In semi-euclidean spaces, conformal mappings are consists of similarities, inversions, and Bateman mapping [14]. In this note, we shall discuss problems whether there exist caloric morphisms with Bateman space mapping for radial semi-euclidean metrics. It is based on the similar arguments as were used in [8], [9], and [10].

A subsemigroup of a torsion-free abelian group is called grading monoid. This is a note on star operations in ideal theory of grading monoids. Explicitly, we study stability, ascents-descents, and Kronecker function rings of semistar operations on grading monoids.

We raise a problem on coextensions and find partial answers.

Let p be a prime number of the form p = 2ℓ+1 with some odd prime number ℓ. For such a prime number p, it is shown that the relative class number h_p^- of the pth cyclotomic field Q(ζ_p) is odd when 2 remains prime in Q(ζ_ℓ)⁺ by Estes [3], Stevenhagen [11] and Metsänkylä [8] using a Bernoulli number associated to Q(ζ_p). In this note, we give an alternative proof of the assertion using a cyclotomic unit of Q(ζ_p)⁺.

Let 1 < p < ∞ and let Ω be a bounded domain of R^N (N ≥ 1). In this paper, we consider a class of second order quasilinear elliptic operators A in Ω including the p-Laplace operator ∆_p. First we establish various type of Kato'’s inequalities for A when Au is a Radon measure. Then we prove the inverse maximum principle and describe the strong maximum principle. For this purpose it is crucial to introduce a notion of admissible class for the operator A and use it effectively.