Show that F2 has an infinite index free subgroup of rank 2
Show the existence of a certain subgroup of F2
/A problem from the 2012 MIT Integration Bee is $$ \int_0^1\frac{x^7-1}{\log(x)}\mathrm dx $$ The answer is $\log(8)$. Wolfram Alpha gives an indefinite form in terms of the logarithmic integral function, but times out doing the computation. Is there a way to do it by hand? If a blocking creature leaves the battlefield, is the attacker still blocked, or will it damage the opponent? Suppose Player 1 has a Dragon on the table and  in hand, while Player 2 has two Spiders on the table.  Player 1 chooses to attack with the Dragon and Player 2 chooses to block with one of the Spiders.  Before combat damage is to be dealt, Player 1 uses Unsummon on the blocking Spider.    Does Player 2 take any life damage from the attacking Dragon? Is there a benchmark tool for Ubuntu? Is there a benchmark tool to measure computer performance for Ubuntu like SuperPI, 3DMark, or PCMark? I was tasked to prove that when given 2 graphs $G$ and $\bar{G}$ (complement), at least one of them is a always a connected graph.  Well, I always post my attempt at solution, but here I'm totally stuck. I tried to do raw algebraic manipulations with # of components, circuit ranks, etc, but to no avail. So I really hope someone could give me a hint on how to approach this problem. Close vote review audit flawed - fails without casting vote Having just  from any of the queues, it was reviewing close votes; however, this audit was flawed in that I didn't even cast a close vote.  Often I press the Close button to preview greater detail and sentiment of the closure, spawning the Why should this question be closed dialog box:    Especially for duplicates, where functionality is solely dependent on spawning this dialog.  From Closing > Duplicate, I can search or view selected duplicates:    Although, I realize general closure reasons are summarized on the page:    This question had 6-upvotes, but seemed very localized referencing subjective query is "taking too long" and even had a sqlfiddle link.  Maybe overly broad regarding performance tuning, the OP was posting like a forum with two answer posts to his own question  and  with one of those answers asking follow-up questions.    Wanting to gauge community response, current closure totals struck my curiosity.  So, I pressed the Close button and boom - oh, the humanity!  I failed, and didn't even cast a close vote.    Votes are not cast until a reason for closure is selected and Vote to Close is pressed.  Taking any review very seriously on this site, it only seems fair that my review quality is tested through actionable steps instead of clicking for more detailed information, leveraging the functional design of the close vote user interface.  Sure, it's complex to seed fake vote totals, and then there's controversy of mindless voting in the majority or people who simply cherry pick 4-vote questions.  Still though, it seems unfair I should hesitate to review further information of vote totals and duplicates before skipping due to the implementation of this audit. How can one prove that $\lim_{n \to \infty}a^{1/n}=1$ for every $a>0$? Problem:  Ajax suggest-search on [n] ingredients in recipes. That is: match recipes against multiple ingredients.  For instance: SELECT Recipes using "flower", "salt" would produce: "Pizza", "Bread", "Saltwater" and so forth.  Tables:  Ingredients [     IngredientsID INT [PK],     IngredientsName VARCHAR ]  Recipes [     RecipesID INT [PK],     RecipesName VARCHAR ]  IngredientsRecipes [     IngredientsRecipesID INT [PK],     IngredientsID INT,     RecipesID INT ]   Query:  SELECT     Recipes.RecipesID,     Recipes.RecipesName,     Ingredients.IngredientsID,     Ingredients.IngredientsName FROM     IngredientsRecipes      INNER JOIN Ingredients     ON IngredientsRecipes.IngredientsID = Ingredients.IngredientsID      INNER JOIN Recipes     ON IngredientsRecipes.RecipesID = Recipes.RecipesID WHERE     Ingredients.IngredientsName IN ('salt', 'water', 'flower')   I am currently constructing my query using ASP.NET C# because of the dynamic nature of the WHERE clause.  I bites that I have to construct the query in my code-layer instead of using a stored procedure/pure SQL, which in theory should be much faster.  Have you guys got any thoughts on how I would move all of the logic from my code-layer to pure SQL, or at least how I can optimize the performance of what I'm doing?  I am thinking along the lines of temporary tables:  Step one: SELECT IngredientsID FROM Ingredients and INSERT INTO temp-table  Step two: SELECT RecipesName FROM Recipes joined with IngredientsRecipes joined with temp-table.IngredientsID How does TikZ sort intersections? I used the sort by key for intersections to sort along a line, expecting the intersections to be numbered 1 through 5. Why were they numbered 4, 5, 3, 1, 2? Playing around with it, I could not seem to figure out how the order is determined. The code and image follow.  \documentclass{standalone} \usepackage{tikz} \usetikzlibrary{intersections}  \begin{document} \begin{tikzpicture}     \draw[name path=grid] [xstep=3,ystep=2] (0,0) grid (9,8);     \draw[-&gt;, name path=line] (2,1) -- (7,7);     \draw[name intersections={of=grid and line, sort by=line, name=i, total=\t}]         \foreach \s in {1,...,\t}{(i-\s) node {\s}}; \end{tikzpicture} \end{document} Tried to install both Windows and Ubuntu, now neither will boot Hardware:  I've got an SSD (c),an internal HDD (d), an external HDD (e) and a USB flash drive (f).   How this all went down:   I downloaded Ubuntu and used a program to install it on my (F) drive. I reset my computer, ran the (F) drive for boot, and installed Ubuntu. I reset my computer, and it forcibly automatically booted into Windows 10 instead of Ubuntu. I went into Windows 10, and used a program called Easy BCD, by NeoSmart. I followed their guide, but after step 6, it wasn't the same for me. Their guide said that upon rebooting, I should see a grub screen that lets me choose between Windows/Linux and a few other options, however my computer would only boot into Windows 10.  I decided to follow the rest of the guide regardless, however by the end it would let me boot 'windows' or 'linux', but the linux option would always break. It went to a screen with lots of commands, that I thought -may- be a barebones Linux or something, but it definitely wasn't Ubuntu.  I decided to reinstall Ubuntu using the USB flash drive. Upon resetting, no matter what hard drive or format I select for booting (besides the F drive), it says An operating system wasn't found. Try disconnecting any drives that don't contain an operating system. I can boot the F drive and run Ubuntu, but it's a one time version that doesn't save any of my data. When I went into the installation, it still says that I already have Ubuntu installed on the 4TB HDD and that I have Windows 10 installed on the SSD, however I cannot seem to boot into either of them.   TLDR: Windows 10/Ubuntu are both installed, Windows on the internal SSD and Ubuntu on the external HDD. Ubuntu's installation program on the (F) drive recognizes them, however, when I try to boot anything it says An operating system wasn't found.   When I press F11 and force it to boot from the SSD it says:  error: no such device: 83b6c64d-9a2c0467b-a02c-6374ee6ac3dc.  error: unknown filesystem. Entering rescue mode. . .  grub rescue&gt; _   and it will allow me to type some commands. $f(x)=x^3$ is convex nor concave without to derivate How can I undo git reset --hard HEAD~1? remove isp-provided dns My isp provides a dns service; but unfortunately they do block certain sites. I would like to use opendns or google; but it seems like it doesn't work.    I have set up a wired connection; given a manual ip and entered opendns-ip's in the nameserver field (all through the gui). Still url's are being resolved through my ISP. I have tried to configure the same on my girlfriends laptop (win 7) and I also can't seem to do the setup. My previous computer also was a win 7 and had no problem. (so I believe the problem is 'me') If it were only one site, i could edit the hosts file; but I prefere to circumvent their whole dns server.  Is there something an isp can do to block the use of other dns-servers than theirs? Is it possible to re-route the ISP dns to the opendns? Online calculators such as  (view page source) use the below formulas to get meters per degree. I understand in general how distance per degree varies depending on latitude location, but I do not understand how that translates to the below. More specifically, where do the constants, the 3 "cos" terms in each formula, and coefficients (2, 4, 6; 3, and 5) for "lat" come from?      // Set up "Constants"     m1 = 111132.92;     // latitude calculation term 1     m2 = -559.82;       // latitude calculation term 2     m3 = 1.175;         // latitude calculation term 3     m4 = -0.0023;       // latitude calculation term 4     p1 = 111412.84;     // longitude calculation term 1     p2 = -93.5;         // longitude calculation term 2     p3 = 0.118;         // longitude calculation term 3      // Calculate the length of a degree of latitude and longitude in meters     latlen = m1 + (m2 * Math.cos(2 * lat)) + (m3 * Math.cos(4 * lat)) +             (m4 * Math.cos(6 * lat));     longlen = (p1 * Math.cos(lat)) + (p2 * Math.cos(3 * lat)) +                 (p3 * Math.cos(5 * lat)); I have had several business loans and understand that this is not by any means typical.  I noticed that for a year my payments were not decreasing the principal on the loan. When I asked the bank why, I found out that at some point my payments were getting applied to principal only. Now I am paying interest off that was not paid for 6 months or so.  My banker states that he can change all the interest payments to principal payments. I am confused to why this would be. If I make all my principal payments first then would the bank lose a substantial amount of interest?  This has happened on 2 of the 3 loans I have looked at so far.  If they are willing to change all the payments to principal only, would this be beneficial to paying less interest ultimately? I assume there would be an interest account accruing somewhere that would need to be paid off at the end of the loan.  I understand that typically you have a loan balance that would include interest and principal. My balance owed did not change when I continued to make same payment? It gets applied to interest. The balance on the statement must not be a true balance. There must be interest owed above the balance showed but why would they do this to start with? They applied full payments to principle only first in order for interest to get to be more than several payments Calendar month view is unresponsive on iOS 9 How to get my hover-menu back? Local html file AJAX Call and jQuery Woes This is problem 31b) from Dummit and Foote chapter 14.2. I am looking for a hint on how to attack the problem, since I have been thinking about it for a couple of hours but I don't even know where to start. The problem says:  Let $K$ be a finite extension of $F$ of degree $n$. Let $\alpha$ be an element of $K$. Prove that the minimal polynomial for $\alpha$ over $F$ is the same as the minimal polynomial for the linear transformation $T_{\alpha}$. In this problem, $T_{\alpha}$ is an $F$-linear transformation of $K$ that arises from $\alpha$ acting by left multiplication on $K$.  I appreciate any helpful suggestions on how to start attacking the problem or any possible hint. Thanks! Graphical glitches using Ubuntu 18.04 on Dell XPS 13 2019 laptop My laptop keeps on having weird screen glitches, sometimes for a second, and that goes away just by moving the cursor, sometimes, making me restart the whole thing.  I could still use the play/pause key and it still played/paused the video that was running even though the screen was completely teared.  Also, printscreens actually print a normal screen and do not show the problem.  Some pictures of the screen when it happens :     Do you guys know what I could do to solve the issue ? How to create a custom "confirm" & pause js execution until user clicks button? Ok, I'm doing a bunch of RIA/AJAX stuff and need to create a "pretty", custom confirm box which is a DIV (not the built-in javascript confirm). I'm having trouble determining how to accomplish a pause in execution to give the user a chance to accept or decline the condition before either resuming or halting execution. (depending upon their answer)   So here's the general flow of logic I'm dealing with:   User selects an item from dropdown and clicks button. In client-side javascript eventhandler for button, I need to check a (this is the key) SERIES of conditions for the item they chose in dropdown.  These conditions could possibly result in not showing any confirmation at all or possibly only some of the conditions may evaluate to true which means I'll need to ask the user to accept or decline the condition before proceeding. Only one confirmation should be show at a time.   To demonstrate the logic:  function buttonEventHandler() {  if (condition1) {   if(!showConfirmForCondition1) // want execution to pause while waiting for user response.      return; // discontinue execution }  if (condition2) {   if (!showConfirmForCondition2) // want execution to pause while waiting for user response.       return; // discontinue execution }  if (condition3) {   if (!showConfirmForCondition3) // want execution to pause while waiting for user response.       return; // discontinue execution }  ...   }    If anybody has dealt with this challenge before and found a solution, help would be greatly appreciated. As a note, I'm also using the MS Ajax and jQuery libraries although I haven't found any functionality that may already be included in those for this problem.