168 research outputs found
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
Relationships between Stomatal Behavior and Internal Carbon Dioxide Concentration in Crassulacean Acid Metabolism Plants
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
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Strategies in nitrogen uptake and use by deciduous and evergreen woody species in a seasonally dry sandhill community
Dry seasonal ecosystems such as the Florida sandhill are defined by a pronounced and consistent dry season that results in both water and nutrient limitation. Deciduous and evergreen species have evolved different leaf phenologies, which change the temporal pattern of nutrient demands for leaf growth.To measure N uptake for deciduous and evergreen species during late dry season deciduous leaf-out, we labeled the shallow soil with 15N by spreading a mixture of native sand and 15N-KNO3 around each tree and measured foliar nitrogen isotopic composition. We also measured foliar N content and nitrogen resorption efficiency (NRE).Evergreen and deciduous species were not consistently different in foliar N content and NRE, but natural foliar δ15N values were consistently higher in deciduous species. Labeling experiments show that uptake rate of 15N- labeled NO3
− was lowest for the deciduous species.Inconsistent results in foliar N content and NRE do not explain the differences in leaf habit in nutrient-poor ecosystems. Low N uptake during leaf-out limits deciduous species to the most fertile regions of the Florida scrub as well as in several other ecosystems, so that they can take up sufficient N to meet the demands of leaf growth
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Linking soil nutrient availability, fine root production and turnover, and species composition in a seasonally dry plant community
We studied the relationship between seasonal nutrient availability and fine root density with soil depth to determine potential nutrient uptake strategies of evergreen and deciduous woody species in an infertile, seasonally dry plant community.PO4
3−, NO3
−, and NH4
+ were measured with soil depth and across seasons, using ion-exchange resins. Fine root density was measured seasonally by counting the first four root terminal orders (root branching from tip to base). Aboveground stem density and species composition were measured.N and P availability were highest in the shallow soil layer and peaked in the late wet season and not during the initial rains as was hypothesized. Substantial N and P were found at deeper soil depths during the early dry season. Fine root density was highest in the shallow soil layer and in the wet season and underwent substantial turnover from dry to wet season. Stem and root area were correlated.Having high fine root densities in the shallow soil layer benefits P uptake more than N or water uptake. Dormancy in deciduous species decreases fine root turnover but continued nutrient uptake in the dry season by the more abundant evergreen species appears to be of greater importance
Spatial D/H Heterogeneity of Leaf Water
The mean δD value of petiole water of Pterocarpus indicus Willd (δD = −9.0 ± 2.5‰, n = 3) was not significantly different from the mean value of stem water (−8.3 ± 2.8‰, n = 3). δD values of main vein water ranged from −11.1 to + 12.0‰ (n = 14) and increased along the main vein from petiole to the tip of leaves. Mesophyll water was highly enriched in deuterium (mean δD = +32.0 ± 2.0‰, n = 19) when compared with stem, petiole, and vein water. δD values of mesophyll water for different areas of the lamina, however, were not homogenous and could differ by as much as 20‰
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