How large are the level sets of the Takagi function?

Let T be Takagi's continuous but nowhere-differentiable function. This paper considers the size of the level sets of T both from a probabilistic point of view and from the perspective of Baire category. We first give more elementary proofs of three recently published results. The first, due to Z. Buczolich, states that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. The second, due to J. Lagarias and Z. Maddock, states that the average number of points in a level set is infinite. The third result, also due to Lagarias and Maddock, states that the average number of local level sets contained in a level set is 3/2. In the second part of the paper it is shown that, in contrast to the above results, the set of ordinates y with uncountably infinite level sets is residual, and a fairly explicit description of this set is given. The paper also gives a negative answer to a question of Lagarias and Maddock by showing that most level sets (in the sense of Baire category) contain infinitely many local level sets, and that a continuum of level sets even contain uncountably many local level sets. Finally, several of the main results are extended to a version of T with arbitrary signs in the summands.Comment: Added a new Section 5 with generalization of the main results; some new and corrected proofs of the old material; 29 pages, 3 figure

Savanna burning and the assessment of long-term fire experiments with particular reference to Zimbabwe

Long-term fire experiments in savannnas are rare, given the difficulties and demands of operation. Controlled fire experiments date from colonial times in West Africa, although the largest and best-known is located in the Kruger National Park, South Africa. The achievements of these experiments are assessed from examples in Africa, South America and Australia. A less well-known experiment in Zimbabwe was sited at the Marondera Grassland Research Station and ran from 1953 to 1991. Some of the preliminary results on the impact of fire on vegetation are analysed and compared with further vegetation surveys in 2007. Studies on tree growth in this miombo savanna woodland indicate that the plots burned at three- and four-year intervals recovered to greater mean heights than the unburned control plots. There was no significant variation between treatments, suggesting that the few trees that did survive in the frequently burned plots were large specimens. Brachystegia and Julbernadia dominated the plots throughout and after the experiment. Basal area and stocking density were highest in the four-yearly burned plots but there was a high variability throughout the experiment, suggesting that many trees may have attained heights and bark thicknesses sufficient to protect from fire damage. Fire also affected the composition of the herbaceous plant community, but not the number of species. By the end of the experiment some grass and sedge species had flourished while others revealed greater susceptibility to fire, and fire-tolerant species predominated in the most frequently burned areas. The experimental design appeared to cope well with the variability between plots and indicated the soundness of the initial design and its implementation

Evaluating herbivore management outcomes and associated vegetation impacts

African savannas are characterised by temporal and spatial fluxes that are linked to fluxes in herbivore populations and vegetation structure and composition. We need to be concerned about these fluxes only when management actions cause the system to shift towards a less desired state. Large herbivores are a key attribute of African savannas and are important for tourism and biodiversity. Large protected areas such as the Kruger National Park (KNP) manage for high biodiversity as the desired state, whilst private protected areas, such as those adjacent to the KNP, generally manage for high income. Biodiversity, sustainability and economic indicators are thus required to flag thresholds of potential concern (TPCs) that may result in a particular set of objectives not being achieved. In large conservation areas such as the KNP, vegetation changes that result from herbivore impact, or lack thereof, affect biodiversity and TPCs are used to indicate unacceptable change leading to a possible loss of biodiversity; in private protected areas the loss of large herbivores is seen as an important indicator of economic loss. Therefore, the first-level indicators aim to evaluate the forage available to sustain grazers without deleteriously affecting the vegetation composition, structure and basal cover. Various approaches to monitoring for these indicators were considered and the importance of the selection of sites that are representative of the intensity of herbivore use is emphasised. The most crucial step in the adaptive management process is the feedback of information to inform management decisions and enable learning. Feedback loops tend to be more efficient where the organisation’s vision is focused on, for example, economic gain, than in larger protected areas, such as the KNP, where the vision to conserve biodiversity is broader and more complex. Conservation implications: In rangeland, optimising herbivore numbers to achieve the management objectives without causing unacceptable or irreversible change in the vegetation is challenging. This manuscript explores different avenues to evaluate herbivore impact and the outcomes of management approaches that may affect vegetation