We have given in [3] the definition of the quasi-flow and discussed the representation and the time-change of the quasi-flow. Further we have defined the TQ-system and studied some properties of them using the time-change and the representation of the quasi-flow. These properties are very useful to study the ergodicity, the entropy and increasing partitions of the automorphism. We are now going to extend the definition of the TQ-system and to study the similar problems as the above.

Throughout this paper A is a commutative Noetherian ring of dimension d with the maximal ideal m and we assume that there exists a regular local ring S such that A is a homomorphic image of S, i.e., A = S/I for some ideal I of S. Furthermore we assume that A is equi-dimensional, i.e., dim A = dim A/ for any minimal prime ideal of A. We put

.

Let k be an algebraic number field of finite degree. We denote the absolute class field of k by and the absolute ideal class group of k by Cl(k).

In [7], R. Nevanlinna gave the following uniqueness theorem of meromorphic functions as an improvement of a result of G. Pólya ([8]).

Theorem A. Let f, g be non-constant meromorphic functions on C. If there are five mutually distinct values a1, …, a5 such that f−1(ai = g−1(ai) (1 ≦ i ≦ 5), then f ≡ g.

Let S, T be positive definite integral symmetric matrices of degree m, n respectively and let us consider the quadratic diophantine equation S[X] = T. We know already [1] that the following assertion (A)m>n is true for m ≥ 2n + 3.

For a given Riemannian manifold M and its submanifold N, one can find various types of geodesies on M starting from any point of N and ending in any point of N. For example, geodesies which start perpendicularly from N and end perpendicularly in N are treated by many mathematicians. K. Grove has stated a condition in a general case for the existence of such a geodesic ([4]), where he has used the method of the infinite dimensional critical point theory. This method is very useful for the study of geodesies and many geometricians have used it successfully. It has two aspects: one is an existence theory and the other is a quantitative theory, which one can find, for instance, in the excellent theory for closed geodesies of W. Klingenberg ([1], [7]) and so on.

The main purpose of this paper is to give a method to evaluate the actual value of the filtering error which arises in the transmission of a signal process, using the optimal coding, over a Gaussian channel. In his earier papers ([4] and [7]), the author has shown a method to construct an optimal causal coding for which the filtering error is minimized and at the same time the mutual information is maximized.

H. P. F. Swinnerton-Dyer determined the structure of the algebra of modular forms mod p for all prime numbers p in elliptic modular case (cf. [10]). Using his result, J.-P. Serre investigated the properties of p-adic modular forms and succeeded to construct the p-adic zeta functions for any totally real number fields (cf. [8]).

This report is a survey of some of the many different ways of characterizing a class of plane quasiconformal mappings. This class was considered by Ahlfors [4] in his treatment of the Teichmiiller problem, and it has been studied rather extensively in the last ten years.

This paper treats a holomorphic self-mapping f: Ω → Ω of a bounded domain Ω in a separable Hilbert space with a fixed point p. In case the domain is convex, we prove an infinite-dimensional version of the Cartan-Carathéodory-Kaup-Wu Theorem. This is basically a rigidity result in the vein of the uniqueness part of the classical Schwarz lemma. The main technique, inspired by an old idea of H. Cartan, is iteration of the mapping f and its derivative. A normality result for holomorphic mappings in the compact-weak-open topology, due to Kim and Krantz, is used.

Let A be a local Noetherian domain of Krull dimension d. Heinzer, Rotthaus, and Sally have shown that if the generic formal fiber of A has dimension d – 1, then A is birationally dominated by a 1-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field of A. We explore further this correspondence between prime ideals in the generic formal fiber and 1-dimensional analytically ramified local rings. Our main focus is on the case where the analytically ramified local rings are stable, and we show that in this case the embedding dimension of the stable ring reflects the embedding dimension of a prime ideal maximal in the generic formal fiber, thus providing a measure of how far the generic formal fiber deviates from regularity. A number of characterizations of analytically ramified local stable domains are also given.

Consider a parabolic end Ω of a Riemann surface in the sence of Heins [2] (cf. Nakai [3]). A density P = P(z)dxdy (z = x + iy) is a 2-form on with nonnegative locally Hölder continuous coefficients P(z). A density P is said to be finite if the integral

G. Mokobodzki proved [5] that on any harmonic space with countable basis satisfying the axioms 1, 2, T+, K0 [2] [1] any equally bounded set of harmonic functions is equicontinuous. P. Loeb and B. Walsh showed [4] that the same property holds on a harmonic space without countable basis, if Brelot’s axiom 3 is fulfilled. The aim of this paper is to generalize these results to a harmonic space X satisfying only the axioms 1, 20, K1, [2] [1] where 20 is a weakened form of axiom 2. As a corollary we get: if any point of X possesses two open neighbourhoods U, V such that the set of harmonic functions on U separates the points of U∩V, then X has locally a countable basis.

Consider an algebraic differential equation F = 0 of the first order. A rigorous definition will be given to the classical concept of “particular solutions” of F = 0. By Ritt’s low power theorem we shall prove that a singular solution of F = 0 belongs to the general solution of F if and only if it is a particular solution of F = 0.

1. Let R and C be a ring and its center, and G an automorphism group of R of order n. By a factor set {cα,τ}, we mean a system of regular elements cα,τ (α,τ ∈ G) in C such that

(1)

We present a reconstruction theorem for Fano vector bundles on projective space which recovers the small quantum cohomology for the projectivization of the bundle from a small number of low-degree Gromov-Witten invariants. We provide an extended example in which we calculate the quantum cohomology of a certain Fano 9-fold and deduce from this, using the quantum Lefschetz theorem, the quantum period sequence for a Fano 3-fold of Picard rank 2 and degree 24. This example is new, and is important for the Fanosearch program.

In this paper, we shall discuss boundedness of solutions of the equation

under suitable conditions. And we shall discuss asymptotic stability of a periodic solution and convergence of solutions for the equation

for a positive constant cand a periodic function e(t)under some restricted conditions.

In this paper R is a commutative noetherian local ring with unit element 1 and M is its maximal ideal. Let K be the residue field R/M and let {t1,t2,…, tn) be a minimal system of generators for M. By a complex R<T1. . ., Tp> we mean an R-algebra* obtained by the adjunction of the variables T1. . ., Tp of degree 1 which kill t1,…, tp. The main purpose of this paper is, among other things, to construct an R-algebra resolution of the field K, so that we can investigate the relationship between the homology algebra H (R < T1,…, Tn>) and the homological invariants of R such as the algebra TorR(K, K) and the Betti numbers Bp = dimk TorR(K, K) of the local ring R. The relationship was initially studied by Serre [5].

Let n = p1p2 … pr be a product of r prime numbers which are not necessarily different. We define then an arithmetic function µm(n) by

where m is a natural number. We further define the function L(s, µm) by the Dirichlet series

and will show that L(s, µm), (m ≥ 3), has an infinitely many valued analytic continuation into the half plane Re s > ½.

The theorem of identity for analytic subsets of a reduced complex space is stated as follows;