We give two types of 3-dimensional CR-submanifolds of the 6-dimensional sphere. First we study whether there exists a 3-dimensinal CR-submanifold which is obtained as an orbit of a 3-dimensional simple Lie subgroup of G2. There exists a unique (up to G2) 3-dimensional CR-submanifold which is obtained as an orbit of reducible representations of SU(2) on R7. As orbits of the subgroup which corresponds to the irreducible representation of SU(2) on R7, we obtained 2-parameter family of 3-dimensional CR-submanifolds. Next we give a generalization of the example which was obtained by K. Sekigawa.

We prove that the averaging formula for Nielsen numbers holds for continuous maps on infra-solvmanifolds of type (R): Let M be such a manifold with holonomy group Ψ and let f: M → M be a continuous map. The averaging formula for Nielsen numbers

is proved. This is a workable formula for the difficult number N(f).

In 1903 H. F. Blichfeldt proved the following brilliant theorem : Let G be a matrix group of order g and of degree n. Let p be a prime divisor of g such that Then G contains the abelian normal p-Sylow subgroup. In 1941 applying his modular theory of the group representation, R. Brauer improved this theorem in the case in which p divides g to the first power only. Further in 1943 H. F. Tuan improved this result of R. Brauer one step more.

The Lefschetz hyperplane section theorem has roots going back at least to Picard, but it was Lefschetz [20] who first stated and proved it in the modern form for integer homology. Later it was improved up to the homotopy level by Andreotti-Frankel [1] and Bott [8] using an idea of Thorn. Numerous generalizations along the same lines have appeared, e.g. [14, Theorem H], [19], [24, App. II] etc.

The proof of Theorem 1 in the paper “On Krull’s conjecture concerning valuation rings” (vol. 4 (1952) of this journal) is not correct. We want to give here a corrected proof of the theorem: From p. 30, l. 14 to p. 31, l. 7 should be changed as follows.

Fiber cones of 0-dimensional ideals with almost minimal multiplicity in Cohen-Macaulay local rings are studied. Ratliff-Rush closure of filtration of ideals with respect to another ideal is introduced. This is used to find a bound on the reduction number with respect to an ideal. Rossi’s bound on reduction number in terms of Hilbert coefficients is obtained as a consequence. Sufficient conditions are provided for the fiber cone of 0-dimensional ideals to have almost maximal depth. Hilbert series of such fiber cones are also computed.

Let M be a connected surface immersed in a Euclidean m-space Em. Let h be the second fundamental form of this immersion it is a certain symmetric bilinear mapping for X ∈ M, where Tx is the tangent space and the normal space of M at x. Let H be the mean curvature vector of M in Em. If there exists a real λ such that for all tangent vectors X, Y in Tx, then ilf is said to be pseudo-umbilical at x.

Let A be a regular local ring of dim A = 3 and p a prime ideal in A of dim A/p = 1. We put Rs(p) = (here t denotes an indeterminate over A) and call it the symbolic Rees algebra of p. With this notation the authors [5, 6] investigated the condition under which the A-algebra Rs(p) is Cohen-Macaulay and gave a criterion for Rs(p) to be a Gorenstein ring in terms of the elements f and g in Huneke’s condition [11, Theorem 3.1] of Rs(p) being Noetherian. They furthermore explored the prime ideals p = p(n1, n2, n3) in the formal power series ring A = k[X, Y, Z] over a field k defining space monomial curves and Z = with GCD(n1, n2, nz) = 1 and proved that Rs(p) are Gorenstein rings for certain prime ideals p = p(n1 n2, n3). In the present research, similarly as in [5, 6], we are interested in the ring-theoretic properties of Rs(p) mainly for p = p(n1 n2) nz) and the results of [5, 6] will play key roles in this paper.

In [2, § 9] there is a general result of Fulton and Lazarsfeld relating the homotopy groups of a subvariety of in a certain range of dimensions with those of its pullback under a holomorphic map in the corresponding range of dimensions. It is asked in [2, § 10] whether here is a corresponding result with replaced by a general rational homogeneous manifold, Y, and with the range of dimensions alluded to above shifted by the ampleness of the holomorphic tangent bundle of Y in the sense of [4]. In this paper we use the techniques of [4, 5, 6, 7] to answer this question in the affirmative.

We show that every Kato surface admits a locally conformally Kähler metric.

In his study of non-linear differential equations of the second order, N. Levinson [3] defined the dissipative systems (D-systems) which arise in many important cases in practice. To a dissipative system a transformation T: R2 → R2 called the Poincaré transformation is associated. Levinson used the Poincaré transformation in the qualitative study of dissipative systems, and he [3] and Massera [5] obtained certain equalities between the number of subharmonic solutions of a dissipative systems under suitable conditions. We call these the Levinson-Massera’s equalities.

We show that the Bergman metric of a bounded homogeneous domain has a potential function whose gradient has a constant norm with respect to the Bergman metric, and further that this constant is independent of the choice of such a potential function.

In [4] Deligne-Mumford introduced a concept of stable curves and proved the existence of the coarse moduli space of stable curves of given genus in the category of algebraic stacks. Thereafter Mumford has shown this coarse moduli space is a protective scheme. We can consider the coarse moduli space of stable curves as a geometric compactification of the coarse moduli space of non-singular curves.

The theory of newforms is very important and useful for arithmetical study of modular forms of integral weight. It is natural to try to extend this theory into the case of modular forms of half-integral weight Until now, several authors have attempted to find a theory of newforms of half-integral weight (cf. [She], [N], [K], [M-R-V], [She-W]). But complete results have not been obtained yet.

L’extension aux fonctions algébroïdes du théorème de Picard est faite par Rémoundos [7] et après quelques années, celle de la théorie de Nevanlinna aux fonctions algébroïdes dans |z| < + ∞ est donnée par Selberg [9], Ullrich [14] et Valiron [16]. Dans ce mémoire, on considère l’allure des fonctions algébroïdes d’après la méthode dernière.

N-ary algebras are modules with a n-fold multiplication which we assume to be associative if nothing else is stated. They are a canonical generalization of binary and ternary associative algebras. Ternary rings were first investigated by Lister [8]. The aim of this note is to show that the Wedderburn structure theory and the usual cohomology for binary associative algebras can be extended to n-ary algebras. For ternary algebras this has been done in [8] and [1]. Moreover analogous results are wellknown for Lie and alternative triple systems, and for ternary Jordan pairs.

Recently one of the writers used, in proving a theorem on the commutativity of certain division rings, the following lemma :

I. Let L be a field and K be its proper subfield. Except either when L is of characteristic p ≠ O and absolutely algebraic or ivhen L is algebraic and purely inseparable over K, there exists a pair of distinct (special exponential) valuations in L which coincide on K.

II. Let K be a field which is either of characteristic 0 or not absolutely algebraic, and L be its separable finite extension. There exist then infinitely many valutions in L which are of 1st degree over K.

Class number one problem for imaginary quadratic fields was solved in 1966 by A. Baker and H.M. Stark independently. However, the problem for real quadratic fields is still unsolved. It seems to us that one of the most essential difficulties of the problem for real quadratic fields comes from deep connection of the class number with the fundamental unit.

Krull’s principal ideal theorm [Krull] states that q elements in the maximal ideal of a local noetherian ring generate an ideal whose minimal components are all of height at most q. Writing R for the ring, we may consider the q elements, x1, · · ·, xq say, as coordinates of an element x ∈ Rq.

In recent papers [10, 11, 12, 13, 14], we defined some new ρ-invariants for any rational prime ρ congruent to 1 mod 4 and D-invariants for any positive square-free integer D such that the fundamental unit εD of real quadratic field Q(√D) satisfies NεD = –1, and studied relationships among these new invariants and already known invariants.

One of our main purposes in this paper is to generalize these D-invariants to invariants valid for all square-free positive integers containing D with NεD = 1. Another is to provide an improvement of the theorem in [14] related closely to class number one problem of real quadratic fields. Namely, we provide, in a sense, a most appreciable estimation of the fundamental unit to be able to apply, as usual (cf. [3, 4, 5, 9, 12, 13]), Tatuzawa’s lower bound of L(l, XD) (Cf[7]) for estimating the class number of Q(√D) from below by using Dirichlet’s classical class number formula.