1. 报告人：AKITOSHI KAWAMURA, 副教授， 日本京都大学
题目：A lower bound on opaque sets
摘要：We proved that the total length of any set of countably many rectifiable curves whose union meets all straight lines that intersect the unit square U is at least 2.00002. This is the first improvement on the lower bound of 2 known since 1964. A similar bound is proved for all convex sets U other than a triangle.
2. 报告人：KENSHI MIYABE, 副教授，日本明治大学
题目：The properties of computable predictions
摘要：We try to predict the next bit from a given finite binary string when the sequence is sampled from a computable probability measure on the Cantor space. There exists the best betting strategy among a class of effective ones up to a multiplicative constant, the induced prediction from which is called algorithmic probability or universal induction by Solomonoff. The prediction converges to the true induced measure for sufficiently random sequences. However, the prediction is not computable.
We propose a framework to study the properties of computable predictions. We prove that all sufficiently general computable predictions also converge to the true induced measure. The class of sequences along which the prediction converges is related to computable randomness.
We also discuss the speed of the convergence. We prove that, even when a computable prediction predicts a computable sequence, the speed of the convergence cannot be bounded by a computable function monotonically decreasing to 0.