Among his various interests in algebra Nakayama also took part in the various researches, published in the early and middle 1950’s, which dealt with the commutativity of rings. This paper, which studies a problem of a related sort, thus seems appropriate in a Journal honoring his memory.

We shall study a certain class of rings which satisfy a weak form of the commutative law and shall show that the structure of such rings can be determined.

Let R be an open Riemann surface. By a density P on R we mean a non-negative and continuously differentiable functions P(z) of local parameters z = x + iy such that the expression P(z)dxdy is invariant under the change of local parameters z. In this paper we always assume that P≢0 unless the contrary is explicitly mentioned. We consider an elliptic partial differential equation

which is invariantly defined on R.

No manifold had been known which can carry two distinct differentiable structures until the recent important contribution due to J. Milnor [7] concerning the 7-sphere appeared.

In connection with his work, there are several problems, for example, about the existence of any other manifold with such property, about the topological invariance of the Pontrjagin classes of manifolds, etc.; some of them will be discussed in the present note.

A compact Riemannian symmetric space admits a canonical complexification. This so called adapted complex manifold structure JA is defined on the tangent bundle. For compact rank-one symmetric spaces another complex structure JS is defined on the punctured tangent bundle. This latter is used to quantize the geodesic flow for such manifolds. We show that the limit of the push forward of JA under an appropriate family of diffeomorphisms exists and agrees with JS.

Even though the logical structure of any formal deduction can be nicely expressed in a tree-form diagram, it is more practical to write it down in a series of propositions. In each step of inference, we usually deduce a proposition on basis of some foregoing propositions. However, global aspects of mathematical theories show us that this is not always the case. For, in mathematical theories, theorems are usually stated before their proofs. In fact, also in proofs of theorems, it is often practical that we prove propositions after stating them. Accordingly, in our real way of thinking, we arrange propositions going back and forth in the logical order.

Gauss conjectured that there are infinitely many real quadratic fields with class number one. Today this is still an open problem. Moreover, as Dorian Goldfeld, one of the recipients of the 1987 Cole prize in number theory (for his work on another problem going back to Gauss) recently stated in his acceptance of the award: “This problem appears quite intractible at the moment.” However there has recently been a search for conditions which are tantamount to class number one for real quadratic fields. This may be viewed as an effort to shift the focus of the problem in order to understand more clearly the inherent difficulties, and to reveal some other beautiful interrelationships.

Let X be a smooth n-dimensional projective variety over an algebraically closed field k such that KX is not nef. We give a characterization of non nef extremal rays of X of maximal length (i.e of length n – 1); in the case of Char(k) = 0 we also characterize non nef rays of length n – 2.

It was observed in [Su5] that the spectrum of a periodic Schrödinger operator on a Riemannian manifold has band structure if the transformation group acting on the manifold satisfies the Kadison property (see below for the definition). Here band structure means that the spectrum is a union of mutually disjoint, possibly degenerate closed intervals, such that any compact subset of R meets only finitely many. The purpose of this paper is to show, by a slightly different method, that this is also true for general periodic elliptic self-adjoint operators.

The object of this paper is an investigation of existence problems and Dirichlet problems on an abstract Riemann surface in the sense of Weyl-Radó or on a covering surface over it, and of boundary correspondence in the conformal mapping of the surface.

Let Sbe a compact complex protective and smooth variety of dimc2, shortly a surface. We employ the following standard notations:

Os: structure sheaf of S.

Ks: the canonical bundle on S.

One purpose of this paper is a purely algebraic study of (systems of) ordinary differential equations of the type

where the coefficients are taken from a fixed associative, commutative, unital ring R, such as the field R of real or C of complex numbers or a commutative, unital Banach algebra. The right hand sides of D are considered to be elements in the polynomial ring R[X1, …, Xn] of associating but non-commuting variables X1, …, Xn. An algebraic study calls for maps between such differential equations and, in fact, morphisms are defined between differential equations having the same arity m but not necessarily the same dimension n. These morphisms are rectangular matrices with entries in R which satisfy certain relations. This leads to a category RDiffm whose objects are precisely the differential equations of arity m and in which the composition of the morphisms is the usual matrix multiplication.

In our earlier paper [8] we discussed Fano manifolds X that are of the form X = ℙ() with a rank-2 vector bundle on a surface S. Here we study a more general situation of Fano manifolds, ruled over the complex projective plane P2 as Pr-1-bundles, i.e., being of the form ℙ() with -a bundle of rank r ≥ 3 on P2.

In this paper we consider the relations between quasilinear wave equations

Let G be a connected Lie group and H a closed subgroup with Lie algebra such that in the Lie algebra g of G there exists a subspace m with (subspace direct sum) and In this case the corresponding manifold M = G/H is called a reductive homogeneous space and (g,) (or (G,H)) a reductive pair. In this paper we shall show how to construct invariant pseudo-Riemannian connections on suitable reductive homogeneous spaces M which make M into an Einstein manifold.

Stochastic processes on a Hilbert space have been discussed in connection with quantum field theory, theory of partial differential equations involving random terms, filtering theory in electrical engineering and so forth, and the theory of those processes has greatly developed recently by many authors (A. B. Balakrishnan [1, 2], Yu. L. Daletskii [7], D. A. Dawson [8, 9], Z. Haba [12], R. Marcus [18], M. Yor [26]).

We consider in this paper eleven conditions on a commutative ring R. The first of these is that R contains an identity. It is well known that each of the other properties is a consequence of the first condition. This paper considers other relations which exist between these properties. A complete diagram of all simple implications which exist between the eleven properties, together with proof of these implications, is given in section 3. Examples illustrating simple implications which do not hold are presented in section 4.

Making use of 1-forms and geometric inequalities we prove the rigidity property of complete submanifolds Mn with parallel mean curvature normal in a complete and simply connected Riemannian (n+p) -manifold Nn+p with positive sectional curvature. For given integers n, p and for a nonnegative constant H we find a positive number T(n,p) ∈ (0,1) with the property that if the sectional curvature of N is pinched in [T(n,p), 1], and if the squared norm of the second fundamental form is in a certain interval, then Nn+p is isometric to the standard unit (n + p)-sphere. As a consequence, such an M is congruent to one of the five models as seen in our Main Theorem.

Discrete linear Weingarten surfaces in space forms are characterized as special discrete $\unicode[STIX]{x1D6FA}$-nets, a discrete analogue of Demoulin’s $\unicode[STIX]{x1D6FA}$-surfaces. It is shown that the Lie-geometric deformation of $\unicode[STIX]{x1D6FA}$-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known Lawson correspondence in the constant mean curvature case.

A Brody curve is a holomorphic map from the complex plane ℂ to a Hermitian manifold with bounded derivative. In this paper we study the value distribution of Brody curves from the viewpoint of moduli theory. The moduli space of Brody curves becomes infinite dimensional in general, and we study its “mean dimension”. We introduce the notion of “mean energy” and show that this can be used to estimate the mean dimension.

We introduce the Hurwitz-type spectral zeta functions for the quantum Rabi models, and give their meromorphic continuation to the whole complex plane with only one simple pole at $s=1$. As an application, we give the Weyl law for the quantum Rabi models. As a byproduct, we also give a rationality of Rabi–Bernoulli polynomials introduced in this paper.