Proof that $M \;$is a field
Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field
jsFiddle: no connection between html and js? Can't call simple function from button? How does one use circular polarizers? In  (2017), Logan seems to be dying from sort sort of ailment, that prevents him to heal properly and seems to be decaying his body in a way that wasn't happening in any of the other films his character appears in.  I don't recall that this is being explained in the film. The Wikipedia page only mentions "adamantium poisoning" and "his healing factor fading", but it is not backed up with citations, so I would love to get an in-depth answer for this question, maybe using excerpts from the comic or original source material.  What is happening to Logan? Square roots not on same height This part C\frac{1}{\sqrt{G}}\sqrt{\rho} doesn't seem to be on the same height as this part \sqrt{b}\sqrt{l} when compiled. What am I doing wrong?  \begin{flalign*} T=C\frac{1}{\sqrt{G}}\sqrt{\rho}\sqrt{b}\sqrt{l} &amp;&amp; \end{flalign*} How to draw neat polygons around scatterplot regions in ggplot2 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. ALTER table - adding AUTOINCREMENT in MySQL How do I efficiently get "the most recent corresponding row"? I have a query pattern that must be very common, but I don't know how to write an efficient query for it.  I want to look up the rows of a table that correspond to "the most recent date not after" the rows of another table.  I have a table, inventory say, which represents the inventory I hold on a certain day.  date       | good | quantity ------------------------------ 2013-08-09 | egg  | 5 2013-08-09 | pear | 7 2013-08-02 | egg  | 1 2013-08-02 | pear | 2   and a table, "price" say, which holds the price of a good on a given day  date       | good | price -------------------------- 2013-08-07 | egg  | 120 2013-08-06 | pear | 200 2013-08-01 | egg  | 110 2013-07-30 | pear | 220   How can I efficiently get the "most recent" price for each row of the inventory table, i.e.  date       | pricing date | good | quantity | price ---------------------------------------------------- 2013-08-09 | 2013-08-07   | egg  | 5        | 120 2013-08-09 | 2013-08-06   | pear | 7        | 200 2013-08-02 | 2013-08-01   | egg  | 1        | 110 2013-08-02 | 2013-07-30   | pear | 2        | 220   I know one way of doing this:  select inventory.date, max(price.date) as pricing_date, good from inventory, price where inventory.date &gt;= price.date and inventory.good = price.good group by inventory.date, good   and then join this query again to inventory. For large tables even doing the first query (without joining again to inventory) is very slow.  However, the same problem is quickly solved if I simply use my programming language to issue one max(price.date) ... where price.date &lt;= date_of_interest ... order by price.date desc limit 1 query for each date_of_interest from the inventory table, so I know there is no computational impediment.  I would, however, prefer to solve the whole problem with a single SQL query, because it would allow me to do further SQL processing on the result of the query.  Is there a standard way to do this efficiently? It feels like it must come up often and that there should be a way to write a fast query for it.  I'm using Postgres, but an SQL-generic answer would be appreciated. "Not with a whimper, but with a roar of triumph!" Looking for the title of an old sci-fi book, pretty sure it is a Baen or Del Rey imprint. Basically, mankind has reached the stars and expanded throughout the galaxy. Suddenly, an unknown race appears and begins a war of extermination against man. Finally, the galaxy is desolate, and one man remains. He is in a starship with an AI and becomes injured in the battle.  The ship's AI takes over and speeds out into the unknown, followed by the last remaining member of the unknown alien race. As all speeds are at tremendous multiples of the speed of light, the man cannot get away, but the alien cannot catch up.  Finally, communication is opened, at which point the man asks "why?" The alien tells him that man had discovered immortality, but that it required the death of a member of a race known as the Flower People for every human who achieved immortality. The alien says that Man would have found the answer eventually, but that genocide was so intolerable that the aliens came from another dimension, to which they could never return, to stop Man, which resulted in the war of extermination.  Finally, the light speeds become so great that the two remaining living beings in the universe approach the Big Crunch. As the Man begins to dive into the singularity, with the alien hot on his heels, he says that maybe there will be room in the next universe for both races to live in peace, and the final line is "not with a whimper, but with a roar of triumph"! Is adjusting p-values in a multiple regression for multiple comparisons a good idea? Lets assume you are a social science researcher/econometrician trying to find relevant predictors of demand for a service. You have 2 outcome/dependent variables describing the demand (using the service yes/no, and the number of occasions). You have 10 predictor/independent variables that could theoretically explain the demand (e.g., age, sex, income, price, race, etc). Running two separate  multiple regressions will yield 20 coefficients estimations and their p-values. With enough independent variables in your regressions you would sooner or later find at least one variable with a statistically significant correlation between the dependent and independent variables.   My question: is it a good idea to correct the p-values for multiple tests if I want to include all independent variables in the regression? Any references to prior work are much appreciated. Including section titles in fancyhf headers Proving an integral equation. My Visual Studio (2008) Editor has stopped to underline Errors (this nifty wavy red lines). I can't really tell when, but it can be related to the installation of .Net Framework 3.5 SP 1 or the MVC Beta (which I guess is unlikely). Furthermore have I installed and uninstalled both CodeRush and Resharper for evaluation purposes (decided not to keep either one of them).  Does anyone know the problem and how to restore this functionality again? Clarification on maps in Salesforce I know this might be a very basic question but i'm pretty confused by this:  Map&lt;Id,Account&gt; accountWithOpportunities =      new Map&lt;Id,Account&gt;([SELECT Id,(SELECT Id FROM Opportunities) FROM Account]);   I have no clue on to where the Opportunities are relevant at all in this map and how they are stored or can be accessed?   Can anyone clear this out for me? Does the Draconic Bloodline sorcerer's Elemental Affinity feature apply to the spell Dragon's Breath? The  feature Elemental Affinity states:     [When] you cast a spell that deals damage of the type associated with your draconic ancestry, you can add your Charisma modifier to one damage roll of that spell.   The  spell (XGtE, p. 154) incorporates a damage roll, but it is not a direct consequence of the spell:     Until the spell ends, the creature can use an action to exhale energy of the chosen type in a 15-foot cone. Each creature in that area must make a Dexterity saving throw, taking 3d6 damage of the chosen type on a failed save, or half as much damage on a successful one.   Would one of these exhalations qualify for the application of the Elemental Affinity feature? If so, is only one eligible or every one throughout the duration of dragon's breath How to make Fedora user a sudoer? Problem with \mkbibdateapalongextra of biblatex-apa biblatex-apa gives me this:  ! Undefined control sequence. &lt;argument&gt; \mkbibdateapalongextra                                {labelyear}{labelmonth}{labelday}\iffieldu...   I'm using the default (english) language, and have done the:  \DeclareLanguageMapping{norsk}{norsk-apa} \DeclareLanguageMapping{english}{american-apa}   thing.   Here's a short example:  \documentclass[english]{memoir} \usepackage{babel} \usepackage{roffe} \usepackage[backend=biber,date=short,maxcitenames=2,style=apa]{biblatex}  \DeclareLanguageMapping{norsk}{norsk-apa} \DeclareLanguageMapping{english}{american-apa} \addbibresource{Papers2.bib} \addbibresource{R.bib}  \begin{document} \cite{R-base}  \printbibliography \end{document}   This one replicates the problem. g++ undefined reference with a shared library Homeomorphism that maps given points to given points Find a recurrence relation for the number of $n$-digit binary sequences with no pair of consecutive $1$s Find a recurrence relation for the number of $n$-digit binary sequences with no pair of consecutive $1$s.  I know my base case:$$a_1 = 2$$ for either $1$ or $0$. Normally to construct the recurrent relation I would say fill in $n-1$ spaces then add a $1$ or $0$, which is simply:$$a_n = 2\cdot a_{n-1}$$However, I don't believe this is right, because this doesn't take into account that there can't be any consecutive $1$s.  Even when I use the base case of $2$ digits:$$a_2 = 3$$for either $01$, $10$, or $00$, I still get the recurrence relation of:$$a_n = 3\cdot a_{n-2}$$which also does not account for $01$ and $10$ occurring consecutively.   So how would I express this?