IDM - A New Parallel Methodology to Calculate the Determinant of Matrices of the Order n, with Computational Complexity O(n)

This paper presents a new parallel methodology for calculating the determinant of matrices of the order n, with computational complexity O(n), using the Gauss-Jordan Elimination Method and Chio's Rule as references. We intend to present our step-by-step methodology using clear mathematical language, where we will demonstrate how to calculate the determinant of a matrix of the order n in an analytical format. We will also present a computational model with one sequential algorithm and one parallel algorithm using a pseudo-code.

Assimilation of Meteosat Second Generation (MSG) satellite data in a regional numerical weather prediction model using a one-dimensional variational approach

The quality of temperature and humidity retrievals from the infrared SEVIRI sensors on the geostationary Meteosat Second Generation (MSG) satellites is assessed by means of a one dimensional variational algorithm. The study is performed with the aim of improving the spatial and temporal resolution of available observations to feed analysis systems designed for high resolution regional scale numerical weather prediction (NWP) models. The non-hydrostatic forecast model COSMO (COnsortium for Small scale MOdelling) in the ARPA-SIM operational configuration is used to provide background fields. Only clear sky observations over sea are processed. An optimised 1D–VAR set-up comprising of the two water vapour and the three window channels is selected. It maximises the reduction of errors in the model backgrounds while ensuring ease of operational implementation through accurate bias correction procedures and correct radiative transfer simulations. The 1D–VAR retrieval quality is firstly quantified in relative terms employing statistics to estimate the reduction in the background model errors. Additionally the absolute retrieval accuracy is assessed comparing the analysis with independent radiosonde and satellite observations. The inclusion of satellite data brings a substantial reduction in the warm and dry biases present in the forecast model. Moreover it is shown that the retrieval profiles generated by the 1D–VAR are well correlated with the radiosonde measurements. Subsequently the 1D–VAR technique is applied to two three–dimensional case–studies: a false alarm case–study occurred in Friuli–Venezia–Giulia on the 8th of July 2004 and a heavy precipitation case occurred in Emilia–Romagna region between 9th and 12th of April 2005. The impact of satellite data for these two events is evaluated in terms of increments in the integrated water vapour and saturation water vapour over the column, in the 2 meters temperature and specific humidity and in the surface temperature. To improve the 1D–VAR technique a method to calculate flow–dependent model error covariance matrices is also assessed. The approach employs members from an ensemble forecast system generated by perturbing physical parameterisation schemes inside the model. The improved set–up applied to the case of 8th of July 2004 shows a substantial neutral impact.

Bestimmung der Ionisationsenergie von Actinium und Ultraspurenanalyse von Plutonium mit resonanter Ionisationsmassenspektrometrie (RIMS)

ZusammenfassungDie Resonanzionisationsmassenspektrometrie (RIMS) verbindet hohe Elementselektivität mit guter Nachweiseffizienz. Aufgrund dieser Eigenschaften ist die Methode für Ultraspurenanalyse und Untersuchungen an seltenen oder schwer handhabbaren Elementen gut geeignet. Für RIMS werden neutrale Atome mit monochromatischem Laserlicht ein- oder mehrfach resonant auf energetisch hoch liegende Niveaus angeregt und anschließend durch einen weiteren Laserstrahl oder durch ein elektrisches Feld ionisiert. Die Photoionen werden in einem Massenspektrometer massenselektiv registriert.Ein Beispiel für die Anwendung von RIMS ist die präzise Bestimmung der Ionisationsenergie als fundamentale physikalisch-chemische Eigenschaft eines bestimmten Elements; insbesondere bei den Actinoiden ist die Kenntnis der Ionisationsenergie von Interesse, da es dort bis zur Anwendung der laser-massenspektroskopischer Methode nur wenige experimentelle Daten gab. Die Bestimmung der Ionisationsenergie erfolgt durch die Methode der Photoionisation im elektrischen Feld gemäß dem klassischen Sattelpunktsmodell. Im Experiment werden neutrale Atome in einem Atomstrahl mittels Laserlicht zunächst resonant angeregt. Die angeregten Atome befinden sich in einem äußeren, statischen elektrischen Feld und werden durch einen weiteren Laserstrahl, dessen Wellenlänge durchgestimmt wird, ionisiert. Das Überschreiten der Laserschwelle macht sich durch einen starken Anstieg im Ionensignal bemerkbar. Man führt diese Messung bei verschiedenen elektrischen Feldstärken durch und erhält bei Auftragen der Ionisationsschwellen gegen die Wurzel der elektrischen Feldstärke durch Extrapolation auf die Feldstärke Null die Ionisationsenergie.Im Rahmen dieser Arbeit wurde die Ionisationsenergie von Actinium erstmalig zu 43398(3) cm-1 º 5,3807(4) eV experimentell bestimmt. Dazu wurden Actiniumatome zunächst einstufig resonant mit einem Laser mit einer Wellenlänge von 388,67 nm auf einen Zustand bei 25729,03 cm-1 angeregt und anschließend mit Laserlicht mit einer Wellenlänge von ca. 568 nm ionisiert. Damit sind die Ionisationsenergien aller Actinoiden bis einschließlich Einsteinium mit Ausnahme von Protactinium bekannt. Als Atomstrahlquelle wird ein spezielles 'Sandwichfilament' benutzt, bei dem das Actinoid als Hydroxid auf eine Tantalfolie aufgebracht und mit einer reduzierenden Deckschicht überzogen wird. Das Actinoid dampft bei Heizen dieser Anordnung atomar ab. Bei den schwereren Actinoiden wurde Titan als Deckschicht verwendet. Um einen Actiniumatomstrahl zu erzeugen, wurde aufgrund der hohen Abdampftemperaturen statt Titan erstmals Zirkonium eingesetzt. Bei Protactinium wurde Thorium, welches noch stärkere Reduktionseigenschaften aufweist, als Deckmaterial eingesetzt. Trotzdem gelang es mit der 'Sandwichtechnik' nicht, einen Protactiniumatomstrahl zu erzeugen. In der Flugzeitapparatur wurde lediglich ein Protactinium-monoxidionensignal detektiert. Um ein erst seit kurzem verfügbares Fest-körperlasersystem zu explorieren, wurden zusätzlich noch die bekannten Ionisations-ener-gien von Gadolinium und Plutonium erneut bestimmt. Die gemessenen Werte stimmen mit Literaturdaten gut überein.Ferner wurde noch ein bestehender Trennungsgang für Plutonium aus Umweltproben auf die Matrices Meerwasser und Hausstaub angepasst und für die Bestimmung von Plutonium und dessen Isotopenzusammensetzung in verschiedenen Probenreihen mittels RIMS eingesetzt. Der modifizierte Trennungsgang ermöglicht das schnelle Aufarbeiten von großen Probenmengen für Reihenuntersuchungen von Plutoniumkontaminationen. Die ermittelten Gehalten an 239Pu lagen zwischen 8,2*107 Atome pro 10 l Meerwasserprobe und 1,7*109Atome pro Gramm Staubprobe.

Perfluorinated compounds in environmental matrices of Lake Victoria Gulf

ABSTRACT One of the major ecological challenges on Lake Victoria resources is the existence of “hot spots”, caused by human waste, urban runoff, and industrial effluents. The lake is tending towards eutrophication which is attributed to the increasing human population in its watershed. A report of the levels of perfluorooctane sulfonate and perfluorooctanoic acid in environmental matrices of Lake Victoria is presented, and the management implication of perfluorinated compounds and similar potential organic pollutants examined. Two widely consumed and economically important fish species namely Lates niloticus (Nile perch) and Oreochromis niloticus (Nile tilapia) were obtained from Winam gulf of Lake Victoria, Kenya, and analysed for perfluorooctane sulfonate and perfluorooctanoic acid in muscles and liver using liquid chromatography coupled with mass spectroscopy. Variability in the concentrations of perfluorooctanoic acid or perfluorooctane sulfonate in river waters (range perfluorooctanoic acid 0.4 – 96.4 ng/L and perfluorooctane sulfonate < 0.4 – 13.2 ng/L) was higher than for Lake waters (range perfluorooctanoic acid 0.4 – 11.7 ng/L and perfluorooctane sulfonate < 0.4 – 2.5 ng/L respectively). Significant correlations were tested between perfluorinated compounds levels in sediments, fish and water. Wastewater treatment plants and other anthropogenic sources have been identified as significant sources or pathways for the introduction of perfluoroalkyl compounds into Lake Victoria ecosystem. In this study, elevated concentrations of perfluorooctanoic acid and perfluorooctane sulfonate was found in two wastewater treatment plants (WWTPs) in Kisumu, City of Kenya. An alternative analytical method to liquid chromatography/ mass spectroscopy for analysis of perfluorocarboxylic acids in abiotic and biotic matrices where high concentrations are expected is also presented. Derivatisation of the acid group to form a suitable alkyl ester provided a suitable compound for mass spectroscopy detection coupled to gas chromatography instrumental analysis. The acid is esterified by an alkyl halide i.e benzyl bromide as the alkylating agent for Perfluorocarboxylic acids quantification. The study also involved degradability measurements of emerging perfluorinated surfactants substitutes. The stability of the substitutes of perfluorinated surfactants was tested by employing advanced oxidation processes, followed by conventional tests, among them an automated method based on the manometric respirometry test and standardized fix bed bioreactor [FBBR] on perfluorobutane sulfonate (PFBS), a fluoroethylene polymer, fluorosurfactant (Zonyl), two fluoraliphaticesters (NOVEC ™ FC4430 and NOVEC ™ FC4432) and 10-(trifluoromethoxy) decane-sulfonate. Most of these emmerging surfactants are well-established in the market and have been used in several applications as alternatives to PFOS and PFOA based surfactants. The results of this study can be used as pioneer information for further studies on the sources, behaviour and fate of PFOA and PFOS and other related compounds in both abiotic and biota compartments of Lake Victoria and other lakes. Further an overview in degradation of emerging perfluorinated compounds substitutes is presented. Contribution in method development especially for acid group based fluorosurfactants is presented. The data obtained in this study can particularly be considered when formulating policies and management measures for preservation and sustainability of Lake Victoria resources.

Original analytical methods for the determination of psychoactive compounds in complex matrices

This thesis work aims to develop original analytical methods for the determination of drugs with a potential for abuse, for the analysis of substances used in the pharmacological treatment of drug addiction in biological samples and for the monitoring of potentially toxic compounds added to street drugs. In fact reliable analytical techniques can play an important role in this setting. They can be employed to reveal drug intake, allowing the identification of drug users and to assess drug blood levels, assisting physicians in the management of the treatment. Pharmacological therapy needs to be carefully monitored indeed in order to optimize the dose scheduling according to the specific needs of the patient and to discourage improper use of the medication. In particular, different methods have been developed for the detection of gamma-hydroxybutiric acid (GHB), prescribed for the treatment of alcohol addiction, of glucocorticoids, one of the most abused pharmaceutical class to enhance sport performance and of adulterants, pharmacologically active compounds added to illicit drugs for recreational purposes. All the presented methods are based on capillary electrophoresis (CE) and high performance liquid chromatography (HPLC) coupled to various detectors (diode array detector, mass spectrometer). Biological samples pre-treatment was carried out using different extraction techniques, liquid-liquid extraction (LLE) and solid phase extraction (SPE). Different matrices have been considered: human plasma, dried blood spots, human urine, simulated street drugs. These developed analytical methods are individually described and discussed in this thesis work.

Formula Zero: Traduzione trilingue inglese-italiano-spagnolo di un policy statement sulla sicurezza stradale della Federazione Internazionale dell'Automobile (FIA)

The aim of this dissertation is to provide a trilingual translation from English into Italian and from Italian into Spanish of a policy statement from the Fédération Internationale de l’Automobile (FIA) regarding road safety. The document, named “Formula Zero: a strategy for reducing fatalities and injuries on track and road”, was published in June 2000 and involves an approach about road safety inspired by another approach introduced in Sweden called ‘Vision Zero’. This work consists of six sections. The first chapter introduces the main purposes and activities of the Federation, as well as the institutions related to it and Vision Zero. The second chapter presents the main lexical, morphosyntactic and stylistic features of the institutional texts and special languages. In particular, the text contains technical nomenclature of transports and elements of sport language, especially regarding motor sport and Formula One. In the third chapter, the methodology is explained, with all the resources used during the preliminary phase and the translation, including corpora, glossaries, expert consultancy and specialised sites. The fourth chapter focuses on the morphosyntactic and terminology features contained in the text, while the fifth chapter presents the source text and the target texts. The final chapter deals with all the translation strategies that are applied, alongside with all the challenging elements detected. Therefore, the dissertation concludes with some theoretical and practical considerations about the role of inverse translation and English as Lingua Franca (ELF), by comparing the text translated into Spanish to the original in English, using Italian as a lingua franca.

The influence of zero-value subtraction on the performance of two laser fluorescence devices for detecting occlusal caries in vitro

BACKGROUND: The aim of this study was to evaluate the influence of zero-value subtraction on the performance of two laser fluorescence (LF) devices developed to detect occlusal caries. METHODS: The authors selected 119 permanent molars. Two examiners assessed three areas (cuspal, middle and cervical) of both mesial and distal portions of the buccal surface and one occlusal site using an LF device and an LF pen. For each tooth, the authors subtracted the value measured in the cuspal, middle and cervical areas in the buccal surface from the value measured in the respective occlusal site. RESULTS: The authors observed differences among the readings for both devices in the cuspal, middle and cervical areas in the buccal surface as well as differences for both devices with and without the zero-value subtraction in the occlusal surface. When the authors did not perform the zero-value subtraction, they found statistically significant differences for sensitivity and accuracy for the LF device. When this was done with the LF pen, specificity increased and sensitivity decreased significantly. CONCLUSIONS: For the LF device, the zero-value subtraction decreased the sensitivity. For this reason, the authors concluded that clinicians can obtain measures with the LF device effectively without using zero-value subtraction. For the LF pen, however, the absence of the zero-value subtraction changed both the sensitivity and specificity, and so the authors concluded that clinicians should not eliminate this step from the procedure. CLINICAL IMPLICATIONS: When using the LF device, clinicians might not need to perform the zero-value subtraction; however, for the LF pen, clinicians should do so.

The influence of zero value subtraction on the performance of a new laser fluorescence device for approximal caries detection

The aim of this study was to assess the influence of the zero value subtraction on the performance of laser fluorescence (LFpen) for approximal caries detection. Three areas (cuspal, middle and cervical) of both mesial and distal buccal surfaces of 78 permanent molars were assessed using both wedge-shaped (WDG) and tapered wedge-shaped (TWDG) tips. With the addition of the average, one cut-off value for each area was obtained and the performance was assessed. The areas under the receiver operating characteristics (ROC) curve, specificity, sensitivity and accuracy with and without the zero value subtraction were calculated. The McNemar test revealed a statistically significant difference for specificity at thresholds D(1), D(2) and D(3) (WDG) and D(1) and D(2) (TWDG) when the zero value subtraction was not performed. Influence of the zero value subtraction on the LFpen performance was observed for approximal caries detection. However, when modified cut-off values were used, the zero value subtraction could be eliminated.

Promoting Pollinating Insects in Intensive Agricultural Matrices: Field-Scale Experimental Manipulation of Hay-Meadow Mowing Regimes and Its Effects on Bees

Bees are a key component of biodiversity as they ensure a crucial ecosystem service: pollination. This ecosystem service is nowadays threatened, because bees suffer from agricultural intensification. Yet, bees rarely benefit from the measures established to promote biodiversity in farmland, such as agri-environment schemes (AES). We experimentally tested if the spatio-temporal modification of mowing regimes within extensively managed hay meadows, a widespread AES, can promote bees. We applied a randomized block design, replicated 12 times across the Swiss lowlands, that consisted of three different mowing treatments: 1) first cut not before 15 June (conventional regime for meadows within Swiss AES); 2) first cut not before 15 June, as treatment 1 but with 15% of area left uncut serving as a refuge; 3) first cut not before 15 July. Bees were collected with pan traps, twice during the vegetation season (before and after mowing). Wild bee abundance and species richness significantly increased in meadows where uncut refuges were left, in comparison to meadows without refuges: there was both an immediate (within year) and cumulative (from one year to the following) positive effect of the uncut refuge treatment. An immediate positive effect of delayed mowing was also evidenced in both wild bees and honey bees. Conventional AES could easily accommodate such a simple management prescription that promotes farmland biodiversity and is likely to enhance pollination services.

Comparison of energy expenditure, eating pattern and physical activity of grazing and zero-grazing dairy cows at different time points during lactation

An experiment was conducted to determine the effect of grazing versus zero-grazing on energy expenditure (EE), feeding behaviour and physical activity in dairy cows at different stages of lactation. Fourteen Holstein cows were subjected to two treatments in a repeated crossover design with three experimental series (S1, S2, and S3) reflecting increased days in milk (DIM). At the beginning of each series, cows were on average at 38, 94 and 171 (standard deviation (SD) 10.8) DIM, respectively. Each series consisted of two periods containing a 7-d adaptation and a 7-d collection period each. Cows either grazed on pasture for 16–18.5 h per day or were kept in a freestall barn and had ad libitum access to herbage harvested from the same paddock. Herbage intake was estimated using the double alkane technique. On each day of the collection period, EE of one cow in the barn and of one cow on pasture was determined for 6 h by using the 13C bicarbonate dilution technique, with blood sample collection done either manually in the barn or using an automatic sampling system on pasture. Furthermore, during each collection period physical activity and feeding behaviour of cows were recorded over 3 d using pedometers and behaviour recorders. Milk yield decreased with increasing DIM (P<0.001) but was similar with both treatments. Herbage intake was lower (P<0.01) for grazing cows (16.8 kg dry matter (DM)/d) compared to zero-grazing cows (18.9 kg DM/d). The lowest (P<0.001) intake was observed in S1 and similar intakes were observed in S2 and S3. Within the 6-h measurement period, grazing cows expended 19% more (P<0.001) energy (319 versus 269 kJ/kg metabolic body size (BW0.75)) than zero-grazing cows and differences in EE did not change with increasing DIM. Grazing cows spent proportionally more (P<0.001) time walking and less time standing (P<0.001) and lying (P<0.05) than zero-grazing cows. The proportion of time spent eating was greater (P<0.001) and that of time spent ruminating was lower (P<0.05) for grazing cows compared to zero-grazing cows. In conclusion, lower feed intake along with the unchanged milk production indicates that grazing cows mobilized body reserves to cover additional energy requirements which were at least partly caused by more physical activity. However, changes in cows׳ behaviour between the considered time points during lactation were too small so that differences in EE remained similar between treatments with increasing DIM.

Search for supersymmetry in events with large missing transverse momentum, jets, and at least one tau lepton in 20 fb⁻¹ of √s= 8 TeV proton-proton collision data with the ATLAS detector

A search for supersymmetry (SUSY) in events with large missing transverse momentum, jets, at least one hadronically decaying tau lepton and zero or one additional light leptons (electron/muon), has been performed using 20.3 fb−1 of proton-proton collision data at √s = 8TeV recorded with the ATLAS detector at the Large Hadron Collider. No excess above the Standard Model background expectation is observed in the various signal regions and 95% confidence level upper limits on the visible cross section for new phenomena are set. The results of the analysis are interpreted in several SUSY scenarios, significantly extending previous limits obtained in the same final states. In the framework of minimal gauge-mediated SUSY breaking models, values of the SUSY breaking scale ʌ below 63TeV are excluded, independently of tan β. Exclusion limits are also derived for an mSUGRA/CMSSM model, in both the R-parity-conserving and R-parity-violating case. A further interpretation is presented in a framework of natural gauge mediation, in which the gluino is assumed to be the only light coloured sparticle and gluino masses below 1090GeV are excluded.

Pauli Spin Matrices

The Pauli Spin Matrix representation of Spin Operators in Quantum Mechanics are explicitly demonstrated and illustrated in detail for one and two spin systems.

Carbonate chemistry, community metabolism, PAR, temperature and salinity of One Tree Island reef

There are few in situ studies showing how net community calcification (Gnet) of coral reefs is related to carbonate chemistry, and the studies to date have demonstrated different predicted rates of change. In this study, we measured net community production (Pnet), Gnet, and carbonate chemistry of a reef flat at One Tree Island, Great Barrier Reef. Diurnal pCO2 variability of 289-724 µatm was driven primarily by photosynthesis and respiration. The reef flat was found to be net autotrophic, with daily production of ? 35 mmol C/m**2/d and net calcification of ? 33 mmol C/m**2/d . Gnet was strongly related to Pnet, which drove a hysteresis pattern in the relationship between Gnet and aragonite saturation state (Omega ar). Although Pnet was the main driver of Gnet, Omega ar was still an important factor, where 95% of the variance in Gnet could be described by Pnet and Omega ar. Based on the observed in situ relationship, Gnet would be expected to reach zero when Omega ar is 2.5. It is unknown what proportion of a decline in Gnet would be through reduced calcification and what would occur through increased dissolution, but the results here support predictions that overall calcium carbonate production will decline in coral reefs as a result of ocean acidification.

Efficient methodologies for system matrix modelling in iterative image reconstruction for rotating high-resolution PET

A fully 3D iterative image reconstruction algorithm has been developed for high-resolution PET cameras composed of pixelated scintillator crystal arrays and rotating planar detectors, based on the ordered subsets approach. The associated system matrix is precalculated with Monte Carlo methods that incorporate physical effects not included in analytical models, such as positron range effects and interaction of the incident gammas with the scintillator material. Custom Monte Carlo methodologies have been developed and optimized for modelling of system matrices for fast iterative image reconstruction adapted to specific scanner geometries, without redundant calculations. According to the methodology proposed here, only one-eighth of the voxels within two central transaxial slices need to be modelled in detail. The rest of the system matrix elements can be obtained with the aid of axial symmetries and redundancies, as well as in-plane symmetries within transaxial slices. Sparse matrix techniques for the non-zero system matrix elements are employed, allowing for fast execution of the image reconstruction process. This 3D image reconstruction scheme has been compared in terms of image quality to a 2D fast implementation of the OSEM algorithm combined with Fourier rebinning approaches. This work confirms the superiority of fully 3D OSEM in terms of spatial resolution, contrast recovery and noise reduction as compared to conventional 2D approaches based on rebinning schemes. At the same time it demonstrates that fully 3D methodologies can be efficiently applied to the image reconstruction problem for high-resolution rotational PET cameras by applying accurate pre-calculated system models and taking advantage of the system's symmetries.

Medidas autosemejantes en el plano, momentos y matrices de Hessenberg

La tesis MEDIDAS AUTOSEMEJANTES EN EL PLANO, MOMENTOS Y MATRICES DE HESSENBERG se enmarca entre las áreas de la teoría geométrica de la medida, la teoría de polinomios ortogonales y la teoría de operadores. La memoria aborda el estudio de medidas con soporte acotado en el plano complejo vistas con la óptica de las matrices infinitas de momentos y de Hessenberg asociadas a estas medidas que en la teoría de los polinomios ortogonales las representan. En particular se centra en el estudio de las medidas autosemejantes que son las medidas de equilibrio definidas por un sistema de funciones iteradas (SFI). Los conjuntos autosemejantes son conjuntos que tienen la propiedad geométrica de descomponerse en unión de piezas semejantes al conjunto total. Estas piezas pueden solaparse o no, cuando el solapamiento es pequeño la teoría de Hutchinson [Hut81] funciona bien, pero cuando no existen restricciones falla. El problema del solapamiento consiste en controlar la medida de este solapamiento. Un ejemplo de la complejidad de este problema se plantea con las convoluciones infinitas de distribuciones de Bernoulli, que han resultado ser un ejemplo de medidas autosemejantes en el caso real. En 1935 Jessen y A. Wintner [JW35] ya se planteaba este problema, lejos de ser sencillo ha sido estudiado durante más de setenta y cinco años y siguen sin resolverse las principales cuestiones planteadas ya por A. Garsia [Gar62] en 1962. El interés que ha despertado este problema así como la complejidad del mismo está demostrado por las numerosas publicaciones que abordan cuestiones relacionadas con este problema ver por ejemplo [JW35], [Erd39], [PS96], [Ma00], [Ma96], [Sol98], [Mat95], [PS96], [Sim05],[JKS07] [JKS11]. En el primer capítulo comenzamos introduciendo con detalle las medidas autosemejante en el plano complejo y los sistemas de funciones iteradas, así como los conceptos de la teoría de la medida necesarios para describirlos. A continuación se introducen las herramientas necesarias de teoría de polinomios ortogonales, matrices infinitas y operadores que se van a usar. En el segundo y tercer capítulo trasladamos las propiedades geométricas de las medidas autosemejantes a las matrices de momentos y de Hessenberg, respectivamente. A partir de estos resultados se describen algoritmos para calcular estas matrices a partir del SFI correspondiente. Concretamente, se obtienen fórmulas explícitas y algoritmos de aproximación para los momentos y matrices de momentos de medidas fractales, a partir de un teorema del punto fijo para las matrices. Además utilizando técnicas de la teoría de operadores, se han extendido al plano complejo los resultados que G. Mantica [Ma00, Ma96] obtenía en el caso real. Este resultado es la base para definir un algoritmo estable de aproximación de la matriz de Hessenberg asociada a una medida fractal u obtener secciones finitas exactas de matrices Hessenberg asociadas a una suma de medidas. En el último capítulo, se consideran medidas, μ, más generales y se estudia el comportamiento asintótico de los autovalores de una matriz hermitiana de momentos y su impacto en las propiedades de la medida asociada. En el resultado central se demuestra que si los polinomios asociados son densos en L2(μ) entonces necesariamente el autovalor mínimo de las secciones finitas de la matriz de momentos de la medida tiende a cero. ABSTRACT The Thesis work “Self-similar Measures on the Plane, Moments and Hessenberg Matrices” is framed among the geometric measure theory, orthogonal polynomials and operator theory. The work studies measures with compact support on the complex plane from the point of view of the associated infinite moments and Hessenberg matrices representing them in the theory of orthogonal polynomials. More precisely, it concentrates on the study of the self-similar measures that are equilibrium measures in a iterated functions system. Self-similar sets have the geometric property of being decomposable in a union of similar pieces to the complete set. These pieces can overlap. If the overlapping is small, Hutchinson’s theory [Hut81] works well, however, when it has no restrictions, the theory does not hold. The overlapping problem consists in controlling the measure of the overlap. The complexity of this problem is exemplified in the infinite convolutions of Bernoulli’s distributions, that are an example of self-similar measures in the real case. As early as 1935 [JW35], Jessen and Wintner posed this problem, that far from being simple, has been studied during more than 75 years. The main cuestiones posed by Garsia in 1962 [Gar62] remain unsolved. The interest in this problem, together with its complexity, is demonstrated by the number of publications that over the years have dealt with it. See, for example, [JW35], [Erd39], [PS96], [Ma00], [Ma96], [Sol98], [Mat95], [PS96], [Sim05], [JKS07] [JKS11]. In the first chapter, we will start with a detailed introduction to the self-similar measurements in the complex plane and to the iterated functions systems, also including the concepts of measure theory needed to describe them. Next, we introduce the necessary tools from orthogonal polynomials, infinite matrices and operators. In the second and third chapter we will translate the geometric properties of selfsimilar measures to the moments and Hessenberg matrices. From these results, we will describe algorithms to calculate these matrices from the corresponding iterated functions systems. To be precise, we obtain explicit formulas and approximation algorithms for the moments and moment matrices of fractal measures from a new fixed point theorem for matrices. Moreover, using techniques from operator theory, we extend to the complex plane the real case results obtained by Mantica [Ma00, Ma96]. This result is the base to define a stable algorithm that approximates the Hessenberg matrix associated to a fractal measure and obtains exact finite sections of Hessenberg matrices associated to a sum of measurements. In the last chapter, we consider more general measures, μ, and study the asymptotic behaviour of the eigenvalues of a hermitian matrix of moments, together with its impact on the properties of the associated measure. In the main result we demonstrate that, if the associated polynomials are dense in L2(μ), then necessarily follows that the minimum eigenvalue of the finite sections of the moments matrix goes to zero.