Famous problems of elementary geometry; the duplication of the cube, the trisection of an angle; the quadrature of the circle;

Lettered on cover: Problems in geometry. Beman and Smith. 1922.

Defining Nutrient and Biochemical Oxygen Demand Baselines for Tropical Rivers and Streams in So Paulo State (Brazil): A Comparison Between Reference and Impacted Sites

Determining reference concentrations in rivers and streams is an important tool for environmental management. Reference conditions for eutrophication-related water variables are unavailable for Brazilian freshwaters. We aimed to establish reference baselines for So Paulo State tropical rivers and streams for total phosphorus (TP) and nitrogen (TN), nitrogen-ammonia (NH(4) (+)) and Biochemical Oxygen Demand (BOD) through the best professional judgment and the trisection methods. Data from 319 sites monitored by the So Paulo State Environmental Company (2005 to 2009) and from the 22 Water Resources Management Units in So Paulo State were assessed (N = 27,131). We verified that data from different management units dominated by similar land cover could be analyzed together (Analysis of Variance, P = 0.504). Cumulative frequency diagrams showed that industrialized management units were characterized by the worst water quality (e.g. average TP of 0.51 mg/L), followed by agricultural watersheds. TN and NH(4) (+) were associated with urban percentages and population density (Spearman Rank Correlation Test, P < 0.05). Best professional judgment and trisection (median of lower third of all sites) methods for determining reference concentrations showed agreement: 0.03 & 0.04 mg/L (TP), 0.31 & 0.34 mg/L (TN), 0.06 & 0.10 mg-N/L (NH(4) (+)) and 2 & 2 mg/L (BOD), respectively. Our reference concentrations were similar to TP and TN reference values proposed for temperate water bodies. These baselines can help with water management in So Paulo State, as well as providing some of the first such information for tropical ecosystems.

Os três problemas clássicos da Matemática grega

Pós-graduação em Matemática em Rede Nacional - IBILCE

Construções euclidianas e o desfecho de problemas famosos da geometria

Construction techniques with ruler and the compasses, fundamental on Euclidean geometry, have been related to modern algebraic theories such as solving equations and extension of bodies from the works by Paolo Ruffini (1765-1822), Niels Henrik Abel (1802-1829) and Evariste Galois (1811-1832). This relation could provide an answer to some famous problems, from ancient Greece, such as doubling the cube, the trisection Angle, the Quadrature of the Circle and the construction of regular polygons, which remained unsolved for over two thousand years. Also important for our purposes are the notions of algebraic numbers, transcendental and the criteria for constructability, of those numbers. The objective of this study is to reconstruct relevant steps of geometric constructions with ruler (unmarked) and the compasses, from the elementary to the outcome buildings, in the nineteenth century, considering those mentioned problems.

Origami: o uso como instrumento alternativo no ensino da geometria

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Nutrient reference concentrations and trophic state boundaries in subtropical reservoirs

Nutrient criteria as reference concentrations and trophic state boundaries are necessary for water management worldwide because anthropogenic eutrophication is a threat to the water uses. We compiled data on total phosphorus (TP), nitrogen (TN) and chlorophyll a (Chl a) from 17 subtropical reservoirs monitored from 2005-2009 in the Sao Paulo State (Brazil) to calculate reference concentrations through the trisection method (United States Environmental Protection Agency). By dividing our dataset into thirds we presented trophic state boundaries and frequency curves for the nutrient levels in water bodies with different enrichment conditions. TP and TN baseline concentrations (0.010 mg/L and 0.350 mg/L, respectively) were bracketed by ranges for temperate reservoirs available in the literature. We propose trophic state boundaries (upper limits for the oligotrophic category: 0.010 mg TP/L, 0.460 mg TN/L and 1.7 mu g Chl a/L; for the mesotrophic: 0.030 mg TP/L, 0.820 mg TN/L and 9.0 mu g Chl a/L). Through an example with a different dataset (from the Itupararanga Reservoir, Brazil), we encouraged the use of frequency curves to compare data from individual monitoring efforts with the expected concentrations in oligotrophic, mesotrophic and eutrophic regional systems. Such analysis might help designing recovery programs to reach targeted concentrations and mitigate the undesirable eutrophication symptoms in subtropical freshwaters.

De undecimo Euclidis axiomate iudicium cui accedunt pauca De trisectione anguli /

"In certamine litterario civium Academiae Ienensis die VI septembris MDCCCXXIII praemio principum munificentia constituto ex sententia illustris philosophorum ordinis ornatum."