1. The classification of Riemann surfaces with respect to the equation Δu = Pu (P ≥ 0, P ≢ 0) was initiated by Ozawa [13] and further developed by L. Myrberg [8, 9], Royden [14], Nakai [10, 11], Sario-Nakai [15], Nakai-Sario [12], Glasner-Katz [3], and Kwon-Sario [7].

p. 187, line 20: Delete “—s” from the formula.

p. 228, line 4: Delete “—1” from the formula.

p. 228, lines 18 and 21: Delete “—s” from the formulas.

In this paper we give a negative answer to the problem which is suggested in [3]: if a nonwandering set Ω is hyperbolic, are the periodic points dense in Ω?

Newhouse and Palis proved that on two dimensional closed manifolds the answer is positive ([1], [2]).

Let C, C′ and C″ be abelian categories where C and C′ have enough injectives and let F: C → C′, G: C′ → C″ be additive covariant functors. Then for an object X of C, let C(X) be the complex associated with an injective resolution of X. Grothendieck gets a first quadrant spectral sequence by taking an injective resolution of the complex F(C(X)) and applying G to the associated double complex.

A function f(z), regular in the unit disk D, is called annular ([1], p. 340) if there is a sequence of closed Jordan curves Jn ⊂ D satisfying

1. (A1) Jn is contained in the interior of Jn+1 for every n,

2. (A2) given ε > 0, there exists a positive number n(ε) such that, for each n > n(ε), Jn lies in the region 1 – ε < | z | < 1 and

3. (A3) lim min {| f(z) |; z ∈ Jn} = + ∞.

In connection with the class field theory a problem concerning p-groups was proposed by W. Magnus: Is there any infinite tower of p-groups G1, G2,…, Gn, Gn+1,…such that G1 is abelian and each Gn is isomorphic to Gn+1/θn(Gn+1), θn(Gn+1) ≠ 1, n = 1,2,…, where θn(Gn+1) denotes the n-th commutator subgroup of Gn+1? The present note is, firstly, to construct indeed such a tower, to settle the problem, and also to refine an inequality for p-groups of P. Hall.

Let us start with a short survey concerning complex white noise (for details see Hida [3]): Let be the space of real valued rapidly decreasing testing functions and its complexification.

The purpose of this paper is to present the results of E. Artin and Furtwängler, with which they proved the principal ideal theorem, as a structure theorem of the idele group of an algebraic number field. Such treatment may be helpful to clarify the Arithmetic nature these results possess.

Cohomology groups of (associative) algebras have been introduced (for higher dimensions) and studied by G. Hochschild in his papers [2], [3] and [4]. 1-, 2-, and 3-dimensional cohomology groups are in closest connection with some classical properties of algebras. In particular, an algebra is absolutely segregated. if and only if its 2-dimensional cohomology groups are all trivial. It is thus of use and importance to determine the structure of algebras with universally vanishing 2-cohomology groups, i.e. absolutely segregated algebras; they form a class which is wider than the class of all algebras with universally vanishing 1-cohomology groups, i.e. separable algebras in the sense of the Dickson-Wed-derburn theorem.

It is an important problem to determine the spectral type of automorphisms or flows on a probability measure space. We shall deal with a unitary operator U and a 1-parameter group of unitary operators {Ut} on a separable Hilbert space H, and discuss their spectral types, although U and {Ut} are not necessarily supposed to be derived from an automorphism or a flow respectively.

The main purpose of this paper is, using the estimates for character sums and the analytic method, to study the 2k-th power mean of the inversion of Dirichlet L-functions with the weight of general quadratic Gauss sums, and give two interesting asymptotic formulas.

In “Seminormal rings and weakly normal varieties” we introduced the notion of a c-regular function on an algebraic variety defined over an algebraically closed field of characteristic zero. Our intention was to describe those k-valued functions on a variety X that become regular functions when lifted to the normalization of X, but without any reference to the normalization in the definition. That is, we aspired to identify the c-regular functions on X with the regular functions on the weak normalization of X

In the following we shall fix an algebraically closed field K of arbitrary characteristic. The term variety will mean an algebraic scheme over K which is integral. In general we shall use the notations and the terminology of Éléments de Géométrie Algébrique of A. Grothendieck and J. Dieudonné. For instance, if Z is a variety and x ∈ Z is a point, then OZ, x means the local ring of Z at x and mx – the maximal ideal of 0Z, x.

A semialgebraic map $f:X\rightarrow Y$ between two real algebraic sets is called blow-Nash if it can be made Nash (i.e., semialgebraic and real analytic) by composing with finitely many blowings-up with nonsingular centers.

We prove that if a blow-Nash self-homeomorphism $f:X\rightarrow X$ satisfies a lower bound of the Jacobian determinant condition then $f^{-1}$ is also blow-Nash and satisfies the same condition.

The proof relies on motivic integration arguments and on the virtual Poincaré polynomial of McCrory–Parusiński and Fichou. In particular, we need to generalize Denef–Loeser change of variables key lemma to maps that are generically one-to-one and not merely birational.

We illustrate the theory of log abelian varieties and their moduli in the case of log elliptic curves.

We construct explicit examples of weak Hopf algebras (actually face algebras in the sense of Hayashi [H]) via vacant double groupoids as explained in [AN]. To this end, we first study the Kac exact sequence for matched pairs of groupoids and show that it can be computed via group cohomology. Then we describe explicit examples of finite vacant double groupoids.

Barth and others [1], [2], [5] have begun the study of stable algebraic vector bundles of rank 2 on projective space. Maruyama [7] has shown that stable rank 2 bundles have a variety of moduli which is the finite union of quasi-projective varieties.

The aim of this paper is to carry out an explicit construction of CAP representations of $\text{GL}(2)$ over a division quaternion algebra with discriminant two. We first construct cusp forms on such a group explicitly by lifting from Maass cusp forms for the congruence subgroup ${\rm\Gamma}_{0}(2)$. We show that this lifting is nonzero and Hecke-equivariant. This allows us to determine each local component of a cuspidal representation generated by such a lifting. We then show that our cuspidal representations provide examples of CAP (cuspidal representation associated to a parabolic subgroup) representations, and, in fact, counterexamples to the Ramanujan conjecture.

About the behaviour of brownian motion at time point ∞ there are many results by P. Levy and A. Khintchine etc. The method of W. Feller is applicable to a similar discussion about a homogeneous differential process. In this paper we shall study, applying his method, the properties of a homogeneous differential process.

Let Ω be a smoothly bounded pseudoconvex domain in Cn and let A(Ω) denote the functions holomorphic on Ω and continuous on . A point p ∈ bΩ is a peak point if there is a function f ∈ A(Ω) such that f (p) = 1, and | f(z) | < 1 for z ∉ Ω − {p}.