When do you use 'that' and 'which'
Why is 'that' sometimes optional before dependent clauses?
If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t Let  $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$.    I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the same as  $g(xy) = g(x) + g(y)$   I looked up the hint and it says let $g(x) = \log f(a^x) $   The wikipedia page for functional equations only states the form of the solutions without proof.  Attempt  Using the hint (which was like pulling a rabbit out of the hat)  Restricting the codomain $f:(0,+\infty)\rightarrow (0,+\infty)$ so that we can define the real function $g(x) =  \log f(a^x)$ and we have $$g(x+y) = g(x)+ g(y)$$   i.e $g(x) = xg(1)$ as $g(x)$ is continuous (assuming $f$ is).  Letting $\log_a f(a) = p$ we get $f(a^x) =a^p $. I do not have a rigorous argument but I think I can conclude that $f(x) = x^p$ (please fill any holes or unspecified assumptions) Different solutions are invited sstatic.net SSL certificate is for an incorrect domain Calculating Optimal Geohash Precision from Bounding Box I'm using Elasticsearch's  to plot clusters on a map (using Leaflet). I understand that for larger areas a lower precision setting should be used to limit the number of buckets created/returned.  How should I determine the appropriate precision value to request?  Is there a standard or recommended formula for calculating the optimal precision based on a bounding box and/or zoom level? Is it better to just map zoom levels to precision values (I know that's probably easiest.) Find the limit: $\displaystyle\lim_{x\to \infty} (\sqrt{x^2+x}-\sqrt{x^2-x})$. So I'm making this summary of a course where I've only been getting information from one book. Sometimes I want to show what page from the book a statement is taken from but I can't figure out a clean way of doing it.  Basically I would like to have something (in function) similar to:  "Statement[Book p123]"  Do you have any suggestions? I can’t for the life of me figure out where to use a and where to use the — and where there is no article at all. Is there a simple rule of thumb to memorize?   The standard rule you always hear:     “If a person knows which item you are talking about then use "the"   . . . doesn’t clear things up for me, as I have no idea whether or not they know. What does "ain't" mean? What does the contraction ain't mean? Is it appropriate to use it in formal settings? How do I define global variables in CoffeeScript? Would someone give a thorough explanation of 1.14 Minecraft's /data modify? What are the downsides of removing ubuntu-desktop metapackage? What is the purpose of the EBP frame pointer register? I'm a beginner in assembly language and have noticed that the x86 code emitted by compilers usually keeps the frame pointer around even in release/optimized mode when it could use the EBP register for something else.  I understand why the frame pointer might make code easier to debug, and might be necessary if alloca() is called within a function.  However, x86 has very few registers and using two of them to hold the location of the stack frame when one would suffice just doesn't make sense to me.  Why is omitting the frame pointer considered a bad idea even in optimized/release builds? Can I enable the Windows 7 Start menu in Windows 10? I've upgraded from Windows 7 to Windows 10 and now I only find this new slower Start menu with all the tiles in it   (picturesource: )  Can I enable the older Windows 7 one again which suites my needs more? I'm trying to install pandas. When I do, it's automatically installing to the python installation for v 2.7.   How do I force pandas to install to the python 3.5 installation? Papers published from an academic project (MSc or PhD) usually have two authors; the first author is the student who mainly conducted the research, and the second author is the professor who supervised the projects.  The corresponding author is the one who take the responsibility of a paper, and thus, some believe that students are not yet prepared to take this responsibility.  Ideally, who should be corresponding author for papers published by MSc or PhD projects?  I understand that it mainly depends on personal agreements and preferences, but I want to know which case is more reasonable from academic logic? Value of having done applied math research when applying to PhD programs in pure math I will be a second year math under-grad in September, and currently doing an internship at applied math research institute. Most probably, I will come up with a research paper at the end of the internship.  However, my interest after my bachelors is to do PhD in pure math from a top math school.My question is this:     Would the admission committee consider my current research since it is in applied math?    In addition, if anyone can suggest me some ideas on how can I bridge the gap between my paper in applied math and my interest in analysis, I would be extremely grateful.  P.S.:  is a related but different question also of mine. 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 Square bracket notation for dimensions and units: usage and conventions One of the most useful tools in  is the use of square brackets around some physical quantity $q$ to denote its dimension as  $$[q].$$  However, the precise meaning of this symbol varies from source to source; there are a few possible interpretations and few strict guidelines. What conventions are there, who uses them, and when am I obliged to follow them? Confirming that my own question is a duplicate by voting to close should have immediate effect Someone voted to close my question as a duplicate. I agreed, so I cast my close vote (I have enough reputation to vote to close).  . So I expected this to happen.  But it didn't. This confused me — is this not a question I asked after all? No, it is. Maybe I need to reload the page? Nope, the question still isn't closed. Is the asker confirmation feature broken? Well, there's the “This question may already have an answer here” banner at the top. What if I click “That solved my problem!”? Ah, that does close the question.  It turns out that only clicking “That solved my problem!” instantly closes the question. Casting a close vote has a different effect, it's just an ordinary close vote.  Please make casting a close vote equivalent to clicking “That solved my problem!”. I get it after updating in Synaptic Manager  I recently did a clean install of Ubuntu 17.04 from 16.10.  error message:-  W: Download is performed unsandboxed as root as file '/var/cache/apt/archives/partial/samba-libs_2%3a4.5.8+dfsg-0ubuntu0.17.04.1_i386.deb' couldn't be accessed by user '_apt'. - pkgAcquire::Run (13: Permission denied) How can I do the following in Git?  My current branch is branch1 and I have made some local changes. However, I now realize that I actually meant to be applying these changes to branch2. Is there a way to apply/merge these changes so that they become local changes on branch2 without committing them on branch1?