Consider $P(n)$ as a number of $n$-permutations, which each cycle have even length, and $N(n)$ as a number of $n$-permutations, which each cycle have odd length. Calculate $P(2n)-N(2n)$
Let $X(n)$ denote number of all permutations of $\left\{1,\ldots,n \right\}$ that have only cycles of even length. Let $Y(n)$ denote number of all permutations of the same set that have only cycles of odd length. Count $X(2n)-Y(2n)$.   I don't know even how to start. I counted this sum for $n=1,2,3$ and it seems that it is always equal to $0$, but it doesn't help with calculating it in general.
Evaluate  $$\int_0^\infty \frac{\log x \; dx}{x^{2} + 2x + 2}$$  by integrating a branch of $(\log z)^{2}/(z^{2} + 2z +2)$ along a keyhole contour.  The thing I have trouble with is why I should be examining the square of the log - I guess it has something to do with ln x in fact NOT being the real part of log z, since log z is (or can be) defined on the entire negative real axis and ln x can't. But looking at the square would definitely not have been my first plan of attack :( By definition a matrix representing a Lorentz transformation is orthogonal, so that its inverse is equal to its transpose.   Consider a pure boost in the t-x plane; $$\Lambda_x=\begin{pmatrix} \cosh(\gamma) &amp;&amp; \sinh(\gamma) &amp;&amp; 0 &amp;&amp; 0\\ \sinh(\gamma) &amp;&amp; \cosh(\gamma) &amp;&amp; 0 &amp;&amp; 0\\ 0 &amp;&amp; 0 &amp;&amp; 1 &amp;&amp; 0\\ 0&amp;&amp;0&amp;&amp;0&amp;&amp;1 \end{pmatrix}.$$ $\Lambda_x$ has inverse  $$\Lambda_x^{-1}=\begin{pmatrix} \cosh(\gamma) &amp;&amp; -\sinh(\gamma) &amp;&amp; 0&amp;&amp;0\\ -\sinh(\gamma) &amp;&amp; \cosh(\gamma) &amp;&amp; 0&amp;&amp;0\\ 0 &amp;&amp; 0 &amp;&amp; 1 &amp;&amp;0\\ 0&amp;&amp;0&amp;&amp;0&amp;&amp;1 \end{pmatrix}$$ but tranpose  $$\Lambda_x^T=\begin{pmatrix} \cosh(\gamma) &amp;&amp; \sinh(\gamma) &amp;&amp; 0&amp;&amp;0\\ \sinh(\gamma) &amp;&amp; \cosh(\gamma) &amp;&amp; 0&amp;&amp;0\\ 0 &amp;&amp; 0 &amp;&amp; 1&amp;&amp;0\\ 0&amp;&amp;0&amp;&amp;0&amp;&amp;1 \end{pmatrix}.$$ These are not equal. Where have I gone wrong? What could cause exposure problems after my camera got wet? I had an unfortunate accident where my DSLR camera (and me) briefly ended up in a river. The camera wasn't fully submerged but it did get very wet. It still works, sort of, although the electrics are doing some random things. I'm hoping that's just a bit of dampness doing that, that will fix itself when it dries out fully.  More concerning is that the documents exposure metering seems to be all wrong when using the viewfinder. Pictures are extremely overexposed and nearly white. But if I use live view, then they are fine.  Why could cause this? I assume the light is routed a different way when using the viewfinder and maybe it is going through something with water/condensation on it? In Star Trek TOS, we see the Defiant disappear to unknown parts  ('The Tholian Web'), and Star Trek: Enterprise revealed that the Defiant went back in time to the mirror universe ('In a Mirror Darkly', parts 1 &amp; 2).   Because of this, one might wonder why (in-universe) the mirror universe didn't obtain (and maintain) technological superiority over the main timeline. (Granted, the out of universe answer is that TOS depicted the two universes as having essentially the same level of tech)  So does any work (whether STO, or a pocket books work) address why the mirror universe didn't simply leap frog the main universe's technology?   (Or to put it another way, why did the mirror universe's technology appear to stagnate for a century after obtaining access to the Defiant's advanced technology?) Formally, is it correct to write:     A number of questions has been asked here.   or:     A number of questions have been asked here.   As a non-native speaker of English, I would prefer the former: the subject seems to be "number", therefore the verb ought to be singular, I'd say. However, the latter seems more common, and therefore I believe that my gut feeling is just plain wrong — but I would really like to have a definite answer.  Moreover, is it the same for "a myriad of", "a plethora of", and so on? Is there a way to have a copy of any message sent from a Gmail account sent to another address?  I've got any messages coming to my address copied to another (see  for one solution, setting up filters is another), but I haven't yet found a way to have outbound messages automatically copied.  Is there a way? Erlang server, Java client - TCP messages get split? Prove $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^n = n!$ and $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^m = 0$ Prove $$\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^n = n! \qquad\text{and}\qquad \sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^m = 0,$$ where $m \in \{0, 1, 2, \cdots, n - 1\}$.  To be honest, I tried to use some combinatorial ideas but was not able to see interconnections. Also, I had an idea to prove it using induction but failed to do that either. Help will be much appreciated! I just got stuck and don't see any continuation of solution for this problem Support anchor names in posts I admit this feature request is probably somewhat limited in useful scope, but I'm throwing it out there anyway. Inspired by , and because I want to use it on , I'm requesting that name be supported on a tags in posts.  On very long answers, such as the closing/migration guidance answer, this would allow direct linking to the specific closure reason. This would then allow us, when someone , to link directly to the appropriate reason and description thereof.  I recognize the limited scope of this, however, I have seen other long answers that could stand to have that kind of "deep" linking ability as well.  (The original incarnation of this post had either name or id, but preferenced name. Per Koper's answer, which I agree with, I took out the idea of supporting id, because Koper's right -- too dangerous.) libsensors5 or libsensors-config upgrade problem How to use an item in my off-hand? Why is the work done on a charge calculated from infinity? Why is the work done on a charge calculated from infinity to a point? Why not from one particular point to other? JavaScriptSerializer - JSON serialization of enum as string Timeout on a function call Why are we scared of singularities? I often hear people say that general relativity predicted its own demise because of the singularities it predicts. If I'm not mistaken, this is also a problem in QFT. I wonder why singularities garner so much hate. To me, it seems like negative or complex numbers: we used to hate these things but now they are more generally accepted. Why can't it be that nature just has singularities? We might not get it, we might not like it, but it may be so. I was practicing with modeling a leaf using a image reference. Everything went fine and I finished modeling my leaf, but I wanted to make it more realistic (I'll add textures later), I saw lots of people using the Subdivision Surface modifier but it doesn't work for me.  My model:   My model when I add the Subdivision Surface modifier Just that. By good I mean something at the level of SolidWorks, AutoCAD, or similar. This is a complete limitation for me (and many others) to use Ubuntu. Kubuntu - can not enter wifi credentials after adding external wifi card My ethernet connection is quite unreliable (It uses powerline adapters) so I bought an external wifi adapter. I have plugged it in and run the command sudo ip address add 192.168.0.39/24 brd + dev enp7s0, and KDE has detected it. Unfortunately, it doesn't seem to realise it is a wireless network and therefore needs credentials. Where I could normally find a place to enter credentials (the network widget, system settings) there are none. KDE reports that the device is connected in the UI. How do I make it work out that it is wireless and therefore needs credentials? Edit: With ethernet plugged out, (USB) adapter plugged in: $ ip address show 1: lo: &lt;LOOPBACK,UP,LOWER_UP&gt; mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000     link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00     inet 127.0.0.1/8 scope host lo        valid_lft forever preferred_lft forever     inet6 ::1/128 scope host         valid_lft forever preferred_lft forever 2: enp7s0: &lt;NO-CARRIER,BROADCAST,MULTICAST,UP&gt; mtu 1500 qdisc fq_codel state DOWN group default qlen 1000     link/ether 04:d9:f5:8a:f5:e0 brd ff:ff:ff:ff:ff:ff  With ethernet plugged in, adapter plugged in: 1: lo: &lt;LOOPBACK,UP,LOWER_UP&gt; mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000     link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00     inet 127.0.0.1/8 scope host lo        valid_lft forever preferred_lft forever     inet6 ::1/128 scope host         valid_lft forever preferred_lft forever 2: enp7s0: &lt;BROADCAST,MULTICAST,UP,LOWER_UP&gt; mtu 1500 qdisc fq_codel state UP group default qlen 1000     link/ether 04:d9:f5:8a:f5:e0 brd ff:ff:ff:ff:ff:ff     inet 192.168.0.51/24 brd 192.168.0.255 scope global dynamic noprefixroute enp7s0        valid_lft 863988sec preferred_lft 863988sec     inet6 fe80::7eb1:eca2:96a0:fe5c/64 scope link noprefixroute         valid_lft forever preferred_lft forever  With just ethernet: 1: lo: &lt;LOOPBACK,UP,LOWER_UP&gt; mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000     link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00     inet 127.0.0.1/8 scope host lo        valid_lft forever preferred_lft forever     inet6 ::1/128 scope host         valid_lft forever preferred_lft forever 2: enp7s0: &lt;BROADCAST,MULTICAST,UP,LOWER_UP&gt; mtu 1500 qdisc fq_codel state UP group default qlen 1000     link/ether 04:d9:f5:8a:f5:e0 brd ff:ff:ff:ff:ff:ff     inet 192.168.0.51/24 brd 192.168.0.255 scope global dynamic noprefixroute enp7s0        valid_lft 863960sec preferred_lft 863960sec     inet6 fe80::7eb1:eca2:96a0:fe5c/64 scope link noprefixroute         valid_lft forever preferred_lft forever  With nothing: 1: lo: &lt;LOOPBACK,UP,LOWER_UP&gt; mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000     link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00     inet 127.0.0.1/8 scope host lo        valid_lft forever preferred_lft forever     inet6 ::1/128 scope host         valid_lft forever preferred_lft forever 2: enp7s0: &lt;NO-CARRIER,BROADCAST,MULTICAST,UP&gt; mtu 1500 qdisc fq_codel state DOWN group default qlen 1000     link/ether 04:d9:f5:8a:f5:e0 brd ff:ff:ff:ff:ff:ff  lsusb, adapter plugged in: Bus 004 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub Bus 003 Device 002: ID 046d:c31c Logitech, Inc. Keyboard K120 Bus 003 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub Bus 001 Device 003: ID 1d57:32da Xenta 2.4GHz Receiver (Keyboard and Mouse) Bus 001 Device 004: ID 0bda:b812 Realtek Semiconductor Corp. 802.11ac NIC Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub  I'm not adding it with it plugged out because it's obviously the Bus 001 Device 004: ID 0bda:b812 Realtek Semiconductor Corp. 802.11ac NIC. There were no restarts between any of these changes. "To work for" vs. "to work in" vs. "to work at" How can I find a saved Wi-Fi password? I have an Android phone which is connected to a secured Wi-Fi hotspot.  How can I find out the password of the hotspot from the phone?