Should it be: "each other's belongings" or "each others' belongings"? In this sentence:     I'm proud we're not one of those couples who have to snoop through   each other's belongings.   Is the above correct? Or should it be "each others' belongings"? Can you ever say each others' in any situation?  Thanks for any advice!
Should "each" be followed by a singular or plural possessive? If a possessive noun, which is plural, is preceded by "each", then should it use the singular or plural possessive form?  For example, which of the following is correct?    spend time in each other's presence spend time in each others' presence
Exterior power "commutes" with direct sum I know that for vector spaces $V, W$ over a field $K$, we have the following identity :  $$ \bigoplus_{k=0}^n \left[ \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \right] \simeq \Lambda^n(V \oplus W) $$ which holds in a canonical way : for $0 \le k \le n$, the maps  $$ \Lambda^k(V) \times \Lambda^{n-k}(W) \to \Lambda^n(V \oplus W), \\ (v_1 \wedge\cdots \wedge v_k,w_{k+1} \wedge \cdots \wedge w_n) \mapsto (v_1 \wedge \cdots \wedge v_k) \wedge (w_{k+1} \wedge \cdots \wedge w_n) $$ factors through the tensor product's universal properties, which gives us a map $\varphi_k : \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \to \Lambda^n(V \oplus W)$, and since the category of $K$-modules has biproducts, we get a canonical map  $$ \varphi = \bigoplus_{k=0}^n \varphi_k : \bigoplus_{k=0}^n \left[ \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \right] \longrightarrow \Lambda^n(V \oplus W) $$  What if I replace $K$ by a (commutative ring with $1$) ring $A$? Does this map have a kernel? I know it is surjective (it suffices to "look at it", i.e generators of the codomain are obviously all hit by $\varphi$). I don't know if it helps, but I am working in integral domains. In the field case, this is obvious if $V$ and $W$ are finitely generated, since one can use dimension arguments. I wouldn't mind if the above identity held only in the finitely generated case either. If it worked in general it would be cool though.   Note that when trying to compute the inverse map with the universal property, there seems to be a problem... my guess would have been $$ (v_1 + w_1,\cdots,v_n + w_n) \mapsto \bigoplus_{k=0}^n (v_1 \wedge \cdots \wedge v_k) \otimes (w_{k+1} \wedge \cdots \wedge w_n) $$ and this map is obviously $A$-multilinear, but there is a problem with the alternating property ; I get some terms left. I'm guessing this is the wrong map... if there is one. Expected number of rolling a pair of dice to generate all possible sums A pair of dice is rolled repeatedly until each outcome (2 through 12) has occurred at least once. What is the expected number of rolls necessary for this to occur?  Notes: This is not very deep conceptually, but because of the unequal probabilities for the outcomes, it seems that the calculations involved are terribly messy.  It must have been done already (dice have been studied for centuries!) but I can't find a discussion in any book, or on line. Can anybody give a reference? Prevent direct access to a php include file I have a php file which I will be using as exclusively as an include. Therefore I would like to throw an error instead of executing it when it's accessed directly by typing in the URL instead of being included.  Basically I need to do a check as follows in the php file:  if ( $REQUEST_URL == $URL_OF_CURRENT_PAGE ) die ("Direct access not premitted");   Is there an easy way to do this? Proof of Steinitz Theorem I want a source containing the proof of Steinitz Isomorphism Theorem stating:     For any Dedekind domain $R$ and any two nonzero ideals $I$ and $J$ of $R$ we have $I⊕J≅R⊕IJ$.   Thanks! When should you use WP_Query vs query_posts() vs get_posts()? It seems like half the tutorials in the  and around the blogosphere use  and half use . They all do similar things, so when should I use one over the others? Hydrostatic pressure - doesn't density vary with depth? Our class is learning about hydrostatic water pressure and we have been told that we can calculate the force of the liquid on an object at any depth using "the density x 9.8 x the depth". However, as the depth increases, wouldn't the density of the liquid increase because of the weight of the liquid above it compressing it? So should't there be something in the equation to account for the varying density? To me, "density x 9.8 x depth" seems like it is saying that the density will be constant... How to redirect apache /var/www to the home directory and access all the directories and files in my home directory? I have 16GB of system RAM. On startup with no applications open except the task-manager Windows is using about 3gb of RAM. I looked in the processes tab, but nothing seems to be out of the ordinary. How can I find out why my Windows is using so much RAM.    all processes from all users      Reading from the poolmon it seems that my wireless broadcom driver is using about 0.4GB of of RAM. Even if I remove  it would still be using  2.6GB on start-up, which is still too much.      After reinstalling the wireless driver associated with the memory leak. I have a new screenshot and would like to confirm that it was indeed a memory leak. Handler frame_change_pre doesn't work in render this works in viewport but not in render.  import bpy from math import *  def frame_handler(scene):     z = scene.objects['Lift'].location[2]     scene.objects['Text'].data.body = str(floor(z / 4.0) - 2)  bpy.app.handlers.frame_change_pre.clear() bpy.app.handlers.frame_change_pre.append(frame_handler)   How to get it working in render? I've read  and changed from "bpy.data.objects" to "scene.objects" as already seen in the script. What am I missing? Blender v. 2.81.16  Edit. Tried that depsgraph thing mentioned in comments and changed to frame_change_post:  import bpy from bpy.app.handlers import persistent from math import *  @persistent  # maybe not relevant but doesn't harm anything? def frame_handler(scene, depsgraph):     z = scene.objects['Lift'].evaluated_get(depsgraph).location[2]     scene.objects['Text'].data.body = str(floor(z / 4.0) - 2)  bpy.app.handlers.frame_change_pre.clear() bpy.app.handlers.frame_change_post.clear() bpy.app.handlers.frame_change_post.append(frame_handler)   Now the Text doesn't show up at all in rendered (but works in viewport). What's the easiest way to use a linked list in python? In scheme, a linked list is defined simply by '(1 2 3 4 5). Python's lists, [1, 2, 3, 4, 5], and tuples, (1, 2, 3, 4, 5), are not, in fact, linked lists, and linked lists have some nice properties such as constant-time concatenation, and being able to reference separate parts of them. Make them immutable and they are really easy to work with! Is there an easy way to get to www.stackexchange.com from any of the SE sites? One thing I've noticed about the relatively new sitewide layout is that I can't find a quick way to get back to www.stackexchange.com without finding StackExchange in the sites list. As a result, I am typing stackexchange in my toolbar a whole lot more often now. In particular, I tend to enjoy browsing the hot questions list and my own personal feed quite often so visit these pages on a daily basis. Is there an easy link to them that I'm missing?  (On the android app, there is a feed in the menu that I really appreciate, something like that on the desktop would be ideal!) TikZ: Intersection of two lines what is the easiest way to determine the intersection of two lines? I tried  \usetikzlibrary{intersections} \begin{tikzpicture}[every node/.style={black,above right}] \draw[name path=line 1] (0,0) -- (2,2); \draw[name path=line 2] (2,0) -- (0,2); \fill[red,name intersections={of=line 1 and line 2}]     (intersection-1) circle (2pt) node {1}     (intersection-2) circle (2pt) node {2} \end{tikzpicture} When a user who approved an edit is removed, an incorrect "-2 reverted" event appears in reputation history I am aware that . But now there is an extraneous "reverted" event to go along with user removal:     Source: .   The edit was not reverted, it's . In any case, I should not lose 2 points twice for the total of -4. And indeed, my reputation only dropped from 131 to 129 despite what the history suggests.   So the line with "-2 reverted" is a bug.     A more extreme example:  got 21 "reverted" events on 2017-06-30, none of which should be there. His rep loss is 42, not 84 as the reputation history says. Close popup doesn't appear after editing the question without refreshing the page Seems like a bug:  The user:   Edits a question Then, he decide to close it. Clicks on the Close button. The popup doesn't appear. Refreshes the page. Click on the close button. The popup appears.   After editing in the link appears a hash: #. But I don't think that this is the problem because if I add the hash without editing the close popup appears.  It's minor, but I think that is good to be reported and solved.  Edit: The close popup doesn't open after we click an edit has been made to this post; click to load. Unfortunately hard disk of my system is damaged, now I recovered my database. But it cannot be attached again to SQL management studio 2005. The error is:  myfile.mdf is not a primary database file. (SQL server , error 5171) Solution  Full solution for producing PDF/X compliant document can be found in MWE and answer to this question:   Adapted MWE and compilation method are taken from Leonid's answer to this question.  Please note that solution requires compilation with TeXLive 2015 (or later).  Original question  I am trying to compile my source with XeLaTeX to produce PDF/X-1a:2001 compliant document. Getting PDF/X-1a:2001 is possible with pdfx package (while using up-to-date TeX distribution), but only when document is compiled with pdflatex.  Since I am using OTF and TTF fonts in my document, I have to use XeLaTeX to compile it. This is MWE:  \documentclass{book} \usepackage[x-1a1]{pdfx} \begin{document} Some text \end{document}   When I try to compile this document with MiKTeX's 2.9 xelatex I get following error and compilation halts:  ! Undefined control sequence. l.111 \pdfminorversion                       =4 %  assumed for PDF/A ;  options may change this for...  ?    Is there any way to compile PDF/X-1a:2001 with XeLaTeX, either using pdfx or different package?  Ps. I used MiKTeX 2.9, because I don't want to update my TeX Live 2013/Debian distribution I normally use. With MiKTeX I get most recent version of pdfx package.  EDIT: Since it isn't obvious from Leonid's answer and is mentioned only in comments, please note that you need TeXLive 2015 in order for Leonid's answer to work fully. TeXLive 2013 (installed by your system's package manager) or MiKTeX 2.9 won't do the trick (this is true as of March 2016; it might change by the time you read this).    I also tried to compile this MWE with LuaLaTeX to see if this works. It didn't. I get following error:  ! Undefined control sequence. \pdfx@findUUID ... \pdfx@tmpstring {\pdfmdfivesum                                                    {#1}} \expandafter \pdfx@e... l.360 \pdfx@findUUID{\jobname.pdf}  ?   In the event version of LuaLaTeX matters: lualatex --version outputs  This is LuaTeX, Version beta-0.76.0-2013062821  (MiKTeX 2.9 64-bit) (rev 4627) I have a collection that contains linked assets. I would like to override their materials and give them a unique material. Is it possible to assign a material to a whole collection, and how ?  Thanks ! As generally acknowledged TeX offers superb formatting capabilities for math in an automated fashion. Nevertheless there are some areas with clear deficiencies  that one either has to accept or manually improve on a case by case basis.  One of the biggies in that respect is TeX's handling of sub-formulas (i.e., material inside a brace group, e.g., and index, but also any group on top-level). TeX sets such sub-formulas always in natural width even if the whole formula is subject to severe stretching or shrinking otherwise. As an example, consider the following (not very sensible) example:  \documentclass{article}  \newlength\x  \begin{document}  \newcommand\formula{$ a+\mathbf{b+c}+d+e+f = \sum_{i=1}^{n-x-y-z} x_{i+j}$}  \settowidth\x{\formula}  \hbox to \x{\formula}    % natural width  \addtolength\x{-20pt}  \hbox to \x{\formula}   % now shrink it a lot  \addtolength\x{100pt}  \hbox to \x{\formula}   % now stretch a lot in the opposite direction  \end{document}   The formula here has a number of obvious sub-formulas in the subscripts and superscript, but for illustration I also added a sub-formula via \mathbf. Of course the usage of \mathbf in this way is wrong!!! (it should not be applied to several symbols), but I'm sure you would find this in documents. In any case just {b+c} would have had the same effect here. Now what do you think happens if we run this?  This:    As one can see the naturally "boxed" sub-formulas are very wrong the moment the rest of the formula is subject to stretching or shrinking.  Until recently no TeX engine successor addressed this issue. With LuaTeX opening up a lot of the internals of TeX I had some hope that this would be different. However, upon studying the manual my conclusion is that this area (sor far it least) has not been addressed (or considered).  As far as I can see the only way something could be done in LuaTeX about this issue would be to  use the mlist_to_hlist callback. However, this would really mean replacing the full math typesetting algorithm, which of course could be a way to solve the problem but ... but what is needed is not that (as 99% of this algorithm is next to perfect) but to add support to not simply box sub-formulas at their natural width.  So long text ... here is the short question:   Is this analysis correct, or did I overlook something? And in case anybody knows: are there plans to look into the sub-formula issue and provide support for it eventually? (it is not listed on the "to-dos" for math)    In case somebody wonders that I talk about "this sub-formula issue" as if it is something like a known thing ... it is, it was already raised way back in  which at the time a some thorough (if not say heated) discussions. How many 5-cards poker hands are there containing at least 3 of the 4 suits?  I don't how know how to properly handle this question.  There 13 ranks of each card, and 4 suits  My Attempt:  Begin with the first 3: C(13,3) X 4 ways to choose suits  Then pick last 3 cards of anything:  C(49,2)  My solution: 1,345,344  Thanks, 6-letter permutations in MISSISSIPPI How many 6-letter permutations can be formed using only the letters of the word, MISSISSIPPI? I understand the trivial case where there are no repeating letters in the word (for arranging smaller words) but I stumble when this isn't satisfied. I'm wondering if there's a simpler way to computing all the possible permutations, as an alternative to manually calculating all the possible 6-letter combinations and summing the corresponding permutations of each. In addition, is there a method to generalize the result based on any P number of letters in a set with repeating letters, to calculate the number of n-letter permutations?  Sorry if this question (or similar) has been asked before, but if it has, please link me to the relevant page. I also apologize if the explanation is unclear, or if I've incorrectly used any terminology (I'm only a high-school student.) If so, please comment. :-)