I. Introduction. The purpose of this paper is to discuss functions defined on the continuous sample paths of Gaussian Markov processes which serve as Green’s functions for pairs of generalized Schroedinger equations. The results extend the author’s earlier paper [2] to a forward time version, and consider different boundary conditions.

In a Bourbaki seminary note, La Théorie des Fonctions Thêta, A. Weil has discussed two fundamental theorems of the general theory of Theta functions. The first, due to H. Poincaré, was proved very skilfully in the note by means of harmonic integrals on a torus and the second, due to Frobenius, was treated by the systematic use of the notion of analytic structure.

The relationship between interpolation and separation properties of hypersurfaces in Bargmann–Fock spaces over $\mathbb{C}^{n}$ is not well understood except for $n=1$. We present four examples of smooth affine algebraic hypersurfaces that are not uniformly flat, and show that exactly two of them are interpolating.

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).

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

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.

Soit f(z) une fonction algébroïde transcendante à n(≧2) branches dans le plan |z| < ∞ définie par une équation irréductible

Matsumoto proved in, [Prime end rotation numbers of invariant separating continua of annular homeomorphisms, Proc. Amer. Math. Soc. 140(3) (2012), 839–845.] that the prime end rotation numbers associated to an invariant annular continuum are contained in its rotation set. An alternative proof of this fact using only simple planar topology is presented.

The lower central series of the absolute Galois group of a field is obtained by iterating the process of forming the maximal central extension of the maximal nilpotent extension of a given class, starting with the maximal abelian extension. The purpose of this paper is to give a cohomological description of this central series in case of an algebraic number field. This description is based on a result of Tate which states that the Schur multiplier of the absolute Galois group of a number field is trivial. We are in a profinite situation throughout which requires some functorial background especially for treating the dual of the Schur multiplier of a profinite group. In a future paper we plan to apply our results to construct a nilpotent reciprocity map.

The purpose of this paper is to determine left-invariant vector fields on a Lie group G with a left-invariant Riemannian metric which induces C- flows on G.

This paper contains part of the author’s Ph.D. dissertation directed by Frederick Bagemihl at Wayne State University. The research was supported by a grant from the Michigan Institute of Science and Technology.

Alice Roth has made an extensive study of entire meromorphic functions with prescribed behavior along half rays emanating from the origin (6). The question arose whether analogous results could be found for functions meromorphic in the unit disk with the same behavior prescribed along an exhaustive class of spirals emanating from the origin. In this paper, I present a class of spirals which satisfactorily fills this role. However, I make no claim to the effect that only this class will suffice.

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.

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.

Let P+(3) and P+(3) be the 3-dimensional space-time Poincaré group and the Poincaré subsemigroup, that is, P(3) = R3 × sSU(1, 1) and P+(3) = V+(3)=SSU(1, 1) where The multiplication is defined by the formula (x, g)(x′, g′) = (x + g*−1x′g−1, gg′) for x, x′ ∈ R3 and g, g′ ∈ SU(l, 1). Here x = (x0, x1, x2) is identified with the matrix

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.

There are various generalizations of Hopf’s brilliant theorem, which may be stated, as newly formulated by Alexandroff; all the homotopy classes of the mappings of a compact Hausdorff space X with dim X≦n into an n-sphere Sn are in a (1-1) -correspondence with the elements of the n-dimensional Čech cohomology group Hn(X) with integer coefficients.

Let be a smooth foliated manifold, and the Lie algebra of all leaf-tangent vector fields on M.

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.

The binary quadratic diophantine equation

is of interest in the class number problem for real quadratic number fields and was studied in recent years by several authors (see [4], [5], [2] and the literature cited there).