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 $\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.

1. Let D be a positive square-free integer. Throughout this note we shall use the following notations;

d = d(D): the discriminant of ,

t0, u0: the least positive solution of Pell’s equation t2 — du2 = 4,

A submanifold M (connected but not necessary compact) of a Euclidean m-space Em is said to be of finite type if each component of its position vector X can be written as a finite sum of eigenfunctions of the Laplacian Δ of M, that is,

where X0 is a constant vector and ΔXt = λtXt, t = 1, 2, · · ·, k. If in particular all eigenvalues {λ1, λ2, · · ·, λk are mutually different, then M is said to be of k-type (cf. [3] for details).

Let k be a positive integer greater than 1, and let X(n) be a real primitive character modulo k, The series

can be divided into groups of k consecutive terms.

We give an explicit construction of the Ree groups of type G2 as groups acting on mixed Moufang hexagons together with detailed proofs of the basic properties of these groups contained in the two fundamental papers of Tits on this subject (see [7] and [8]). We also give a short proof that the norm of a Ree group is anisotropic.

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

The holomorphy conjecture roughly states that Igusa’s zeta function associated to a hypersurface and a character is holomorphic on $\mathbb{C}$ whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article, we prove the holomorphy conjecture for surface singularities that are nondegenerate over $\mathbb{C}$ with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volumes (which appear in the formula of Varchenko for the zeta function of monodromy) of the faces in a simplex in arbitrary dimension. We then study some specific character sums that show up when dealing with false poles. In contrast to the context of the trivial character, we here need to show fakeness of certain candidate poles other than those contributed by $B_{1}$-facets.

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 consider in the 2 dimensional space with the coordinate (x,y). Let Γ be a segment of the y-axis containing the origin in its interior and let Ω be a domain whose boundary contains Γ.

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).

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 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

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.

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.

Resolutive compactification and harmonic measures. Let R be an open Riemann surface. A compact Hausdorff space R* containing R as its dense subspace is called a compactification of R and the compact set Δ = R* -R is called an ideal boundary of R. Hereafter we always assume that R does not belong to the class OG. Given a real-valued function f on Δ, we denote by the totality of lower bounded superharmonic (resp. upper bounded subharmonic) functions sonis satisfying

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.

To a pair $P$ and $Q$ of finite posets we attach the toric ring $K[P,Q]$ whose generators are in bijection to the isotone maps from $P$ to $Q$. This class of algebras, called isotonian, are natural generalizations of the so-called Hibi rings. We determine the Krull dimension of these algebras and for particular classes of posets $P$ and $Q$ we show that $K[P,Q]$ is normal and that their defining ideal admits a quadratic Gröbner basis.