Let $n$ and $k$ be positive integers. Show that $\gcd(n, nk + 1) = 1$
Why $\gcd(qb+r,b)=\gcd(b,r)$?
If $a_{1},a_{2},a_{3},a_{4}\in \mathbb{R}$ and $a_{1}+a_{2}+a_{3}+a_{4}=0$ and $a^2_{1}+a^2_{2}+a^2_{3}+a^2_{4}=1.$ Then Range of $$E =(a_{1}-a_{2})^2+(a_{2}-a_{3})^2+(a_{3}-a_{4})^2+(a_{4}-a_{1})^2$$ is  Try: From $$(a_{1}-a_{2})^2+(a_{2}-a_{3})^2+(a_{3}-a_{4})^2+(a_{4}-a_{1})^2$$ $$=2(a^2_{1}+a^2_{2}+a^2_{3}+a^2_{4})+2(a_{1}a_{2}+a_{2}a_{3}+a_{3}a_{4}+a_{4}a_{1})$$ $$=2-2(a_{1}+a_{3})(a_{2}+a_{4})=2+2(a_{1}+a_{3})^2\geq 2$$ and equality hold when $\displaystyle a_{1}=-a_{3}$ and $a_{2}=-a_{4}$ Could some help me how to find its upper bound, Thanks Too many password attempts While I was out, someone attempted to guess my phone password. They couldn't guess it and now it displays the 'too many password attempts' screen, which needs to be unlocked with a google account. However, my google account is not working. I enter the username and password correctly, and have attempted to do so about 10 times. So why is it not unlocking? Do I need an internet connection to unlock it? When should I use an em-dash, an en-dash, and a hyphen? I generally know how to use a hyphen, but when should I use an en-dash (–) instead of an em-dash, or when should I use a hyphen (-) instead of an em-dash (—)? Adding a comment should not make me fail the review audit If $d(x,y)$ is a metric, then $\frac{d(x,y)}{1 + d(x,y)}$ is also a metric Differences Between Row Echelon and Reduced Row Echelon Let $$A =\begin{bmatrix} 1&amp;3&amp;-6&amp;2\\5&amp;12&amp;-24&amp;7\\9&amp;15&amp;-30&amp;6\end{bmatrix}$$  $$RREF(A) =\begin{bmatrix} 1&amp;0&amp;0&amp;-1\\0&amp;1&amp;-2&amp;-1\\0&amp;0&amp;0&amp;0\end{bmatrix}$$ $$REF(A) =\begin{bmatrix} 1&amp;3&amp;-6&amp;2\\0&amp;1&amp;-2&amp;1\\0&amp;0&amp;0&amp;0\\\end{bmatrix}$$  Now, provided this information, it can be found that the basis for the nullspace of $A$ using the $RREF(A)$ is: $$\begin{pmatrix} \begin{bmatrix} 0\\2\\1\\0 \end{bmatrix}, \begin{bmatrix} 1\\-1\\0\\1 \end{bmatrix}\end{pmatrix}$$  Using the $REF(A)$ the "basis" for the nullspace is:  $$\begin{pmatrix} \begin{bmatrix} 6\\2\\1\\0 \end{bmatrix}, \begin{bmatrix} -2\\-1\\0\\1 \end{bmatrix}\end{pmatrix}$$  Is the "basis" that I provided for the $REF(A)$ actually a basis? How does the  $REF$ and $RREF$ of a matrix differ? Calculate the limit including binomial coefficients $\lim\limits_{n\to\infty}{\sum\limits_{k=n}^{5n}{k-1 \choose n-1}(\frac{1}{5})^{n}(\frac{4}{5})^{k-n}}$  It's clear that we can simplify the limit a little bit, after which we get:  $\lim\limits_{n\to\infty}{(\frac{1}{4})^{n}\sum\limits_{k=n}^{5n}{k-1 \choose n-1}(\frac{4}{5})^{k}}$  I could further simplify the expression, but I feel like there's a more elegant solution.   Give me a hint, please My wifi connection doesnt work properly; it stays connected only for around 5 minutes after I turn on the PC. I'm using Ubuntu 16.04.  I've tried   sudo service network-manager restart   which makes no difference  Same with   killall nm-applet &amp;&amp; nm-applet &amp;   After that I tried   sudo gedit /etc/pm/config.d/config   where I insert this string  SUSPEND_MODULES = "drivername logicalchipname"   But since I don't have permission to modify that file I tried to use chmod on it, which led to  but left me still in the same situation.  $ lspci -knn | grep Net -A3; rfkill list 03:00.0 Network controller [0280]: Realtek Semiconductor Co., Ltd. RTL8821AE 802.11ac PCIe Wireless Network Adapter [10ec:8821] Subsystem: XAVi Technologies Corp. RTL8821AE 802.11ac PCIe Wireless Network Adapter [1b9a:2482]  Kernel driver in use: rtl8821ae  Kernel modules: rtl8821ae  0: hci0: Bluetooth  Soft blocked: yes   Hard blocked: no 1: phy0: Wireless LAN   Soft blocked: no    Hard blocked: no   What do you suggest I do? I would like some help with this exercise.  Suppose that $f_1,\ f_2 \in V^*$ and that $\text{Ker} f_1 = \text{Ker} f_2$. Show that $f_1 = k f_2$ for some scalar $k$.  I expect your suggestions.  Thanks! How can I make my AES encryption identical between Java and Objective-C (iPhone)? I was wondering why a laser beam diverges. If all the photons are in the same direction, I would imagine that it would stay that way over a long distance. I am aware that a perfectly collimated beam with no divergence cannot be created due to diffraction, but I am looking for an explanation based on  photons rather than wave physics. I imagine a point-like particle can only experience the local properties of spacetime. But locally there is no curvature and no gravity, as it is often stated that     Locally, as expressed in the equivalence principle, spacetime is Minkowskian, and the laws of physics exhibit local Lorentz invariance. ()   But if this is the case how does a point-like particle know, which direction it should follow in a gravitational field?   Can this be a seen as a hint that point-like particles cannot be used as underlying concept for a field theory like general relativity? Getting text from last marker to EOF in POSIX.2 I have a text with marker lines like:  aaa --- bbb --- ccc   I need to get a text from the last marker (not inclusive) to EOF. In this case it will be  ccc   Is there an elegant way within POSIX.2? Right now I use two runs: first with nl and grep for the last occurrence with respective line number. Then I extract the line number and use sed to extract the chunk in question.  The text segments may be quite large, so I'm afraid to use some text-adding method like we add the text to a buffer, if we encounter the marker we empty the buffer, so that at EOF we have our last chunk in the buffer. What exactly is the cause of RAM incompatibility? I'm looking at rather short QVL lists from manufacturers, and I'm wondering what is the underlying cause of RAM incompatibility aside from the obvious: Obvious Reasons for Incompatibility  RAM Generation (DDR, DDR2, DDR3, DDR4) Form Factor / pin count RAM Type (ECC vs Non-ECC, Registered vs not, Buffered vs not) Minimum speeds on a board that cannot adjust the memory frequency Non-standard voltage range on a board that cannot adjust the memory voltage Channel count (single vs dual vs quad) Bad or inoperable module Mismatched set  Not Obvious Reasons  Brand SKU Max Capacity Speed too high  As in, why can't motherboard manufacturers just list the specifications of the RAM they will accommodate, and support every memory chip of that category instead of listing specific SKUs from specific brands? What happens electrically/physically that causes boards to fail to POST / operate with RAM that is otherwise 'compatible' with all of the specifications of RAM that is on the QVL? after upgrading to 19.04 the Grub menu have become hidden. It still works and I can boot to both Windows 10 and Ubuntu by blindly finding the entry and pressing enter. I have tried to edit the /etc/default/grub and running sudo update-grub afterward but no change. Here is how it looks now:  # If you change this file, run 'update-grub' afterwards to update # /boot/grub/grub.cfg. # For full documentation of the options in this file, see: #   info -f grub -n 'Simple configuration'  GRUB_DEFAULT=0 GRUB_TIMEOUT=10 GRUB_DISTRIBUTOR=`lsb_release -i -s 2&gt; /dev/null || echo Debian` GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" GRUB_CMDLINE_LINUX="" GRUB_TIMEOUT_STYLE=menu GRUB_HIDDEN_TIMEOUT_QUIET=false  # Uncomment to enable BadRAM filtering, modify to suit your needs # This works with Linux (no patch required) and with any kernel that obtains # the memory map information from GRUB (GNU Mach, kernel of FreeBSD ...) #GRUB_BADRAM="0x01234567,0xfefefefe,0x89abcdef,0xefefefef"  # Uncomment to disable graphical terminal (grub-pc only) #GRUB_TERMINAL=console  # The resolution used on graphical terminal # note that you can use only modes which your graphic card supports via VBE # you can see them in real GRUB with the command `vbeinfo' #GRUB_GFXMODE=640x480  # Uncomment if you don't want GRUB to pass "root=UUID=xxx" parameter to Linux #GRUB_DISABLE_LINUX_UUID=true  # Uncomment to disable generation of recovery mode menu entries #GRUB_DISABLE_RECOVERY="true"  # Uncomment to get a beep at grub start #GRUB_INIT_TUNE="480 440 1"   I can't use either Shift or Esc to make it visible too. After pressing either of these, Grub doesn't work anymore and I need to turn my laptop off and on in order to boot.  Edit: Not sure if it helps or not, but I just realized that if I go to the BIOS when I exit BIOS the grub menu shows up with no problem. The next time I boot, the grub menu is invisible again. SQL Server default collation vs database with different collation - potential problems? I'm struggling with the design of new database that has to use a different collation (Y Collation) than the SQL Server Instance default collation (X Collation). I'm looking for potential problems.  Temp Tables by default will be created with X Collation so any joins on text data types will fail unless I create temp table with Y collations on columns.  What about SQL Server string functions: REPLACE, SUBSTRING, etc.? Will they always be returned in X collation? What about when I use REPLACE(...) COLLATE Y? Do you see any potential data loss or maybe any different problems? Windows shortcut (.lnk) parser in Java? I'm currently using Win32ShellFolderManager2 and ShellFolder.getLinkLocation to resolve windows shortcuts in Java. Unfortunately, if the Java program is running as a service under Vista, getLinkLocation, this does not work. Specifically, I get an exception stating "Could not get shell folder ID list".  Searching the web does turn up mentions of this error message, but always in connection with JFileChooser. I'm not using JFileChooser, I just need to resolve a .lnk file to its destination.  Does anyone know of a 3rd-party parser for .lnk files written in Java I could use?  I've since found unofficial documentation for the .lnk format , but I'd rather not have to do the work if anyone has done it before, since the format is rather scary. I am using gedit text editor with embedded terminal in ubuntu 12.04. I'm trying to search for some text using grep. I want to search for this line of code   'type' =&gt; 'select'   I tried:  grep -r '\'type\' =&gt; \'select\''  But grep didn't return any results.  So can someone kindly tell me how to search for the code above? Wiener Process $dB^2=dt$ Why is $dB^2=dt$? Every online source I've come across lists this as an exercise or just states it, but why isn't this ever explicitly proved? I know that $dB=\sqrt{dt}Z$, but I don't know what squaring a Gaussian random variable means. Why is the picture not inserted? When I pdflatex the text below it doesn't insert the picture, but writes the name of the file instead (pic).  And the geometry line gives the error  ! LaTeX Error: Missing \begin{document}.   The picture I am using is at  and the floatrow package .  Question  Can anyone see what is wrong?  Text  \documentclass[a4paper]{article} \pdfpagewidth=\paperwidth \pdfpageheight=\paperheight \usepackage{aeguill} \pdfcompresslevel=9 \usepackage[pdftex,colorlinks]{hyperref}  \usepackage{a4} \usepackage{fixltx2e} \usepackage[danish]{babel} \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc}  \usepackage[draft]{graphicx} \usepackage{floatrow}  \usepackage{geometry}  \geometry{margin=10pt,paperwidth=\textwidth+20pt,paperheight=\textheight+20pt}  \begin{document}  The first paragraph.  \begin{figure}   \includegraphics{pic} \end{figure}  And the second.  \end{document}