In 2019, Adams, Ehlting, and Edwards developed a four-variable system of ordinary differential equations modelling phenylalanine metabolism in plants according to Michaelis-Menten kinetics. Analysis of the model suggested that when a series of reactions known as the Shikimate Ester Loop (SEL) is included, phenylalanine flux into primary metabolic pathways is prioritized over flux into secondary metabolic pathways when the availability of shikimate, a phenylalanine precursor, is low. Adams et al. called this mechanism of metabolic regulation the Precursor Shutoff Valve (PSV). Here, we attempt to simplify Adams and colleagues’ model by reducing the system to three variables and replacing the Michaelis-Menten terms with piecewise-defined approximations we call ramp functions. We examine equilibria and stability in this simplified model, and show that PSV-type regulation is still present in the version with the SEL. Then, we define a class of systems structurally similar to the simplified Adams model called biochemical ramp systems. We study the properties of the Jacobian matrices of these systems and then explore equilibria and stability in systems of n ≥ 2 variables. Finally, we make several suggestions regarding future work on biochemical ramp systems. / Graduate

Let but de cette thèse est de proposer différents modèles mathématiques de neurones pour l'Optogénétique et d'étudier leur contrôle optimal. Nous définissons d'abord une version contrôlée des modèles déterministes de dimension finie, dits à conductances. Nous étudions un problème de temps minimal pour un système affine mono-entrée dont nous étudions les singulières. Nous appliquons une méthode numérique directe pour observer les trajectoires et contrôles optimaux. Le contrôle optogénétique apparaît comme une nouvelle façon de juger de la capacité des modèles à conductances de reproduire les caractéristiques de la dynamique du potentiel de membrane, observées expérimentalement. Nous définissons ensuite un modèle stochastique en dimension infinie pour prendre en compte le caractère aléatoire des mécanismes des canaux ioniques et la propagation des potentiels d'action. Il s'agit d'un processus de Markov déterministe par morceaux (PDMP) contrôlé, à valeurs dans un espace de Hilbert. Nous définissons une large classe de PDMPs contrôlés en dimension infinie et prouvons le caractère fortement Markovien de ces processus. Nous traitons un problème de contrôle optimal à horizon de temps fini. Nous étudions le processus de décision Markovien (MDP) inclus dans le PDMP et montrons l'équivalence des deux problèmes. Nous donnons des conditions suffisantes pour l'existence de contrôles optimaux pour le MDP, et donc le PDMP. Nous discutons des variantes pour le modèle d'Optogénétique stochastique en dimension infinie. Enfin, nous étudions l'extension du modèle à un espace de Banach réflexif, puis, dans un cas particulier, à un espace de Banach non réflexif. / The aim of this thesis is to propose different mathematical neuron models that take into account Optogenetics, and study their optimal control. We first define a controlled version of finite-dimensional, deterministic, conductance based neuron models. We study a minimal time problem for a single-input affine control system and we study its singular extremals. We implement a direct method to observe the optimal trajectories and controls. The optogenetic control appears as a new way to assess the capability of conductance-based models to reproduce the characteristics of the membrane potential dynamics experimentally observed. We then define an infinite-dimensional stochastic model to take into account the stochastic nature of the ion channel mechanisms and the action potential propagation along the axon. It is a controlled piecewise deterministic Markov process (PDMP), taking values in an Hilbert space. We define a large class of infinite-dimensional controlled PDMPs and we prove that these processes are strongly Markovian. We address a finite time optimal control problem. We study the Markov decision process (MDP) embedded in the PDMP. We show the equivalence of the two control problems. We give sufficient conditions for the existence of an optimal control for the MDP, and thus, for the initial PDMP as well. The theoretical framework is large enough to consider several modifications of the infinite-dimensional stochastic optogenetic model. Finally, we study the extension of the model to a reflexive Banach space, and then, on a particular case, to a nonreflexive Banach space.

Childbearing is often associated with employment interruptions in women’s careers. Since 2005, the German federal government has implemented childcare reforms aiming at expanding the suitable infrastructure for children under the age of three, which should facilitate and accelerate the return to employment. The reforms have been a paradigm shift, because they show a shift from a traditional breadwinner family model to a dual earner-carer model. Despite federal leadership in childcare reforms, the characteristics of the care infrastructure in Germany vary by state and over time, which may contribute to different employment-interruption lengths. The study at hand evaluates Germany’s recent childcare reforms regarding the impact on maternal employment by examining relationships between childcare-characteristics -namely quality and availability- and mothers’ employment interruptions. A piecewise-constant exponential model is used to capture the cross-state and over time differences in childcare and their impact on the timing of women’s return to employment within the first three years after birth of their first child. The study uses individual data from the Pairfam 10.0 study and childcare indicators, which are collected by the federal and state’s statistical bureaus. The risk population includes 927 first-time mothers who gave birth between March 2006 and March 2018. Within this period, 525 first-time mothers return to employment within the first three years after childbirth.   A significant positive effect of the childcare reform on maternal employment is revealed. Both the availability expansion and the quality improvements are associated with earlier returns to employment, establishing both institutional and cultural effects of childcare policies. An educational gradient of the effect of childcare quality on maternal employment was tested, but the results were not significant.

Eine Klasse von Markovprozessen mit deterministischem Pfaden und zufälligen Sprüngen wird unter Zuhilfenahme von Martingalen und des erweiterten infinitesimalen Generators untersucht. Dabei steht die Berechnung des Erwartungswertes und der Laplacetransformierten bestimmter Stoppzeiten im Vordergrund. Des weiteren wird die Frage untersucht, wann die in Frage kommenden Prozesse über stationäre Verteilungen verfügen und wie diese im Existenzfall beschaffen sind. Die Methoden werden am Beispiel eines verallgemeinerten Altersprozesses und eines Lager- bzw. Dammprozesses vorgeführt.

Les lois de commande affines par morceaux ont attiré une grande attention de la communauté d'automatique de contrôle grâce à leur pertinence pour des systèmes contraints, systèmes hybrides; également pour l'approximation de commandes nonlinéaires. Pourtant, leur mise en oeuvre est soumise à quelques difficultés. Motivé par l'intérêt à cette classe de commandes, cette thèse porte sur leur analyse, mise en oeuvre et synthèse.La première partie de cette thèse a pour but le calcul de la marge de robustesse et de la marge de fragilité pour une loi de commande affine par morceaux donnée et un système linéaire discret. Plus précisément, la marge de robustesse est définie comme l'ensemble des systèmes linéaires à paramètres variants que la loi de commande donnée garde les trajectoires dans de la région faisable. D'ailleurs, la marge de fragilité comprend toutes les variations des coefficients de la commande donnée telle que l'invariance de la région faisable soit encore garantie. Il est montré que si la région faisable donnée est un polytope, ces marges sont aussi des polytopes.La deuxième partie de ce manuscrit est consacrée au problème de l'optimalité inverse pour la classe des fonctions affines par morceaux. C'est-à-dire, l'objective est de définir un problème d'optimisation pour lequel la solution optimale est équivalente à la fonction affine par morceaux donnée. La méthodologie est fondée sur le convex lifting, i.e., un variable auxiliaire, scalaire, qui permet de définir un ensemble convex à partir de la partition d'état de la fonction affine par morceaux donnée. Il est montré que si la fonction affine par morceaux donnée est continue, la solution optimale de ce problème redéfini sera unique. Par contre, si la continuité n'est pas satisfaite, cette fonction affine par morceaux sera une solution optimale parmi les autres du problème redéfini.En ce qui concerne l’application dans la commande prédictive, il sera montré que n'importe quelle loi de commande affine par morceaux continue peut être obtenue par un autre problème de commande prédictive avec l'horizon de prédiction au plus égal à 2. A côté de cet aspect théorique, ce résultat sera utile pour faciliter la mise en oeuvre des lois de commandes affines par morceaux en évitant l'enregistrement de la partition de l'espace d'état. Dans la dernière partie de ce rapport, une famille de convex liftings servira comme des fonctions de Lyapunov. En conséquence, ce "convex lifting" sera déployé pour synthétiser des lois de commande robustes pour des systèmes linéaires incertains, également en présence de perturbations additives bornées. Des lois implicites et explicites seront obtenues en même temps. Cette méthode permet de garantir la faisabilité récursive et la stabilité robuste. Cependant, cette fonction de Lyapunov est limitée à l'ensemble λ −contractive maximal avec une constante scalaire 0 ≤ λ < 1 qui est plus petit que l'ensemble contrôlable maximal. Pour cette raison, une extension de cette méthode pour l'ensemble contrôlable de N − pas, sera présentée. Cette méthode est fondée sur des convex liftings en cascade où une variable auxiliaire sera utilisée pour servir comme une fonction de Lyapunov. Plus précisément, cette variable est non-négative, strictement décroissante pour les N premiers pas et égale toujours à 0 − après. Par conséquent, la stabilité robuste est garantie. / Piecewise affine (PWA) feedback control laws have received significant attention due to their relevance for the control of constrained systems, hybrid systems; equally for the approximation of nonlinear control. However, they are associated with serious implementation issues. Motivated from the interest in this class of particular controllers, this thesis is mostly related to their analysis and design.The first part of this thesis aims to compute the robustness and fragility margins for a given PWA control law and a linear discrete-time system. More precisely, the robustness margin is defined as the set of linear time-varying systems such that the given PWA control law keeps the trajectories inside a given feasible set. On a different perspective, the fragility margin contains all the admissible variations of the control law coefficients such that the positive invariance of the given feasible set is still guaranteed. It will be shown that if the given feasible set is a polytope, then so are these robustness/fragility margins.The second part of this thesis focuses on inverse optimality problem for the class of PWA controllers. Namely, the goal is to construct an optimization problem whose optimal solution is equivalent to the given PWA function. The methodology is based on emph convex lifting: an auxiliary 1− dimensional variable which enhances the convexity characterization into recovered optimization problem. Accordingly, if the given PWA function is continuous, the optimal solution to this reconstructed optimization problem will be shown to be unique. Otherwise, if the continuity of this given PWA function is not fulfilled, this function will be shown to be one optimal solution to the recovered problem.In view of applications in linear model predictive control (MPC), it will be shown that any continuous PWA control law can be obtained by a linear MPC problem with the prediction horizon at most equal to 2 prediction steps. Aside from the theoretical meaning, this result can also be of help to facilitate implementation of PWA control laws by avoiding storing state space partition. Another utility of convex liftings will be shown in the last part of this thesis to be a control Lyapunov function. Accordingly, this convex lifting will be deployed in the so-called robust control design based on convex liftings for linear system affected by bounded additive disturbances and polytopic uncertainties. Both implicit and explicit controllers can be obtained. This method can also guarantee the recursive feasibility and robust stability. However, this control Lyapunov function is only defined over the maximal λ −contractive set for a given 0 ≤ λ < 1 which is known to be smaller than the maximal controllable set. Therefore, an extension of the above method to the N-steps controllable set will be presented. This method is based on a cascade of convex liftings where an auxiliary variable will be used to emulate a Lyapunov function. Namely, this variable will be shown to be non-negative, to strictly decrease for N first steps and to stay at 0 afterwards. Accordingly, robust stability is sought.

This paper examines childbearing among Polish migrant women and their descendants in Sweden. While many studies have focused on immigrants' childbearing in relation to women in the destination country, this study uses a country-of-origin and a country-of-destination approach in order to more thoroughly examine the socialization, selection and adaptation hypotheses. Using a piecewise-exponential model, the transitions to first and second births are analyzed using Swedish register data and the Polish Generations and Gender survey (GGS). The results show that the Polish stayers and the first-generation have relatively similar fertility behavior in the transition to first birth but not in the transition to second birth. However, parts of the similarity in the transition to first birth can be attributed to marital status selection. By examining the 1.5-generation and the second-generation in relation to Swedish natives, it is possible to see fertility convergence across generations, both when it comes to timing and quantum. This study also shows that family migrants have higher risk of having a first child compared to migrants moving for other reasons. However, in the transition to second birth, there is no difference.

[pt] Erdal Arıkan introduziu os códigos polares em 2009. Trata-se de uma nova classe de códigos de correção de erros capaz de atingir o limite de Shannon. Usando decodificação de cancelamento sucessivo em lista, concatenada por verificação de redundância cíclica e a construções rápida de código, os códigos polares tornaram-se um código de correção de erros atraente e de alto desempenho para uso prático. Recentemente, códigos polares foram adotados para o padrão de geração 5th para sistemas celulares, mais especificamente para as informações de controle dos canais reverso e direto para os serviços de comunicação eMBB. No entanto, os códigos polares são limitados a comprimentos de bloco a potências de dois, devido a um produto Kronecker recursivo do kernel polarizador 2x2. Para aplicações práticas, é necessário fornecer técnicas de construção de código polar de comprimento flexível. Outro aspecto a ser analisado é o obtenção de uma técnica de construção de códigos polares de baixa complexidade e que tenha um ótimo desempenho em canal de ruído aditivo gaussiano branco, principalmente para blocos longos, inspirada na otimização da construção da aproximação gaussiana. Outro aspecto relevante é o poder de decodificação paralela do decodificador de propagação de crenças. Esta é uma alternativa para atender aos novos critérios de velocidade e latência previstos para o padrão de próxima geração para sistemas celulares. No entanto, ele precisa de melhorias de desempenho para tornar-se operacionalmente viável, tanto para 5G quanto para as gerações futuras. Nesta tese, três aspectos dos códigos polares são abordados: a construção de códigos com comprimentos arbitrários que visam maximizar a flexibilidade e eficiência dos códigos polares, o aprimoramento do método de construção por métodos gaussianos aproximação e a decodificação de códigos usando um algoritmo adaptativo de propagação de crenças reponderadas, bem como analisar quaisquer compromissos que afetem o desempenho da correção de erros. / [en] Erdal Arikan introduced the polar codes in 2009. This is a new class of error correction codes capable of reaching the Shannon limit. Using cyclic redundancy check concatenated list successive cancellation decoding and fast code constructs, polar codes have become an attractive, high-performance error correction code for practical use. Recently, polar codes have been adopted for the 5th generation standard for cellular systems, more specifically for the uplink and downlink control information for the extended Mobile Broadband (eMBB) communication services. However, polar codes are limited to block lengths to powers of two, due to a recursive Kronecker product of the 2x2 polarizing kernel. For practical applications, it is necessary to provide flexible length polar code construction techniques. Another aspect analyzed is the development of a technique of construction of polar codes of low complexity and that has an optimum performance on additive white Gaussian noise channels, mainly for long blocks, inspired by the optimization of the Gaussian approximation construction. Another relevant aspect is the parallel decoding power of the belief propagation decoder. This is an alternative to achieve the new speed and latency criteria foreseen for the next generation standard for cellular systems. However, it needs performance improvements to become operationally viable, both for 5G and for future generations. In this thesis, three aspects of polar codes are addressed: the construction of codes with arbitrary lengths that are intended for maximizing the flexibility and efficiency of polar codes, the improvement of the construction method by Gaussian approximation and the decoding of codes using an adaptive reweighted belief propagation algorithm, as well as the analysis of trade-offs affecting error correction performance.

The far-reaching successes of deep neural networks in a wide variety of learning tasks have prompted research on how model properties account for high network performance. For a specific class of models whose activation functions are piecewise linear, one such property of interest is the number of linear regions that the network generates. Such models themselves define piecewise linear functions by partitioning input space into disjoint regions and fitting a different linear function on each such piece. It would be expected that the number or configuration of such regions would describe the model’s ability to fit complicated functions. However, previous works have shown difficulty in identifying linear regions as satisfactory predictors of model success. In this thesis, the question of whether the generation of linear regions due to training encode the properties of the learning problem is explored. More specifically, it is investigated whether change in linear region density due to model fitting is related to the geometric properties of the training data. In this work, data geometry is characterized in terms of the curvature of the underlying manifold. Models with ReLU activation functions are trained on a variety of regression problems defined on artificial manifolds and the change in linear region density is recorded along trajectories in input space. Learning is performed on problems defined on curves, surfaces and for image data. Experiments are repeated as the data geometry is varied and the change in density is compared with the manifold curvature measure used. In no experimental setting, was the observed change in density found to be clearly linked with curvature. However, density was observed to increase at points of discontinuity. This suggests that linear regions can in some instances model data complexities, however, the findings presented here do not support that data curvature is encoded by the formation of linear regions. Thus, the role that linear regions play in controlling the capacity of piecewise linear networks remains open. Future research is needed to gain further insights into how data geometry and linear regions are connected. / De breda framgångar som djupa neurala nätverk har uppvisat i en mängd olika inlärningsproblem har inspirerat ny forskning med syfte att förklara vilka modellegenskaper som resulterar i högpresterande nätverk. För neurala nätverk som använder styckvis linjära aktiveringsfunktioner är en intressant egenskap att studera de linjära regioner som nätverket genererar i det vektorrum som utgör träningsdatans definitionsmängd. Nätverk med styckvis linjära aktiveringsfunktioner delar upp definitionsmängden i distinkta regioner på vilka olika linjära funktioner avbildas. Dessa nätverk avbildar själva styckvis linjära funktioner. Genom att anpassa flera skilda linjära avbildningar går det att approximera funktioner som är icke-linjära. Därför skulle man kunna förvänta sig att antalet linjära regioner som en modell genererar och hur de är fördelade i rummet kunde fungera som mått på modellens förmåga att lära sig komplicerade funktioner. Tidigare efterforskingar inom detta område har dock inte kunnat demonstrera ett samband mellan antalet eller fördelningen av linjära regioner och modellens prestanda. I den här avhandlingen undersöks det vilken roll linjära regioner spelar i att förklara en modells kapacitet och vad den lär sig. Fångar de linjära regioner som ett nätverk lär sig de underliggande egenskaperna hos träningsdatan? Mer specifikt så studeras huruvida den lokala förändringen i antalet linjära regioner efter modellträning korrelerar med träningsdatans geometri. Träningsdata genereras från syntetiska mångfalder och datageometrin beskrivs i termer av mångfaldens krökning. På dessa mångfalder definieras regressionsproblem och träning upprepas för topologier av olika form och med olika krökning. Skillnaden i antalet linjära regioner efter träning mäts längs banor i definitionsdomänen och jämförs med datans krökning. Ingen av de experiment som utfördes lyckades påvisa något tydligt samband mellan förändring i antal regioner och datans krökning. Det observerades dock att antalet linjära regioner ökar i närheten av punkter som utgör diskontinuiteter. Detta antyder att linjära regioner under vissa omständigheter kan modellera komplexitet. Således förblir rollen som linjära regioner har i att förklara modellförmåga diffus.

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