Prove that $~a,~b~$ are integers and $~a \ge2~$, then either $~b~$ is not divisible by $~a~$ or $~b+1~$ is not divisible by $~a~$.   Should I use contradiction of uniqueness of prime factorisation to prove or simply prove by mathematical induction?
This one is killing me, any help is greatly appreciated!
Converting a Uniform Distribution to a Normal Distribution How can I convert a uniform distribution (as most random number generators produce, e.g. between 0.0 and 1.0) into a normal distribution? What if I want a mean and standard deviation of my choosing? ''More than one row returned by a subquery used as an expression'' postgres error How to make skill use on investigations "important" but failure not derail the plot? Well guys, following previous recommendations here I was able to build a Horror Themed Savage Worlds campaign that has made my players excited. However they're still getting used to the style, as well as I am. In order for them to hurt the major evils of the adventures they have to find their Form (physical appearance, name), Truth (How the creature came to be) and Reason (What the creature wants to), so they have to do research, kill lesser creatures, and run away from the big baddie of the adventure as they seek what it is.  Well, this is my problem:  How can I make them use skills significantly?  I'm used to railroading my players, even with mysteries, and I don't know how to put their skills for significant use. An example is, they usually require Streetwise or Investigation to gather clues and such, however, I wonder what to do if they fail the roll... If they fail they're not supposed to find anything, so chances they never finish the adventure are well, high, that and I have no idea how to give them additional, interesting ways for them to find the truth. So do I give it no matter what they rolled? How do I make use of their skills important and not just busy-work without having a failure on a roll completely stop their progress in the investigation? "That... be" construction Correctly typesetting a tilde How do I set the tilde (~) character correctly in LaTeX?  There’s a  on Stack Overflow, with a lot of answers. But dare I say it? They all suck:   \textasciitilde is too high, \texttildelow is too low. $\sim$ can be tweaked to look more or less acceptable: {\raise.17ex\hbox{$\scriptstyle\sim$}} – but even that is a hack and uses the wrong symbol.   The point is – the tilde character is definitely an existing glyph in any font I am tempted to use. How can I just access and typeset that character, please? Without resorting to some other, similar-looking glyph that first needs to be nudged into place using a cunning combination of font size and raised boxes?  There must be a possibility to access that glyph directly from the font definition.  To demonstrate, look at this screenshot which attempts simply to display a C++ destructor of a type T (~T):    Compare this with the symbol we get in the browser: ~ – that’s what I want. Live refresh shows "modified" on tagged questions even when unmodified Reference undefined in \section with amsbook I have a section that is devoted to the proof of a theorem, and hence I used  \section{Proof of Theorem~\ref{thm}}   Of course, because of the headers that are capitalized, it asks for \ref{THM} when producing the header, and I get an obvious error. What is the proper way to handle such a case? Below is a MWE:  \documentclass{amsbook} \usepackage{lipsum} \newtheorem{thm}{Theorem} \begin{document} \chapter{First Chapter} \section{The Theorem} \begin{thm}[Lipsum Theorem] \label{thm} \lipsum[1] \end{thm} \section{Proof of Theorem~\ref{thm}} \lipsum[1-23] \end{document}   When compiling I get:  LaTeX Warning: Reference `THM' on page 3 undefined on input line 14.   (each time it produces a header), and the header shows     `2. PROOF OF THEOREM ??' wpa_supplicant store password as hash (WPA-EAP with phase2="auth=PAP") Is there a way to store my password in /etc/wpa_supplicant/wpa_supplicant.conf as some hash instead of plaintext?  By "password" I refer here to the password used for phase2 authentification. I do not refer to the Preshared Key (PSK) which could be hashed using wpa_passphrase.  For phase2 MSCHAPv2 or MSCHAP authentification I could store the password as MD4 hash using nt_password_hash (see example wpa_supplicant.conf line 659).  Is there any way to store my PAP password as hash?  Or: Is there a way to store the password in some sort of external storage? The example wpa_supplicant.conf suggests the use of such an external storage (using ext:????) but I could not find any documentation about it.  I am aware that storing the password as hash does not increase wifi security. But as the password MUST be the same as the password for other services (account management, subscriptions, payments ...) I don't want it to be stored as plaintext. Using natural deduction to prove that $p \implies q \vdash \lnot p \lor q$ Using only rules of natural deduction, I am trying to prove that $$p \implies q \vdash \lnot p \lor q$$ but am having a lot of difficulty. I was able to prove the other direction.   Could anyone show me or point me to the right direction? Get number of frames in scene from the command line? I'd like to know if there is a command-line option (or combination, or a way!) to get the number of frames that are included in a specified blender scene; for example I'd like to use is like this:  $ blender &lt;option&gt; file.blend $ Number of frames in blender scene: xxxx Why "be king", not "be a king"? I've heard people say "be king" (as in "I can't wait to be king") in movies and TV. Why don't they say "be a king"? Which is correct? Subject-verb inversion / verb-subject-object -- is this correct? I recently read the following in a schoolbook:     Wrote the researchers, "[...]"   I wonder if this is correct English. I have seen it a couple of more times. Is this just a matter of preference? English doesn't really have a very liberal word order, unless you're writing poetry or the like.  Can anyone tell me what this is called? I found some information on subject-verb inversion, but it only seems to be when the object is in front, but in this case, the word order is verb-subject-object. A virtual file containing the concatenation of other files Is there a way of creating a filesystem object akin to this:  mknod files p cat file1 file2 ... fileN &gt; files   but such that it can be seeked in, as if it were a regular file? Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$ Unable to install Cisco Packet Tracer 7.3 on Ubuntu 20.04 I am trying to install Packet Tracer 7.3 (latest version) on Ubuntu 20.04 but do not succeed to so so because of two dependencies which are not installable ("libdouble-conversion1" and "qt-at-spi").  However I successfully installed the previous version (7.2.2) which was available in an Ubuntu installer format (.run file) but I nothing happens when I run it from my gnome desktop.  I am quite a newbie in Linux so I appologies if answers are obvious...  Thanks in advance ;-)  Denis I bought an Acer laptop and installed Ubuntu 14.04LTS but WiFi drivers are not available for it. So I am trying to install a windows driver using wifidocs/driver/ndiswrapper.  On doing lspci -vvnn, it gives:  Network controller ［0280: Qualcomm Atheros Device ［168c:0042］ (rev 30)   Then I also disabled the atheros drivers. Using lspci and lspci -n, I can say PCI id is 168c:0042. But Acer website has three atheros drivers. I could not figure out which one to download but downloaded one on a windows PC and pasted the zip file on my desktop then unzipped it from the terminal using unzip &lt;filename&gt;.zip. The page says you have to go to control panel system hardware and all that on the windows system. I did not do it. Windows computer not being mine, I am afraid I should not install any additional driver on it. After unzipping it has one .inx file and one .bin file but no .sys file. I don't know whether I am completely right till here or have I missed something? And if I am right then how to proceed from here? I also made a new directory and copied the unzipped .inx and .bin files in it. Then in that directory I ran the ndiswrapper -i &lt;filename&gt;.inf but it says ndiswrapper not installed and when I do sudo apt-get install ndiswrapper-common, I get the reply that ndiswrapper-common is already the latest version. What to do? here is my dmesg | grep ath10k result  [   7.882758] ath10k_pci 0000:03:00.0: pci irq msi-x interrupts 8 irq_mode 0 reset_mode 0 [    8.190016] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/cal-pci-0000:03:00.0.bin failed with error -2 [    8.334587] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/board-2.bin failed with error -2 [   10.140677] ath10k_pci 0000:03:00.0: qca9377 hw1.0 (0x05020000, 0x003820ff sub 105b:e09a) fw WLAN.TF.1.0-00267-1 fwapi 5 bdapi 1 htt-ver 3.1 wmi-op 4 htt-op 3 cal otp max-sta 32 raw 0 hwcrypto 1 features ignore-otp [   10.140682] ath10k_pci 0000:03:00.0: debug 0 debugfs 0 tracing 0 dfs 0 testmode 0 Here is the output of lspci -nnk | grep -iA2 net  02:00.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller [10ec:8168] (rev 15)     Subsystem: Acer Incorporated [ALI] Device [1025:098a]     Kernel driver in use: r8169 03:00.0 Network controller [0280]: Qualcomm Atheros Device [168c:0042] (rev 30)     Subsystem: Foxconn International, Inc. Device [105b:e09a]     Kernel driver in use: ath10k_pci Is it possible to configure a TP_Link ADSL router to work with a cable connection? Local colormaps in global coordinate system I am trying to get a surface graph with two colormaps.  Instead of using the full Z range i want to give the first surface its own colormap  and the second surface should receive its own colormap too (it may be even the same)... however the current version takes the full range of Z-Data and calculates the colormap. Is there any way?  \begin{tikzpicture}     \begin{axis}[grid = major, colormap={bw}{gray(0cm)=(0); gray(1cm)=(1)}, view={-30}{15}]   \addplot3[surf]    table     {DATA1.dat};   \addplot3[surf]    table     {DATA2.dat};  \end{axis} \end{tikzpicture}   Right now it looks like that:   I want each of the surface to be scaled individually... just to see more clearly how the amplitude changes :)  Any ideas?  Thanks, Alex When I play Black Ops 2, it seems that my xbox controller is inverted. Example: if I press up, my character looks down and vice versa. It was not like this before. Does anyone know how to reset the controller back to its original settings? Distribution of sum of exponentials Let $X_1$ and $X_2$ be independent and identically distributed exponential random variables with rate $\lambda$. Let $S_2 = X_1 + X_2$.  Q: Show that $S_2$ has PDF $f_{S_2}(x) = \lambda^2 x \text{e}^{-\lambda x},\, x\ge 0$.  Note that if events occurred according to a Poisson Process (PP) with rate $\lambda$, $S_2$ would represent the time of the 2nd event.   Alternate approaches are appreciated. The approaches provided are commonly used when learning queueing theory &amp; stochastic processes.     Recall the Exponential distribution is a special case of the Gamma distribution (with shape parameter $1$). I've learned there is a more general version of this  that can be applied.