And, on a personal note, there is Jacob Victor, a.k.a. Yasha, born in August 2012
Weak Nullstellensatz says, that for an algebraically closed field each maximal ideal in has form , for , i.e. , the ideal of an one-element subset of . Its proof in this generality needs quite a bit of commutative algebra. However, if we futher assume that is uncountable (thus covering a very important case ) we can give a much quicker proof.
Theorem 1 Let be an uncountable algebraically closed field, e.g. , and a proper maximal ideal in . Then there exists such that .
Proof: The first step is to show that . To see this, we will show that every is algebraic, i.e. a root a nonzero polynomial . Note that the dimension of as a vectorspace over is at most countable, as is generated by the images of the monomials under the ring homomorphism , and the exponents form a countable set. Thus for the set
is linearly dependent, i.e. there exist such that Thus
where and . As is algebraically closed, we have that is linear, i.e. .
Next, we observe that maps to , and set , for . As , we see that , for . By maximality of , we obtain , as claimed.
Here one can find reduction of the general case (not assuming non-countability of ) to this one.
One cannot always triangulate a non-convex polyhedron using only its vertices, sometimes one need to add more of them. A simple example of this phenomenon is Schonhardt polyhedron. Here is a picture illustrating how one builds it that I drew for a forthcoming paper, using Tikz LaTeX package, which is awesome, but totally overwhelming.
It fact, it’s easy to see that the 6 vertices and 12 edges it has are not enough. Indeed, each pair of non-intersecting edges determines a simplex, but it’s easy to observe that any such selection will include one the forbidden pairs of vertices AC, A’B, or B’C’. (The LaTeX source of the picture is here).
The paper I mention is related to a topic of the program on inverse moment problems at IMS (NUS/Singapore) in late 2013-early 2014 which I co-organize.
The battle is heating up! Now Elsevier, Springer and a smaller third publisher are suing a major university in Switzerland, the Eidgenössische Technische Hochschule Zürich, or ETH Zürich for short. Why? Because this university's library is distributing copies of their journal articles at a lower cost than the publishers themselves.
Aren't university libraries supposed to make journal articles available? Over on Google+, Willie Wong explains:
We are to embark upon teaching a 2nd year undergraduate course Experimental Mathematics, which will cover computer algebra basics, and refresh concepts from 1st year linear algebra, calculus, and combinatoris, with few more advanced things thrown in. The course will be based on Sage, with the actual software running on dedicated servers, and accessible via Sage web notebook interface (i.e. basically nothing but a web browser running on the student’s computer/laptop/ipad, etc).
Given the enrollment of about 120, this will be interesting…
PS. Most students did not appreciate the freedom given, and complained, complained, complained…
I was very glad today to receive an email from Akihiro Munemasa (from Tohoku university at Sendai), my long-term collaborator and friend. He says he’s OK, but has no electricity/water/gas…
And, as we see, they have internet, at least some kind of service (email went via me.com, a web-mail service run by Apple).
Well, I can only wish a lot of strength to Akihiro, and all the people affected by this disaster!