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For any r-order {0,1}-tensor A with e ones, we prove that the spectral radius of A is at most exp((r-1)/r) with the equality holds if and only if e=k^r for some integer k and all ones forms a principal sub-tensor 1_{k x k x ... x k}. We also prove a stability result for general tensor A with e ones where e = k^r+l with relatively small l. Using the stability result, we completely characterized the tensors achieving the maximum spectral radius among all r-order {0,1}-tensor A with k^r+l ones, for -r-1 ≤ l ≤ r, and k sufficiently large.

This paper summarizes most of the main developments in the cryptanalysis of multivariate cryptosystems and discuss some problems that remain open. A strong emphasis is put on the symbolic computation tools that have been used to achieve these advances.

We derive a formula for the virtual class of the moduli space of rubber maps to [P^1/G] pushed forward to the moduli space of stable maps to BG. As an application, we show that the Gromov-Witten theory of [P^1/G] relative to 0 and ∞ are determined by known calculations.

This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in $T^{2n}$ is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold. We also discuss various conditions on the potential function that guarantee global existence and convergence.