Affine Reductive Spaces of Small Dimension and Left A-Loops

In this paper we determine the at least 4-dimensional affine reductive homogeneous manifolds for an at most 9-dimensional simple Lie group or an at most 6-dimensional semi-simple Lie group. Those reductive spaces among them which admit a sharply transitive differentiable section yield local almost differentiable left A-loops. Using this we classify all global almost differentiable left A-loops L having either a 6-dimensional semi-simple Lie group or the group as the group topologically generated by their left translations. Moreover, we determine all at most 5-dimensional left A-loops L with as the group topologically generated by their left translations.     

Spaces of analytic functions of Hardy-Bloch type

For 0<p≤∞ and 0<q≤∞, the space of Hardy-Bloch type ℬ(p,q) consists of those functionsf which are analytic in the unit diskD such that (1−r)M _p (r,f′)⊂L ^q (dr/(1−r)). We note that ℬ(∞,∞) coincides with the Bloch space ℬ and that ℬ⊂ℬ(p,∞) for allp. Also, the space ℬ(p,p) is the Dirichlet spaceD _p−1 ^p . We prove a number of results on decomposition of spaces with logarithmic weights which allow us to obtain sharp results about the mean growth of the ℬ(p,q). In particular, we prove that iff is an analytic function inD and 2≤p<∞, then the conditionM _p (r,f′)=O((1−r)⁻¹), asr→1, implies that

Monotonicity properties of operations on generalized topologies

Summary In the paper [5], several operations on generalized topologies are considered. They are not monotone in general, but an old result on monotonicity may be sharpened.     

On Dupin Hypersurfaces in R5 Parametrized by Lines of Curvature

We provide a complete local characterization of Dupin hypersurfaces in R⁵, with four distinct principal curvatures, parametrized by lines of curvature. Such hypersurfaces are given in terms of the principal curvatures and vector valued functions that describe plane curves. We include explicit examples of such Dupin hypersurfaces which are irreducible and have nonconstant Lie curvature.     

Radii of starlikeness and convexity of a cross-product of Bessel functions

To illustrate vertical and horizontal equidistributions of Kloosterman sums, we prove two relevant central limit theorems by some analytic and combinatorial properties of Chebyshev polynomials and independence of monodromy groups of Kloosterman sheaves. The first one supports a short interval version of Katz’s vertical Sato–Tate distribution theorem, and the second one coincides with the horizontal conjecture of Katz in the statistical sense.     

A simple method to find out when an ordinary differential equation is separable

In many statistical discussions, especially in data analysis, the idea of polynomials plays a key role. For example, Dwyer [1] employed polynomials to express factorial moments of discrete distribution in terms of cumulative totals. Traditionally, polynomials are derived using the difference operator method (see [2], p. 134]). In this article, using the differential equation approach as an alternative method, we obtain generalized exponential and logarithmic polynomials, and find their special cases appearing in statistical signal‐noise models.     

Generalized theta functions and the Deuring polynomial

In this paper we present a calculation to study a relationship between the generalized theta function and the Deuring polynomial.     

Commutative orders revisited

This article studies commutative orders, that is, commutative semigroups having a semigroup of quotients. In a commutative order \(S\) , the square-cancellable elements \(\mathcal {S}(S)\) constitute a well-behaved separable subsemigroup. Indeed, \(\mathcal {S}(S)\) is also an order and has a maximum semigroup of quotients \(R\) , which is Clifford. We present a new characterisation of commutative orders in terms of semilattice decompositions of \(\mathcal {S}(S)\) and families of ideals of \(S\) . We investigate the role of tensor products in constructing quotients, and show that all semigroups of quotients of \(S\) are homomorphic images of the tensor product \(R\otimes _{\mathcal {S}(S)} S\) . By introducing the notions of generalised order and semigroup of generalised quotients, we show that if \(S\) has a semigroup of generalised quotients, then it has a greatest one. For this we determine those semilattice congruences on \(\mathcal {S}(S)\) that are restrictions of congruences on \(S\) .