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Column-partitioned matrices over rings without invertible transversal submatrices ; Lehtonen, Erkko in Ars Combinatoria (2010), 97 Let the columns of a p×q matrix M over any ring be partitioned into n blocks, M = [M1,...,Mn]. If no p×p submatrix of M with columns from distinct blocks Mi is invertible, then there is an invertible p×p ... [more ▼] Let the columns of a p×q matrix M over any ring be partitioned into n blocks, M = [M1,...,Mn]. If no p×p submatrix of M with columns from distinct blocks Mi is invertible, then there is an invertible p×p matrix Q and a positive integer m ≤ p such that QM = [QM1,...,QMn] is in reduced echelon form and in all but at most m - 1 blocks QMi the last m entries of each column are either all zero or they include a non-zero non-unit. [less ▲] Detailed reference viewed: 15 (1 UL)Composition of Post classes and normal forms of Boolean functions Couceiro, Miguel ; ; Lehtonen, Erkko in Discrete Mathematics (2006), 306(24), 3223-3243 The class composition CK of Boolean clones, being the set of composite functions f(g1,...,gn) with f∈C, g1,...,gn∈K, is investigated. This composition CK is either the join C∨K in the Post Lattice or it ... [more ▼] The class composition CK of Boolean clones, being the set of composite functions f(g1,...,gn) with f∈C, g1,...,gn∈K, is investigated. This composition CK is either the join C∨K in the Post Lattice or it is not a clone, and all pairs of clones C,K are classified accordingly. Factorizations of the clone Ω of all Boolean functions as a composition of minimal clones are described and seen to correspond to normal form representations of Boolean functions. The median normal form, arising from the factorization of Ω with the clone SM of self-dual monotone functions as the leftmost composition factor, is compared in terms of complexity with the well-known DNF, CNF, and Zhegalkin (Reed–Muller) polynomial representations, and it is shown to provide a more efficient normal form representation. [less ▲] Detailed reference viewed: 87 (0 UL)On the complexity of representing sets of vertices in the n-cube Couceiro, Miguel ; ; Lehtonen, Erkko in Tsitouras, C.; Simos, T. E.; Psihoyios, G. (Eds.) ICNAAM 2005 (2005) Representations of an arbitrary set of vertices of an n-dimensional cube in terms of convex sets are compared with respect to their complexity. The notion of complexity used in the representations is ... [more ▼] Representations of an arbitrary set of vertices of an n-dimensional cube in terms of convex sets are compared with respect to their complexity. The notion of complexity used in the representations is based on the union, symmetric difference and ternary median operations. Convexity of a set of vertices refers to Hamming distance or, equivalently, to the intersection of cubes with supporting hyperplanes. [less ▲] Detailed reference viewed: 29 (1 UL)On compositions of clones of Boolean functions Couceiro, Miguel ; ; Lehtonen, Erkko in Simos, T.; Maroulis, G. (Eds.) International Conference of Computational Methods in Sciences and Engineering 2004 (ICCMSE 2004) (2004) We present some compositions of clones of Boolean functions that imply factorizations of ?, the clone of all Boolean functions, into minimal clones. These can be interpreted as representation theorems ... [more ▼] We present some compositions of clones of Boolean functions that imply factorizations of ?, the clone of all Boolean functions, into minimal clones. These can be interpreted as representation theorems, providing representations of Boolean functions analogous to the disjunctive normal form, the conjunctive normal form, and the Zhegalkin polynomial representations. [less ▲] Detailed reference viewed: 81 (0 UL) |
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