This glab post is essentially about running a certain shell command remotely on multiple systems within a network that has been set up for public-key authentication. It's a standard task for experienced system administrators I am sure, but for me it was a little adventure in bash scripting — and I wanted to share it with you.
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I wrote a little text that outlines why vector bundles and locally free sheaves are the same thing. This approach is very messy with a lot of gluing, mostly because I did not look at Exercise II.5.18 in Hartshorne right away. The construction given there is much more canonical and preferable over mine. However, I decided to put this online simply because it is different and personally, it gave me a better feeling for why the two notions coincide.
I was riding the backseat of a car, a pal of mine with a large Sudoku book on the seat beside me. I glared over at him and remarked that I find Sudokus utterly boring and would feel that my time is wasted on any of them. He looked up at me, clearly demanding an explanation for that statement. I continued to explain that a computer program could solve a Sudoku with such ease that there is no need for humans to do it. He replied that something similar could be said about chess, but still it's an interesting game. And it was then, that I realized why Sudoku is so horribly boring, and chess is not.
It was the fact that I could code a Sudoku solver and solve the Sudoku he was puzzling about, and I would be able to do it faster than it would take him to solve the entire thing by hand. This does not apply to chess, evidently. Of course, I confidently explained this to him. »Prove it.«, he said. So I did.
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Zariski's proof ((Oscar Zariski. A new proof of Hilbert's Nullstellensatz, Bulletin of the Ameican Mathematical Society Volume 53, Number 4 (1947), 362-368.)) of the Hilbert Nullstellensatz makes use of the ineffable Rabinowitch Trick ((J. L. Rabinowitsch, Zum Hilbertschen Nullstellensatz, Mathematische Annalen Volume 102, No. 1 (1929), 520.)) (check it out, that has got to be the shortest paper ever). But who is that awesome guy Rabinowitsch? I found out today, and the answer is basically in in this MO post:
> Rainich was giving a lecture in which he made use of a clever trick which he had discovered. Someone in the audience indignantly interrupted him pointing out that this was the famous Rabinowitsch trick and berating Rainich for claiming to have discovered it. Without a word Rainich turned to the blackboard, picked up the chalk, and wrote RABINOWITSCH. He then put down the chalk, picked up an eraser and began erasing letters. When he was done what remained was RABINOWITSCH. He then went on with his lecture.
Apparently, George Yuri Rainich is the mysterious stranger that went by the name of Rabinowitsch, which was his birthname ((Bruce P. Palka, Editor's Endnotes (May 2004), The American Mathematical Monthly 111 (5): 456–460)) ((Bruce P. Palka, Editor's Endnotes (December 2004). The American Mathematical Monthly 111 (10): 927–929)). I even updated the wikipedia page. Oh right, the reason this even caught my attention: Daniel R. Grayson has a really sweet, short proof of the Nullstellensatz, also using the Rainich Trick.
I have a folder
data with stuff in it. I am not using an Administrator account on my Windows machine, and that folder is supposed to be owned by my user. When you search for how to take ownership of all files and folders in a certain directory recursively, you get told to use TAKEOWN, a lot. Well, it has a little catch: You can only really use it to take ownership. Not to give it. A much more giving command is
ICACLS C:\data /setowner rattle /T
/T switch that makes it recursive.
Harm Derksen brachte uns schon im Jahre 1998 die weltschnellste Methode zur Berechnung irreduzibler Charaktere der symmetrischen Gruppe. Aber in seinem preprint, Computing with Characters of the Symmetric Group, bleiben einige Fragen offen, die ich hiermit zu klären versuche.
» The code to end all codes «
Die Welt ist einen halben Herzschlag alt. Der unsterbliche Prophet meditiert seit Anbeginn der Laufzeit auf dem তাজমহল und studiert die heiligen 12 Zeilen C-Code, die jede SAT Instanz in $\mathcal{O}(n^{9699690})$ lösen. Wenn der Puls eine Ganzzahl wird, so wird er erwachen und die Wahrheit verkünden, und Lob wird gepriesen, und Licht wird ewig scheinen auf die Kinder des Schaltkreises. Denn er ist der Taktgeber, und an der Spitze des Signals wird es sein, wenn er uns erscheint. Buch Arithmæl Circulæ, Vers 21:7 Wollen Sie mehr wissen?
Suppose you have a Laptop, which therefore must be a [ThinkPad](http://www.lenovo.com/products/us/laptop/thinkpad/). Let's also assume that you [use Windows 2008 as a workstation](http://www.win2008workstation.com/), just like [I do](/2012/03/22/turning-windows-2008-r2-into-a-workstation/). Then, you might run into some issues with [Lenovo System Update](http://support.lenovo.com/de_DE/detail.page?LegacyDocID=TVSU-UPDATE). To be precise, it tells you:
> An error occurred while gathering user information.
Then it stops. I will provide a solution to this problem which is not specific to Windows 2008 R2, and therefore, I hope that my solution will also work for future versions of Windows that are not out-of-the-box supported by SystemUpdate.
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Trust me, I am not one to enjoy deeply nested case distinctions in a mathematical proof. In fact, I'd rather there were no case distinctions necessary at all. However, to quote the famous mathematician Nikolai Nowaczyk:
> Some theorems have beautiful proof - and some theorems are worth fighting for.
Anyway. If you have to do it, you should do it in beautiful $\KaTeX$. Do you want to know more?
If you like electronic music, you obviously lurk getworkdonemusic.com all the time with jDownloader running, so you can copy the link to any nice track on the air and download it, slowly aquiring a groovy selection of electronic music for the lonely train hours where you have to get work done, offline style. Sometimes, however, these tracks are in wave format, for reasons that are completely beyond me. A quick google search for »convert WAV to MP3« points you towards a huge selection of shitware, when it could be so much easier: Fire up VLC, the best media player in the world ((This post refers to VLC media player 2.0.5 Twoflower.)) and go
* Media, Convert/Save (Ctrl+R)
* Add your file to the list.
* Click the button that says Convert/Save.
* Select a destination file.
* Set the profile to Audio - MP3 and hit Start.
Also, make sure you don't have VLC set to loop, because otherwise it'll spend the next few hours overwriting that file with the same converted MP3 again and again. True story.
After discovering the toggle command available in MathJax, I immediately went to asked the capable people of tex.stackexchange whether this could be done inside a PDF file. And indeed: It can be done!
\documentclass{article}
\usepackage{animate}
\begin{document}
\begin{animateinline}[step]{1}
\strut$x=1$
\newframe
\strut$x=2$
\newframe
\strut$x=3$
\end{animateinline}
\end{document}
Looking at Roman's beautiful blog, I really couldn't take it no more. I finally removed that scourge of a tiny font size from the blag, now it is actually possible to read everything on a commonly shaped screen. At this point I wished this had been done in LESS, because then I would probably only have to change two variables in order to achieve, what now took me, approximately one hour.
Let $K\subseteq L$ be a field extension, and let $V$ be a $K$-vector space. The extension of $V$ by scalars in $L$ is the tensor product $E=V\otimes_KL$. I will prove that every $L$-vector spaced is obtained as some extension in this way and that $\dim_L(E)=\dim_K(V)$.
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Most people probably know about the stacks project, but fewer might have heard about the CRing project. Now, there might be certain disadvantages to having a large number of contributors to a single document, but I would like to put some emphasis on the one, big advantage: You actually end up with a very complete reference, not just the narrower standpoint of one or two authors. If done correctly, group efforts of this sort are the perfect way to cover all basic (whatever that means) knowledge in some field. I really hope to see more areas pick up on the idea.