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

Determinants of flammability in savanna grass species

1. Tropical grasses fuel the majority of fires on Earth. In fire-prone landscapes, enhanced flammabil-ity may be adaptive for grasses via the maintenance of an open canopy and an increase in spa-tiotemporal opportunities for recruitment and regeneration. In addit ion, by burning intensely butbriefly, high flammability may protect resprouting buds from lethal temperatures. Despite thesepotential benefits of high flammability to fire-prone grasses, variation in flammability among grassspecies, and how trait differences underpin this variation, remains unknown.2. By burning leaves and plant parts, we experimentally determined how five plant traits (biomassquantity, biomass density, biomass moisture content, leaf surface-area-to-volume ratio and leaf effec-tive heat of combustion) combined to determine the three components of flammability (ignitability,sustainability and combustibility) at the leaf and plant scales in 25 grass species of fire-pr one SouthAfrican grasslands at a time of peak fire occurrence. The influence of evolutionary history onflammability was assessed based on a phylogeny built here for the study species.3. Grass speci es differed significantly in all components of flammability. Accounting for evolution-ary history helped to explain patterns in leaf-scale combustibility and sustainability. The five mea-sured plant traits predicted components of flammability, particularly leaf ignitability and plantcombustibility in which 70% and 58% of variation, respectively, could be explained by a combina-tion of the traits. Total above-ground biomass was a key drive r o f combustibility and sustainabi litywith high biomass species burning more intensely and for longer, and producing the highest pre-dicted fire spread rates. Moisture content was the main influence on ignitability, where speci es withhigher moisture conten ts took longer to ignite and once alight burnt at a slower rate. Bioma ss den-sity, leaf surface-area-to-volume ratio and leaf effective heat of combustion were weaker predictorsof flammability components.4. Synthesis. We demonstrate that grass flammability is predicted from easily measurable plant func-tional traits and is influenced by evolutionary history with some components showing phylogeneticsignal. Grasses are not homogenous fuels to fire. Rather, species differ in functional traits that inturn demonstrably influence flammability. This diver sity is consistent with the idea that flammabilitymay be an adaptive trait for grasses of fire-prone ecosystems

Biomass burning fuel consumption rates: a field measurement database

Landscape fires show large variability in the amount of biomass or fuel consumed per unit area burned. Fuel consumption (FC) depends on the biomass available to burn and the fraction of the biomass that is actually combusted, and can be combined with estimates of area burned to assess emissions. While burned area can be detected from space and estimates are becoming more reliable due to improved algorithms and sensors, FC is usually modeled or taken selectively from the literature. We compiled the peerreviewed literature on FC for various biomes and fuel categories to understand FC and its variability better, and to provide a database that can be used to constrain biogeochemical models with fire modules. We compiled in total 77 studies covering 11 biomes including savanna (15 studies, average FC of 4.6 t DM (dry matter) ha 1 with a standard deviation of 2.2),tropical forest (n = 19, FC = 126 +/- 77),temperate forest (n = 12, FC = 58 +/- 72),boreal forest (n = 16, FC = 35 +/- 24),pasture (n = 4, FC = 28 +/- 9.3),shifting cultivation (n = 2, FC = 23, with a range of 4.0-43),crop residue (n = 4, FC = 6.5 +/- 9.0),chaparral (n = 3, FC = 27 +/- 19),tropical peatland (n = 4, FC = 314 +/- 196),boreal peatland (n = 2, FC = 42 [42-43]),and tundra (n = 1, FC = 40). Within biomes the regional variability in the number of measurements was sometimes large, with e. g. only three measurement locations in boreal Russia and 35 sites in North America. Substantial regional differences in FC were found within the defined biomes: for example, FC of temperate pine forests in the USA was 37% lower than Australian forests dominated by eucalypt trees. Besides showing the differences between biomes, FC estimates were also grouped into different fuel classes. Our results highlight the large variability in FC, not only between biomes but also within biomes and fuel classes. This implies that substantial uncertainties are associated with using biome-averaged values to represent FC for whole biomes. Comparing the compiled FC values with co-located Global Fire Emissions Database version 3 (GFED3) FC indicates that modeling studies that aim to represent variability in FC also within biomes, still require improvements as they have difficulty in representing the dynamics governing FC

Degradation of communal rangelands in South Africa: towards an improved understanding to inform policy

In South Africa, the relative extent of range degradation under freehold compared to communal tenure has been strongly debated. We present a perspective on the processes that drive rangeland degradation on land under communal tenure. Our findings are based on literature as well as extensive field work on both old communal lands and ‘released’ areas, where freehold farms have been transferred to communal ownership. We discuss the patterns of degradation that have accompanied communal stewardship and make recommendations on the direction policy should follow to prevent further degradation and mediate rehabilitation of existing degraded land.Keywords: communal rangelands, land degradation, rehabilitation, social systemsAfrican Journal of Range & Forage Science 2013, 30(1&2): 57–6

Understandings of reproductive tract infections in a peri-urban pueblo joven in Lima, Peru

BACKGROUND: Control programs for Reproductive Tract Infections (RTIs) typically focus on increasing awareness of risks associated with different forms of sexual contact, and pay little attention to how or why people may link RTIs to other features of their physical or social environments. This paper describes how women in a peri-urban pueblo joven located in the coastal desert surrounding Lima, Peru conceptualize the links between RTIs, sexual behaviour, personal hygiene, and the adverse environment in which they live. METHODS: We combined qualitative interviews and a participatory voting exercise to examine social and physical environmental influences on RTIs and gynaecologic symptom interpretation. RESULTS: Knowledge of RTIs in general was limited, although knowledge of AIDS was higher. Perceived causes of RTIs fell into three categories: sexual contact with infected persons, personal hygiene and exposure to the contaminated physical environment, with AIDS clearly related to sexual contact. The adverse environment is thought to be a major contributor to vaginal discharge, "inflamed ovaries" and urinary tract infection. The more remote parts of this periurban squatter settlement, characterized by blowing sand and dust and limited access to clean water, are thought to exhibit higher rates of RTIs as a direct result of the adverse environment found there. Stigma associated with RTIs often keeps women from seeking care or obtaining information about gynaecologic symptoms, and favours explanations that avoid mention of sexual practices. CONCLUSION: The discrepancy between demonstrated disease risk factors and personal explanations influenced by local environmental conditions and RTI-related stigma poses a challenge for prevention programs. Effective interventions need to take local understandings of RTIs into account as they engage in dialogue with communities about prevention and treatment of RTIs

Victimisation in the lives of lesbian-identified women in South Africa : implications for clinical assessment and treatment

Few clinical studies have examined victimisation in the lives of lesbian women in South Africa and whether there are distinct implications for psychological treatment. This paper presents the assessment and treatment of a lesbian-identified South African survivor of childhood sexual abuse who, as an adult, was raped and later gang raped. Her victimisation in adulthood represented ‘corrective rape’ motivated by the prejudiced assumption that the sexuality of lesbian women is pathological and should be ‘corrected’ through rape. This paper lends insights into the role of heterosexism in shaping vulnerability to victimisation and the process of recovery. It provides recommendations for work with sexual minority clients and highlights the implications when there is an absence of safety and support in the external environment

Facilitation or Competition? Tree Effects on Grass Biomass across a Precipitation Gradient

Savanna ecosystems are dominated by two distinct plant life forms, grasses and trees, but the interactions between them are poorly understood. Here, we quantified the effects of isolated savanna trees on grass biomass as a function of distance from the base of the tree and tree height, across a precipitation gradient in the Kruger National Park, South Africa. Our results suggest that mean annual precipitation (MAP) mediates the nature of tree-grass interactions in these ecosystems, with the impact of trees on grass biomass shifting qualitatively between 550 and 737 mm MAP. Tree effects on grass biomass were facilitative in drier sites (MAP≤550 mm), with higher grass biomass observed beneath tree canopies than outside. In contrast, at the wettest site (MAP = 737 mm), grass biomass did not differ significantly beneath and outside tree canopies. Within this overall precipitation-driven pattern, tree height had positive effect on sub-canopy grass biomass at some sites, but these effects were weak and not consistent across the rainfall gradient. For a more synthetic understanding of tree-grass interactions in savannas, future studies should focus on isolating the different mechanisms by which trees influence grass biomass, both positively and negatively, and elucidate how their relative strengths change over broad environmental gradients. © 2013 Moustakas et al