Prove that the set of polynomials with rational coefficients is countable. (Doubt)
Show that the set of polynomials with rational coefficients is countable.
Dot Product Intuition I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ).      For example, suppose I have vector $\mathbf{a} = (a_1,a_2)$ and vector $\mathbf{b}=(b_1,b_2)$.  What's the physical or geometrical meaning that   $$a_1b_1 + a_2b_2 = |\mathbf{a}||\mathbf{b}|\cos(\theta)\;?$$  Why is multiplying $|\mathbf{b}|$ times $|\mathbf{a}|$ in direction of $\mathbf{b}$ the same as multiplying the first and second components of $\mathbf{a}$ and $\mathbf{b}$ and summing ?       I know this relationship comes out when we use the law of cosines to prove, but even then i cant get a intuition in this relationship.  This image clarifies my doubt:    Thanks Invalid SSL certificate when pushing to Git server Basic desktop actions are not available on Ubuntu 20.04 In Ubuntu 20.04, on the desktop, there is a number of actions I wish I could do, such as, by order of importance :  use keyboard shortcuts such as select all, copy / paste. use drag &amp; drop with icons deactivate the &quot;mouse over&quot; highlight, and restrict the clicking area to only the icon, not the surrounding square  To put it simply, I'd like to have the exact same features as offered by the desktop of Ubuntu 18.04 (probably related to a specific version of GNOME, and not Ubuntu itself). Am I missing something obvious here? I actually see no reason why those actions are removed or deactivated on the desktop, since they all work in any Nautilus window. Proving that $a \mathbb{Z} \cap b\mathbb{Z} = \operatorname{lcm}(a,b)\mathbb Z$ Given $m \in \mathbb{Z}$, let $m\mathbb{Z}$ denote the set of integer multiples of $m$, i.e. $m\mathbb{Z} := \{mk\mid k \in \mathbb{Z}\}$.   Now let $a,b \in \mathbb{Z}$ with $a,b$ not both $0$. Prove that $a\mathbb{Z} \cap b\mathbb{Z} = \operatorname{lcm}(a,b)\mathbb{Z}$.   I am trying to write a proof for this, but I am unsure of what method to use. Also I am confused by $mk\mid k$, because wouldn't $m=1$ for this to be true. Ways to update Ubuntu (debian-based distros) offline What is the easiest way to update/upgrade an offline Ubuntu machine with the help of a Windows machine that has internet access?  This question also applies to installing new packages offline and not only updates/upgrades.  The Linux system shall remain completely cut off from the Windows machine, meaning there should be no ethernet connection or the likes between the two computers.  Update:  Please have patience, since I haven't updated this since two weeks. There are some complications getting apt-offline to work properly on Lubuntu and all other methods seem to be too old and deprecated:  AptOff - last update 2006 AptMedium - last update 2007 Keryx - last update 2011, looks like this project was discontinued  I hope I will be able to somehow update the offline machine, finally.  While I was still not able to update/upgrade the offline system due to authentication errors, I accepted an answer, because it's taking too long. Troubleshoot High CPU usage by the "System" process I have noticed that from some time my system is freezing and its probably caused by the high CPU usage which is caused by the system process.  All applications I'm running is the Skype, TeamSpeak and Chrome so it definitely shouldnt eat that amount of CPU.  You can see the problem itself and running processes in the screenshot below:    Sometimes CPU usage is reaching 90%, but the average usage is like 40-65%.  My PC parameters:   Windows 8 (customer preview) Intel Core i3 - 2350M 8 GB RAM   I'd appreciate any help attempt! Regards.  --UPDATE--  As the user below posted a great answer, I have noticed that the process that is eating the most CPU in the system is called Arthurx.sys, simple google tells that it's a TPLink driver (an wifi adapter, I have bought like 2 weeks ago!) drivers has been installed from the Windows MSDN, but also tried to install the drivers from the attached CD, but it doesn't help. From the system start, it is using like 5% of the CPU only, but after 2-4 hours of working it is growing up and reaching 40-60% of the CPU usage.  Device name: TPLink WN722N Take $\mathbb{R}$ as a vector space over $\mathbb{Q}$, then a basis for $\mathbb{R}$, then $|B|=2^{\aleph_0}$ I am trying to prove that Given $\mathbb{R}$ as a vector space over $\mathbb{Q}$, and a basis for $\mathbb{R}$, then $|B|=2^{\aleph_0}$.  Proof: Suppse that $|B|&lt;2^{\aleph_0}$, then $\mathbb R = span\{ b_i ; b_i \in B \}$ So $ \mathbb R = \{ \Sigma_{i=1}^{\infty}\alpha_i b_i ; \alpha_i \in \mathbb Q, b_i \in B, i \in \mathbb N \} = \bigcup_{n=1}^{\infty}\{\Sigma_{i=1}^n \alpha_ib_i\}$. Which is a contradiction since a countable union of countable union of countable sets is countable and $\mathbb R$ is not.  What do you think?  Thank you, Shir Integrating $\int\frac{x^3}{\sqrt{9-x^2}}dx$ via trig substitution What I have done so far: Substituting $$x=3\sin(t)\Rightarrow dx=3\cos(t)dt$$ converting our integral to  $$I=\int\frac{x^3}{\sqrt{9-x^2}}dx=\int \frac{27\sin^3(t) dt}{3\sqrt{\cos^2(t)}}3\cos(t)dt\\ \Rightarrow \frac{I}{27}=\int \sin^3(t)dt=-\frac{\sin^2(t)\cos(t)}{3}+\frac{2}{3}\int\sin(t)dt\\ \Rightarrow I=18\cos(t)-9\sin^2(t)\cos(t)+c$$ Which is where I'm stuck. I can substitute back in for the $\sin(t)$ terms but don't know hot to deal with the $\cos(t)$ terms and back out information about $x$.   edited: because I miswrote the problem, I am sorry. Unable to install Ubuntu on Lenovo Y500 I am totally unable to install any flavor of Linux on my lenovo Y500. TL;DR:  with EFI on -&gt; cannot even access installer with legacy - &gt; installer runs fine, but installed OS never boots  Long Version: It is going to be hard to recap what I did, because I tried so many things, but here is the gist of it: EFI mode (secure boot disabled): Trial #1  ubuntu 12.10 x64 &amp; Mint 14 x64 boot from usb see screen with choices pick &quot;try ubuntu&quot; get black screen  Trial #2  ubuntu 12.10 x64 &amp; Mint 14 x64 boot from usb see screen with choices press &quot;e&quot; to enter edit mode remove &quot;quiet splash&quot; replace with &quot;nomodeset&quot; end up on the command line with an error message from x server: &quot;no screen found&quot; try fiddling around with xorg.conf to no avail  Trial #3  Arch Linux Live USB EFI mode no boot  Trial #4  ubuntu 12.10 x64 &amp; Mint 14 x64 set &quot;e&quot; to enter edit mode in menu add acpi=off end up with &quot;(initramfs) unable to find a medium&quot; message  Trial #5  same as above set SATA mode to legacy (it was AHCI) error loop that says something along the lines of &quot;ata1.00: hard resetting link&quot;  Legacy Mode Trial #1  ubuntu 12.10 x64 &amp; Mint 14 x64 USB boot Everything seems to work Proceed to normal install (picks SSD drive by default) Everything seems to work reboot blank screen reboot with &quot;shift&quot; pressed see message &quot;grub loading.&quot; and nothing As per advised on Ubuntu's community EFI page, run boot-repair same results run boot-repair again same results  Trial #2  ubuntu 12.10 x64 &amp; Mint 14 x64 same as above, but pick 1T HDD as target install same results  Trial #3  create a /boot partition on the SSD install on SSD same results (&quot;grub loading.&quot;) run boot repair twice same results  Trial #4  create a GPT filesystem on HDD create a /boot partition on the HDD install on HDD same results as above  Trial #5  install ubuntu on HDD install boot on all drives (HDD &amp; SSD) same results  Trials #6  install Arch on SSD no boot loader found  Notes:  Every step has been tried with multiple USBs. All isos' MD5Sums have been verified LiveUSBs have been created both through ubuntu's embedded app and unetbootin (save for Arch's live USB which was created by DDing the iso as per advised in the Arch wiki) I have wiped out windows in my first trials, and I don't care, I am never going to use it. I don't want to dual-boot. I just want Ubuntu or Mint or Arch as far as I can remember (it gets hard after 50+ trials), I tried every option with sata mode IDE or normal. Not sure though.  I've also tried numerous other things that I can't recall right now. All in all I've been giving this 3 to 5 hours a day for a week and a half to absolutely no avail. I am extremely disappointed as this is my first Lenovo and word is Lenovo is Linux-friendly...I have never spent more than an hour installing linux on any machine. What's even more frustrating is that it seems to be working for , and I can't find any hint telling me how they managed. I wrote to one guy who posted a serie of youtube videos about how to get everything working perfectly on the Y500, and he told me he just installed mint without any problems. Might be I am a particular case because I have the SLI version of the Y500. If anyone has pointers, incomplete solutions, ideas, black magic, I'll be forever in debt. Thank you P.S: This question, afaik, is not a duplicate of an existing question. I've scoured all questions containing &quot;lenovo&quot; and &quot;install&quot;, and none applies to me. What tool do you use to monitor your servers? For a more comprehensive list of monitoring tools and their features, check out .   As the question states, what are the most commonly used tools used for this task and what are their strengths and weaknesses? How to Set Firefox 4 to only store History for a set period of time, i.e. one day? I want to setup Firefox 4 to only store my history for 1 day (or some other set period of time), but I can't seem to find an option to do that in the options dialog, all I can see is "Remember History", "Never Remember History" and "Custom History Settings". "Custom History Settings" only lets me select what to keep, not for how long.  Is there anyway I can do this? How many RSA keys before a collision? I was wondering how many possible private/public keys exist? If a million people – for whatever reason – would try to generate 5 keys each in the same minute (on the same date and time) is there a high chance of collision? I believe GUID would suffer from that problem as many bits are reversed for date/time (and GUID version) and isn't meant to be used in that way.  Would RSA suffer from collisions if many keys were to be generated in the same moment? Is the amount of possible keys known? I know RSA is based on prime numbers and small numbers are to be rejected. I’m sure values above a certain amount of digits/bits are rejected because software may not be able to support those large values?  So: How many RSA keys before a collision? And if you would try to make many at the same time, would that give you a high chance of collision? Can we have a way to edit bounty custom message? I put up  over on SO with the assumption that the text would be formatted like it is for a question or answer.  I'm not really sure if I was incorrect in my assumption, or if I just formatted my text improperly, as I can't go back and edit what I put in.  Am I missing an obvious feature?  If not, would it be reasonable to request bounty-text edits—perhaps with moderator-approval to prevent abuse?  Edit: I was trying to use markdown to produce an unordered list as part of my bounty.  What you see in the link is a blob of text with dashes separating phrases Consider the following MWE:  \documentclass{article} \usepackage{pstricks} \begin{document} A \end{document}   It produces the error  Runaway argument? ] {pgffor.code.tex}  ! Paragraph ended before \ProvidesPackageRCS@i was complete. &lt;to be read again&gt;                     \par  l.11   How do I fix this? I have set theme handler for a form in Drupal and this handler has a template file. What is the easiest way to send a value from PHP to JavaScript in that include file? Hypothetically, should I pay off the smaller + higher interest rate loan first, or the larger + higher absolute interest cost? I have this hypothetical situation on my mind and I am not sure which is the best solution. Given:  Loan 1: $340,000 at an interest rate of 1.69% per year Loan 2: $40,000 at an interest rate of  3.6% per year  I have a fixed amount available to pay these, let's say, $1,500 monthly and I can decide how much I want to pay each one (I could pay 750/750, 1000/500, 1500/0 for instance). I do not have a time limit to pay these. My question: What is the optimal way to pay these? What I always heard is: pay the debt that has the highest interest first. This would lead me to a solution: pay loan 2 as fast as possible. But the total amount of interest on 340,000 is higher (in this case, $5780 annually) than the amount over $40,000 (1440, in this case), so, to me, it feels I should pay the Loan 1 first. I would love to run some simulations, but I have no idea how to calculate these things (I don't even have a clue on how to research it). ps: this is really a hypothetical situation, I am not getting such loans without being able to calculate it, don't worry ;) I've got a road bike with brake lever shifters, and it's developed a small problem with age: sometimes, the shifter doesn't catch the first time or two I push it. It'll move easily, with no resistance, since it's not actually pulling on the cable, then after a few tries, catch and work normally. The colder the temperature, the more often this happens.  I don't know how the mechanism works internally, so I'm curious what would cause this, and also whether it's (as I suspect) gradual wear of some part of the mechanism, and if it's likely to fail catastrophically, or gradually worsen. Story about astronauts who set off an alien warning beacon on the moon (or Mars?) We are currently working on a windows based application where users capture data. Some data is captured in a table where other data is captured in text blocks. Im currently faced with the dilemma of how to align contents specifically decimal values.  Editable cells in a grid are indicated with a pencil icon which disappears once in edit mode. You will notice that the table consists of various field types, decimals and text. So what should I do and why?  A> right align all cell content  B> left align all cell content  C> right align only decimal values in cells  D> Whatever else you can suggest Removing elements with Array.map in JavaScript