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Pattern recognition on oriented matroids: Layers of Tope Committees
, 2006
"... A tope committee K ∗ for a simple oriented matroid M is a subset of its maximal covectors such that every positive halfspace of M contains more than a half of the maximal covectors from K ∗. The structures of the family of all committees for M, and of the family of its committees that contain no pa ..."
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A tope committee K ∗ for a simple oriented matroid M is a subset of its maximal covectors such that every positive halfspace of M contains more than a half of the maximal covectors from K ∗. The structures of the family of all committees for M, and of the family of its committees that contain no pairs of opposites, are described. Farey subsequences associated to the elements of the mth layer of the Boolean
A note on Boolean lattices and Farey sequences
 II, Integers
"... Abstract. We establish monotone bijections between the Farey sequence of order m and the halfsequences of a Farey subsequence associated to the rank m elements of the Boolean lattice of subsets of a 2mset; we also present a few related combinatorial identities. 1. ..."
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Abstract. We establish monotone bijections between the Farey sequence of order m and the halfsequences of a Farey subsequence associated to the rank m elements of the Boolean lattice of subsets of a 2mset; we also present a few related combinatorial identities. 1.
Neighboring fractions in Farey subsequences
"... Abstract. We present explicit formulas for computation of the neighbors of several elements in Farey subsequences. 1. ..."
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Abstract. We present explicit formulas for computation of the neighbors of several elements in Farey subsequences. 1.
A NOTE ON BOOLEAN LATTICES AND FAREY SEQUENCES. II
, 2008
"... We establish monotone bijections between subsequences of the Farey sequences and the halfsequences of Farey subsequences associated with elements of the Boolean lattices. ..."
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We establish monotone bijections between subsequences of the Farey sequences and the halfsequences of Farey subsequences associated with elements of the Boolean lattices.
Pattern recognition on oriented matroids: . . .
, 2009
"... Oriented matroids can serve as a tool of modeling of collective decisionmaking processes in contradictory problems of pattern recognition. We present a generalization of the committee techniques of pattern recognition to oriented matroids. A tope committee for an oriented matroid is a subset of it ..."
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Oriented matroids can serve as a tool of modeling of collective decisionmaking processes in contradictory problems of pattern recognition. We present a generalization of the committee techniques of pattern recognition to oriented matroids. A tope committee for an oriented matroid is a subset of its maximal covectors such that every positive halfspace contains more than half of the covectors from this subset. For a large subfamily of oriented matroids their committee structure is quite rich; for example, any maximal chains in their tope posets provide