In this paper we discuss work on graphs defined in terms of alliances between countries. We will use scan statistics to investigate years in which there are an unusual number of agreements, not just between one country and its allies, but amongst the allies themselves. This is related to work on email “chatter” discussed in Priebe et al. [2006]. In this section we will lay out the basic graph t...

I state without proof a few results concerning the local properties of smooth maps. Section 11, beginning at page 89, of our class notes contains more information, examples, and proofs. All functions and maps will be assumed here to be smooth (infinitely continuously differentiable) but the discussion in the notes is more general. By a neighborhood of a point I mean an open set containing the p...

I. Contractible manifolds and aspherical manifolds. Suppose that M n is a compact, contractible n-manifold with boundary. These assumptions imply that the boundary of M has the homology of an (n-l)-sphere; however, they do not imply that it is simply connected. If n ~ 3, then for M n to be homeomorphic to a disk it is obviously necessary that ~M be simply connected. For n ~ 5 this condition is ...

The paper puts forth a theory of choice functions in a neat way connecting it to a theory of extensive operators and neighborhood systems. We consider four classes of heritage choice functions satisfying the conditions M, N, W, and C.

A possible attack on the fifth problem of Hilbert is to demonstrate the existence of one-parameter subgroups in any locally Euclidean group. I t is known that, provided there are no "small" subgroups, some one-parameter subgroups exist. One would like to prove, however, that in a suitable neighborhood of the identity, every element is on one and only one one-parameter subgroup. If this is true,...

In 1979 Wertheimer and Leeper developed their W-L wire code, based on power line construction, size, and distance from a residence, as a surrogate for residential magnetic field exposure (1). This W-L wire code was applied to data from a 1988 Denver study, and an association was reported between "high" W-L wire code and childhood cancer (2). The authors of "Childhood Cancer in Relation to a Mod...

Let M be a compact C1 manifold which is invariant and normally hyperbolic with respect to a C1 semiflow in a Banach space. Then in an -neighborhood of M there exist local C1 center-stable and center-unstable manifolds W cs( ) and W cu( ), respectively. Here we show that W cs( ) and W cu( ) may each be decomposed into the disjoint union of C1 submanifolds (leaves) in such a way that the semiflow...

The concept of a homoclinic tube was introduced by Silnikov 14) in a study on the structure of the neighborhood of a homoclinic tube asymptotic to an invariant torus σ under a diffeomorphism F in a finite dimensional phase space. The asymptotic torus is of the saddle type. The homoclinic tube consists of a doubly infinite sequence of tori {σj , j = 0,±1,±2, · · ·} in the transversal intersectio...

The aim of this study was to examine the effect built environment (BE) elements on older people’s (OP) leisure-time physical activity (LTPA) and walking (LTW) level in city Ningbo. Older participants numbering 312 were randomly recruited using cross-sectional survey 2019. International Physical Activity Questionnaire-short version (IPAQ-S) Neighborhood Environment Walkability Scale—abbreviated ...