For a smooth variety proper over a curve having a fibre with isolated ordinary quadratic singularities, it is well-known that we have the vanishing cycles associated to the singularities in the étale cohomology of the geometric generic fibre. The base-change by a double cover of the base curve ramified at the image of the singular fibre has singularities corresponding to the singularities in the fibre. In this note, we show that in the even relative-dimensional case, there exist elements of the bivariant Chow group of the base-change with supports in the singularities and hence their images in the bivariant Chow group with supports in the special fibre and that the usual cohomological vanishing cycles are obtained as their images by a natural map, a kind of “cycle map” so that the elements in the bivariant Chow groups can be regarded as the vanishing cycles. The bivariant Chow group with supports in the special fibre has a ring structure and the natural map is a ring homomorphism to the cohomology ring of the geometric generic fibre. Also discussed is the relation of the bivariant Chow group with supports in the special fibre to the specialization map of Chow groups.

1. Let V be a free Z-module of rank 2n. Let G=Sp(2n, Z) be the symplectic modular group and let Φ be the non-singular alternating bilinear form on V left invariant by G. Let p ∊ Z be a prime and let X be the set of all endomorphisms ξ of V such that

Let A and B be densely defined closed linear operators in complex Banach spaces X, Y, respectively, with nonempty resolvent sets.

In his paper [4] or lecture note [3], Eichler asked the problem when the ring of modular forms is Cohen-Macaulay. We shall try to investigate it for the Hilbert or Siegel modular case.

The main purpose of the present paper is to introduce a new understanding of formal theories.

It has been a traditional pattern of formal theories to presuppose a logic and an axiom system for each formal theory. The axiom system of any formal theory consists of a finite number of axiom schemata in general, but occasionally it can be regarded as consisting of a finite number of axioms. I will call any formal theory of this kind an axiomatic theory or an axiom-schematic theory according as its axiom system is regarded as consisting of a finite number of axioms or as consisting of a finite number of axiom schemata.

Let ξ(t), t ∈ T, be a Gaussian process on a set T, and H = H(ξ) be the closed linear manifold generated by all values ξ(t), t ∈ T, with the inner product

A formal quantization of Poisson algebras was discussed by several authors (see for instance Drinfel’d [D]). A formal Lie algebra generated by homotopy classes of loops on a Riemann surface ∑ was obtained by W. Goldman in [G], and its Poisson algebra was quantized, in the sense of Drinfel’d, by Turaev in [T].

Let k denote the quotient field of a complete discrete rank one valuation ring R. The purpose of this paper is to establish a relationship between the Brauer group of k and the set of maximal orders over R which are equivalent to crossed products over tamely ramified extensions of R.

Let E be a compact set of logarithmic capacity zero in the complex plane. Then the following is well-known as Evans-Selberg’s theorem [1] [8]: there is a measure with support contained in E such that its logarithmic potential is positively infinite at each point of E. But such a potential does not exist for E of logarithmic positive capacity. Now suppose that E is contained in the circumference of the unit disc |z| < 1 and is of linear measure zero.

Let X = (X(t); t ∈ R) be a real stationary mean continuous Gaussian process with expectation zero which is purely nondeterministic. In this paper we shall investigate the structure of splitting fields of X having finite multiple Markovian property using the results in [6]. We follow the notations and terminologies in [6].

By an algebraic homogeneous space, we mean the factor space X = G/P, where G is a simply-connected, complex, semi-simple Lie group and P is a parabolic subgroup of G. Many typical manifolds such as the projective spaces and the Grassmann varieties belong to this class of manifolds. For instance, the Grassmann variety G(k, n) can be expressed as SL(n + 1, C)/P, where P is a maximal parabolic subgroup of SL(n + 1, C) leaving a suitable k + 1 dimensional subspace invariant. In this paper, we devote ourselves to study the Bruhat decomposition of an algebraic homogeneous space X = G/P.

Let C be an affine curve, contained on a non-singular surface X as a closed 1-dimensional subscheme. If P is a closed point on C, the blowing-up C′ of C with center P (induced by the blowing-up of X with center P) is an affine curve. It is known that there is a sequence:

where C is the normalization of C, and each Ci + 1 is the blowing-up of Ci with center a singular point Pt on Ci (i = 0, …, k – 1).

In my joint work with J. Ito [7], we have pointed out that the following property (called ASSUMPTION REMOVABILITY) is characteristic of positive logics LO, LP, and LQ:

ASSUMPTION REMOVABILITY. If → is provable for any pair of propositions andhaving no primitive notions (proposition-, predicate-, and relation-symbols) in common, thenis also provable.

In [0], the writer proved some theorems of Hasse type for two orthogonal groups which operate on the same vector space. In this paper, we shall further generalize those results in two directions. One is to consider the propositions of that type for two orthogonal groups which operate respectively on two vector spaces whose dimensions are different from each other, and the other is to deal with some conspicuous subgroups of an orthogonal group simultaneously which play important roles in the structure theory for orthogonal groups. For this reason, the present paper consists of three steps §1, §2 and §3 which give the generalizations in the above sense of the results in the corresponding sections of [0].

In this paper we show, for an example of hyperbolic system, that the wave front set in Gevrey class of a certain solution propagates precisely along “generalized broken null-bicharacteristic flow” defined by Kumano-go-Taniguchi [7] and Wakabayashi [10] [11].

A domain D ⊂ Cz admits the circular slit mapping P(z) for a, b ∈ D such that P(z) – 1/(z – a) is regular at a and P(b) = 0. We call p(z) = log|P(z)| the Li-principal function and α = log |P′(b)| the L1-constant, and similarly, the radial slit mapping Q(z) implies the L0-principal function q(z) and the L0-constant β. We call s = α – β the harmonic span for (D, a, b). We show the geometric meaning of s. Hamano showed the variation formula for the L1-constant α(t) for the moving domain D(t) in Cz with t ∈ B:= {t ∈ C: |t| < ρ}. We show the corresponding formula for the L0-constant β (t) for D(t) and combine these to prove that, if the total space D = ∪t∈B(t, D (t)) is pseudoconvex in B × Cz, then s(t) is subharmonic on B. As a direct application, we have the subharmonicity of log cosh d(t) on B, where d(t) is the Poincaré distance between a and b on D(t).

Consider the polar coordinate differentials (dr, dθ) on a hyperbolic Riemann surface R with center z0 ∈ R which are given by

where GR(z, ζ) is the Green’s function on R with pole ζ ∈ R.

For a character of the absolute Galois group of a complete discrete valuation field, we define a lifting of the refined Swan conductor, using higher dimensional class field theory.

The many unanswerable questions (1) which arise in the study of finite groups have lead to a review of fundamental ideas, e.g. the Theorem of Burnside (3, p. 299; 2, 6) that if λ be any faithful irreducible representation of G over a field K, then every irreducible representation of G over K is contained in some tensor power of λ.

In this paper we shall discuss nonlinear eigenvalue problems for the equations of the form

where L is a linear operator on a real Banach space X with non-zero kernel, K(-) is a linear or nonlinear operator on X and M(·, ·) is an operator from X X R into X. Equations of the form (1) arise in various fields of physics and engineering.