Quantitative K-Theory Related to Spin Chern Numbers

We examine the various indices defined on pairs of almost commuting unitary matrices that can detect pairs that are far from commuting pairs. We do this in two symmetry classes, that of general unitary matrices and that of self-dual matrices, with an emphasis on quantitative results. We determine which values of the norm of the commutator guarantee that the indices are defined, where they are equal, and what quantitative results on the distance to a pair with a different index are possible. We validate a method of computing spin Chern numbers that was developed with Hastings and only conjectured to be correct. Specifically, the Pfaffian-Bott index can be computed by the "log method" for commutator norms up to a specific constant

Principal angles and approximation for quaternionic projections

We extend Jordan's notion of principal angles to work for two subspaces of quaternionic space, and so have a method to analyze two orthogonal projections in M_n(A) for A the real, complex or quaternionic field (or skew field). From this we derive an algorithm to turn almost commuting projections into commuting projections that minimizes the sum of the displacements of the two projections. We quickly prove what we need using the universal real C*-algebra generated by two projections.Comment: 11 pages, 4 figures, 4 auxiliary Matlab file

Estimating Norms of Commutators

We find estimates on the norms commutators of the form [f(x), y] in terms of the norm of [x, y] assuming that x and y are contractions in a C*-algebra A, with x normal and with spectrum within the domain of f. In particular we discuss [x^2, y] and [x^(1/2), y] for 0 <=, x <=, 1. For larger values of \delta = \|[x; y]\| we can rigorous calculate the best possible upper bound \|[f(x), y]\| for many f. In other cases we have conducted numerical experiments that strongly suggest that we have in many cases found the correct formula for the best upper bound.Comment: We are posting the next version of this paper at : http://repository.unm.edu/handle/1928/23462. Also posted at http://repository.unm.edu is theMatlab code used to generate example

Incorporating glass transition concepts to explain rice milling-quality reductions during the drying process

Previous research has indicated that while drying rough rice using air temperatures above the glass transition temperature (Tg), head rice yield (HRY) reductions are incurred if a state transition occurs when severe intra-kernel moisture content (MC) gradients are present. State transitions can occur by extended drying using high-temperature air or by cooling kernels below Tg before sufficient tempering has occurred. The objectives of this experiment were to determine the maximum MC removal per initial drying pass and the associated tempering durations required to prevent HRY reduction. Two long-grain cultivars, ‘Francis’ and ‘Wells’, at two harvest moisture contents (HMC) were used. Samples were dried with air conditions of either 60°C/17% RH or 50°C/28% RH for various durations to create a range of intra-kernel MC gradients and were subsequently tempered in sealed bags for durations ranging from 0 to 160 min. After tempering, samples were cooled to cause a state transition, and then slowly dried to 12.2% MC. Samples were then milled to determine HRY. Control samples were dried at 21°C/60% RH. Results showed that the amount of moisture that could be removed in the initial drying pass was directly related to the HMC and the drying air condition. The tempering duration required to prevent HRY reductions increased with the amount of MC removed from the kernel in a drying pass. The HRY reduction patterns concur with a hypothesis that explains fissure formation during the drying process based on the Tg of rice kernels