I have been given this question from my university to complete and I am having some trouble. The lecturer did not do a very good job of explaining the concept and I could not find any materials online to help so I came here. The question is-  What happens if you apply the method of steepest descent to  $f(x)=(x_1^2)+(x_2^2)+(x_3^2)$  Thank you.
I am totally stumped for what to do here, any help would be appreciated.   1) What happens if you apply the method of steepest descent to $f(x)=x_1^2+x_2^2+x_3^2$?  2)Lagrange multipliers method- Find the dimensions of the rectangular box open at the top of greatest internal volume, given the surface area of the five faces is $108 \, \mathbb{cm}^2$.  Thanks
Yesterday, one of our computers dropped to grub shell or honestly, I am unsure what shell it was when we turned on the machine.  It showed that it can't mount the root filesystem or something in this sense, because of inconsistencies.  I ran, I believe:  fsck -fy /dev/sda2   Rebooted and the problem was gone.  Here comes the question part:  I already have in her root's crontab:   @reboot /home/ruzena/Development/bash/fs-check.sh    while the script contains:  #!/bin/bash touch /forcefsck   Thinking about it, I don't know, why I created script file for such a short command, but anyways...  Further, in the file:  /etc/default/rcS   I have defined:   FSCKFIX=yes    So I don't get it. How could the situation even arise?    What should I do to force the root filesystem check (and optionally a fix) at boot?  Or are these two things the maximum, that I can do?  OS: Linux Mint 18.x Cinnamon 64-bit.  fstab:  cat /etc/fstab | grep ext4   shows:   UUID=a121371e-eb12-43a0-a5ae-11af58ad09f4    /    ext4    errors=remount-ro    0    1    grub:   fsck.mode=force    was already added to the grub configuration. How do you set, clear, and toggle a bit? Subsubsection: remove the newline (like paragraph) Using the article document class, how can one remove the newline at the end of a subsubsection title?  I am using \documentclass{article}; I do not currently use titlesec or any of the alternatives; I do use fancyhdr. How do you maximize the cycle count of your battery?  There seems to be a debate between:   Leave plugged in 24/7 but do a full cycle once a month Charge to ~100%, drain to ~10%, repeat.   NOTE: Taking your   Which method is better and why? How much of a difference does it really make?  Here are some of the better sources I've found about battery behavior so far. Even still I feel they don't conclusively answer the question above. See my elaborations below:           Elaborating on Option 1  Now IF Apple engineers were smart and optimized for AC draw when plugged in to not use up your charge cycles, then it seem logical to assume that you are using effectively none of your 1000 charge cycles while plugged in. If that's the case, it would seem that the LiPo drop from holding at 100% would be vastly outweighed by the fact you're not using any charge cycles.  However, this is a purely speculative assumption and I could find no evidence to either confirm or deny.  If this is NOT true, and the AC adapter only goes to charging the battery, then there would be no difference. Your battery would drain to 99%, then back up to 100%, then back down to 99%, etc. Those micro-charge cycles would add up at the same rate as 100% -> 0% -> 100%, and you would get no gain. In this case the negative effect of holding LiPo batteries at 100% would outweigh everything else.  Elaborating on Option 2  There are lots of good reasons why Option 2 is the way it is:   Because of known LiPo chemistry issues, holding a 100% charge for a long time causes the battery to degrade Apple specifically recommends not to leave it plugged in all the time  Draining a battery all the way to zero all the time is bad for the cell (Which is why ~10% is used) Heat is a killer, and when plugged in AND charging, you get extra heat that causes damage.   There are a couple reasons why I'm challenging Option 2:   It seems silly that the most intuitive use case for thousands of people that use their macs as primary computers is wrong. If the hypothesis in option 1 is true, you're just burning unnecessarily through your finite charge cycles. It's a pain to have to remember to keep plugging in and plugging out and be worried about whether or not your device is plugged in. Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$ For $a\ge 0$ let's define $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right)dx.$$ Find explicit formula for $I(a)$.   My attempt: Let  $$\begin{align*} f_n(x) &amp;= \frac{\ln\left(1-2 \left(a+\frac{1}{n}\right)\cos x+\left(a+\frac{1}{n}\right)^2\right)-\ln\left(1-2a\cos x+a^2\right)}{\frac{1}{n}}\\ &amp;=\frac{\ln\left(\displaystyle\frac{1-2 \left(a+\frac{1}{n}\right)\cos x+\left(a+\frac{1}{n}\right)^2}{1-2a\cos x+a^2}\right)}{\frac{1}{n}}\\ &amp;=\frac{\ln\left(1+\dfrac{1}{n}\left(\displaystyle\frac{2a-2\cos x+\frac{1}{n}}{1-2a\cos x+a^2}\right)\right)}{\frac{1}{n}}. \end{align*}$$ Now it is easy to see that $f_n(x) \to \frac{2a-2\cos x}{1-2a\cos x+a^2}$ as $n \to \infty$. $|f_n(x)|\le \frac{2a+2}{(1-a)^2}$ RHS is integrable so $\lim_{n\to\infty}\int_0^\pi f_n(x)dx = \int_0^\pi \frac{2a-2\cos x}{1-2a\cos x+a^2} dx=I'(a)$. But  $$\int_0^\pi \frac{2a-2\cos x}{1-2a\cos x+a^2}=\int_0^\pi\left(1-\frac{(1-a)^2}{1-2a\cos x+a^2}\right)dx.$$ Consider $$\int_0^\pi\frac{dx}{1-2a\cos x+a^2}=\int_0^\infty\frac{\frac{dy}{1+t^2}}{1-2a\frac{1-t^2}{1+t^2}+a^2}=\int_0^\infty\frac{dt}{1+t^2-2a(1-t^2)+a^2(1+t^2)}=\int_0^\infty\frac{dt}{(1-a)^2+\left((1+a)t\right)^2}\stackrel{(*)}{=}\frac{1}{(1-a)^2}\int_0^\infty\frac{dt}{1+\left(\frac{1+a}{1-a}t\right)^2}=\frac{1}{(1-a)(1+a)}\int_0^\infty\frac{du}{1+u^2}=\frac{1}{(1-a)(1+a)}\frac{\pi}{2}.$$ So $$I'(a)=\frac{\pi}{2}\left(2-\frac{1-a}{1+a}\right)\Rightarrow I(a)=\frac{\pi}{2}\left(3a-2\ln\left(a+1\right)\right).$$  It looks too easy, is there any crucial lack?  $(*)$ — we have to check $a=1$ here by hand and actually consider $[0,1), (1,\infty)$ but result on these two intervals may differ only by constant - it may be important but in my opinion not crucial for this proof. How can I get LaTeX to warn about unreferenced figures and unused labels in general? When writing documents, I consider it to be necessary that all figures and tables are referenced (\ref) in the text, whether or not I've remembered to \label them. LaTeX does not currently warn if a figure or table is not referenced. Is there a way to ensure that this is flagged up as a warning?  Note that I am specifically concerned about figures and tables, not chapter or section headings; it will be no good looking just at \labels that don't have an associated \ref.  In other words, it should report boo and not report bla in the following MWE:  \documentclass{article} \begin{document} \section{BLA}\label{bla} \section{BOO}\label{boo} In Section~\ref{bla} we do something. \end{document} Split Data Based on Year in a Visual force Page PDF Let $m \in \mathbb{Z^+} , n \in \mathbb{Z^+}$ and let $d=\gcd(m,n)$. Prove that  $$ m\mathbb{Z}+n\mathbb{Z}=d\mathbb{Z}. $$  My attempt is use inclusion to show. Let $a \in m\mathbb{Z}+n\mathbb{Z}$. Then we have $a=ms+nt$. Since $d=\gcd(m,n)$, by Bézout's Lemma, we have $d=ms+nt$ for some integers $s$, $n$. Hence, we have $a=ms+nt=d \in d\mathbb{Z}$. Let $a \in d \mathbb{Z}$. Then we have $a=db=msb+ntb \in m\mathbb{Z}+n\mathbb{Z}$. Is my proof valid?  Remark: the following is the part 2 of the question. Prove that $m\mathbb{Z}\cap n\mathbb{Z}=\frac{mn}{d}\mathbb{Z}$. This one I have no idea how to start The definitive guide to ConTeXt MkIV documentation My primary router has trouble extending wifi coverage to some bedrooms in the house so I am planning to add a second router to extend the wifi network using the wired Ethernet ports in one of the areas of poor coverage. Basic layout -   The ISP's router is in my family room downstairs (fiber connection) and the internet connects to the fiber via a box in an upstairs closet.   I purchased and installed an unmanaged Ethernet switch/splitter. In the upstairs closet where all of the Ethernet wires meet, the family room Ethernet cable was plugged into the ISP's box. I disconnected this wire and replaced it with a CAT5 cable running from the ISP's box in the closet to the Ethernet switch's port 1.  I think connected the family room Ethernet cable to port 2 and did the same in order for all the remaining Ethernet cables to various bedrooms.   So far, the switch is working fine.  I have wifi working with internet coming off of the router in the living room so I believe the switch is installed correctly otherwise that wifi should not have internet coming off of the living room jack.   Here is where things go awry - I took my second wifi router and connected the WAN/Yellow port to the Ethernet wall jack in another bedroom.  I then connected a lap top to one of the Ethernet ports on this router and found "no internet" after connecting.  On a whim, I changed the connection so that the router was connected to the Ethernet wall jack in one of the white ports but with the same results.   I don't think this counts as cascading and I don't really need a new network since I do not share the connection with any non-family members.  The router has been reset to factory settings to clear out any old instructions - but I still can't get it to connect to the internet or access the router to change the settings even - it's a netgear and the routerlogin.net will not let me access the router even when I am on the working WIFI for internet and have a cable running from the laptop to the netgear router.   I could use some help - just trying to extend the wifi range and it seems like it should be much less complicated. Tried following instructions in several other posts and they did not work for me - because I cannot access the router to change any settings. Custom right-click action in Unity Launcher keeps cursor busy for 20 seconds To have a "minimize window" option through right click on an icon in the Unity launcher (Ubuntu 14.04), I followed the detailed instructions  about changing a .desktop file and made a custom Firefox launcher in ~/.local/share/applications/ folder. The relevant part of the .desktop file is:  Actions=NewWindow;NewPrivateWindow;Minimize  [Desktop Action NewWindow] Name=Open a New Window Exec=firefox -new-window OnlyShowIn=Unity;  [Desktop Action NewPrivateWindow] Name=Open a New Private Window Exec=firefox -private-window OnlyShowIn=Unity;  [Desktop Action Minimize] Name=Minimize Windows Exec=sh /home/myusername/Documents/xdotool_sh/minimize.sh firefox OnlyShowIn=Unity;   The desktop action "Minimize" invokes a simple shell script, minimize.sh that has the following content:  #/bin/bash name=$1 for i in $(xdotool search --class "$name"); do     xdotool windowminimize $i done   The script uses xdotool, that can be installed from the official repositories, to find all the firefox windows, iterate over them and minimize them.  The script works and the launcher right menu option "Minimize Windows" works as well, but as soon as the windows are minimized, the mouse pointer gets in "busy" mode and stays like this for about 20 seconds (although the mouse actions are still responsive).   Does anyone know why starting a shell script from a right menu option in Unity could lead to this behaviour?  EDIT: Apparently the wait period is inevitable, as explained in Jacob Vlijm's answer. Since the mouse remains responsive, avoiding the transformation of the pointer in the spinning wheel is a partial esthetic workaround, as explained on .  EDIT2: giving the system a fake window is a better solution, as explained by Jacob below. What's the difference between the simple past tense and the present perfect? The simple past tense indicates that an activity or situation began   and ended at a particular time in the past.      The present perfect expresses the idea that something happened (or   never happened) before now, at an unspecified time in the past.The   exact time it happened is not important. It also expresses the   repetition of an activity before now. The exact time of each   repetition is not important.   Even though I understand the explanation above, I can't figure out: If the present perfect states the idea that something happened in the past, why not use the simple past?  For example, would you please explain the following sentence to tell the difference between these two tenses?         The plane crashed there.    The plane has crashed there.      What is the biggest difference between them? I've been reading a Wikipedia article that describes the symptoms of rabies in various species, and I want to add a clarification note about which species (or species) a specific paragraph refers to. How can I make it clear that I'm referring either to the singular or plural of the word "species" when the singular of this word is the same as its plural? "Species or species" is the only grammatically correct way that I can think of, but it seems redundant and confusing. Is there any better way to contrast the singular of "species" from the plural of "species"? Allow moderators to see delete/undelete activity on users' profiles We get auto flags for 'possible vandalism' related to deletes, and these can be helpful at times in finding 'rage quit' users, or other potential problems.  But often, I would like to know if there is a prior pattern with the user doing this. While we can see posts of theirs which have been flagged, and perhaps find previous auto flags this way, it would be nice to see an overall view of their self-deletions and undeletions; mostly just to identify patterns.  Such a feature would be nice if it was activity centered, rather than post-centered. That is; each delete/undelete action should be shown. So we could easily see, for example, if someone went back-and-forth on a single post.  The "activity" tab in the profile would be a logical place to show these events.  I think this would be best to be limited to instant self-deletions, not deletion votes. What are the effects of exceptions on performance in Java? Question:  Is exception handling in Java actually slow?  Conventional wisdom, as well as a lot of Google results, says that exceptional logic shouldn't be used for normal program flow in Java.  Two reasons are usually given,   it is really slow - even an order of magnitude slower than regular code (the reasons given vary),     and    it is messy because people expect only errors to be handled in exceptional code.     This question is about #1.  As an example,  describes Java exception handling as "very slow" and relates the slowness to the creation of the exception message string - "this string is then used in creating the exception object that is thrown. This is not fast."  The article  says that "the reason for this is due to the object creation aspect of exception handling, which thereby makes throwing exceptions inherently slow".  Another reason out there is that the stack trace generation is what slows it down.  My testing (using Java 1.6.0_07, Java HotSpot 10.0, on 32 bit Linux), indicates that exception handling is no slower than regular code.  I tried running a method in a loop that executes some code.  At the end of the method, I use a boolean to indicate whether to return or throw.  This way the actual processing is the same. I tried running the methods in different orders and averaging my test times, thinking it may have been the JVM warming up.  In all my tests, the throw was at least as fast as the return, if not faster (up to 3.1% faster).  I am completely open to the possibility that my tests were wrong, but I haven't seen anything out there in the way of the code sample, test comparisons, or results in the last year or two that show exception handling in Java to actually be slow.  What leads me down this path was an API I needed to use that threw exceptions as part of normal control logic.  I wanted to correct them in their usage, but now I may not be able to.  Will I instead have to praise them on their forward thinking?  In the paper , the authors suggest that the presence of exception handlers alone, even if no exceptions are thrown, is enough to prevent the JIT compiler from optimizing the code properly, thus slowing it down.  I haven't tested this theory yet. What factors influence the spring constant? For certain springs we have been taught that Hookes law, $F=-kx$ works. For such springs, what factors determine k. For instance, k is proportional to length. Can k be expressed in terms of dimensions, mass and other fundemental quantities, or is k a fundemental property of the spring itself? I know that the Linux kernel supports TRIM as of version 2.6.33, so there should be TRIM support in ubuntu.  Is TRIM enabled by default or do I need to change some settings or install additional software for it to work? If so, how? Is there any tweak to bring back emblems in Nautilus? Zip folder without including path to folder and require only one argument I'm trying to make a zip function that'll do all the steps I typically do when zipping a file or folder. Say my current directory is ~, and I want to zip a folder named Folder1, whose file path is ~/Folder2/Folder1.  Issuing  cd ~/Folder2 &amp;&amp; zip -r Folder1 Folder1 &amp;&amp; cd -   does exactly what I want. Namely, it zips so that the output Folder1.zip does not contain folders that lead up to the folder I'm zipping (namely, /home/user/Folder2), it zips recursively, and it takes in the first argument of zip with the exact name as the file or folder I'm zipping (Folder1.zip).  So I want to make a function in my .zshrc something akin to  coolerzip() { cd /path/to/file &amp;&amp; zip -r file file &amp;&amp; cd - }   where it need only take in one argument, which in this case, it will be ~/Folder2/Folder1. Then Folder1 will be the name for both files in coolerzip, and I will cd right into ~/Folder2 so Folder1 lies in my current directory before zipping. Is there any way I can achieve this so that coolerzip only requires one argument rather than three? Does entropy increase or decrease as our Universe is expanding? Scientists say that entropy of our universe is increasing as it is expanding and our universe is cooling down gradually from the time of its birth. If something is getting cooler and cooler, then how can it become more random (entropy increase) with the passage of time? According to laws of thermodynamics, at absolute zero temperature, the entropy is zero. That means as we go down and down to the temperature scale entropy must decrease, but why is it not happening in the case of our universe?