The boundary components of an abstract Riemann surface were defined by B. v. Kérékjértó [7] and utilized in the book [14] written by S. Stoïlow. It is the purpose of the present paper to investigate their images under conformal mapping and to solve the Dirichlet problem with boundary values distributed on them.

For an integer N let X1(N) be the modular curve defined over Q which corresponds to the modular group Γ1(N) To each primitive cusp form f ═ Σ amqm, a1═1, (Γ normalized new form in the sense of [1]) on Γ1(N) of weight 2, there corresponds a factor Jf of the jacobian variety of X1(N) (cf. Shimura [19]).

In this paper we shall investigate the structure of complex Lie groups from function theoretical points of view. A. Morimoto proved in [10] that every connected complex Lie group G has the smallest closed normal connected complex Lie subgroup Ge, such that the factor group G/Ge is Stein. On the other hand there hold the following two basic structure theorems (A1) and (A2) for a connected algebraic group G (cf. [12]). (A1): G has the smallest normal algebraic subgroup N such that the factor group G/N is an affine algebraic group. Moreover N is a connected central subgroup. (A2): G has the unique maximal connected affine algebraic subgroup L, where L is normal and the factor group G/L is an abelian variety.

We consider the moduli space of genus 4 curves endowed with a $g^1_3$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that it defines a degree $\frac {1}{2}(3^{10}-1)$ cover of the nine-dimensional Deligne–Mostow ball quotient such that the natural divisors that live on that moduli space become totally geodesic (their normalizations are eight-dimensional ball quotients). This isomorphism differs from the one considered by S. Kondō, and its construction is perhaps more elementary, as it does not involve K3 surfaces and their Torelli theorem: the Deligne–Mostow ball quotient parametrizes certain cyclic covers of degree 6 of a projective line and we show how a level structure on such a cover produces a degree 3 cover of that line with the same discriminant, yielding a genus 4 curve endowed with a $g^1_3$.

We describe a general method for expanding a truncated $G$-iterative Hasse–Schmidt derivation, where $G$ is an algebraic group. We give examples of algebraic groups for which our method works.

Building on the embedding of an n-abelian category $\mathscr {M}$ into an abelian category $\mathcal {A}$ as an n-cluster-tilting subcategory of $\mathcal {A}$ , in this paper, we relate the n-torsion classes of $\mathscr {M}$ with the torsion classes of $\mathcal {A}$ . Indeed, we show that every n-torsion class in $\mathscr {M}$ is given by the intersection of a torsion class in $\mathcal {A}$ with $\mathscr {M}$ . Moreover, we show that every chain of n-torsion classes in the n-abelian category $\mathscr {M}$ induces a Harder–Narasimhan filtration for every object of $\mathscr {M}$ . We use the relation between $\mathscr {M}$ and $\mathcal {A}$ to show that every Harder–Narasimhan filtration induced by a chain of n-torsion classes in $\mathscr {M}$ can be induced by a chain of torsion classes in $\mathcal {A}$ . Furthermore, we show that n-torsion classes are preserved by Galois covering functors, thus we provide a way to systematically construct new (chains of) n-torsion classes.

Quasi-socle ideals, that is, ideals of the form I = Q: mq (q ≥ 2), with Q parameter ideals in a Buchsbaum local ring (A,m), are explored in connection to the question of when I is integral over Q and when the associated graded ring G(I) ⊕ n≥0In/In+1 of I is Buchsbaum. The assertions obtained by Wang in the Cohen-Macaulay case hold true after necessary modifications of the conditions on parameter ideals Q and integers q. Examples are explored.

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.

Montel [10] proved in 1912 the following theorem: Let w = f(z) be an analytic function in the horizontal strip B : 0 < x < + ∞, 0 < y < 1 (z = x + iy) which is continuous on 0 < x < + ∞, 0 ≦ y < 1 and omits at least two values. If f (x) converges to a value w0 as x → + ∞, then f(z) converges to W0 as z tends to in 0 < x < + ∞, 0 ≦ y < 1 − ε for any ε such that 0 < ε < 1.

In the present note we associate a certain algebra of finite rank over Q to each non-singular polarized algebraic variety defined over C. For a surface the algebra is a Jordan algebra with identity, and for an abelian variety A the algebra is canonically isomorphic to the Jordan algebra of symmetric elements in EndQ(A) with respect to the involution induced by the polarization. This algebra may be important for a polarized algebraic variety as much as EndQ(A) for an abelian variety A.

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):

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.

Let f(z) be entire i.e. analytic in the finite whole plane Z. The order of f(z) is defined as

where A ray χ(θ) = {z = r·eiθ : 0 < r < + ∞} is called a Julia direction of f(z) if, in any open sector containing the ray, f(z) takes all values of Z, with at most one finite exceptional value, infinitely often.

Let n > 1, m ≥ 1, g ≥ 3 and γ be given integers. The purpose of this paper is to determine the relations of n, m, g and γ for the existence of the symmetric Riemann surfaces S of type (n, m) with genus g and species γ. If n is an odd prime, the relations are known in [3]. In the case that n is odd, we shall show the analogous result when E(S) is isomorphic to a cyclic group Z2n and when the quotient space S/E(S) is orientable.

This paper is devoted to the problem of the realisation of homology classes of a PL-manifold by PL-submanifolds.

The present study is founded on the consideration of Thorn complexes M(PLk), M(SPLk) for PL-microbundles which is defined by R. Williamson [5]. We shall apply Thom’s method [4] to PL-manifolds.

The author is grateful to Professors R. Shizuma and K. Shiraiwa for their kind criticisms.

It is well known that the quasi-exellent property (i.e. G-ring and J-II) ascends by finite type morphisms. On the other hand Greco showed in [2, Proposition 4.1 and Proposition 2.3] that G-property (i.e. formal fibers are geometrically regular) does not descend by finite type morphisms, although J-II property (i.e. any finite type algebra has regular locus open) does.

It is a privilege to share in the honoring of Professor Katuzi Ono on this occasion, It would be enough to be simply putting in evidence my high regard for a good and dear friend. But there is so very much more than that here. It is an opportunity for me to declare my esteem for the kind of mathematician who looks at his field from the outside as well as the inside; who sees the richness of all experience and seeks its instruction to know better and better the meaning of mathematics ; who by the example of his own life urges young, coming scientists to the more productive path of an uncompartmentalized personality. Katuzi Ono is all of these.

In the pairwise spectral analysis of two projections F and G (see [1, §4] of the References) there are determined two resolutions of the identity, EF-and EG, which are spectral measures on the Borel interval [0, 1], and which are, in fact, the spectral measures of the Hermitian operators SF and SG of (1. 6) below.

We give pluriharmonic and harmonic representations of prehomogeneous vector spaces of regular classical parabolic type (L = Γ×M, V) througt the commutative quotients V ‖ M = spec(C[V]M) of them.

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.