Quantum Einstein gravity - advancements of heat kernel-based renormalization group studies

The asymptotic safety scenario allows to define a consistent theory of quantized gravity within the framework of quantum field theory. The central conjecture of this scenario is the existence of a non-Gaussian fixed point of the theory's renormalization group flow, that allows to formulate renormalization conditions that render the theory fully predictive. Investigations of this possibility use an exact functional renormalization group equation as a primary non-perturbative tool. This equation implements Wilsonian renormalization group transformations, and is demonstrated to represent a reformulation of the functional integral approach to quantum field theory.rnAs its main result, this thesis develops an algebraic algorithm which allows to systematically construct the renormalization group flow of gauge theories as well as gravity in arbitrary expansion schemes. In particular, it uses off-diagonal heat kernel techniques to efficiently handle the non-minimal differential operators which appear due to gauge symmetries. The central virtue of the algorithm is that no additional simplifications need to be employed, opening the possibility for more systematic investigations of the emergence of non-perturbative phenomena. As a by-product several novel results on the heat kernel expansion of the Laplace operator acting on general gauge bundles are obtained.rnThe constructed algorithm is used to re-derive the renormalization group flow of gravity in the Einstein-Hilbert truncation, showing the manifest background independence of the results. The well-studied Einstein-Hilbert case is further advanced by taking the effect of a running ghost field renormalization on the gravitational coupling constants into account. A detailed numerical analysis reveals a further stabilization of the found non-Gaussian fixed point.rnFinally, the proposed algorithm is applied to the case of higher derivative gravity including all curvature squared interactions. This establishes an improvement of existing computations, taking the independent running of the Euler topological term into account. Known perturbative results are reproduced in this case from the renormalization group equation, identifying however a unique non-Gaussian fixed point.rn

La rovina del giocatore: Analisi matematica di alcune dinamiche del gioco d'azzardo

La tesi introduce il problema matematico della rovina del giocatore dopo un'introduzione storica relativa allo sviluppo del gioco d'azzardo ed alla sua concezione sociale. Segue un'analisi di alcuni particolari giochi con un approfondimento sul calcolo della probabilità di vittoria e sul gioco equo. Infine sono introdotti alcuni preliminari matematici relativi alla misura, alle successioni di Bernoulli ed alle leggi dei grandi numeri, necessari per comprendere il problema.

Chargaff symmetric stochastic processes

Scopo della modellizzazione delle stringhe di DNA è la formulazione di modelli matematici che generano sequenze di basi azotate compatibili con il genoma esistente. In questa tesi si prendono in esame quei modelli matematici che conservano un'importante proprietà, scoperta nel 1952 dal biochimico Erwin Chargaff, chiamata oggi "seconda regola di Chargaff". I modelli matematici che tengono conto delle simmetrie di Chargaff si dividono principalmente in due filoni: uno la ritiene un risultato dell'evoluzione sul genoma, mentre l'altro la ipotizza peculiare di un genoma primitivo e non intaccata dalle modifiche apportate dall'evoluzione. Questa tesi si propone di analizzare un modello del secondo tipo. In particolare ci siamo ispirati al modello definito da da Sobottka e Hart. Dopo un'analisi critica e lo studio del lavoro degli autori, abbiamo esteso il modello ad un più ampio insieme di casi. Abbiamo utilizzato processi stocastici come Bernoulli-scheme e catene di Markov per costruire una possibile generalizzazione della struttura proposta nell'articolo, analizzando le condizioni che implicano la validità della regola di Chargaff. I modelli esaminati sono costituiti da semplici processi stazionari o concatenazioni di processi stazionari. Nel primo capitolo vengono introdotte alcune nozioni di biologia. Nel secondo si fa una descrizione critica e prospettica del modello proposto da Sobottka e Hart, introducendo le definizioni formali per il caso generale presentato nel terzo capitolo, dove si sviluppa l'apparato teorico del modello generale.

Storia, sviluppi e applicazioni dei logaritmi.

Scopo della tesi è la trattazione dei logaritmi a partire dalla storia di quest'ultimi, al loro sviluppo, fino ad arrivare alle diverse applicazioni dei logaritmi in svariate discipline. La tesi è strutturata in quattro capitoli, nel primo dei quali si parte analizzando quali istanze teoriche e necessità pratiche abbiano preparato la strada all'introduzione dei logaritmi. Vengono riportati alcuni passi del testo più importante dedicato da Nepero ai logaritmi, Mirifici Logarithmorum Canonis Constructio, la modifica ad opera di Henry Briggs e la diffusione dei logaritmi in gran parte dell' Europa. Nel secondo capitolo viene evidenziato il legame tra i logaritmi e la geometria dell'iperbole per poi passare alla trattazione dei primi studi sulla curva logaritmica. Nel terzo capitolo viene esaminata la controversia tra Leibniz e Bernoulli sul significato da attribuire ai logaritmi dei numeri negativi soffermandosi su come Eulero uscì da una situazione di stallo proponendo una teoria dei logaritmi dei numeri complessi. Nel quarto ed ultimo capitolo vengono analizzati i diversi utilizzi della scala logaritmica ponendo soprattutto l'attenzione sul regolo calcolatore, arrivando infine a mostrare le applicazioni dei logaritmi in altre discipline.

Auditory motion direction encoding in auditory cortex and high-level visual cortex

The aim of this functional magnetic resonance imaging (fMRI) study was to identify human brain areas that are sensitive to the direction of auditory motion. Such directional sensitivity was assessed in a hypothesis-free manner by analyzing fMRI response patterns across the entire brain volume using a spherical-searchlight approach. In addition, we assessed directional sensitivity in three predefined brain areas that have been associated with auditory motion perception in previous neuroimaging studies. These were the primary auditory cortex, the planum temporale and the visual motion complex (hMT/V5+). Our whole-brain analysis revealed that the direction of sound-source movement could be decoded from fMRI response patterns in the right auditory cortex and in a high-level visual area located in the right lateral occipital cortex. Our region-of-interest-based analysis showed that the decoding of the direction of auditory motion was most reliable with activation patterns of the left and right planum temporale. Auditory motion direction could not be decoded from activation patterns in hMT/V5+. These findings provide further evidence for the planum temporale playing a central role in supporting auditory motion perception. In addition, our findings suggest a cross-modal transfer of directional information to high-level visual cortex in healthy humans.

A brief update on lung stereology

Lung stereology has a long and successful tradition. From mice to men, the application of new stereological methods at several levels (alveoli, parenchymal cells, organelles, proteins) has led to new insights into normal lung architecture, parenchymal remodelling in emphysema-like pathology, alveolar type II cell hyperplasia and hypertrophy and intracellular surfactant alterations as well as distribution of surfactant proteins. The Euler number of the network of alveolar openings, estimated using physical disectors at the light microscopic level, is an unbiased and direct estimate of alveolar number. Surfactant-producing alveolar type II cells can be counted and sampled for local size estimation with physical disectors at a high magnification light microscopic level. The number of their surfactant storage organelles, lamellar bodies, can be estimated using physical disectors at the EM level. By immunoelectron microscopy, surfactant protein distribution can be analysed with the relative labelling index. Together with the well-established classical stereological methods, these design-based methods now allow for a complete quantitative phenotype analysis in lung development and disease, including the structural characterization of gene-manipulated mice, at the light and electron microscopic level.

Effect of voluntary exercise on number and volume of cardiomyocytes and their mitochondria in the mouse left ventricle

Voluntary exercise (VE) has a beneficial influence on the heart and mean lifespan. The present study evaluates structural adaptations of cardiomyocytes and their mitochondria due to VE by new, unbiased stereological methods. Female, 7-9-week-old mice were randomly assigned to a control (CG, n = 7) or VE group (EG, n = 7). EG animals were housed in cages with free access to a running wheel and had a mean running distance of 6.7 (1.8) km per day. After 4 weeks, the hearts of all mice were processed for light and electron microscopy. We estimated the number and volume of cardiomyocytes by the disector method and the number and volume of mitochondria by estimation of the Euler number. In comparison to CG, VE did not have an effect on the myocardial volume of the left ventricle (CG: 93 (10), EG: 103 (17) (mm(3))), the number of cardiomyocytes (CG: 2.81 (0.27), EG: 2.82 (0.43) (x10(6))) and their number-weighted mean volume. However, the composition of the cardiomyocytes changed due to VE. The total volume of mitochondria (CG: 21.8 (4.9), EG: 32.2 (4.3) (mm(3)), P < 0.01) and the total number (CG: 3.76 (0.44), EG: 7.02 (1.13) (x10(10)), P < 0.001) were significantly higher in EG than in CG. The mean number-weighted mitochondrial volume was smaller in EG than in CG (P < 0.05). In summary, VE does not alter ventricular volume nor cardiomyocyte volume or number but the oxidative capacity of cardiomyocytes by an increased mitochondrial number and total volume in the left ventricle. These structural changes may participate in the beneficial effects of VE.

Contractile performance of adult ventricular rat cardiomyocytes is not directly jeopardized by NO/cGMP-dependent induction of pro-apoptotic pathways

The activation of NO/cGMP pathways can induce pro-apoptotic pathways in cardiomyocytes although only a small number of cardiomyocytes fulfill the criteria of apoptosis. The same pathways reduce the contractile performance of cardiomyocytes. In the present study, we tested the hypothesis that exposure of cells to NO/cGMP for 24 h decrease their contractile performance due to an activation of pro-apoptotic pathways. Experiments were performed on freshly isolated and cultured adult ventricular rat cardiomyocytes. Cells were incubated with 8-bromo-cyclo-GMP (100 nmol/L-1 micromol/L), the NO donor SNAP (1 nmol/L-100 micromol/L), or the guanylyl cyclase activator YC-1 (3 micromol/L). Cell shortening, contraction and relaxation velocities, and diastolic cell lengths were determined at beating frequencies of 0.5, 1, and 2 Hz 24 h later. The activation of pro-apoptotic pathways was determined by staining of cardiomyocytes with an antibody directed against active caspase-3 and quantification of the number of apoptotic cells (annexin staining). Caspase-3 activation and an increase in the number of apoptotic cells was observed, but only at the highest concentrations tested (8-bromo-cyclo-GMP: 1-10 mmol/L; SNAP: 1-100 micromol/L). At these concentrations, none of the drugs decreased the mean cell shortening of cardiomyocytes. However, at concentrations lower than those required for induction of apoptotic cell death, the diastolic cell lengths and sarcomere lengths increased but cell shortening decreased. In conclusion, low concentrations of either NO or cGMP cause a desensitization of myofibrils, as indicated by elongated cell shapes, increased sarcomere lengths and reduced load-free cell shortening. High concentrations of NO/cGMP induce caspase-3 activation and increase the number of cells fulfilling the criteria of apoptotic cell death but did not impair cell function. Therefore, induction of apoptotic cell death per se seems not to contribute to the loss of contractile efficiency on the cellular level.