Optimal strong stationary times for random walks on the chambers of a hyperplane arrangement

This paper studies Markov chains on the chambers of real hyperplane arrangements, a model that generalizes famous examples, such as the Tsetlin library and riffle shuffles. We discuss cutoff for the Tsetlin library for general weights, and we give an exact formula for the separation distance for the hyperplane arrangement walk. We introduce lower bounds, which allow for the first time to study cutoff for hyperplane arrangement walks under certain conditions. Using similar techniques, we also prove a uniform lower bound for the mixing time of Glauber dynamics on a monotone system.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1605.0833

Cutoff for the cyclic adjacent transposition shuffle

We study the cyclic adjacent transposition (CAT) shuffle of $n$ cards, which is a systematic scan version of the random adjacent transposition (AT) card shuffle. In this paper, we prove that the CAT shuffle exhibits cutoff at $\frac{n^3}{2 \pi^2} \log n$, which concludes that it is twice as fast as the AT shuffle.Comment: 26 pages, 3 figure

Cutoff for random to random card shuffle

In this paper, we use the eigenvalues of the random to random card shuffle to prove a sharp upper bound for the total variation mixing time. Combined with the lower bound due to Subag, we prove that this walk exhibits cutoff at $\frac{3}{4} n \log n - \frac{1}{4}n\log\log{n}$ with window of order $n$, answering a conjecture of Diaconis

The study of aluminium anodes for high power density AL-air batteries with brine electrolytes

In this thesis aluminium alloys containing small additions of both tin (~ 0.1 wt %) and gallium (~ 0.05 wt %) dissolve anodically at high rates in brine media; at room temperature, current densities > 0.2 A cm-2 can be obtained at potentials close to the open circuit potential, ~ -1.5 V vs SCE. Alloys without both tin and gallium do not dissolve at such a negative potential. The tin exists in the alloys as a second phase, typically as ~ 1 ?m inclusions throughout the aluminium structure. Anodic dissolution leads to rounded pits around the tin inclusions. The pits are different in structure from the crystallographic pits observed with Al and other alloys. Clearly, the AlMgSnGa alloys dissolve by a different mechanism. Although the distribution of the gallium in the alloy could not be established, it is essential to the formation of these pits and maintaining dissolution. In addition to the composition, mechanical working and heat treatment influence both the stability of the alloys to open circuit corrosion and the overpotential for high rate dissolution, factors critical to battery performance. The correlation between performance and alloy microstructure has been investigated. Imaging with a high speed camera with a resolution of 10 – 20 ?m was used to observe the dissolution of AlMgSnGa alloys. Using microelectrodes with only a few Sn inclusions in their surface, allows confirmation that hydrogen evolution occurs only from the Sn inclusions and also showed that the evolution of H2 is not continuous. Therate of H2 evolution correlates with shifts in potential between - 1.5 V and much less negative potentials. The performance of a laboratory Al-air battery with 2 M NaCl electrolyte was limited by both the performance of the O2 cathode and the extent of dissolution of the alloy. Using a cell with a low electrolyte volume/surface area ratio, dissolution of the anode stopped after the passage of 1000 C cm-2 due to a high impedance, thick film of crystals clinging to the surface. Removal of this film allowed the dissolution to recommence. The charge limitation depends on cell design but a high charge density would be difficult to achieve with a low volume battery