Sticky footer & 100% height - force div to get 100% height
How to make a div 100% height of the browser window
Stone-Weierstrass theorem with $p(1/x)$ Capturing screenshots on the Mac has gotten very slow recently.  It used to be nearly instantaneous but now it is taking more than 10 seconds for the image to appear on the desktop.  I seem to recall having this problem years ago. The workaround was to delete a large accumulation of log files that the os had built up. After I deleted these log files, the screen capture worked very fast again.  Problem is that I cannot remember where these log files are. Searched for quite some time and could not find the information. Let $P(x)$ be a degree $n$ polynomial with distinct roots $r_1, r_2, \cdots r_n$. Prove that   $$\displaystyle\sum_{i=1}^{n}\frac{P''(r_i)}{P'(r_i)}=0$$  My proof: We can equivalently rewrite the polynomial as $(x-r_1)(x-r_2)\cdots(x-r_n)$ and if we plug in the roots in the polynomial, we get that all of them are equal to $0$ so their first derivatives are equal to $1$ and then their second derivatives are equal to $0$ which means that all quotients are equal to $0$ and there for the sum is as well, $Q.E.D$  Don't show the solution please, but if you have a hint, feel free. Thank you. How to calculate the exact probability that the second player wins? Consider a game that uses a generator which produces independent random integers between 1 and 100 inclusive. The game starts with a sum S = 0. The first player adds random numbers from the generator to S until S > 100 and records her last random number 'x'. The second player, continues adding random numbers from the generator to S until S > 200 and records her last random number 'y'. The player with the highest number wins, i.e. if y > x the second player wins. Is this game fair? Write a program to simulate 100,000 games. What is the probability estimate, based on your simulations, that the second player wins? Give your answer rounded to 3 places behind the decimal. For extra credit, calculate the exact probability (without sampling).     import random  CONST_TIMES = 100000 CONST_SMALL = 100 CONST_LARGE = 200  def playGame():     s = 0     while s &lt;= CONST_SMALL:          x = random.randint(1, CONST_SMALL)         s = s + x;     while s &lt;= CONST_LARGE:         y = random.randint(1, CONST_SMALL)         s = s + y     if x &lt; y:         return 's'     elif x == y:         return 'm'     else:         return 'f'  fst = sec = 0 for i in range(CONST_TIMES):     winner = playGame()     if winner == 'f':         fst = fst + 1     elif winner == 's':         sec = sec + 1 secWinPro = round(float(sec) / CONST_TIMES, 3)  print secWinPro   The simulation probability is about 0.524. I want to know how to calculate the exact probability. How can I delete an email address from Outlook's email history? How to reduce memory usage on a Unix webserver I'm currently using a Joyent Accelerator to host my webapps, and it's working fine, however I need to reduce costs so I'm downgrading my current plan and that imposes some new memory limits (256M rss, 512M swap). I wasn't too far over them yesterday, but after restarting Apache several times today, I'm now 411M rss, 721M swap (prstat -Z -s cpu).  Searching in Server Fault only gives me lots of ways and specific tools to monitor the server, but no advice on how to reduce/optimize it's memory usage. I've also seen , but I don't think it's good for this particular (or may I say generic?) situation.  The server is running Solaris on a shared CPU, and I'm using a Apache + MySQL + PHP stack.  I'm interested in knowing the steps one can take to troubleshot this and solve the issues. However, I'm also running out of time to lower my memory foot print and downgrade the plan before the current ends, so anything that can make magic and save the day is welcome as well :) If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$ If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$. Here is my attempt at a proof. It feels very elaborate and I am not sure if it is correct. Can someone please point out any mistakes or places where I may improve. Thanks!  By the definition of continuity at $a$ of $f,g$ we have that $\lim\limits_{x\to a}f(x)=f(a)$ and $\lim\limits_{x\to a}g(x)=g(a)$. Suppose that $f(a)&gt;g(a)$. Then there exists a $\delta&gt;0$ such that $f&gt;g$ for all $x$ satisfying $|x-a|&lt;\delta$. Then for $x$ satisfying $|x-a|&lt;\delta$ we have $h(x)=f(x)$ and $k(x)=g(x)$ and thus $h$ and $k$ are continuous at $a$ because $f$ and $g$ are continuous at $a$. Now if $f(a)&lt;g(a)$ we simply relabel $f=\tilde{g}$ and $g=\tilde{f}$, so that $\tilde{f}(a)&gt;\tilde{g}(a)$ and we apply the previous result to show that again $h$ and $k$ are continuous at $a$.\  Now if $f(a)=g(a)$ then we can distinguish three cases. $f\geq g$ in a small neighborhood about $a$, $f\leq g$  in a small neighborhood about $a$ or $f\geq g$ on one side of $a$ and $f\leq g$ on the other side. In the first case we assume that $f\geq g$ in some small neighborhood about $a$. Then, since $f(a)=g(a)$ we have $h(x) = f(x)$ in this neighborhood and $k(x) = g(x)$ and again we see that $h$ and $k$ are continuous at $a$ due to the continuity of $f$ and $g$ at $a$. Similarly if $f\leq g$ in a neighborhood around $a$ then $h(x)=g(x)$ and $k(x)=f(x)$ in this neighborhood and $h$ and $k$ are continuous at $a$.   Suppose that just to the left of $a$ we have $f\geq g$ and just to the right of $a$ we have $f\leq g$. Then $\lim\limits_{x\to a^-}h(x)=\lim\limits_{x\to a^-}f(x)=f(a)=h(a)=g(a)=\lim\limits_{x\to a^+}g(x)=\lim\limits_{x\to a^+}h(x)$. Similarly $\lim\limits_{x\to a^-}k(x)=\lim\limits_{x\to a^-}g(x)=g(a)=k(a)=f(a)=\lim\limits_{x\to a^+}f(x)=\lim\limits_{x\to a^+}k(x)$. So $\lim\limits_{x\to a}h(x)=h(a)$ and $\lim\limits_{x\to a}k(x)=k(a)$ which means that $h$ and $k$ are continuous at $a$. Lastly if $f\leq g$ just to the left of $a$ and $f\geq g$ just to the right of $a$ we can relabel $f=\tilde{g}$ and $g=\tilde{f}$ and apply the last result to show $h$ and $k$ are continuous at $a$.   Again, please point out any mistakes I may have made. Thanks!! Integer coefficient polynomial $p(x)$ has no integer roots of $\,p(0)$ and $p(1)$ are odd [Parity Root Test] This came up an a training piece for the Putnam Competition and also in Ireland and Rosen.  The question posed was basically:  Let $p(x)$ be a polynomial with integer coefficients satisfying that $p(0)$ and $p(1)$ are odd. Show that $p$ has no integer zeros.  I&amp;R give an example:  $p(x) = x^2 - 117x + 31$  and show (no problem) that for any $n$ whether even or odd, $p(n)$ will be odd. And claim that this shows $p(n)$ will never be $0$.  I can see, e.g., that $x^2 + 2x + 1$ will be odd substituting an even $n$ and even for an odd $n$.   But would appreciate help in understanding the underlying math and what is happening here.  Also, as a second part, can a general statement about the existence of an integer solution be made if $n$, even and odd, generates an even and an odd as in the last example.  I can see that if you look at these equations (mod $2$), you can distinguish whether there is an integer solution. And I would guess this is intimately connected with the question.  Thanks as always. Why is "cupboard" pronounced with a silent "p"? How to kill the process using the name of the program instead of PID? I started my Python program in the background using nohup as mentioned below -   nohup zook.py &amp;   Now I am trying to kill this process so I did the ps command as mentioned below  root@phxdbx1145:/home/david/zook# ps ax | grep zook.py 16352 pts/6    S+     0:00 grep --color=auto zook.py   But somehow, everytime its PID getting changed, I don't know why. Whenever I do like this -   kill -9 16352   It always say, No Such Process.  And when I do px command again, I see that PID got changed automatically..  So I am not sure how do I kill this process?  Is there any way I can kill the process with the name somehow?  UPDATE:-  This is what I am getting. I did pkill -9 zook.py and then I did ps command as mentioned below and it is shwoing zook.py constantly?  root@dbx1145:/home/david/zook# pkill -9 zook.py  root@dbx1145:/home/david/zook# ps ax | grep zook.py 23870 pts/6    S+     0:00 grep --color=auto zook.py  root@dbx1145:/home/david/zook# ps ax | grep zook.py 23872 pts/6    S+     0:00 grep --color=auto zook.py  root@dbx1145:/home/david/zook# ps ax | grep zook.py 23874 pts/6    S+     0:00 grep --color=auto zook.py  root@dbx1145:/home/david/zook# ps ax | grep zook.py 23876 pts/6    S+     0:00 grep --color=auto zook.py How to both display a command line's output on console and save the output into a text file? Custom attributes - Yea or nay? How do I clone a single branch in Git? Install windows 7 through virtual box Is the axiom of choice needed in proving that metric spaces in which every infinite subset has a limit point are compact? In WPF, what are the differences between the x:Name and Name attributes? Prove that a finite union of closed sets is also closed Let $X$ be a metric space.  If $F_i \subset X$ is closed for $1 \leq i \leq n$, prove that $\bigcup_{i=1}^n F_i$ is also closed.  I'm looking for a direct proof of this theorem.  (I already know a proof which first shows that a finite intersection of open sets is also open, and then applies De Morgan's law and the theorem "the complement of an open set is closed.")  Note that the theorem is not necessarily true for an infinite collection of closed $\{F_\alpha\}$.  Here are the definitions I'm using:  Let $X$ be a metric space with distance function $d(p, q)$.  For any $p \in X$, the neighborhood $N_r(p)$ is the set $\{x \in X \,|\, d(p, x) &lt; r\}$.  Any $p \in X$ is a limit point of $E$ if $\forall r &gt; 0$, $N_r(p) \cap E \neq \{p\}$ and $\neq \emptyset$.  Any subset $E$ of $X$ is closed if it contains all of its limit points. I have some stuff in settings.py that I'd like to be able to access from a template, but I can't figure out how to do it. I already tried  {{CONSTANT_NAME}}   but that doesn't seem to work. Is this possible? Should I keep 32bit or 64 bit Windows 10 on my x64 based Intel Atom Z3735F cpu on 2gb ram non-upgradable laptop I have an Acer Aspire Switch 10 SW5-012 detachable notebook which has 2GB of RAM based on an x64 based processor. I got it with a 32 bit version of windows 8 but upgraded to windows 10 recently. But before I do anything else, I was considering if I'd be having Better performance with the 64 bit version of windows 10. Would it make my notebook perform better since it's based on an x64 based Intel Atom Z3735F? Or should I just stick to the 32 bit version of windows 10? Prove that there exists such a set of $n$ positive integers Prove that for any positive integers $m$ and $n$ , there exists a set of $n$ consecutive positive integers each of which is divisible by a number of the form $d^m$, where $d$ is some positive integer not equal to $1$.  I don't know how to approach this question.