Should there be a comma in this:  providing young adults access to drugs grants younger teens complete access, which can not be accepted given the consequences that would follow
I'm not sure which of the following sentences is correct. Is the comma mandatory, optional or unnecessary? I've googled a bit about this topic, but I got confused.     Our mapping contains 2000 words, which map to more than one lemma.   or     Our mapping contains 2000 words which map to more than one lemma.   Context: In total, there are more than 2000 words.  I've read  . If I understand the answers correctly, my "which map to more than one lemma" is a defining clause and no comma is required. But is it wrong to add one?
Can I stay overnight [15-hour layover] in Haneda airport without a Japan visa? I’m a Filipino citizen. My flight goes from Vancouver to Manila via Tokyo Haneda. I will arrive in Japan at 6pm then depart at 9am the next day. I'm not planning to go out of the airport during my layover, so I don't see a need to have a transit visa. However, I called the Consulate-General of Japan here in Canada, and they asked me to get a transit visa.  Then I called the Ministry of Foreign Affairs in Japan and asked them the same question.. at first, they told me I don’t need a visa as long as I stay inside the airport, but when I told them that the Consulate-General of Japan in Canada asked me to get a visa, they also asked me to get one.  I'm so confused. I would like to know your knowledge or experience regarding this. Can I sync bookmarks between Firefox and Chromium? So, Firefox has turned so resource intensive that my netbook can't run it without making me want to pull my hair out. Thus, I have been trying Chromium with quite some success (I have managed to find proper equivalents for my firefox plugins), the only thing I'm lacking is my bookmarks, which are stored in my Firefox Sync account. Point is I don't want to use Google Bookmarks, mostly because of the whole thing with encryption.  Is there any way to use Firefox Sync with Chromium?, or is there any secure alternative? Difference between alphabet $a$ and "a" I have a very simple problem. I want to write "a" in an equation, however output is .... Please see output. How i can overcome this problem? Prove that subsequence converges to limsup Given a sequence of real numbers, $\{ x_n \}_{n=1}^{\infty}$, let $\alpha =$ limsup$x_n$ and $\beta = $ liminf$x_n$.   Prove that there exists a subsequence $\{ x_{n_k}\}$ that converges to $\alpha$ as $k \rightarrow \infty$.   Not sure how to start this without since I'm not given that the subsequence is bounded.. Proof that the set of all possible curves is of cardinality $\aleph_2$? I am reading George Gamow's book  and a certain assertion Gamow makes seems a little startling considering what I already know of the subject of infinite cardinality.     But the number of all geometrical points, though larger than the number of integer and fractional numbers, is not the largest one known to mathematicians. In fact it was found the variety of all possible curves, including those of the most unusual shapes, has a larger membership than the collection of all geometric points, and thus has to be described by the third number in the infinite sequence. [Gamow, 22]   He later does explicitly say the number of all possible curves is $\aleph_2$. But nowhere else can I find reference to this being true, nor does he provide any sort of reasoning to its validity.  My question is how can you prove the set of all possible curves is of cardinality $\aleph_2$ and not $\aleph_1$ or $\aleph_0$? True Ubuntu on Chromebook ARM I have installed Ubuntu 13.04 on my desktop (runs like a charm). Then I used Crouton to install Ubuntu 12. 04 on my chromebook samsung (ARM). It also works quite well.  Here comes my question. Can we have a true Ubuntu (preferably 13.04) on my ARM chromebook?  It seems that the Crouton/Ubuntu is not 100% the real thing.  The best option would be to have the full Ubuntu 13.04 and remove the Chrome OS. In H. P. Lovecraft's The Case of Charles Dexter Ward, Dr. Willett      Enters Joseph Curwen's dungeon and resurrects a dead person identified as No. 118 by reciting the incantation of the Ascending Node over his salts.   Dr. Willett is told by      The resurrected Joseph Curwen, impersonating Charles Dexter Ward, who he secretly murdered,   that   " 'twas Number 118, and I conceive you would have shook had you looked it up in my list in t'other room."  No. 118 himself gives Dr. Willett a note asking the doctor to      kill Joseph Curwen.   It soon becomes clear that No. 118 is opposed to the nefarious dealings that have been transpiring and desirous of being an ally of Dr. Willett.  Who is No. 118?  Earlier in the story,      The grave of Ezra Weeden, a primary instigator in Joseph Curwen's initial downfall and the romantic rival of Curwen, is robbed. There doesn't seem to be any other mention of Weeden in the 20th century occurrences, so without a further connection the passage seems superfluous.   The shared goals of these two people seem to indicate that No. 118 may, in fact, have been that person, but the      Note given by No. 118 to Dr. Willett in eighth century Latin seems to indicate someone much older than Ezra Weeden.   It also occurred to me that No. 118 could be      The recently murdered Charles Dexter Ward himself. Ward's reference to killing Dr. Allen (who is actually an alias of Curwen) and dissolving his body with acid and No. 118's mention of doing the same to Curwen seem to support the idea, but that still doesn't explain the archaic note. Ward is described as writing in a somewhat modern style, at least as opposed to Curwen. It also doesn't explain why Curwen would want to murder Ward if he was going to resurrect him later anyway. Evaluate the line segment intergal Evaluate the line integral  $$\int_C xe^{y}\, {\rm d}s,$$  where $C$ is the line segment from $(-1,2)$ to $(1,1)$.  I do not get this part of calculus at all please show me how this is solved and if you give me hints is ok too. Please help me. Thanks ..and I don't mean finding out why it doesn't compile, I mean serious debugging: breakpoints, figuring out value of variables at a given moment during run/compile, stepping, examining a stack (is there one?), etc. All the things that we take for granted when debugging a "real" programming language. Since TeX is "turing-complete", I would like to be able to really see what is going on internally.   I'm guessing that a GUI interface for the debugger would be too much to ask, but if it exists, I'd love to hear about it. Density of random variables Let $X$ and $Y$ be independent random variables with respective densities: $f_X (x) = xe^{-\frac{x^2}{ 2}} 1_{]0,+\infty[} (x)$ and $ f_Y (y) =  \frac{1}{\pi \sqrt{1-y^2}} 1_{]-1,1[} (y)$  Let $U:=XY$ and $V:=X\sqrt{1-Y^2}$  I try to check whether U and V are independent and which law they follow.  What I did:  After some calculations, I found the density of $(U,V)$:  $ f_{(U,V)}(u,v)=\frac{1}{\pi } e^{-\frac{u^2}{ 2}}1_{\mathbb{R}}e^{-\frac{v^2}{ 2}}1_{\mathbb{R}_+^* }$ Which mean that $(U,V)$ follows a $ \mathcal{N}(0,1)$ law (because $u^2+v^2=x^2$) and that  U and V are independent because $f_{(U,V)}$ can be factored as the products of densities of $U$ and $V$  My question:  There is a factor $\frac{1}{\pi}$ and I don't know to which density it will belong: to $U$ density or to $V$ density?  Many thanks! Let $\theta$ be a root of $p(x)=x^3+9x+6$, find the inverse of $1+\theta$ in $\mathbb{Q(\theta)}$.  So problems like this really annoy me but I did crappy on the last homework after making a lot of arithmetic mistakes so I want to run everything by you guys. This one isn't actually for homework it's just a suggested practice problem but okay lets go!!!  So I'm not sure if the method I used to do this was standard or not, I don't remember my professor showing me, but what I did first was used the canonical euclidean algorithm in $\mathbb{Q[x]}$ and wrote:  $x^3+9x+6$  $=(1+x)(x^2-x+10-\frac{4}{x+1})$  $=(1+x)(x^2-x+10)-4$  the reduced modulo  the minimal polynomial of $\theta$ i.e. $x^3+9x+6$ and so  $4=(1+x)(x^2-x+10)$  $\rightarrow$  $1= \frac{1}{4}(1+x)(x^2-x+10)$ and sooo ah $(\theta^2-\theta+10)\frac{1}{4}$ is what I believe is the inverse of $(1+\theta)$ in the quotient field... Like i said I hate these problems but ehh is this correct? On a lighter note it's finally getting cool enough for me to go on proper runs here and so that's pretty bomb! how to disable system apps without root? How to disable system apps without root?  My phone model is vivo x5 max+ and android 4.4.4  I have tried Debloater but it failed to connect (I am sure I've allowed USB debugging). When I press Ctrl+Alt+T the Terminal opens after around 30 seconds. That's pretty annoying. If I open it with Alt+F2 and enter gnome-terminal the window appears instantly as usual.  I'm using Ubuntu 16.04 with gnome3.  Thank you for your help in advance! How to sum 2 vectors in spherical coordinate system I have 2 vectors, and each defined by 3 coordinates: radius, azimuth and zenith How i can sum it? Help me pls guys. What i have done: I found azimuth and zenith of resultant vector, by decomposition to 2 polar vectors And it gave me 2 radiuses: radius of resultant vector for azimuth polar sum and radius of resultant vector of zenith polar sum. What can i do with them? Please give me feedback on my answer to this question. Question: For all $ n\geq1:\binom{2n}{0}+\binom{2n}{2}+\binom{2n}{4}+\cdots+\binom{2n}{2k}+\cdots+\binom{2n}{2n} $   is equal to $ \binom{2n}{1}+\binom{2n}{3}+\binom{2n}{5}+\cdots+\binom{2n}{2k+1}+\cdots+\binom{2n}{2n-1} $. Answer: False, since;  Let $a=1,  b=-1$  and $n=1$  ,$k=0$   , using Binomial expression.  $$(a+b)^n=  \sum   \binom{n}{k}  a^{k}b^{n-k}$$  Then $(1+-1)^{2(1)}=\binom{2}{0}1^{0}(-1)^{2-0}+\binom{2}{1}1^{1}(-1)^{2-1}+\binom{2}{2}1^{2}(-1)^{2-2}=\binom{2}{0}-\binom{2}{1}+\binom{2}{2}$  Thus :For all $n\geq1:  \binom{2n}{0}+\binom{2n}{2}+\binom{2n}{4}+\cdots+\binom{2n}{2k}+\cdots+\binom{2n}{2n}   \ne \binom{2n}{1}+\binom{2n}{3}+\binom{2n}{5}+\cdots+\binom{2n}{2k+1}+\cdots+\binom{2n}{2n-1}$  . what is a thyristor As part of my 2330 lv3 am required to know about thyristors. I just want to confirm my understanding is correct. A thyristor is basically similar to a transistor, only it stays switched on when it's trigged for as long as there isn't a break in the circuit it's switching. Am I correct? What is the rationale for all comparisons returning false for IEEE754 NaN values? Curve Object Deformed by Curve - Distortions (Artefacts) Shaping curve object by curve (modifier) results in a terrible shape with cracked surface. Here is an extreme situation, but good visible issue.  Perspective   Top   How can I curve it smootly without adding more control points or remeshing? What is the sum of the 'second half' of the harmonic series?  $$\lim_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} =~ ?$$  More precisely, what is the limit of the above sequence of partial sums? Possible to bake texture to new UV map? I have a finished model where the UV maps overlap - lots of wood that isn't detailed to be particular to the model.    I'd like to re-map the texture so that specific details can be added. I'd like to rebuild the UV map into non-overlapping islands, without losing the current texture application, and preferably without manually pulling apart the islands.  Is this possible?