Cancellation property in groups
How to prove $b=c$ if $ab=ac$ (cancellation law in groups)?
Is it permissible to use somebody else's words (without quotations) to describe a concept once their work is cited? Quite often one comes across a really succinct and precise explanation of a concept. I'm wondering if it is permissible to use this explanation verbatim, if the source work is cited in your paper (next to the explanation)? What's in question here is at most 3 sentences. Let $A$ and $B$ be two fixed points on a fixed straight line. Two circles touch this line at $A$ and $B$ respectively and tangent to each other at $M$. When the circles vary, what does the locus of $M$ form? Oscillation of air particles and speed of sound wave A sound wave is essentially air particles oscillating parallel to the direction of travel of the wave.  We learnt that $v = f\lambda$, where $v$ is the speed of the wave, $f$ is the frequency of the wave and $\lambda$ is the wavelength of the wave.  Suppose the air particles oscillate with a frequency of $f_{particle}$. Is there any relationship between $f_{particle}$ and $v, f, \lambda$ of the entire sound wave?   I thought about this because intuitively, if the air particles vibrated faster, then the wave should travel faster, but I am unable to come up with any formula that describe this relationship (if it is even true!). Prove that in a triangle $$r^2+r_1^2+r_2^2+r_3^2=16R^2-(a^2+b^2+c^2)$$ where the symbols have their usual meanings. I am looking for a smaller or elegant proof using trigonometry. A geometric proof would be super cool. I did prove it but it took me ages to do the calculation. Thanks. Is it standard to say $-i \log(-1)$ is $\pi$? I typed $\pi$ into Wolfram Alpha and in the short list of definitions there appeared  $$ \pi = -i \log(-1)$$  which really bothered me. Multiplying on both sides by $2i$:  $$ 2\pi i = 2 \log(-1) = \log(-1)^2 = \log 1= 0$$  which is clearly false.  I guess my error is $\log 1 = 0$ when $\log$ is complex-valued.  I need to use $1 = e^{2 \pi i}$ instead.  So my question: is it correct for WA to say $\pi = -i \log(-1)$?  Or should be they specifying "which" $-1$ they mean?  Clearly $ -1 = e^{i \pi}$ is the "correct" value of $-1$ here. What is this triangle icon [Samsung Galaxy S9+]? I have just had my broadband upgraded to  which means my router has had to change. Previously I was using a Thomson TG585v7 ADSL thingy provided by my ISP. One nice thing about that router was that it would act as a local DNS and allow me to find my other machines on the network by using &lt;hostname&gt;.lan instead of IP addresses.  I've now switched to a Linksys WRT320N ethernet router which connects to my fibre modem, and which I've updated to the latest firmware (1.0.04). I have 2 machines on the network which I regularly want to connect to each other; one is running Windows 7 and the other is running OS X Lion.  The problem I'm having is that I'm unable to connect to my Win7 machine from the OS X machine using the Win7 machine's name. However, I'm able to ping the OS X machine from my Win7 machine using the hostname. Switching from MTB to road handlebars So, I'm working creating a lamp... yes, newbie here.  After modeling for a bit, I noticed that my point of origin was no longer the center of my object:    How to I reset an object origin point to the center of it's geometry? Isomorphisms preserving integral domains and fields Let $R \simeq S$ be isomorphic commutative rings with unity. Prove the following:  a). If $R$ is an integral domain then $S$ is an integral domain.  For this, I said that If I let $f: R \to S$ be an isomorphism and $(R, + , \circ), (S, \oplus, \ast)$, then let $x, y\in R$,  such that $x \circ y = 0_R$, because $f$ is an isomorphism, $f(x \circ y) = f(x) \ast f(y) = f(0_R)$. If I let $R$ be an integral domain then, as per the definition $R$ is commutative, has unity, and no zero divisors.   So $f(x \circ y) = f(x) \ast f(y) = f(0_R)$ where we have either $f(x \circ y) = f(0_R) \ast f(y) = f(0_R)$ or $f(x \circ y) = f(x) \ast f(0_R) = f(0_R)$.   Since $f$ is an isomomorphism, let $s = f(x), t = f(y)| s,t \in S$. Then $s \ast t = f(0_R)$. By definition, $S$ also has unity so $0_S$ exists. If we map $0_R \to 0_S$. Then either $s = 0_S$ or $t = 0_S$.   So then, since $S$ has unity, is commutative, and has a no zero divisors as we showed above, it is also an Integral domain.  I'm not sure I have this correctly.  b). If $R$ is a field then $S$ is a field. I don't really understand how to do this one. Since $S$ is an integral domain, it has a $0_S$. So I need to show that $S$ distributes $\ast$ over $\oplus$.  Any help with these is appreciated I like some of the features provided by the Vim-LaTeX suite, but not all of them.  In particular, I dislike all of the mappings like [[ = \left[ \right] and the automatic placeholder stuff with &lt;++&gt;. I am a very fast typist but with that comes more typing mistakes, which I correct almost instantly by pressing backspace. Unfortunately, if something has automatically added a bunch of place-holders or expanded some expression, I then have to move around the document and remove more characters than necessary. For example, if I accidentally press [ twice, instead of only having to delete one character, I have to delete 10.  Having already looked through the documentation, and some of the source, I have managed to disable some of these "features", but not all of them:  In particular:  let g:Tex_SmartKeyBS = 0 let g:Tex_SmartKeyQuote = 0 let g:Tex_SmartKeyDot = 0   (Turn off maps for backspace, double quote and triple dot.)  let g:Imap_UsePlaceHolders = 0   (Turn off &lt;++&gt; placeholder insertion.)  But I can't seem to find a way of turning off the other stuff. Anyone have any suggestions?  Edit: I was able to "disable" some more things by setting:  let g:Tex_Leader = '`tex' let g:Tex_Leader2 = ',tex'   (Which stops some expansions but still not all. It's not really disabling it but just making it less likely that you're going to type in `texa as opposed to the potentially more common `a, for example.)  PS: Personally, I think it is very poor practice to make an optional package that, when installed, greatly changes the default handling of your system. I don't mind some intelligent source highlighting etc, or some new commands (that don't clobber any old ones), but it seems unhelpful to enable this drastic kind of key remapping by default. Is  free for business platforms?  Has Ubuntu changed its conditions about using the OS for profit purposes? Are there rules about using "that" to join two clauses? He will understand that I was not joking.   He will understand I was not joking.  Which of the sentences is correct? Are there any specific rules about the use of &quot;that&quot; in the sentences I reported as example? Trying to figure what language this PDA (pushdown automata) accepts I have the following PDA and I can't figure our what words it accepts, would like to get some help with figuring this out.    Of course it only accepts if that stack is empty in the accepting state. When electric dipole placed in non uniform electric field, what is the approach to calculate torque acting on it? Can it be zero? QuickStart basic layout/template for web development? I'm looking for a quick start or template, or something, to get a web site's basic layout done quickly and looking fairly professional, and then let me do the coding.  I don't want a CMS, as this is for a highly customised reporting application, but I need something like frame header and two columns and frame footer.  I can design the pretty reports for the wide right hand column myself, as the left hand column menu. Run a batch file in a completely hidden way This question is inspired by    I see these constructs  for file in `find . -type f -name ...`; do smth with ${file}; done   and  for dir in $(find . -type d -name ...); do smth with ${dir}; done   being used here almost on a daily basis even if some people take the time to comment on those posts explaining why this kind of stuff should be avoided... Seeing the number of such posts (and the fact that sometimes those comments are simply ignored) I thought I might as well ask a question:  Why is looping over find's output bad practice and what's the proper way to run one or more commands for each file name/path returned by find ? With no good reason I had a look at my computer's storage going on the apple and About this Mac. Here is the display:    The pink bar doesn't make much sense. It obviously does not represent the 80.67GB of free storage. Two sections correspond to pink; "Photos" and "Backups". I obviously don't have 140.76TB of backups. And probably not on this computer that has only 320GB of storage.  What is going on here? How do I select all the columns in a table that only contain NULL values for all the rows?  I'm using MS SQL Server 2005.  I'm trying to find out which columns are not used in the table so I can delete them.