Are the apostrophe placements correct in every example?  • two weeks' vacation • four days' personal time • three months' sabbatical time • ten nights' accommodations • three hours' sleep • eight months' pregnancy • eight months pregnant • five minutes' time • a penny's worth • two pennies' worth • a nickel's worth • two nickels' worth • a dollar's worth • five dollars' worth • two seconds' time • five years' probation  Thank you kindly.
I’ve always understood that the phrase two weeks usually turns into two weeks’ when used as a modifier as in   I’m giving my two weeks’ notice.  I get two weeks’ vacation. (“two weeks’ holiday” for Brits)   with an apostrophe on the word weeks’, indicating that the vacation “belongs” to the weeks.   One way to explain this is the phrase “two weeks of vacation” being contracted to “two weeks’ vacation” – the vacation is “of” the weeks; that is, it is possessed by them.  But I’ve seen a lot of people omit the apostrophe in casual writing, and thinking about it, it seems plausible that the noun vacation would be plainly modified by the adjectival phrase two weeks.  But on the other hand, shouldn’t adjectival phrases be hypenated, namely two-weeks vacation?  Yet that seems wrong, too: the hyphenate should be singular, two-week vacation (like two-tone shoes) because .   touches on this (and references a good ) but doesn’t explain why the plural must be possessive in this case.   suggests that two weeks might be adverbial rather than adjectival, and thus resists being used attributively. But that still doesn’t quite explain this particular muddle of pluralization, possession, and attribution.  Given all this, why should “two weeks vacation” be written “two weeks’ vacation” with a possessive apostrophe?
Check Constraint with Case Statement I want to add a check constraint on a column if entered value for column is greater than 3 than it should be saved as 1 else entered value. For that scenario I am using below query but on insertion of data it shows an error message.  create table tblTestCheckConstraint  ( id int, NewColumn int )   alter table tblTestCheckConstraint add constraint chk_tblTestCheckConstraint_NewColumn1 CHECK   (     CASE         WHEN NewColumn &gt;4 THEN 1         ELSE NewColumn     END = 1 )   Insert Statement:   insert into tblTestCheckConstraint values ( 1,5) insert into tblTestCheckConstraint values ( 1,2)   Error Message:   Msg 547, Level 16, State 0, Line 1 The INSERT statement conflicted with the CHECK constraint "chk_tblTestCheckConstraint_NewColumn1". The conflict occurred in database "DBName" , table "dbo.tblTestCheckConstraint", column 'NewColumn'. The statement has been terminated. Unity-like Dashboard for Gnome fallback? Is there any way I can run the Unity dashboard (or something similar, like Bolt for Gnome 3) in Gnome Fallback? I have an input type="image". This acts like the cell notes in Microsoft Excel. If someone enters a number into the text box that this input-image is paired with, I setup an event handler for the input-image. Then when the user clicks the image, they get a little popup to add some notes to the data.  My problem is that when a user enters a zero into the text box, I need to disable the input-image's event handler. I have tried the following, but to no avail.  $('#myimage').click(function { return false; }); MongoDB could not connect local server I had mongdb installed and evrything was working fine. I wanted to delete the mongodb lock. By mistake the directory /var/lib/mongodb/ got deleted.  So, I uninstalled mongo db and installed again. But now i always keep getting this error:  Error: couldn't connect to server 127.0.0.1:27017 at src/mongo/shell/mongo.js:145 exception: connect fail   The error remains same even if I try to login from terminal or rockmongo or my application. I am using ubuntu12.04 and have latest version of mongo. Badge progress reports Is there a way to view your own progress towards certain badges? For example:   Tumbleweed - What value constitutes low views for a week? Fanatic  – Is there a way to view how many days in a row you currently have? Beta – Is it even possible to get this badge anymore? In my game, XP needed to reach the next level is Current Level × Level Threshold.  Given that, how can I get my current level from the total XP ever earned?    For example:  Level Threshold = 50 Current Level = 1   Starting at level 1, I would need (1 × 50) = 50 XP to get to level 2, and so on.  Level 1: 50 XP needed to reach level 2 Level 2: 100 more XP needed to reach level 3 Level 3: 150 more XP needed to reach level 4   In other words, the progression table is as follows:  Level 1: 0 XP to 49 XP Level 2: 50 XP to 149 XP  Level 3: 150 XP to 299 XP Level 4: 300 XP to 499 XP   If I have 300 XP, I'll just reached level 4. How can I calculate that generally? $4$ letter words taken from the letters of CONCENTRATIONS How many words with or without meaning can be made from the letters of the word CONCENTRATIONS by taking $4$ letters at a time?  There are  $14$ letters total present in the word. $4$ letters can be picked in $14\choose 4$ ways. $4$ letters can be arranged in $4!$ ways. So ideally answer has to be ${14\choose 4} * 4!$. But I know this answer is wrong because some letters are repeating. How do I solve this then? I appreciate any help. I would like to highlight specific digits in a number. Since the number is large I would like to use \num{} from . Of course I can do it manually as in the MWE below, but was hoping that there might be an easy way to use the commented out syntax to accomplish this:  Code: \documentclass{article} \usepackage{xcolor} \usepackage{siunitx}  \newcommand*{\Color}[2][red]{\textcolor{#1}{#2}}%  \begin{document}  \num{99 976}  88\,9\Color{76} %\num{88 9\Color{76}}  77\,\Color{976} %\num{77 \Color{976}}  \end{document} Ubuntu 17.10 Keyboard not working after login following upgrade from 17.04 (Gnome version) Boot drops to a (initramfs) prompts/busybox I am running an HP pavilion dv6000 dual boot win7 and Ubuntu 12.04. (well, up until today).  After a reboot, the boot process drops to the BusyBox shell and I end up at the prompt:  BusyBox v1.18.5 (Ubuntu 1:1.18.5-1ubuntu4) built-in shell (ash) Enter 'help' for a list of built-in commands.  (initramfs)   I've been researching others who have had this same problem, but haven't been able to find any of those solutions to work for me.  I tried the method described , and after the final command   mount -t ntfs-3g /dev/sda1 /root -o force   it does nothing and gives me another (initramfs) prompt.  I can boot to a live CD (USB) and get to a terminal, but it doesn't seem to do much good, as I can see the /dev/sda1 in the ls command, but it doesn't recognize it when I try to cd to it.  One more question: using the command fdisk -l how can I tell which mount point (sda1/sda2) is my windows partition and which one is Ubuntu? Inequality vs. Inequation, Equation vs. Equality When talking about statements that two terms are mathematically (un-)equal (e.g., 1 = 1, 1 &lt; 2, or 1 != 2), what is the correct notion for such mathematical statements? Are there special cases w.r.t. "mixed" relations like x &lt;= y? Can some notions be used interchangeably?  It seems many online sources use "equation" and "inequality" but there is also a . Apart from personal posts in forums, I have not found any sources using "equality" in such a situation, yet. Systematically, I would think that "equation" and "inequation" would make more sense than "equation" and "inequality", but not everything (probably very few things) in natural languages have developed according to a purely logical system.  Note that this question is more general than  for that it particularly asks for the relation between the notion for equations and the one for inequalities. Equations are not part of the existing question at all. Issue with particles in blender donut tutorial I have been recently doing the blender donut tutorial and ran into a problem regarding the sprinkle particles. The particles (hair) keep appearing on the underside of the icing. As seen in the image below, most of the hair particles only come out from the bottom of the icing. I tried to readjust the normals but I realised the normals were already correct. (Blue colour donut in image) Someone suggested following through the weight painting part of the tutorial. However, the sprinkles still remain on the underside of the icing. What is the actual problem and how do I solve this?  Particle settings below  Edit: @moonboots Thanks the hair particle problem has been solved. How do I prevent the submersion of the sprinkle particles? The following definition is needed for my actual question:  For $A, B \in \mathcal{M}_{n \times n}$ define $$  \langle A, B \rangle = \text{Trace}(B^{T}A) = \sum_{j=1}^{n}\sum_{i=1}^{n}b_{ij} \, a_{ij} $$  And here is the actual question:  For $B \in \mathcal{M}_{n \times n}$ show that $$ \text{Trace}(B)^{2} \:\leq\: n \,\text{Trace}(B^{T}B) $$  I know I have to use the Cauchy-Schwarz Inequality, but I wasn't exactly sure how to.  Here is my attempt:  Since I only have one matrix I decided to use the identity matrix $$ \begin{align} \text{Trace}(B)^{2} \:&amp;\leq\: n \,\text{Trace}(B^{T}B) \\ \text{Trace}(I^{T}B)^{2} \:&amp;\leq\: n \, \langle B, B \rangle \\ \langle I_{n}, B \rangle^{2} \:&amp;\leq\: n \, \Vert B \Vert^{2} \\ \langle I_{n}, B \rangle \:&amp;\leq\: \sqrt{n} \, \Vert B \Vert \end{align} $$  I feel like I either did it incorrect, or that I'm close but am missing something. is there any diffrence to use nonlinear or liner activation function in single hidden layer network Discrepancy in using adjective or adverb with “taste” This is a question from the free Harvard online abstract algebra .  I'm posting my solutions here to get some feedback on them.  For a fuller explanation, see  post. This problem is from assignment 5.  a) Prove that the relation $x$ conjugate to $y$ in a group $G$ is an equivalence relation on $G$. b) Describe the elements $a$ whose conjugacy class (= equivalence class) consists of the element $a$ alone.  a) Let $G$ be a group and $R$ be a relation on $G$ defined by $a\sim b$ if $a$ is conjugate to $b$.  Then $a\sim b$ if there is a $g\in G$ such that $a = gbg^{-1}$.  Let $a$ be an element of $G$.  Then $a=eae^{-1}$.  So $a\sim a$ and $R$ is reflexive.  Let $a$ and $b$ be elements of $G$ such that $a\sim b$.  Then there is a $g\in G$ such that $a=gbg^{-1}$.  Then $b=g^{-1}ag$.  Since $g^{-1}\in G$, $b\sim a$.  Hence, $R$ is symmetric.  Let $a,b$, and $c$ be elements of $G$ such that $a\sim b$ and $b\sim c$. Then there are elements $g,g^\prime\in G$ such that $a=gbg^{-1}$ and $b=g^\prime cg^{\prime -1}$.  Then $a=g(g^\prime c g^{\prime -1})g^{-1}=(gg^\prime)c(g^{\prime -1}g^{-1})=(gg^\prime)c(gg^\prime)^{-1}$. Since $gg^\prime\in G$, $a\sim c$.  Hence $R$ is transitive.  Therefore $R$ is an equivalence relation on $G$. b) Let $S$ be the set of elements of $G$ such that, for $s\in S$, $s=gsg^{-1}$ for any $g\in G$.  Then $sg=gs$.  So $S$ is the set of elements that commute with every element of $G$.  In other words, $S$ is the center of $G$. Again, I welcome any critique of my reasoning and/or my style as well as alternative solutions to the problem. Thanks. Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even I have a question:     Suppose $f$ is continuous and even on $[-a,a]$, $a&gt;0$ then prove that   $$\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$$   How can I do this? Don't know how to start. Finding the ID of a process and killing it When ever I need to kill a background process I do ps -e | grep &lt;process_name&gt;  Which prints something like this 1766 ? 00:00:13 conky , Then I use the process ID to kill it like so kill 1766 .   Is there any way I can simplify this ? Make it quicker ? reduce the amount of typing ? Getting Acquaintances to become Friends English equivalent for the Persian idiom "send someone out in pursuit of black chickpeas" "To send somebody out in pursuit of some black " is a Persian idiom that implies 'to make or ask someone to run an errand so that you be able to have/ buy some time in order to deal with your private affairs in their absence', like having or continuing a (secret) conversation with someone else.   According to its etymology, since 'black garbanzos/chickpeas' were rarely available in the markets, when someone wanted to buy them, they had to search many stores for a long time, and most of time they couldn't find them at all.   Today, its meaning has changed somehow, and it doesn't necessarily mean that we ask someone to run an impossible errand, but it means we use that errand as an excuse to send them away for a while just to have time to do something in their absence.   Example scenario:   Suppose your classmate has come to your house and you are talking about some girly issues in the sitting room. Now you want to tell her a big secret about one of your classmates, but not in the presence of your younger sister who is sitting there too. So you ask her to go to the market and buy some snacks for you. In this way, you will have some time to continue your conversation until she is back. So you might say to your friend:     Well, I sent my sister out in pursuit of some black chickpeas.  I'm sure that she won't be back for another 15 minutes, so we can talk more freely now."   And your sister, before leaving the house, might sarcastically say to you:     Okay, but I know this is just an excuse, and you are just sending me in pursuit of some black chickpeas!!! so don't think that you can fool me!   Is there any idiom, expression or proverb that conveys the same meaning or connotation in English (i.e. to make or ask someone to do an errand just in order to buy time so that you can deal with some of your private affairs)?   I have found to be/go on  but I'm not sure if I can use it or not.