If there a formal definition for the sum of two irrationals is rational if it is only the additive inverse plus some rational?
Jutifying: irrational + rational = irrational
I am wondering if we should use  or  for buttons and headings in websites and web/mobile apps? Title Case: Manage my Account Delete my Account  Sentence case: Manage my account Delete my account  In  I found the recommendation 'choose your style and be consistent'. This is it? Aren't there any other (additional) recommendations or sources. Like for specific use cases or apps where the one style would be preferred over the other? Edit: The question refers primarily to English, but differences in other languages are also good to know. What is this "answer your own question" jazz? $M$ finitely generated if submodule and quotient are finitely generated. I know this must be easy. I'm new to modules, so perhaps I must be missing  something important.     Let $M$ be an $R$-module. Show that if there exists a submodule $N$ such that $N$ and $M/N$ are finitely generated, then $M$ is finitely generated.   If the exact sequence $0 \longrightarrow N \longrightarrow M \longrightarrow M/N \longrightarrow 0 $ (where the second and third arrows are inclusion and projection respectively) splits then $M$ would be isomorphic to the direct sum of $M$ and $M/N$ and the conclusion would follow (I think). But I can't show that it splits.  Thank you. How many attacks can a fighter with two-weapon fighting make at 5th level? How can I stop Steam from loading when I start my computer? Every time I boot into OSX or Windows, Steam automatically starts.  Is there a way to stop Steam starting whenever I boot up my computer? Regarding the advanced edition of FTL:  If all of your crew dies but you have a cloning bay on your vessel, is it immediately game over, or does the cloning bay prevent that?  Is the game smart enough to know that a damaged cloning bay and a fully dead crew means you cannot recover, but a functional one (even if temporarily without power) could revive your crew?  Also, does this answer differ between hostile ships and your own ship?  (That is, do you get to capture and loot enemy vessels that have all dead crew but have a functional cloning bay?) Test whether a glob has any matches in bash Name for “having no concern about the truthfulness of one’s own remarks” Take the following statement:     I hate the ease with which X makes disparaging remarks about their rivals without the least concern as to whether they are truthful or fair.   Now I would like a word expressing this lack of concern, so that I could just plug it in the blank below and keep the meaning of the sentence above:     I hate the _________ with which X makes disparaging remarks about their rivals.   This is an all too common attitude and I think it deserves a name of its own. The word that came to my mind was levity, but I haven’t come across this word very often and I’m not sure of its particular tones and shades of meaning.      Simple definition   : a lack of seriousness   : an amusing quality   Full definition   1 : excessive or unseemly frivolity   2 : lack of steadiness: changeableness     The attitude I’m trying to name certainly displays lack of seriousness as to the truthfulness and fairness of the remarks, but I do not want to imply they are made in jest or that X is being silly or thoughtless. X could be very serious about their aims: “I’m going to say this; don’t give a dam whether that’s true or not; even if it’s not true I’ll get away with it.” This excludes, I think, all the words Merriam-Webster lists as related to levity, which probably means it excludes levity too:     facetiousness, flightiness, flippancy, frivolousness, frothiness, frivolity, light-headedness, light-mindedness, lightness, silliness   So I’ll accept an answer that despite my reservations shows that one of these words is indeed the one that works best in the sentence above; or gives a more fitting word; or shows that no word really fits (I am of course aware of the difficulties of proving a negative).  I prefer a single word but I’m willing to trade singleness for meaning: so I would take a short phrase it fits better than any single word.  Update  I’m spoilt for choice. But I wonder whether the various suggestions implying lack of concern would in my sentence imply lack of concern specifically for truth, which is what I’m after, or could instead be interpreted as lack of regard for norms of polite behaviour, rival’s feelings, or consequences (say, libel lawsuits, rivals reacting with violence, people feeling encouraged to attack the rivals, etc.)  Tweak the example sentence if it helps (say, I hate the ____ of X’s remarks). How can I reset my menus? I accidentally opened too many menus (or windows) and I want to reset them to default.  It is really annoying, and effecting my animation.  Thanks MySQL Results as comma separated list I need to run a query like:  SELECT p.id, p.name,         (SELECT name            FROM sites s           WHERE s.id = p.site_id) AS site_list   FROM publications p   But I'd like the sub-select to return a comma separated list, instead of a column of data.  Is this even possible, and if so, how? How can momentum but not energy be conserved in an inelastic collision? Why is the Earth so fat? I made a naive calculation of the height of Earth's equatorial bulge and found that it should be about 10km.  The true height is about 20km.  My question is: why is there this discrepancy?  The calculation I did was to imagine placing a ball down on the spinning Earth.  Wherever I place it, it shouldn't move.  The gravitational potential per unit mass of the ball is $hg$, with $h$ the height above the pole-to-center distance of the Earth (call that $R$) and $g$ gravitational acceleration.  Gravity wants to pull the ball towards the poles, away from the bulge.  It is balanced by the centrifugal force, which has a potential $-\omega^2R^2\sin^2\theta/2$ per unit mass, with $\omega$ Earth's angular velocity and $\theta$ the angle from the north pole.  This comes taking what in an inertial frame would be the ball's kinetic energy and making it a potential in the accelerating frame.  If the ball doesn't move, this potential must be constant, so  $$U = hg - \frac{(\omega R \sin\theta)^2}{2} = \textrm{const}$$  we might as well let the constant be zero and write  $$h = \frac{(\omega R \sin\theta)^2}{2g}$$  For the Earth,   $$R = 6.4*10^6m$$  $$\omega = \frac{2\pi}{24\ \textrm{hours}}$$  $$g = 9.8\ m/s^2$$  This gives 10.8 km when $\theta = \pi/2$, so the equatorial bulge should be roughly that large.    , the Earth is 42.72 km wider in diameter at the equator than pole-to-pole, meaning the bulge is about twice as large as I expected.  (Wikipedia cites diameter; I estimated radius.)  Where is the extra bulge coming from?  My simple calculation uses $g$ and $R$ as constants, but neither varies more than a percent or so.  It's true the Earth does not have uniform density, but it's not clear to me how this should affect the calculation, so long as the density distribution is still spherically-symmetric (or nearly so).  (Wikipedia also includes , without derivation, that agrees with mine.) Can markdown icons be available on the iPad? As you can see by the screenshot below, neither buttons nor even the live-preview is available on the iPad.    StackExchange is great on mobile devices, maybe it can be full available in mobile safari as well.  The same thing happens with the tags autocomplete, there is no options for it! we need to check previous questions to know what tags are available to use. Simple explanation to the induction from the slowly changing $\vec B$ of a solenoid in the region of $0$ magnetic field equivalent norms in Banach spaces of infinite dimension When I gain control of a creature, do I also gain control of any auras or equipment attached to it? What would happen if I 'd a creature that had been enchanted by a bestow creature and cast a spell to give it protection from the enchantment's color on it to remove the bestowed creature?  For instance:  They have just bestowed a  on a  and passes the turn. I use Portent of Betrayal to take control of the Asphodel Wanderer, attack and in the second main I cast Gods Willing to give the Asphodel Wanderer protection from black. Nighthowler detaches and where does it go, my control, opponents control or graveyard?  As an additional question, in the same vein:  What if I destroyed the creature while it was on my side? So change the gods willing to a hero's downfall in the example.  Relevent (under state-based actions):     704.5n If an Aura is attached to an illegal object or player, or is not attached to an object or player, that Aura is put into its owner's graveyard. One of my coworkers has provided me with a Bash syntax that I am unfamiliar with.  My Google foo has failed me on figuring out what it does and why/when I should use it.    The command that he sent me was of this form:  someVariable=something command   Initially, I thought that this was equivalent to the following:   someVariable=something ; command   Or  someVariable=something  command   But this doesn't appear to the be case.  Examples:   [Jan-03 11:26][~]$ # Look at the environment variable BAZ. It is currently empty [Jan-03 11:26][~]$ echo $BAZ  [Jan-03 11:27][~]$ # Try running a command of the same format     [Jan-03 11:27][~]$ BAZ=jake echo $BAZ  [Jan-03 11:27][~]$     [Jan-03 11:27][~]$ # Now, echo BAZ again. It is still empty:     [Jan-03 11:27][~]$ echo $BAZ  [Jan-03 11:27][~]$     [Jan-03 11:28][~]$     [Jan-03 11:28][~]$ # If we add a semi-colon to the command, we get dramatically different results: [Jan-03 11:28][~]$ BAZ=jake ; echo $BAZ jake [Jan-03 11:28][~]$ [Jan-03 11:28][~]$ # And we can see that the variable is actually set: [Jan-03 11:29][~]$ echo $BAZ jake [Jan-03 11:29][~]$   What does this syntax do?  What happens to the variable that has been set?  Why does this work? Prove that $\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2$ If $x_1,...,x_n$ are positive real numbers and if $y_k=1/x_k$, prove that $$\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2.$$   I've been learning induction, and I've come across this problem that I really can't even break down and begin to think about. I've been told it has something to do with Cauchy-Schwarz, but I cannot figure out how to apply it. I would appreciate help figuring out how to go about and formulate this proof. Thanks! Suppose we have (some categories and some functors such that) $F_1$ is left adjoint to $G_1$, $G_1$ left adjoint to $F_2$, $F_2$ left adjoint to $G_2$. Will $F_1$ then be equal to $F_2$ (and $G_1$ to $G_2$)?  Or can there exist arbitrarily long 'chains' of adjoint functors that do not trivialize? I'm planning to visit a summer school in the Schengen zone. What kind of visa should I apply for? I will be attending a summer school which will last 10 days.  I want to know: for applying for a Schengen visa, is the purpose of my trip considered as touristic or study?