SECTION OF MATHEMATICS: ALGEBRAIC AND COMBINATORIAL ASPECTS OF COHERENT STRUCTURES*

Generalisation of Dedekind's problem of the enumeration of coherent structures

“…See the numerical results of Table 2. For n ig 15 the bound (10) is somewhat better than the bound obtained from (9). However the recursive nature of (9) means it can be used in conjunction with (10) to obtain a systematic improvement on Hansel's bound for (n +1) even.…”

“…For the case when / = 2, i.e. when components and systems can be in only one of two states (operational or failed) (8) becomes n+1 U 2 = n U 2 ( n U 2 + l)l2 (9) which allows us to calculate an upper bound for the number of semi-coherent functions for n +1 components provided we are given an upper bound (or the actual value) for the number of such functions for n components. Methods for obtaining sharp upper bounds for the number of semi-coherent functions of n components with this dichotomic behaviour has long been of interest; see for example Dedekind The sharpest explicit and non-asymptotic bound to data is due to Hansel who proved that…”

Voting Fairly: Transitive Maximal Intersecting Families of Sets

“…Dihedral group A n Alternating group M 11 Matthieu group G_H Direct product of groups G and H G " H Wreath product of G and H A more intuitive characterization which highlights the 2-transitivity of the underlying automorphism group was found by Dmitri Zvonkin. Consider an icosahedron and identify opposite vertices (the resulting map is K 6 drawn on the projective plane!).…”

The Fundamental Theorem of Voting Schemes

“…They can be interpreted as voting schemes or social choice functions [6], as ipsodual elements of the free distributive lattice [19], as members of the free median set [15,16,14], as self-dual anti-chains (or clutters) [13], as maximal intersecting families of sets [5,12], as (ultra)-filters [4], as non-dominated coteries [20], or as critical tripartite hypergraphs [1,2]. Games are thought of in reliability theory as semi-coherent structure functions [17,7]. Linear programmers tend to think of games as boolean functions; threshold functions (or majority games or quota games) in particular are of special interest [12,8].…”