Help with complicated system of coupled differential equations with a parameter
DSolve gets stuck on system of differential equations with unassigned variable
Find and replace text within a file using commands How can I find and replace specific words in a text file using command line? A quick search for currency regex brings up . The problem I have in choosing one is that regex is difficult to verify without testing all the edge cases. Does anyone have a regex for U.S. currency that has been thoroughly tested? My only requirement is that the matched string is U.S. currency and parses to System.Decimal:  [ws][sign][digits,]digits[.fractional-digits][ws]   Elements in square brackets ([ and ]) are optional.  The following table describes each element.   ELEMENT             DESCRIPTION ws                  Optional white space. sign                An optional sign. digits              A sequence of digits ranging from 0 to 9. ,                   A culture-specific thousands separator symbol. .                   A culture-specific decimal point symbol. fractional-digits   A sequence of digits ranging from 0 to 9. Find the limit of $\log\Big(\frac{(n^2+n)!}{n^2!n^{2n}}\Big)$ I have to prove that      $$\Big[\log\Big(1+\frac 1{n^2}\Big)+ \log\Big(1+\frac 2{n^2}\Big)+...+ \log\Big(1+\frac n{n^2}\Big)\Big] \to \frac 12$$   I know that $$\log\Big(1+\frac 1{n^2}\Big)+ \log\Big(1+\frac 2{n^2}\Big)+...+ \log\Big(1+\frac n{n^2}\Big)= \\ =\log\Big(\frac {n^2+1}{n^2}\Big)+ \log\Big(\frac {n^2+2}{n^2}\Big)+...+ \log\Big(\frac {n^2+n}{n^2}\Big)=\\ =\log\Big(n^2+1\Big)+ \log\Big(n^2+2\Big)+...+ \log\Big(n^2+n\Big)-2n\log n$$  Doing some manipulation we get:           $$\lim_{n \to +\infty}\log\Big(\frac{(n^2+n)!}{n^2!n^{2n}}\Big)=\frac 12$$      How can I prove this limit? Properly discontinuous action: equivalent definitions Let us define a properly discontinuous action of a group $G$ on a topological space $X$ as an action such that every $x \in X$ has a neighborhood $U$ such that $gU \cap U \neq \emptyset$ implies $g = e$. I would like to prove that this property is equivalent to, having given $G$ the discrete topology and in the $X$ locally compact Hausdorff case, the map $G \times X \rightarrow X \times X$ given by $(g, x) \mapsto (x, gx)$ being proper (i.e. closed and preimage of compact sets is compact) plus the action being free.  I have managed to prove one direction, that is, if the action is proper and free with $G$ having the discrete topology then it is properly discontinuous. I'm having trouble though with the other direction. Here is an attempt: let's denote by $\rho : G \times X \rightarrow X \times X$ the map $\rho(g, x) = (x, gx)$. Suppose $K \subset X \times X$ is compact. We wish to show $\rho^{-1}(K)$ is compact. Let $(g_i, x_i)$ be a net in $\rho^{-1}(K)$. Then $\rho(g_i, x_i) = (x_i, g_i x_i)$ admits a convergent subnet, so passing to it we may assume $x_i \rightarrow x$ and $g_i x_i \rightarrow y$. Essentially we must now find a way to prove $g_i$ converges, but I can't seem to do this. Any hints? Has anyone had any success using twitters typeahead.js within visualforce pages? How do I give text or an image a transparent background using CSS? The kernel and range of the powers of a self-adjoint operator Is it possible to recover Classical Mechanics from Schrödinger's equation? Let me explain in details. Let $\Psi=\Psi(x,t)$ be the wave function of a particle moving in a unidimensional space. Is there a way of writing $\Psi(x,t)$ so that $|\Psi(x,t)|^2$ represents the probability density of finding a particle in classical mechanics (using a Dirac delta function, perhaps)? ubuntu 18.04 LTS bluetooth [0cf3:3004] discovery not working "I", "me" and "myself" Possible Duplicate:           What is correct? We are a family of four: my father, my mother, my brother and me. or We are a family of four: my father, my mother, my brother and I. or We are a family of four: my father, my mother, my brother and myself. The question is, what elements on a DM screen are essential? I'm new at playing and DMing, altough a lot of experience watching games, just got a group to play with now. Started with the starter set in 5E and the Lost Mine of Phandelver adventure, however I've noticed the game can come to a stop when checking the rules or how to do something, while getting the books and stuff like that. So I thought to make a DM screen with some cheat sheets, of the most usual things new players, and new DM's have doubts on, and how to make the game flow well without long rule-checking breaks. What are the essential things to have on that screen? So far this is what I found  but not sure if it has all I need or so much stuff that'll also make the game stop a lot while cheking it out. $f(z)$ and $\overline{f(\overline{z})}$ simultaneously holomorphic Find a formula for a sequence $\{\sqrt{3},\sqrt{3\sqrt{3}},\sqrt{3\sqrt{3\sqrt{3}}},...\}$ Consider the following Gaussian Integral $$I = \int_{-\infty}^{\infty} e^{-x^2} \ dx$$  The usual trick to calculate this is to consider $$I^2 = \left(\int_{-\infty}^{\infty} e^{-x^2} \ dx \right) \left(\int_{-\infty}^{\infty} e^{-y^{2}} \ dy \right)$$  and convert to polar coordinates. We get $\sqrt{\pi}$ as the answer.   Is it possible to get the same answer by considering $I^{3}, I^{4}, \dots, I^{n}$? As human beings, we observe the world in which we live in three dimensions. However, it is certainly theoretically possible that more dimensions exist.  Is there any direct or indirect evidence supporting a fourth dimension? How would one write object-oriented code in C? What are some ways to write object-oriented code in C? Especially with regard to polymorphism.    See also this Stack Overflow question . Allow suppression of automatic linking of web addresses I'm new to the ICS os, coming from Gingerbread from another phone. The phone's quite nice and fast when I first used it, but just this morning it started exhibiting weird actions. Camera360 doesn't open up automatically when I click it, the Camera closes for no reason, Pocket restarts when an article has an image, and TTPod resets the song to the previous song on my widget in the middle of a new song I'm playing.  I checked out the ram, and and was surprised to see 500+ mb used, 0 mb available. I looked at the list of programs and I see the usual culprits like Facebook, but the ram they're using doesn't add up to 500+ mb. Yeah, I know that the whole ram indicated isn't usable by the apps, but I don't think I've done anything to warrant the 0 mb. I close my apps. There are times when there's free 4 to 5 mb, but I think that too shouldn't be the case. I tried installing Advance Task Killer, it listed less than 10 apps, killed it, still the ram is what it is above.   I did a reboot, and now it reads 300+ mb usage, 200 something free.   What can I do? How do I ensure an app's closed on ICS? I installed GoLaucher because it has a handy task killer and I can see the ram freed in an instant, but I would like to know if there's a better way to handle the RAM. Resize multiple layers around their respective anchor points I have 50 or so layers (imported images as smart objects) that I would like to scale down, while they stay in place. Is there a way of doing this? I know After Effects has, is there a similar Scale property in photoshop? How do you pull first 100 characters of a string in PHP