The ramification theory in commutative rings, as a generalization of the classical one in maximal orders over a Dedekind domain, was established in [1], [13], [15] and etc.. For non-commutative algebras this was also studied in [3], [4], [9], [18] and etc.. However, the different theorem, which is a central part of ramification theory, has not been given in those, except for some special cases (cf. [12], [18]). The main object of this paper is to give the discriminant theorem and the different theorem for projective orders in a (non-commutative) separable algebra, in the most general form.

L’objet de ce mémoire est essentiellement la recherche et l’étude de structures algébriques permettant de traduire le Calcul propositionnel et le Calcul des prédicats restreint du premier ordre. Nous avons guidé et limité cette recherche en imposant aux ensembles algébriques introduits de satisfaire simultanément aux trois conditions:

• — donner une représentation du Calcul Logique sous sa forme la plus classique, c’est à dire avant l’obtention d’un anneau booléien par passage au quotient. Ce sera le rôle des structures “prébooléiennes”.

• — posséder un ensemble d’opérations traduisant un procédé de décision tel que celui des tableaux sémantiques, qui correspondent ici aux “tableaux de réduction”. —avoir un caractère universel dans leur construction à partir des atomes.

We show that if is a frame and {ψＱ}Ｑ∈Q ∈ ∩ Mα(ℝn) is its dual frame (for the definition of Mα(ℝn), see Definition 2.1), where Q is the collection of dyadic cubes, then for any f ∈ S′(ℝn), there exists a sequence of polynomials, PL,L′,L″, such that

(0.1)

in the topology of S′(ℝn), where δ(i) = max(2i, 1). We prove this result by explicitly constructing the polynomials PL,L′,L″. Furthermore, using the above result, we assert that the linear span of the one-dimensional wavelet system is dense in a function space if and only if the dual space of this function space has an trivial intersection with the set of polynomials. This is proved by using the annihilator of the one-dimensional wavelet system.

If U=U(0), U(i) is a maximal subgroup of U(i + 1) and U(n) = V, then we term the U(i) a densest chain connecting U and V; and n is the length of this chain. The subgroup U of V will furthermore be termed an n-uniserial subgroup of V if there exists one and only one densest chain connecting U and V and if its length is n. The principal aim of this note is to give characterizations of dedekind groups [ = groups all of whose subgroups are normal] in terms of the normality of uniserial subgroups. We quote one of our results:

In the previous paper [3], we introduced new coordinates to the Schottky space, and defined the augmented Schottky spaces . Here, in § 1, we will define fiber spaces over the augmented Schottky spaces. In §2, we will consider a property of the new coordinates, namely, we will state a relation between limits of sequences of elements of the new coordinates and limits of sequences of length of loops on Riemann surfaces.

We describe the divisor class group and the graded canonical module of the multisection ring T (X;D1,…, Ds) for a normal projective variety X and Weil divisors D1,…, Ds on X under a mild condition. In the proof, we use the theory of Krull domain and the equivariant twisted inverse functor.

Let f(z) be a function analytic and bounded, |f(z)| < 1, in |z| < 1. Then, by Fatou’s theorem the radial limit f*(eiθ) = limr→1f(reiθ) exists almost everywhere on |z| = 1. Seidel [8, p. 208] and Calderón, González- Domínguez, and Zygmund [1] (see also [9, pp. 281-282]) proved the following: if f*(eiθ) is of modulus 1 almost everywhere on an arc a < θ < b of |z| = 1, then either f(z) is analytically continuable across this arc or the values f*(eiθ), a < d < b, cover the circumference |w| = 1 infinitely many times.

Let Γ be a ring with unit element and let Λ be the Ore extension of Γ with respect to a derivation d of Γ [4, 3]. It is shown in [3] that l.gl. dim Λ ≦ 1 + l.gl. dim Γ. It is not in general possible to replace this inequality by equality.

We defined a prime decomposition symbol [d1 d2, p] in a previous paper [3], and characterized in [4] the set of rational primes p which are decomposed completely in a non-abelian central extension which is of degree 8 in substance. The explicit value of the symbol was determined by using a solution of certain ternary quadratic diophantine equation. The solution corresponds to a square root of an ideal class of the principal genus of a quadratic field. This is translated to a problem in classes of integral quadratic forms, namely to find a form whose duplication is a given one contained in a principal genus. An explicit method to find the form is given by Gauss in [5, Art. 286], which is due to his ternary form reduction.

The purpose of the present paper is to give a new proof to the triad excision theorem of Blakers and Massey [1], in case m≦2 and n≦2, by the aid of path spaces and in connection with a recent work of J. P. Serre [2].

In [5], R. Thorn has started the study of the foliated structures by using the Morse theory. Recently K. Yamato [7] has studied the topological properties of leaves of a codimension one foliated manifold by investigating the “critical points” of variation equation of the given one-form.

In this paper we investigate the boundedness of Cauchy kernels. The Cauchy kernel associated with a locally integrable real-valued function θ(x) is defined by

We prove in this paper a theorem on analytic mappings of the complex space Cn into the complex projective space Pn. The theorem is closely related to that of S. S. Chern in [1], and the main idea of the proof is the same with the latter, though the calculations are rather different. The background of our calculation is the normal contact metric structure which was found by S. Sasaki and Y. Hatakeyama [4].

In this short note, we would like to point out that the following property (called ASSUMPTION REMOVABILITY in the present paper) is characteristic of the positive logics, the primitive logic LO, the positive predicate logics LP (intuitionistic) and LQ (classical):

Recently H. Hasse has given an interesting theory of Galois algebras, which generalizes the well known theory of Kummer fields; an algebra over a field Ω is called a Galois algebra with Galois group G when possesses G as a group of automorphisms and is (G, Ω)-operator-isomorphic to the group ring G(Ω) of G over Ω. On assuming that the characteristic of Ω does not divide the order of G and that absolutely irreducible representations of G lie in Ω, Hasse constructs certain Ω-basis of , called factor basis, in accord with Wedderburn decomposition of the group ring and shows that a characterization of is given by a certain matrix factor system which defines the multiplication between different parts of the factor basis belonging to different characters of G. Now the present work is to free the theory from the restriction on the characteristic. We can indeed embrace the case of non-semisimple modular group ring G(Ω).

In this paper, we will prove that any $\mathbb{A}^{3}$-form over a field $k$ of characteristic zero is trivial provided it has a locally nilpotent derivation satisfying certain properties. We will also show that the result of Kambayashi on the triviality of separable $\mathbb{A}^{2}$-forms over a field $k$ extends to $\mathbb{A}^{2}$-forms over any one-dimensional Noetherian domain containing $\mathbb{Q}$.

N. Aronszajn introduced in [4] an abstract Frechét space R (0<R≤∞), which is isomorphic to the space of analytic solutions of the heat equation in if 0 < R ∞, and in if R = ∞, and called it the space of traces of analytic solutions of the heat equation. Hereafter, we call it the space of traces, shortly.

The purpose of this paper is to study the dimension formula for cusp forms of weight one, following the series of Hiramatsu [2] and Hiramatsu-Akiyama [3]. We define as usual the subgroup Γ0(N) of SL2(Z) by

.

On the half space Rn × R+, it has been known that harmonic Bergman space bp can contain a positive function only if . Thus, for , Poisson integrals can be bp-functions only by means of their boundary cancellation properties. In this paper, we describe what those cancellation properties explicitly are. Also, given such cancellation properties, we obtain weighted norm inequalities for Poisson integrals. As a consequence, under weighted integrability condition given by our weighted norm inequalities, we show that our cancellation properties are equivalent to the bp-containment of Poisson integrals for p under consideration. Our results are sharp in the sense that orders of our weights cannot be improved.

An adjoint variety X(g)associated to a complex simple Lie algebra is by definition a projective variety in ℙ*(g) obtained as the projectivization of the (unique) non-zero, minimal nilpotent orbit in g. We first describe the tangent loci of X(g) in terms of triples. Secondly for a graded decomposition of contact type we show that the intersection of X(g) and the linear subspace ℙ*(g1) in ℙ*(g) coincides with the cubic Veronese variety associated to g.