The following is a well-known result due to A.P. Calderon [2], asserting the existence of non-tangential limits of multiply harmonic functions.

Let k be a field, and X = [xij] be an n × (n + m) matrix whose elements are algebraically independent over k.

We shall study the canonical module of the graded ring R, which is a quotient ring of the polynomial ring A = k[X] by the ideal αn(X) generated by all the n × n minors of X.

The main purpose of this paper is to investigate the parallelism of vectors in homogeneous spaces. The definition of a vector and the condition for spaces under which a covariant differential of a vector is also a vector were given by E. Cartan [4] in a very intuitive way. Here I formulate this in a stricter way by his method of moving frame. Even if a homogeneous space has the property that the covariant differential of a vector is of the same kind, another definition of covariant differential may also have the required property. I will give a necessary and sufficient condition under which the definition of covariant differential is unique. Once the covariant differential has been defined it is easy to introduce a parallelism of vectors in the space. But the parallelism depends in general on the path along which we translate a vector. The condition for the spaces with an absolute parallelism can be obtained.

Let p be a prime. Let k be a -adic number field and be the ring of all integers of k. Let K/k be a cyclic totally ramified extension of degree pn with Galois group G. Clealy the ring of all integers of K is an [G]-module, and the purpose of this paper is to give a necessary and sufficient condition for the [G]-module to be indecomposable.

Let f: X → Y be a finite type morphism of locally noetherian schemes. It is well known ([3, IV, 7.8.6]) that the excellent property ascends from Y to X. On the other side there are counter-examples where X is excellent and Y is not. First of all it is easy to show that the condition on chains of prime ideals does not descend (see [3, IV, 7.8.4]), even by finite morphisms. Secondly in [2] it is produced an example where X is excellent while Y is not a G-scheme (i.e. it has not the good properties of formal fibers). However in [2] it is also proved that the property concerning the openness of regular loci (the so called “J-2”) descends by finite type surjective morphisms. Therefore we are led to the following question: When does the G-scheme property descend? I.e. what conditions do we need on f? A reasonable condition is conjectured (in [2]) as the following: f is proper surjective. The aim of the present paper is precisely to give an answer to such a question. What we really prove is the following. If X is a G-scheme and J-2 (quasi-excellent), then the same is true for Y, provided that f is proper surjective and moreover all the residue fields of Y have characteristic 0. We remark that the result is strongly based on Hironaka’s desingularization for quasi-excellent schemes defined over a field of characteristic 0.

Für jede Riemannsche Fläche R bezeichnen wir mit M(R) die Algebra von Royden auf R und mit R* (bzw. ΔR)- die Kompaktifizierung (bzw. den idealen Rand) von Royden von R ([5]). Für jede Funktion ξ ∊ M(R) sei Jξ das kompakte Segment und X(R) der kompakte Raum . Laut der Definition von R* ist jede Funktion ξ∊M(R) auf R* (in eindeutiger Weise) stetig fortsetzbar. Bezeichnet man ebenfalls mit ξ die so fortgesetzte Funktion und mit φR die Abbildung von R*→X(R), so ist φR ein Homeomorphismus von R* auf φR(R*).

For an integer m > 2, we denote by C(m) and H(m) the ideal class group and the class-number of the field

K = Q(ζm + ζm−1)

respectively, where ζm is a primitive m-th root of unity. Let q be a prime and /Q be a real cyclic extension of degree q. Let C() and h() be the ideal class group and the class-number of . In this paper, we give a relation between C() (resp. h()) and C(m) (resp. H(m)) in the case that m is the conductor of (Main Theorem). As applications of this main theorem, we give the following three propositions. In the previous paper [4], we showed that there exist infinitely many square-free integers m satisfying n|H(m) for any given natural number n. Using the result of Nakahara [2], we give first an effective sufficient condition for an integer m to satisfy n|H(m) for any given natural number n (Proposition 1). Using the result of Nakano [3], we show next that there exist infinitely many positive square-free integers m such that the ideal class group C(m) has a subgroup which is isomorphic to (Z/nZ)2 for any given natural number n (Proposition 2). In paper [4], we gave some sufficient conditions for an integer m to satisfy 3|H(m) and m≡l (mod 4). In this paper, using the result of Uchida [5], we give moreover a sufficient condition for an integer m to satisfy 4|H(m) and m ≡ 3 (mod 4) (Proposition 3). Finally, we give numerical examples of some square-free integers m satisfying 4|H(m) and m ≡ 3 (mod 4).

The analytic capacity γ(E) of a compact set E in the complex plane C is defined by γ(E) = sup , where — f′(∞) is the 1/z-coeffieient of f(ζ) at infinity and the supremum is taken over all bounded analytic functions f(ζ) outside E with supremum norm less than or equal to 1. Analytic capacity γ(·) plays various important roles in the theory of bounded analytic functions.

A nonnegative locally Hölder continuous second order differential P = P(z)dxdy (z = x + iy) on a Riemann surface R is referred to as a density on R. A density P is said to be finite if P is integrable over R, i.e.

(1) ∫ R P(z)dxdy < ∞.

We prove that for $n\geqslant 4$, every knot has infinitely many conjugacy classes of $n$-braid representatives if and only if it has one admitting an exchange move.

Let X be a locally compact and σ-compact abelian group and let denote the dual group of X. We denote by ξ a fixed Haar measure on X and by the Haar measure associated with ξ. In [2], we show the following

Theorem. Let (αt)t≧0 be a sub-Markov convolution semi-group on X and let ψ be the negative definite function associated with (αt)t≧0. Then (αt)t≧0 is transient if and only if Re (1/ψ) is locally -summable.

Cet article fait suite à [3c]; dans [3c] étaient étudiées des généralisations éventuelles du principe de minimum fin de Fuglede (voir [2] théo. 9.1. et [3a] théo. 6.) et le cardre était celui des processus de Markov en dualité au sens de Kunita-Watanabe (voir [4b]). Le présent article a le même but, si dans [3c] on a supposé que la limite inférieure fine de la fonction finement hyperharmonique donnée doit être non négative en tout point de la fontière fine de l’ensemble de définition, on se permet dans cet article de supposer que cette lim. inf. fine pourra avoir des valeurs négatives (même égales à — ∞) sur un ensemble polaire de la fontière fine.

Let k be an algebraic number field of finite degree, and p be a fixed rational prime. We denote the set of all the non-Archimedian prime divisors of k by S0(k) and the set of all the real Archimedian ones by (k). Put and S = S0 ∪ S∞, and define a subgroup of the unit group (k) of k by

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 Hällström [3], S. Kametani [4], K. Noshiro [13], K. Matsumoto [8], [9], [10], [11], [12], S. Toppila [15], etc..

Hara and Yoshida introduced a notion of α-tight closure in 2003, and they proved that the test ideals given by this operation correspond to multiplier ideals. However, their operation is not a true closure. The alternative operation introduced here is a true closure. Moreover, we define a joint Hilbert-Kunz multiplicity that can be used to test for membership in this closure. We study the connections between the Hara-Yoshida operation and the one introduced here, primarily from the point of view of test ideals. We also consider variants with positive real exponents.

L’étude sur les directions de Julia des fonctions méromorphes dans le plan fini est fait présque complètement par Ostrowski [5] et après Rauch [6] etc. l’ont développé.

In the derived category of a commutative noetherian ring, we explicitly construct a silting object associated with each sp-filtration of the Zariski spectrum satisfying the “slice” condition. Our new construction is based on local cohomology and it allows us to study when the silting object is tilting. For a ring admitting a dualizing complex, this occurs precisely when the sp-filtration arises from a codimension function on the spectrum. In the absence of a dualizing complex, the situation is more delicate and the tilting property is closely related to the condition that the ring is a homomorphic image of a Cohen–Macaulay ring. We also provide dual versions of our results in the cosilting case.

In their paper [11], Kubota-Leopoldt constructed the p-adic L-function Lp(s; χ) for each Dirichlet character χ. This function Lp(s; χ) is a mero-morphic function on the open dies {|s − 1| < |<q-1 p1/(p-1)|} with a possible simple pole at s = 1.

In this paper we prove that a normal Gorenstein surface dominated by P2 is isomorphic to a quotient P2/G, where G is a finite group of automorphisms of P2 (except possibly for one surface ). We can completely classify all such quotients. Some natural conjectures when the surface is not Gorenstein are also stated.

Let k be any integer, k > 0. The k-th Bergman measure on unit ball B of Cn, μk, is given by