We bootstrap N=(1,0) superconformal field theories in six dimensions, by analyzing the four-point function of flavor current multiplets. Assuming E_8 flavor group, we present universal bounds on the central charge C_T and the flavor central charge C_J . Based on the numerical data, we conjecture that the rank-one E-string theory saturates the universal lower bound on C_J , and numerically determine the spectrum of long multiplets in the rank-one E-string theory. We comment on the possibility of solving the higher-rank E-string theories by bootstrap and thereby probing M-theory on AdS_7×S^4/Z_2.

A product formula is proved which involves the unitary group generated by a semibounded self-adjoint operator and an orthogonal projection P. It gives a partial answer to the question about existence of the limit which describes quantum Zeno dynamics in the subspace Ran P.

In contrast to the generalized FischerBurmeister function that is a natural extension of the popular FischerBurmeister function NCP-function, the generalized natural residual NCP-function based on discrete extension, recently proposed by Chen, Ko, and Wu, does not possess symmetric graph. In this paper we symmetrize the generalized natural residual NCP-function, and construct not only new NCP-functions and merit functions for the nonlinear complementarity problem, but also provide parallel functions to the generalized FischerBurmeister function.

Despite its usefulness, the Kalman-Bucy filter is not perfect. One of its weaknesses is that it needs a Gaussian assumption on the initial data. Recently Yau and Yau introduced a new direct method to solve the estimation problem for linear filtering with non-Gaussian initial data. They factored the problem into two parts: (1) the on-line solution of a finite system of ordinary differential equations (ODEs), and (2) the off-line calculation of the Kolmogorov equation. Here we derive an explicit closed-form solution of the Kolmogorov equation. We also give some properties and conduct a numerical study of the solution.

We give exact formulas for the growth of the ideal Aλ for λ a quadratic element of the algebra of Boolean functions over the Galois field GF(2). That is, we calculate dim A_k λ where A_k is the subspace of elements of degree less than or equal to k. These results clarify some of the assertions made in the article of Yang, Chen and Courtois [22,23] concerning the efficiency of the XL algorithm.