We develop further the consequences of the irreducible-Boolean classification established by Zertuche, [“On the robustness of NK-Kauffman networks against changes in their connections and Boolean functions,” J. Math. Phys. 50, 043513 (2009)10.1063/1.3116166] which have the advantage of allowing strong statistical calculations in disordered Boolean function models, such as the NK-Kauffman networks. We construct a ring-isomorphism \documentclass[12pt]{minimal}\begin{document}$\mathfrak {R}_K \left\lbrace i_1, \dots , i_\lambda \right\rbrace \cong \mathcal {P}^2 \left[ K \right]$\end{document}RKi1,⋯,iλ≅P2K of the set of reducible K-Boolean functions that are reducible in the Boolean arguments with indexes {i1, …, iλ}, and the double power set \documentclass[12pt]{minimal}\begin{document}$\mathcal {P}^2 \left[ K \right]$\end{document}P2K of the first K natural numbers. This allows us, among other things, to calculate the number ϱK(λ, ω) of K-Boolean functions which are λ-irreducible with weight ω. ϱK(λ, ω) is a fundamental quantity in the study of the stability of NK-Kauffman networks against changes in their connections between their Boolean functions, as well as in the mean field study of their dynamics when Boolean irreducibility is taken into account.
In this work we present an algorithm to construct sparse-paving matroids over finite set S. From this algorithm we derive some useful bounds on the cardinality of the set of circuits of any Sparse-Paving matroids which allow us to prove in a simple way an asymptotic relation between the class of Sparse-paving matroids and the whole class of matroids. Additionally we introduce a matrix based method which render an explicit partition of the r-subsets of S, S r = ⊔ γ i=1 Ui such that each Ui defines a sparse-paving matroid of rank r.
Dedicated to Claus M. Ringel on the occasion of his 60th birthday.Abstract. If a convex body C has modular and irreducible face lattice (and is not strictly convex), there is a face-preserving homeomorphism from C to a section of a cone of hermitian matrices over R, C, or H, or C has dimension 8, 14 or 26.1991 Mathematics Subject Classification. 52A20, 06C05, 51A05, 15A48.Recall that a projective space consists of a set P (the points) and a set L (the lines) so that (a) Each pair of points is contained in a unique line, (b) If a, b, c, d are distinct points and the lines ab and cd intersect, then the lines ac and bd intersect (c) Each line contains at least 3 points and there are at least 2 lines (d) Every chain of subspaces (also called flats) has finite length. The maximum length of a chain starting with a point is the projective dimension of the space.The flats of a projective space form an algebraic, atomic, irreducible, modular lattice. Conversely, any lattice with these properties is the lattice of flats of a
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