Mathematical aspects of Feynman integrals

In the present dissertation we consider Feynman integrals in the framework of dimensional regularization. As all such integrals can be expressed in terms of scalar integrals, we focus on this latter kind of integrals in their Feynman parametric representation and study their mathematical properties, partially applying graph theory, algebraic geometry and number theory. The three main topics are the graph theoretic properties of the Symanzik polynomials, the termination of the sector decomposition algorithm of Binoth and Heinrich and the arithmetic nature of the Laurent coefficients of Feynman integrals.rnrnThe integrand of an arbitrary dimensionally regularised, scalar Feynman integral can be expressed in terms of the two well-known Symanzik polynomials. We give a detailed review on the graph theoretic properties of these polynomials. Due to the matrix-tree-theorem the first of these polynomials can be constructed from the determinant of a minor of the generic Laplacian matrix of a graph. By use of a generalization of this theorem, the all-minors-matrix-tree theorem, we derive a new relation which furthermore relates the second Symanzik polynomial to the Laplacian matrix of a graph.rnrnStarting from the Feynman parametric parameterization, the sector decomposition algorithm of Binoth and Heinrich serves for the numerical evaluation of the Laurent coefficients of an arbitrary Feynman integral in the Euclidean momentum region. This widely used algorithm contains an iterated step, consisting of an appropriate decomposition of the domain of integration and the deformation of the resulting pieces. This procedure leads to a disentanglement of the overlapping singularities of the integral. By giving a counter-example we exhibit the problem, that this iterative step of the algorithm does not terminate for every possible case. We solve this problem by presenting an appropriate extension of the algorithm, which is guaranteed to terminate. This is achieved by mapping the iterative step to an abstract combinatorial problem, known as Hironaka's polyhedra game. We present a publicly available implementation of the improved algorithm. Furthermore we explain the relationship of the sector decomposition method with the resolution of singularities of a variety, given by a sequence of blow-ups, in algebraic geometry.rnrnMotivated by the connection between Feynman integrals and topics of algebraic geometry we consider the set of periods as defined by Kontsevich and Zagier. This special set of numbers contains the set of multiple zeta values and certain values of polylogarithms, which in turn are known to be present in results for Laurent coefficients of certain dimensionally regularized Feynman integrals. By use of the extended sector decomposition algorithm we prove a theorem which implies, that the Laurent coefficients of an arbitrary Feynman integral are periods if the masses and kinematical invariants take values in the Euclidean momentum region. The statement is formulated for an even more general class of integrals, allowing for an arbitrary number of polynomials in the integrand.

Phylogeny and historical biogeography of the Australian Camphorosmeae (Chenopodiaceae)

Diese Studie befasst sich mit der Phylogenie und Biogeographie der australischen Camphorosmeae, die ein wichtiges Element der Flora arider Gebiete Australiens sind. Die molekularen Phylogenien wurden mit Hilfe Bayes’scher Statistik und „maximum likelihood”berechnet. Um das Alter der Gruppe und interner Linien abzuschätzen, wurden die Methoden „Nonparametric rate smoothing” und “penalized likelihood” benutzt. Morphologische Merkmale wurden nach Kriterien der Parsimonie auf den molekularen Baum aufgetragen. „Brooks parsimony analysis”, „cladistic analysis of distributions and endemism”, „dispersal-vicariance analysis”,„ancestral area analysis” und „weighted ancestral area analysis” wurden angewandt, um Abfolge und Richtungen der Ausbreitung der Gruppe in Australien zu analysieren.Von sieben getesteten Markern hatten nur die nukleären ETS und ITS genügend Variation für die phylogenetische Analyse der Camphorosmeae. Die plastidären Marker trnL-trnF spacer,trnP-psaJ spacer, rpS16 intron, rpL16 intron und trnS-trnG spacer zeigten kein ausreichendes phylogenetisches Signal. Die gefundenen phylogenetischen Hypothesen widersprechen der jetzigen Taxonomie der Gruppe. Neobassia, Threlkeldia, Osteocarpum und Enchylaena sollten den Gattungen Sclerolaena bzw. Maireana zugeordnet werden. Die kladistische Analyse der Fruchtanhängsel unterstützt die taxonomischen Ergebnisse der auf DNA basierenden Phylogenie. Allerdings hat die Behaarung, die bei anderen Gruppen der Chenopodiaceae als wichtiges taxonomisches Merkmal herangezogen wird, die Phylogenie nicht unterstützt. Vorfahren der heutigen Camphorosmeen sind im Miozän, vor ca. 8-14 Millionen Jahren, durch Fernausbreitung vermutlich aus Asien in Australien eingewandert. Anfängliche Diversifizierung fand während des späten Miozäns bis in das frühe Pliozän vor ca. 4-7 Millionen Jahren statt. Am Ende des Pliozäns existierten schon 45% - 72% der Abstammungslinien der jetzigen Camphorosmeen. Dies weist auf eine schnelle Ausbreitung hin. Das Alter stimmt mit dem Einsetzen der Aridisierung Australiens überein, und deutet darauf hin, dass die Ausbreitung der ariden Gebiete eine große Rolle bei der Diversifizierung der Gruppe spielte. Die Vorfahren der australischen Camphorosmeae scheinen die Südküste Australiens zuerst besiedeln zu haben. Dies geschah vor dem Einsetzen der Aridisierung des Kontinents. Die anschließende Ausbreitung erfolgte in verschiedene Richtungen und folgte der fortschreitenden Austrocknung im späten Tertiär und im ganzen Quartär. Durch ihre Anpassung an Trockenheit ist der Erfolg der Camphorosmeae in den ariden Gebieten zu erklären.Die Abwesenheit von klaren phylogenetischen und artspezifischen Signalen zwischen Arten der australischen Camphorosmeae ist auf das junge Alter und die schnelle Diversifizierung der Gruppe zurückzuführen, welche die Häufung von Mutationen und eine starke morphologische Differenzierung nicht zugelassen haben.

Non-parametric estimation of the diffusion coefficient of a branching diffusion with immigration

Wir betrachten Systeme von endlich vielen Partikeln, wobei die Partikel sich unabhängig voneinander gemäß eindimensionaler Diffusionen [dX_t = b(X_t),dt + sigma(X_t),dW_t] bewegen. Die Partikel sterben mit positionsabhängigen Raten und hinterlassen eine zufällige Anzahl an Nachkommen, die sich gemäß eines Übergangskerns im Raum verteilen. Zudem immigrieren neue Partikel mit einer konstanten Rate. Ein Prozess mit diesen Eigenschaften wird Verzweigungsprozess mit Immigration genannt. Beobachten wir einen solchen Prozess zu diskreten Zeitpunkten, so ist zunächst nicht offensichtlich, welche diskret beobachteten Punkte zu welchem Pfad gehören. Daher entwickeln wir einen Algorithmus, um den zugrundeliegenden Pfad zu rekonstruieren. Mit Hilfe dieses Algorithmus konstruieren wir einen nichtparametrischen Schätzer für den quadrierten Diffusionskoeffizienten $sigma^2(cdot),$ wobei die Konstruktion im Wesentlichen auf dem Auffüllen eines klassischen Regressionsschemas beruht. Wir beweisen Konsistenz und einen zentralen Grenzwertsatz.

Climate change in Libya and desertification of Jifara Plain

The study was arranged to manifest its objectives through preceding it with an intro-duction. Particular attention was paid in the second part to detect the physical settings of the study area, together with an attempt to show the climatic characteristics in Libya. In the third part, observed temporal and spatial climate change in Libya was investigated through the trends of temperature, precipitation, relative humidity and cloud amount over the peri-ods (1946-2000), (1946-1975), and (1976-2000), comparing the results with the global scales. The forth part detected the natural and human causes of climate change concentrat-ing on the greenhouse effect. The potential impacts of climate change on Libya were ex-amined in the fifth chapter. As a case study, desertification of Jifara Plain was studied in the sixth part. In the seventh chapter, projections and mitigations of climate change and desertification were discussed. Ultimately, the main results and recommendations of the study were summarized. In order to carry through the objectives outlined above, the following methods and approaches were used: a simple linear regression analysis was computed to detect the trends of climatic parameters over time; a trend test based on a trend-to-noise-ratio was applied for detecting linear or non-linear trends; the non-parametric Mann-Kendall test for trend was used to reveal the behavior of the trends and their significance; PCA was applied to construct the all-Libya climatic parameters trends; aridity index after Walter-Lieth was shown for computing humid respectively arid months in Libya; correlation coefficient, (after Pearson) for detecting the teleconnection between sun spot numbers, NAOI, SOI, GHGs, and global warming, climate changes in Libya; aridity index, after De Martonne, to elaborate the trends of aridity in Jifara Plain; Geographical Information System and Re-mote Sensing techniques were applied to clarify the illustrations and to monitor desertifi-cation of Jifara Plain using the available satellite images MSS, TM, ETM+ and Shuttle Radar Topography Mission (SRTM). The results are explained by 88 tables, 96 figures and 10 photos. Temporal and spatial temperature changes in Libya indicated remarkably different an-nual and seasonal trends over the long observation period 1946-2000 and the short obser-vation periods 1946-1975 and 1976-2000. Trends of mean annual temperature were posi-tive at all study stations except at one from 1946-2000, negative trends prevailed at most stations from 1946-1975, while strongly positive trends were computed at all study stations from 1976-2000 corresponding with the global warming trend. Positive trends of mean minimum temperatures were observed at all reference stations from 1946-2000 and 1976-2000, while negative trends prevailed at most stations over the period 1946-1975. For mean maximum temperature, positive trends were shown from 1946-2000 and from 1976-2000 at most stations, while most trends were negative from 1946-1975. Minimum tem-peratures increased at nearly more than twice the rate of maximum temperatures at most stations. In respect of seasonal temperature, warming mostly occurred in summer and au-tumn in contrast to the global observations identifying warming mostly in winter and spring in both study periods. Precipitation across Libya is characterized by scanty and sporadically totals, as well as high intensities and very high spatial and temporal variabilities. From 1946-2000, large inter-annual and intra-annual variabilities were observed. Positive trends of annual precipi-tation totals have been observed from 1946-2000, negative trends from 1976-2000 at most stations. Variabilities of seasonal precipitation over Libya are more strikingly experienced from 1976-2000 than from 1951-1975 indicating a growing magnitude of climate change in more recent times. Negative trends of mean annual relative humidity were computed at eight stations, while positive trends prevailed at seven stations from 1946-2000. For the short observation period 1976-2000, positive trends were computed at most stations. Annual cloud amount totals decreased at most study stations in Libya over both long and short periods. Re-markably large spatial variations of climate changes were observed from north to south over Libya. Causes of climate change were discussed showing high correlation between tempera-ture increasing over Libya and CO2 emissions; weakly positive correlation between pre-cipitation and North Atlantic Oscillation index; negative correlation between temperature and sunspot numbers; negative correlation between precipitation over Libya and Southern Oscillation Index. The years 1992 and 1993 were shown as the coldest in the 1990s result-ing from the eruption of Mount Pinatubo, 1991. Libya is affected by climate change in many ways, in particular, crop production and food security, water resources, human health, population settlement and biodiversity. But the effects of climate change depend on its magnitude and the rate with which it occurs. Jifara Plain, located in northwestern Libya, has been seriously exposed to desertifica-tion as a result of climate change, landforms, overgrazing, over-cultivation and population growth. Soils have been degraded, vegetation cover disappeared and the groundwater wells were getting dry in many parts. The effect of desertification on Jifara Plain appears through reducing soil fertility and crop productivity, leading to long-term declines in agri-cultural yields, livestock yields, plant standing biomass, and plant biodiversity. Desertifi-cation has also significant implications on livestock industry and the national economy. Desertification accelerates migration from rural and nomadic areas to urban areas as the land cannot support the original inhabitants. In the absence of major shifts in policy, economic growth, energy prices, and con-sumer trends, climate change in Libya and desertification of Jifara Plain are expected to continue in the future. Libya cooperated with United Nations and other international organizations. It has signed and ratified a number of international and regional agreements which effectively established a policy framework for actions to mitigate climate change and combat deserti-fication. Libya has implemented several laws and legislative acts, with a number of ancil-lary and supplementary rules to regulate. Despite the current efforts and ongoing projects being undertaken in Libya in the field of climate change and desertification, urgent actions and projects are needed to mitigate climate change and combat desertification in the near future.