Optimization of Rational Approximations by Continued Fractions

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006.

Polynomials of Pellian Type and Continued Fractions

We investigate infinite families of integral quadratic polynomials {fk (X)} k∈N and show that, for a fixed k ∈ N and arbitrary X ∈ N, the period length of the simple continued fraction expansion of √fk (X) is constant. Furthermore, we show that the period lengths of √fk (X) go to infinity with k. For each member of the families involved, we show how to determine, in an easy fashion, the fundamental unit of the underlying quadratic field. We also demonstrate how the simple continued fraction ex- pansion of √fk (X) is related to that of √C, where √fk (X) = ak*X^2 +bk*X + C. This continues work in [1]–[4].

Molecular characterization of Cladium peat from the Florida Everglades: biomarker associations with humic fractions

The accumulation and preservation of peat soils in Everglades freshwater marshes and mangrove swamps is an essential process in the ecological functioning of these ecosystems. Human intervention and climate change have modified nutrient dynamics and hydroperiod in the Everglades and peat loss due to such anthropogenic activities is evident. However, not much is known on the molecular level regarding the biogeochemical characteristics, which allow peat to be preserved in the Everglades. Lipid biomarkers trapped within or bound to humic-type structures can provide important geochemical information regarding the origin and microbial transformation of OM in peat. Four lipid fractions obtained from a Cladium peat, namely the freely extractable fraction and those associated with humin, humic acid, and fulvic acid fractions, showed clear differences in their molecular distribution suggesting different OM sources and structural and diagenetic states of the source material. Both, higher plant derived and microbial lipids were found in association with these humic-type substances. Most biomarker distributions suggest an increment in the microbial/terrestrial lipid ratio from the free to humin to humic to fulvic fractions. Microbial reworking of lipids, and the incorporation of microbial biomarkers into the humic-type fractions was evident, as well as the preservation of diagenetic byproducts. The lipid distribution associated with the fulvic acids suggests a high degree of microbial reworking for this fraction. Evidence for this 3D structure was obtained through the presence of the relatively high abundance of α,ω-dicarboxylic acids and phenolic and benzenecarboxylic compounds. The increment in structural complexity of the phenolic and benzencarboxylic compounds in combination with the reduction in the carbon chain length of the dicarboxylic acids from the free to fulvic fraction suggests the latter to be structurally the most stable, compacted and diagenetically altered substrate. This analytical approach can now be applied to peat samples from other areas within the Everglades ecosystem, affected differently by human intervention with the aim to assess changes in organic matter preservation.

What Teaching Strategies Can I Employ to Improve Student Achievement in the Addition of Fractions with Different Denominators?

The purpose of this action research was to determine what instructional strategies could be used to improve student achievement in fraction addition. An eighth grade intensive math class practiced multiplication facts and hands-on applications of fractions concepts for 2 months. Pretests/posttests were used to measure improvement in computation and understanding.

Analysis of Students’ Misconceptions and Error Patterns in Mathematics: The Case of Fractions

This study analyzed three fifth grade students’ misconceptions and error patterns when working with equivalence, addition and subtraction of fractions. The findings revealed that students used both conceptual and procedural knowledge to solve the problems. They used pictures, gave examples, and made connections to other mathematical concepts and to daily life topics. Error patterns found include using addition and subtraction of numerators and denominators, and finding the greatest common factor.

Geochemical data and radiocarbon ages of organic matter fractions and bacteriohopanepolyol data of sediment core GeoB6518-1 from the Congo River basin

The age of organic material discharged by rivers provides information about its sources and carbon cycling processes within watersheds. While elevated ages in fluvially-transported organic matter are usually explained by erosion of soils and sediments, it is commonly assumed that mainly young organic material is discharged from flat tropical watersheds due to their extensive plant cover and high carbon turnover. Here we present compound-specific radiocarbon data of terrigenous organic fractions from a sedimentary archive offshore the Congo River in conjunction with molecular markers for methane-producing land cover reflecting wetland extent in the watershed. We find that the Congo River has been discharging aged organic matter for several thousand years with increasing ages from the mid- to the Late Holocene. This suggests that aged organic matter in modern samples is concealed by radiocarbon from nuclear weapons testing. By comparison to indicators for past rainfall changes we detect a systematic control of organic matter sequestration and release by continental hydrology mediating temporary carbon storage in wetlands. As aridification also leads to exposure and rapid remineralization of large amounts of previously stored labile organic matter we infer that this process may cause a profound direct climate feedback currently underestimated in carbon cycle assessments.

(Table 5, page 626), Tin and main metal elements composition of ferromanganese concretions (acid-soluble fractions)

Concentrations of tin in sea water decreased from estuarine and shelf (0.02-0.04 µg/kg) to surface Atlantic waters (0.009 µg/kg). Mean contents (ppm) in other materials included: ultramafic rocks, 0.8; basalts, 1.7; silicic rocks, 2.5; red clays, 3.4; amphibolites, 1.2. Oceanic ferromanganese deposits contained from 0.2 to 5.8 ppm; tin and cobalt contents were correlated.

(Table 1) Mineral composition of sand- and silt-sized fractions of bottom sediments from the Romanche Trench

A study was made of mineral composition of sand- and silt-sized fractions of recent clastic (riftogenic) sediments and solidified deposits collected from the bottom of the Romanche Trench during the first voyage of R/V Akademik Kurchatov. Similarity between mineral compositions of sediments and bedrocks (ultrabasites, gabbroids, diabases) was established. This similarity is a basis for considering the mineral complex of the deposits that have been derived from the bedrocks of the trench slopes, and have formed due to their submarine denudation accompanied by tectonic crushing. The same mineral composition was found in pieces of older consolidated deposits; this suggests that conditions of sedimentation similar to those at recent times have existed for a long time in the Romanche Trench.