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.

In this paper, we present a new algorithm to solve algebraically the following lattice-related problems: 1) the small integer solution (SIS) problem under the condition: if the solution is bounded by an integer β in l_∞ norm, which we call a bounded SIS (BSIS) problem, and if the difference between the row dimension n and the column dimension m of the corresponding basis matrix is relatively small with respect the row dimension m; 2) the learning with errors (LWE) problems under the condition: if the errors are bounded - the errors do not span the whole prime finite field F_q but a fixed known subset of size D (D < q), which we call a learning with bounded errors (LWBE) problem. We will show that we can solve these problems with polynomial complexity.

A not necessarily continuous, linear or multiplicative function from an algebra A into itself is called a 2-local automorphism if agrees with an automorphism of A at each pair of points in A. In this paper, we study when a 2-local automorphism of a C-algebra, or a standard operator algebra on a locally convex space, is an automorphism.

By studying modular invariance properties of some characteristic forms, we prove some new anomaly cancellation formulas which generalize the Han-Zhang and Han-Liu-Zhang anomaly cancellation formulas

Recent proposals for implementation of kernel-based nonparametric curve estimators are seen to be faster than naive direct implementations by factors up into the hundreds. The main ideas behind the two different approaches are made clear. Careful speed comparisons in a variety of settings and using a variety of machines and software are done. Various issues on computational accuracy and stability are also discussed. Our speed tests show that the fast methods are as fast or somewhat faster than methods traditionally considered very fast, such as LOWESS and smoothing splines.