For any odd prime $p≥5$, $(-3/p) = 1$ if $p ≡ 1,-5\pmod{12}$ and $(-3/p) = -1$ if $p ≡ -1,5 \pmod{12}$
When is $-3$ a quadratic residue mod $p$?
Enlarged partition, filesystem still reports old size I resized the root partition with GParted, from 20 GB to 40 GB, but the filesystem still reports a size of 20 GB.  How can I use all 40 GB?    Update  The resized partition was an LVM Physical Volume (/dev/sda2), and the boot partition is ext4 (/dev/sda1). Integral of periodic function over the length of the period is the same everywhere Assume that $(\text{X}, T)$ is compact and Hausdorff. Prove that a comparable but different topological space $(\text{X},T')$ is not. Say that a topological space is CH if it is both compact and Hausdorff.   Let $T$ and $T'$ be two topologies on the same set X that are comparable   but different, i.e., $T$ is either strictly smaller or strictly larger than $T'$.   Assume that $(\text{X}; T )$ is CH. Prove that $(\text{X}; T')$ is not CH.   I just can seem to get started, so I was wondering if anybody could give me a hint...  If $T' \subsetneqq T$, then the identity $id: (\text{X},T) \rightarrow (\text{X}, T')$ is continuous and surjective. This implies that $id(\text{X},T) = (\text{X},T')$ is also compact. So in order to show that $(\text{X},T')$ is not CH, I need to show that it is not Hausdorff, right? But I'm stuck... Why is my starting gravity low? In the fall of 2011 I made the leap from extract/partial kits to all-grain kits. I'm enjoying AG, but my SG is continuously below the expected. I have brewed 5 AG kits and 4 have had a low SG.   My assumption is my sugar extraction during mashing (or sparging) isn't optimal. I heat my mash water to 165 and mash the grains for 60 minutes. Mashing levels out at 152. For Lautering I heat the water to 175-180. After I drain the initial mash I add the lautering water and let that soak for 10 minutes. the temperature levels out around 170-172. Then I drain into the kettle and being the boil.  I aim for a gentle boil just enough to break the surface tension. I boil for 60 minutes, chill, rack, take a reading and pitch yeast.  Here are some of the numbers  type            expected     actual  kit st paul porter  1.052        1.039   http://www.northernbrewer.com/shop/st-paul-porter-all-grain-kit.html belgium strong  1.067        1.055   http://www.midwestsupplies.com/superior-strong-ale-all-grain-kit.html troegs hop back 1.063        1.049   http://www.byo.com/stories/recipeindex/article/recipes/90-american-amber-a-pale-ale/2196-troeegs-hopback-amber-ale-clone   The fuorth was the Phat Tyre kit from Northern Brewer. Can't remember the SG though. It was low. They taste great, but the numbers weren't "right". What can I adjust so the numbers are more inline with the recipe?   Update Thank you for all the feedback. There are a number of factors for me to review and adjust. Given my current setup I will follow @mdma advice and recirculate the wort once and extend lautering to 30 minutes.  It will be a week or two before I can attempt this, so it will be awhile before I report back again.  Update #2 I now have some awesome local contacts who are helping me out with technique and trouble shooting. Another (major) issue I could be experiencing is draining the wort way too fast. According to my friend. The drain should be low and slow after cycling a gallon or 2 of wort. sparging should take ~2 hours. for each gallon drained plato/brix should drop 3 points and go no lower than 3-5 points at the end.  the second recommendation is to heat up 3-5 gallons of water to 180 and poor it in prior to/while sparging.  my first attempt using this technique proved successful with the gravity being a lot closer to the anticipated OG. I had some temp issues with the batch, but I was so focused on sparging that I ignored the temp. My guess is I would have been right on if my temp didn't fall so low.  Update #3 I had the chance to brew with my mentor this weekend and he pointed out 2 glaring problems with my mashing technique.   Starting mash temp is too low. I was heating my water to 10-12 degrees above target temp. He recommends 17 degrees. between grain absorption &amp; the tun itself most of the heat dissipates within the first few minutes. This does go against what I originally said. That I was hitting my temp marks. Looking back I was hasty with my readings. I should not dump my sparge water (or lauter?) into the tun while draining the wort. The idea is to compact the liquid into the bottom of the tun thus compacting the sugars. I am stirring up the mash when I dump all sparge water in. We kept our sparge water 1-2 inches above the grain bed and our final results we excellent.   A 3rd recommendation he made was to lower the pH of the water. tap water is ~7 and the ideal mash pH is 5.2. He added a solvent, which I cannot remember the name of, which did this for us. I'll be picking up some of that this weekend.  Thanks to all who helped me out with this problem. "Leave a comment" link even when you can't On  I disabled  comments for "USDCAD options vs CADUSD options arbitrage?", but it  still has a "Leave a Comment" link.   Clicking the link takes you to    where it, of course, shows that you can't leave a comment.   The timeline page correctly does not show the 'comment' link:    How to fix? I have a Windows 7 64-bit installation on my current PC, and I want to move it to a new PC .. I do NOT have a problem with licensing, as the new system already has its own new Windows 7 license, which I intend to use.  I want to use my existing installation, because it has 3 years worth of installed office related software development software, which installing again could take weeks !  I'm assuming that I can use Windows 7's Backup and Restore feature to backup a system image to a network location, then restore that network stored system image on the new PC ?  Again, like I said, I don't have a WIndows 7 licensing issue, as both machines are corporate provided and come with their own licences. Changing "index.php" Within The Address iPhone 7 Plus takes days to turn on showing Apple logo or black screen I’ve had this issue for months but in the last couple of days it’s got very very bad.  When my phone runs out of battery it needs like 2–3 days to turn back on. At the beginning of this problem the phone would need about 1h. Every time when my phone goes off again the time to switching on goes up.  What does this all look like? Well, when my phone died I put it on the charger, then the logo appears and goes off. When it goes off, a black screen pops out with a little white line on in it, then it turns on back to Apple logo. It keeps doing this until it turns on which is 1 or 2 even now 3 days.  I tried to restore it and also update it but nothing happens, literally nothing happened. Prove convergence of the sequence $(z_1+z_2+\cdots + z_n)/n$ of Cesaro means A person who cannot understand that there may be other perspectives Outcast boy meets girl underwater Here's another one I can't find, but somebody will probably recognize right away. A young man has been thrown out of respectable society. To make sure he stays away from people (and they stay away from him), those who have condemned him alter his body so that he always smells bad. Of course, no one can smell underwater - and that's where he meets a girl, while they're both swimming underwater in a swimming pool. I don't remember a lot of details about this story - but I remember the main character was not very nice - he was often mean to other people. Whether he'd always been like that, or whether his nastiness was because everyone had rejected him, I can't remember. I would have read this story in the mid-seventies, but it was probably written earlier. When does $P(x)=\sum_{n=0}^{\infty}{\frac{n!}{n^n}\cdot x^n}$ with $x\in \mathbb{C}$ converge? These wonderful icon overlays that give you useful information in the Windows Explorer are not showing up.   It's the most annoying with Dropbox (I already read ). I just tried reinstalling Dropbox - no help. I have this problem also with the  so I don't think that it is a dropbox problem specifically; and something is wrong in my OS. The OS is actually a pretty fresh clean-install...  OS: Win8 Pro 64-bit. data & synchronization - manually sync mail. calender and contacts What is the simplest SQL Query to find the second largest value? Modified Hamilton's Principle overconstraining a system by imposing too many boundary conditions In Hamiltonian Mechanics, a version of Hamilton's principle is shown to evolve a system according to the same equations of motion as the Lagrangian, and therefore Newtonian formalism. In particular, letting $\delta$ indicate a variation of the path through phase space, $$ \delta \int_{t_1}^{t_2} \big(p_i \dot{q}_i - H(p_i,q_i,t)\big)dt = 0 $$ is shown to generate the same motion equations as those found by the Legendre transform of the Lagrangian equations of motion. Essentially, when we calculate the Euler-Lagrange equations for the above integrand, we find $$ \dot{p}_i + \frac{\partial H}{\partial q_i} = 0\text{, } \dot{q}_i - \frac{\partial H}{\partial p_i} = 0. $$ Now, since these are both first-order equations of motion, we require $2n$ boundary conditions to obtain a solution, where $n$ is the number of particles. This is perfectly consistent with the Lagrangian formalism which had $2n$ boundary conditions due to its $n$ distinct second-order motion equations.   An important thing to note is that since $p_i\dot{q}_i - H(p_i,q_i,t)$ contains no $\dot{p}_i$ dependence, the variation in $p_i(t)$ need not be zero at the endpoints of the path. This is not the case with $q_i(t)$ because the $\dot{q}_i$ dependance results in the appearance of boundary terms that must be set to zero in order to obtain the motion equations.  Now, as far as the boundary conditions are concerned, this makes sense. For the path to be uniquely specified given Newtonian mechanics, we require $2n$ boundary conditions. These can either be the usual Newtonian choice of initial positions and velocities/momenta, or they can be the initial and final positions. Both are perfectly acceptable mathematically speaking. If we were required, in the modified Hamilton's principle variation, to enforce that the variations in the momenta also be zero at the initial and final times, this would mean $2n$ additional boundary conditions, which would generally over constrain a Newtonian system.   However, my confusion is this: when textbooks (Goldstein in particular) consider generating functions, they require that the integrand be invariant to the addition of a total time derivative of a function $F(q_i,p_i,t)$ of the phase space coordinates. However, adding such a function will in general add some $\dot{p}_i$ dependence to the integrand, which will then add boundary terms to the equations of motion unless we require that the $p_i(t)$ variations be zero at the boundaries. This is fine as far as it goes. We can always define the variation however we like. The point is just that it spits out the right motion equations at the end of the day. But my confusion lies in how this doesn't generally over constrain the system. If we specify all the positions and momenta at both the initial and the final times, might not the required path to connect those points in phase space be non-Newtonian? Does this only matter if you're actually trying to use the stationary action principle to find the paths, rather than just using it to find the motion equations? Why does local fn=$(...) mask the $? status code Two function definitions, the only difference is that the first combines the local storage keyword with the assignment, while the second separates them:  function foo {     local fn=$(mktemp -p /path/does/not/exist 2&gt;/dev/null)     echo $? }  function bar {     local fn     fn=$(mktemp -p /path/does/not/exist 2&gt;/dev/null)     echo $? }  foo bar   This echoes "0" then "1". I expect it to echo "1" then "1". It seems like the value of $? is the result of the assignment to local, rather than the result of the command substitution.  Why does bash 4.2.46(1)-release behave this way? "All our X" vs. "all of our X" Is the word "of" optional in this instance? Is either of these considered preferable to the other?        Taste all our delicious treats.   Taste all of our delicious treats. Where is attribute table in ArcGIS Desktop when not visible? I've just updated to ArcGIS 10.1 for Desktop and when I go to open an attribute table, I can not see it anywhere.   I'm wondering if it is off my screen somewhere or if its just super tiny that I can't see it.   I'm using dual monitors and it's not popping up on either.  Is there a way I can find my attribute table? Copy current terminal prompt to clipboard Is there a way to copy the currently written content of the bash prompt? Say I wrote:  ping www.google.com   so that the lines in terminal looks like:  petr@sova:~$ ping www.google.com   and didn't hit return yet. I want to copy that text to clipboard. Is that possible?  The usual usecase would be hitting up several times and finding a command which I want to copy.