(Georgia Institute of Technology, 2022-05)
Raghavan, Aaditya
It is well-known that the game of \textit{Lights Out} on a graph $G$ with $|V(G)| = n$ is universally solvable if and only if the sum of the adjacency matrix of $G$ and the identity matrix $I_{n}$ is invertible over $\Z_2$. Denoting $G(n, 1/2)$ to be a graph on $n$ vertices chosen uniformly at random (each edge is chosen to be included or not independently with probability $1/2$), Forrest and Manno conjectured that as $n$ goes to infinity, the probability that $G(n, 1/2)$ is universally solvable is roughly $0.419$. We derive an explicit formula for this probability which, as a consequence, yields a more precise asymptotic probability. Moreover, we show that the probability that a random, symmetric matrix $M$ in $\Z_2^{n \times n}$ is invertible does not depend on the distribution of 0's and 1's along the main diagonal: we prove that the probabilities are identical for any given 0-1 distribution along the main diagonal over all matrices in $\Z_2^{n \times n}$ if $n$ is even, and over all matrices in $\Z_2^{n \times n}$ with at least one 1 along the main diagonal if $n$ is odd. We also develop expressions for the probability that, for $r \in \N$, $M$ has nullity $2r+1$ in the case that $n$ is odd, and $2r$ in the case that $n$ is even, in terms of closely related quantities.
(Georgia Institute of Technology, 2013-05-08)
Middlemas, Eleanor
Global climate changes during the Last Glacial Maximum (LGM) have been ascribed to a combination of lower atmospheric CO2, increased albedo, large continental ice sheets, and altered ocean circulation. The effect of lowered sea level on LGM climate is typically considered negligible, despite the exposure of a continent-scale landmass, the Sunda Shelf, in the tropical Indo-Pacific. A recent investigation of coupled atmosphere-ocean general circulation model (AOGCMs) simulation of LGM climate suggests that the emergence of the Sunda Shelf alone may have reshaped tropical Indo-Pacific climate, contributing to a significant reduction in the strength of the Walker circulation (DiNezio et al, 2011). If true, this would have profound implications for the attribution of LGM paleoclimate signals of proxy data from the Indo-Pacific region. The only modeling experiment that isolated the effects of lowered LGM sea level on climate documented significant atmospheric circulation changes spanning the entire tropical Pacific (Bush, et. al., 2003). Here we present the results of lowered sea level simulations with the AGCM SPEEDY (Molteni, 2003), designed to gauge the sensitivity of tropical Pacific climate to an LGM sea level drop of roughly 120m. Modifying the model’s land/sea mask, we replaced ocean gridpoints with land gridpoints to simulate an exposed Sunda Shelf. Both in the control run and the exposed land run, sea surface temperatures were prescribed along a tropical strip (-10-S to 10-N) to capture the atmospheric responses to realistic tropical SST variability associated with the El Nino-Southern Oscillation. Fifty ensembles of 60 years each were run for the control and for the Sunda Shelf experiment. Our results show that the presence of the Sunda Shelf has the largest impacts in the vicinity of the new landmass, but impacts on large-scale atmospheric circulation variables are observed across the Pacific and Indian Ocean basins and extend into southern Asia. In the Sunda Shelf experiment, we observe changes in both near-surface winds and aloft wind fields, implying shifts in both the Walker and Hadley circulations. Precipitation decreases along the western portion of the Pacific Intertropical Convergence Zone, reflecting large-scale changes in wind fields, and accompany large-scale changes in sea-level pressure.