Computational homology of n types

... permutations and convexset, and has a nonempty interior set Ω0 Let ϕ : Ω → R be continuous on Ω and continuously differentiable in Ω0 Then, ϕ is a Schur-convex (Schur-concave) function if and only ... Shandong Science and Technology Press, Jinan, China, 3rd edition, 2004 [14] D S Mitrinovi´ , Analytic Inequalities, vol 1965 of Die Grundlehren der mathematischen Wisenc schaften, Springer, New ... and I Olkin, Inequalities: Theory of Majorization and Its Applications, vol 143 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1979 [13] J.-C Kuang, Applied Inequalities,...

... bar all unbarred elements in b and to unbar all barred elements in b We denote this by ¯ To unbar a block b is to unbar all barred b elements of < /b> b We denote this by b For < /b> example, 3¯¯ = ¯ ¯ and ... xj whenever i and j are in the same block of < /b> π and both are barred or both are unbarred, • xi = −xj whenever i and j are in the same block of < /b> π and one is barred and the other is unbarred, • xi ... forms a basis for < /b> Hn−2 (ΠDB (T )) n Proof There are two types < /b> of < /b> regions of < /b> the hyperplane arrangement DBn (T ), namely Rω, for < /b> (ω, ) ∈ Bn and ω(1) ∈ T , and Rω, for < /b> (ω, ) ∈ Dn and ω(1) ∈ T By Propositions...