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

Caracterização etnopedológica de Planossolos utilizados em cerâmica artesanal no Agreste Paraibano Ethnopedological studies on solonetz and Planosols used in pottery craftwork in the Agreste region, State of Paraiba

Técnicas adaptadas da etnociência clássica foram utilizadas para descrever e avaliar os conhecimentos de um grupo de artesãos camponeses do Agreste Paraibano sobre alguns solos que eles usam como recurso cerâmico. Cinco perfis de solo foram descritos por agrônomos-pesquisadores (abordagem eticista) e por camponeses (abordagem emicista), em locais onde a população local extrai material para cerâmica. Amostras coletadas em ambas as abordagens foram usadas para caracterização morfológica e analítica desses solos. Os camponeses pesquisados foram capazes de distinguir, identificar e nomear diversos materiais de solo, arranjados em estratos ao longo dos perfis de solo, de modo comparável ao arranjo dos horizontes pedogenéticos. A visão, o tato e o paladar foram empregados pelos artesãos na avaliação da qualidade do solo para cerâmica. Quatro perfis descritos junto às fontes de material cerâmico foram classificados como Planossolo Nátrico e um como Planossolo Háplico. A realização de pesquisas etnopedológicas em diferentes contextos sociais e pedológicos pode contribuir para o avanço da ciência do solo, sendo também uma oportunidade para melhor compreender as formas camponesas de conhecimento e manejo de solos.<br>Classic ethnoscientific techniques were adapted to describe and analyze the knowledge of peasant potters about soils they use for making pottery in a rural village in the Agreste region, State of Paraiba, northeastern Brazil. Five soil profiles from pits where local people obtain pottery clay were formally described by researchers (etic approach) and by peasant artisans (emic approach). Samples collected during both the emic and etic approaches were used for the morphological and analytical soil characterization. The peasant potters were able to distinguish, identify, and name soil materials arranged in layers along the soil profiles, somehow comparable to the arrangement of the pedogenetic horizons. Vision, touch and taste are used by local potters to evaluate the quality of pottery clay. Among the soil profiles described near clay pits, four were classified as Haplic Solonetz, and one as Eutric Planosol, according to the FAO/UNESCO legend. Ethnopedological studies in different social and pedological environments could contribute to the advancement of soil science and are an opportunity for an improvement in the understanding and appreciation of soil knowledge and management by peasant potters

The comP locus of Neisseria gonorrhoeae encodes a type IV prepilin that is dispensable for pilus biogenesis but essential for natural transformation

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/74520/1/j.1365-2958.1999.01269.x.pd