Triangles III: Complex triangle functions

Summary This paper is the third in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first two papers, we looked at triangle shapes and triangle coordinates. In this paper, we look at the triangle coordinates of the special points of a triangle, and show that they are functions of its shape. We then show how these functions can be used to prove theorems about triangles, and to gain some insight into what makes a special point of a triangle a centre.     

Comparison of agarose and cross-linked protein gels as magnetic resonance imaging phantoms

Measurements of the magnetic field dependence of spin-lattice relaxation rates and the response of the water-proton signal intensity to off-resonance radio frequency fields show that the commonly used agarose phantom provides a less faithful representation for the magnetic response of tissue than does a cross-linked protein system. The origin of these differences lies in the structure and intramolecular dynamics of the macromolecular system used to make the gel. These distinctions will also cause differences in the magnetic response of the water spin system when paramagnetic relaxation agents or contrast agents are incorporated. Use of a thermally cross-linked bovine serum albumin phantom is suggested.     

Casimir self-stress on a perfectly conducting spherical shell

The Casimir stress on a perfectly conducting uncharged sphere, due to occurrence of fluctuations in the electromagnetic field, is calculated using a source theory formulation. Two independent methods are employed: we compute (1) the total Casimir energy inside and outside the sphere, and (2) the radial component of the stress tensor on the surface. It is necessary to exercise care in allowing the field points to overlap; a correct limiting procedure supplies a “cutoff” in the frequency integration. In spite of numerous technical improvements, the result of Boyer, that the self-stress is repulsive (and not attractive as Casimir hoped), is confirmed unambiguously. The magnitude of the Casimir energy of a sphere of radius a is found, by numerical and analytic techniques, to be $\text{E = (}\text{h}\text{?}\text{c}\text{2a}\text{)(0.09235)}$, also in agreement with the very recent result of Balian and Duplantier.     

Inverse vibrational distributions of photofragments

X-ray diffraction data from a Cr₂(SO₄)₃ solution were examined. A peak at 3.25 Å in the correlation function reveals the formation of inner chromium-sulphate complexes. Least-squares refinements of the i(s) curve are consistent with a structural unit in which the CrOS angle is 136°. Each sulphate ion interacts with approximately eight water molecules.     

A study of nitric oxide adsorbed on nickel oxide,cobalt oxide,and graphite by X-ray photoelectron spectroscopy

The nitrogen 1s binding energy of nitric oxide adsorbed on nickel oxide, cobalt oxide and graphite was measured at several temperatures by X-ray photoelectron spectroscopy. For the oxides, a temperature dependent range of binding energies was observed, but for graphite the nitrogen 1s binding energy was independent of temperature. The values obtained suggest that significant back donation of electrons occurs from the oxides to the adsorbed nitric oxides, but no back donation occurs from graphite to adsorbed nitric oxide. However, adsorption did not cause changes in the binding energy of substrate core electrons. It is believed that unless the mean free path of electrons is short, the total photoelectron signal will not reveal changes in binding energy of electrons from substrate atoms at the surface.     

Calogero-Moser and Toda systems for twisted and untwisted affine Lie algebras

The elliptic Calogero-Moser Hamiltonian and Lax pair associated with a general simple Lie algebra G are shown to scale to the (affine) Toda Hamiltonian and Lax pair. The limit consists in taking the elliptic modulus τ and the Calogero-Moser couplings m to infinity, while keeping fixed the combination M = m e^iδθτ for some exponent δ. Critical scaling limits arise when 1/δ equals the Coxeter number or the dual Coxeter number for the untwisted and twisted Calogero-Moser systems respectively; the limit consists then of the Toda system for the affine Lie algebras G⁽¹⁾ and (G⁽¹⁾)^V. The limits of the untwisted or twisted Calogero-Moser system, for δ less than these critical values, but non-zero, consists of the ordinary Toda system, while for δ = 0, it consists of the trigonometric Calogero-Moser systems for the algebras G and G^V respectively.