Abstract The traditional methods of analysing heart rate variability based on means and variance are unable to detect subtle but potentially important changes in interbeat heart rate behaviour. This research was designed to evaluate the clinical applicability and prognostic significance of new dynamical methods of analysing heart rate behaviour derived from nonlinear mathematics. The study covered four different patient populations, their controls and one general population of elderly people. The first patient group consisted of 38 patients with coronary artery disease without previous myocardial infarction, the second of 40 coronary artery disease patients with a prior Q-wave myocardial infarction, and the third of 45 patients with a history of ventricular tachyarrhythmia. The fourth group comprised 10 patients with a previous myocardial infarction who had experienced ventricular fibrillation during electrocardiographic recordings. The fifth group comprised a random sample of 347 community-living elderly people invited for a follow-up of 10 years after electrocardiographic recordings. Heart rate variability was analysed by traditional time and frequency domain methods. The new dynamical measures derived from nonlinear dynamics were: 1) approximate entropy, which reflects the complexity of the data, 2) detrended fluctuation analysis, which describes the presence or absence of fractal correlation properties of time series data, and 3) power-law relationship analysis, which demonstrates the distribution of spectral characteristics of RR intervals, but does not reflect the magnitude of spectral power in different spectral bands. Approximate entropy was higher in postinfarction patients (1.17 ± 0.22), but lower in coronary artery disease patients without myocardial infarction (0.93 ± 0.17) than in healthy controls (1.03 ± 014, p < 0.01, p < 0.05 respectively). It did not differ between patients with and without ventricular arrhythmia. The short term fractal-like scaling exponent of the detrended fluctuation analysis was higher in coronary artery disease patients without myocardial infarction (1.34 ± 0.15, p < 0.001), but not in postinfarction patients without arrhythmia (1.06 ± 0.13) compared with healthy controls (1.09 ± 0.13). The short term exponent was markedly reduced in patients with life-threatening arrhythmia (0.85 ± 0.25 ventricular tachycardia patients, 0.68 ± 0.18 ventricular fibrillation patients, p < 0.001 for both). The long term power-law slope of the power-law scaling analysis was lower in the ventricular fibrillation group than in postinfarction controls without arrhythmia risk (-1.63 ± 0.24 vs. -1.33 ± 0.23, p < 0.01) and predicted mortality in a general elderly population with an adjusted relative risk of 1.74 (95% CI 1.42–2.13). The present observations demonstrate that dynamic analysis of heart rate behaviour gives new insight into analysis of heart rate dynamics in various cardiovascular disorders. The breakdown of the normal fractal-like organising principle of heart rate variability is associated with an increased risk of mortality and vulnerability to life-threatening arrhythmias.

Given a graph G, a K_3-decomposition of G, also called a triangle decomposition, is a set of subgraphs isomorphic to K_3 whose edges partition the edge set of G. Further, a rational K_3-decomposition of G is a non-negative rational weighting of the copies of K_3 in G such that the total weight on any edge of G equals one. In this thesis, we explore the problem of rational triangle decompositions of dense graphs. We start by considering necessary conditions for a rational triangle decomposition, which can be represented by facets of a convex cone generated by a certain incidence matrix. We identify several infinite families of these facets that represent meaningful obstructions to rational triangle decomposability of a graph. Further, we classify all facets on up to 9 vertices and check all 8-vertex graphs of degree at least four for rational triangle decomposability. As the study of graph decompositions is closely related to design theory, we also prove the existence of certain types of designs. We then explore sufficient conditions for rational triangle decomposability. A famous conjecture in the area due to Nash-Williams states that any sufficiently large graph (satisfying some divisibility conditions) with minimum degree at least 3/4v is K_3-decomposable; the same conjecture stands for rational K_3-decomposability (no divisibility conditions required). By perturbing and restricting the coverage matrix of a complete graph, we show that minimum degree of at least 22/23v is sufficient to guarantee that the given graph is rationally triangle decomposable. This density bound is a great improvement over the previously known results and is derived using estimates on the matrix norms and structures originating from association schemes. We also consider applications of rational triangle decompositions. The method we develop in the search for sufficient conditions provides an efficient way to generate certain sampling plans in statistical experimental design. Furthermore, rational graph decompositions serve as building blocks within certain design-theoretic proofs and we use them to prove that it is possible to complete partial designs given certain constraints. / Graduate / 0405

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Many fields in science and engineering require large-scale numerical simulations of complex systems described by differential equations. These systems are typically multi-physics (they are driven by multiple interacting physical processes) and multiscale (the dynamics takes place on vastly different spatial and temporal scales). Numerical solution of such systems is highly challenging due to the dimension of the resulting discrete problem, and to the complexity that comes from incorporating multiple interacting components with different characteristics. The main contributions of this dissertation are the creation of new families of time integration methods for multiscale and multiphysics simulations, and the development of industrial-strengh tools for sensitivity analysis. This work develops novel implicit-explicit (IMEX) general linear time integration methods for multiphysics and multiscale simulations typically involving both stiff and non-stiff components. In an IMEX approach, one uses an implicit scheme for the stiff components and an explicit scheme for the non-stiff components such that the combined method has the desired stability and accuracy properties. Practical schemes with favorable properties, such as maximized stability, high efficiency, and no order reduction, are constructed and applied in extensive numerical experiments to validate the theoretical findings and to demonstrate their advantages. Approximate matrix factorization (AMF) technique exploits the structure of the Jacobian of the implicit parts, which may lead to further efficiency improvement of IMEX schemes. We have explored the application of AMF within some high order IMEX Runge-Kutta schemes in order to achieve high efficiency. Sensitivity analysis gives quantitative information about the changes in a dynamical model outputs caused by caused by small changes in the model inputs. This information is crucial for data assimilation, model-constrained optimization, inverse problems, and uncertainty quantification. We develop a high performance software package for sensitivity analysis in the context of stiff and nonstiff ordinary differential equations. Efficiency is demonstrated by direct comparisons against existing state-of-art software on a variety of test problems. / Ph. D.

Cette thèse présente trois essais en macroéconométrie appliquée. Leur dénominateur commun est l’emploi conjoint de méthodes non-linéaires et bayesiennes afin de rendre compte de cycles économiques. Le choix de ces méthodes s’appuie sur deux constats fondamentaux. Premièrement, la plupart des séries temporelles macroéconomiques et financières présentent de soudains changements dans leur comportement résultant d’évènements tels que les crises financières, les changements brutaux de politiques fiscales et monétaires, l’alternance de phases d’expansion et de récession, etc. La prise en compte de ces changements discontinus et occasionnels nécessite une modélisation non-linéaire, c’est-à-dire la conception de modèles dont les paramètres évoluent au cours du temps. Deuxièmement, l’analyse économétrique moderne des modèles à vecteur autorégressif (VAR) et des modèles dynamiques et stochastiques d’équilibre général (DSGE) soulève de nombreux problèmes auxquels peut répondre un cadre bayesien. Tout d’abord, les modèles DSGE correspondent à une représentation partielle et simplifiée de la réalité, cette dernière étant généralement trop compliquée pour être formalisée ou trop coûteuse en termes de ressources computationnelles ou intellectuelles. Cette mauvaise spécification, inhérente aux modèles DSGE, s’ajoute en général à une pénurie de données informatives nécessaires à l’obtention de réponses précises. Dans un cadre bayesien, le praticien introduit une information supplémentaire, une distribution a priori, qui rend l’inférence des paramètres du modèle plus accessible aux macroéconomistes. S’agissant des modèles DSGE, la distribution a priori, construite à partir d’informations microéconomiques telles que les élasticités agrégées ou les taux de croissance moyens des variables macroéconomiques à long terme, permet de déplacer la fonction de vraisemblance du modèle dans les régions économiquement interprétables de l’espace de paramètres. Ceci, en vue de parvenir à une interprétation raisonnable des paramètres structurels du modèle, rendant ainsi l’inférence beaucoup plus précise. [...] / This dissertation presents three essays in applied macroeconometrics. Their common denominator is the use of Bayesian and non-linear methods to study of business cycle fluctuations. The first chapter of this dissertation revisits the issue of whether business cycles with financial crises are different, in the euro area since 1999. To do so, I fit a vector autoregression in which equation coefficients and structural disturbance variances are allowed to change over time according to Markov-switching processes. I show that financial crises have been characterized by changes not only in the variances of structural shocks, but also in the predictable and systematic part of the financial sector. By predictable and systematic part of the financial sector, I mean equation coefficients that describe the financial behavior of the system. I then examine the role of financial sector in financial crises and standard business-cycle fluctuations. The evidence indicates that the relative importance of financial shocks (“non-systematic part”) is significantly higher in periods of financial distress than in non-distress periods, but the transmission of these shocks to the economy appears linear over time. Counterfactual analyses suggest that the systematic part of financial sector accounted for up to 2 and 4 percentage points of output growth drops during the downturn in 2001-2003 and the two recessions, respectively. The second chapter examines the quantitative sources of changes in the macroeconomic volatility of the euro area since 1985. To do so, I estimate a variety of large-scale Dynamic Stochastic General Equilibrium (DSGE) models in which structural disturbance variances are allowed to change according to a Markov-switching process. The empirical results show that the best-fit model is one in which all shock variances are allowed to switch between a low- and a high-volatility regime, where regime changes in the volatilities of structural shocks are synchronized. The highvolatility regime was in place during the pre-euro period, while the low-volatility regime has been prevailed since the euro introduction. Although the size of different types of shock differs between the two shock regimes, their relative importance remains unchanged. Neutral technology shocks and shocks to the marginal efficiency of investment are the dominant sources of business cycle fluctuations. Moreover, the decline in the variance of investment shocks coincide remarkably well with the development of the European financial market that has increased access to credit by firms and households, suggesting that investment shocks reflect shocks originating in the financial system. [...]

Este artigo tem como objetivo verificar se há assimetria na rigidez de preços e de salários na economia brasileira; ou seja, se os agentes da economia são mais rígidos para baixo ou para cima para ajustarem seus preços e salários. Além disso, realiza-se a análise dos efeitos da política monetária e fiscal na dinâmica da economia. Para isso, utiliza-se um modelo Dinâmico Estocástico de Equilíbrio Geral (DSGE) não linear com custos de ajustamento assimétricos de preços e de salários com base no trabalho de Aruoba, Bocola e Schorfheide (2013). Esse modelo pode gerar rigidez de preços e de salários para baixo (ou para cima) que podem gerar não linearidades fortes. Diante da não linearidade gerada por esses aspectos, o modelo é solucionado através de um método de solução não linear e os seus parâmetros são estimados com a ajuda do Filtro de Partículas. O resultado encontrado é que tanto os preços quanto os salários nominais são mais rígidos para baixo e essas assimetrias na rigidez influenciam a dinâmica da economia quando esta sofre choques de política monetária e fiscal. / The objective of this article is to verify if there is asymmetry in the rigidity of prices and wages for the Brazilian economy; i.e. if the economic agents are more rigid downward or upward when adjusting their prices and wages. In addition, it performs the analysis of the effects of monetary and fiscal policy in the dynamics of the economy. For this, it uses a nonlinear model of Dynamic Stochastic General Equilibrium (DSGE) with asymmetric adjustment costs in prices and wages based on the work of Aruoba, Bocola and Schorfheide (2013). This model can generate prices and wages rigidity downward (or upward) that can produce strong nonlinearities. Considering the non-linearity generated by these aspects, the model is solved through a non-linear solution method and its parameters are estimated with the help of Particle Filter. The obtained result is that both prices and nominal wages are more rigid downwards and these asymmetries in rigidity influence the dynamics of the economy when it suffers shocks from monetary and fiscal policies.

Στην παρούσα εργασία ασχοληθήκαμε με την παρατηρησιμότητα σε μη-γραμμικά συστήματα, αφού πρώτα εξετάσαμε τα γραμμικά συστήματα μέσω μίας γεωμετρικής προσέγγισης. Αρχικώς, ασχοληθήκαμε με την παρατηρησιμότητα affine συστημάτων υπό τμηματικά σταθερές εισόδους. Στη συνέχεια μελετήσαμε τα ίδια συστήματα υπό διαφορίσιμες εισόδους. Κατόπιν, αποδείξαμε ένα θεώρημα που θέτει ένα άνω φράγμα στο πλήθος των Lie παραγωγίσεων της συνάρτησης εξόδου. Τέλος, εξετάσαμε τη μη-γραμμική μη-παρατηρησιμότητα σε γενικότερες κατηγορίες μη-γραμμικών συστημάτων με χρήση γραμμικών μεθόδων και logic dynamic approach. / This thesis is focused on the problem of observability in non-linear systems. A brief examination of linear systems via a geometric approach is initially given. First, observability of affine systems under piecewise constant controls is examined. Second, the same examination is repeated for the same systems under differentiable inputs. Moreover, a theorem that sets an upper bound on the number of Lie derivatives of the output functions is proved and explained. Finally, the non-linear unobservability of non-linear systems in general is presented under the scope of linear methods and logic dynamic approach.

Este artigo tem como objetivo verificar se há assimetria na rigidez de preços e de salários na economia brasileira; ou seja, se os agentes da economia são mais rígidos para baixo ou para cima para ajustarem seus preços e salários. Além disso, realiza-se a análise dos efeitos da política monetária e fiscal na dinâmica da economia. Para isso, utiliza-se um modelo Dinâmico Estocástico de Equilíbrio Geral (DSGE) não linear com custos de ajustamento assimétricos de preços e de salários com base no trabalho de Aruoba, Bocola e Schorfheide (2013). Esse modelo pode gerar rigidez de preços e de salários para baixo (ou para cima) que podem gerar não linearidades fortes. Diante da não linearidade gerada por esses aspectos, o modelo é solucionado através de um método de solução não linear e os seus parâmetros são estimados com a ajuda do Filtro de Partículas. O resultado encontrado é que tanto os preços quanto os salários nominais são mais rígidos para baixo e essas assimetrias na rigidez influenciam a dinâmica da economia quando esta sofre choques de política monetária e fiscal. / The objective of this article is to verify if there is asymmetry in the rigidity of prices and wages for the Brazilian economy; i.e. if the economic agents are more rigid downward or upward when adjusting their prices and wages. In addition, it performs the analysis of the effects of monetary and fiscal policy in the dynamics of the economy. For this, it uses a nonlinear model of Dynamic Stochastic General Equilibrium (DSGE) with asymmetric adjustment costs in prices and wages based on the work of Aruoba, Bocola and Schorfheide (2013). This model can generate prices and wages rigidity downward (or upward) that can produce strong nonlinearities. Considering the non-linearity generated by these aspects, the model is solved through a non-linear solution method and its parameters are estimated with the help of Particle Filter. The obtained result is that both prices and nominal wages are more rigid downwards and these asymmetries in rigidity influence the dynamics of the economy when it suffers shocks from monetary and fiscal policies.

Este artigo tem como objetivo verificar se há assimetria na rigidez de preços e de salários na economia brasileira; ou seja, se os agentes da economia são mais rígidos para baixo ou para cima para ajustarem seus preços e salários. Além disso, realiza-se a análise dos efeitos da política monetária e fiscal na dinâmica da economia. Para isso, utiliza-se um modelo Dinâmico Estocástico de Equilíbrio Geral (DSGE) não linear com custos de ajustamento assimétricos de preços e de salários com base no trabalho de Aruoba, Bocola e Schorfheide (2013). Esse modelo pode gerar rigidez de preços e de salários para baixo (ou para cima) que podem gerar não linearidades fortes. Diante da não linearidade gerada por esses aspectos, o modelo é solucionado através de um método de solução não linear e os seus parâmetros são estimados com a ajuda do Filtro de Partículas. O resultado encontrado é que tanto os preços quanto os salários nominais são mais rígidos para baixo e essas assimetrias na rigidez influenciam a dinâmica da economia quando esta sofre choques de política monetária e fiscal. / The objective of this article is to verify if there is asymmetry in the rigidity of prices and wages for the Brazilian economy; i.e. if the economic agents are more rigid downward or upward when adjusting their prices and wages. In addition, it performs the analysis of the effects of monetary and fiscal policy in the dynamics of the economy. For this, it uses a nonlinear model of Dynamic Stochastic General Equilibrium (DSGE) with asymmetric adjustment costs in prices and wages based on the work of Aruoba, Bocola and Schorfheide (2013). This model can generate prices and wages rigidity downward (or upward) that can produce strong nonlinearities. Considering the non-linearity generated by these aspects, the model is solved through a non-linear solution method and its parameters are estimated with the help of Particle Filter. The obtained result is that both prices and nominal wages are more rigid downwards and these asymmetries in rigidity influence the dynamics of the economy when it suffers shocks from monetary and fiscal policies.

Bei der Modellierung elektrischer Schaltungen ergeben sich Algebro-Differentialgleichungen (DAEs) mit proper formuliertem Hauptterm. Diese Gleichungen müssen z.B. bei der transienten Schaltungssimulation numerisch gelöst werden. Bei den klassischen Ansätzen der Linearen Mehrschrittverfahren oder der Runge-Kutta Verfahren ergeben sich Nachteile, die durch Verwendung von Allgemeinen Linearen Verfahren vermieden werden können. Sowohl Lineare Mehrschrittverfahren als auch Runge-Kutta Verfahren sind als Spezialfälle in dieser allgemeineren Klasse enthalten. Darüberhinaus sind aber neue Verfahren mit verbesserten Eigenschaften möglich. In dieser Arbeit werden DAEs der Schaltungssimulation eingehend studiert und Allgemeine Lineare Verfahren für solche Gleichungen untersucht. Die Verfahrenskonstruktion und Implementierungsfragen werden ausführlich diskutiert. Diese Arbeit erscheint im Logos Verlag Berlin (www.logos-verlag.de, ISBN 3-8325-1353-1). / Modelling electrical circuits leads to differential algebraic equations (DAEs) having a properly stated leading term. These equations need to be solved numerically, e.g. in case of a transient analysis of the given circuit. Classical methods such as linear multistep methods or Runge-Kutta schemes suffer from disadvantages that can be overcome by studying general linear schemes. Both Runge-Kutta methods and linear multistep schemes belong to this class as special cases, but there is plenty of room for new methods with improved properties. This work presents both a detailed study of DAEs in the framework of integrated circuit design and a thorough analysis of general linear methods for these kind of equations. The construction and implementation of general linear methods for DAEs is discussed in detail. This work is published by Logos Verlag Berlin (www.logos-verlag.de, ISBN 3-8325-1353-1).