In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on.  The first issue I have is that my book began with the statement that given  $$L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  the problem $L[y] \ = \ f$ can be re-cast in the form $L[y] \ = \ \lambda y$.  Now it could be a typo on their part but I see no justification for the way you can just do that!  More importantly though is the motivation for Sturm-Liouville theory in the first place. The story as I know it is as follows:  Given a linear second order ode   $$F(x,y,y',y'') = L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  it is an exact equation if it is derivable from a differential equation of one order lower, i.e.  $$F(x,y,y',y'') = \frac{d}{dx}g(x,y,y').$$  The equation is exact iff  $$a''(x) - b'(x) + c(x) = 0. $$  If $F$ is not exact it can be made exact on multiplication by a suitable integrating factor $\alpha(x)$.  This equation is exact iff  $$(\alpha(x)a(x))'' - (\alpha(x)b(x))' + \alpha(x)c(x) = 0 $$  If you expand this out you get the Adjoint operator  $$L^*[\alpha(x)] \ = \ (\alpha(x)a(x))'' \ - \ (\alpha(x)b(x))' \ + \ \alpha(x)c(x) \ = 0 $$  If you expand $L^*$ you see that we can satisfy $L \ = \ L^*$ if $a'(x) \ = \ b(x)$ &amp; $a''(x) \ = \ b'(x)$ which then turns $L[y]$ into something of the form  $$L[y] \ = \ \frac{d}{dx}[a(x)y'] \ + \ c(x)y \ = \ f(x).$$  Thus we seek an integratiing factor $\alpha(x)$ so that we can satisfy this &amp; the condition this will hold is that $\alpha(x) \ = \ \frac{1}{a(x)}e^{\int\frac{b(x)}{a(x)}dx}$  Then we're dealing with:  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \alpha(x)f(x)$$  But again, by what my book said they magically re-cast this problem as  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \lambda \alpha(x) y(x)$$  Then calling  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ ( \alpha(x)c(x)y \ - \ \lambda \alpha(x) )y(x) \ = \ 0$$  a Sturm-Liouville problem.  My question is, how can I make sense of everything I wrote above? How can I clean it up &amp; interpret it, like at one stage I thought we were turning our 2nd order ode into something so that it reduces to the derivative of a first order ode so we can easily find first integrals then the next moment we're pulling out eigenvalues &amp; finding full solutions - what's going on? I want to be able to look at $a(x)y'' \ + \ b(x)y' \ + \ c(x)y \ = \ f(x)$ &amp; know how &amp; why we're turning this into a Sturm-Liouville problem in a way that makes sense of exactness &amp; integrating factors, thanks for reading!
In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on.  The first issue I have is that my book began with the statement that given  $$L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  the problem $L[y] \ = \ f$ can be re-cast in the form $L[y] \ = \ \lambda y$.  Now it could be a typo on their part but I see no justification for the way you can just do that!  More importantly though is the motivation for Sturm-Liouville theory in the first place. The story as I know it is as follows:  Given a linear second order ode   $$F(x,y,y',y'') = L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  it is an exact equation if it is derivable from a differential equation of one order lower, i.e.  $$F(x,y,y',y'') = \frac{d}{dx}g(x,y,y').$$  The equation is exact iff  $$a''(x) - b'(x) + c(x) = 0. $$  If $F$ is not exact it can be made exact on multiplication by a suitable integrating factor $\alpha(x)$.  This equation is exact iff  $$(\alpha(x)a(x))'' - (\alpha(x)b(x))' + \alpha(x)c(x) = 0 $$  If you expand this out you get the Adjoint operator  $$L^*[\alpha(x)] \ = \ (\alpha(x)a(x))'' \ - \ (\alpha(x)b(x))' \ + \ \alpha(x)c(x) \ = 0 $$  If you expand $L^*$ you see that we can satisfy $L \ = \ L^*$ if $a'(x) \ = \ b(x)$ &amp; $a''(x) \ = \ b'(x)$ which then turns $L[y]$ into something of the form  $$L[y] \ = \ \frac{d}{dx}[a(x)y'] \ + \ c(x)y \ = \ f(x).$$  Thus we seek an integratiing factor $\alpha(x)$ so that we can satisfy this &amp; the condition this will hold is that $\alpha(x) \ = \ \frac{1}{a(x)}e^{\int\frac{b(x)}{a(x)}dx}$  Then we're dealing with:  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \alpha(x)f(x)$$  But again, by what my book said they magically re-cast this problem as  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \lambda \alpha(x) y(x)$$  Then calling  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ ( \alpha(x)c(x)y \ - \ \lambda \alpha(x) )y(x) \ = \ 0$$  a Sturm-Liouville problem.  My question is, how can I make sense of everything I wrote above? How can I clean it up &amp; interpret it, like at one stage I thought we were turning our 2nd order ode into something so that it reduces to the derivative of a first order ode so we can easily find first integrals then the next moment we're pulling out eigenvalues &amp; finding full solutions - what's going on? I want to be able to look at $a(x)y'' \ + \ b(x)y' \ + \ c(x)y \ = \ f(x)$ &amp; know how &amp; why we're turning this into a Sturm-Liouville problem in a way that makes sense of exactness &amp; integrating factors, thanks for reading!
Few font size-related layout issues I am using a default font size of 26px and stackoverflow's layout seems to not work well with such a large sizes.  The most important issue is that many inputs tend to overflow and partially get hidden below side text. This includes one with login, ‘Title’ and — the most important issue — the editor, which makes writing posts a little painful.  Another important one is that sometimes 'search' field doesn't fit in topbar and jumps lower, being partially hidden by logo. I met that issue on stackoverflow itself only while serverfault &amp; meta display fine (but that may be related to rep or sth).  There's one minor issue too — navbar is too narrow and ‘Ask Question’ jumps below — but this one looks quite nice (maybe it's even intended?).  Screenshot: (click to enlarge) Is $$d_1\mid n,d_2\mid n\iff [d_1,d_2]\mid n$$ true ? And if yes, how can I prove it ? I recall that $[d_1,d_2]$ is the least common multiple.  My tries  For the implication : Let $d_1\mid n$ and $d_2\mid n$. Then, $d_1d_2\mid n^2$. By the way, since $d_1,d_2\mid n$, we have that $(d_1,d_2)\mid n$ where $(d_1,d_2)=:\gcd(d_1,d_2)$  and thus $$[d_1,d_2](d_1,d_2)\mid n^2.$$  Question: How can I get $[d_1,d_2]\mid n$ from this?  For the converse, since $d_1,d_2\mid [d_1,d_2]\mid n$, the claim follow. I'm playing around with a native (non-web) single-player game I'm writing, and it occured to me that having a daily/weekly/all-time online highscore list (think Xbox Live Leaderboard) would make the game much more interesting, adding some (small) amount of community and competition. However, I'm afraid people would see such a feature as an invitation to hacking, which would discourage regular players due to impossibly high scores.  I thought about the obvious ways of preventing such attempts (public/private key encryption, for example), but I've figured out reasonably simple ways hackers could circumvent all of my ideas (extracting the public key from the binary and thus sending fake encrypted scores, for example).  Have you ever implemented an online highscore list or leaderboard? Did you find a reasonably hacker-proof way of implementing this? If so, how did you do it? What are your experiences with hacking attempts? Proof of the square root inequality $2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}$ An awful lot of people seem to use the phrase "substitute X for Y" to mean "replace X with Y", while I've always used and understood it as "replace Y with X". This makes sense to me, given that a substitute is the replacement, not the thing which has been replaced.  I thought I might be able to figure things out from a dictionary, but now after seeing two sorts of definitions, I'm even more confused:   to put or use in the place of another to take the place of; replace   The first one suggests that the object will be the substitute (replacement), the thing put into place, while the second suggests that the object will be the thing which has been replaced.  Is one of these usages more correct in any sense? (I know this is probably a pointless question, since people will continue to speak the way they speak.) Is there perhaps a difference between American and British English?   And most importantly, is there any reliable way to tell what someone means when they ask how to substitute honey for sugar, or is the verb simply guaranteed to cause confusion? I have this tex: \(\mu_{A}\left(x\right)\in \[0,1  \right\]\)  The issue here is that [ and ] are not showing around 0,1, why is that?  Thanks. I am debating whether or not to get the Potato Sack (yes, I realize I've been debating for several weeks now) bundle through steam.  I already own: Defense Grid: The Awakening, Super Meat Boy, and Audiosurf, all of which are part of the sack pack. What happens if I buy the Potato Sack?  Do I get 2nd copies of the three above mentioned games to gift elsewhere?  Is the price of the bundle reduced by the games I already have?  Do I get nothing special for already owning those 3 games? How to implement Enums in Ruby? What's the best way to implement the enum idiom in Ruby? I'm looking for something which I can use (almost) like the Java/C# enums. The Center of $\operatorname{GL}(n,k)$ The given question:  Let $k$ be a ﬁeld and $n \in \mathbb{N}$. Show that the centre of $\operatorname{GL}(n, k)$ is $\lbrace\lambda I\mid λ ∈ k^∗\rbrace$.  I have spent a while trying to prove this and have succeeded if $ k \subseteq \mathbb{R}$. So I imagine there is a nicer way to go about this. I have seen people saying take matrices where every element except one is zero in proving similar results but such matrices are not in $\operatorname{GL}(n,k)$ as they are not invertible.  What I have done.  I have said let $B \in$ Center then $B$ commutes with everything in $\operatorname{GL}(n,k)$. So take a permutation matrix, $P$ which swaps rows $i,j$ $P \neq I$. From this you can deduce that $B^T = B$ and that $B_{ii} = B_{jj}$. We can use the fact that $B$ is symmetric to find an orthogonal diagonalisation of $B$ by the spectral theorem.  This gives $QBQ^T = D \rightarrow B=D$. And as $B_{ii} = B_{jj}$ we can say $B = k * I$.   But the spectral theorem requires $B$ to be a real matrix.   How do I prove this for a general $k$?. It's big and empty and I can't get up to the top, what should I do? I can't jump high enough, anti-gravity doesn't seem to do anything either.  My first tactic was to use the octopus summoning crown to create a pile of octopi that I could climb up, but when I teleported to the back of the room they followed me back, trapping me in the entrance. Is it possible to embed a scrollable frame inside a PDF? (This is an end-of-year crazy question)   Is it possible to embed a (scrollable) frame or a PDF inside a PDF? If so, how to do it in (pdf)LaTeX?  This is something pretty common in HTML (see for example ). Here it is a webpage with PDF frames   Here it is a fake screenshot so you get the idea.  Portable and non-portable (e.g. only-Acrobat) answers are welcomed.     Example of a down arrow below a frame. The up arrow can be on top of the frame and the triangles point up. Achievements tab isn't working in Area 51 When I try to bring down the Achievements tab in Area 51, it's showing an error dialogue.  Strangely enough, the global inbox dropdown works just fine.  Could be connected to .  Screenshot: I did a fresh install of Ubuntu 16.04. /dev/md1 contains an ext4 filesystem that is mounted at /:  /dev/md1 on / type ext4 (rw,relatime,data=ordered)   Now I want to set the automatic filesystem check interval to one month, but I'm getting a strange error that I find no useful information about:  &gt; tune2fs -i 1m /dev/md1 tune2fs 1.42.13 (17-May-2015) tune2fs: Filesystem has unsupported read-only feature(s) while trying to open /dev/md1 Couldn't find valid filesystem superblock.   Rebooting didn't help. On another machine running Ubuntu 12.04, the same command works like a charm.  Does anyone have an idea what's wrong?  The command tune2fs -l /dev/md1 works fine, by the way. Here is its output:  tune2fs 1.42.13 (17-May-2015) Filesystem volume name:   &lt;none&gt; Last mounted on:          / Filesystem UUID:          38366a40-bbe1-476b-acfe-d271be6335eb Filesystem magic number:  0xEF53 Filesystem revision #:    1 (dynamic) Filesystem features:      has_journal ext_attr resize_inode dir_index filetype needs_recovery extent 64bit flex_bg sparse_super large_file huge_file dir_nlink extra_isize metadata_csum Filesystem flags:         signed_directory_hash Default mount options:    user_xattr acl Filesystem state:         clean Errors behavior:          Continue Filesystem OS type:       Linux Inode count:              8380416 Block count:              33521664 Reserved block count:     1676083 Free blocks:              32419430 Free inodes:              8312762 First block:              0 Block size:               4096 Fragment size:            4096 Group descriptor size:    64 Reserved GDT blocks:      1024 Blocks per group:         32768 Fragments per group:      32768 Inodes per group:         8192 Inode blocks per group:   512 Flex block group size:    16 Filesystem created:       Thu May 31 10:01:27 2018 Last mount time:          Thu May 31 10:49:41 2018 Last write time:          Thu May 31 10:15:14 2018 Mount count:              7 Maximum mount count:      -1 Last checked:             Thu May 31 10:01:27 2018 Check interval:           0 (&lt;none&gt;) Lifetime writes:          5782 MB Reserved blocks uid:      0 (user root) Reserved blocks gid:      0 (group root) First inode:              11 Inode size:               256 Required extra isize:     32 Desired extra isize:      32 Journal inode:            8 First orphan inode:       4325378 Default directory hash:   half_md4 Directory Hash Seed:      9d55d989-2c2b-4ad8-a29e-53c8798ec68e Journal backup:           inode blocks Checksum:                 0x3d9f8d23 Miktex Package Installation I recently installed MiKTeX with the texmaker editor and cannot figure out how to update the packages. When I attempt to use the subfigure package with the following preamble  \documentclass[a4paper,12pt]{article} \usepackage[english]{babel} \usepackage{graphicx} \usepackage{subfigure}   I then get a prompt to install a new package, which I then set to the online directory at ctan. I then get the following error:   ! LaTeX Error: File `subfig.sty' not found.Type X to quit or &lt;RETURN&gt; to proceed,or enter new name. (Default extension: sty)Enter file name:! Emergency stop.&lt;read &gt;   How do I properly install packages to miktex? I'm absolutely confused. Thanks in advance for any help. How do I erase an iPhone? I had an iPhone on loan from a friend to try it out and now I need to give it back. How do I make sure that everything is erased from it? SMS, Calendar, Phone Logs, contacts? What is the easiest way to find and replace text in 20 text files in a directory? I have 20 text files with various extensions in a directory.  I want to replace "nnn" with "ooo" in all files.  I could import them into a project in Eclipse or Visual Studio to replace them all but that is overkill and would add other meta files etc.  What is a simple way to do this, I'm thinking: select all files, right click, choose some tool called (e.g. "QuickReplacer") and fill in find and replace, click and done.  Does anyone know of a tool like this? In an Integral Domain is it true that $\gcd(ac,ab) = a\gcd(c,b)$? I've been wondering about this one for a while; I find it a little weird how abruptly it happens. Basically, why do we need just three uniforms for $Z_n$ to smooth out like it does? And why does the smoothing-out happen so relatively quickly?  $Z_2$:     $Z_3$:    (images shamelessly stolen from John D. Cook's blog: )  Why doesn't it take, say, four uniforms? Or five? Or...? How to suppress scientific notation when printing float values? Here's my code:  x = 1.0 y = 100000.0     print x/y   My quotient displays as 1.00000e-05.  Is there any way to suppress scientific notation and make it display as 0.00001? I'm going to use the result as a string. How to reinstall Ubuntu keeping my data intact?