We show an equivalence of Drinfeld's pentagon equation and Hirose-Sato's confluence relations. As a corollary, we obtain `pentagon-free' presentation the Grothendieck-Teichm\"{u}ller group $GRT_1$ associators.

This year’s AGI took place in Arlington Virginia at the Crowne Plaza Hotel, just across the river from Washington D.C. and a stone’s throw from the Pentagon. This was only the second year for this conference, but even so it boasted at least 100 participants. The attendees represented a broad spectrum of backgrounds, from academic researchers to industry professionals to interested laymen. AGI s...

The alternating group of degree 6 is located at the junction of three series of simple non-commutative groups : simple sporadic groups, alternating groups and simple groups of Lie type. It plays a very special role in the theory of finite groups. We shall study its new roles both in a finite geometry of certain pentagon in the Leech lattice and also in a complex algebraic geometry of K3 surfaces.

No countable C,-free graph contains every countable C,-free graph as a subgraph, for n 2 4. For n = 4, this was proved earlier by J. Pach.

We propose the decay modes B → a0(→ ηπ)π to determine the CKM phase α. One can analyze these modes through (i) the B → a0π isospin pentagon, (ii) the time dependent Dalitz plot of B0(t) → a±0 π → ηπ+π−, and (iii) the time dependence of B0(t) → a0(→ ηπ0)π0. We show that the a0π modes have certain advantages as compared to the ρπ modes, and strongly recommend the time dependent Dalitz plot analys...

This paper contains some results of pentagons which can be determined by two X-rays. The results reveal this problem is more complicated.