GROUPS, GRAPHS, AND THE HANNA NEUMANN CONJECTURE

Submultiplicativity, an analytic property generalizing the Strengthened Hanna Neumann Conjecture (SHNC) to complexes was proved in [2] assuming the deep-fall property. This in particular implied SHNC. The purpose of this note is to write the proof of the original SHNC and purely in terms of groups and graphs. We also give explicit examples showing that the upper bound in SHNC is sharp.

Property

This article argues that despite its seeming disintegration, property is more vibrant than ever — it is a field that has focused on understanding the formal and informal institutions by which society channels decision-making for scarce resources. Many exciting recent innovations in property theory have arisen through dialogue between US and Commonwealth scholars and legislatures. The article is organized as follows. The first part explains the focus on analytic property theory, which is posed in distinction to a jurisprudential approach. The second part introduces the familiar division of ownership into a trilogy of ideal types: private, commons, and state. The next three parts use this trilogy to show how defining, integrating, and constructing these ideal types can lead to useful innovation in property theory. In sum, property theory scholarship seems to work cyclically — reasoning from real-world contests over scarce resources, to analytic tools that translate these struggles into useful conceptual terms, to jurisprudential debates regarding the rightness of resulting allocations, to practical politics that implement one property regime or another, and then back to new on-the-ground struggles.

On Singular Potential of the Schrödinger Equation

We study the asymptotic behavior of a singular potential that arises under several frequently occurring analytic behaviors of the eigenfunctions (of the Schrödinger eigenvalue problem) within the C2-class of functions. We find that the asymptotic behavior of the singular potential crucially depends on the analytic property of the eigenfunction near the singular point.

Nonreflecting stratifications

Electromagnetic, acoustic, and particle waves are specularly reflected by planar inhomogeneities (stratifications). A brief review is given of the cases in which zero reflection is known to occur. In general these are rare, but stratifications that are symmetric with respect to inversion about their midplane have an analytic property of the reflection amplitude that makes zero reflection a frequent occurrence. Examples are given of the locii of zero reflection in the angle of incidence – thickness plane of some symmetric stratifications. For uniform layers the zero reflection locii can be classified into Malus–Brewster–Green type and interference type, but this distinction does not hold in general.

Bishop's Condition (β)

In 1959, Bishop [4] published a seminal paper in which he studied various types of spectral decompositions or “duality theories” that an arbitrary bounded linear operator on a reflexive Banach space might have. In the course of his investigations, he isolated the following analytic property which he called condition (β).