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Let's say I am missing my novel and I think that my friend might have taken it after I left the place.  1: By the time I left my apartment, my friend was still there, so he might have taken it. 2:When I left the apartment, my friend was still there, so he might have taken it.  I know what 2nd expression means, but not so sure about the first expression. Does it convey the same meaning as the 2nd expression? Novel from 50s or 60s - Good vs. Evil I'm not sure I recall enough of this story for a good identification, and the below is definitely out of story order, but...  The POV character goes to his home town and finds it different. He later finds that he's trapped in the town; there are two young people who turn out later on to be ... avatars ... of Ormazd and Ahriman, and the town is a battleground; what he is seeing of the town is Ahriman's effect on the town, and he turns out to have some limited ability to see the original town, which is more-or-less Ormazd's version of reality. In the end, Ahriman's avatar is destroyed, and the town restored.  I read this in the '80s or earlier; it was part of my father's collection, and the book was showing its age; I think it more likely to be the '60s rather than the '50s, and was probably by a prominent name in SF, but not one of the Masters. Here's a corollary of AC stating that: if there exists an onto mapping $f:A\to B$, then there exists a 1-1 mapping $g:B\to A$. What I'm confused about is that whether or not  AC can be derived from this proposition, in return? I have tried but haven't found a proof. Any help would be appreciated. Should I use 'or' or 'nor' after a negative statement? How do I calculate the CRC32 (Cyclic Redundancy Checksum) of a string in .NET? Hamiltonian Monte Carlo for dummies Could you provide a step-by-step for dummies explanation of how Hamiltonian Monte Carlo work?  PS: I've already read the answers here, , and here, , and here,  and they do not address it in a step-by-step way. Problem Statement  Let $f(x) = x^3 + ax^2 + bx + c$ and $g(x) = x^3 + bx^2 + cx + a$ where $a,b,c$ are integers with $c\not=0$   Suppose that the following conditions hold:   $f(1)=0$ the roots of $g(x)=0$ are the squares of the roots of $f(x)=0$      $$\text{Find the value of} \: \: a^{2013} + b^{2013} + c^{2013}$$   My attempt  Let $f(x) = (x-p)(x-q)(x-r) \: \:$ and $g(x) = (x - p^2)(x-q^2)(x-r^2)$  Expanding we get $$f(x) = x^3 - (p + q + r)x^2 + (pq + pr + qr)x - pqr$$  $$g(x) = x^3 - (p^2 + q^2 + r^2)x^2 + (p^2q^2 + p^2r^2 + r^2q^2)x - p^2q^2r^2$$  And so we have $$a = -(p+q+r) = -p^2q^2r^2$$ $$b = pq + pr + qr = p^2 + q^2 + r^2$$ $$c = -pqr = p^2q^2 + p^2r^2 + r^2q^2$$  Then we get the following relations  $$a = -c^2$$ $$b = a^2 - 2b \implies 3b = a^2 \implies b = \frac{c^4}{3}$$  From $f(1) = 0$ we have $a + b + c = -1 \implies \dfrac{c^4}{3} + c - c^2 = -1$  However, upon graphing this function, I saw that there are no integer solutions for $c$. Have I done something wrong?  Edit  For future readers, my approach can simplified alot if you replace $r$ with $1$ since $f(1) = g(1) = 0$ Computing dominant area in polygon using ArcGIS Desktop? Video editing. Audio and Video strip have different length Term for group of guys attempting to show who is alpha? Similar to pissing contest - but not in an aggressive manner. Just each guy attempting to out talk one another to show they are the leader / in charge.   Sample sentence:     Well, that meeting was more of a X than being productive with Greg trying to show he was in control. I'm trying to prove that the converse of Lagrange's theorem is true for finite abelian groups (i.e. "given an abelian group $G$ of order $m$, for all positive divisors $n$ of $m$, $G$ has a subgroup of order $n$"). This is an exercise from a book, and it is in the section on finite abelian groups, so I know I have to use the fundamental theorem of finite abelian groups. I have come up with a proof, but it seems a bit messy, and I'm not entirely sure if it's correct. It is given below.     Let the order of $G$ be $m$ = $p_1^{\alpha_1} \ldots p_k^{\alpha_k}$. It is known that $G$ is a direct product of $p$-groups, say:      $$G = G_1 \times \ldots \times G_k$$      where each $G_i$ is a $p_i$-group. By the fundamental theorem of finite abelian groups, each $G_i$ is isomorphic to a direct product of cyclic groups of the form      $$\mathbb{Z}_{{p_i}^{\beta_1}} \times \ldots \times \mathbb{Z}_{{p_i}^{\beta_l}},$$      where $\beta_1, \ldots, \beta_l$ are positive integers such that $\sum_{j=1}^l \beta_j = \alpha_i$. Now if $n$ divides $m$, then we must have      $$n = p_1^{\gamma_1} \ldots p_k^{\gamma_k}$$      for some $\gamma_1, \ldots, \gamma_k$ with $0 \leq \gamma_i \leq \alpha_i$.      Claim: Each $G_i$ has a subgroup of order $p_i^{\gamma_i}$      Proof: As above, we have that       $$ G_i \cong \mathbb{Z}_{{p_i}^{\beta_1}} \times \ldots \times \mathbb{Z}_{{p_i}^{\beta_l}} $$      where $\beta_1, \ldots, \beta_l$ are positive integers such that $\sum_{j=1}^l \beta_j = \alpha_i$.      Now since $0 \leq \gamma_i \leq \alpha_i$, we can find   $l$ numbers $\delta_1, \ldots , \delta_l$ such that $\gamma_i = \sum_{j=1}^l \delta_j$, and $0 \leq \delta_j \leq \beta_j$. (This choice of numbers is not necessarily unique).      Then $p_i^{\delta_j} | p_i^{\beta_j}$ for each $j = 1, \ldots , l$. Hence, for each factor $\mathbb{Z}_{{p_i}^{\beta_j}}$, there exists a subgroup of order $p_i^{\delta_j}$, namely $\mathbb{Z}_{{p_i}^{\delta_j}}$ (using the fact that the converse of Lagrange's theorem is true for finite cyclic groups). Taking the direct product of each of these subgroups, we get a new subgroup $G_i'$ of $G_i$:      $$G_i' \cong \mathbb{Z}_{{p_i}^{\delta_1}} \times \ldots \times \mathbb{Z}_{{p_i}^{\delta_l}}$$      The order of this subgroup is $p_i^{\delta_1} \times \ldots \times p_i^{\delta_l} = p_i^{\delta_1 + \ldots + \delta_l} = p_i^{\gamma_i} $. So we have found a subgroup of $G_i$ of order $p_i^{\gamma_i}$, as required.      So each factor $G_i$ in the product $G = G_1 \times \ldots \times G_k$ has a subgroup $G_i'$ of order $p_i^{\gamma_i}$.       Therefore, $G$ has a subgroup   $$G_1' \times G_2' \times \ldots \times G_k'$$       of order $p_1^{\gamma_i}...p_k^{\gamma_k} = n$, which completes the proof.   I have two questions about this: firstly, does this proof seem to work? Secondly, is there a way to make the proof more concise (e.g. a way to prove the statement without using all these indices)? Is there a definition of  and proper explanation of it?  It is said "Electric charge is the physical property of matter that causes it to experience a force when close to other electrically charged matter."  How is it though that matter can get charged?  Defining charge as the property of feeling a force with other charged matter seems circular.  What is charge?  Is there a non-circular definition / explanation? I'm looking for mathematics books, courses + exercises for high school level. My daughter was high school certificated but she has stopped her studies for 5 years. She would like to be involved in new educational program but she needs to restore her mathematic capabilities.  Could you help me to identify which mathematics books can help her?  Thanks a lot for your reply,  FR. Mile or miles? Singular or plural? Could you explain me please, why sometimes the word "mile" is used in singular, sometimes in plural, while in both cases it refers to several and not 1 mile.  Example:     They took part in a 26 mile long swimming race.   After 12 miles he got tired.   Thanks in advance to those who can enlighten me! Clean URLs and php extension on nginx I've looked at dozens of other questions and references on the web - and by all my calculations, my setup should work, but it doesn't.  I have nginx installation with php-fpm.  If I try to access a .php file, it runs correctly and I get the correct results.  I got this in my config file:  location ~ \.php$ {     try_files $uri =404;     fastcgi_pass unix:/var/run/php5-fpm.sock;     include fastcgi_params;     fastcgi_param SCRIPT_FILENAME  $document_root$fastcgi_script_name; }   Now, I want to setup my web app so that /somedir/file automatically executes /somdir/file.php while still displaying /somdir/file in the browser's address bar.  So I modified my config to contain the following:  location / {     try_files $uri $uri/ $uri.php?query_string }   This kind of works, that is, the server does access the .php file.  Yet, instead of executing it using the existing location ~ \.php$ block above, it simply spits the php source code as the download into the browser.  If I append the .php manually to the requested URL, then the php is executed.  It feels as if once the server matches try_files to $uri.php, it then does not do another pass at locations to see that what it needs to do with the php files.  I tried putting the php block above and below the location /, but it makes no difference.  How can I get the php to be executed? Show that $\mathbb{E}(T) = \sum\limits^\infty_{k=1}\mathbb{P}(T \geq k)$ for $T$ nonnegative integer valued and $E[T] < \infty$ Let T be a non-negative integer-valued random variable with $\mathbb{E}(T) &lt; \infty $. Prove that $\mathbb{E}(T) = \sum^\infty_{k=1}\mathbb{P}(T \geq k)$.  Had a few attempts, haven't really got anywhere. I'm wondering as I'm typing this if proof by induction is a good way to go.  Edit: One major thing I forgot to add, am I correct in thinking that also, $\mathbb{E}(T) = \sum^\infty_{k=1}k\mathbb{P}(T = k)$? What is the proper way to change the DNS IP? I suspect that the name server provided by my ADSL modem/router is buggy. Whenever I browse to a website for the first time in ubuntu, resolving the domain name takes at least 15 seconds.  To work around that problem, I changed the nameserver configuration in /etc/resolv.conf from 192.168.1.1 (my ADSL modem) to 8.8.8.8 (google's primary DNS). This seems to fix the problem, but unfortunately my changes to /etc/resolv.conf are overwritten by "NetworkManager" at each startup.  What is the proper way to configure the name server IP in ubuntu 10.4? I'm trying to enable alsa on different prefix than default through winetricks.  More specifically, the prefix that has Rosetta Stone on it, as it cannot see my microphone.  When I open winetricks, it only lets me select default.  I tried anyway, on default, and it did not resolve the issue.  I saw some answers online, through the terminal, but that didn't work either.  Maybe I have the syntax wrong?  steve@stevelap:~$ WINEPREFIX=~/.PlayOnLinux/wineprefix/Rosetta sh winetricks alsa sh: 0: Can't open winetricks steve@stevelap:~$ sudo WINEPREFIX=~/.PlayOnLinux/wineprefix/Rosetta sh winetricks alsa [sudo] password for steve:  sh: 0: Can't open winetricks   Could anyone please offer assistance?  Thanks! I would like to configure Apache 2 running on Kubuntu to execute Perl CGI scripts. I've tried some steps that I came across by googling, but nothing seems to work.  What is the right way of achieving this? "Hot" tab in is badly skewed and misleading - time to replace it with actual hot questions? Each site got a &quot;hot&quot; tab, where its contents are calculated based on . However take a look on the screenshots below and see for yourself that &quot;hot&quot; got nothing to do with many of the questions listed there. :    :    As you can see, there are questions with negative score, no answers, and even closed. I won't be surprised to see spam in there as well. This can't be any worse. What I suggest, instead of fixing the formula or patching it, simply have the hot tab show the same questions as the Hot Network Questions sidebar. (from the specific site only, of course.) Since the hot questions &quot;pool&quot; (as seen ) is limited to 100 for the whole network, either expand it or have the per-site list show questions with less hotness points - it would still be better than what we have now.