In the traditional study of Łukasiewicz propositional logic, the finite-valued or infinite-valued linearly ordered model exists at the start, and then the axiomatization of the set of all formulas valid in its model are studied. On the other hand, we are in a point of view such that the set of provable formulas is important and models are no more than means to characterize the set.

Our aim in this note is to generalize some topological results of commutative noetherian rings to non-commutative rings. As a supplemental remark of [2] we prove in § 1 that any right ideal of a complete right semi-local ring is closed, and that

for any finitely generated right module M over a complete right semi-local ring Λ where J is the Jacobson radical of Λ.

A polynomial P(w) is called a uniqueness polynomial (or a uiqueness polynomial in a broad sense) if P(f) = cP(g) (or P(f) = P(g)) implies f = g for any nonzero constant c and nonconstant meromorphic functions f and g on C. We consider a monic polynomial P(w) without multiple zero whose derivative has mutually distinct k zeros ej with multiplicities qj. Under the assumption that P(el) ≠ P(em) for all distinct l and m, we prove that P(w) is a uniqueness polynomial in a broad sense if and only if ∑l<mqlqm > ∑l ql. We also give some sufficient conditions for uniqueness polynomials.

In this paper, we study topologically stable homeomorphisms of the circle. Our results are the following:

Theorem 1. A homeomorphism of the circle is topologically stable if and only if it is topologically conjugate to some Morse-Smale diffeomorphism.

Theorem 2. There exists a homeomorphism of the circle which has the pseudo-orbit-tracing-property but is not topologically stable.

Suppose Ω is a simply connected domain which is mapped conformally onto a disk. A much studied problem is the behavior of the mapping function at an accessible boundary point P of Ω, in particular the question, under what conditions the map is ‘ “conformai” at such a point (a) in the sense that angles are preserved as P is approached from Ω (“semi-conformality” at P) and (b) the dilatation at P is finite and positive. In his fundamental paper [8] in 1936, A. Ostrowski established a necessary and sufficient condition (depending on the geometry of the domain only) for the validity of the first property which subsumes all previous results and establishes a definitive solution of this problem.

Let L, M be regular quadratic lattices over Zp. The local density αp(L, M) is an important invariant in the theory of representation of quadratic forms and they appear naturally in Fourier coefficients of Eisenstein series. In spite of the importance we knew little except the case when either L = M or rk L = 1 and M is unimodular. Evaluating them is a laborious task.

We study the Yablonskii-Vorob’ev polynomial associated with the second Painlevé equation. To study other special polynomials (Okamoto polynomials, Umemura polynomials) associated with the Painlevé equations, our purely algebraic approach is useful.

Let D be a bounded pseudoconvex domain in Cn, and let KD (z, w) be the Bergman kernel function of D. The boundary behavior of KD (z, w), or that of KD (z, z), has attracted a lot of attention because it is closely related to the pseudoconformal geometry of D and ∂D.

In the previous paper [3], the author generalized the uniqueness theorems of meromorphic functions given by G. Pólya in [5] and R. Nevanlinna in [4] to the case of meromorphic maps of Cn into the N- dimensional complex projective space PN(C).

The aim of this paper is to discuss the multiplicity of the sum of two independent Gaussian processes

where x1(t) is a Wiener process and is a simple Markov process.

Recently Prof. Chevalley in Nagoya suggested to the author the following problem: Let k be a field, K5 = k(x1, x2, x3, x4, x5) be a purely transcendental extension field (of transcendental degree 5) of k, s5 be the cyclic permutation of x: S5X1 = x2s5x2 = x3s5x3 = x4s5x4 = x5s5x5 = x1, and let L5 be the field of invariants of s5 in K5. Is L5 then purely transcendental over k or not? When the characteristic p of k is not equal to 5, it is answered in the following positively. When the characteristic p of k is equal to 5, it is answered also positively by Mr. Kuniyoshi’s result in [2].

Fix a complex number q with |q| < 1. Let T1…, Tn be n-commuting q-difference operators defined by

for a function f(x), x = (x1,…,xn) ε (C*)n. Consider a system of linear q-difference equations in several variables for a matrix valued function on (C*)n as follows:

This work grew out of an attempt to prove the false result that an n-ple Markov process in the sense of Hida (1960) or Lévy (1956a) has multiplicity one. Instead we proved the representation theorem (Theorem III. 1.) that a centered Gaussian process x(t) is n-ple Markov iff it can be written in the form

(I.1)

where is a Gaussian martingale with

(I.2) sp {x(s): s ≤ t} ≡ sp {ai(s): s ≤ t and 1 ≤ i ≤ n}

and A(t) and {ei(t)} satisfy some non-degeneracy condition. We also show (Corollary IV. 13.) that for any Gaussian martingale A(t) with simple left innovation spectrum, continuous ei(t) may be found so that the process x(t) given in (I.1) will satisfy (I.2).

The present paper is concerned with the stability of diffeomorphisms of C∞ closed manifolds. Let M be a C∞ closed manifold and Diffr(M) be the space of Cr diffeomorphisms of M endowed with the Cr topology (in this paper we deal with only the case r = 0 or 1). Let us define

The celebrated Ito formula for the n-dimensional Brownian motion Xt and for u ∈ C2(Rn) runs as follows:

(0.1)

In § 6 of this paper we extend this to the case where u is any element of the Sobolev space H1R(n) and accordingly Δu is a tempered distribution which is not even a signed measure in general. As a consequence the second term of the right hand side of (0.1) may not be of bounded variation in t.

Painlevé studied the differential equations y″ = R(y′ y, x) without moving critical point, where R is a rational function of y′ y, x. Most of them are integrated by the so far known functions. There are 6 equations called Painlevé’s equations which seem to be irreducible or seem to define new transcendental functions. The simplest one among them is y″ = 6y2 + x. Painlevé declared on Comptes Rendus in 1902-03 that y″ = 6y2 + x is irreducible. It seems that R. Liouville pointed out an error in his argument. In fact there are discussions on this subject between Painlevé and Liouville on Comptes Rendus in 1902-03. In 1915 J. Drach published a new proof of the irreducibility of the differential equation y″ = 6y2 + x. The both proofs depend on the differential Galois theory developed by Drach. But the differential Galois theory of Drach contains errors and gaps and it is not easy to understand their proofs. One of our contemporaries writes in his book: the differential equation y″ = 6y2 + x seems to be irreducible dans un sens que on ne peut pas songer à préciser. This opinion illustrates well the general attitude of the nowadays mathematicians toward the irreducibility of the differential equation y″ = 6y2 + x. Therefore the irreducibility of the differential equation y″ = 6y2 + x remains to be proved. We consider that to give a rigorous proof of the irreducibility of the differential equation y″ = 6y2 + x is one of the most important problem in the theory of differential equations.

Let S be the integral closure of a discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and denote the Galois group of the quotient field extension by G. It has been proved by Auslander and Rim in [4] that the trivial crossed product Δ(l, S, G) is an hereditary order for tamely ramified extensions S of R and that Δ(l, S, G) is a maximal order if and only if S is an unramified extension of R. The purpose of this paper is to study the crossed product Δ(f, S, G) where [f] is any element of H2(G, U(S)) and S is a tamely ramified extension of R with multiplicative group of units U(S).

A ring R is said to be right (left) hereditary if every right (left) ideal in R is projective, that is, a direct summand of a free R-module. Cartan and Eilenberg [3, p. 15] ask whether there exists a right hereditary ring which is not left hereditary. The answer: yes.

This paper is concerned with a finitely generated module M over a (commutative Noetherian) local ring R. In the case when R is a homomorphic image of a Gorenstein local ring, one can use the well-known associativity formula for multiplicities, together with local duality and Matlis duality, to produce analogous associativity formulae for the local cohomology modules of M with respect to the maximal ideal. The main purpose of this paper is to show that these formulae also hold in the case when R is universally catenary and such that all its formal fibres are Cohen–Macaulay.

These formulae involve certain subsets of the spectrum of R called the pseudosupports of M; these pseudo-supports are closed in the Zariski topology when R is universally catenary and has the property that all its formal fibres are Cohen–Macaulay. However, examples are provided to show that, in general, these pseudo-supports need not be closed. We are able to conclude that the above-mentioned associativity formulae for local cohomology modules do not hold over all local rings.

Let J be a directed set and let {Λj, φij} be a direct system of rings indexed by J and with limit Λ. Let {Aj, φij} be a direct system of groups indexed by J. Assume that each Aj is a left Λj-module and that φij(λa) = φij(λ) φij(a) for each λ ∈ Λj, a ∈ Aj. Then the limit A of (Aj, φij) is a left Λ-module.