MarriageTheoremのこと

_ Twitterでも似たことを書いたのだけど、研究者を目指している数学科の大学院生には、「数学が」好きな人と、数学の「研究が」好きな人がいるという印象である。前者のタイプの人が数学の職を得られなかった場合にはとても大変なことになるかもしれないが、一方、後者は数学で職が見つからなくても周辺分野の研究者としてやっていける可能性が高いと思われる。

そこで問題になるのが、当の本人であっても、自分がどちらのタイプの人間であるかの見極めがそれほど自明な問題ではないことである。私自身の経験からいっても、特に所謂純粋数学中心の大学院にいると、何となく「前者のタイプでなければならない」という思い込みが形成されやすく、本当は後者のタイプである人が「自分は「数学が」好きなのだ」と思い込んでしまう事例が少なくないのではないか、と推測している。

そのように、本来なら数学自体でなくてもその周辺分野の研究者として活き活きと活動できる素養のある学生さんが、上記のような思い込みから数学のポストを得ることに拘って、結果として「討ち死に」してしまうとしたらそれは非常に悲しいことである。そのようなことが少しでも減るようにと願って止まない。

_ プレプリント確認状況：arXiv:math 5月8日分まで、IACR ePrint 2012/531まで

_ 気になった論文１：On the longest length of arithmetic progressions (MinZhi Zhao, Huizeng Zhang, arXiv:1204.1149)

Suppose that $\xi^{(n)}_1,\xi^{(n)}_2,...,\xi^{(n)}_n$ are i.i.d with $P(\xi^{(n)}_i=1)=p_n=1-P(\xi^{(n)}_i=0)$. Let $U^{(n)}$ and $W^{(n)}$ be the longest length of arithmetic progressions and of arithmetic progressions mod $n$ relative to $\xi^{(n)}_1,\xi^{(n)}_2,..., \xi^{(n)}_n$ respectively. Firstly, the asymptotic distributions of $U^{(n)}$ and $W^{(n)}$ are given. Simultaneously, the errors are estimated by using Chen-Stein method. Next, the almost surely limits are discussed when all $p_n$ are equal and when considered on a common probability space. Finally, we consider the case that $\lim_{n\to\infty}p_n=0$ and $\lim_{n\to\infty}{np_n}=\infty$. We prove that as $n$ tends to $\infty$, the probability that $U^{(n)}$ takes two numbers and $W^{(n)}$ takes three numbers tends to 1.

_ 気になった論文２：Density-based group testing (Dániel Gerbner, Balázs Keszegh, Dömötör Pálvölgyi, Gábor Wiener, arXiv:1204.1464)

In this paper we study a new, generalized version of the well-known group testing problem. In the classical model of group testing we are given n objects, some of which are considered to be defective. We can test certain subsets of the objects whether they contain at least one defective element. The goal is usually to find all defectives using as few tests as possible. In our model the presence of defective elements in a test set Q can be recognized if and only if their number is large enough compared to the size of Q. More precisely for a test Q the answer is 'yes' if and only if there are at least \alpha |Q| defective elements in Q for some fixed \alpha.

_ 気になった論文３：The minimal degree of permutation representations of finite groups (Oren Becker, arXiv:1204.1668)

In this thesis we study the following property of a finite group G: the minimal number n such that G embeds in Sn. We start with an explicit formula for the number n for abelian groups. Then, we study the behavior of this group property in respect to direct products. Finally, we define and explore the "compression ratio" of a finite group G which measures how much better the best embedding is relative to the embedding given by Cayley's theorem.

_ 気になった論文４：On convex hulls of orbits of Coxeter groups and Weyl groups (Georg Hofmann, Karl-Hermann Neeb, arXiv:1204.2095)

The notion of a linear Coxeter system introduced by Vinberg generalizes the geometric representation of a Coxeter group. Our main theorem asserts that if $v$ is an element of the Tits cone of a linear Coxeter system and $\cW$ is the corresponding Coxeter group, then $\cW v \subeq v - C_v,$ where $C_v$ is the convex cone generated by the coroots $\check \alpha$, for which $\alpha(v) > 0$. This implies that the convex hull of $\cW v$ is completely determined by the image of $v$ under the reflections in $\cW$. We also apply an analogous result for convex hulls of $\cW$-orbits in the dual space, although this action need not correspond to a linear Coxeter system. Motivated by the applications in representation theory, we further extend these results to Weyl group orbits of locally finite and locally affine root systems. In the locally affine case, we also derive some applications on minimizing linear functionals on Weyl group orbits.

_ 気になった論文５：A Secret Sharing Scheme Based on Group Presentations and the Word Problem (Maggie Habeeb, Delaram Kahrobaei, Vladimir Shpilrain, arXiv:1205.0157)

A (t,n)-threshold secret sharing scheme is a method to distribute a secret among n participants in such a way that any t participants can recover the secret, but no t-1 participants can. In this paper, we propose two secret sharing schemes using non-abelian groups. One scheme is the special case where all the participants must get together to recover the secret. The other one is a (t,n)-threshold scheme that is a combination of Shamir's scheme and the group-theoretic scheme proposed in this paper.