Normal curvature

As stated before, very small droplets tend to be more spherical than large ones, especially if the capillary length of the liquid is large. The case of very large drops is also relevant and allows the simplification that all curvatures normal to the plane of the paper may be neglected in fig. 1.3 2 S [1.2.10) the... 

Spherical UMEs can be made for gold (16), but are difficult to realize for other materials. Hemispherical UMEs can be achieved by plating mercury onto a microelectrode disk. In these two cases the critical dimension is the radius of curvature, normally symbolized by ro- The geometry of these two types is simpler to treat than that of the disk, but in many respects behavior at a disk is similar to that at a spherical or hemispherical UME with the same tq. 

A crack is a 2-dimensional flaw an area across which the bonds are broken. The boundary of this area is called the crack tip. The curvature (normal to the plane of the crack) at the tip is assumed to be inflnitely sharp in the continuum models but is of atomistic dimensions in real materials. The detailed atomistic structure of a crack tip is unresolved at present (Lawn, 1993). In silicate glasses, a crack tip has a radius of curvature on the order of 0.3 nm which is approximately the size of a single siloxane bridge [=Si-0-Si=]. 

This estimate is remarkably simple, given the complexity of the underlying boundary value problem. The left sides of the expressions in (3.114) are the curvatures normalized by the effective Stoney curvature based on the average film thickness hfb/p. Furthermore, the right sides of (3.114) do not involve the parameter p. In other words, p enters the estimate for curvature only through the effective Stoney curvature based on the amount of film material involved. When b = p, the response is identical to that for a periodic array of parallel cracks. 

In general, this matched curvature normal will not be a suitable candidate density, because we determined a global property of the density (the variance) from a local property (the curvature of the density at a single point). The spread may be much too small. However, it will provide us the basis for finding a suitable density. 

We see that the matched curvature normal is heavier on the lower tail, but it b much lighter on the upper tail. We shall see that this will make it a very bad candidate distribution. 

Use Student s t candidate density that is similar to the matched curvature normai. The way forward is to find a candidate density that is similar shape to the matched curvature normal, but with heavier tails than the target. The Student s t with low degrees of freedom will work very well. The Student s t with 1 degree of freedom will dominate all target distributions. 

Figure 7.1 Four steps of the Gauss-Newton algorithm showing convergence to matched-curvature normal distribution. The left-hand panels are the logarithm of the target and the quadratic that matches the first two derivatives at the value 0 i, and the right-hand panels are the corresponding target and normal approximation.

Figure 7 The traceplot and the histogram (with the taiget density) for the first 1000 draws from the chain using the matched curvature normal candidate distribution.

Figure 8.4 Histograms for the thinned posterior sample of 0o and 0 together with their matched curvature normal approximations.

Figure 8.8 Histograms for the posterior sample of intercept 0o and the slope coefficients /3i,..., 010 together with matched curvature normal approximation.

---