Mathematics Scholarly Works
https://hdl.handle.net/1808/264
2024-03-23T21:18:37ZGeneral polygonal line tilings and their matching complexes
https://hdl.handle.net/1808/34162
General polygonal line tilings and their matching complexes
Bayer, Margaret; Milutinović, Marija Jelić; Vega, Julianne
A (general) polygonal line tiling is a graph formed by a string of cycles, each intersecting the previous at an edge, no three intersecting. In 2022, Matsushita proved the matching complex of a certain type of polygonal line tiling with even cycles is homotopy equivalent to a wedge of spheres. In this paper, we extend Matsushita's work to include a larger family of graphs and carry out a closer analysis of lines of triangles and pentagons, where the Fibonacci numbers arise.
2023-03-31T00:00:00ZDispersal limitation and fire feedbacks maintain mesic savannas in Madagascar
https://hdl.handle.net/1808/34060
Dispersal limitation and fire feedbacks maintain mesic savannas in Madagascar
Goel, Nikunj; Van Vleck, Erik S.; Aleman, Julie C.; Staver, A. Carla
Madagascar is regarded by some as one of the most degraded landscapes on Earth, with estimates suggesting that 90% of forests have been lost to indigenous Tavy farming. However, the extent of this degradation has been challenged: paleoecological data, phylogeographic analysis, and species richness indicate that pyrogenic savannas in central Madagascar predate human arrival, even though rainfall is sufficient to allow forest expansion into central Madagascar. These observations raise a question—if savannas in Madagascar are not anthropogenic, how then are they maintained in regions where the climate can support forest? Observation reveals that the savanna–forest boundary coincides with a dispersal barrier—the escarpment of the Central Plateau. Using a stepping-stone model, we show that in a limited dispersal landscape, a stable savanna–forest boundary can form because of fire–vegetation feedbacks. This phenomenon, referred to as range pinning, could explain why eastern lowland forests have not expanded into the mesic savannas of the Central Highlands. This work challenges the view that highland savannas in Madagascar are derived by human-lit fires and, more importantly, suggests that partial dispersal barriers and strong nonlinear feedbacks can pin biogeographical boundaries over a wide range of environmental conditions, providing a temporary buffer against climate change.
2020-09-02T00:00:00ZThe hyperbolic Anderson model: moment estimates of the Malliavin derivatives and applications
https://hdl.handle.net/1808/33955
The hyperbolic Anderson model: moment estimates of the Malliavin derivatives and applications
Balan, Raluca M.; Nualart, David; Quer-Sardanyons, Lluís; Zheng, Guangqu
In this article, we study the hyperbolic Anderson model driven by a space-time colored Gaussian homogeneous noise with spatial dimension d=1,2. Under mild assumptions, we provide Lp-estimates of the iterated Malliavin derivative of the solution in terms of the fundamental solution of the wave solution. To achieve this goal, we rely heavily on the Wiener chaos expansion of the solution. Our first application are quantitative central limit theorems for spatial averages of the solution to the hyperbolic Anderson model, where the rates of convergence are described by the total variation distance. These quantitative results have been elusive so far due to the temporal correlation of the noise blocking us from using the Itô calculus. A novel ingredient to overcome this difficulty is the second-order Gaussian Poincaré inequality coupled with the application of the aforementioned Lp-estimates of the first two Malliavin derivatives. Besides, we provide the corresponding functional central limit theorems. As a second application, we establish the absolute continuity of the law for the hyperbolic Anderson model. The Lp-estimates of Malliavin derivatives are crucial ingredients to verify a local version of Bouleau-Hirsch criterion for absolute continuity. Our approach substantially simplifies the arguments for the one-dimensional case, which has been studied in the recent work by [2].
2022-01-18T00:00:00ZCounting Matrices Over Finite Fields
https://hdl.handle.net/1808/33724
Counting Matrices Over Finite Fields
Critzer, Geoffrey
This project was submitted to the Mathematics department in partial fulfillment of the requirements for the degree of Master of Arts.
2022-12-07T00:00:00Z