In semi-euclidean spaces, conformal mappings are consists of similarities, inversions, and Bateman mapping . 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 , , and .
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.
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 hp- of the pth cyclotomic field Q(ζp) is odd when 2 remains prime in Q(ζℓ)+ by Estes , Stevenhagen  and Metsänkylä  using a Bernoulli number associated to Q(ζp). In this note, we give an alternative proof of the assertion using a cyclotomic unit of Q(ζp)+.
Let 1 < p < ∞ and let Ω be a bounded domain of RN (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.