Sobre a integração indefinida de funções racionais complexas: teoria e implementação de algoritmos racionais

We present indefinite integration algorithms for rational functions over subfields of the complex numbers, through an algebraic approach. We study the local algorithm of Bernoulli and rational algorithms for the class of functions in concern, namely, the algorithms of Hermite; Horowitz-Ostrogradsky; Rothstein-Trager and Lazard-Rioboo-Trager. We also study the algorithm of Rioboo for conversion of logarithms involving complex extensions into real arctangent functions, when these logarithms arise from the integration of rational functions with real coefficients. We conclude presenting pseudocodes and codes for implementation in the software Maxima concerning the algorithms studied in this work, as well as to algorithms for polynomial gcd computation; partial fraction decomposition; squarefree factorization; subresultant computation, among other side algorithms for the work. We also present the algorithm of Zeilberger-Almkvist for integration of hyperexpontential functions, as well as its pseudocode and code for Maxima. As an alternative for the algorithms of Rothstein-Trager and Lazard-Rioboo-Trager, we yet present a code for Benoulli’s algorithm for square-free denominators; and another for Czichowski’s algorithm, although this one is not studied in detail in the present work, due to the theoretical basis necessary to understand it, which is beyond this work’s scope. Several examples are provided in order to illustrate the working of the integration algorithms in this text

(Table 1) Age determination of sediment core KY07-04-PC1

Variations in Mg/Ca-based sea surface temperature and oxygen isotope ratio (d18O) of the surface water in the northern East China Sea (ECS) were reconstructed with high resolution during the last 18 kyr using planktic foraminifera. Millennial-scale variations between warmer, more saline surface water and cooler, less saline surface water were recognized during the early deglacial period and the Holocene, suggesting changes in the mixing ratio between the Kuroshio Water and the Changjiang Diluted Water. Stronger East Asian summer monsoon (EASM) precipitation events in south China are identified at 10.5, 8.8, 7.0, 5.3, 4.7, 2.9, 1.7, and 0.5 ka, based on sea surface salinity (SSS) records of the northern ECS. Weaker EASM precipitation events are also detected at 9.3, 8.3, 7.3, 6.0, 3.3, 2.3, 0.7, and 0.4 ka during the Holocene. These events agree with the maxima in d18O records of stalagmites from various parts of the Changjiang (Yangtze) River drainage. This agreement supports that our SSS record properly captures the millennial-scale dry (less EASM precipitation) events over the drainage basin of the Changjiang River during the Holocene. These dry events are also in good agreement with North Atlantic ice-rafted events, suggesting a teleconnection between North Atlantic climate and the EASM during the Holocene.

Estimation of sea surface temperatures and salinity of ODP Site 177-1089 (Appendix A)

ODP Site 1089 is optimally located in order to monitor the occurrence of maxima in Agulhas heat and salt spillage from the Indian to the Atlantic Ocean. Radiolarian-based paleotemperature transfer functions allowed to reconstruct the climatic history for the last 450 kyr at this location. A warm sea surface temperature anomaly during Marine Isotope Stage (MIS) 10 was recognized and traced to other oceanic records along the surface branch of the global thermohaline (THC) circulation system, and is particularly marked at locations where a strong interaction between oceanic and atmospheric overturning cells and fronts occurs. This anomaly is absent in the Vostok ice core deuterium, and in oceanic records from the Antarctic Zone. However, it is present in the deuterium excess record from the Vostok ice core, interpreted as reflecting the temperature at the moisture source site for the snow precipitated at Vostok Station. As atmospheric models predict a subtropical Indian source for such moisture, this provides the necessary teleconnection between East Antarctica and ODP Site 1089, as the subtropical Indian is also the source area of the Agulhas Current, the main climate agent at our study location. The presence of the MIS 10 anomaly in the delta13C foraminiferal records from the same core supports its connection to oceanic mechanisms, linking stronger Agulhas spillover intensity to increased productivity in the study area. We suggest, in analogy to modern oceanographic observations, this to be a consequence of a shallow nutricline, induced by eddy mixing and baroclinic tide generation, which are in turn connected to the flow geometry, and intensity, of the Agulhas Current as it flows past the Agulhas Bank. We interpret the intensified inflow of Agulhas Current to the South Atlantic as responding to the switch between lower and higher amplitude in the insolation forcing in the Agulhas Current source area. This would result in higher SSTs in the Cape Basin during the glacial MIS 10, due to the release into the South Atlantic of the heat previously accumulating in the subtropical and equatorial Indian and Pacific Ocean. If our explanation for the MIS 10 anomaly in terms of an insolation variability switch is correct, we might expect that a future Agulhas SSST anomaly event will further delay the onset of next glacial age. In fact, the insolation forcing conditions for the Holocene (the current interglacial) are very similar to those present during MIS 11 (the interglacial preceding MIS 10), as both periods are characterized by a low insolation variability for the Agulhas Current source area. Natural climatic variability will force the Earth system in the same direction as the anthropogenic global warming trend, and will thus lead to even warmer than expected global temperatures in the near future.

Swath sonar bathymetry during POLARSTERN cruise ARK-VII/3a (PS17) with links to multibeam raw data files

In multibeam echosounder and subbottom profiler data acquired during R/V Polarstern cruise ARK-VII/3a from the Hovgaard Ridge (Fram Strait), we found evidence for very deep (>1200 m) iceberg scouring. Five elongated seafloor features have been detected that are interpreted to be iceberg scours. The scours are oriented in north-south/south-north direction and are about 15 m deep, 300 m wide, and 4 km long crossing the entire width of the ridge. They are attributed to multiple giant paleo-icebergs that most probably left the Arctic Ocean southward through Fram Strait. The huge keel depths are indicative of ice sheets extending into the Arctic Ocean being at least 1200 m thick at the calving front during glacial maxima. The deep St. Anna Trough or grounded ice observed at the East Siberian Continental Margin are likely source regions of these icebergs that delivered freshwater to the Nordic Seas.

N2 fixation in eddies of the eastern tropical South Pacific Ocean

Mesoscale eddies play a major role in controlling ocean biogeochemistry. By impacting nutrient availability and water column ventilation, they are of critical importance for oceanic primary production. In the eastern tropical South Pacific Ocean off Peru, where a large and persistent oxygen-deficient zone is present, mesoscale processes have been reported to occur frequently. However, investigations into their biological activity are mostly based on model simulations, and direct measurements of carbon and dinitrogen (N2) fixation are scarce. We examined an open-ocean cyclonic eddy and two anticyclonic mode water eddies: a coastal one and an open-ocean one in the waters off Peru along a section at 16°S in austral summer 2012. Molecular data and bioassay incubations point towards a difference between the active diazotrophic communities present in the cyclonic eddy and the anticyclonic mode water eddies. In the cyclonic eddy, highest rates of N2 fixation were measured in surface waters but no N2 fixation signal was detected at intermediate water depths. In contrast, both anticyclonic mode water eddies showed pronounced maxima in N2 fixation below the euphotic zone as evidenced by rate measurements and geochemical data. N2 fixation and carbon (C) fixation were higher in the young coastal mode water eddy compared to the older offshore mode water eddy. A co-occurrence between N2 fixation and biogenic N2, an indicator for N loss, indicated a link between N loss and N2 fixation in the mode water eddies, which was not observed for the cyclonic eddy. The comparison of two consecutive surveys of the coastal mode water eddy in November 2012 and December 2012 also revealed a reduction in N2 and C fixation at intermediate depths along with a reduction in chlorophyll by half, mirroring an aging effect in this eddy. Our data indicate an important role for anticyclonic mode water eddies in stimulating N2 fixation and thus supplying N offshore.

Sedimentology of the Okhotsk Sea

On the basis of two sedimentary records from the central Sea of Okhotsk, we reconstruct the closely coupled glacial/interglacial changes in terrigenous flux, marine productivity, and sea ice coverage over the past 1.1 Myr. The correspondance of our sedimentary records to the China loess grain size record (China loess particle timescale, CHILOPARTS) suggests that environmental changes in both the Sea of Okhotsk area and in SE Asia were closely related via the Siberian atmospheric high-pressure cell. During full glacial times our records point to a strong Siberian High causing northerly wind directions, the extension of the sea ice cover, and a reduced Amur River discharge. Deglacial maxima of terrigenous flux were succeeded by or synchronous to high-productivity events. Marine productivity was strengthened during glacial terminations because of an effective nutrient utilization at times of enhanced water column stratification and high nutrient supply from fluvial runoff and sea ice thawing. During interglacials, SE monsoonal winds prevailed, analogous to today's summer situation of a pronounced Mongolian Heat Low and a strong Hawaiian High. Strong freshwater discharge induced by high precipitation rates in the Amur drainage area and a seasonally reduced and mobile sea ice cover favored marine productivity (although being considerably lower than during the terminations) and a lowered flux of ice-rafted detritus.

Stable carbon and oxygen isotope ratios of planktonic foraminifera from the equatorial Atlantic

Carbon isotopic records of nutrient-depleted surface water place constraints on the past fertility of the oceans and on past atmospheric pCO2 levels. The best records of nutrient-depleted delta13C are obtained from planktonic foraminifera living in the thick mixed layers of the western equatorial and tropical Atlantic Ocean. We have produced a composite, stacked Globigerinoides sacculifer delta13C record from the equatorial Atlantic, which exhibits significant spectral power at the 100,000- and 41,000-year Milankovitch periods, but no power at the 23,000-year period. Similar to the record presented by Shackleton and Pisias [1985], surface-deep ocean Delta delta13C produced with the G. sacculifer record leads the delta18O ice volume record. However, the glacial-interglacial amplitudes of Delta delta13C differ between our record and Shackleton and Pisias [1985] record. Although large changes in Delta delta13C occur in the equatorial Atlantic during early stages of the last three glacial cycles, surface-deep Delta delta13C at glacial maxima (18O stage 2, late stage 6, and late stage 8) was only about 0.2? greater than during the subsequent interglacial. Our results imply that nutrient-driven pCO2 changes account for about one third of the pCO2 decrease observed in ice cores, and consequently, Delta delta13C should not be used as a proxy pCO2 index. Enough variance in the ice core pCO2 records remains to be explained that conclusions about pCO2 and ice volume phase relationships should also be reexamined. As much as 40 ppm pCO2 change still has not been accounted for by models of past physics and chemistry of the ocean.

Age determination and paleotemperatures of western Mediterranean Sea sediments

Sea surface temperature (SST) profiles over the last 25 kyr derived from alkenone measurements are studied in four cores from a W-E latitudinal transect encompassing the Gulf of Cadiz (Atlantic Ocean), the Alboran Sea, and the southern Tyrrhenian Sea (western Mediterranean). The results document the sensitivity of the Mediterranean region to the short climatic changes of the North Atlantic Ocean, particularly those involving the latitudinal position of the polar front. The amplitude of the SST oscillations increases toward the Tyrrhenian Sea, indicating an amplification effect of the Atlantic signal by the climatic regime of the Mediterranean region. All studied cores show a shorter cooling phase (700 years) for the Younger Dryas (YD) than that observed in the North Atlantic region (1200 years). This time diachroneity is related to an intra-YD climatic change documented in the European continent. Minor oscillations in the southward displacement of the North Atlantic polar front may also have driven this early warming in the studied area. During the Holocene a regional diachroneity propagating west to east is observed for the SST maxima, 11.5-10.2 kyr B.P. in the Gulf of Cadiz, 10-9 kyr B.P. in the Alboran Sea, and 8.9-8.4 kyr B.P. in the Thyrrenian Sea. A general cooling trend from these SST maxima to present day is observed during this stage, which is marked by short cooling oscillations with a periodicity of 730±40 years and its harmonics.

Cyclostratigraphy and eccentricity tuning of the early Oligocene through early Miocene record from Walvis Ridge

Few astronomically calibrated high-resolution (<=5 kyr) climate records exist that span the Oligocene-Miocene time interval. Notably, available proxy records show responses varying in amplitude at frequencies related to astronomical forcing, and the main pacemakers of global change on astronomical time-scales remain debated. Here we present newly generated X-ray fluorescence core scanning and benthic foraminiferal stable oxygen and carbon isotope records from Ocean Drilling Program Site 1264 (Walvis Ridge, southeastern Atlantic Ocean). Complemented by data from nearby Site 1265, the Site 1264 benthic stable isotope records span a continuous ~13-Myr interval of the Oligo-Miocene (30.1-17.1 Ma) at high resolution (~3.0 kyr). Spectral analyses in the stratigraphic depth domain indicate that the largest amplitude variability of all proxy records is associated with periods of ~3.4 m and ~0.9 m, which correspond to 405- and ~110-kyr eccentricity, using a magnetobiostratigraphic age model. Maxima in CaCO3 content, d18O and d13C are interpreted to coincide with ~110 kyr eccentricity minima. The strong expression of these cycles in combination with the weakness of the precession- and obliquity-related signals allow construction of an astronomical age model that is solely based on tuning the CaCO3 content to the nominal (La2011_ecc3L) eccentricity solution. Very long-period eccentricity maxima (~2.4-Myr) are marked by recurrent episodes of high-amplitude ~110-kyr d18O cycles at Walvis Ridge, indicating greater sensitivity of the climate/cryosphere system to short eccentricity modulation of climatic precession. In contrast, the responses of the global (high-latitude) climate system, cryosphere, and carbon cycle to the 405-kyr cycle, as expressed in benthic d18O and especially d13C signals, are more pronounced during ~2.4-Myr minima. The relationship between the recurrent episodes of high-amplitude ~110-kyr d18O cycles and the ~1.2-Myr amplitude modulation of obliquity is not consistent through the Oligo-Miocene. Identification of these recurrent episodes at Walvis Ridge, and their pacing by the ~2.4-Myr eccentricity cycle, revises the current understanding of the main climate events of the Oligo-Miocene.

Total mercury and total methylmercury during the POLARSTERN cruise ARK-XXVI/3 (TransArc)

Methylmercury (MeHg) is a neurotoxic compound that threatens wildlife and human health across the Arctic region. Though much is known about the source and dynamics of its inorganic mercury (Hg) precursor, the exact origin of the high MeHg concentrations in Arctic biota remains uncertain. Arctic coastal sediments, coastal marine waters and surface snow are known sites for MeHg production. Observations on marine Hg dynamics, however, have been restricted to the Canadian Archipelago and the Beaufort Sea (<79°N). Here we present the first central Arctic Ocean (79-90°N) profiles for total mercury (tHg) and MeHg. We find elevated tHg and MeHg concentrations in the marginal sea ice zone (81-85°N). Similar to other open ocean basins, Arctic MeHg concentration maxima also occur in the pycnocline waters, but at much shallower depths (150-200 m). The shallow MeHg maxima just below the productive surface layer possibly result in enhanced biological uptake at the base of the Arctic marine food web and may explain the elevated MeHg concentrations in Arctic biota. We suggest that Arctic warming, through thinning sea ice, extension of the seasonal sea ice zone, intensified surface ocean stratification and shifts in plankton ecodynamics, will likely lead to higher marine MeHg production.

Biogenic components, major, trace and rare earth elements of equatorial Pacific Ocean surface sediments (Table 1)

We have analyzed the major, trace, and rare earth element composition of surface sediments collected from a transect across the Equator at 135°W longitude in the Pacific Ocean. Comparing the behavior of this suite of elements to the CaCO3, opal, and Corg fluxes (which record sharp maxima at the Equator, previously documented at the same sampling stations) enables us to assess the relative significance of the various pathways by which trace elements are transported to the equatorial Pacific seafloor. The 1. (1) high biogenic source at the Equator, associated with equatorial divergence of surface water and upwelling of nutrient-rich water, and 2. (2) high aluminosilicate flux at 4°N, associated with increased terrigenous input from elevated rainfall at the Intertropical Convergence Zone (ITCZ) of the tradewinds, are the two most important fluxes with which elemental transport is affiliated. The biogenic flux at the Equator transports Ca and Sr structurally bound to carbonate tests and Mn primarily as an adsorbed component. Trace elements such as Cr, As, Pb, and the REEs are also influenced by the biogenic flux at the Equator, although this affiliation is not regionally dominant. Normative calculations suggest that extremely large fluxes of Ba and P at the Equator are carried by only small proportions of barite and apatite phases. The high terrigenous flux at the ITCZ has a profound effect on chemical transport to the seafloor, with elemental fluxes increasing tremendously and in parallel with Ti. Normative calculations, however, indicate that these fluxes are far in excess of what can be supplied by lattice-bound terrigenous phases. The accumulation of Ba is greater than is affiliated with biogenic transport at the Equator, while the P flux at the ITCZ is only 10% less than at the Equator. This challenges the common view that Ba and P are essentially exclusively associated with biogenic fluxes. Many other elements (including Mn, Pb, As, and REEs) also record greater accumulation beneath the ITCZ than at the Equator. Thus, adsorptive scavenging by terrigenous paniculate matter, or phases intimately associated with them, appears to be an extremely important process regulating elemental transport to the equatorial Pacific seafloor. These findings emphasize the role of vertical transport to the sediment, and provide additional constraints on the paleochemical use of trace elements to track biogenic and terrigenous fluxes.

Dissolved Zinc in the Southern Ocean measured on water bottle samples during POLARSTERN cruise ANT-XXIV/3

The distribution of dissolved zinc (Zn) was investigated in the Atlantic sector of the Southern Ocean in the austral autumn of 2008 as part of the IPY GEOTRACES expedition ZERO & DRAKE. Research focused on transects across the major frontal systems along the Zero Meridian and across the Drake Passage. There was a strong gradient in surface zinc concentrations observed across the Antarctic Polar Front along both transects and high zinc levels were found in surface waters throughout the Southern Ocean. Vertical profiles for dissolved Zinc showed the presence of local minima and maxima in the upper 200 m consistent with significant uptake by phytoplankton and release by zooplankton grazing, respectively. Highest deep water zinc concentrations were found in the centre of the Weddell Gyre associated with Central Intermediate Water (CIW), a water mass which is depleted in O2, elevated in CO2 and is regionally a CFC minimum. Our data suggests that the remineralization of sinking particles is a key control on the distribution of Zn in the Southern Ocean. Disappearance ratios of zinc to phosphate (Zn:P) in the upper water column increased southwards along both transects and based on laboratory studies they suggest slower growth rates of phytoplankton due to iron or light limitation. Zinc and silicate were strongly correlated throughout the study region but the disappearance ratio (Zn:Si) was relatively uniform overall except for the region close to the ice edge on the Zero Meridian.

Radiolarian abundance of sediment core Y8

A high-resolution record of radiolarian faunal changes from Site Y8 south of the Subtropical Front (STF), offshore eastern New Zealand, provides insight into the paleoceanographic history of the last 265 kyrs. Quantitative analysis of radiolarian paleotemperature indicators and radiolarian-based sea surface temperature (SST) estimates reveal distinct shifts during glacial-interglacial (G-I) climate cycles encompassing marine isotope stages (MIS) 8-1. Faunas at Site Y8 are abundant and diverse and consist of a mixture of species typical of the subantarctic, transitional and subtropical zones which is characteristic of subantarctic waters just south of the STF. During interglacials, diverse radiolarian faunas have increased numbers of warm-water taxa (not, vert, similar 15%) while cool-water taxa decrease to not, vert, similar 11% of the assemblage. Warmest climate conditions occurred during MIS 5.5 and the early Holocene Climatic Optimum (HCO) at the onset of MIS 1 where SSTs reach maxima of 12.8 and 12.9 °C, respectively. This suggests that temperatures during the HCO were comparable to the Eemian, one of the warmest interglacial intervals of the Late Quaternary. Glacials are characterized by less diverse radiolarian faunas with cool-water taxa increasing to 49% of the assemblage. Coolest climate conditions occurred in MIS 4 and 2 where SSTs are reduced to 5.4 °C and 4.3 °C, respectively. Radiolarian faunal changes and SST estimates clearly identify major water masses and oceanic fronts in the offshore eastern New Zealand area. During warmest MIS 5.5 and early MIS 1 substantial influence of northern-sourced Subtropical Surface Water (STW) is evident at Site Y8. This implies southward incursions of STW around the eastern crest of Chatham Rise with the STF displaced towards higher latitudes and spinning off eddies as far south as Campbell Plateau. Additionally, increased flow of the Southland Current (SC) might have enhanced the local occurrence of warm-water radiolarians derived from the subtropical Tasman Sea. Coolest glacials are marked by a strong inflow of cool, southern-sourced waters at Site Y8 indicating a more vigorous flow along the Subantarctic Front (SAF).

Magnetic suceptibility, carbon and oxygen stable isotope ratios and reflectance from the Bottaccione Gorge and Furlo section, Italy

The oceans at the time of the Cenomanian-Turonian transition were abruptly perturbed by a period of bottom-water anoxia. This led to the brief but widespread deposition of black organic-rich shales, such as the Livello Bonarelli in the Umbria-Marche Basin (Italy). Despite intensive studies, the origin and exact timing of this event are still debated. In this study, we assess leading hypotheses about the inception of oceanic anoxia in the Late Cretaceous greenhouse world, by providing a 6-Myr-long astronomically-tuned timescale across the Cenomanian-Turonian boundary. We procure insights in the relationship between orbital forcing and the Late Cretaceous carbon cycle by deciphering the imprint of astronomical cycles on lithologic, geophysical, and stable isotope records, obtained from the Bottaccione, Contessa and Furlo sections in the Umbria-Marche Basin. The deposition of black shales and cherts, as well as the onset of oceanic anoxia, is related to maxima in the 405-kyr cycle of eccentricity-modulated precession. Correlation to radioisotopic ages from the Western Interior (USA) provides unprecedented age control for the studied Italian successions. The most likely tuned age for the Livello Bonarelli base is 94.17 ± 0.15 Ma (tuning #1); however, a 405-kyr older age cannot be excluded (tuning #2) due to uncertainties in stratigraphic correlation, radioisotopic dating, and orbital configuration. Our cyclostratigraphic framework suggests that the exact timing of major carbon cycle perturbations during the Cretaceous may be linked to increased variability in seasonality (i.e. a 405-kyr eccentricity maximum) after the prolonged avoidance of seasonal extremes (i.e. a 2.4-Myr eccentricity minimum). Volcanism is probably the ultimate driver of oceanic anoxia, but orbital periodicities determine the exact timing of carbon cycle perturbations in the Late Cretaceous. This unites two leading hypotheses about the inception of oceanic anoxia in the Late Cretaceous greenhouse world.