Axioms

A fundamental objective in the study of convex cones is the description and analysis of their extreme rays. In the case of the copositive cone, these rays are generated by extremal copositive matrices, which encode the boundary structure of the cone and are closely related to challenging instances of copositive and completely positive programming. In this work, we propose a constructive framework for generating copositive matrices from a given extremal copositive matrix of a smaller order and establish conditions under which copositivity and extremality are preserved. This approach highlights the interplay between the zero structure of a copositive matrix, its minimal zeros, and the facial geometry of the copositive cone. The results obtained allow one to generate new families of extremal copositive matrices in higher dimensions.

This paper investigates the structure of fuzzy Lie subalgebras, with particular emphasis on isomorphisms and nilpotency. Building on two prior conference contributions, one of which established foundational results on fuzzy bases of Lie algebras, we develop here a more complete and unified treatment of these themes. We introduce a notion of isomorphism between fuzzy Lie subalgebras based on the transfer principle via t-cut sets, and we prove that isomorphic fuzzy Lie subalgebras necessarily share the same nilpotency measure. The central contribution of the paper is a fuzzy measure of nilpotency , defined for any non-constant fuzzy Lie subalgebra $\mu$ of a Lie algebra $\mathfrak{g}$. This invariant equals 1 precisely when $\mu$ is fuzzy nilpotent, and decreases as the subalgebra departs from nilpotency. We show that nilpotency of the underlying Lie algebra implies , but that the converse fails in general, as witnessed by an explicit counterexample.

This paper aims to introduce a real four-component integrable extension of the complex Kaup–Newell soliton hierarchy. Following a general idea for extending the standard Kaup–Newell spectral matrix, we propose a specific matrix eigenvalue problem involving four real potentials and construct the corresponding integrable Hamiltonian hierarchy via the zero-curvature formulation. A recursion operator and a bi-Hamiltonian structure are presented to demonstrate the Liouville integrability of the resulting hierarchy. As an illustrative example, we derive an integrable system of four real derivative nonlinear Schrödinger equations, each containing two linear dispersion terms and generalizing the standard complex derivative nonlinear Schrödinger equations.