Towards a splitter theorem for internally 4-connected binary matroids III

This paper proves a preliminary step towards a splitter theorem for internally 4-connected binary matroids. In particular, we show that, provided M   or M^?$M^{?}$ is not a cubic Möbius or planar ladder or a certain coextension thereof, an internally 4-connected binary matroid M with an internally 4-connected proper minor N   either has a proper internally 4-connected minor M^′$M^{\prime }$ with an N  -minor such that |E(M)−E(M^′)|?3$|E(M)-E(M^{\prime })|?3$ or has, up to duality, a triangle T and an element e of T   such that M\e$M\backslash e$ has an N-minor and has the property that one side of every 3-separation is a fan with at most four elements.     

Nanoelectromechanical systems

Nanoelectromechanical systems (NEMS) are nano-to-micrometer scale mechanical resonators coupled to electronic devices of similar dimensions. NEMS show promise for fast, ultrasensitive force microscopy and for deepening our understanding of how classical dynamics arises by approximation to quantum dynamics. This article begins with a survey of NEMS and then describes certain aspects of their classical dynamics. In particular, we show that for weak coupling the action of the electronic device on the mechanical resonator can be effectively that of a thermal bath, this despite the device being a driven, far-from-equilibrium system.     

On the intersections of circuits and cocircuits in matroids

Seymour has shown that a matroid has a triad, that is, a 3-element set which is the intersection of a circuit and a cocircuit, if and only if it is non-binary. In this paper we determine precisely when a matroidM has a quad, a 4-element set which is the intersection of a circuit and a cocircuit. We also show that this will occur ifM has a circuit and a cocircuit meeting in more than four elements. In addition, we prove that if a 3-connected matroid has a quad, then every pair of elements is in a quad. The corresponding result for triads was proved by Seymour.     

Atomic-resolution electron energy loss spectroscopy imaging in aberration corrected scanning transmission electron microscopy

The "delocalization" of inelastic scattering is an important issue for the ultimate spatial resolution of innershell spectroscopy in the electron microscope. It is demonstrated in a nonlocal model for electron energy loss spectroscopy (EELS) that delocalization of scanning transmission electron microscopy (STEM) images for single, isolated atoms is primarily determined by the width of the probe, even for light atoms. We present experimental data and theoretical simulations for Ti L-shell EELS in a [100] SrTiO3 crystal showing that, in this case, delocalization is not significantly increased by dynamical propagation. Issues relating to the use of aberration correctors in the STEM geometry are discussed.     

On Crapo's Beta Invariant for Matroids

In this paper we characterize the matroids for which Crapo's beta invariant does not exceed 4. In addition, we relate the beta invariant of a matroid to its chromatic number and to its connectivity.     

Anatomy of a polarization switch of a vertical-cavity semiconductor laser

Using a streak camera we have measured the three Stokes polarization parameters during a polarization switch of a vertical-cavity semiconductor laser. The switch occurs along a corkscrew path on the Poincare sphere and takes on average a few nanoseconds; this value agrees with a theoretical treatment based upon the Fokker-Planck equation.     

Strain-hardening extrusion—II. Analysis of slip-line fields based on experimental flow patterns

A method of constructing slip-line fields based on experimental extrusion flow patterns was described in a previous paper. Such fields provide a basis for constructing representative solutions of strain-hardening flow problems. In this paper, after proving some theoretical results on the structure of such solutions, a method of analyzing slip-line fields is described and some typical results presented. The analysis comprises computation of the velocity solution and construction from it of a set of streamlines for comparison with the original flow pattern; computation of the total strain by integration along the streamlines, its graduation and adjustment to meet the boundary conditions; estimation of the stress field (shear stress k and mean pressure p) and check for consistency; and computation of the forces on the boundary of the plastic region and their resultant.     

Highly selective synthesis of oxabicycloalkanes by indium tribromide-mediated cyclization reactions of epoxyalkenes

The cyclization of epoxyalkenes to oxabicycloalkanes is catalyzed by stoichiometric quantities of indium tribromide which exhibits excellent selectivity giving the oxabicyclic product in high yield in preference to other cyclized or rearrangement products.     

On removable circuits in graphs and matroids

Mader proved that every 2‐connected simple graph G with minimum degree d exceeding three has a cycle C, the deletion of whose edges leaves a 2‐connected graph. Jackson extended this by showing that C may be chosen to avoid any nominated edge of G and to have length at least d − 1. This article proves an extension of Jackson's theorem. In addition, a conjecture of Goddyn, van den Heuvel, and McGuinness is disproved when it is shown that a natural matroid dual of Mader's theorem fails. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 51–66, 1999