Showing papers in "Nagoya Mathematical Journal in 1990"
TL;DR: In this article, a-functions of prehomogeneous vector spaces were introduced in the original way of M. Sato and a proof of the structure theorem of them was given.
Abstract: The purpose of this paper is to introduce a-functions and b-functions of prehomogeneous vector spaces in the original way of M. Sato and give a proof of the structure theorem of them. All the results were obtained by M. Sato when he constructed the theory of prehomogeneous vector spaces in 60’s. However he did not write a paper on his outcomes at that time. His theory was distributed through his lectures and informal seminars. Only small number of people could know it. The only publication left for us is a mimeographed note of his lecture [Sa-Sh1] written by T. Shintani in Japanese.
110 citations
TL;DR: In this paper, a simplification of the proof of the General Neron Desingularization is given. But the result is not affected in characteristic zero (they use just iii1) from [4] (9.5) from the condition iii2) (this is true because the changing from line 5 from down the page 123 may not preserve iii2).
Abstract: This letter concerns our papers [4], [5] and its aim is to give a simplification to the proof of the General Neron Desingularization (see [5] (2.4) or here below) together with a small reparation; as T. Ogoma pointed out in [3], our Lemma (9.5) from [4] does not hold in the condition iii2) (this is true because the “changing” from line 5 from down the page 123 [4] may not preserve iii2)). However our results were not affected in characteristic zero (they use just iii1) from [4] (9.5)). In [3] Ogoma gives a nice simplification of our proof.
83 citations
59 citations
TL;DR: A Kummer surface is the minimal desingularization of the surface T/i, where T is a complex torus of dimension 2 and i the involution automorphism on T as discussed by the authors.
Abstract: A Kummer surface is the minimal desingularization of the surface T/i, where T is a complex torus of dimension 2 and i the involution automorphism on T. T is an abelian surface if and only if its associated Kummer surface is algebraic. Kummer surfaces are among classical examples of K3-surfaces (which are simply-connected smooth surfaces with a nowhere-vanishing holomorphic 2-form), and play a crucial role in the theory of K3-surfaces. In a sense, all Kummer surfaces (resp. algebraic Kummer surfaces) form a 4 (resp. 3)-dimensional subset in the 20 (resp. 19)-dimensional family of K3-surfaces (resp. algebraic K3 surfaces).
53 citations
TL;DR: In this paper, the authors consider the problem of proving the irreducibility of the differential equation y = 6y2 + x, which is the most important problem in the theory of differential equations.
Abstract: Painleve 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 Painleve’s equations which seem to be irreducible or seem to define new transcendental functions. The simplest one among them is y″ = 6y2 + x. Painleve declared on Comptes Rendus in 1902-03 that y″ = 6 y2 + x is irreducible. It seems that R. Liouville pointed out an error in his argument. In fact there are discussions on this subject between Painleve 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 a preciser. 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.
47 citations
TL;DR: In this paper, a harmonic analysis on infinite dimensional spaces by means of the Levy Laplacian and infinite dimensional rotation groups is presented, which is inspired by the work of H. Yoshizawa and T. Hida.
Abstract: P. Levy introduced, in his celebrated books [21] and [22], an infinite dimensional Laplacian called the Levy Laplacian in connection with a number of interesting topics in variational calculus. One of the most significant features of the Levy Laplacian is observed when it acts on the singular part of the second functional derivatives. For this reason the Levy Laplacian has become important also in white noise analysis initiated by T. Hida [12]. On the other hand, as was pointed out by H. Yoshizawa [29], infinite dimensional rotation groups are profoundly concerned with the structure of white noise, and therefore, play essential roles in certain problems of stochastic calculus. Motivated by these works, we aim at developing harmonic analysis on infinite dimensional spaces by means of the Levy Laplacian and infinite dimensional rotation groups.
43 citations
TL;DR: The Siegel cusp form is a holomorphic function on the Siegel upper half plane which satisfies functional equations relative to Γn and vanishes at the cusps.
Abstract: Let Gn (resp. Γn ) be the real symplectic (resp. Siegel modular) group of degree n. The Siegel cusp form is a holomorphic function on the Siegel upper half plane which satisfies functional equations relative to Γn and vanishes at the cusps.
33 citations
TL;DR: The connection between the two problems is given by a result of Cowsik which says that the Noetherian property of S(I) implies that I is a set-theoretic complete intersection as discussed by the authors.
Abstract: Let S be a three-dimensional regular local ring and let I be a prime ideal in S of height two. This paper is motivated by the question of when I is a set-theoretic complete intersection and when the symbolic Rees algebra S(I) = ⊕ n ≥0 I (n) t n is Noetherian. The connection between the two problems is given by a result of Cowsik which says that the Noetherian property of S(I) implies that I is a set-theoretic complete intersection ([1]).
31 citations
31 citations
TL;DR: In this article, the authors studied a general situation of Fano manifolds, ruled over the complex projective plane P 2 as P r-1 -bundles, being of the form ℙ( ) with -a bundle of rank r ≥ 3 on P 2.
Abstract: 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 P 2 as P r-1 -bundles, ie, being of the form ℙ( ) with -a bundle of rank r ≥ 3 on P 2
31 citations
TL;DR: In this paper, the problem of finding a minimal free resolution of S/I t, over arbitrary base ring R = Z, the ring of integers, has been studied in the Noetherian commutative ring setting.
Abstract: Let R be a Noetherian commutative ring with, unit element, and X ij be variables with 1 ≤ i ≤ m and 1 ≤ j ≤ n . Let S = R[x ij ] be the polynomial ring over R , and I t be the ideal in S , generated by the t × t minors of the generic matrix (x ij ) ∈ M m, n (S) . For many years there has been considerable interest in finding a minimal free resolution of S/I t , over arbitrary base ring R . If we have a minimal free resolution P. over R = Z, the ring of integers, then R′ ⊗ z P . is a resolution of S/I t over the base ring R′ .
TL;DR: In this paper, the authors extend and unify several results concerning complete ideals in 2-dimensional regular local rings by using the theory of joint reductions and mixed multiplicities, which was developed by Zariski in 1938.
Abstract: The aim of this paper is to extend and unify several results concerning complete ideals in 2-dimensional regular local rings by using the theory of joint reductions and mixed multiplicities. The theory of complete ideals in a 2-dimensional regular local ring was developed by Zariski in his 1938 paper [Z]. This theory is presented in a simpler and general form in [ZS, Appendix 5] and [H2].
TL;DR: In this article, a system of spin configurations S = {S(t, x); t ≥ 0, x ∊ ℤ} on a one-dimensional lattice was investigated.
Abstract: We shall investigate a system of spin configurations S = {S(t, x); t ≥ 0, x ∊ ℤ} on a one-dimensional lattice ℤ changing randomly in time. The evolution law is described by an infinite-dimensional stochastic differential equation (SDE): where {β(t, x); t > 0, x ∊ Z} is a family of independent standard Wiener processes and U′ is the derivative of a self-potential U: R → R .
TL;DR: The class number one problem for real quadratic fields was solved in 1966 by A. Baker and H.M. Stark independently as mentioned in this paper, however, the problem is still unsolved.
Abstract: 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.
TL;DR: In this article, the authors characterize the class of hitting time distributions of single points of one dimensional generalized diffusion processes and give their tail behaviors in terms of speed measures of the generalized diffusion process.
Abstract: In this paper, we will characterize the class of (conditional) hitting time distributions of single points of one dimensional generalized diffusion processes and give their tail behaviors in terms of speed measures of the generalized diffusion processes.
TL;DR: In this paper, the authors studied Banach space objects in the Scott-Solovay Boolean valued universe and provided some useful transfer principles from theorems of Banach spaces to Banach modules over commutative AW*-algebras.
Abstract: Recently, systematic applications of the Scott-Solovay Boolean valued set theory were done by several authors; Takeuti [25, 26, 27, 28, 29, 30], Nishimura [13, 14] Jech [8] and Ozawa [15, 16, 17, 18, 19, 20] in analysis and Smith [23], Eda [2, 3] in algebra. This approach seems to be providing us with a new and powerful machinery in analysis and algebra. In the present paper, we shall study Banach space objects in the Scott-Solovay Boolean valued universe and provide some useful transfer principles from theorems of Banach spaces to theorems of Banach modules over commutative AW*-algebras. The obtained machinery will be applied to resolve some problems concerning the module structures of von Neumann algebras.
TL;DR: In this article, the ideal boundary of a Hadamard manifold is defined to be the equivalence classes of rays, defined by Busemann [Bu], which is not symmetric in general.
Abstract: In this paper we study the ideal boundaries of surfaces admitting total curvature as a continuation of [Sy2] and [Sy3]. The ideal boundary of an Hadamard manifold is defined to be the equivalence classes of rays. This equivalence relation is the asymptotic relation of rays, defined by Busemann [Bu]. The asymptotic relation is not symmetric in general. However in Hadamard manifolds this becomes symmetric. Here it is essential that the manifolds are focal point free.
TL;DR: In this article, a trace formula for the case of weight one was introduced and applied to computations of traces of Hecke operators acting on the space of cusp forms of weight greater than or equal to two.
Abstract: In Selberg [11], he introduced the trace formula and applied it to computations of traces of Hecke operators acting on the space of cusp forms of weight greater than or equal to two. But for the case of weight one, the similar method is not effective. It only gives us a certain expression of the dimension of the space of cusp forms by the residue of the Selberg type zeta function. Here the Selberg type zeta function appears in the contribution from the hyperbolic conjugacy classes when we write the trace formula with a certain kernel function ([3J, [4], [7], [8], [9], [12]).
TL;DR: In this article, a commutative local Noetherian ring with maximal ideal m and M a finitely generated A -module with dim (M ) = d is defined. And the ideal of A generated by x 1, …, x d.
Abstract: Throughout this note, A denotes a commutative local Noetherian ring with maximal ideal m and M a finitely generated A -module with dim ( M ) = d . Let x 1 , …, x d be a system of parameters (s.o.p. for short) for M and I the ideal of A generated by x 1 , …, x d .
TL;DR: In this article, the authors present some basic properties of minimal fine derivatives which seems to be a new concept (or at least a new combination of well-known ones) and why is it worthwhile to study this new concept? Why hasn't it been done earlier?
Abstract: The paper will present some basic properties of minimal fine derivatives which seems to be a new concept (or at least a new combination of well-known ones). Why is it worthwhile to study this new concept? Why hasn’t it been done earlier?
TL;DR: The defect relation gives the best-possible estimate, where f is a linearly non-degenerate holomorphic curve in P n (C) and H 1, H q are hyperplanes in general position as discussed by the authors.
Abstract: The defect relation gives the best-possible estimate, where f is a linearly non-degenerate holomorphic curve in P n (C) and H 1 , …, H q are hyperplanes in P n (C) which are in general position. However, the case of moving hyperplanes has ever got only n(n + 1) instead of n + 1 (Stoll [4]) and it has not yet been known whether this bound is best-possible or not. In this paper we shall give some particular cases which have the bound n + 1.
TL;DR: In this article, a relative version of the notion of ray class fields is defined for every holomorphy ring of the field and the prototype for this new notion is M Rosen's definition of a Hilbert class field for function fields.
Abstract: The notion of a ray class field, which is fundamental in Takagi’s class field theory, has no immediate analogon in the function field case The reason for this lies in the lacking of a distinguished maximal order In this paper I overcome this difficulty by a relative version of the notion of ray class fields to be defined for every holomorphy ring of the field The prototype for this new notion is M Rosen’s definition of a Hilbert class field for function fields [6]
TL;DR: In this paper, the authors investigate the dependency of Gaussian random fields X(D) indexed by a domain D in d-dimensional Euclidean space Rd with white noise analysis.
Abstract: The purpose of this paper is to investigate way of dependency of Gaussian random fields X(D) indexed by a domain D in d-dimensional Euclidean space Rd . Our main tool is variational calculus, where the boundary of a domain varies and deforms and we appeal to the white noise analysis. We therefore assume that X(D) is expressed white noise integral of the form (0.1) X(D) = X(D, W)=∫D F(D, u)W(u)du, where W is the Rd -parameter white noise and the kernel F(D, u) is a square integrable function over Rd , and where D is a bounded domain with smooth boundary.
TL;DR: In this article, it was shown that the following properties are equivalent for a function ψ:]0, ∞ [→ R (which then is called completely monotone):
Abstract: Bernstein’s theorem states that the following properties are equivalent for a function ψ:]0, ∞ [→ R (which then is called completely monotone): moreover, the measure σ in iii) is uniquely determined
TL;DR: In this paper, the authors investigated injective cyclic modules over twisted polynomial rings with coefficients in a commutative field and showed that every simple module is injective and only a single isomorphism class of simple modules is independent.
Abstract: Differential polynomial rings over a universal field and localized twisted polynomial rings over a separably closed field of non-zero characteristic twisted by the Frobenius endomorphism were the first domains not divisions rings that were shown to have every simple module injective (see [C] and [C-J]). By modifying the separably closed condition for the polynomial rings twisted by the Frobenius, the conditions of every simple being injective and only a single isomorphism class of simple modules were shown to be independent (see [O]). In this paper we continue the investigation of injective cyclic modules over twisted polynomial rings with coefficients in a commutative field.
TL;DR: In this paper, the authors studied holomorphic mappings from a connected complex manifold M of dimension m to a projective algebraic manifold N of dimension n. Assume first that N is of general type, i.e. where KN → N is the canonical bundle of N.
Abstract: We will study holomorphic mappings from a connected complex manifold M of dimension m to a projective algebraic manifold N of dimension n. Assume first that N is of general type, i.e. where KN →N is the canonical bundle of N. If KN is positive, then N is of general type.