusals or removals.  In essence they can become 'reincarnated' as a first-time traveller and hence avoid the unpleasantness of revealing their prior interactions with the Schengen authorities.  The relevant parts of the application form are shown here...    Nothing asks about a prior removal or refusal, and the applicant is left with simple denials for 3 questions in order to present themselves as a first-time traveller.  Assuming the person is happy to set aside the moral issues associated with denying their history, are there any holes to this strategy?  This question is scoped to Schengen visa applications so that it will not be unmanageably broad, but answers from other regimes (particularly the affluent Commonwealth and the USA) are acceptable as long as the main points are relatively consistent with Schengen. Q&A blank in Stack Exchange Android app v1.0.95 - S10e/Android 10