We show that every complete nontrivial gradient Yamabe soliton admits a special global warped product structure with a one-dimensional base. Based on this, we obtain a general classification theorem for complete nontrivial locally conformally flat gradient Yamabe solitons.

In this paper, we present an efficient attack on the multivariate Quadratic Quasigroups (MQQ) public key cryptosystem. Our cryptanalysis breaks the MQQ cryptosystem by solving a system of multivariate quadratic polynomial equations using both the MutantXL algorithm and the F4 algorithm. We present the experimental results that show that MQQ systems is broken up to size n equal to 300. Based on these results we show also that MutantXL solves MQQ systems with much less memory than the F4 algorithm implemented in Magma.

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Suppose M is a m-dimensional submanifold without umbilic points in the (m+ p)-dimensional unit sphere S m+ p. Four basic invariants of M m under the Mbius transformation group of S m+ p are a symmetric positive definite 2-form g called the Mbius metric, a section B of the normal bundle called the Mbius second fundamental form, a 1-form called the Mbius form, and a symmetric (0, 2) tensor A called the Blaschke tensor. In the Mbius geometry of submanifolds, the most important examples of Mbius minimal submanifolds (also called Willmore submanifolds) are Willmore tori and Veronese submanifolds. In this paper, several fundamental inequalities of the Mbius geometry of submanifolds are established and the Mbius characterizations of Willmore tori and Veronese submanifolds are presented by using Mbius invariants.