We establish the second part of Milnor's conjecture on the volume of simplexes in hyperbolic and spherical spaces. A characterization of the closure of the space of the angle Gram matrices of simplexes is also obtained.

We study Vasiliev’s system of higher spin gauge fields coupled to massive scalars in AdS_3, and compute the tree level two and three point functions. These are compared to the large N limit of the W_N minimal model, and nontrivial agreements are found. We propose a modified version of the conjecture of Gaberdiel and Gopakumar, under which the bulk theory is perturbatively dual to a subsector of the CFT that closes on the sphere.

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One of the theorems of Bengt Stolt’s article “On the Diophantine Equation$u$^{2}−$Dv$^{2}=4$N$” is not quite correct in its entirety. A counter-example will be given to show this. A modification of the theorem which he was trying to prove will be given for certain special cases.

We define type A, type B, type C as well as C*-semi-finite C*-algebras.