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].

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.

Contributions of humic substances to the dissolved organic carbon pool in wetlands from different climates

Wetlands are an important source of DOM. However, the quantity and quality of wetlands’ DOM from various climatic regions have not been studied comprehensively. The relationship between the concentrations of DOM (DOC), humic substances (HS) and non-humic substances (NHS) in wetland associated sloughs, streams and rivers, in cool temperate (Hokkaido, Japan), sub-tropical (Florida, USA), and tropical (Sarawak, Malaysia) regions was investigated. The DOC ranged from 1.0 to 15.6 mg C L−1 in Hokkaido, 6.0–24.4 mg C L−1 in Florida, and 18.9–75.3 mg C L−1 in Sarawak, respectively. The relationship between DOC and HS concentrations for the whole sample set was regressed to a primary function with y-intercept of zero (P < 0.005) and a slope value of 0.841. A similar correlation was observed between DOC and NHS concentrations, with a smaller slope value of 0.159. However, the correlation coefficient of the latter was much larger when the data was regressed to a logarithmic curve. These observations suggest the presence of a general tendency that the increased DOC in the river waters was mainly due to the increased supply of HS from wetland soils, whereas the rate of the increase in the NHS supply has an upper limit which may be controlled by primary productivity.

Fluorescence Characteristics of Size-Fractionated Dissolved Organic Matter: Implications for a Molecular Assembly Based Structure?

Surface freshwater samples from Everglades National Park, Florida, were used to investigate the size distributions of natural dissolved organic matter (DOM) and associated fluorescence characteristics along the molecular weight continuum. Samples were fractionated using size exclusion chromatography (SEC) and characterized by spectroscopic means, in particular Excitation-Emission Matrix fluorescence modeled with parallel factor analysis (EEM-PARAFAC). Most of the eight components obtained from PARAFAC modeling were broadly distributed across the DOM molecular weight range, and the optical properties of the eight size fractions for all samples studied were quite consistent among each other. Humic-like components presented a similar distribution in all the samples, with enrichment in the middle molecular weight range. Some variability in the relative distribution of the different humic-like components was observed among the different size fractions and among samples. The protein like fluorescence, although also generally present in all fractions, was more variable but generally enriched in the highest and lowest molecular weight fractions. These observations are in agreement with the hypothesis of a supramolecular structure for DOM, and suggest that DOM fluorescence characteristics may be controlled by molecular assemblies with similar optical properties, distributed along the molecular weight continuum. This study highlights the importance of studying the molecular structure of DOM on a molecular size distribution perspective, which may have important implications in understanding the environmental dynamics such materials.

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.

Water temperature, salinity, DOC content and THNS concentration in samples taken during POLARSTERN cruise to the Fram Strait and Greenland Sea

Dissolved organic matter (DOM) was isolated with XAD-2 and 4 resins from different water masses of the Greenland Sea and Fram Strait. The contribution of XAD-extractable dissolved organic carbon (DOC), operationally defined as 'recalcitrant' or humic substances, to total DOC was in the range of 45 ± 9% in surface waters and 60 ± 6% in deep waters. The carbohydrate concentration and composition were determined using the l-tryptophan/sulfuric acid method (for the bulk carbohydrate concentration, TCHO) and high performance anion-exchange chromatography after sulfuric acid hydrolysis (for the distribution of total hydrolysable neutral sugars, THNS). Carbohydrates contributed up to 6.8% to both total and recalcitrant DOC. TCHO contribution to total DOC decreased with depth from on average 4.1 ± 1.2% in surface waters to 2.2 ± 1.0% in deep waters, whereas the THNS contribution was similar in both layers, accounting for 2.5 ± 1.6% (surface) and 2.4 ± 0.2% (at depth). TCHO contribution to XAD-extractable DOC also decreased with depth from 4.5 ± 1.7% to 2.1 ± 1.0%, whereas THNS contribution was almost constant, with yields of 0.5 ± 0.3% for surface samples and 0.6 ± 0.1% at depth. The molecular size distribution of the recalcitrant DOM showed for all fractions a clear trend towards small molecules in the deep sea. More than half of the XAD-extractable carbohydrates of surface samples and more than 70% of deep sea samples were found in the nonpolar fraction from XAD, which was eluted with methanol. Glucose was the dominant carbohydrate in the surface water samples, whereas in the deep sea the composition was more uniform. In the XAD extracts, the compositions were less variable than in the original samples. The neutral sugar composition, in particular glucose and the deoxysugars, is indicative of the diagenetic state of the extracted DOM. The molar ratio (fucose + rhamnose)/(arabinose + xylose) was lowest for deep sea extractable DOM, indicating a high contribution of material modified by microorganisms. The THNS composition and distribution reveal that "recalcitrant" carbohydrates are heteropolysaccharides, carbohydrate units incorporated into a framework of a highly nonpolar structure with a lack of functional groups.

(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.