Serie di Puiseux

La tesi è incentrata sullo studio dei punti di singolarità di una curva nel piano proiettivo complesso. Nel caso in cui il punto sia regolare possiamo sfruttare il teorema delle funzioni implicite che ci permette di esplicitare il luogo di zeri di un'equazione implicita rispetto a una variabile. Quando questa ipotesi di regolarità viene meno per avere un risultato analogo diventa necessario utilizzare le serie di Puiseux. L'interpretazione algebrica del teorema di Puiseux risponde alla domanda di trovare un'estensione del campo delle serie di Laurent che sia algebricamente chiuso; prendendo un polinomio di grado positivo in K(x)*[y], mostreremo che esiste sempre una radice del polinomio appartenente a K(x)*. Il legame con l’interpretazione analitica risulta ora evidente: data infatti una curva nel piano complesso la sua equazione può essere vista come un particolare polinomio in K(x)*[y], esplicitare la y in funzione della x equivale appunto a trovare una radice in K(x)*. Nel primo capitolo abbiamo in primo luogo richiamato il risultato di Dini e parlato del luogo singolare di una curva, mostrando che quest'ultimo è un numero finito di punti. In seguito abbiamo introdotto il poligono di Newton, il quale è un insieme convesso del piano associato ad un polinomio in due variabili. Nel secondo capitolo abbiamo visto due formulazioni del teorema di Puiseux, entrambe le dimostrazioni di questo risultato sono costruttive; per renderle più scorrevoli abbiamo ritenuto opportuno costruire degli esempi che evidenziassero i vari passi.

Coordinate isoterme

Questa tesi si propone di verificare l'esistenza di coordinate isoterme su una superficie. Le coordinate isoterme danno localmente una mappa conforme da una varietà riemanniana bidimensionale al piano Euclideo. Se la superficie è orientabile, allora si può dare un atlante di carte isoterme, cioè le cui coordinate associate siano isoterme. Queste coordinate esistono a patto che vengano soddisfatte certe condizioni. Il risultato nelle classi di Holder è dovuto a Korn e Lichtensten. Chern ha notevolmente semplificato la loro dimostrazione.

Applications of elliptic functions to solve differential equations

In questa tesi si mostrano alcune applicazioni degli integrali ellittici nella meccanica Hamiltoniana, allo scopo di risolvere i sistemi integrabili. Vengono descritte le funzioni ellittiche, in particolare la funzione ellittica di Weierstrass, ed elenchiamo i tipi di integrali ellittici costruendoli dalle funzioni di Weierstrass. Dopo aver considerato le basi della meccanica Hamiltoniana ed il teorema di Arnold Liouville, studiamo un esempio preso dal libro di Moser-Integrable Hamiltonian Systems and Spectral Theory, dove si prendono in considerazione i sistemi integrabili lungo la geodetica di un'ellissoide, e il sistema di Von Neumann. In particolare vediamo che nel caso n=2 abbiamo un integrale ellittico.

Detection of complex point targets with distributed assets in a MIMO radar system

The report explores the problem of detecting complex point target models in a MIMO radar system. A complex point target is a mathematical and statistical model for a radar target that is not resolved in space, but exhibits varying complex reflectivity across the different bistatic view angles. The complex reflectivity can be modeled as a complex stochastic process whose index set is the set of all the bistatic view angles, and the parameters of the stochastic process follow from an analysis of a target model comprising a number of ideal point scatterers randomly located within some radius of the targets center of mass. The proposed complex point targets may be applicable to statistical inference in multistatic or MIMO radar system. Six different target models are summarized here – three 2-dimensional (Gaussian, Uniform Square, and Uniform Circle) and three 3-dimensional (Gaussian, Uniform Cube, and Uniform Sphere). They are assumed to have different distributions on the location of the point scatterers within the target. We develop data models for the received signals from such targets in the MIMO radar system with distributed assets and partially correlated signals, and consider the resulting detection problem which reduces to the familiar Gauss-Gauss detection problem. We illustrate that the target parameter and transmit signal have an influence on the detector performance through target extent and the SNR respectively. A series of the receiver operator characteristic (ROC) curves are generated to notice the impact on the detector for varying SNR. Kullback–Leibler (KL) divergence is applied to obtain the approximate mean difference between density functions the scatterers assume inside the target models to show the change in the performance of the detector with target extent of the point scatterers.

Variations of the FEAST Eigenvalue Algorithm

FEAST is a recently developed eigenvalue algorithm which computes selected interior eigenvalues of real symmetric matrices. It uses contour integral resolvent based projections. A weakness is that the existing algorithm relies on accurate reasoned estimates of the number of eigenvalues within the contour. Examining the singular values of the projections on moderately-sized, randomly-generated test problems motivates orthogonalization-based improvements to the algorithm. The singular value distributions provide experimentally robust estimates of the number of eigenvalues within the contour. The algorithm is modified to handle both Hermitian and general complex matrices. The original algorithm (based on circular contours and Gauss-Legendre quadrature) is extended to contours and quadrature schemes that are recursively subdividable. A general complex recursive algorithm is implemented on rectangular and diamond contours. The accuracy of different quadrature schemes for various contours is investigated.

Chronology of Lake El'gygytgyn sediments – a combined magnetostratigraphic, palaeoclimatic and orbital tuning study based on multi-parameter analyses

A 318-metre-long sedimentary profile drilled by the International Continental Scientific Drilling Program (ICDP) at Site 5011-1 in Lake El’gygytgyn, Far East Russian Arctic, has been analysed for its sedimentologic response to global climate modes by chronostratigraphic methods. The 12 km wide lake is sited off-centre in an 18 km large crater that was created by the impact of a meteorite 3.58 Ma ago. Since then sediments have been continuously deposited. For establishing their chronology, major reversals of the earth’s magnetic field provided initial tie points for the age model, confirming that the impact occurred in the earliest geomagnetic Gauss chron. Various stratigraphic parameters, reflecting redox conditions at the lake floor and climatic conditions in the catchment were tuned synchronously to Northern Hemisphere insolation variations and the marine oxygen isotope stack, respectively. Thus, a robust age model comprising more than 600 tie points could be defined. It could be shown that deposition of sediments in Lake El’gygytgyn occurred in concert with global climatic cycles. The upper �160m of sediments represent the past 3.3 Ma, equivalent to sedimentation rates of 4 to 5 cm ka−1, whereas the lower 160m represent just the first 0.3 Ma after the impact, equivalent to sedimentation rates in the order of 45 cm ka−1. This study also provides orbitally tuned ages for a total of 8 tephras deposited in Lake El’gygytgyn.

A Bayesian approach to estimating the regression coefficients of a multinomial logit model

A Bayesian approach to estimation of the regression coefficients of a multinominal logit model with ordinal scale response categories is presented. A Monte Carlo method is used to construct the posterior distribution of the link function. The link function is treated as an arbitrary scalar function. Then the Gauss-Markov theorem is used to determine a function of the link which produces a random vector of coefficients. The posterior distribution of the random vector of coefficients is used to estimate the regression coefficients. The method described is referred to as a Bayesian generalized least square (BGLS) analysis. Two cases involving multinominal logit models are described. Case I involves a cumulative logit model and Case II involves a proportional-odds model. All inferences about the coefficients for both cases are described in terms of the posterior distribution of the regression coefficients. The results from the BGLS method are compared to maximum likelihood estimates of the regression coefficients. The BGLS method avoids the nonlinear problems encountered when estimating the regression coefficients of a generalized linear model. The method is not complex or computationally intensive. The BGLS method offers several advantages over Bayesian approaches. ^

Ay Pitágoras! : acerca de las determinaciones sociales en el surgimiento de las matemáticas

El artículo pretende mostrar a partir del ejemplo del descubrimiento del Teorema de Pitágoras, cómo las ideas sociales de una época determinada, para nuestro caso las de la sociedad griega del siglo V a.C., intervienen en el proceso de formación científica ya sea para facilitar o bloquear su desarrollo. El caso de las matemáticas pitagóricas es significativo al respecto; su análisis no deja de ser un punto paradigmático en la historia de las ciencias desde el punto de vista de la sociología.

Late Neogene biostratigraphy of ODP Leg 135 sites

Integration of biostratigraphic and magnetostratigraphic results from Leg 135 sites has given additional information as to the position and reliability of various bioevents compared with previously published results. Two sites (834: Gilbert to Brunhes; and 836: Brunhes) provided excellent magnetic and biostratigraphic data. From these it is suggested that some bioevents are older than previously recorded: the first appearances (FAs) of Emiliania huxleyi (within the Brunhes Chron, at the same level as the FA of Helicosphaera inversa) and Globorotalia (Truncorotalia) truncatulinoides (within the upper Gauss Chron), and the last appearance (LA) of Gr. (Tr.) tosaensis (upper Matuyama Chron). The FA of Gr. (Tr.) crassaformis hessi is variable, but the oldest occurrence is just below the Cobb Mountain Subchron. Other key bioevents, such as the LAs of Discoaster pentaradiatus (just above the Réunion Subchron), D. tamalis (within the lower reversed part of the Matuyama Chron), Sphenolithus (lower Gauss Chron), and Amaurolithus primus (topmost Gilbert Chron) appear higher than previously recorded. Some key biostratigraphic taxa, such as Globigerinoides quadrilobatus fistulosus, Pulleniatina finalis, P. primalis, and Sphaeroidinella dehiscens, are either rare or their distribution is sporadic to the extent that they are unsuitable for biostratigraphic use in the area studied. Because of the rarity of P. primalis, the FA of Globorotalia (Globorotalia) multicamerata has been used to mark the base of Zone N17B. Though levels are present at most sites in which populations of Pulleniatina are sinistrally coiled, it is difficult to equate these coiling changes with previous records.

Organic geochemistry of sediments of the Gulf of California

The effects of intrusive thermal stress have been studied on a number of Pleistocene sediment samples obtained from Leg 64 of the DSDP-IPOD program in the Gulf of California. Samples were selected from Sites 477, 478 and 481 where the organic matter was subjected to thermal stress from sill intrusions. For comparison purposes, samples from Sites 474 and 479 were selected as representative of unaltered material. The GC and GC-MS data show that lipids of the thermally unaltered samples were derived from microbial and terrestrial higher-plant detritus. Samples from sill proximities were found to contain thermally-derived distillates and those adjacent to sills contained essentially no lipids. Curie point pyrolysis combined with GC and GC-MS was used to show that kerogens from the unaltered samples reflected their predominantly autochthonous microbial origin. Pyrograms of the altered kerogens were much less complex than the unaltered samples, reflecting the thermal effects. The kerogens adjacent to the sills produce little or no pyrolysis products since these intrusions into unconsolidated, wet sediments resulted in in situ pyrolysis of the organic matter. Examination of the kerogens by ESR showed that spin density and line width pass through a maximum during the course of alteration but ESR g-values show no correlation with maturity. Stable carbon isotope (d13C) values of kerogens decrease by 1-1.5 per mil near the sills at Sites 477 and 481 and the atomic N/C decreases slightly with proximity to a smaller sill at Site 478. Differences in maturation behavior between Site 477 and 481 and Site 478 are attributed to dissimilarities in thermal stress and to chemical and isotopic heterogeneity of Guaymas Basin protokerogen.

Biogenic and mineral composition and foraminiferal stratigraphy of sediment cores obtained in northern Germany

The Ratekau boring ended in clays of the so-called Asterigerina-Zone; these clays have shallow-water features in the uppermost samples. The clays are overlain by deep-water clays with pteropods; this formation is split into two parts by a shallow-water deposit. The fossiliferous series ends upward in sandy deposits with shallow-water fossils. The question is raised whether the two deep-water deposits might correspond to the Lower Doberg Beds (Eochattian) and the Upper Doberg Beds (Neochattian) at the Doberg hill, closer to the rim of the basin. All fossiliferous samples from this boring are thought to be of Late Oligocene age; the boundary towards the Middle Oligocene, however, could not be ascertained. The Vaale boring ended in rather typical Septaria clay of the Middle Oligocene. This clay is capped by some metres of unfossiliferous glauconite clays, which in turn are overlain by silts and silty clays with planktonic fossils identical to those found at Dingden locality. These deposits are tentatively dated as Early Miocene. The next higher series of samples consists of sands and clays deposited in shallower waters. They contain a rich fauna of benthic molluscs, which, according to the current notion in stratigraphy, would have a Reinbek Age. In addition, they contain a set of planktonic fossils which differs from the 'Lower Miocene' assemblages. These sands and clays are overlain by a thick series of marine sands very poor in fossils. Finally, four metres of clay with foraminifera, having Younger Miocene affinities, form the top of the fossiliferous sequence. The borings at Wulksfelde and Langenhorn were not far apart and their sediments are easily correlated. Both wells start below in continental 'Lignite Sands' and contain overlying shallow water sands and clays. These yielded Hemmoor benthic mollusca, supposed to indicate Lower Miocene in the relevant literature; however, we encountered their planktonic foraminifera in the uppermost Miocene as well. The same planktonic species were found in all samples of both borings. These deposits under discussion furthermore contain a particular pteropod species. They are overlain by a thick series of gypsiferous clays, with scarce fossils. The uppermost fossiliferous clays (probably Langenfelde Age) contain another pteropod species, not met with in other samples. The discrepancies between the plankton zonation and the traditional subdivision according to benthic molluscs in the borings of Vaale, Wulksfelde and Langenhorn (and in samples from Twistringen, Dingden and Antwerp localities as well) renders the time-stratigraphic value of the denominations Reinbek and Hemmoor rather doubtful. The samples of the Westerland boring can be placed in the Gram and Sylt stages of local chronostratigraphy on the strength of the Astarte series established by HINSCH. The Gram samples contain a typical pteropod species; both groups of samples contain the same planktonic foraminifera as the borings Wulksfelde and Langenhorn. Our material did not bring the problem of the Miocene-Pliocene boundary in this region any closer to a solution. In conclusion, it can be claimed that this investigation provides strong arguments that the usual recognition of Hemmoor and Reinbek does not correspond to well-defined chronostratigraphical units. A better chronostratigraphic subdivision has to be based on the examination of many more samples, and on a better understanding of the paleoecology of the fossils involved.

Middle Miocene to Quaternary diatom biostratigraphy of DSDP Site 90-594

A diatom biostratigraphy is presented for middle Miocene through Quaternary sediments recovered from the Chatham Rise east of New Zealand's South Island. The upper 590 m of the 639.5-m composite-section Site 594 represents approximately 16 m.y. and is characterized by moderately to very poorly preserved diatoms of antarctic to temperate affinity. Pliocene through Quaternary assemblages are poorly preserved and dominated by antarctic-subantarctic species which provide detailed biostratigraphic control. Recognized are 11 of 14 zones of the middle upper Miocene to Quaternary Neogene Southern Ocean diatom zonation (NSD 7-NSD 20) of Ciesielski (1983; this chapter). Four Neogene Southern Ocean diatom zones (NSD 3-NSD 6) are recognized in the lower middle Miocene to middle upper Miocene of Site 594. Assemblages of this interval have a mixed high-latitude and temperate affinity; however, poor preservation limits correlation to high- and temperate-latitude zonal schemes. Neogene North Pacific diatom zones and subzones of NNPD 3 through NNPD 5 (Barron, in press, b) are correlated to Neogene Southern Ocean diatom zones NSD 3 through NSD 7: the upper portions of the Actinocyclus ingens Zone (NNPD 3) is correlative to the upper Nitzschia maleinterpretaria Zone (NSD 3); the Denticulopsis lauta Zone (NNPD 4) and Subzones a and b are correlative to the lower Coscinodiscus lewisianus Zone (NSD 4); and the D. hustedtü-D. lauta Zone (NNPD 5) and its Subzones a through d encompass the upper C. lewisianus Zone (NSD 4), N. grossepunctata Zone (NSD 5), N. denticuloides Zone (NSD 6), and the lower D. hustedtii-D. lauta Zone (NSD 7). A major disconformity spans the late Gilbert to early Gauss Chron (3.9-2.8 Ma). A second disconformity brackets the Miocene/Pliocene boundary; the section missing covers late Chron 5 and the early Gilbert chron (5.5-4.6 Ma). The remainder of the siliceous-fossil-bearing Miocene sediments at Site 594 appear to be correlative to lower paleomagnetic Chronozone 5 through upper Chronozone 16. Uppermost lower Miocene or lowermost middle Miocene sediments in the basal 50 m of Hole 594A are barren of diatoms.