A Method for Accurate Calculation ofB1Fields in Three Dimensions. Effects of Shield Geometry on Field Strength and Homogeneity in the Birdcage Coil

A method for calculatingB₁field strength and homogeneity as functions of radiofrequency shield geometry is presented. The method requires use of three-dimensional finite-element analysis, birdcage-coil theory, and antenna-array theory. Calculations were performed for a 12-element birdcage coil (19 cm diameter, 21 cm length) at 125 MHz. CalculatedB₁field strengths and homogeneities for the coil in 25 different shields and in no shield are given. For configurations where the shield is longer than the coil, bothB₁field strength and homogeneity decrease as shield diameter decreases or as shield length increases. In configurations where the shield is shorter than the coil and has a diameter of 25.6 cm,B₁homogeneity is greater than in an unshielded coil.B₁field strength was measured experimentally at 125 MHz in a birdcage coil of the same geometry as the model within shields of four different diameters. Calculated results very closely matched experimental measurement.     

Conceptual Limitations in Curricular Presentations of Area Measurement: One Nation’s Challenges

Research has found that elementary students face five main challenges in learning area measurement: (1) conserving area as a quantity, (2) understanding area units, (3) structuring rectangular space into composite units, (4) understanding area formulas, and (5) distinguishing area and perimeter. How well do elementary mathematics curricula address these challenges? A detailed analysis of three U.S. elementary textbook series revealed systematic deficits. Each presented area measurement in strongly procedural terms using a shared sequence of procedures across grades. Key conceptual principles were infrequently expressed and often well after related procedures were introduced. Particularly weak support was given for understanding how the multiplication of lengths produces area measures. The results suggest that the content of written curricula contributes to students’ weak learning of area measurement.     

Causal Algebras on Chain Event Graphs with Informed Missingness for System Failure

Graph-based causal inference has recently been successfully applied to explore system reliability and to predict failures in order to improve systems. One popular causal analysis following Pearl and Spirtes et al. to study causal relationships embedded in a system is to use a Bayesian network (BN). However, certain causal constructions that are particularly pertinent to the study of reliability are difficult to express fully through a BN. Our recent work demonstrated the flexibility of using a Chain Event Graph (CEG) instead to capture causal reasoning embedded within engineers’ reports. We demonstrated that an event tree rather than a BN could provide an alternative framework that could capture most of the causal concepts needed within this domain. In particular, a causal calculus for a specific type of intervention, called a remedial intervention, was devised on this tree-like graph. In this paper, we extend the use of this framework to show that not only remedial maintenance interventions but also interventions associated with routine maintenance can be well-defined using this alternative class of graphical model. We also show that the complexity in making inference about the potential relationships between causes and failures in a missing data situation in the domain of system reliability can be elegantly addressed using this new methodology. Causal modelling using a CEG is illustrated through examples drawn from the study of reliability of an energy distribution network.     

Experimental observation of lattice distortions due to a flux line lattice in niobium

A polarized neutron scattering investigation of the flux line lattice in the type-II superconductor niobium is reported. A modulation of the nuclear lattice has been detected, and the magnitude of the first Fourier component of the lattice distortion established relative to the magnitude of the magnetic scattering. This constitutes the first experimental observation of lattice distortions due to the presence of magnetic flux lines within the bulk of a type-II superconductor. Using a simple microscopic model the lattice distortion in niobium is estimated. A new mechanism is suggested for the coupling of the flux line lattice to the crystallographic lattice. The experimental technique opens up the possibility of investigating the microscopic mechanism of flux line - nuclear lattice interactions, in particular the pinning of flux lines within the bulk of a type-II superconductor. Received: 8 August 1997 / Accepted: 16 October 1997     

Effective pair potentials in fluids in the presence of three-body forces. II

Earlier calculations are extended to the second order in density thus providing the non-additive contributions to the fourth virial coefficient in the presence of a weak three-body force given by the triple-dipole dispersion potential of Axilrod and Teller. We compare the Percus-Yevick approximations of Rushbrooke and Silbert and of Rowlinson at the level of the fourth virial coefficient and find that the former is more accurate. We also compare our calculations of the effective pair potential of liquid argon with the results of Mikolaj and Pings obtained from X-ray diffraction measurements.     

Perturbation theory for the radial distribution function

Perturbation theory is used to consider expansions for the radial distribution function, g ₂(r), of a fluid with a soft core. We consider the Lennard-Jones (12, 6) potential and divide it into repulsive and attractive regions. In the repulsive region we expand the function exp (β u(r))g ₂(r) about a hard sphere value. For the first-order contribution of the attractive region we consider a simple approximation to the exact analytical expression. The resulting g ₂(r) is accurate at densities below about ρσ ³=0·5.     

Atomic interactions in neon and helium

The first two quantum corrections to the second virial coefficients of the Smith-Thakkar potential are calculated. Parameters for neon and helium, gases in which quantum effects are important, are then determined by fitting to semiempirical dispersion coefficients and experimental second virial coefficients. Viscosity coefficients for both gases and vibrational energy level spacings for the neon dimer are calculated as independent tests of the potentials. Overall agreement with experiment is excellent for neon and moderate for helium. The previously determined parameters for argon are found to be only very slightly perturbed by the inclusion of quantum corrections in the calculated second virial coefficients.     

The numerical approximation of a class of finite integrals

The object of this paper is to derive a method for the numerical approximation of integrals of the form $$\int_{ - 1}^1 {w(x)f(x)(x^2 + \mathop {a^2 }\limits^ - )^{ - 1} } dx$$ where w(x)=(1?x ²)^±1/2 or ((1?x)/(1+x))^1/2.     

On subgroups of finite index in compact Hausdorff groups

Let G be a compact Hausdorff group and n a positive integer. It is proved that all subnormal subgroups of G of index dividing n are open if and only if there are only finitely many such subgroups, and that all subgroups of finite index in G are open if and only if there are only countably many such subgroups.