Using the method of Lagrange multipliers (or otherwise) find the maximum possible volume of a (right circular) cone which can be inscribed in a sphere of radius a.
I checked this  but didn't fully understand it.   I know that the volume of a right circular cone is $V = \frac{1}{3}\pi x^2h$  I know that I must take the first derivative and set it equal to zero, which will find the maximum.   My problem is how to deal with the variable $h$, the height?   How can I rewrite that in terms of $x$ or $r$.      I have tried to implicitly differentiate this equation, but that wasn't helpful for me.
Positive semidefinite cone is generated by all rank-$1$ matrices. I know that there is a simple way to shutdown the computer in command prompt, but is it possible to do system updates and then shutdown the computer in command prompt?   For example, when Vista needs to install updates, it will let me install them and shut down by clicking the Shut Down button (while relegating the normal shut down button into the menu). So is there a command to do this in command prompt? How do I find duplicate values in a table in Oracle? What are the alternatives to OS X's Spotlight? Why don't we add a question with the same tag on multiple Stack Exchange sites? After the training phase, is it better to run neural networks on a GPU or CPU? My understanding is that GPUs are more efficient for running neural nets, but someone recently suggested to me that GPUs are only needed for the training phase. Once trained, it's actually more efficient to run them on CPUs.  Is this true? How to get a list of all the labels in a LaTeX document in WinEdt I am writing a large document using LaTeX and WinEdt.  I have labeled each chapter, section and so on.  I would like to print out a list of just the labels (so that I don't duplicate one later on, and so that I have it handy in order to write \refs).    Is there a way to do this? Suspending with root on USB I have a laptop running Ubuntu 14.04 from a root file system on USB storage. This is not working well, because after wake up from suspend, ext4 will frequently try to write to the root file system before USB is ready.  Here is what I see in the kernel log when this happens, notice how I get a bunch of I/O errors on sda1, and then one second later the USB storage drive is finally detected by the kernel.  [ 2826.517419] wlan0: associated [ 2826.517452] IPv6: ADDRCONF(NETDEV_CHANGE): wlan0: link becomes ready [ 2827.575371] EXT4-fs warning (device sda1): ext4_end_bio:317: I/O error -5 writing to inode 1733735 (offset 0 size 0 starting block 12629950) [ 2827.575380] Buffer I/O error on device sda1, logical block 12629694 [ 2827.575400] EXT4-fs warning (device sda1): ext4_end_bio:317: I/O error -5 writing to inode 3148603 (offset 0 size 8192 starting block 12844470) [ 2827.575404] Buffer I/O error on device sda1, logical block 12844212 [ 2827.575411] Buffer I/O error on device sda1, logical block 12844213 [ 2827.575448] EXT4-fs warning (device sda1): ext4_end_bio:317: I/O error -5 writing to inode 3015015 (offset 0 size 90112 starting block 6588832) [ 2827.575453] Buffer I/O error on device sda1, logical block 6588576 [ 2827.575461] Buffer I/O error on device sda1, logical block 6588577 [ 2827.575465] Buffer I/O error on device sda1, logical block 6588578 [ 2827.575469] Buffer I/O error on device sda1, logical block 6588579 [ 2827.575473] Buffer I/O error on device sda1, logical block 6588580 [ 2827.575477] Buffer I/O error on device sda1, logical block 6588581 [ 2827.575481] Buffer I/O error on device sda1, logical block 6588582 [ 2828.857284] sd 0:0:0:0: [sda] No Caching mode page found [ 2828.857293] sd 0:0:0:0: [sda] Assuming drive cache: write through   At first there is no visible indication outside of the kernel log, that the problem has triggered, but if I let Ubuntu stay running beyond this point, then the file system will get errors and eventually switch to read only mode. At that point I have to reboot into recovery mode and run fsck.ext4 manually from a root shell in order to repair the file system.  Is there some setting I can change such that access to the root device after waking from suspend can be delayed until the USB drive is ready? Guess the formula for $\sum\frac 1{(4n-3)(4n+1)}$ and prove by induction What does `exec {STDIN}>&0` do? I'm taking a complex analysis course and doing some practice computing residues &amp; evaluating integrals. I pulled out an old book called     On page 196-197, there are some interesting integrals to evaluate. I'm at 5.4.3.10.: I was able to do question 1, but was stumped at how to even begin with question 2:     Evaluate the integral   $$\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx ,$$   where $n$ is an natural number.   The answer is simply $\frac{2 \pi}{n}$. Any hints? How to approach a student with bad hygiene? I teach engineering at a community college in the US. I currently have a student with very poor hygiene. He stinks pretty badly, has obviously greasy hair, doesn’t change his clothes frequently, and I usually seat him in the back of the room (during exams and labs, when I have power over seating arrangements) so I don’t have to smell his (relatively strong) body odor.  This student has also spoken with a colleague of mine about his low self-esteem. He’s worried that he isn’t making friends, and doesn’t know if the people he hangs out with are hanging out with him because they want something from him (help with classwork) or because they are truly friends.   In my opinion, this student follows me around like a puppy, and he tries too hard to impress people (students and faculty alike). I am a young-ish female faculty member and I try to create very large boundaries, especially with my male students. As an example, he frequently finds the need to explain to me (in way too much detail) why he gets questions wrong on exams (I don’t care, and I don’t need to know, which I’ve told him several times with no success).  So, I would like to mention to this student that his hygiene is off-putting and isn’t helping him out in the friends department. However, because he already follows me around like a puppy dog, I don’t want to encourage any more interaction from him. Additionally, it frankly embarrasses me to have to bring up hygiene issues with somebody in their late teens / early twenties. How can I bring this issue up while still maintaining my boundary as a female faculty member who doesn’t want her male students following her around and asking her for advice all the time?  I was hoping that I would be finished up with this student by now, but I am the only professor who teaches Electrical Engineering courses, so I can’t recommend that he take classes with someone else, or ask another colleague to discuss this with him. He will be taking another of my courses next semester and I don’t personally know any of his other intended professors for next semester to ask them to bring it up. I'm self-studying a bit of complex analysis, and I'm attempting to figure out the following.   Suppose $\lim_{n\to\infty}z_n=A$. How can I show that $$ \lim_{n\to\infty}\frac{1}{n}(z_1+\cdots+z_n)=A. $$  Is there a clever way to write the limit to make it more approachable? Thank you. Meta's tag pages inaccessible due to self-redirect loop Tag pages (route /questions/tagged/tagname) redirect to themselves indefinitely:  $ curl -D - http://meta.stackexchange.com/questions/tagged/featured HTTP/1.1 302 Found Cache-Control: private Content-Type: text/html; charset=utf-8 Location: /questions/tagged/featured X-Frame-Options: SAMEORIGIN Date: Fri, 05 Jul 2013 16:47:42 GMT Content-Length: 143  &lt;html&gt;&lt;head&gt;&lt;title&gt;Object moved&lt;/title&gt;&lt;/head&gt;&lt;body&gt; &lt;h2&gt;Object moved to &lt;a href="/questions/tagged/featured"&gt;here&lt;/a&gt;.&lt;/h2&gt; &lt;/body&gt;&lt;/html&gt;   This only happens on Meta, so far, but only applies to any meta-tag that doesn't have a dash in it; those with a dash seem to be exempt, as do invalid tags:    redirects.  does not.  does not. How do I solve $$x'=e^{it}\overline{x}?$$  This is a complex differential equation, but I don't see how to solve it.  Edit: the original ODE is given by $$(x', y')=\begin{pmatrix}\cos t&amp; \sin t\\ \sin t&amp;-\cos t\end{pmatrix}(x,y)$$ I want to show that solutions of this ODE do not remain bounded for all $t$, and the idea was to solve the complex equation. I'm attempting to get a better understanding of cryptography and have run into some questions about bcrypt.   Bcrypt relies on the blowfish cipher, which is a reversible method of encryption. But bcrypt is not reversible. How is that possible? Or am I mistaken? What does bcrypt do on top of blowfish to make it irreversible? Why does bcrypt use blowfish instead of twofish or threefish? Assuming I'm paying long term capital gains tax, what is the break even point where it is more worthwhile to switch from one fund to another?  For (a hypothetical) example, if I have a fund with an expected return of 3.5%, what would be the minimum expected return of another fund to make it worthwhile to pay the capital gains tax and switch to the other fund?  (I'm assuming the answer to this question is an equation involving investment horizon, capital gains tax rate, and my current taxable gains) Why $\dim U+\dim U^\perp=\dim V$? Why does connecting a high-current device to my digital circuit cause weird behavior? Let $G$, $H$, and $K$ be abelian groups. Assume $G \oplus H \cong G \oplus K$. If each are finitely generated, then the structure theorem easily gives us that $H \cong K$. However, there are counterexamples in the general case, namely $H=\mathbb{Z}$, $K=\mathbb{Z} \oplus \mathbb{Z}$, and $G=\bigoplus_{i=1}^\infty\mathbb{Z}$.   I am wondering what goes wrong with the following argument:   Form the split short exact sequence  $$0 \to G \xrightarrow{i} G \oplus H \xrightarrow{\pi} H \to 0$$  where $i$ and $\pi$ are the inclusion and projection, respectively. Then don't we immediately get from the first isomorphism theorem that $H \cong (G \oplus H)/G$, which would contradict the counterexample above?   If we run this for the counterexample given, we get the sequence   $$0 \to \bigoplus_{i=1}^\infty\mathbb{Z} \to \bigoplus_{i=1}^\infty\mathbb{Z} \to \mathbb{Z} \to 0$$  which seems to still be short exact, even though here $G \cong G \oplus H$. But of course here we can see directly that $(G \oplus H)/G \cong G/G$, which is trivial. So what went wrong?   It seems to me to be something along these lines: In this case, the kernel of the projection is not the same as the original group, it is just, say, $0 \oplus \mathbb{Z} \oplus \mathbb{Z} \oplus \dots$, i.e. nontrivial from the second coordinate on, which just happens to be isomorphic to the original group. So if we form the other short exact sequence with $K$ in place of $H$, then the difference seems to be that we have two different projection maps with different (but isomorphic) kernels.   But I'm not really sure how to explain this in the general case, if my reasoning is correct to begin with.   Thanks in advance for any help!