Predictable trajectories of the reduced Collatz iteration and a possible pathway to the proof of the Collatz conjecture (Version 2)

I show here that there are three different kinds of iterations for the reduced Collatz algorithm; depending on whether the root of the number is odd or even. There is only one kind of iteration if the root is odd and two kinds if the root is even. I also show that iterations on numbers with odd roots will cause an increase in value and eventually lead to an even rooted number. The iterations on even rooted numbers will subsequently cause a decrease in values. Because increase in values during the odd root iterations are bounded, I conclude that the Collatz iteration cannot veer to infinity. Since the sequence generated by the Collatz iteration is infinite and the values of the numbers do not veer to infinity it must either cycle and/or converge. I postulate that any cycling must occur with alternating types of iterations: e.g. odd rooted iterations which cause the values of the numbers to increase followed by even rooted iterations which causes the values to decrease. I show here that for simpler types of cycles, valid values of odd rooted or even rooted numbers are only found in a narrow gap which closes as the number of iterations increase. I further generalize to all types of odd-even and even-odd iterations. Given that previous work has shown that only very large non-trivial cycles are feasible during the Collatz iteration and this study shows the low probability of large simple cycles, leads us to the conclusion most likely cycles other than the trivial cycle are not possible during the Collatz iteration

Shrimp farming: Where does the carbon go?

Abstract The muscle tissues of the Litopenaeus vannamei shrimp grown in ponds through organic and traditional (intensive) management show that δ13C values were similar amongst the shrimp. Shrimp grown in the traditional pond were enriched in 13C by 7‰ relative to the carbon isotope ratios of their feed. The differences in the carbon isotope ratios of shrimp and feed in the traditional pond shows that the feed is not the main carbon source for shrimp grown in the traditional intensive management. Using mass balance we calculate that feed in traditional culture contributes at most 13% of the shrimp's carbon biomass

Tree Islands in Everglades Landscapes: Current Status, Historical Changes, and Hydrologic Impacts on Population Dynamics and Moisture Relations, First Annual Report

In 2005 we initiated a project designed to better understand tree island structure and function in the Everglades and the wetlands bordering it. Focus was on the raised portions at the upstream end of the islands, where tropical hardwood species adapted to well-drained conditions usually are the most prominent component of the vegetation. The study design is hierarchical, with four levels; in general, a large number of sites is to be surveyed once for a limited set of parameters, and increasingly small sets of islands are to be sampled more intensively, more frequently, and for more aspects of ecosystem function. During the first year of the 3-year study, we completed surveys of 41 Level 1 (i.e., the least intensive level) islands, and established permanent plots in two and three islands of Levels 2 and 4 intensity, respectively. Tree species richness and structural complexity was highest in Shark Slough “hammocks”, while islands in Northeast Shark Slough and Water Conservation Area 3B, which receive heavy human use, were simpler, more park-like communities. Initial monitoring of soil moisture in Level 4 hammocks indicated considerable local variation, presumably associated with antecedent rainfall and current water levels in the adjacent marsh. Tree islands throughout the study area were impacted significantly by Hurricanes Katrina and Wilma in 2005, but appear to be recovering rapidly. As the project continues to include more islands and repeated measurements, we expect to develop a better grasp of tree island dynamics across the Everglades ecosystem, especially with respect to moisture relations and water levels in the adjacent marsh. The detailed progress report which follows is also available online at http://www.fiu.edu/~serp1/projects/treeislands/tree_islands_2005_annual_report.p

Divergent Biochemical Fractionation, Not Convergent Temperature, Explains Cellulose Oxygen Isotope Enrichment across Latitudes

Recent findings based on the oxygen isotope ratios of tree trunk cellulose indicate that the temperature of biomass production in biomes ranging from boreal to subtropical forests converge to an average leaf temperature of 21.4°C. The above conclusion has been drawn under the assumption that biochemically related isotopic fractionations during cellulose synthesis are not affected by temperature. Here we test the above assumption by heterotrophically generating cellulose at different temperatures and measuring the proportion of carbohydrate oxygen that exchange with water during cellulose synthesis and the average biochemical fractionation associated with this exchange. We observed no variation in the proportion of oxygen that exchange with different temperatures, which averaged 0.42 as it has been observed in other studies. On the other hand, the biochemical oxygen isotope fractionation during cellulose synthesis is affected by temperature and can be described by a 2nd order polynomial equation. The biochemical fractionation changes little between temperatures of 20 and 30°C averaging 26‰ but increases at lower temperatures to values of 31‰. This temperature sensitive biochemical fractionation explains the pattern of cellulose oxygen isotope ratios of aquatic plants encompassing several latitudes. The observed temperature sensitive biochemical fractionation also indicates that divergent biochemical fractionation and not convergent leaf temperature explains the increase in oxygen isotope enrichment of cellulose across several biomes

Differential Response to Soil Salinity in Endangered Key Tree Cactus: Implications for Survival in a Changing Climate

Understanding reasons for biodiversity loss is essential for developing conservation and management strategies and is becoming increasingly urgent with climate change. Growing at elevations <1.4 m in the Florida Keys, USA, the endangered Key tree cactus (Pilosocereus robinii) experienced 84 percent loss of total stems from 1994 to 2007. The most severe losses of 99 and 88 percent stems occurred in the largest populations in the Lower Keys, where nine storms with high wind velocities and storm surges, occurred during this period. In contrast, three populations had substantial stem proliferation. To evaluate possible mortality factors related to changes in climate or forest structure, we examined habitat variables: soil salinity, elevation, canopy cover, and habitat structure near 16 dying or dead and 18 living plants growing in the Lower Keys. Soil salinity and elevation were the preliminary factors that discriminated live and dead plants. Soil salinity was 1.5 times greater, but elevation was 12 cm higher near dead plants than near live plants. However, distribution-wide stem loss was not significantly related to salinity or elevation. Controlled salinity trials indicated that salt tolerance to levels above 40 mM NaCl was related to maternal origin. Salt sensitive plants from the Lower Keys had less stem growth, lower root:shoot ratios, lower potassium: sodium ratios and lower recovery rate, but higher δ 13C than a salt tolerant lineage of unknown origin. Unraveling the genetic structure of salt tolerant and salt sensitive lineages in the Florida Keys will require further genetic tests. Worldwide rare species restricted to fragmented, low-elevation island habitats, with little or no connection to higher ground will face challenges from climate change-related factors. These great conservation challenges will require traditional conservation actions and possibly managed relocation that must be informed by studies such as these

GROWTH FORMS OF LARREA TRIDENTATA

Volume: 23Start Page: 408End Page: 41

Growth and gas exchange responses of Brazilian pepper (Schinus terebinthifolius) and native South Florida species to salinity

Schinus terebinthifolius Raddi (Schinus) is an invasive exotic species widely found in disturbed and native communities of Florida. This species has been shown to displace native species as well as alter community structure and function. The purpose of this study was to determine if the growth and gas exchange patterns of Schinus, under differing salinity conditions, were different from native species. Two native upland glycophytic species (Rapanea punctata and Randia aculeata) and two native mangrove species (Rhizophora mangle and Laguncularia racemosa) were compared with the exotic. Overall, the exotics morphologic changes and gas exchange patterns were most similar to R. mangle. Across treatments, increasing salinity decreased relative growth rate (RGR), leaf area ratio (LAR) and specific leaf area (SLA) but did not affect root/shoot ratios (R:S). Allocation patterns were however significantly different among species. The largest proportion of Schinus biomass was allocated to stems (47%), resulting in plants that were generally taller than the other species. Schinus also had the highest SLA and largest total leaf area of all species. This meant that the exotic, which was taller and had thinner leaves, was potentially able to maintain photosynthetic area comparable to native species. Schinus response patterns show that this exotic exhibits some physiological tolerance for saline conditions. Coupled with its biomass allocation patterns (more stem biomass and large area of thin leaves), the growth traits of this exotic potentially provide this species an advantage over native plants in terms of light acquisition in a brackish forested ecosystem