Recently this author studied several merit functions systematically for the second-order cone complementarity problem. These merit functions were shown to enjoy some favorable properties, to provide error bounds under the condition of strong monotonicity, and to have bounded level sets under the conditions of monotonicity as well as strict feasibility. In this paper, we weaken the condition of strong monotonicity to the so-called uniform <i>P</i> ^*-property, which is a new concept recently developed for linear and nonlinear transformations on Euclidean Jordan algebra. Moreover, we replace the monotonicity and strict feasibility by the so-called <i>R</i> ₀₁ or <i>R</i> ₀₂-functions to keep the property of bounded level sets.

In this paper, we will introduce a precise classification of characteristic functions in the Fourier space according to the moment constraint in the physical space of any order. Based on this, we construct measure-valued solutions to the homogeneous Boltzmann equation with the exact moment condition as the initial data.

Let R be a not necessarily commutative ring with 1. In the present paper we first introduce a notion of quasi-orderings, which axiomatically subsumes all the orderings and valuations on R. We proceed by uniformly defining a coarsening relation ≤ on the set Q(R) of all quasi-orderings on R. One of our main results states that (Q(R),≤′) is a rooted tree for some slight modification ≤′ of ≤, i.e. a partially ordered set admitting a maximum such that for any element there is a unique chain to that maximum. As an application of this theorem we obtain that (Q(R),≤′) is a spectral set, i.e. order-isomorphic to the spectrum of some commutative ring with 1. We conclude this paper by studying Q(R) as a topological space.

In this paper, we present an experimental analysis of HFE Challenge 2 (144 bit) type systems. We generate scaled versions of the full challenge fixing and guessing some unknowns. We use the MXL_3 algorithm, an efficient algorithm for computing Gröbner basis, to solve these scaled versions. We review the MXL_3 strategy and introduce our experimental results.

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