Let w = f(z) be a quasiconformal mapping, in the sense of Pfluger [5]-Ahlfors [1], with maximal dilatation K, which will be simply referred to a K-QC mapping. As is well known, any K-QC mapping w = f(z) of Im z > 0 onto Im w > 0 can be extended to a homeomorphism from Im z ≧ 0 onto Im w ≧ 0 and hence it transforms any set of logarithmic capacity zero on Im z = 0 into a set with the same property on Im w = 0.

Let X be a complex n-dimensional reduced analytic space with isolated singular point x0, and with a strongly plurisubharmonic function ρ : X → [0, ∞) such that ρ(x0) = 0. A smooth Kähler form on X \ {x0} is then defined by i∂∂ρ. The associated metric is assumed to have -curvature, to admit the Sobolev inequality and to have suitable volume growth near x0. Let E → X \ {x0} be a Hermitian-holomorphic vector bundle, and ξ a smooth (0, 1)-form with coefficients in E. The main result of this article states that if ξ and the curvature of E are both then the equation has a smooth solution on a punctured neighbourhood of x0. Applications of this theorem to problems of holomorphic extension, and in particular a result of Kohn-Rossi type for sections over a CR-hypersurface, are discussed in the final section.

In this paper we compute and continue meromorphically to the entire complex plane the zeta functions of twisted quaternionic Shimura varieties of dimension 3. The twist of the quaternionic Shimura varieties is done by a mod ℘ representation of the absolute Galois group.

In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables by starting from the data of an internally Calabi–Yau algebra, which becomes the endomorphism algebra of a cluster-tilting object in the resulting category. In this paper, we construct appropriate internally Calabi–Yau algebras for cluster algebras with polarized principal coefficients (which differ from those with principal coefficients by the addition of more frozen variables) and obtain Frobenius categorifications in the acyclic case. Via partial stabilization, we then define extriangulated categories, in the sense of Nakaoka and Palu, categorifying acyclic principal coefficient cluster algebras, for which Frobenius categorifications do not exist in general. Many of the intermediate results used to obtain these categorifications remain valid without the acyclicity assumption, as we will indicate, and are interesting in their own right. Most notably, we provide a Frobenius version of Van den Bergh’s result that the Ginzburg dg-algebra of a quiver with potential is bimodule $3$-Calabi–Yau.

S. C. Kleene developed the theory of recursive functionals of finite types in Kleene [5]. He proved that a set A of natural numbers is recursive in E if and only if A is hyperarithmetical, where E is the type 2 object defined by

We consider a general twisted shift-invariant system, $V^{t}(\mathcal {A})$, consisting of twisted translates of countably many generators and study the problem of obtaining a characterization for the system $V^{t}(\mathcal {A})$ to form a frame sequence or a Riesz sequence. We illustrate our theory with some examples. In addition to these results, we study a dual twisted shift-invariant system and also obtain an orthonormal sequence of twisted translates from a given Riesz sequence of twisted translates.

Let $\Lambda $ be a finite-dimensional algebra. A wide subcategory of $\mathsf {mod}\Lambda $ is called left finite if the smallest torsion class containing it is functorially finite. In this article, we prove that the wide subcategories of $\mathsf {mod}\Lambda $ arising from $\tau $-tilting reduction are precisely the Serre subcategories of left-finite wide subcategories. As a consequence, we show that the class of such subcategories is closed under further $\tau $-tilting reduction. This leads to a natural way to extend the definition of the “$\tau $-cluster morphism category” of $\Lambda $ to arbitrary finite-dimensional algebras. This category was recently constructed by Buan–Marsh in the $\tau $-tilting finite case and by Igusa–Todorov in the hereditary case.

We establish certain deLeeuw type theorems for Littlewood-Paley functions. By these theorems, we know that the boundedness of a Littlewood-Paley function on ℝn is equivalent to the boundedness of its corresponding Littlewood-Paley function on the torus Tn.

The following small remark on the relationship of the (quasi-)regularity and the zero-divisor property may be more or less known but does not seem to the writer to have been explicitly stated in a literature.

This article is a continuation and completion of our study in the previous one [1] in which we gave the defining equations of abelian varieties projectively embedded by theta functions of level three. These defining equations are nothing else the canonical generators of theta relations for theta functions with ⅓—characteristics. In the present article the canonical generators of 3 theta relations will be given for the wider class of theta functions with rational characteristics.

We prove a multiplicity result of 2π-periodic solutions for certain weakly coupled system of ordinary differential equations with real parameters. The proofs are based on differential inequalities and coincidence degree.

Using Frobenius automorphisms ingeniouslly, S. Lang has established an elegant theory of unramified class fields of function fields in several variables over finite fields [2]. As an application of class field theory and theory of reduction he has proved that any separable unramified abelian extension of a function field of one variable comes from a pull back of a separable ingeny of its jacobian variety [3].

In this paper we prove the fundamental exact sequence in (co)homology for non-normal subgroups announced in our previous note [8]: Let H be a subgroup of a group G. If M is a G-module and if, for a natural number n, Hm(U, M) = 0 for m = 1, …, n - 1 and for every subgroup U of H which is an intersection of conjugates of H in G, then we have an exact sequence

the significance of the maps and the group Hn(H, M, M)1 will be explained below. (We have a dual result for cohomology groups).

Let w = w(z) be any K-quasiconformai mapping (in the sense of Pfluger-Ahlfors) of a domain D in the z-plane into the w-plane. Since w = w(z) is a measurable mapping (vid. Bers [1]), it transforms any set of Hausdorffs 2-dimensional measure zero in D into such another one. However, A. Mori [5] showed that for 0 < α ≤ 2, any set of -dimensional measure zero in a Jordan domain D is transformed by w = w(z)into a set of α-dimensional measure zero.

The well-known Thullen-Remmert-Stein’s theorem ([9], [7]) asserts that, for a domain D in CN and an n-dimensional irreducible analytic set S in D, a purely n-dimensional analytic set A in D — S has an essential singularity at any point in 5 if A has at least one essential singularity in S. In [1], E. Bishop generalized this to the case that A has the boundary of capacity zero in his sense. Afterwards, in [8], W. Rothstein obtained more precise informations on the essential singularities of A under the assumption dim A = 1. The main purpose in this paper is to generalize these Rothstein’s results to the case of arbitrary dimensional analytic sets.

In our previous paper [1], we proved that any Cantor set E with successive ratios ξn satisfying the condition

is a Picard set, that is, each single-valued meromorphic function with E as the set of essential singularities takes on every value infinitely often in any neighborhood of each singularity with the possible exception of two values.

Let G be a locally compact connected group, and let A (G) be the group of all continuous automorphisms of G. We shall introduce a natural topology into A(G) as previously (i.e. the topology of uniform convergence in the wider sense.) When the component of the identity of A(G) coincides with the group of inner automorphisms, we shall call G complete. The purpose of this note is to prove the following theorem and give some applications of it.

Generalizing the concepts of Stanley-Reisner and affine monoid algebras, one can associate to a rational pointed fan Σ in ℝd the ℤd-graded toric face ring K[Σ]. Assuming that K[Σ] is Cohen-Macaulay, the main result of this paper is to characterize the situation when its canonical module is isomorphic to a ℤd-graded ideal of K[Σ]. From this result several algebraic and combinatorial consequences are deduced. As an application, we give a relation between the cleanness of K[Σ] and the shellability of Σ.

We prove an analog of the classical Hartogs extension theorem for CR L2 functions defined on boundaries of certain (possibly unbounded) domains on coverings of strongly pseudoconvex manifolds. Our result is related to a question formulated in the paper of Gromov, Henkin and Shubin [GHS] on holomorphic L2 functions on coverings of pseudoconvex manifolds.

We treat the sum , where α and γ are real with α positive. This sum was treated first by Hardy and Littlewood [4], and after them, by Behnke [1] and [2], Mordell [9], Wilton [11] and others. The reader will find its history in [7] and in the comments of the Collected Papers [4]. Here we show that the sum can be expressed explicitly, together with an error term O(N1/2), using the regular continued fraction expansion of α. As the statements have complications we will divide them into two theorems. In the followings all letters except ϑ, i, σ, ζ, χ and those in 3° are real, N is a positive real, and always k, n, a, A, B, C, D and E denote integers. The author expresses his thanks to Professor Tikao Tatuzawa and Professor Tomio Kubota for their encouragements.