Let R be an integral domain with quotient field K, let h (resp., g, f) be the non-zero R-submodules of K (resp., the non-zero fractional ideals of R, the finitely generated non-zero fractional ideals of R), and let {x, y} be a subset of the set {f, g, h} of symbols. For a semistar operation ★ on R, if (EE₁)^★ = (EE₂)^★ implies E₁^★ = E₂^★ for every E ∈ x and every E₁, E₂ ∈ y, then ★ is called xy-cancellative. Let ★ be a gg-cancellative semistar operation on R which is an extension of a star operation on R. In this paper, we show that ★ need not be gh-cancellative.

Let Ω be a smooth bounded domain of R^N. In this paper, we study the equivalences among three capacities and Hausdorff measure. First we present the equivalence between p-capacity C_p(K) and p-Laplace-capacitiy C_Δp(K) relative to Ω for any compact set K ⊂ Ω. Secondly we establish the equivalence between p-Laplace capacity C_p(K, ∂Ω) relative to ∂Ω and Hausdorff measure ℌ^N-1(K) for any compact set K ⊂ ∂Ω.

Let n ≥ 1 be an integer and let 2^e be the highest power of 2 dividing n. For a prime number p = 2nl + 1 with an odd prime number l, let N be the imaginary abelian field of conductor p and degree 2^e+1l over Q. We show that for n ≤ 30, the relative class number h_N^- of N is odd when 2 is a primitive root modulo l except for the case where (n, l) = (27, 3) and p = 163 with the help of computer.

In our paper [2], we have classified simple regular polyhedral BP-complexes, which are polyhedral complexes satisfying certain natural conditions on their vertex structures. As an addendum of this classification, we prove that 2-skeletons of higher dimensional regular polytopes are simple regular polyhedral complexes, but not polyhedral BP-complexes. The proof is done by a detailed investigation of their vertex structures.

In [8] Matsuda has investigated stability of a semistar operation. In this paper we extend the notion of stability of a semistar operation to the presemistar operation case and we shall study stability properties of presemistar operations.