Showing papers in "Nagoya Mathematical Journal in 1982"
88 citations
TL;DR: In this paper, the initial value problems for quasilinear wave equations with dissipative terms with initial conditions where the initial values of the wave equations are assumed to be constant.
Abstract: In this paper we consider the initial value problems for the following quasilinear wave equations with dissipative terms with initial conditions where
79 citations
TL;DR: In this article, the Douady space of compact complex subspaces of a complex space is defined and the corresponding universal family of subspace families of complex spaces of the same complex space are defined.
Abstract: Let X be a complex space. Let Dx be the Douady space of compact complex subspaces of X [6] and px : Zx→ Dx the corresponding universal family of subspaces of X.
56 citations
TL;DR: In this paper, the Laplacian of an n-dimensional, complete, connected and locally compact Riemannian manifold is defined and g is its metric. And ΔM is the LAP on M.
Abstract: Let M be an n-dimensional, complete, connected and locally compact Riemannian manifold and g be its metric. Denote by ΔM the Laplacian on M.
47 citations
TL;DR: The Lefschetz hyperplane section theorem has roots going back at least to Picard as discussed by the authors, who first stated and proved it in the modern form for integer homology and later improved up to the homotopy level by Andreotti-Frankel [1] and Bott [8] using an idea of Thorn.
Abstract: 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, eg [14, Theorem H], [19], [24, App II] etc
42 citations
TL;DR: In this article, the Laplacian of an n-dimensional, complete, connected and non-convex Riemannian manifold is defined and g is its metric.
Abstract: Let M be an n-dimensional, complete, connected and non compact Riemannian manifold and g be its metric. ΔM denotes the Laplacian on M.
37 citations
TL;DR: In this paper, the authors give explicit Z-basis of certain maximal orders of definite quaternion algebras over the rational number field Q (see Theorems below).
Abstract: In this paper, we shall give explicit Z-basis of certain maximal orders of definite quaternion algebras over the rational number field Q (See Theorems below). We shall also give some remarks on symmetric maximal orders in Ponomarev [9] and Hashimoto [6] (Proposition 4.3). More precise contents are as follows. Let D be a division quaternion algebra over Q.
37 citations
TL;DR: In this article, it was shown that all homogeneous bounded domains of dimension ≤ 3 are symmetric, and that all such domains have an unbounded realization of a certain type, as a so-called Siegel domain.
Abstract: In 1935 E. Cartan classified all symmetric bounded domains [6]. At that time he proved that a bounded symmetric domain is homogeneous with respect to its group of holomorphic automorphisms. Thus the more general problem of investigating homogeneous bounded domains arose. It was known to E. Cartan that all homogeneous bounded domains of dimension ≤3 are symmetric [6]. For domains of higher dimension little was known. The first example of a 4-dimensional, homogeneous, non-symmetric bounded domain was provided by I. Piatetsky-Shapiro [41]. In several papers he investigated homogeneous bounded domains [20], [21], [41], [42], [43]. One of the main results is that all such domains have an unbounded realization of a certain type, as a so-called Siegel domain. But many questions still remained open. Amongst them the question for the structure and explicit form of the infinitesimal automorphisms of a homogeneous Siegel domain.
29 citations
TL;DR: In this paper, a generalization of Abel Theorem for integrals of rational forms in terms of hyper-logarithms is proposed, where the integrals can be described as local relations among rational differential forms.
Abstract: Several kinds of generalizations of classical Abel Theorem in algebraic curves are known, for example see [12] and [13]. It seems to the author these are all regarded as local relations among rational differential forms. In this article we shall try to generalize Abel Theorem for integrals of rational forms in some specific cases where these can be described in terms of hyper-logarithms (for the definition see [3] and [4], Theorem 2). Trigonometric functions have been generalized to higher dimensional cases by L. Schlafli who has obtained a very important variational formula related to the volume of a spherical simplex [16].
26 citations
TL;DR: In this article, a general method to study algebraic subgroups in the Cremona group of n variables was proposed and a new proof of Enriques Theorem (2.25) was given.
Abstract: This paper is a continuation of the two preceding papers [12], [13] where the classification of the de Jonquieres type subgroups in the Cremona group of 3 variables is promised. However the classification of such subgroups is postponed until the article in preparation 'On the maximal connected algebraic subgroups of the Cremona group Π\". The purpose of this paper is to establish a general method to study algebraic subgroups in the Cremona group of n variables and to illustrate how it works and leads to the classification of Enriques (Theorem (2.25)) when applied to the 2 variable case. This method gives us also the classification of the maximal connected algebraic subgroups of the Cremona group of 3 variables. The reason why we dare to write a new proof of the notorious Enriques Theorem is as follows. The case of 3 variables is rather complicated and we are sometimes obliged to indicate only the results and the way how to prove them without going into the detailed calculations if they are done quite similarly as in the 2 variable case. Hence we consider the best way to understand our classification in the 3 variable case is to read a complete proof in the 2 variable case beforehand. This is a raison d'etre of a new proof of the Enriques Theorem and hence of this paper. Our method will be applied for the 4 variable case too. As in the preceding papers, we work over an algebraically closed field of characteristic 0. All algebraic groups are connected and when we speak of a law chunk of algebraic operation (G, -X\"), X is irreducible. 1 denotes either the unit element of a group or the group consisting only of the unit element.
TL;DR: In this article, the authors studied the exceptional imprimitive groups of the 4-dimensional Cremona group Cr4 and gave modern and rigorous proofs to the results of Enriques and Fano.
Abstract: Enriques and Fano [4], [5] classified all the maximal connected algebraic subgroups of Cr3. Our aim is to give modern and rigorous proofs to their results. In [10], we studied the primitive subgroups. In this paper, we deal with exceptional imprimitive groups. The imprimitivity is an analytic notion. The natural translation of the imprimitivity in algebraic geometry is the de Jonquieres type operation (definition (2.1)). Every de Jonquieres type operation is imprimitive. However, the difference of these notions is subtle. The imprimitive algebraic operations in Cr3 which are not of de Jonquieres type are rather exceptional; there are only 3 such operations (theorem (3.26)). This paper together with [10] recovers all the results on Cr3 of Enriques and Fano [4]. It remains only to reconstruct Fano's classification [5] of the de Jonquieres type operations, which shall be done in our forthcoming paper. Our technique is rather old; the classification of 3 dimensional primitive operations, a very easy part of invariant theory which are of 19th century, combined with the theory of algebraic groups and transformation spaces of A. Weil. As the 4 dimensional primitive law chunks of analytic operations are classified, our method can be applied to the 4 dimensional Cremona group Cr4. We use the notations and the conventions of [10]. Therefore all manifolds, analytic groups, algebraic varieties, etc. are defined over C. The transformation spaces X of analytic or algebraic law chunk or operation (G, X) are connected. However, a differences lies in the language that we employ. Here is our French-English dictionary:
TL;DR: In this paper, a field-theoretic interpretation for the various partitions of the genus into half-genera by certain splitting integers is presented, which can be used to provide an invariant classification of forms up to spinor-equivalence.
Abstract: Let f(xί9 , xm) be a quadratic form with integer coefficients and ceZ. If f(x) = c has a solution over the real numbers and if f(x) = c (mod N) is soluble for every modulus N, then at least some form h in the genus of / represents c. If m 5> 4 one may further conclude that h belongs to the spinor genus of /. This does not hold when m = 3. However, in that situation there is a so-called \"75% Theorem\" which asserts that either every spinor genus in the genus of / represents c (i.e., there is a form in each spinor genus representing c) or else precisely half of all the spinor genera do. See [JW], [K], [H]. The theory of spinor exceptional representations is concerned with resolving the remaining 25% ambivalence. This we discuss in §§3, 4. We show in § 1 a field-theoretic interpretation for the various partitions of the genus into half-genera by certain \"splitting integers\", and in § 2 how this splitting feature can be exploited in certain cases to provide an invariant classification of forms up to spinor-equivalence, which may be viewed as a kind of a partial \"spinor character theory\", yielding in these instances an alternative to the algorithmic process of determining spinor-equivalence expounded recently by Cassels in [C], [CJ.
TL;DR: In this article, the cardinal number and the intersection of a subset of positive integers with a semi-group of integers is given. But this is not the case for all positive integers, and it is not always possible to identify a subset E of Z + with a sequence.
Abstract: We denote by Z + the semi-group of positive integers. For a subset E of Z + , we denote by | E | (≦ + ∞) its cardinal number and by E(n) the intersection of E and an interval [1, n ) ( n ≧ 1). We shall identify a subset E of Z + with a sequence, arranging elements of E according to their order.
TL;DR: In this article, the authors showed that the field of moduli for a non-hyperelliptic curve over a field coincides with the one for its canonically polarized jacobian variety.
Abstract: Throughout the paper, a scheme means a noetherian scheme. By a curve C over a scheme S of genus g, we mean a proper and smooth S-scheme with irreducible curves of genus g as geometric fibres. In the previous paper [15], the author showed that the field of moduli for a non-hyperelliptic curve over a field coincides with the one for its canonically polarized jacobian variety, and in [16], he gave a partial result on the coincidence of the fields of rationality for a hyperelliptic curve and for its canonically polarized jacobian variety. In the present paper, we will discuss the isomorphy of the isomorphism schemes of two curves over a scheme and of their canonically polarized jacobian schemes, by using Oort-Steenbrink’s result [12].
TL;DR: Chevalley and Bourbaki as discussed by the authors showed that the invariants of a finite group H ⊂ GLk (S 1) generated by pseudo-reflections are equivalent to the natural representation of H in the regular representation.
Abstract: In a classical paper [C] Chevalley considered the invariants of a finite group H ⊂ GLk (S 1) generated by pseudo-reflections, acting on the graded polynomial ring S = k[X 1,…,X n] over a field k of characteristic zero. He proved that S is free as a graded SH -module, hence SH is a graded polynomial ring (Theorem A), and that the natural representation of H in is equivalent to the regular representation (Theorem B). On the other hand, a theorem of Shephard and Todd shows that when SH is a polynomial ring, the (finite) group H is generated by pseudo-reflections. These results have been extended by Bourbaki [Bo2] to fields whose characteristic may be positive, but does not divide the order |H| of the group.
TL;DR: In this paper, it was shown that every distal homeomorphism of a compact connected metric space has not the pseudo-orbit tracing property, and the authors also showed that the same is true for the case of a Riemannian manifold of positive dimension.
Abstract: Recently, A. Morimoto [5] proved that every isometry of a compact Riemannian manifold of positive dimension has not the pseudo-orbit tracing property, and that if a homeomorphism of a compact metric space has the pseudo-orbit tracing property then Eφ — Oψ (see § 1 for definition). The purpose of this paper is to show that every distal homeomorphism of a compact connected metric space has not the pseudo-orbit tracing property. The author benefited from reading the papers by A. Morimoto [5, 6].
TL;DR: In this paper, the number of linearly independent automorphic forms of weight 1 for a fuchsian group of the first kind not containing the element is given, and a formula of d 0 is given.
Abstract: Let Γ be a fuchsian group of the first kind not containing the element . We shall denote by d 0 the number of linearly independent automorphic forms of weight 1 for Γ . It would be interesting to have a certain formula for d 0 . But, Hejhal said in his Lecture Notes 548, it is impossible to calculate d 0 using only the basic algebraic properties of Γ . On the other hand, Serre has given such a formula of d 0 recently in a paper delivered at the Durham symposium ([7]). His formula is closely connected with 2-dimensional Galois representations.
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TL;DR: In this article, the behaviour of Fourier series in an interval [0, 2K] is studied, in particular the behavior of so-called lacunary series, where the spectrum of a Taylor series is defined by Spec (F ) =, where Z + denotes the semi-group of positive integers.
Abstract: We are concerned with the behaviour of Fourier series in an interval [0, 2K ) and, in particular, interested in that of so-called lacunary series. The spectrum of a Taylor series is defined by Spec ( F ) = , where Z + denotes the semi-group of positive integers.
TL;DR: In this paper, it was shown that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.
Abstract: Around the beginning of this century G. Humbert ([9]) made a detailed study of the properties of compact complex surfaces which can be parametrized by singular abelian functions. A surface parametrized by singular abelian functions is the image under a holomorphic map of a singular abelian surface (i.e. an abelian surface whose endomorphism ring is larger than the ring of rational integers). Humbert showed that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and he constructed an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.
TL;DR: In this article, the Euclidean variance of a mean zero Gaussian random field (n ⋜ 2 ) was shown to be invariant under the Euclidian motion.
Abstract: Let be a mean zero Gaussian random field ( n ⋜ 2). We call X Euclidean if the probability law of the increments X(A ) − X(B ) is invariant under the Euclidean motions. For such an X , the variance of X(A ) − X(B ) can be expressed in the form r (| A − B |) with a function r(t ) on [0, ∞) and the Euclidean distance |A − B| .
TL;DR: In this article, it was shown that some unitary representations of the Poincare group are irreducible, even if they are restricted to the poincare semigroup.
Abstract: Since E. Wigner set up a framework of the relativistically covariant quantum mechanics, several aspects of unitary representations of the Poincare group have been investigated (see [8], [16]). In this paper it will be shown that some unitary representations of the Poincare group are irreducible, even if they are restricted to the Poincare semigroup (Theorem 1, 2 and 3).
TL;DR: Theorem 1.3 as discussed by the authors shows that for 2-dimensional rings, the answer is intimately connected with the structure of projective modules, and the main result in the positive direction.
Abstract: An R -module M is a generator (of the category of modules) provided every module is a homomorphic image of a suitable direct sum of copies of M . Equivalently, some M (k) has R as a summand. Except in the last section, all rings are assumed to be commutative, Noetherian domains, and modules are usually finitely generated. In this context generators are exactly those modules that have non-zero free summands locally . Of course, generators can fail to have free summands (e.g., over Dedekind domains), and we ask whether they necessarily have non-zero projective summands. The answer is “yes” for rings of dimension 1, as we point out in § 3, and for the polynomial ring in one variable over a Dedekind domain. In § 1 we show that for 2-dimensional rings the answer is intimately connected with the structure of projective modules. Our main result in the positive direction, Theorem 1.3, grew out of the attempt, in conversations with T. Stafford, to understand the case R = k[x, y] . In § 2 we give examples of rings having generators with no projective summands. The last section contains miscellaneous observations, some of them on rings without chain conditions.
TL;DR: In this article, the authors studied the behaviour of meromorphic functions with non-isolated essential singularities and generalizations of the Gross' result, and showed that such functions cannot be rammed.
Abstract: In the complex function theory, Picard's Great Theorem plays an essential and important role. It is well-known as generalizations of this theorem that in a neighborhood of an isolated essential singularity, a meromorphic function cannot be exceptionally ramified (see W. Gross [2]) and that even it cannot be normal (see O. Lehto and K. I. Virtanen [7]). We are therefore interested in the behaviour of meromorphic functions with non-isolated essential singularities as well as in generalizations of the Gross' result. Several approaches in this direction have been made by G. af Hallstrδm [3], S. Kametani [4], K. Noshiro [13], K. Matsumoto [8], [9], [10], [11], [12], S. Toppila [15], etc.. As for the functions with \"more than two Picard exceptional values\", K. Matsumoto ([10], [11]) has given sufficient conditions on Cantor sets E whose complements do not admit such functions. One of his basic results is
TL;DR: In this article, the cohomology at infinity of a congruence subgroup of the symplectic group G = Sp (2 l, R ) is studied, where G is the subgroup consisting of matrices g satisfying t gJg = J where
Abstract: This paper is concerned with the cohomology at “infinity” (in the sense of Harder [4], [5]) of a congruence subgroup of the symplectic group G = Sp (2 l , R). G is the subgroup of GL (2 l, R ) consisting of matrices g satisfying t gJg = J where