What is meant by a stiff/rigid ODE?
What's the intuition behind "stiff systems"?
SQL Server 2012 and 2016 Standard:  If I put if-else logic in a stored procedure to execute one of two branches of code, depending on the value of a parameter, does the engine cache the latest version?  And if on the following execution, the value of the parameter changes, will it re-compile and re-cache the stored procedure, since a different branch of the code must be executed? (This query is quite expensive to compile.) I am trying to solve an integral in $k$-space where I want to calculate the ratio of the average energy of an arbitrary electronic band ($E(k)=k^n)$ to the square of average group velocity of electrons($v_g   = k^{n-1})$ approximately in the form of this expression given by $$L=\frac{\frac{\sum_{\mathbf{k}}k^{n}}{\sum_{\mathbf{k}}1}}{\left(\frac{\sum_{\mathbf{k}}k^{n-1}}{\sum_{\mathbf{k}}1}\right)^2}\;.$$  Now, I want to do this for $d$-dimensions, so I convert this sum to integral.  $$L=\frac{\dfrac{\int_{0}^{k_f}k^{n} d^d k}{\int_{0}^{k_f}1d^d k}}{\left(\dfrac{\int_{0}^{k_f}k^{n-1}d^d k}{\int_{0}^{k_f}1d^d k}\right)^2}$$ where $d^dk  = \dfrac{d \pi^{d/2}}{\Gamma(\frac{d}{2}+1)}k^{d-1}dk$ is the volume element in $d$-dimensional space.  I want to get a simple expression for $L$ and how it depends with $k_f,n$ and $d$. any physical insight into this integral could also be helpful. The Problem For a small mass a distance $R_i$ away from the center of the Earth, how long would it take for the object to fall to the surface of the Earth, assuming that the only force acting upon the object is the Earth's gravitational force?  Relevant Information The following discussion seems to have solved exactly the same problem:   However, upon working out the mathematics, I'm not exactly sure how to evaluate the constant of integration.   A Partial Solution $$ F=\frac {-GMm}{s^2} $$ $$ a=\frac {-GM}{s^2} $$ $$ \frac {dv}{dt} = \frac {-GM}{s^2} $$ Multiplying by $v$ and then integrating by $dt$ on both sides, we have $$\frac {1}{2} v^2=\frac {GM}{s} +c_1$$ where $c_1$ is a constant of integration. Substituting initial conditions of $v=0, s=R$, we have $$\frac {1}{2} v^2=GM(\frac {1}{s}-\frac{1}{R})$$  At this point of time, when I use Wolfram Alpha, I get  $$c_2+\sqrt{\frac{2}{R}}t=\frac{\sqrt{s}(s-R)+R\sqrt{R-s}\times{\tan^{-1}(\sqrt{\frac{s}{R-s}})}}{\sqrt{GM(R-s)}}$$  where $c_2$ is a constant of integration. Substituting initial conditions of s=R, t=0, we find that the term $$\tan^{-1}(\sqrt{\frac{s}{R-s}})$$ is undefined. At this point, I'm stuck. Any ideas on where I've made the mistake here?  (For those interested, this question was inspired by the Greek myth which states that a bronze hammer dropped from heaven would take 9 days to hit the Earth and would reach on the tenth). A secant inequality for convex functions Groups of prime squared order I will be attending a summer school which will last 10 days.  I want to know: for applying for a Schengen visa, is the purpose of my trip considered as touristic or study? Polygon label repeated for each tile How do you integrate $e^{x^2}$? I know that $\int{\frac{1}{x}}dx$ is simply $\ln{(x)}+c$ (-which is clearly unrelated to the problem but I just thought I would share anyway) but I am not sure how to approach $e^{x{^2}}$. Perhaps a substitution? Overheating results in system shutdown Are there uncountably many surjections from $\mathbb N$ to $\mathbb N$? Statement: There are uncountably many surjections from $\mathbb N$ to $\mathbb N$.  Is the following proof valid:  The real numbers are uncountable.   There exists a bijection between the set of real numbers and the set of all functions from $\mathbb N$ to $\{0,1\}$.   Given a function, $f$, from $\mathbb N$ to $\{0,1\}$, construct a surjection from $\mathbb N$ to $\mathbb N$, denoted $g$, as follows:  We map the $n$th composite number to $n$, and the nth prime, to $f(n)$.   Clearly this is a surjection since the composite numbers are countable. Clearly different functions $f$ lead to different surjections $g$. Since there are uncountably many $f$s, there are uncountably many surjections $g$. How to find an ellipse , given 2 passing points and the tangents at them? Please answer to a question , how to find an ellipse which passes the 2 given points and has the given tangents at them. And one related question is that the given condition can decide just one ellipse which satisfies it?  Thank you in advance.  Edit : May be , I can say the first answer (by Mr. André Nicolas) is for the general case. Is there no special case where only a finite set of ellipses satisfies the condition?  Edit : According to the answers and comments , I can compute an arbitrary chosen ellipse for my condition (using the method by Mr.Patrick Da Silva). But there are possibly many others which satisfy my conditions. Am I right? WinForms DataGridView font size How do I change font size on the DataGridView? I want to perform performance testing on my test server.  How can I capture workload from production and replay on my test server?   Basically I'm looking for detailed steps so that I can replay the workload from production to my test environment. tikz relative coordinates Suppose one defines coordinates a and b. One may draw a line connecting these two coordinates (\draw (a) -- (b)). Suppose one desires to draw a line that is shifted upward 1 unit relative to a and b. A first guess is the command \draw (a){}+(0,1) -- (b){}+(0,1). This command, however, connects the point a+(0,1) to the point b, presumably because addition is subsequent to the draw command. How does one obtain the desired result (drawing from a+(0,1) to b+(0,1)?  MWE:  \documentclass{article} \usepackage{tikz} \usetikzlibrary{calc} \begin{document} \begin{tikzpicture}  \draw[fill] (0,0) circle (2pt) coordinate (a);  \draw[fill] (5,0) circle (2pt) coordinate (b);  \draw (a) -- (b);  \draw (a){}+(0,1) -- (b){}+(0,1); \end{tikzpicture} \end{document}     Edit: How does one draw coordinates at a point shifted relative to the arithmetic average of the coordinates of two points (something akin to \node at ($(((s1)+(s2))/2)+(0,1)$)). I installed Ubuntu 12.04 a month ago and am using it till now. I failed to notice that all this time there was no sound at all while running Ubuntu, even while playing a game in Wine. The weird thing is that only the startup sound comes when I log in (Indian/African drum tone), then comes the utter silence.  I tested both Digital Output (S/PDIF) and the speakers in the sound settings but can hear nothing.  Any help? I cannot find the Polygon Neighbors tool. I have searched for it as I usually do and I have also browsed the proximity toolset but it is not there. The other Tools, e.g. Buffer and Near are there and work fine. Here is a screenshot: How can I get a debug log for the sites guest user/public profile? [updated for Spring 18] How many days a year is a U.S. citizen allowed to stay in Morocco? I arrived in Morocco in October of 2014 and renewed my visa by traveling to Spain by ship for a day and getting my passport stamped again in January of 2015. My question is, am I allowed to leave the country and repeat this process this year by traveling again to another country, other than my own in order to return to Morocco for another 90 days? And if so, how many times a year is this allowed? I have read 180 days on some websites but there is not any clear answers anywhere about this that I can find. The browser's "Back" button cause problems with the review buttons This is an unofficial list/changelog of new features and various changes to Stack Overflow and the Stack Exchange network.  It is maintained by the community, while a Stack Exchange employee changes the accepted answer to ensure that the latest changes remain on top (given default user settings).  To see the list in reverse chronological order (latest entry on top), .  For those adding new features here: please post only feature changes that affect the whole network and have some global impact. i.e. bug fixes don't belong here, nor do mobile app updates.