Today, this video tutorial led me to the application verifier, which is pure awesome, as explained here in part. I had never heard of this thing before, and I am having the time of my life with it. If you like Windows hax, check this out.
Let $R$ be a ring and $f\in R[X]$ a polynomial with infinitely many zeros in $R$. You might think that $f$ is the zero polynomial, but that is not true if $R$ is not commutative, as this example of the quaternions shows.
What about if $R$ is commutative? I didn't find a counterexample online, but it's easy to give one, and I found this somewhat enlightening. Consider the field $\mathbb{F}_2=\{0,1\}$ with two elements. The polynomial $f=X^2-X\in\mathbb{F}_2[X]$ has two zeros, namely $0$ and $1$. Now consider the Ring $R=\mathbb{F}_2^{\mathbb{N}}=\{ \mathbb{N}\to\mathbb{F}_2\}$. We can think of elements of $R$ as sequences $(0,1,1,0,1,\ldots)$. Now clearly, any such sequence is also a zero of $f$. So $f$ actually vanishes everywhere on $R$, which has infinite size, but $f$ is not the zero polynomial.
The statement is of course true if $R$ is a commutative integral domain.
Hello, fellow applied mathematicians and computer scientists, hello also to all the brave physicists who use the [arXiv](http://www.arxiv.org/). Did you know that you can [publish source code and other ancillary files on the arXiv](http://arxiv.org/help/ancillary_files), along with your preprint? If you didn't, this must be great news for you. However, if you ever tried to actually do this, you *might* have been just as confused as me. It's actually quite likely that you were, because as soon as this blogpost has vanished from the front page, I am pretty sure that a google search is what led you here.
> Ancillary files are included with an arXiv submission by placing them in a directory
anc at the root of the submission package.
If you are a novice to uploading files to the arXiv, like me, this might be confusing. What is the *submission package*? I only ever submitted a single $\KaTeX$ file! Well, let me put it straight for you.
- In the directory with your .tex file(s), make a directory called anc.
- Place all your source code and stuff in that directory.
- Make a zip file containing all your LaTeX sources and the folder anc.
- Upload that zip file to the arXiv.
Trust me - everything will be fine.
As simple as that. I registered at the website g2play.de some time back to buy a computer game cheap, and since I use fresh eMail addresses for each service where I register, I can say without a doubt that somehow, g2play leaked the eMail address I provided. I received targeted spam to that particular address today and there is no way this eMail address could have gotten into the hands of that spammer by any other means. Shame on you, g2play.
Even though this does not really constitute a post with substantial content, this is a blog after all, so I thought I'd let the 8 people who read it know that for the next six weeks, I will be attending a special semester on Algorithms and Complexity in Algebraic Geometry at the Simons Institute for the Theory of Computing in Berkeley. So. If you happen to be in the bay area, give me a shout.
In the past weeks I've been visiting quite the number of jobs fairs, networking events, trainings on how to hunt for jobs and the like. I certainly learned a lot, albeit mostly things that are obvious after a moment of contemplation or come to you by common sense. Yet a simple reminder and a bit of practice are surely beneficial. High in demand are all these abstract skills we all have been told one too many times to include on your CV (complex and analytical thinking ...) and programming skills (although many employers seem to be happy to train on the job). What somehow fell of my radar and which came up more than once, is a basic knowledge of statistics.
This leaves us in a bit of a pickle because most available material seems to fall into one of the two following categories:
1. Statistic for statisticians: Something we all have very actively decided against studying. All the proofs and constructions and endless excursions into theoretical concepts that haven't yet and maybe never will make their way into practice. Scientific exploration that is fun if and only if you have a thing for statistics.
2. Statistics for undergrads who don't even know if they need it, lectured by junior-professors or assistants who rather spend their time on research than on preparations: What you get is a incoherent bunch of powerpoint slides that seem to live more through examples than concise explanations and miss out on the most important part, namely a self-contained overview over available methods.
To the rescue comes Arnaud Delorme, a computational neuroscientist from the University San Diego. His paper on statistical methods does basically everything you want on 23 pages: A quick recap of the necessary definition, a clean-cut overview over the most common methods and a short concise explanation for each of them with just the right amount of theoretical background and a short demonstrating example.
Sure, towards the end it can be a bit tiresome to go through method after method and you won't want to read it front to back in one go. Of course I also can't guarantee for its exhaustiveness, but it seems like we have a winner when it comes to ratio of usefulness over required time-investment.
I decided to switch to the new design made by Roman for us, even though he protests that it is not finished. After a long sequence of emails, I managed to pester him into letting me fix the remaining technical wordpress issues with several dirty hacks myself, and there we are. This layout is responsive and will work properly on smartphones. Just two months ago, I could not have cared less about such things, but now I have a smartphone myself.
As some of you might know, I got my first mobile device now and started playing around with it. Today, I need to rant, because I lost 79 cents. When you buy an app on the amazon app store, you can no longer use it after you uninstall the amazon app store. Other android users also hate the amazon app store because it is an obnoxious app which is mean and does not let you use apps which you paid for with money after you get rid of the data-leaking abomination it is. Do not use it. I simply bought ownCloud on the humble, trustworthy google play store again. I have no time to deal with amazon over those 79 cents. But I have time to write this scornful post.
My colleague and me were confronted with a question concerning the introductory chapter of a text on algebraic geometry. The scenario of a plane curve $C=Z(f)\subseteq \mathbb A_{\Bbbk}^2$ over a field $\Bbbk$ is considered, for a nonconstant polynomial $f\in\Bbbk[x,y]$. It was stated that the function field $\mathbb K:=\mathrm{Quot}(\Bbbk[x,y]/\langle f\rangle)$ is obviously of transcendence degree $1$ over $\Bbbk$ because $f(x,y)=0$ is a new relation between $x$ and $y$. The prerequisites to this text are basic undergraduate knowledge of algebra and topology. The question was about the obviousness of the above statement. We came up with a proof that only really requires some linear algebra. I found it rather cute. Do you want to see it?
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.
Do you want to know more?
The idea behind groupon is to make it easy (and cheaper) for new customers to try out a service or product. The whole site really looks like racket but I used it some time ago. I have the common practice that I create an e-mail address for every single service I register to. So I used an e-mail address of the style "mymail-groupon@mydomain.tld" for it. Today I got an e-mail with the subject "Abmahnung Ihrer aussehender Rechnung über 262,00 Euro" (impolite german for "Reminder for a payment about 262 euros") from an debt collecting agency.
Do you want to know more?
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.