A differential equation is  if a numerical scheme requires a very small time-step in order to be stable for that equation. However I don't understand why it is called stiff (sometimes rigid). Even the wikipedia page says it's more of a phenomenon than a mathematically definable property. One characterization is the stiffness ratio, defined on the wikipedia page as $$ \frac{|Re\bar{\lambda}|}{|Re\underline{\lambda}|}$$ where I think $\bar{\lambda}$ and $\underline{\lambda}$ refer to the eigenvalues with maximum and minimum real parts. So why is it called this way? I don't know how to interpret this stiffness ratio. I was thinking of stiff as similar to &quot;stringent&quot; or &quot;demanding&quot; as a stiff person would be, which makes sense for the &quot;requiring a small time step&quot; definition. How is the requirement of a small time-step related to the stiffness ratio?
One of the mysteries that my numerics class left me with are "stiff systems". Our prof didn't really manage to explain what they are. Now that the class (and the test) are over, I'm still curious.   The most concise definition about "stiff systems" I've found so far is: "There is no universally accepted definition of stiffness". So, if there is none, I don't expect one.   I'd be thankful about information on both of the following points:   From what kind of (real-world) problems do stiff systems arise? What's so "stiff" about them? What's the good thing - or the bad thing - that I'd worry about when I have to numerically solve such a real-world problem?
(A kind soul at  suggested I post here as well, sorry if out of bounds.)  I'm trying to programmatically model a damped harmonic spring for use in mobile UI animations (physics mathematics isn't my background, please pardon any misconceptions).  Having derived the parameters for the general case equation, I can iteratively calculate values until I reach a suitable threshold, though because this is bound to "simple" trigonometric and $e^{x}$ functions on the CPU, the 4000-some-odd steps can cause about 0.25 seconds lag on slow devices while it calculates. I'd like to speed this up using my platform's super-optimized vector and BLAS/LAPACK variants.  The requirement for doing this is precalculating the number of steps necessary to reach my threshold value.    In the underdamped case, where the roots of the characteristic function of the differential equation are non-real, I can use algebraic tricks to get my values:  $$x(t) = c_{1}e^{r_{1}}\cos(i_{1}t) + c_{2}e^{r_{2}}\sin(i_{2}t)$$  (Given $r_{1}$, $i_{1}$, $r_{2}$, and $i_{2}$ are the real and irrational components of my two roots, respectively.)  Knowing that $r_{1} = r_{2}$ and $i_{1} = -i_{2}$, I can simplify to:  $$x(t) = c_{1}e^{r_{1}}\cos(i_{1}t)$$  And get my desired value of $t$ for my threshold $a$:  $$t = \arccos(a / c_{1} / e^{r_{1}}) / i_{1}$$    When the roots are real, the equation looks a lot simpler:  $$x(t) = c_{1}e^{r_{1}} + c_{2}e^{r_{2}}$$  However, I don't have my trig functions floating around to help me solve it (even if I did, the irrational components being 0 would cause problems, of course).  Take the concrete example on pages 3-4 of  (my bible during this process), since they at least solve cleanly:  $$x(t) = 1.5e^{-t} - 0.5e^{-3t}$$  I know how I would solve for $t$ to get my for when $x(t) = a$ on paper, by setting $x=e^{t}$, solving, and back substituting, but I don't have that luxury here.  I can make a few assumptions: the roots and constants are all real. I'm always going to be looking for the smallest, first, positive value of $t$. Obviously, the iterative solution is the simplest for this case, but in the end that would involve more steps and therefore be slower no matter what my other optimizations would be. How, then, would I go about solving for my threshold value algorithmically in this (supposedly) simplified case?  Addendum  The underdamped solution presents an extra requirement. The motion curve will oscillate back and forth a few times across the endpoint. Therefore, "first and lowest" $t$ requirement is not necessarily true. In my current, iterative code, the threshold value is both checked against the distance from the current $x(t)$ to the endpoint, as well as to the distance from the previous $x(t)$ as well to allow for a number of oscillations. This might make a more efficient solution nearly impossible. What's the Best Way to Shuffle an NSMutableArray? If you have an NSMutableArray, how do you shuffle the elements randomly?  (I have my own answer for this, which is posted below, but I'm new to Cocoa and I'm interested to know if there is a better way.)    Update: As noted by @Mukesh, as of iOS 10+ and macOS 10.12+, there is an -[NSMutableArray shuffledArray] method that can be used to shuffle. See  for details. (But note that this creates a new array, rather than shuffling the elements in place.) Is it possible to be admitted to a masters in Germany with a 3-year bachelors from another country? In Book 1, Artemis et al escaped from the time stop by drugging themselves to sleep.  In the same time stop, Julius Root "accidentally" plugged Briar Cudgeon with a tranquilizer dart.  Cudgeon fell asleep, but did not disappear from the time stop.  Why not? I won a Samsung Galaxy S6 Edge, signed into it with my Google account, played around with it, wiped it, and sold it.  Now the person can't get past setup because it's saying he must login with an account previously synced to the phone.   Is there anything I can do from my side to remove this lock? Pointwise convergent and total variation Are there any distinct finite simple groups with the same order? In finite group theory, it's often quite easy to show that there are no simple groups of a given order $n$.  My question is different: is there some natural number $n$ such that there are two non-isomorphic simple groups of order $n$?  The two easiest families of finite simple groups - $\mathbb{Z}_p$ for $p$ a prime and $A_n$ for $n\ge5$ - clearly don't yield any examples.    It should be possible to check this using the , but I don't know enough about the fabled 'sixteen infinite families of groups of Lie type' to do the check myself.  Also, if the answer is 'no', I'd be much more interested to see an elementary(ish) proof than I would be to see a proof that relies on over 3,000 pages of non-stop hard group theory. Ubuntu 14.04 System Logging I've set up an Ubuntu 14.04 Server (from an Ubuntu 12.04 Server) but today I wanted to look in the auth.log but this file is not used. It seems that ´syslog´ was deleted.  How does Ubuntu 14.04 log? I need this for a server.  Thank you. Internet Accounts spontaneously forgets passwords and I can't reauthorize them I've had this problem before, but I was able to resolve it that time. macOS 10.13.4, everything up to date. Yesterday, Internet Accounts forgot my LinkedIn password and would not reauthenticate. Today, it forgot my Facebook password.   I've tried:   App specific password. Removing the account and re-adding it. Turning off 2-factor authentication. Deleting associated entries from Keychain. Logging out of iCloud, deleting Keychain entries, removing and re-adding the account, then logging back in.   First of all, what the hell—why does this keep happening?! Secondly, what else can I do to fix this short of reinstalling macOS? Question statement is as in the title:     $\varphi:G\to H$ is a Lie Group Homomorphism, then $\ker \varphi$ has Lie algebra $\ker D_e \varphi$   Since $\varphi$ is a Lie group homomorphism, in particular it is continuous. As $H$ must be Hausdorff, every point set is closed, and hence $\{e\}$ is closed in $H$, which implies that $\ker\varphi = \varphi^{-1}(e)$ is closed as well.  So $N=\ker\varphi$ is an embedded Lie subgroup of $G$. Therefore, the inclusion $j:N\to G$ is a Lie Group homomorphism, and induces a Lie algebra homomorphism $D_e j :T_e N \to T_e G$. So I only need to show that $D_e j$ is in fact an isomorphism between $T_e N$ and $\ker D_e \varphi$.  But I'm stuck at this point. I know that for $X\in T_e N$, we have, for $f\in C^{\infty}(H)$,  $$ D_e\varphi (D_e j(X))(f)=D_e j (X)(f\circ \varphi)=X(f\circ \varphi\circ j)=0$$  So $D_ej(T_e N) \subset \ker D_e \varphi$.  At this point, I can either show inverse inclusion: that is, for every $X\in \ker D_e \varphi$, we can find $Y\in T_e N$ such that $D_e j (Y)=X$, or for all $f\in C^{\infty}(G)$, $Y(f\circ j)=X(f)$  Or I can somehow show that dimensions of the two spaces are the same.   But I can't proceed from here.  It'd be great if someone could help me Is it possible to dynamically set Views' pager settings? Say for instance, you have a view that displays when viewing a certain node type, but you want the pager settings to be different for each individual node. I was thinking of making a CCK field like "pager setting" and then give it an integer option of x-y. But I don't know if it's possible to somehow dynamically plug that field into the Views' settings. Or is there another way to do this maybe? Clarify or visualize the parameters involved for receiving the Curious badge How do I test if all minecraft players on a server are in a radius? This has got me. I need a command that tells me if all players on a server are in a radius of a command block. If one of them isn't in the radius, the command will be false. Any ideas? Thanks! I have defined my own language whose syntax is very close to the one of Pascal. I would like to type the following program for instance. Many people use the packages algorithmicx or algorithm2e to print programs, but it seems that we have to use their specific keywords of its syntax like procedure, end while.  I just want to indent spaces when needed, and bold some words when I want (sorry that I could not bold letters in the following figure). Do I have to use  texttt for each line? Also, with verbatim, I could not bold words...  Also, as the program is not large, I would need to put it on the left side of the page, and leave space for some explanasion text on its right hand side. So I  guss I need to get them all together, and use multicols.  Could anyone help?  program test                                      some text here var   i : integer;                                    ...    b : boolean; begin    i := 1;                                        ...    b := true;    while i &lt; 15 do       i := i  + 1;       b := not b                                  some text here    od end   Edit1: Following @Werner's tabbing solution, I have made the code a minipage:  \begin{minipage}[t]{0.48\linewidth}% \begin{tabbing} Werner's solution \end{tabbing} \end{minipage}\hfill \begin{minipage}[t]{0.48\linewidth} a paragraph \end{minipage}   Actually as the code shows, I want the texts to be another separated minipage on the right hand side of the code. I have then 2 questions:   is it reasonable to make 0.48\linewidth instead of 0.5\linewidth? 0.48\linewidth is a conventional choice? is it possible to make a frame around the code, and use \caption and \label to make it a reference (I prefer figures)? Restoring MySQL database from physical files Is it possible to restore a MySQL database from the physical database files. I have a directory that has the following file types:  client.frm client.MYD client.MYI    but for about 20 more tables.  I usually use mysqldump or a similar tool to get everything in 1 SQL file so what is the way to deal with these types of files? How do I eliminate the vertical realignment when using arrayrulecolor with cmidrule? I'm using consecutive \cmidrule commands in conjunction with \arrayrulecolor to alternately hide and unhide horizontal rules in a table.  However, when \arrayrulecolor is used with \cmidrule, some odd vertical spacing is introduced. I'd like to eliminate it.  (I'm using longtable, so a colorbox, minipage, or anything else tied to a page won't work.)  Below is an image that introduces the challenge. The first block shows a \toprule.  My goal is to create a shaded border around the table, so I've added a dummy column.  The second rule shows a \cmidrule that skips the first column.  The third block shows how I can use a \cmidrule to fill in the whitespace with a shorter \cmidrule.  Finally, I want to recolor this short \cmidrule to the background color and restore the color later.  Note the jagged vertical skipping in case 4 compared to case 3.  Why would \arrayrulecolor change vertical alignment?  How can I eliminate the jagged edges?    Here is the code:   \documentclass[11pt]{article}  \usepackage{longtable} \usepackage{booktabs} \usepackage{color} \usepackage{xcolor} \usepackage{colortbl}  \setlength{\aboverulesep}{0pt} \setlength{\belowrulesep}{0pt} \setlength{\extrarowheight}{0.25ex}  \newcolumntype{L}{&gt;{\columncolor[gray]{0.8}}l} \newcolumntype{C}{&gt;{\columncolor[gray]{0.8}}c} \newcolumntype{R}{&gt;{\columncolor[gray]{0.8}}r}  \begin{document}  \textsf{ \begin{longtable}{LLLLL}  &amp; \multicolumn{3}{@{}L}{\small {\tablename\ \thetable{}} - demonstration of odd vertical alignment with arrayrulecolor} &amp; \\%  &amp; \multicolumn{3}{@{}L}{\small {case 1: the line below is a toprule}} &amp; \\% \toprule  &amp; \multicolumn{3}{@{}L}{\small {case 2: the line below is a cmidrule that spans the middle 3 columns.  Note the whitespace!}} &amp; \\%  &amp; \multicolumn{3}{@{}L}{\small {The challenge: hide the widespace giving the appearance of a spacing around a longtable.}} &amp; \\% \cmidrule[\heavyrulewidth]{2-4}  &amp; \multicolumn{3}{@{}L}{\small {case 3: the line below is 3 cmidrules, 1 each for the first and last columns and 1 for the middle}} &amp; \\%  &amp; \multicolumn{3}{@{}L}{\small {note how the cmidrules align vertically ... perfect!}} &amp; \\% \cmidrule[\heavyrulewidth]{1-1}\cmidrule[\heavyrulewidth]{2-4}\cmidrule[\heavyrulewidth]{5-5}  &amp; \multicolumn{3}{@{}L}{\small {case 4: Now, lets try recoloring the first and last cmidrules}} &amp; \\%  &amp; \multicolumn{3}{@{}L}{\small {how do I eliminate the odd vertical alignments?}} &amp; \\% \arrayrulecolor[gray]{0.80}\cmidrule[\heavyrulewidth]{1-1}\arrayrulecolor[gray]{0.0}\cmidrule[\heavyrulewidth]{2-4}\arrayrulecolor[gray]{0.80}\cmidrule[\heavyrulewidth]{5-5}\arrayrulecolor[gray]{0.00} \endfirsthead  &amp;  Arizona State University &amp;  &amp; ASU &amp;\\   &amp;   Baylor University &amp;  &amp; BAYLOR &amp;\\   &amp;   Boston University &amp; 164988 &amp; BU &amp;\\   &amp;   Brown University &amp; 217156 &amp; BROWN &amp;\\   &amp;   California Institute of Technology &amp; 110404 &amp; CALTECH &amp;\\   &amp;   Carnegie Mellon University &amp; 211440 &amp; CMU &amp;\\   &amp;   Case Western Reserve University &amp; 201645 &amp; CASE &amp;\\   &amp;   Columbia University &amp; 190150 &amp; COLUMBIA &amp;\\   &amp;   Cornell University &amp; 190415 &amp; CORNELL &amp;\\      &amp; Yale University &amp; 130794 &amp; YALE &amp; % \label{tab:schools} \end{longtable} }  \end{document} Finding $\int x^xdx$ I'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this:  Let $u=x^x$.  $$\begin{align} \int x^x \, dx&amp;=\int u \, du\\[6pt] &amp;=\frac{u^2}{2}\\[6pt] &amp;=\dfrac{\left(x^x\right)^2}{2}\\[6pt] &amp;=\frac{x^{2x}}{2} \end{align}$$  But it's certain that this isn't the correct way to evaluate that, and the answer must be wrong. How to set date format when manually inserting date on excel, without changing regional settings? How to calculate $(G/H)/(K/H)$? I have a group $G=\mathbb{Z}_{36}$ with normal subgroups $K$ and $H$, where $H$ is also a normal subgroup of $K$. What I'm really supposed to find is a bijection between $G/K$ and $(G/H)/(K/H)$ by using the third isomorphism theorem. I have already calculated $G/K$, $G/H$ and $K/H$, but I do not really know how to calculate $(G/H)/(K/H)$. Say that one of the elements in $G/H$ is $\{0,18\}$ and one of the elements in $K/H$ is $\{0,18\}$. Will one of the elements in $(G/H)/(K/H)$ be as follows? $$\{0,18\}+\{0,18\} = \{0+0,18+18\}=\{0,36\}=\{0\}$$ Another way I'm thinking it's possible to caluclate is $$\{0,18\}+\{0,18\} = \{0+0,0+18,18+0,18+18\}=\{0,18,18,36\}=\{0,18\}$$ Is any of these a correct way of calculating an element in $(G/H)/(K/H)$? There was a great SF story I read some time ago (probably 25+ years ago) where there was a well known treasure trove, guarded by a killer robot that would ask all comers a series of difficult questions from all areas (math, physics, literature, history, etc.)  But everyone eventually failed and died... until the last person managed to make it past the robot:     By just answering gibberish to the robot.  Unfortunately, after collecting a ton of treasure, the robot casually asked him why he wanted it, he answered in a sensible way (something like "because it's valuable"), and the robot killed him, because he failed the final test.   Any ideas who wrote this?  What it was titled?