Let p be a fixed prime integer, and G a finite p-group. For a subgroup H of G, we denote the centralizer of H in G by CG(H). The commutator subgroup of G is denoted by [G, G].

It is well known that the quasi-exellent property (i.e. G-ring and J-II) ascends by finite type morphisms. On the other hand Greco showed in [2, Proposition 4.1 and Proposition 2.3] that G-property (i.e. formal fibers are geometrically regular) does not descend by finite type morphisms, although J-II property (i.e. any finite type algebra has regular locus open) does.

This paper is devoted to the problem of the realisation of homology classes of a PL-manifold by PL-submanifolds.

The present study is founded on the consideration of Thorn complexes M(PLk), M(SPLk) for PL-microbundles which is defined by R. Williamson [5]. We shall apply Thom’s method [4] to PL-manifolds.

The author is grateful to Professors R. Shizuma and K. Shiraiwa for their kind criticisms.

In this short note, we would like to point out that the following property (called ASSUMPTION REMOVABILITY in the present paper) is characteristic of the positive logics, the primitive logic LO, the positive predicate logics LP (intuitionistic) and LQ (classical):

Let n > 1, m ≥ 1, g ≥ 3 and γ be given integers. The purpose of this paper is to determine the relations of n, m, g and γ for the existence of the symmetric Riemann surfaces S of type (n, m) with genus g and species γ. If n is an odd prime, the relations are known in [3]. In the case that n is odd, we shall show the analogous result when E(S) is isomorphic to a cyclic group Z2n and when the quotient space S/E(S) is orientable.

Let S be a locally compact (not compact) Hausdorff space satisfying the second axiom of countability and let ℬ be the σ field of all Borel subsets of S and let A be the σ-field of all the subsets of S which, for each finite measure μ defined on (S, A), are in the completed σ field of ℬ relative to μ. We denote by C0 the Banach space of continuous functions vanishing at infinity with the uniform norm and Bk the space of bounded A-measurable functions with compact support in S.

In the previous paper [6] we have shown the examples of hyperelliptic Riemann surfaces of infinite genus such that the Riemann’s theta functions associated with them are absolutely convergent.

We prove that a four-dimensional Hermitian Einstein space is weakly *-Einsteinian and use this result to show that all geodesic symmetries are volume-preserving (up to sign) if and only if it is local symmetric.

Let k be a -adic number field with the ring 0 of all integers in k, and K be a finite normal extension with Galois group G. ∏ denotes a prime element of the ring of all integers in K. Then, an ideal (∏a) of is an 0G-module. E. Noether showed that if K/k is tamely ramified, is a free 0G-module. A. Fröhlich generalized E. Noether’s theorem as follows: is relatively projective with respect to a subgroup S of G if and only if S ⊇ G1 where G1 is the first ramification group of K/k. Now we define the vertex V(∏a) of (∏a) as the minimal normal subgroup S of G such that (∏a) is relatively projective with respect to a subgroup S of G (cf. § 1). Then, the above generalization by A. Fröhlich implies V() = G1. In the previous paper, we proved G1 ⊇ V(∏a) ⊇ G2, (where G2 is the second ramification group of K/k (cf. Theorem 5). Further, we dealt with the case where G = G1 is of order p2, and proved that if V(∏a) ≠ G1 then a ≡ 1(p2) and t2 ≡ 1(p2) for the second ramification number t2 of K/k (cf. Theorems 15 and 21). The purpose of this paper is to prove the similar theorem for the wildly ramified p-extension of degree pn (Theorem 7).

Notations. The following notations will be used throughout this paper.

˛: the identity of a finite group.

Q: the rational number field.

P: an algebraic number field of finite degree, fixed as the ground field.

l: a prime number.

ζl: a primitive l-th root of unity.

An adjoint variety X(g)associated to a complex simple Lie algebra is by definition a projective variety in ℙ*(g) obtained as the projectivization of the (unique) non-zero, minimal nilpotent orbit in g. We first describe the tangent loci of X(g) in terms of triples. Secondly for a graded decomposition of contact type we show that the intersection of X(g) and the linear subspace ℙ*(g1) in ℙ*(g) coincides with the cubic Veronese variety associated to g.

The infinite dimensional Laplacian operator can be discussed in connection with the infinite dimensional rotation group ([1]). Our interest centers entirely on observing how each one-parameter subgroup of the infinite dimensional rotation group contributes to the determination of the Laplacian operator.

In this paper we consider the following nonlinear parabolic variational inequality; u(t) ∈ D(Φ) for all where Δp is the so-called p-Laplace operator and Φ is a proper, lower semicontinuous functional. We have obtained two results concerning to solutions of this problem. Firstly, we prove a few regularity properties of solutions. Secondly, we show the continuous dependence of solutions on given data u0 and f.

A. Friedman, [5] has constructed the fundamental solutions for a class of degenerate parabolic equations of the second order by making use of a probabilistic method and obtained the estimates for the fundamental solutions, especially near the degenerating manifolds (obstacles) of given operators, where a probabilistic method plays an important role.

On the half space Rn × R+, it has been known that harmonic Bergman space bp can contain a positive function only if . Thus, for , Poisson integrals can be bp-functions only by means of their boundary cancellation properties. In this paper, we describe what those cancellation properties explicitly are. Also, given such cancellation properties, we obtain weighted norm inequalities for Poisson integrals. As a consequence, under weighted integrability condition given by our weighted norm inequalities, we show that our cancellation properties are equivalent to the bp-containment of Poisson integrals for p under consideration. Our results are sharp in the sense that orders of our weights cannot be improved.

Recently H. Hasse has given an interesting theory of Galois algebras, which generalizes the well known theory of Kummer fields; an algebra over a field Ω is called a Galois algebra with Galois group G when possesses G as a group of automorphisms and is (G, Ω)-operator-isomorphic to the group ring G(Ω) of G over Ω. On assuming that the characteristic of Ω does not divide the order of G and that absolutely irreducible representations of G lie in Ω, Hasse constructs certain Ω-basis of , called factor basis, in accord with Wedderburn decomposition of the group ring and shows that a characterization of is given by a certain matrix factor system which defines the multiplication between different parts of the factor basis belonging to different characters of G. Now the present work is to free the theory from the restriction on the characteristic. We can indeed embrace the case of non-semisimple modular group ring G(Ω).

A (folklore?) conjecture states that no holomorphic modular form $F(\tau )=\sum _{n=1}^{\infty } a_nq^n\in q\mathbb Z[[q]]$ exists, where $q=e^{2\pi i\tau }$ , such that its anti-derivative $\sum _{n=1}^{\infty } a_nq^n/n$ has integral coefficients in the q-expansion. A recent observation of Broadhurst and Zudilin, rigorously accomplished by Li and Neururer, led to examples of meromorphic modular forms possessing the integrality property. In this note, we investigate the arithmetic phenomenon from a systematic perspective and discuss related transcendental extensions of the differentially closed ring of quasi-modular forms.

Let f(z) be a function analytic and bounded, |f(z)| < 1, in |z| < 1. Then, by Fatou’s theorem the radial limit f*(eiθ) = limr→1f(reiθ) exists almost everywhere on |z| = 1. Seidel [8, p. 208] and Calderón, González- Domínguez, and Zygmund [1] (see also [9, pp. 281-282]) proved the following: if f*(eiθ) is of modulus 1 almost everywhere on an arc a < θ < b of |z| = 1, then either f(z) is analytically continuable across this arc or the values f*(eiθ), a < d < b, cover the circumference |w| = 1 infinitely many times.

We prove in this paper a theorem on analytic mappings of the complex space Cn into the complex projective space Pn. The theorem is closely related to that of S. S. Chern in [1], and the main idea of the proof is the same with the latter, though the calculations are rather different. The background of our calculation is the normal contact metric structure which was found by S. Sasaki and Y. Hatakeyama [4].