The first part of this Thesis deals with the analysis of piecewise-smooth mechanical systems and the definition of special stability criteria in presence of non-conservative follower forces. To illustrate the peculiar stability properties of this kind of dynamical system, a reference 2 d.o.f. structure has been considered, composed of a rigid bar, with one and constrained to slide, without friction, along a curved profile, whereas the other and is subject to a follower force. In particular, the curved constraint is assumed to be composed of two circular profiles, with different and opposite curvatures, defining two separated subsystems. Due to this jump in the curvature, located at the junction point between the curved profiles, the entire mechanical structure can be modelled by discontinuous equations of motion, the differential equations valid in each subsystem can be combined, leading to the definition of a piecewise-smooth dynamical system. When a follower force acts on the structure, an unexpected and counterintuitive behaviour may occur: although the two subsystems are stable when analysed separately, the composed structure is unstable and exhibits flutter-like exponentially-growing oscillations. This special form of instability, previously known only from a mathematical point of view, has been analysed in depth from an engineering perspective, thus finding a mechanical interpretation based on the concept of non-conservative follower load. Moreover, the goal of this work is also the definition of some stability criteria that may help the design of these mechanical piecewise-smooth systems, since classical theorems cannot be used for the investigation of equilibrium configurations located at the discontinuity. In the literature, this unusual behaviour has been explained, from a mathematical perspective, through the existence of a discontinuous invariant cone in the phase space. For this reason, starting from the mechanical system described above, the existence of invariant cones in 2 d.o.f. mechanical systems is investigated through Poincaré maps. A complete theoretical analysis on piecewise-smooth dynamical systems is presented and special mathematical properties have been discovered, valid for generic 2~d.o.f. piecewise-smooth mechanical systems, which are useful for the characterisation of the stability of the equilibrium configurations. Numerical tools are implemented for the analysis of a 2~d.o.f. piecewise-smooth mechanical system, valid for piecewise-linear cases and extendible to the nonlinear ones. A numerical code has been developed, with the aim of predicting the stability of a piecewise-linear dynamical system a priori, varying the mechanical parameters. Moreover, “design maps” are produced for a given subset of the parameters space, so that a system with a desired stable or unstable behaviour can easily be designed. The aforementioned results can find applications in soft actuation or energy harvesting. In particular, in systems devoted to exploiting the flutter-like instability, the range of design parameters can be extended by using piecewise-smooth instead of smooth structures, since unstable flutter-like behaviour is possible also when each subsystem is actually stable. The second part of this Thesis deals with the numerical analysis of an elastic cloak for transient flexural waves in Kirchhoff-Love plates and the design of special metamaterials for this goal. In the literature, relevant applications of transformation elastodynamics have revealed that flexural waves in thin elastic plates can be diverted and channelled, with the aim of shielding a given region of the ambient space. However, the theoretical transformations which define the elastic properties of this “invisibility cloak” lead to the presence of a strong compressive prestress, which may be unfeasible for real applications. Moreover, this theoretical cloak must present, at the same time, high bending stiffness and a null twisting rigidity. In this Thesis, an orthotropic meta-structural plate is proposed as an approximated elastic cloak and the presence of the prestress has been neglected in order to be closer to a realistic design. With the aim of estimating the performance of this approximated cloak, a Finite Element code is implemented, based on a sub-parametric technique. The tool allows the investigation of the sensitivity of specific stiffness parameters that may be difficult to match in a real cloak design. Moreover, the Finite Element code is extended to investigate a meta-plate interacting with a Winkler foundation, to analyse how the substrate modulus transforms in the cloak region. This second topic of the Thesis may find applications in the realization of approximated invisibility cloaks, which can be employed to reduce the destructive effects of earthquakes on civil structures or to shield mechanical components from unwanted vibrations.

This thesis examines the optimal scheduling of a hydroelectric power plant with cascaded reservoirs each with multiple generating units under uncertainty after testing three linearization methods. These linearization methods are Successive Linear Programming, Piecewise Linear Approximations, and a Hybrid of the two together. There are two goals of this work. The first goal of this work aims to replace the nonconvex mixed-integer nonlinear program (MINLP) with a computationally efficient linearized mixed-integer linear program (MILP) that will be capable of finding a high quality solution, preferably the global optimum. The second goal is to implement a stochastic approach on the linearized method in a pseudo-rolling horizon method which keeps the ending time step fixed. Overall, the Hybrid method proved to be a viable replacement and performs well in the pseudo-rolling horizon tests. / Thesis / Master of Applied Science (MASc)

Available security standards for RFID networks (e.g. ISO/IEC 29167) are designed to secure individual tag-reader sessions and do not protect against active attacks that could also compromise the system as a whole (e.g. tag cloning or replay attacks). Proper traffic characterization models of the communication within an RFID network can lead to better understanding of operation under “normal” system state conditions and can consequently help identify security breaches not addressed by current standards. This study of RFID traffic characterization considers two piecewise-constant data smoothing techniques, namely Bayesian blocks and Knuth’s algorithms, over time-tagged events and compares them in the context of rate-based anomaly detection. This was accomplished using data from experimental RFID readings and comparing (1) the event counts versus time if using the smoothed curves versus empirical histograms of the raw data and (2) the threshold-dependent alert-rates based on inter-arrival times obtained if using the smoothed curves versus that of the raw data itself. Results indicate that both algorithms adequately model RFID traffic in which inter-event time statistics are stationary but that Bayesian blocks become superior for traffic in which such statistics experience abrupt changes.

Nous y introduisons une nouvelle classe de distributions bivariées de type Marshall-Olkin, la distribution Erlang bivariée. La transformée de Laplace, les moments et les densités conditionnelles y sont obtenus. Les applications potentielles en assurance-vie et en finance sont prises en considération. Les estimateurs du maximum de vraisemblance des paramètres sont calculés par l'algorithme Espérance-Maximisation. Ensuite, notre projet de recherche est consacré à l'étude des processus de risque multivariés, qui peuvent être utiles dans l'étude des problèmes de la ruine des compagnies d'assurance avec des classes dépendantes. Nous appliquons les résultats de la théorie des processus de Markov déterministes par morceaux afin d'obtenir les martingales exponentielles, nécessaires pour établir des bornes supérieures calculables pour la probabilité de ruine, dont les expressions sont intraitables. / In this contribution, we introduce a new class of bivariate distributions of Marshall-Olkin type, called bivariate Erlang distributions. The Laplace transform, product moments and conditional densities are derived. Potential applications of bivariate Erlang distributions in life insurance and finance are considered. Further, our research project is devoted to the study of multivariate risk processes, which may be useful in analyzing ruin problems for insurance companies with a portfolio of dependent classes of business. We apply results from the theory of piecewise deterministic Markov processes in order to derive exponential martingales needed to establish computable upper bounds of the ruin probabilities, as their exact expressions are intractable.

Dans cette thèse, nous étudions trois types de problèmes stochastiques : les problèmes avec contraintes probabilistes, les problèmes distributionnellement robustes et les problèmes avec recours. Les difficultés des problèmes stochastiques sont essentiellement liées aux problèmes de convexité du domaine des solutions, et du calcul de l’espérance mathématique ou des probabilités qui nécessitent le calcul complexe d’intégrales multiples. A cause de ces difficultés majeures, nous avons résolu les problèmes étudiées à l’aide d’approximations efficaces.Nous avons étudié deux types de problèmes stochastiques avec des contraintes en probabilités, i.e., les problèmes linéaires avec contraintes en probabilité jointes (LLPC) et les problèmes de maximisation de probabilités (MPP). Dans les deux cas, nous avons supposé que les variables aléatoires sont normalement distribués et les vecteurs lignes des matrices aléatoires sont indépendants. Nous avons résolu LLPC, qui est un problème généralement non convexe, à l’aide de deux approximations basée sur les problèmes coniques de second ordre (SOCP). Sous certaines hypothèses faibles, les solutions optimales des deux SOCP sont respectivement les bornes inférieures et supérieures du problème du départ. En ce qui concerne MPP, nous avons étudié une variante du problème du plus court chemin stochastique contraint (SRCSP) qui consiste à maximiser la probabilité de la contrainte de ressources. Pour résoudre ce problème, nous avons proposé un algorithme de Branch and Bound pour calculer la solution optimale. Comme la relaxation linéaire n’est pas convexe, nous avons proposé une approximation convexe efficace. Nous avons par la suite testé nos algorithmes pour tous les problèmes étudiés sur des instances aléatoires. Pour LLPC, notre approche est plus performante que celles de Bonferroni et de Jaganathan. Pour MPP, nos résultats numériques montrent que notre approche est là encore plus performante que l’approximation des contraintes probabilistes individuellement.La deuxième famille de problèmes étudiés est celle relative aux problèmes distributionnellement robustes où une partie seulement de l’information sur les variables aléatoires est connue à savoir les deux premiers moments. Nous avons montré que le problème de sac à dos stochastique (SKP) est un problème semi-défini positif (SDP) après relaxation SDP des contraintes binaires. Bien que ce résultat ne puisse être étendu au cas du problème multi-sac-à-dos (MKP), nous avons proposé deux approximations qui permettent d’obtenir des bornes de bonne qualité pour la plupart des instances testées. Nos résultats numériques montrent que nos approximations sont là encore plus performantes que celles basées sur les inégalités de Bonferroni et celles plus récentes de Zymler. Ces résultats ont aussi montré la robustesse des solutions obtenues face aux fluctuations des distributions de probabilités. Nous avons aussi étudié une variante du problème du plus court chemin stochastique. Nous avons prouvé que ce problème peut se ramener au problème de plus court chemin déterministe sous certaine hypothèses. Pour résoudre ce problème, nous avons proposé une méthode de B&B où les bornes inférieures sont calculées à l’aide de la méthode du gradient projeté stochastique. Des résultats numériques ont montré l’efficacité de notre approche. Enfin, l’ensemble des méthodes que nous avons proposées dans cette thèse peuvent s’appliquer à une large famille de problèmes d’optimisation stochastique avec variables entières. / In this thesis, we studied three types of stochastic problems: chance constrained problems, distributionally robust problems as well as the simple recourse problems. For the stochastic programming problems, there are two main difficulties. One is that feasible sets of stochastic problems is not convex in general. The other main challenge arises from the need to calculate conditional expectation or probability both of which are involving multi-dimensional integrations. Due to the two major difficulties, for all three studied problems, we solved them with approximation approaches.We first study two types of chance constrained problems: linear program with joint chance constraints problem (LPPC) as well as maximum probability problem (MPP). For both problems, we assume that the random matrix is normally distributed and its vector rows are independent. We first dealt with LPPC which is generally not convex. We approximate it with two second-order cone programming (SOCP) problems. Furthermore under mild conditions, the optimal values of the two SOCP problems are a lower and upper bounds of the original problem respectively. For the second problem, we studied a variant of stochastic resource constrained shortest path problem (called SRCSP for short), which is to maximize probability of resource constraints. To solve the problem, we proposed to use a branch-and-bound framework to come up with the optimal solution. As its corresponding linear relaxation is generally not convex, we give a convex approximation. Finally, numerical tests on the random instances were conducted for both problems. With respect to LPPC, the numerical results showed that the approach we proposed outperforms Bonferroni and Jagannathan approximations. While for the MPP, the numerical results on generated instances substantiated that the convex approximation outperforms the individual approximation method.Then we study a distributionally robust stochastic quadratic knapsack problems, where we only know part of information about the random variables, such as its first and second moments. We proved that the single knapsack problem (SKP) is a semedefinite problem (SDP) after applying the SDP relaxation scheme to the binary constraints. Despite the fact that it is not the case for the multidimensional knapsack problem (MKP), two good approximations of the relaxed version of the problem are provided which obtain upper and lower bounds that appear numerically close to each other for a range of problem instances. Our numerical experiments also indicated that our proposed lower bounding approximation outperforms the approximations that are based on Bonferroni's inequality and the work by Zymler et al.. Besides, an extensive set of experiments were conducted to illustrate how the conservativeness of the robust solutions does pay off in terms of ensuring the chance constraint is satisfied (or nearly satisfied) under a wide range of distribution fluctuations. Moreover, our approach can be applied to a large number of stochastic optimization problems with binary variables.Finally, a stochastic version of the shortest path problem is studied. We proved that in some cases the stochastic shortest path problem can be greatly simplified by reformulating it as the classic shortest path problem, which can be solved in polynomial time. To solve the general problem, we proposed to use a branch-and-bound framework to search the set of feasible paths. Lower bounds are obtained by solving the corresponding linear relaxation which in turn is done using a Stochastic Projected Gradient algorithm involving an active set method. Meanwhile, numerical examples were conducted to illustrate the effectiveness of the obtained algorithm. Concerning the resolution of the continuous relaxation, our Stochastic Projected Gradient algorithm clearly outperforms Matlab optimization toolbox on large graphs.

Au cours des deux dernières décennies, les signaux chaotiques ont été de plusen plus pris en compte dans les télécommunications, traitement du signal ou transmissionssécurisées. De nombreux articles ont été publiés qui étudient la densitéspectrale de puissance (DSP) des signaux générés par des transformations spécifiques.La concentration sur la DSP est due à l’importance de la fréquence dans lestélécommunications et la transmission sécurisée. Grâce au grand nombre de systèmessans fil, la disponibilité des fréquences de transmission et de réception est de plus enplus rare pour les communications sans fil. Aussi, les médias guidés ont des limitationsliées à la bande passante du signal. Dans cette thèse, nous étudions certainespropriétés associées à la bifurcation collision de frontière pour une transformationunidimensionnelle linéaire par morceaux avec trois pentes et deux paramètres. Nouscalculons les expressions analytiques de l’autocorrélation et de la densité spectralede puissance des signaux chaotiques générés par les transformations linéaires parmorceaux. Nous montrons l’existence d’une forte relation entre les différents typesde densité spectrale de puissance (passe-bas, passe-haut ou coupe-bande) et les paramètresde bifurcation. Nous notons également en évidence une relation entre le typede spectre et l’ordre des cycles attractifs. Le type du spectre dépend de l’existencedes orbites périodiques au-delà de la bifurcation de collision de frontière qui a donnénaissance au chaos. Nous utilisons ensuite les transformations chaotiques pour étudierla fonction d’ambiguïté. Nous combinons quelques transformations chaotiquesbien déterminées pour obtenir un spectre large bande avec une bonne fonction d’ambiguïtéqui peut être utilisée en système radar / During the two last decades, chaotic signals have been increasingly consideredin telecommunications, signal processing or secure transmissions. Many papers haveappeared which study the power spectral density (PSD) of signals issued from somespecific maps. This interest in the PSD is due to the importance of frequency in thetelecommunications and transmission security. With the large number of wirelesssystems, the availability of frequencies for transmission and reception is increasinglyuncommon for wireless communications. Also, guided media have limitations relatedto the bandwidth of a signal. In this thesis, we investigate some properties associatedto the border-collision bifurcations in a one-dimensional piecewise-linear map withthree slopes and two parameters. We derive analytical expressions for the autocorrelationsequence, power spectral density of chaotic signals generated by our piecewiselinearmap. We prove the existence of strong relation between different types of thepower spectral density (low-pass, high-pass or band-stop) and the parameters. Wealso find a relation between the type of spectrum and the order of attractive cycleswhich are located after the border collision bifurcation between chaos and cycles.We use the chaotic transformations to study the ambiguity function. We combinesome chaotic transformations well determined to obtain a broadband spectrum witha good ambiguity function that can be used in radar systems

Diese Dissertation beschäftigt sich mit der rigorosen Herleitung effektiver Modelle zur Beschreibung von Schädigungsprozessen. Diese effektiven Modelle werden für verschiedene raten-unabhängige Schädigungsmodelle linear elastischer Materialien hergeleitet. Den Ausgangspunkt stellt dabei ein unidirektionales Mikrostrukturevolutionsmodell dar, dessen Fundament eine Familie geordneter zulässiger Mikrostrukturen bildet. Jede Mikrostruktur dieser Familie besitzt die gleiche intrinsische Längenskala. Zur Herleitung eines effektiven Modells wird das asymptotische Verhalten dieser Längenskala mittels Techniken der Zwei-Skalen-Konvergenz untersucht. Um das Grenzmodell zu identifizieren, bedarf es einer Mikrostrukturregularisierung, die als diskreter Gradient für stückweise konstante Funktionen aufgefasst werden kann. Die Mikrostruktur des effektiven Modells ist punktweise durch ein Einheitszellenproblem gegeben, welches die Mikro- von der Makroskala trennt. Ausgehend vom Homogenisierungsresultat für die unidirektionale Mikrostrukturevolution werden effektive Modelle für Zwei-Phasen-Schädungsprozesse hergeleitet. Die aus zwei Phasen bestehende Mikrostruktur der mikroskopischen Modelle ermöglicht z.B. die Modellierung von Schädigung durch das Wachstum von Inklusionen aus geschädigtem Material verschiedener Form und Größe. Außerdem kann Schädigung durch das Wachstum mikroskopischer Hohlräume und Mikrorissen betrachtet werden. Die Größe der Defekte skaliert mit der intrinsischen Längenskala und die unidirektionale Mikrostrukturevolution verhindert, dass bei fixierter Längenskala die Defekte für fortlaufende Zeit schrumpfen. Das Material des Grenzmodells ist dann in jedem Punkt als Mischung von ungeschädigtem und geschädigtem Material durch das Einheitszellenproblem gegeben. Dabei liefert das Einheitszellenproblem nicht nur das Mischungsverhältnis sondern auch die genaue geometrische Mischungsverteilung, die dem effektiven Material des jeweiligen Materialpunktes zugrunde liegt. / This dissertation at hand deals with the rigorous derivation of such effective models used to describe damage processes. For different rate-independent damage processes in linear elastic material these effective models are derived as the asymptotic limit of microscopic models. The starting point is represented by a unidirectional microstructure evolution model which is based on a family of ordered admissible microstructures. Each microstructure of that family possesses the same intrinsic length scale. To derive an effective model, the asymptotic behavior of this intrinsic length scale is investigated with the help of techniques of the two-scale convergence. For this purpose, a microstructure-regularizing term, which can be understood as a discrete gradient for piecewise constant functions, is needed to identify the limit model. The microstructure of the effective model is given pointwisely by a so-called unit cell problem which separates the microscopic scale from the macroscopic scale. Based on these homogenization results for unidirectional microstructure evolution models, effective models for brutal damage processes are provided. There, the microstructure consists of only two phases, namely undamaged material which comprises defects of damaged material with various sizes and shapes. In this way damage progression can be modeled by the growth of inclusions of weak material, the growth of voids, or the growth of microscopic cracks. The size of the defects is scaled by the intrinsic length scale and the unidirectional microstructure evolution prevents that, for a fixed length scale, the defects shrink for progressing time. According to the unit cell problem, the material of the limit model is then given as a mixture of damaged and undamaged material. In a specific material point of the limit model, that unit cell problem does not only define the mixture ratio but also the exact geometrical mixture distribution.

Le problème de séparation pour une classe de langages S est le suivant : étant donnés deux langages L1 et L2, existe-t-il un langage appartenant à S qui contient L1, en étant disjoint de L2 ? Si les langages à séparer sont des langages réguliers, le problème de séparation pour la classe S est plus général que le problème de l'appartenance à cette classe, et nous fournit des informations plus détaillées sur la classe. Ce problème de séparation apparaît dans un contexte algébrique sous la forme des parties ponctuelles, et dans un contexte profini sous la forme d'un problème de séparation topologique. Pour quelques classes de langages spécifiques, ce problème a été étudié en utilisant des méthodes profondes de la théorie des semigroupes profinis.Dans cette thèse, on s'intéresse, dans un premier temps, à la décidabilité de ce problème pour plusieurs sous-classes des langages réguliers. Dans un second temps, on s'intéresse à obtenir un langage séparateur, s'il existe, ainsi qu'à la complexité de ces problèmes.Nous établissons une approche générique pour prouver que le problème de séparation est décidable pour une classe de langages donnée. En utilisant cette approche, nous obtenons la décidabilité du problème de séparation pour les langages testables par morceaux, les langages non-ambigus, les langages localement testables, et les langages localement testables à seuil. Ces classes correspondent à des fragments de la logique du premier ordre, et sont parmi lesclasses de langages réguliers les plus étudiées. De plus, cette approche donne une description d'un langage séparateur, pourvu qu'il existe. / The separation problem, for a class S of languages, is the following: given two input languages, does there exist a language in S that contains the first language and that is disjoint from the second langage ?For regular input languages, the separation problem for a class S subsumes the classical membership problem for this class, and provides more detailed information about the class. This separation problem first emerged in an algebraic context in the form of pointlike sets, and in a profinite context as a topological separation problem. These problems have been studied for specific classes of languages, using involved techniques from the theory of profinite semigroups.In this thesis, we are not only interested in showing the decidability of the separation problem for several subclasses of the regular languages, but also in constructing a separating language, if it exists, and in the complexity of these problems.We provide a generic approach, based on combinatorial arguments, to proving the decidability of this problem for a given class. Using this approach, we prove that the separation problem is decidable for the classes of piecewise testable languages, unambiguous languages, and locally (threshold) testable languages. These classes are defined by different fragments of first-order logic, and are among the most studied classes of regular languages. Furthermore, our approach yields a description of a separating language, in case it exists.

可轉換公司債兼具了選擇權以及債券的性質，價值又會受到股價之影響，以傳統的方法定價十分不易。由於蒙地卡羅模擬能解決定價問題上狀態變數或許為多維度及路徑相依的問題，Kind 與Wilde 在2004 年提出以蒙地卡羅模擬對可轉債定價，且以最小平方迴歸法估計繼續持有價值，並在僅考慮轉換及還本兩種選擇權及沒有違約風險之下，以數值範例呈現單一迴歸模式無法適當估計繼續持有價值。然而，他們並未進行實證。本研究乃以民國99 年台灣發行的可轉債為研究對象，除考慮發行時的合約條件外，另加上信用評等的考量以將違約機率透過現金流量套入定價過程中，並分別以兩段迴歸及單一迴歸估計繼續持有價值以結合蒙地卡羅模擬，實證結果顯示就可轉債之起始定價的偏差比而言，兩段迴歸得到的結果優於單一迴歸。惟在兩段迴歸之下，超過八成的可轉債其模擬價格依然高於市場價格。實證結果也顯示價性(moneyness)及擔保狀況與定價的偏差有關。 / Convertible bonds (CBs) possess features of both bonds and options, and their prices are affected by the underlying stocks, which make the pricing problem an uneasy task for traditional methods. Since Monte Carlo simulation can handle the problems of path-dependence and multivariate dimensions faced by pricing, Kind and Wilde (2004) suggested to price CBs via least-squares Monte Carlo simulations (LSM), which estimate the continuation values by least squares regression. They also demonstrated that a single regression line could not appropriately estimate the continuation value even only conversion and redemption were allowed and the CB was free of default. So the idea of piecewise regression was recommended to improve the estimation process. However, they didn’t apply piecewise regression to real data. Therefore, piecewise regression together with Monte Carlo simulation were employed to investigate the pricing issue of Taiwan’s CBs. CBs issued on 2010 were selected, besides reviewing the contents of CB’s contracts, default risks based on credit ratings were taken into account to evaluate the discounted cash flows in the pricing procedure. Comparing the estimated model prices of LSM with initial selling prices, the mispricing rates of single regression model and piecewise regression model were obtained for further analysis. Result shows that the modified piecewise regression method performs better in mispricing rate. However, similar to previous findings, 80% of the estimated model prices based on piecewise regressions are still higher than market prices. It also shows that moneyness and guaranteed condition will relate to mispricing rate.

Nous y introduisons une nouvelle classe de distributions bivariées de type Marshall-Olkin, la distribution Erlang bivariée. La transformée de Laplace, les moments et les densités conditionnelles y sont obtenus. Les applications potentielles en assurance-vie et en finance sont prises en considération. Les estimateurs du maximum de vraisemblance des paramètres sont calculés par l'algorithme Espérance-Maximisation. Ensuite, notre projet de recherche est consacré à l'étude des processus de risque multivariés, qui peuvent être utiles dans l'étude des problèmes de la ruine des compagnies d'assurance avec des classes dépendantes. Nous appliquons les résultats de la théorie des processus de Markov déterministes par morceaux afin d'obtenir les martingales exponentielles, nécessaires pour établir des bornes supérieures calculables pour la probabilité de ruine, dont les expressions sont intraitables. / In this contribution, we introduce a new class of bivariate distributions of Marshall-Olkin type, called bivariate Erlang distributions. The Laplace transform, product moments and conditional densities are derived. Potential applications of bivariate Erlang distributions in life insurance and finance are considered. Further, our research project is devoted to the study of multivariate risk processes, which may be useful in analyzing ruin problems for insurance companies with a portfolio of dependent classes of business. We apply results from the theory of piecewise deterministic Markov processes in order to derive exponential martingales needed to establish computable upper bounds of the ruin probabilities, as their exact expressions are intractable.