Some Variants of Strong Normality in Closure Spaces Generated via Relations

In this paper, some variants of strongly normal closure spaces obtained by using binary relation are introduced, and examples in support of existence of the variants are provided by using graphs. The relationships that exist between variants of strongly normal closure spaces and covering axioms in absence/presence of lower separation axioms are investigated. Further, closure subspaces and preservation of the properties studied under mapping are also discussed.

On Strong Normality

We introduce the concept of strong normality by defining strong normal numbers and provide various properties of these numbers, including the fact that almost all real numbers are strongly normal.

Kernel Density Estimation and Metropolis-Hastings Sampling in Process Capability Analysis of Unknown Distributions

Strong normality assumption is associated with widely used process capability indices such as cp, cpk. Violation of the assumption will mislead the interpretation in applications. A nonparametric method is proposed for density estimation of any unknown distribution. Kernels are used for density estimation and metropolis-hastings (M-H) algorithm is adopted to generate samples from the density. M-H sampling provides a tool to accommodate different kernel functions and flexibility of future extension to multivariate cases. Conformity (yield) based indices (yp, y) are adopted to replace cp, cpk. These indices can be conveniently assessed by the proposed kernel density based M-H algorithm (K-M-H). The method is validated by several simulation case studies.

On Strongly Normal Functions

Loosely speaking, a function (meromorphic or harmonic) from the hyperbolic disk of the complex plane to the Riemann sphere is normal if its dilatation is bounded. We call a function strongly normal if its dilatation vanishes at the boundary. A sequential property of this class of functions is proved. Certain integral conditions, known to be sufficient for normality, are shown to be in fact sufficient for strong normality.

On normal and strongly normal lattices

In this paper we will investigate the properties of normality and strong normality of lattices and their relationships to zero-one measures. We will eventually establish necessary and sufficient conditions for lattices to be strongly normal. These properties are then investigated in the case of separated lattices.

General solutions depending algebraically on arbitrary constants

In his famous lectures [7] Painlevé investigates general solutions of algebraic differential equations which depend algebraically on some of arbitrary constants. Although his discussions are beyond our understanding, the rigorous and accurate interpretation to make his intuition true would be possible. Successful accomplishments have been done by some authors, for example, Kimura [1], Umemura [8, 9]. From differential algebraic viewpoint in [5] the author introduces the notion of rational dependence on arbitrary constants of general solutions of algebraic differential equations, and in [6] clarifies the relation between it and the notion of strong normality. Here we aim at generalizing to higher order case the result in [4] that in the first order case solutions of equations depend algebraically on those of equations free from moving singularities which are determined uniquely as the closest ones to the given. Part of our result can be seen in [7].