The matrix canonical realizations of the Lie algebra of pseudo-orthogonal group O(m, n) described in the first part of this paper are further investigated. The explicit formulae for values of the Casimir operators (which are multiples of identity in these realizations) are obtained.

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We consider a Hamiltonian with N point interactions in {\bb R}^ d,\: d= 2, 3, all with the same coupling constant, placed at vertices of an equilateral polygon {\cal P} _N. It is shown that the ground-state energy is locally maximized by a regular polygon. The question whether the maximum is global is reduced to an interesting geometric problem.

In this paper, we present an efficient attack to the multivariate Quadratic Quasigroups (MQQ) cryptosystem. Our cryptanalysis breaks MQQ cryptosystems by solving systems of multivariate quadratic polynomial equations using a modified version of the MutantXL algorithm. We present experimental results comparing the behavior of our implementation of MutantXL to Magma's implementation of F_4 on MQQ systems (\ge 135 bit). Based on our results we show that the MutantXL implementation solves with much less memory than Magma's implementation of F_4 algorithm.