A study of saturation number

Thesis (M.S.) University of Alaska Fairbanks, 2017This paper seeks to provide complete proofs in modern notation of (early) key saturation number results and give some new results concerning the semi-saturation number. We highlight relevant results from extremal theory and present the saturation number for the complete graph Kk; and the star K₁,t, elaborating on the proofs provided in the 1964 paper A Problem in Graph Theory by Erdos, Hajnal and Moon and the 1986 paper Saturated Graphs with Minimal Number of Edges by Kászonyi and Tuza. We discuss the proof of a general bound on the saturation number for a family of target graphs provided by Kászonyi and Tuza. A discussion of related results showing that the complete graph has the maximum saturation number among target graphs of the same order and that the star has the maximum saturation number among target trees of the same order is included. Before presenting our result concerning the semi-saturation number for the path Pk; we discuss the structure of some Pk-saturated trees of large order as well as the saturation number of Pk with respect to host graphs of large order.Chapter 1: Introduction -- 1.1 Basic definitions -- 1.2 Saturation number -- 1.3 Chapter overview. Chapter 2: A brief history of saturation number -- 2.1 Extremal theory -- 2.2 The minimal Kk-saturated Graph: Ak(n). Chapter 3: General saturation number results -- 3.1 General bounds for sat(n,F) -- 3.2 Stars. Chapter 4: Saturation numbers for paths and other families of trees -- 4.1 Isolated edges -- 4.2 Paths -- 4.3 Trees of minimum saturation number -- 4.4 Other tree saturation number results -- 4.4.1 Properties of subtrees and saturation number bounds -- 4.4.2 Trees T for which there exists a minimal T-saturated forest. Chapter 5: Semi-saturation number -- 5.1 Motivation -- 5.2 The semi-saturation number for Pk. Chapter 6: Further questions -- References

Enhancing speed and scalability of the ParFlow simulation code

Regional hydrology studies are often supported by high resolution simulations of subsurface flow that require expensive and extensive computations. Efficient usage of the latest high performance parallel computing systems becomes a necessity. The simulation software ParFlow has been demonstrated to meet this requirement and shown to have excellent solver scalability for up to 16,384 processes. In the present work we show that the code requires further enhancements in order to fully take advantage of current petascale machines. We identify ParFlow's way of parallelization of the computational mesh as a central bottleneck. We propose to reorganize this subsystem using fast mesh partition algorithms provided by the parallel adaptive mesh refinement library p4est. We realize this in a minimally invasive manner by modifying selected parts of the code to reinterpret the existing mesh data structures. We evaluate the scaling performance of the modified version of ParFlow, demonstrating good weak and strong scaling up to 458k cores of the Juqueen supercomputer, and test an example application at large scale.Comment: The final publication is available at link.springer.co

Modeling long-term carbon accumulation of tropical peat swamp forest ecosystems

Peatlands play an important role in the global climate system and carbon cycle; their large carbon stocks could be released to the atmosphere due to climate change or disturbance, resulting in increased climate forcing. I modified the Holocene Peat Model (HPM), a process-based model coupling water and carbon balance for simulating carbon dynamic over millennia, to be applicable for tropical peatlands. HPMTrop outputs are generally consistent with the field observations from Indonesia. The simulated long-term carbon accumulation rate for coastal and inland peatlands were 0.63 and 0.26 Mg C ha-1 y -1, and the resulting peat carbon stocks at the end of the simulations were 3,150 Mg C ha-1 and 3,270 Mg C ha-1, respectively. The simulated carbon loss for the coastal scenario caused by forest conversion to oil palm plantation with periodic burning was 1,500 Mg C ha-1 y-1over 100 years, which is equivalent to ∼3,000 years of peat accumulation

Modelling rainforests

In this dissertation, we develop a competition-colonisation model to describe the dynamics of interactions between tropical rainforest tree species.\ud \ud There is a great deal of interest in modelling rainforest diversity. Understanding the natural processes that maintain diversity is essential so that sustainable management systems can attempt to replicate important processes.\ud \ud We find, through numerical investigation and analysis, that with constant colonisation rates, $c_i$, we cannot predict multiple species coexistence. The inclusion of decaying colonisation rates, describing the seedling population decay over time, and random mass fruiting events allows coexistence of species, but using unrealistic parameter values. Finally we investigate a mathematical model without any competition between species and find that, using realistic parameter values, our results qualitatively mimic observations of rainforest dynamics. The results of the no competition model support Hubbell's null\ud hypothesis [17]

N2O, NO, N2, and CO2 emissions from tropical savanna and grassland of Northern Australia: an incubation experiment with intact soil cores

Strong seasonal variability of hygric and thermal soil conditions are a defining environmental feature in northern Australia. However, how such changes affect the soil-atmosphere exchange of nitrous oxide (N2O), nitric oxide (NO) and dinitrogen (N2) is still not well explored. By incubating intact soil cores from four sites (three savanna, one pasture) under controlled soil temperatures (ST) and soil moisture (SM) we investigated the release of the trace gas fluxes of N2O, NO and carbon dioxide (CO2). Furthermore, the release of N2 due to denitrification was measured using the helium gas flow soil core technique. Under dry pre-incubation conditions NO and N2O emissions were very low (<7.0 ± 5.0 müg NO-N m-2 h-1; <0.0 ± 1.4 müg N2O-N m-2 h-1) or in the case of N2O, even a net soil uptake was observed. Substantial NO (max: 306.5 müg N m-2 h-1) and relatively small N2O pulse emissions (max: 5.8 ± 5.0 &müg N m-2 h-1) were recorded following soil wetting, but these pulses were short lived, lasting only up to 3 days. The total atmospheric loss of nitrogen was generally dominated by N2 emissions (82.4-99.3% of total N lost), although NO emissions contributed almost 43.2% to the total atmospheric nitrogen loss at 50% SM and 30 °C ST incubation settings (the contribution of N2 at these soil conditions was only 53.2%). N2O emissions were systematically higher for 3 of 12 sample locations, which indicates substantial spatial variability at site level, but on average soils acted as weak N2O sources or even sinks. By using a conservative upscale approach we estimate total annual emissions from savanna soils to average 0.12 kg N ha-1 yr-1 (N2O), 0.68 kg N ha-1 yr-1 (NO) and 6.65 kg N ha-1 yr-1 (N2). The analysis of long-term SM and ST records makes it clear that extreme soil saturation that can lead to high N2O and N2 emissions only occurs a few days per year and thus has little impact on the annual total. The potential contribution of nitrogen released due to pulse events compared to the total annual emissions was found to be of importance for NO emissions (contribution to total: 5-22%), but not for N2O emissions. Our results indicate that the total gaseous release of nitrogen from these soils is low and clearly dominated by loss in the form of inert nitrogen. Effects of seasonally varying soil temperature and moisture were detected, but were found to be low due to the small amounts of available nitrogen in the soils (total nitrogen <0.1%)

Rainbow Turan Methods for Trees

The rainbow Turan number, a natural extension of the well-studied traditionalTuran number, was introduced in 2007 by Keevash, Mubayi, Sudakov and Verstraete. The rainbow Tur ́an number of a graph F , ex*(n, F ), is the largest number of edges for an n vertex graph G that can be properly edge colored with no rainbow F subgraph. Chapter 1 of this dissertation gives relevant definitions and a brief history of extremal graph theory. Chapter 2 defines k-unique colorings and the related k-unique Turan number and provides preliminary results on this new variant. In Chapter 3, we explore the reduction method for finding upper bounds on rainbow Turan numbers and use this to inform results for the rainbow Turan numbers of specific families of trees. These results are used in Chapter 4 to prove that the rainbow Turan numbers of all trees are linear in n, which correlates to a well-known property of the traditional Turan numbers of trees. We discuss improvements to the constant term in Chapters 4 and 5, and conclude with a discussion on avenues for future work

Deforestation in central Saskatchewan : effects on landscape structure and ecosystem carbon densities

Deforestation is recognized as a serious global problem that contributes to biodiversity loss, soil degradation and atmospheric change. This thesis is an investigation of deforestation in central Saskatchewan. The purposes are: to quantify the extent and rates of deforestation and associated changes in spatial structure for multi-jurisdictional boreal landscapes; and to estimate the magnitude of carbon losses associated with agriculture-induced deforestation at sites within one of these landscapes. Deforestation was analyzed using topographic map chronosequences for two 460 000 ha landscapes in central Saskatchewan. An estimated 16 400 ha was deforested between 1963 and 1990 within the Waskesiu Hills landscape (53° 45' N, 106° 15' W) and 371 000 ha was deforested between 1957 and 1990 within the Red Deer River landscape (52° 45' N, 103° 00' W). Federal and provincial legislation establishing publicly owned parks and forests served to inhibit deforestation within portions of these landscapes. On agricultural lands within the two landscapes, where private holdings dominate and forests are not protected under federal or provincial law, deforestation occurred at rates exceeding 1.2 % yr-1 over the time periods examined even though human populations declined. Within the two study areas, extant forests that are unprotected by legislation remain vulnerable to deforestation. Spatial structure was analyzed for portions of these two landscapes using landscape metrics. A positive correlation between largest patch size index and proportion of land area wooded was evident for both 1975/76 (r2 = 0.99, p < 0.01) and 1990 (r2 = 0.99, p < 0.05). Since past deforestation disproportionately reduced the sizes of the largest wooded patches, future reforestation efforts should be aimed at expanding large patches. Reforestation with large patches contiguous to protected forests may initiate a reversal of the process of fragmentation that has impaired forest wildlife and ecosystem processes. Vegetation carbon densities were compared at six forest sites, six pasture sites and six cultivated sites on hummocky glacial till landforms across three townships within the Waskesiu Hills landscape. Medians for aboveground biomass (60 Mg C ha-1 for forests, 1 Mg C ha-1 for pastures and 4 Mg C ha-1 for cultivated sites) were significantly different (p < 0.15). Including estimated losses of coarse roots, deforestation and subsequent agricultural land use led to losses of approximately 70 Mg C ha-1 for vegetation. Statistically significant losses of soil organic carbon were not detected between the forest sites and the agricultural sites. The experimental design accounted for land use and topographic landform effects, but these were small ( 50 Mg C ha-1 for natural forest sites). Across all sites regardless of land use, there was a positive correlation (rs = 0.76,