I know that "children" is a plural noun and "birthday" is a countable noun, but I saw this sentence in a book:     Children have a birthday cake at their birthday.    Why is "children" (which is a plural noun) followed by "a birthday cake" instead of by "birthday cakes," and why is "their" followed by "birthday" instead of by "birthdays"?
In a situation where say a group (or at least a plurality) of men is being addressed — for example on a sign passed by many married men — which is correct?  &quot;Remind your wife.&quot;  or  &quot;Remind your wives.&quot;
I have a document I am trying to copy to learn TeX. Here is what I have encountered: How can I have the "or" in the equation. This is what I have right now:  \[ f = ma; \]  But $a$ is the change in velocity,  \[ f = m \frac{dv}{dt};\] \[ f = m  \frac{d^2y}{dt^2};\]   Edit: Trying out Stefan's answer Computing the Galois group of polynomials $x^n-a \in \mathbb{Q}[x]$ I have some problems with this exercise. I don't know if it can be done. Consider the polynomial $ x^n - a \in \mathbb{Q} $ Can I compute the Galois group of this over $\mathbb{Q}$? Maybe having a nice "basis.  The splitting field is given by $\mathbb{Q}(\zeta_n,\alpha)$ , where $\zeta_n$ is a primitive root of unity , and $\alpha$ is some number such that $\alpha^n = a $. Well first of all, I want a $\underline{"good basis"}$ for the splitting field. In the sense that the minimal polynomials, of the adjoined elements, are different (in this case the computation of the galois group is very simple).  For example one easy case, it's when $a&gt;0$, then $(a)^{\frac{1}{n}} \in \mathbb{R}$ , so clearly the minimal polynomial of $(a)^{\frac{1}{n}} , \zeta_n$ are distincs, and I'm done. If $n$ is odd then , it's also easy, since one root it's also real, for example $x^3-3 $, the real root is $ \root 3 \of { - 3}  =  - \root 3 \of 3  $ , so I can consider the splitting field as $\mathbb{Q}(-\root 3 \of 3 , \zeta_3 )=\mathbb{Q}(\root 3 \of 3 , \zeta_3 )$. The difficult case is when $n$ is even and $a&lt;0$ , for example  $x^8+20$ or $x^4+20$ in some cases as in the second, there are particular cases since there exist algorithms for the Galois group of quartics, but in general. It can be done?  $\underline{Remark}$ I'm searching a $\underline{"good basis"}$ for the splitting field. In the sense that the minimal polynomials, of the adjoined elements, are different since in this case the computation of the galois group is very simple. I'm doing a bit of self study, but I'm uncomfortable with a certain idea. I want to show that $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is analytic, and by analytic I mean differentiable at each point. Here $f$ is a complex valued function.  What I do is write $f(z)=u(x,y)+iv(x,y)$, where $u$ and $v$ are real functions of two variables. Then $\overline{f(\bar{z})}=u(x,-y)-iv(x,-y)$.  These two forms look very similar, in the sense the one function being differentiable should immediately imply the other is differentiable, since the only thing really changing might be a $-$ sign popping out due to the chain rule.   How can I more rigorously express that $f(z)$ is analytic iff $\overline{f(\bar{z})}$ using this? Many thanks. Is a DNA molecule a single strand of polynucleotide or two of them linked together? Coconut Water from White vs Brown Coconuts How should duplicate questions be handled? What should I do when I see a question that is a duplicate of another one?   Should I answer it? Should I downvote it? Should I comment? Should I edit the question to indicate it's a duplicate? If I can, should I (vote to) close the question? What happens when I vote to close as duplicate? Or: Why did a comment with my name on it just appear? Should I flag it for moderator attention? What about similar or related questions? House computer with AI takes over home, tries to procreate with woman What is the difference between Reduce and Solve? Is my sugar daddy scamming me I've heard that in the study of finite fields, and other concepts related to finite fields, mathematicians have found a sort of gap: there are various results and things that seem like they correspond to a field with one element, $\mathbf{F}_1$, even though there is no such field. Of course, there is an object which seems similar to a field with one element: the zero ring, $\mathbf{Z}_1$, defined as having one element, $0$, obeying the equations $$0 + 0 = 0 - 0 = 0 \cdot 0 = 1 = 0.$$ The zero ring seems to behave like a field in every respect, except for the fact that it fails to satisfy $0 \ne 1$. (But the importance of the axiom $0 \ne 1$ isn't clear to me.) However, in any case, there seems to be consensus that the zero ring is not the field with one element. People say that the zero ring &quot;does not behave like a finite field&quot; () or that it &quot;does not have the features that mathematicians need&quot; (). I'm not familiar enough with algebraic geometry to understand the properties that $\mathbf{F}_1$ is expected to have. Is there an elementary, undergraduate-level explanation of  why it seems like a field with one element &quot;ought to exist&quot; in the first place, what properties we expect it to have and why, and how the zero ring fails to satisfy these properties?  Or do I need to study algebraic geometry if I want to have an understanding of any of this? (A side question: do we have a good answer to the question of whether the single element of $\mathbf{F}_1$ ought to be $0$, $1$, both, or neither?) Strictly diagonally dominant matrices are non singular Decrease available inventory item on Cognito Form, removing it when zero I want to limit inventory quantities on a sales form, so that it can start at say 100 and count backwards as applicants submit orders and then shut down that particular option when it reaches 0.    Is this possible with Cognito? How do I bake multiple texture maps into 1 UV, In Blender Cycles? Using:  Blender V2.79 Cycles Render Engine  After creating a complex interior scene consisting of 3 different texture types, each having 3 associated maps (Diffuse, Gloss, Normal) I noticed the texture and baking options will not successfully combine to create a texture for the exported mesh once I bring it into Unity 3D. The mesh is a single object with the different textures assigned to specified faces.  I've played around with a default cube mesh for testing purposes, using the same 3 textures split among the 6 sides, but have only been able to successfully pull over the color info by baking a diffuse map to my UV and saving the image. When using the other baking options, Blender does not successfully create a map for either my Glossy textures, nor my normal map. Even using the "Combined" baking option, I am provided a flat dull diffuse image for my mesh, no gloss or normal info is saved.  My Node process for textures is displayed here  Here is the result of baking my textures as a Normal  The result of adding other textures made from Blender's baking, results in flat gloss at best.  What can I do differently in Blender, to have my assigned UV Diffuse, Normal, and Gloss maps all appear in a single object in Unity? Follow a question I've come across a few interesting questions over the last few months on Stack Overflow. They're not necessarily questions I need an answer to, but the questions are nonetheless very interesting, and now I want to know what the answers are.  I don't want to add this question to my favourites because I have my really handy posts in there.   Ideally I would like to have another kind of shortlist to which I can add these posts. I would get notifications just as if it were my own post, without having these as favourites.  What do you think of that? It would be available only for questions, and you would receive notifications just like the author, for new answers and comments on the question (not comments on the answers).  You could browse the posts you're following, as well as stop following them.   The thing is sometimes I come across a semi interesting question and upvote and maybe even comment, but I never hear about it again, because sometimes it's just not interesting enough to remember.  Like I said in the comments, I don't feel like the favourite list is really for this. I want in-your-face red dot notifications for new answers on brand new interesting questions I find. The favourites is for that post with an answer that you know you'll need again. git-upload-pack: command not found, when cloning remote Git repo I have been using git to keep two copies of my project in sync, one is my local box, the other the test server. This is an issue which occurs when I log onto our remote development server using ssh;  git clone me@me.mydevbox.com:/home/chris/myproject Initialized empty Git repository in /tmp/myproject/.git/ Password: bash: git-upload-pack: command not found fatal: The remote end hung up unexpectedly fetch-pack from 'me@me.mydevbox.com:/home/chris/myproject' failed.   (the file-names have been changed to protect the guilty... !)   Both boxes run Solaris 10 AMD. I have done some digging,  if I add --upload-pack=$(which git-upload-pack) the command works, (and proves that $PATH contains the path to 'git-upload-pack' as per the RTFM solution)  but this is really annoying, plus 'git push' doesn't work, because I don't think there is a --unpack= option.    Incidentally, all the git commands work fine from my local box, it is the same version of the software (1.5.4.2), installed on the same NFS mount at /usr/local/bin.    Can anybody help? Is there anything in RAW that supports spell level as a character concept, or alternatively that supports it as solely a player concept?  On the one hand, I see nothing that explicitly states that a character knows that there are spell levels. On the other hand, I don't see how a spellcaster could manage spell slots without knowing about spell levels. Possible Duplicate:          As a English non-native speaker it is difficult for me to understand when I must use present perfect or past simple because in my official language there isn't the present perfect tense. I know that present perfect is related to something that happened in the past but its result is important now. But for instance:        The message has been sent.   The message was sent.      Please help me point me out any examples/contexts where I should use 1. instead of 2. Is the empty set a group? Do sets, whose power sets have the same cardinality, have the same cardinality? Bring all planes in mesh to same level I have added a plane in top view. I have then extruded the plane according to a floorplan in topview.  I don't know what happened, but when I switched my view, I could see that some parts of it are for some reason high up in the air.  I would like to ask how I can bring everything back down onto the grid / floor.  And if somebody could tell me how this might have happened, I would also be glad.