Ugly dark leaf stains on a concrete patio - any modern 2015-ish way to treat them?
What is the best way to remove and prevent leaf stains on concrete pavers?
Automorphism group of the quaternion group Area of intersection between two circles Suppose you have 2 circles that intersect each other in such a way that each circle passes through the other's center. What is the area between the circle(or common area) i.e. area between the centres of the circles? If I change the password for the root user is that going to automatically change the password for the user account?   I found this link for root:    How do I change the user account password? $\operatorname{rank}(T^3) + \operatorname{rank}(T) \geq 2\operatorname{rank}(T^2)$ for finite-dimensional $V$ Is the W statistic output by wilcox.test() in R the same as the U statistic? I've recently been reading about the Mann-Whitney U test. It turns out that to carry out this test in R you actually need to run a Wilcoxon test!  My question: is the W statistic of wilcox.test in R identical to the U statistic? Atiyah-Macdonald, Exercise 8.3: Artinian iff finite k-algebra. Atiyah Macdonald, Exercise 8.3. Let $k$ be a field and $A$ a finitely generated $k$-algebra. Prove that the following are equivalent:   (1) $A$ is Artinian.   (2) $A$ is a finite $k$-algebra.   I have a question in the proof of (1$\Rightarrow$2): By using the structure theorem, we may assume that $(A,m)$ is an Artin local ring. Then $A/m$ is a finite algebraic extension of $k$ by . Since $A$ is Artinian, $m$ is the nilradical of $A$ and thus $m^n=0$ for some $n$. Thus we have a chain $A \supseteq m \supseteq m^2 \supseteq \cdots \supseteq m^n=0$. Since $A$ is Noetherian, $m$ is finitely generated and hence each $m^i/m^{i+1}$ is a finite dimensional $A/m$-vector space, hence a finite dimensional $k$-vector space.  But now how can I deduce that $A$ is a finite dimensional $k$-vector space? How do I fix my USB drive to get its original 8GB size back? My USB drive used to be originally 8GB when I bought it.  I'm trying to reformatted in Windows 7 by right clicking on the drive and selecting Format.... But the capacity only shows 250MB.  Is there something I can do to get the original size back? Maybe it got partitioned in a weird way?   The  flash drive is a SanDisk Cruzer Micro 8GB. I got it from Wal-Mart but this is the same drive. Derivative of $(y-XB)' h(y-XB)$ with respect to $B$ One of the assumptions I make as GM is that the players should always be in the spotlight.  Sometimes the players find themselves in a situation where they watch NPCs speak to each other.  I don't like using this kind of situation, because it forces the players to watch me talk to myself and that turns them from active participants into an audience.  Usually I handle it by avoiding the situation.  I try not to write scenes that the players watch.  Instead the players speak to the NPCs one at a time.  When I do come across an exchange between NPCs I opt to summarize the conversation instead of acting it out.    I don't like this limitation though.  It gets a little contrived when all conversations in the game feature 5 PCs and 1 NPC.  Any suggestions? Either how to avoid doing it, or how to effectively role play a conversation between two (or more) NPCs that the characters may or may not be interacting with?   How do you keep clear which character is speaking? How do you make sure that the conversation feels like a real one and not like one person playing both roles? Can you raise a number to an irrational exponent? When using the web interface, there is a favorite tags section where you can select a list of tags that you view most frequently.  This has become a cornerstone of my normal  on stack overflow, but it doesn't look like this is possible in the Android app.  There should be a mechanism by which you can add some favorite tags to the nav bar. Clicking these tags will take you to a list of similarly tagged questions.  Perhaps also the feed should include a section for each favorite tag.  These would be presented on their own, much like how "hot new questions" and "new unanswered questions" appear in the feed.  Perhaps something like this: I get Chrome notifications from several websites and they work when I don't even have any tabs open for those sites.  But sometimes I get these Chrome notifications when I am not at my desk or looking at my laptop.  As some of these notifications are useful to me, I would like to be able to access any that I have missed.  Is there a way to view the notification history? It seems syntax highlighting is not enabled on every SE site, but later on turns out to be rather advantageous. As an example,  may not have sounded like needing it, yet there is , and I think it's justified since many answers have code in them, e.g. JS for InDesign scripts, some GREP-regex and (rather obviously) HTML and CSS. While I'd appreciate having syntax highlighting activate on that specific site (so please upvote ), this question is more general about two points:   Why is syntax highlighting disabled on some sites at all? Are the additional resource costs really so high? Couldn't caching improve that then? How to actually proceed when it should become activated on a specific site? It seems feature requests on the specific meta site are not exactly handled quickly (the request mentioned above is from 2013!), and also appear overkill for a simple "please switch that site's config bool for syntax highlighting". Can the default font be changed? I have been playing with the settings for the text object recently, and every time I add a new text object, I need to change the font to my favorite one. (Eurostile LT ExtendedTwo) Is there any way to change the default font so I do not need to change it every time I add a new text object?   Example: How is the equation of motion on an ellipse derived? I would like to show that a particle orbiting another will follow the trajectory   \begin{equation} r = \frac{a(1-e^2)}{1 + e \cos(\theta)}. \end{equation}  I would like to do this with minimal assumptions. Any pointers? I've read that this is nothing but the equation for an ellipse, as defined from the focus of the ellipse. But I'm not satisfied in just taking the result, I mean why should I assume that orbits are elliptical? I guess it should be derivable from energy and angular momentum considerations... Calculate Limit 0f nested square roots I'm studying machine learning and every book I open I bump into chi-squared distribution, gamma-function, t-distribution, Gaussian, etc.   Every book I have opened so far only defines what the distributions are: they don't explain or give the intuition on where the specific formulas for the functions come from.   For example, why is chi-squared distribution the way it is? What is the t-distribution? What is the intuition behind the distribution? Proofs? etc.   I would like to have a clear and fundamental understanding of the most commonly used distributions so that every time later on when I see them, I truly understand what is a t-distribution, what is a Gaussian distribution and most importantly why are they the way they are.   It would be nice if the books / tutorials  can explain the concepts to a layman so that in order to understand them you don't already need to understand them x) Many books are like this, they don't fit for beginners :( What does "Change" represent in the User Leagues? What does the Change field represent on the  for SO? Refer to the highlighted areas in the image below:   Rank: the user's ranking for the selected time period (week, month, quarter, year) Change: +/- change representing... ??? Total Reputation: the user's total reputation [Time Period] Reputation: the user's reputation for the selected time period (week, month, quarter, year)  So what is the Change field measuring? When should static_cast, dynamic_cast, const_cast and reinterpret_cast be used? How to delete directories with path/names too long for normal delete