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1
Farlow, Kasie Geralyn. "Max-Plus Algebra". Thesis, Virginia Tech, 2009. http://hdl.handle.net/10919/32191.
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In max-plus algebra we work with the max-plus semi-ring which is the set ℝmax=[-∞)∪ℝ together with operations 𝑎⊕𝑏 = max(𝑎,𝑏) and 𝑎⊗𝑏= 𝑎+𝑏. The additive and multiplicative identities are taken to be ε=-∞ and ε=0 respectively. Max-plus algebra is one of many idempotent semi-rings which have been considered in various fields of mathematics. Max-plus algebra is becoming more popular not only because its operations are associative, commutative and distributive as in conventional algebra but because it takes systems that are non-linear in conventional algebra and makes them linear. Max-plus algebra also arises as the algebra of asymptotic growth rates of functions in conventional algebra which will play a significant role in several aspects of this thesis. This thesis is a survey of max-plus algebra that will concentrate on max-plus linear algebra results. We will then consider from a max-plus perspective several results by Wentzell and Freidlin for finite state Markov chains with an asymptotic dependence.Master of Science
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Tam, Kin Po. "Optimizing and approximating eigenvectors in max-algebra". Thesis, University of Birmingham, 2010. http://etheses.bham.ac.uk//id/eprint/858/.
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This thesis is a reflection of my research in max-algebra. The idea of max-algebra is replacing the conventional pairs of operations (+,x) by (max, +). It has been known for some time that max-algebraic linear systems and eigenvalue-eigenvector problem can be used to describe industrial processes in which a number of processors work interactively and possibly in stages. Solutions to such max-algebraic linear system typically correspond to start time vectors which guarantee that the processes meet given deadlines or will work in a steady regime. The aim of this thesis is to study such problems subjected to additional requirements or constraints. These include minimization and maximization of the time span of completion times or starting times. We will also consider the case of minimization and maximization of the time span when some completion times or starting times are prescribed. The problem of integrality is also studied in this thesis. This is finding completion times or starting times which consists of integer values only. Finally we consider max-algebraic permuted linear systems where we permute a given vector and decide if the permuted vector is a satisfactory completion time or starting time. For some of these problems, we developed exact and efficient methods. Some of them turn out to be hard. For these we have proposed and tested a number of heuristics.
3
Jones, Daniel Lewis. "Special and structured matrices in max-plus algebra". Thesis, University of Birmingham, 2017. http://etheses.bham.ac.uk//id/eprint/7527/.
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The aim of this thesis is to present efficient (strongly polynomial) methods and algorithms for problems in maxalgebra when certain matrices have special entries or are structured. First, we describe all solutions to a one-sided parametrised system. Next, we consider special cases of two-sided systems of equations/inequalities. Usually, we describe a set of generators of all solutions but sometimes we are satisfied with finding a non-trivial solution or being able to say something meaningful about a non-trivial solution should it exist. We look at special cases of the generalised eigenproblem, describing the full spectrum usually. Finally, we prove some results on 2x2 matrix roots and generalise these results to a class of nxn matrices. Main results include: a description of all solutions to the two-dimensional generalised eigenproblem; observations about a non-trivial solution (should it exist) to essential/minimally active two-sided systems of equations; the full spectrum of the generalised eigenproblem when one of the matrices is an outer-product; the unique candidate for the generalised eigenproblem when the difference of two matrices is symmetric and has a saddle point and finally we explicitly say when a 2x2 matrix has a kth root for a fixed positive integer k.
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Imaev, Aleksey A. "Hierarchical Modeling of Manufacturing Systems Using Max-Plus Algebra". Ohio University / OhioLINK, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1257871858.
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Tschaikowski, Max [Verfasser] y Mirco [Akademischer Betreuer] Tribastone. "Fluid aggregations for Markovian process algebra / Max Tschaikowski. Betreuer: Mirco Tribastone". München : Universitätsbibliothek der Ludwig-Maximilians-Universität, 2014. http://d-nb.info/1053618638/34.
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Maia, Carlos Andrey. "Identificação e controle de sistemas a eventos discretos na algebra (Max,+)". [s.n.], 2003. http://repositorio.unicamp.br/jspui/handle/REPOSIP/260337.
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Orientadores: Rafael Santos Mendes, Jean Louis Boimond, Laurent HardouinTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação
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Doutorado
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Patlola, Phanindher R. "Efficient Evaluation of Makespan for a Manufacturing System Using Max-Plus Algebra". Ohio University / OhioLINK, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1304980385.
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Murfitt, Louise. "Discrete event dynamic systems in max-algebra : realisation and related combinatorial problems". Thesis, University of Birmingham, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.368451.
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Brat, Guillaume Philippe. "A (max,+) algebra for non-stationary and non-deterministic periodic discrete event systems /". Digital version accessible at:, 1998. http://wwwlib.umi.com/cr/utexas/main.
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優樹, 西田 y Yuki Nishida. "Studies on linear systems and the eigenvalue problem over the max-plus algebra". Thesis, https://doors.doshisha.ac.jp/opac/opac_link/bibid/BB13158503/?lang=0, 2021. https://doors.doshisha.ac.jp/opac/opac_link/bibid/BB13158503/?lang=0.
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Max-plus代数は，実数全体に無限小元を付加した集合に，加法として最大値をとる演算，乗法として通常の加法を考えた代数系である．本論文では，max-plus線形方程式に対するCramerの公式の類似物を用いて，線形方程式の解空間の基底が構成できることを示した．さらに固有値問題に関連して，max-plus行列の固有ベクトルの概念を2通りの観点から拡張した．The max-plus algebra is the semiring with addition "max" and multiplication "+". In the present thesis, the author gives a combinatorial characterization of solutions of linear systems in terms of the max-plus Cramer's rule. Further, the author extends the concept of eigenvectors of max-plus matrices from two different perspectives.
博士(理学)
Doctor of Philosophy in Science
同志社大学
Doshisha University
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Cândido, Renato Markele Ferreira 1988. "Filtros de partículas aplicados a sistemas max plus". [s.n.], 2013. http://repositorio.unicamp.br/jspui/handle/REPOSIP/259747.
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Orientador: Rafael Santos MendesDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação
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Resumo: A principal contribuição desta dissertação é a proposta de algoritmos de filtragem por partículas em sistemas a eventos discretos nos quais predominam os problemas de sincronização. Esta classe de sistemas pode ser descrita por meio de equações lineares em uma álgebra não convencional usualmente conhecida como álgebra Max Plus. Os Filtros de Partículas são algoritmos Bayesianos sub-ótimos que realizam uma amostragem sequencial de Monte Carlo para construir uma aproximação discreta da densidade de probabilidade dos estados baseada em um conjunto de partículas com pesos associados. É apresentada uma revisão de sistemas a eventos discretos, de filtragem não linear e de filtros de partículas de um modo geral. Após apresentar esta base teórica, são propostos dois algoritmos de filtros de partículas aplicados a sistemas Max Plus. Em seguida algumas simulações foram apresentadas e os resultados apresentados mostraram a eficiência dos filtros desenvolvidos
Abstract: This thesis proposes, as its main contribution, particle filtering algorithms for discrete event systems in which synchronization phenomena are prevalent. This class of systems can be described by linear equation systems in a nonconventional algebra commonly known as Max Plus algebra. Particles Filters are suboptimal Bayesian algorithms that perform a sequential Monte Carlo sampling to construct a discrete approximation of the probability density of states based on a set of particles with associated weights. It is presented a review of discrete event systems, nonlinear filtering and particle filters. After presenting this theoretical background, two particle filtering algorithms applied to Max Plus systems are proposed. Finally some simulation results are presented, confirming the accuracy of the designed filters
Mestrado
Automação
Mestre em Engenharia Elétrica
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Nascimento, Tadeu Matos Henriques. "Álgebra tropical: uma abordagem introdutória". Universidade Federal de Sergipe, 2016. https://ri.ufs.br/handle/riufs/6492.
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Often mathematics is seen by high school students as a science restricted to memorizing formulas and
concepts. Therefore limiting in its essence. The work seeks to reverse that view by submitting a new
eld of study: Tropical Algebra. Relatively new area of mathematics that keeps the curious feature
to handle the operations of addition and multiplication di erently from traditional, already presents
interesting practical results. Tropical algebra will be presented in a didactic way, comparing it with
the traditional algebra, showing the consequences of tropical operations in the study of polynomials,
matrices and geometry, and presenting some practical applications.Frequentemente a matemática é vista pelos alunos do ensino médio como uma ciência restrita à memorização de fórmulas e conceitos. Portanto, limitante em sua essência. O trabalho busca reverter tal visão através da apresentação de um novo campo de estudos: A Àlgebra Tropical. Área relativamente nova da matemática que guarda a curiosa característica de tratar as operações de adi- ção e multiplicação de forma diferente da tradicional, já apresenta resultados práticos interessantes. A Àlgebra Tropical será apresentada de forma didática, comparando-a com a álgebra tradicional e mostrando as consequências das operações tropicais no estudo dos polinômios, matrizes e geometria, além de apresentar algumas aplicações práticas.
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David-Henriet, Xavier. "Discrete Event Systems with Standard and Partial Synchronizations". Thesis, Angers, 2015. http://www.theses.fr/2015ANGE0010/document.
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De nombreux systèmes de transport peuvent être modélisées par des synchronisations ordinaires (pour tout k>=l, l'occurrence k de l'événement B se produit au moins t unités de temps après l'occurrence k-l de l'événement A). Ces systèmes sont linéaires dans l'algèbre (max,+). Pour certaines applications, il est primordial de modéliser la simultanéité entre événements. Comme la synchronisation ordinaire ne suffit pas à exprimer ce phénomène, nous introduisons la synchronisation partielle (l'événement B ne peut se produire que quand l'événement A se produit). Dans ce mémoire, des méthodes développées pour la modélisation et le contrôle de systèmes linéaires dans l'algèbre (max,+) sont étendues à des systèmes régis par des synchronisations ordinaires et partielles. Nous considérons uniquement des systèmes divisés en un système principal et un système secondaire et gouvernés par des synchronisations ordinaires entre événements dans le même système et des synchronisations partielles d'événements dans le système secondaire par des événements dans le système principal. Nous introduisons une commande optimale et une commande prédictive pour cette classe de systèmes par analogie avec les résultats disponibles pour les systèmes linéaires dans l'algèbre (max,+). En considérant un comportement donné pour le système principal, il est aussi possible de représenter le système secondaire par une fonction de transfert et de modifier sa dynamique pour suivre un modèle de référenceMany transportation networks can be modeled by (max,+)-linear systems, i.e., discrete event systems ruled by standard synchronizations (conditions of the form: "for all k>=l, occurrence k of event B is at least t units of time after occurrence k-l of event A"). In some applications, it is also necessary to model simultaneity between events (e.g., for a road equipped with traffic lights, a vehicle can cross an intersection only when the associated traffic light is green). Such conditions cannot be expressed using standard synchronizations. Hence, we introduce the partial synchronization (condition of the form: "event B can only occur when event A occurs"). In this thesis, we consider a class of discrete event systems ruled by standard and partial synchronizations, called (max,+)-systems with partial synchronization. Such systems are split into a main system and a secondary system such that there exist only standard synchronizations between events in the same system and partial synchronizations of events in the secondary system by events in the main system. We adapt some modeling and control approaches developed for (max,+)-linear systems to (max,+)-systems with partial synchronization. Optimal feedforward control and model predictive control for (max,+)-linear systems are extended to (max,+)-systems with partial synchronization. Furthermore, transfer relation and model reference control are provided for the secondary system under a predefined behavior of the main system
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Grimsman, David R. "The Asynchronous t-Step Approximation for Scheduling Batch Flow Systems". BYU ScholarsArchive, 2016. https://scholarsarchive.byu.edu/etd/5957.
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Heap models in the max-plus algebra are interesting dynamical systems that can be used to model a variety of tetris-like systems, such as batch flow shops for manufacturing models. Each heap in the model can be identified with a single product to manufacture. The objective is to manufacture a group of products in such an order so as to minimize the total manufacturing time. Because this scheduling problem reduces to a variation of the Traveling Salesman Problem (known to be NP-complete), the optimal solution is computationally infeasible for many real-world systems. Thus, a feasible approximation method is needed. This work builds on and expands the existing heap model in order to more effectively solve the scheduling problems. Specifically, this work:1. Further characterizes the admissible products to these systems.2. Further characterizes sets of admissible products. 3. Presents a novel algorithm, the asynchronous $t$-step approximation, to approximate these systems.4. Proves error bounds for the system approximation, and show why these error bounds are better than the existing approximation.
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Singh, Manjeet. "Mathematical Models, Heuristics and Algorithms for Efficient Analysis and Performance Evaluation of Job Shop Scheduling Systems Using Max-Plus Algebraic Techniques". Ohio University / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1386087325.
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Faccin, Paolo. "Computational problems in algebra: units in group rings and subalgebras of real simple Lie algebras". Doctoral thesis, Università degli studi di Trento, 2014. https://hdl.handle.net/11572/368142.
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In the first part of the thesis I produce and implement an algorithm for obtaining generators of the unit group of the integral group ring ZG of finite abelian group G. We use our implementation in MAGMA of this algorithm to compute the unit group of ZG for G of order up to 110. In the second part of the thesis I show how to construct multiplication tables of the semisimple real Lie algebras. Next I give an algorithm, based on the work of Sugiura, to find all Cartan subalgebra of such a Lie algebra. Finally I show algorithms for finding semisimple subalgebras of a given semisimple real Lie algebra.
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Cardenas, Lucena Carolina. "Contribución al control geométrico de sistemas de eventos discretos en el álgebra max-plus". Thesis, Ecole centrale de Nantes, 2016. http://www.theses.fr/2016ECDN0004/document.
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Le travail présenté s'inscrit dans le contexte de la théorie des systèmes linéaires dans les dioïdes. La motivation initiale de cette étude a été de contribuer à l'analyse et la commande de systèmes linéaires dans max-plus en utilisant spécifiquement une approche géométrique. La contribution de cette thèse est centrée sur deux problèmes. La première partie est dédiée à l'étude de la relation entre les notions d'invariance contrôlée et d'invariance contrôlée par retour d'état dynamique dans un semi-anneau. Cette relation permet de montrer l'équivalence de ces deux notions. La deuxième partie concerne un problème original dans la théorie des systèmes linéaires dans max-plus, il s'agit de la synthèse d'une loi de commande par retour d'état, qui permette de satisfaire un ensemble de spécifications exprimées sous la forme de restrictions sur l'état du système, avec une approche géométrique. Il s'agit plus précisément de commander des systèmes à événements discrets décrits par un modèle linéaire dans max-plus. Nous définissons et caractérisons l'ensemble des conditions initiales admissibles, lesquelles sont à l'origine de solutions non décroissantes. Les restrictions temporelles imposées à l'espace d'état du système sont décrites par le semi-module défini par l'image de l'étoile de Kleene de la matrice associée aux restrictions temporelles. Les propriétés géométriques de ce semi-module permettant de garantir que l'évolution du système en boucle fermée satisfasse les restrictions sont étudiées. Des conditions suffisantes concernant l'existence d'une loi de commande causale par retour d'état statique sont présentées. Le calcul des lois de commande causales est également présenté. Pour illustrer l'application de cette approche, deux problèmes de commande sont présentésThis work is in the context of the theory of linear Systems in the dioids. The initial motivation of this study was to contribute to the analysis and control of max-plus linear systems, specifically using a geometric approach. The contribution of this thesis focuses on two issues. The first part is dedicated to study of the relationship between the concepts of controlled invariance and dynamic state feedback controlled invariance in a semi-ring. This relationship allows us to show the equivalence of these two concepts. The second part relates to a new problem in the theory of max-plus linear systems, it is the synthesis, with a geometric approach, of a static state feedback control law, in order to satisfy a set of specifications that apply to the state space of the system. This is specifically to control of discrete event systems described by a linear model in max-plus. We define and characterize the set of admissible initial conditions, which are the cause of non-decreasing solutions. Temporal restrictions on the system state space are described by the semi-module defined by the image of the Kleene star of the matrix associated with time restrictions. The geometric properties of this semi-module are studied. Sufficient conditions for the existence of a causal control law by static feedback are presented. Calculating causal control laws is also presented. To illustrate the application of this approach, two control problems are presented
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Faccin, Paolo. "Computational problems in algebra: units in group rings and subalgebras of real simple Lie algebras". Doctoral thesis, University of Trento, 2014. http://eprints-phd.biblio.unitn.it/1182/1/PhdThesisFaccinPaolo.pdf.
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In the first part of the thesis I produce and implement an algorithm for obtaining generators of the unit group of the integral group ring ZG of finite abelian group G. We use our implementation in MAGMA of this algorithm to compute the unit group of ZG for G of order up to 110. In the second part of the thesis I show how to construct multiplication tables of the semisimple real Lie algebras. Next I give an algorithm, based on the work of Sugiura, to find all Cartan subalgebra of such a Lie algebra. Finally I show algorithms for finding semisimple subalgebras of a given semisimple real Lie algebra.
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Seedat, Ebrahim. "A study of maximum and minimum operators with applications to piecewise linear payoff functions". Thesis, Rhodes University, 2013. http://hdl.handle.net/10962/d1001457.
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The payoff functions of contingent claims (options) of one variable are prominent in Financial Economics and thus assume a fundamental role in option pricing theory. Some of these payoff functions are continuous, piecewise-defined and linear or affine. Such option payoff functions can be analysed in a useful way when they are represented in additive, Boolean normal, graphical and linear form. The issue of converting such payoff functions expressed in the additive, linear or graphical form into an equivalent Boolean normal form, has been considered by several authors for more than half-a-century to better-understand the role of such functions. One aspect of our study is to unify the foregoing different forms of representation, by creating algorithms that convert a payoff function expressed in graphical form into Boolean normal form and then into the additive form and vice versa. Applications of these algorithms are considered in a general theoretical sense and also in the context of specific option contracts wherever relevant. The use of these algorithms have yielded easy computation of the area enclosed by the graph of various functions using min and max operators in several ways, which, in our opinion, are important in option pricing. To summarise, this study effectively dealt with maximum and minimum operators from several perspectives
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Addad, Boussad. "Evaluation analytique du temps de réponse des systèmes de commande en réseau en utilisant l’algèbre (max,+)". Thesis, Cachan, Ecole normale supérieure, 2011. http://www.theses.fr/2011DENS0023/document.
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Les systèmes de commande en réseau (SCR) sont de plus en plus répandus dans le milieu industriel. Ils procurent en effet de nombreux avantages en termes de coût, de flexibilité, de maintenance, etc. Cependant,l’introduction d’un réseau, qui par nature est composé de ressources partagées, impacte considérablement les performances temporelles des systèmes de commande. Un signal de commande par exemple n’arrive à destination qu’après un certain délai. Pour s’assurer que ce délai soit inférieur à un certain seuil de sécurité ou du respect d’autres contraintes temps réels de ces systèmes, une évaluation au préalable, avant la mise en service d’un SCR, s’avère donc nécessaire. Dans nos travaux de recherche, nous nous intéressons à la réactivité des SCR client/serveur et évaluons leur temps de réponse.Notre contribution dans ces travaux est d’adopter une approche analytique à base de l’algèbre (Max,+) et remédier aux problèmes des méthodes existantes comme l’explosion combinatoire de la vérification formelle ou de la non exhaustivité des approches par simulation. Après modélisation des SCR client/serveur à l’aide de Graphe d’Evénements Temporisés puis représentation de leurs dynamiques à l’aides d’équations (Max,+) linéaires, nous obtenons des formules de calcul direct du temps de réponse. Plus précisément, nous adoptons une analyse déterministe pour calculer les bornes, minimale et maximale, du temps de réponse puis une analyse stochastique pour calculer la fonction de sa distribution. De plus, nous prenons en compte dans nos travaux tous les délais élémentaires qui composent le temps de réponse, y compris les délais de bout-en-bout, dus à la traversée du seul réseau de communication. Ce dernier étant naturellement composé de ressources partagées, rendant l’utilisation des modèles (Max,+) classiques impossibles, nous introduisons une nouvelle approche de modélisation à base du formalisme (Max,+) mais prenant en compte le concept de conflit ou ressource partagée.L’exemple d’un réseau de type Ethernet est considéré pour évaluer ces délais de bout-en-bout. Par ailleurs, cette nouvelle méthode (Max,+) est assez générique et reste applicable à de nombreux systèmes impliquant des ressources partagées, au delà des seuls réseaux de communication. Enfin, pour vérifier la validité des résultats obtenus dans nos travaux, notamment la formule de la borne maximale du temps de réponse, une compagne de mesures expérimentales sont menées sur une plateforme dédiée. Différentes configurations et conditions de trafic dans un réseau Ethernet sont considéréesNetworked automation systems (NAS) are more and more used in industry, given the several advantages they provide like flexibility, low cost, ease of maintenance, etc. However, the use of a communication network in SCR means in essence sharing some resources and therefore strikingly impacts their time performances. For instance, a control signal does get to its destination (actuator) only after a non zero delay. So, to guarantee that such a delay is shorter than a given threshold or other time constraints well respected, an a priori evaluation is necessary before operating the SCR. In our research activities, we are interested in client/server SCR reactivity and the evaluation of their response time.Our contribution in this investigation is the introduction of a (Max,+) Algebra-based analytic approach to solve some problems, faced in the existing methods like state explosion of model checking or the non exhaustivity of simulation. So, after getting Timed Event Graphs based models of the SCR and their linear state (Max,+) representation, we obtain formulae that enables to calculate straightforwardly the SCR response times. More precisely, we obtain formulae of the bounds of response time by adopting a deterministic analysis and other formulae to calculate the probability density of response time by considering a stochastic analysis. Moreover, in our investigation we take into account every single elementary delay involved in the response time, including the end-to-end delays, due exclusively to crossing the communication network. This latter being however constituted of shared resources, making by the way the use of TEG and (Max,+) Algebra impossible, we introduce a novel approach to model the communication network. This approach brings to life a new class of Petri nets, called Conflicting Timed Event Graphs (CTEG), which enables us to solve the problem of the shared resources. We also manage to represent the CTEG dynamics using recurrent (Max,+) equations and therefore calculate the end to-end delays. An Ethernet-based network is studied as an example to apply this novel approach. Note by the way that the field of application of this approach borders largely communication networks and is quite possible when dealing with other systems.Finally, to validate the different results of our research activities and the related hypotheses, especially the maximal bound of response time formula, we carry out lots of experimental measurements on a lab facility. We compare the measures to the formula predictions and check their agreement under different conditions
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Cerulli, Irelli Giovanni. "Structural theory of rank three cluster algebras of affine type". Doctoral thesis, Università degli studi di Padova, 2008. http://hdl.handle.net/11577/3425220.
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Cardenas, Lucena Carolina. "Contribución al control geométrico de sistemas de eventos discretos en el álgebra max-plus". Electronic Thesis or Diss., Ecole centrale de Nantes, 2016. http://www.theses.fr/2016ECDN0004.
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Le travail présenté s'inscrit dans le contexte de la théorie des systèmes linéaires dans les dioïdes. La motivation initiale de cette étude a été de contribuer à l'analyse et la commande de systèmes linéaires dans max-plus en utilisant spécifiquement une approche géométrique. La contribution de cette thèse est centrée sur deux problèmes. La première partie est dédiée à l'étude de la relation entre les notions d'invariance contrôlée et d'invariance contrôlée par retour d'état dynamique dans un semi-anneau. Cette relation permet de montrer l'équivalence de ces deux notions. La deuxième partie concerne un problème original dans la théorie des systèmes linéaires dans max-plus, il s'agit de la synthèse d'une loi de commande par retour d'état, qui permette de satisfaire un ensemble de spécifications exprimées sous la forme de restrictions sur l'état du système, avec une approche géométrique. Il s'agit plus précisément de commander des systèmes à événements discrets décrits par un modèle linéaire dans max-plus. Nous définissons et caractérisons l'ensemble des conditions initiales admissibles, lesquelles sont à l'origine de solutions non décroissantes. Les restrictions temporelles imposées à l'espace d'état du système sont décrites par le semi-module défini par l'image de l'étoile de Kleene de la matrice associée aux restrictions temporelles. Les propriétés géométriques de ce semi-module permettant de garantir que l'évolution du système en boucle fermée satisfasse les restrictions sont étudiées. Des conditions suffisantes concernant l'existence d'une loi de commande causale par retour d'état statique sont présentées. Le calcul des lois de commande causales est également présenté. Pour illustrer l'application de cette approche, deux problèmes de commande sont présentésThis work is in the context of the theory of linear Systems in the dioids. The initial motivation of this study was to contribute to the analysis and control of max-plus linear systems, specifically using a geometric approach. The contribution of this thesis focuses on two issues. The first part is dedicated to study of the relationship between the concepts of controlled invariance and dynamic state feedback controlled invariance in a semi-ring. This relationship allows us to show the equivalence of these two concepts. The second part relates to a new problem in the theory of max-plus linear systems, it is the synthesis, with a geometric approach, of a static state feedback control law, in order to satisfy a set of specifications that apply to the state space of the system. This is specifically to control of discrete event systems described by a linear model in max-plus. We define and characterize the set of admissible initial conditions, which are the cause of non-decreasing solutions. Temporal restrictions on the system state space are described by the semi-module defined by the image of the Kleene star of the matrix associated with time restrictions. The geometric properties of this semi-module are studied. Sufficient conditions for the existence of a causal control law by static feedback are presented. Calculating causal control laws is also presented. To illustrate the application of this approach, two control problems are presented
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at, michor@esi ac. "The Generalized Cayley Map from an Algebraic Group to its Lie Algebra". ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1066.ps.
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Sharify, Meisam. "Algorithmes de mise à l'échelle et méthodes tropicales en analyse numérique matricielle". Phd thesis, Ecole Polytechnique X, 2011. http://pastel.archives-ouvertes.fr/pastel-00643836.
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L'Algèbre tropicale peut être considérée comme un domaine relativement nouveau en mathématiques. Elle apparait dans plusieurs domaines telles que l'optimisation, la synchronisation de la production et du transport, les systèmes à événements discrets, le contrôle optimal, la recherche opérationnelle, etc. La première partie de ce manuscrit est consacrée a l'étude des applications de l'algèbre tropicale à l'analyse numérique matricielle. Nous considérons tout d'abord le problème classique de l'estimation des racines d'un polynôme univarié. Nous prouvons plusieurs nouvelles bornes pour la valeur absolue des racines d'un polynôme en exploitant les méthodes tropicales. Ces résultats sont particulièrement utiles lorsque l'on considère des polynômes dont les coefficients ont des ordres de grandeur différents. Nous examinons ensuite le problème du calcul des valeurs propres d'une matrice polynomiale. Ici, nous introduisons une technique de mise à l'échelle générale, basée sur l'algèbre tropicale, qui s'applique en particulier à la forme compagnon. Cette mise à l'échelle est basée sur la construction d'une fonction polynomiale tropicale auxiliaire, ne dépendant que de la norme des matrices. Les raciness (les points de non-différentiabilité) de ce polynôme tropical fournissent une pré-estimation de la valeur absolue des valeurs propres. Ceci se justifie en particulier par un nouveau résultat montrant que sous certaines hypothèses faites sur le conditionnement, il existe un groupe de valeurs propres bornées en norme. L'ordre de grandeur de ces bornes est fourni par la plus grande racine du polynôme tropical auxiliaire. Un résultat similaire est valable pour un groupe de petites valeurs propres. Nous montrons expérimentalement que cette mise à l'échelle améliore la stabilité numérique, en particulier dans des situations où les données ont des ordres de grandeur différents. Nous étudions également le problème du calcul des valeurs propres tropicales (les points de non-différentiabilité du polynôme caractéristique) d'une matrice polynômiale tropicale. Du point de vue combinatoire, ce problème est équivalent à trouver une fonction de couplage: la valeur d'un couplage de poids maximum dans un graphe biparti dont les arcs sont valués par des fonctions convexes et linéaires par morceaux. Nous avons développé un algorithme qui calcule ces valeurs propres tropicales en temps polynomial. Dans la deuxième partie de cette thèse, nous nous intéressons à la résolution de problèmes d'affectation optimale de très grande taille, pour lesquels les algorithms séquentiels classiques ne sont pas efficaces. Nous proposons une nouvelle approche qui exploite le lien entre le problème d'affectation optimale et le problème de maximisation d'entropie. Cette approche conduit à un algorithme de prétraitement pour le problème d'affectation optimale qui est basé sur une méthode itérative qui élimine les entrées n'appartenant pas à une affectation optimale. Nous considérons deux variantes itératives de l'algorithme de prétraitement, l'une utilise la méthode Sinkhorn et l'autre utilise la méthode de Newton. Cet algorithme de prétraitement ramène le problème initial à un problème beaucoup plus petit en termes de besoins en mémoire. Nous introduisons également une nouvelle méthode itérative basée sur une modification de l'algorithme Sinkhorn, dans lequel un paramètre de déformation est lentement augmenté. Nous prouvons que cette méthode itérative(itération de Sinkhorn déformée) converge vers une matrice dont les entrées non nulles sont exactement celles qui appartiennent aux permutations optimales. Une estimation du taux de convergence est également présentée.
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Ruzmetov, Azizbek. "Approches distribuées et adaptatives pour la gestion de l'énergie". Thesis, Belfort-Montbéliard, 2015. http://www.theses.fr/2015BELF0271/document.
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Au cours des dernières décennies, de grands efforts en recherche et développement ont été faits pour développer et promouvoir les véhicules électriques (VEs). La plupart de ces recherches portent essentiellement sur le développement des moteurs électriques de ces véhicules et des technologies de batteries de recharge. Cependant, un des obstacles majeurs pour le déploiement des VEs à grande échelle réside dans l'incertitude d’assister et de guider les conducteurs de ce type de véhicule d’une façon appropriée pour atteindre les stations de recharge tout en satisfaisant leurs souhaits (points de recharge disponibles, moins d’attente possible, proposition d’autres points d’intérêts : restaurant, shopping, etc.). Afin de remédier à ce manque, nous proposons dans ce travail de thèse une approche distribuée et adaptative orientée modèles pour la gestion de l'énergie pour la recharge des VEs. Pour ce faire, nous nous somme focalisés sur la modélisation des processus de recharge en utilisant une approche formelle basée sur des outils de systèmes à événements discrets, à savoir l'algèbre (max, +) et les réseaux de Petri. Les modèles développés ont permis d’étudier, d’analyser et d’évaluer le comportement du système de recharge. De plus, une approche d'optimisation basée sur la programmation linéaire est proposée afin d’affecter et d’orienter d'une façon optimale les VEs vers les stations de recharge appropriées et ordonnancer leurs opérations de recharge. Afin de prédire le taux et la durée de recharge moyens des VEs compte tenu des dates d’arrivée des demandes de recharge et l'état de recharge de chaque véhicule, une approche dédiée basée sur une fonction prédictive est proposée. En utilisant cette approche, les opérations de recharge pourraient être planifiées en minimisant les temps d'attente des VEs au sein des stations de recharge et en assurant un taux de recharge acceptable pour chaque demande. Les résultats d’analyse et de simulations obtenus ont montré que les approches de modélisation, d’optimisation et de prédiction proposées permettent d’affecter de façon adéquate et optimale les VEs aux stations de recharge tout en satisfaisant toutes les contraintes du processus de rechargeIn the last decades, very great research and development efforts have been made to develop and promote electric vehicles (EVs). Most efforts have been made to further develop the power engine of these vehicles and batteries technologies. However, one of the major obstacles to the large deployment of EVs is the uncertainty of drivers to get a suitable and vacant place at a charging station (CS). In this manuscript, we focus on the charging process modelling using formal approaches based on discrete event system tools namely (max,+) algebra and Petri nets. In addition, an optimization approach based on linear programming is proposed to optimally assign and reroute EVs to the suitable CSs and schedule their charging operations. In order to predict, manage and handle charging needs of EVs, a dedicated model based on a predictive function is introduced. The aim is to predict the average charging rate and time while considering the inter-arrival of charging requests and the state of charging of EVs. Using this approach, charging operations could be planned while minimizing waiting times of EVs and avoiding queuing situations within CSs. Simulation results showed that the proposed approaches allow assigning adequately and optimally EVs to CSs while satisfying all process constraints
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Calzi, Mattia. "Functional Calculus on Homogeneous Groups". Doctoral thesis, Scuola Normale Superiore, 2019. http://hdl.handle.net/11384/85740.
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In the first part of the thesis, we consider the following problem.
Let G be a homogeneous group, and let (L_1,...,L_n) be a jointly hypoelliptic commutative finite family of formally self-adjoint, homogeneous, left-invariant differential operators without constant terms.
Then, the operators L_j are essentially self-adjoint as operators on L^2(G) with domain C^infty_c(G), and their closures commute emph{as self-adjoint operators}.
Therefore, one may consider the joint functional calculus associated with the family (L_1,...,L_n).
More precisely, for every bounded Borel measurable function $m$ on $R^n$, the corresponding operator m(L_1,...,L_n) commutes with left translations, so that it admits a unique right convolution kernel K(m).
The so-defined kernel transform K then maps S(R^n) continuously into S(G), and L^2(eta) isometrically into L^2(G) for some uniquely determined positive Radon measure eta on R^n; this latter property can be considered as an analogue of the Plancherel isomorphism.
In addition, K maps L^1(eta) continuously into C_0(G), and this property can be considered as an analogue of the Riemann--Lebesgue lemma.
We focus on the following properties of K:
(RL) if K(m)in L^1(G), then m can be taken in C_0(R^n): this is again an analogue of the Riemann--Lebesgue lemma;
(S) if K(m)in S(G), then m can be taken in S(R^n).
We prove that properties (RL) and (S) are compatible with products, and we characterize the Rockland operators which satisfy property (S) when the underlying group G is abelian.
We then consider the case of 2-step stratified groups, and families whose elements are either sub-Laplacians or vector fields of homogeneous degree 2. In this setting, we prove several sufficient conditions, as well as some necessary ones, for properties (RL) and (S); we even characterize them in some more specific settings.
In addition, we study the case of general (that is, not necessarily homogeneous) sub-Laplacians on 2-step stratified groups, and prove that they always satisfy properties (RL) and (S).
We also prove that, under some mild assumptions, a multiplier m can be taken so as to satisfy Mihlin--Hormander conditions of order infinity if and only if the corresponding kernel K(m) satisfies Calderon--Zygmund conditions of order infinity.
In the second part of the thesis, we present some results which are joint work with T. Bruno.
We fix the standard sub-Laplacian on an H-type group, and consider its heat kernel (p_s)_{s>0}. We provide sharp asymptotic estimates at $infty$ for basically all the derivatives of p_1. Because of the homogeneity of the family (p_s), these estimates can also be considered as short-time asymptotics.
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Andriamasy, Rabaozafy Louisa. "Les technologies de production tropicales et leurs champs d'applications en économie". Thesis, Perpignan, 2018. http://www.theses.fr/2018PERP0021.
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Les mathématiques tropicales sont une branche des mathématiques correspondant à l'étude d'une algèbre modifiée grâce à la redéfinition de l'addition et de la multiplication. Les mathématiques tropicales sont généralement définies grâce au minimum et à l'addition (algèbre min-plus) mais le terme est parfois utilisé pour désigner l'algèbre max-plus, définie grâce au maximum et à l'addition. Briec et Horvath ont introduit une notion de convexité très proche qui apparait comme un cas limite d’opérateurs utilisés en théorie de l’optimisation par Avriel (1972) et de Ben-Tal (1977). En suivant cette ligne d’investigation, nous allons proposer, dans le domaine de l’économie de la production et de l’optimisation de portefeuille, une certaine classe de modèles économiques à élaborés à partir de ces notions. Pour ce faire, nous introduisons une nouvelle classe de technologie de production permettant de prendre en compte les structures d’homothe´tie-translation dans la mesure de productivité au travers du concept de la Convexité Max-Plus. Ensuite, nous allons établir une relation topologique entre plusieurs classes de modèles convexes généralisés connus. Nous analysons pour cela la limite de Painlevé-Kuratowski des modèles CES-CET et des technologies non paramétriques satisfaisant une hypothèse de rendements d’échelle alpha. On montre que leurs limites topologiques convergent vers les modèles de production B-convexe et Cobb-Douglas. Enfin, nous allons montrer que l'amélioration de l'efficacité technique d’une coalition d’entreprises s'avère compatible avec les technologies de semi-treillis dans un jeu coopératif. Nous introduisons ensuite, le concept d’écart absolu moyen dans la sélection du portefeuille en utilisant le « Shortage Function » qui prend en compte simultanément la réduction des inputs et l’augmentation des outputs comme dans la théorie de la production. Enfin, nous allons étendre le concept de B-convexité et de l’inverse B-convexité en se concentrant sur le calcul des mesures d’efficacité technique dans le grapheTropical algebra is the tropical analogue of linear algebra by redefining the usual operation addition by the maximization operation and the usual addition operation as multiplication. Briec and Horvath introduced a concept of convexity very close to this concept quoted above which appears as one of the limits of use of the theory of optimization by Avriel (1972) and Ben-Tal (1977). Following this line of investigation, we give an overview of contributions involving a semilattice structure of production technologies and an optimization portfolio. To do that, firstly, we propose a framework allowing to consider both semilattice structure and translation homothetic properties in productivity measurement. We introduce the concept of Max-Plus convexity which combine both an upper semilattice structure and an additivity assumption. We establish a topological relation between several classes of known generalized convex models using some basic algebraic convex structures. We analyze the Painlevé-Kuratowski limit of the CES-CET and Alpha-returns to scale models. It is shown that their topological limits yield the B-convex and Cobb-Douglas production models. Moreover, we show that the improvement of technical efficiency is compatible with semilattice technologies in a cooperative game. Then, we introduce a criterion to measure portfolio efficiency based upon the minimization of the maximum absolute deviation and minimum absolute deviation from the expected return using the Shortage function. Finally, we derive simple closed-form expressions to calculate the hyperbolic measure in the case of inverse and B-Convexity that evaluates technical efficiency in the full input-output space
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Bahalkeh, Esmaeil. "Efficient Algorithms for Calculating the System Matrix and the Kleene Star Operator for Systems Defined by Directed Acyclic Graphs over Dioids". Ohio University / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1440116216.
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Trunk, Johannes. "On the modeling and control of extended Timed Event Graphs in dioids". Thesis, Angers, 2019. http://www.theses.fr/2019ANGE0068.
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De nombreux systèmes de production peuvent être modélisés et analysés à l’aide de graphes d’événements temporisés (GET). Les GET forment une classe de systèmes à événements discrets temporisés (SEDT), dont la dynamique est définie uniquement par des phénomènes de synchronisation et de saturation. Un avantage majeur des GET par rapport à d’autres classes de SEDT est qu’ils admettent, sous certaines conditions, un modèle linéaire dans des espaces algébriques particuliers : les dioïdes. Ceci a conduit au développement d’une théorie des systèmes linéaires dans les dioïdes, grâce à laquelle de nombreux concepts de l’automatique classique ont été adaptés aux GET. Par exemple, l’algèbre (max,+) (i.e., le dioïde basé sur les opérations (max,+)) offre des techniques élégantes pour l’analyse et le contrôle de GET. Cependant, les conditions nécessaires pour modéliser un système à événements discrets par un GET sont très restrictives. Pour élargir la classe de systèmes concernés, deux extensions principales ont été développées. D’une part, les GET valués ont été introduits pour décrire des phénomènes d’assemblage et de séparation dans les systèmes de production. Cette extension se traduit par l’association de coefficients entiers aux arrêtes d’un graphe d’événements. Contrairement aux GET, ces systèmes ne sont pas invariants par rapport aux événements et ne peuvent donc pas être décrits par des équations linéaires dans l’algèbre (max,+). D’autre part, la synchronisation partielle (PS) a été introduite pour modéliser des systèmes dans lesquels certains événements ne peuvent se produire que pendant des intervalles prédéfinis. Par exemple, dans une intersection réglée par un feu tricolore, une voiture peut traverser l’intersection lorsque le feu est vert. Contrairement aux GET, ces systèmes ne sont pas invariants dans le domaine temporel et ne peuvent donc pas être décrits par des équations linéaires dans l’algèbre (max,+). Dans cette thèse, une modélisation des GET valués et des GET avec PS dans des dioïdes adaptés est présentée. A l’aide de ces dioïdes, une décompostion pour les GET valués (resp. GET avec PS) en un GET et une partie non-invariante dans le domaine des événements (resp. dans le domaine temporel) est introduite. Sous certaines conditions, la partie invariante est invertible. Dans ce cas, les modèles et contrôleurs pour le GET valué ou le GET sous PS peuvent être directement dérivés des modèles et contrôleurs obtenus pour le GET associéVarious kinds of manufacturing systems can be modeled and analyzed by Timed EventGraphs (TEGs). These TEGs are a particular class of timed Discrete Event Systems (DESs), whose dynamic behavior is characterized only by synchronization and saturation phenomena. A major advantage of TEGs over many other timed DES models is that their earliest behavior can be described by linear equations in some tropical algebra structures called dioids. This has led to a broad theory for linear systems over dioids where many concepts of standard systems theory were introduced for TEGs. For instance, with the (max,+)-algebra linear state-space models for TEGs were established. These linear models provide an elegant way to do performance evaluation for TEGs. Moreover, based on transfer functions in dioids severalcontrol problems for TEGs were addressed. However, the properties of TEGs, and thus the systems which can be described by TEGs, are limited. To enrich these properties, two main extensions for TEGs were introduced. First,Weighted Timed Event Graphs (WTEGs) which, in contrast to ordinary TEGs, exhibit event-variant behaviors. InWTEGs integer weights are considered on the arcs whereas TEGs are restricted to unitary weights. For instance, these integer weights make it straightforward to model a cutting process in a production line. Second, a new kind of synchronization called partial synchronization (PS) was introduced forTEGs. PS is useful to model systems where specific events can only occur in a particular time window. For example, consider a crossroad controlled by a traffic light: the green phase of the traffic light provides a time window in which a vehicle is allowed to cross. Clearly, PS leads to time-variant behavior. As a consequence, WTEGs and TEGs under PS are not (max,+)-linear anymore.In this thesis, WTEGs and TEGs under PS are studied in a dioid structure. Based on these dioid models forWTEGs a decomposition of the dynamic behavior into an event-variant and an event-invariant part is proposed. Under some assumptions, it is shown that the eventvariant part is invertible. Hence, based on this model, optimal control and model reference control, which are well known for ordinary TEGs, are generalized to WTEGs. Similarly, a decomposition model is introduced for TEGs under PS in which the dynamic behavior is decomposed into a time-variant and time-invariant part. Again, under some assumptions, it is shown that the time-variant part is invertible. Subsequently, optimal control, as well as model reference control for TEGs under PS is addressed
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Schadeck, Laurent. "On the K-theory of tame Artim stacks". Doctoral thesis, Scuola Normale Superiore, 2019. http://hdl.handle.net/11384/85745.
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This thesis pertains to the algebraic K-theory of tame Artin stacks. Building on earlier work of Vezzosi and Vistoli in equivariant K-theory, which we translate in stacky language, we give a description of the algebraic K-groups of tame quotient stacks. Using a strategy of Vistoli, we recover Grothendieck-Riemann-Roch-like formulae for tame quotient stacks that refine Toën’s
Grothendieck-Riemann-Roch formula for Deligne-Mumford stacks (as it was realized that the latter pertains to quotient stacks since it relies on the resolution property). Our formulae differ from Toën’s in that, instead of using the standard inertia stack, we use the cyclotomic inertia stack introduced by Abramovich, Graber and Vistoli in the early 2000s. Our results involve the rational part of the K'-theory of the object considered. We establish a few conjectures, the main one (Conjecture 6.3) pertaining to the covariance of our Lefschetz-Riemann-Roch map for proper morphisms of tame stacks (not necessarily representable). Other future works might be dedicated to the study of torsion in K'-groups as well as more general Artin stacks.
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Martini, Alessio. "Algebras of differential operators on Lie groups and spectral multipliers". Doctoral thesis, Scuola Normale Superiore, 2010. http://hdl.handle.net/11384/85663.
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Let (X, μ) be a measure space, and let L1, . . . ,Ln be (possibly unbounded) selfadjoint
operators on L2(X, μ), which commute strongly pairwise, i.e., which
admit a joint spectral resolution E on Rn. A joint functional calculus is then
defined via spectral integration: for every Borel function m : Rn → C,
m(L) = m(L1, . . . ,Ln) =
∫ Rn m(λ) dE(λ) is a normal operator on L2(X, μ), which is bounded if and only if m - called
the joint spectral multiplier associated to m(L) - is (E-essentially) bounded.
However, the abstract theory of spectral integrals does not tackle the following
problem: to find conditions on the multiplier m ensuring the boundedness of
m(L) on Lp(X, μ) for some p ≠ 2.
We are interested in this problem when the measure space is a connected Lie
group G with a right Haar measure, and L1, . . . ,Ln are left-invariant differential
operators on G. In fact, the question has been studied quite extensively in the
case of a single operator, namely, a sublaplacian or a higher-order analogue.
On the other hand, for multiple operators, only specific classes of groups and
specific choices of operators have been considered in the literature.
Suppose that L1, . . . ,Ln are formally self-adjoint, left-invariant differential
operators on a connected Lie group G, which commute pairwise (as operators
on smooth functions). Under the assumption that the algebra generated by
L1, . . . ,Ln contains a weighted subcoercive operator --- a notion due to [ER98],
including positive elliptic operators, sublaplacians and Rockland operators---we
prove that L1, . . . ,Ln are (essentially) self-adjoint and strongly commuting on
L2(G). Moreover, we perform an abstract study of such a system of operators,
in connection with the algebraic structure and the representation theory of G,
similarly as what is done in the literature for the algebras of differential operators
associated with Gelfand pairs.
Under the additional assumption that G has polynomial volume growth,
weighted L1 estimates are obtained for the convolution kernel of the operator
m(L) corresponding to a compactly supported multiplier m satisfying some
smoothness condition. The order of smoothness which we require on m is related
to the degree of polynomial growth of G. Some techniques are presented,
which allow, for some specific groups and operators, to lower the smoothness
requirement on the multiplier.
In the case G is a homogeneous Lie group and L1, . . . ,Ln are homogeneous
operators, a multiplier theorem of Mihlin-H\"ormander type is proved, extending
the result for a single operator of [Chr91] and [MM90]. Further, a product theory
is developed, by considering several homogeneous groups Gj , each of which with
its own system of operators; a non-conventional use of transference techniques
then yields a multiplier theorem of Marcinkiewicz type, not only on the direct
product of the Gj , but also on other (possibly non-homogeneous) groups, containing
homomorphic images of the Gj . Consequently, for certain non-nilpotent
groups of polynomial growth and for some distinguished sublaplacians, we are
able to improve the general result of [Ale94].
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Boito, Paola. "Structured matrix based methods for approximate polynomial GCD". Doctoral thesis, Scuola Normale Superiore, 2008. http://hdl.handle.net/11384/85672.
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Bambozzi, Federico. "On a generalization of affinoid varieties". Doctoral thesis, Università degli studi di Padova, 2014. http://hdl.handle.net/11577/3423509.
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In this thesis we develop the foundations for a theory of analytic geometry over a valued field, uniformly encompassing the case when the base field is equipped with a non-archimedean valuation and the case when it has an archimedean one. We will use the efficient language of bornological algebras to reach this goal. Since, at our knowledge, there is not a strong literature on bornological algebras on which we can base our results, we need to start from scratch with theory of bornological algebras and we develop it as far as we need for our scope. In this way
we can construct our theory of analytic spaces taking as building blocks affinoid dagger algebras, i.e. bornological algebras isomorphic to quotients of the algebras of germs of analytic functions on polycylinders. It turns out that we can show for the category of dagger affinoid algebras the analogous of all the main results of the category of affinoid algebras and hence we obtain a good theory of dagger affinoid spaces, which behave very similarly to affinoid spaces used to construct rigid analytic spaces.
We underline that the archimedean case has special features that we study in the first section of the fifth chapter, and that these special properties enable us to prove the generalization, in archimedean context, of the main result of affinoid algebras theory: the Gerritzen-Grauert theorem (we remark that our proof is inspired by the new proof of Gerritzen-Grauert theorem in Berkovich geometry given by Temkin).
With a good affine theory at our disposal we end this thesis by constructing the global theory of dagger analytic spaces. We use the theory of Berkovich nets and hence a Berkovich-like approach to the globalization. In this way we obtain the category of dagger analytic spaces and we study the relations of the spaces we found with the ones already present in literature. In particular we see that our spaces are very strongly related to dagger spaces of Grosse-Klonne, and that, in the archimedean case, the category of classical complex analytic spaces embeds fully faithfully in the category of complex dagger analytic spaces.
In conclusion, in this work we obtain a complete affinoid theory of dagger spaces over any valued field and we show that on this theory can be developed a good theory of dagger analytic spaces, which deserves to be strengthened in the future.In questa tesi vengono sviluppate le basi per una teoria degli spazi analitici su campi valutati che comprende in modo uniforme il caso in cui il campo base sia archimedeo o non-archimedeo. Per raggiungere questo obiettivo useremo il linguaggio efficiente delle algebre bornologiche. Siccome non sembra essere disponibile una amplia letteratura riguardo le algebre bornologiche, le prima parte della tesi si occupa di stabilire alcuni risultati fondamentali a riguardo. Con questi risultati a disposizione, è possibile costruire una teoria degli spazi analitici prendendo come mattoni fondamentali le algebre dagger affinoidi, cioè algebre bornologiche isomorfe a quozienti delle algebre dei germi di funzioni analitiche su policilindri. Si ottiene quindi che la categoria delle algebra affinoidi dagger così definite soddisfa l'analogo di tutte le proprietà più importanti della categoria delle algebre affinoidi e che quindi otteniamo una buona teoria degli spazi affinoidi dagger che è del tutto analoga alla categoria degli spazi affinoidi usati in geometria rigida. C'è da sottolineare che il caso in cui il campo base è archimedeo ha proprietà particolari, studiate all'inizio del quinto capitolo, e che queste proprietà permettono di ottenere anche per spazi affinodi dagger archimedei il principale risultato strutturale della teoria delle algebre affinoidi: il teorema di Gerritzen-Grauert (la dimostrazione qui data di questo teorema è ispirata dalla nuova dimostrazione data da Temkin valida in geometria di Berkovich). Dopo aver ottenuto una buona teoria affine la tesi si conclude con la costruzione della teoria globale degli spazi analitici dagger. Verrà utilizzato un approccio analogo a quello usato da Berkovich per fare questo. Dopo aver definito questa nuova categoria di spazi verranno studiate le relazioni tra questi e gli spazi analitici presenti in letteratura. In particolare si vedrà che questi spazi sono fortemente legati agli spazi dagger di Grosse-Klonne e che nel caso archimedeo la categoria degli spazi analitici complessi classica si immerge in modo pienamente fedele nella categoria degli spazi analitici dagger complessi. In conclusione, in questo lavoro si ottiene una teoria affinoide completa per spazi dagger su ogni campo valutato sulla base della quale viene proposta una teoria degli spazi dagger globali la quale dovrà essere approfondita in futuro.
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Daviaud, Bérangère. "Méthodes formelles pour les systèmes réactifs, applications au live coding". Electronic Thesis or Diss., Angers, 2024. http://www.theses.fr/2024ANGE0032.
Texto completoResumen
Le formalisme des systèmes à événements discrets et des systèmes réactifs offre un cadre abstrait efficace pour représenter et étudier de nombreux systèmes. Dans cette thèse, nous exploitons ce formalisme afin de modéliser une partition de live coding dont l’interprétation est conditionnée par l’occurrence d’événements spécifiques. Cette approche nous a conduits à étudier les méthodes formelles des systèmes à événements discrets qui permettent de les modéliser, de les analyser et de concevoir des stratégies de contrôle adaptées. Cette étude a donné lieu à plusieurs contributions, notamment en ce qui concerne l’expressivité des automates pondérés, la vérification formelle de propriétés temporelles, et sur l’existence d’une simulation pondérée. La dernière partie de ce mémoire présente le formalisme de la partition interactive ainsi que de la librairie Troop Interactive, développée pour rendre accessible l’écriture de partition interactive et la réalisation de performances sonores interactives reposant sur la pratique du live codingThe formalism of discrete event systems and reactive systems provides an effective abstract framework for representing and studying a wide range of systems. In this thesis, we leverage this formalism to model a live coding score whose interpretation is conditioned by the occurrence of specific events. This approach led us to investigate formal methods for discrete event systems that enable their modeling, analysis, and the design of appropriate control strategies. This study resulted in several contributions, particularly regarding the expressiveness of weighted automata, the formal verification of temporal properties, and the existence of weighted simulation. The final part of this dissertation introduces the formalism of the interactive score, as well as the \textit{Troop Interactive} library, developed to make interactive score writing and the realization of interactive sound performances based on live coding practices more accessible
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TERRAGNI, TOMMASO. "Hecke algebras associated to coxeter groups". Doctoral thesis, Università degli Studi di Milano-Bicocca, 2012. http://hdl.handle.net/10281/29634.
Texto completoResumen
In this thesis, we study cohomological properties of Hecke algebras $H_q(W,S)$ associated with arbitrary Coxeter groups $(W,S)$. Under mild conditions, it is possible to canonically define the Euler characteristic of such an algebra. We define an almost-canonical complex of $H$-modules that allows one to compute the Euler characteristic of $H$. It turns out that the Euler characteristic of the algebra has an interpretation as a combinatorial object attached to the Coxeter group: indeed, for suitable choices of the base ring, it is the inverse of the Poincaré series.
Some other results about Coxeter groups are proved, in particular one new characterization of minimal non-spherical, non-affine types is given.
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Lockner, Micaela. "Alger - framtidens mat. : En studie om algers potential som framtida livsmedel". Thesis, Mittuniversitetet, Institutionen för design, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:miun:diva-42433.
Texto completoResumen
This study aims to account for the potential of algae as future innovative foods in the Swedish market. The methods for collecting the empirical material were quantitative content analysis and mapping of how many food companies there are in the Swedish market that have algae in their products. The second method was a qualitative interview survey in which the respondents were allowed to express their opinions and perceptions about circular design, food technology and algae as alternative foods. Through results and analysis, it has been shown that there are only a couple of food companies with a focus on algae on the Swedish market, but that the market is in its infancy. A lot of research is done on the subject and large investments are made on algae research in Sweden. This in turn shows that there is interest and potential for these innovative and more climate-friendly foods. Unfortunately, it has been difficult to establish innovative food companies, as Sweden is a country that has valued the safety and development of existing foods. On the other hand, Swedes are proving to be the most open consumers to new trends and embrace new foods. This is good for the development of testing new foods in the Swedish market before export. Through the Government's action plan for the 2030 agenda, innovation is a cornerstone that opens up better conditions for food innovation. It is obvious that it is necessary to invest in innovation in the food chain in order for Sweden to be globally competitive and achieve these goals to reach a sustainable future. The global goals that will form this study are the following, 2 - zero hunger, 12 - responsible consumption and production, 13 - climate action, 14 - life below water.
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Ait-Cheik-Bihi, Wafaa. "Approche orientée modèles pour la vérification et l'évaluation de performances de l'interopérabilité et l'interaction des services". Thesis, Belfort-Montbéliard, 2012. http://www.theses.fr/2012BELF0182/document.
Texto completoResumen
De nos jours, les services Web sont très utilisés notamment par les entreprises pour rendre accessibles leurs métiers, leurs données et leurs savoir-faire via le Web. L'émergence des services Web a permis aux applications d'être présentées comme un ensemble de services métiers bien structurés et correctement décrits, plutôt que comme un ensemble d'objets et de méthodes. La composition automatique de services est une tâche complexe mais qui rend les services interopérables, ainsi leur interaction permet d’offrir une valeur ajoutée dans le traitement des requêtes des utilisateurs en prenant en compte des critères fonctionnels et non fonctionnels de la qualité de service. Dans ce travail de thèse, nous nous intéressons plus précisément aux services à base de localisation (LBS) qui permettent d'intégrer des informations géographiques, et de fournir des informations accessibles depuis des appareils mobiles via, les réseaux mobiles en faisant usage des positions géographiques de ces appareils. L'objectif de ce travail est de proposer une approche orientée modèles pour spécifier, valider et mettre en œuvre des processus de composition automatique de services à des fins de sécurité routière dans les transports. Cette approche est basée sur deux outils formels à savoir les Réseaux de Petri (RdP) et l'algèbre (max,+). Pour cela, nous préconisons l'utilisation des workflow patterns dans la composition, où chaque pattern est traduit par un modèle RdP et ensuite par une équation mathématique dans l'algèbre (max,+). Les modèles formels développés ont conduits, d'une part, à la description graphique et analytique des processus considérés, et d'autre part, à l'évaluation et la vérification quantitatives et qualitatives de ces processus. Une plateforme, appelée TransportML, pour la collaboration et l'interopérabilité de services à base de positionnement a été implémentée. Les résultats obtenus par la simulation des modèles formels sont comparés à ceux issus des simulations du fonctionnement de la plateforme et des expérimentations sur le terrain.Cette thèse est effectuée dans le cadre des projets Européens FP7 ASSET (2008-2011) et TeleFOT (2008-2012)Web services are widely used by organizations to share their knowledge over the network and facilitate business-to-business collaboration. The emergence of Web services enabled applications to be presented as a set of business services well structured and correctly described. However, combining Web services and making them interoperable, to satisfy user requests taking into account functional and non-functional quality criteria, is a complex process. In this work, we focus specifically on location-based services (LBS) that integrate geographic information and provide information reachable from mobile devices, through wireless network by making use of the geographical positions of the devices. The aim of this work is to develop a model driven approach to specify, validate and implement service composition process in an automatic fashion for road security. This approach is based on two formal tools namely Petri nets (PN) and (max, +) algebra used to model, to verify and to evaluate the performance of service composition process. Workflow patterns are used to represent service composition processes. The behavior of each pattern is modeled by a PN model and then by a (max,+) state equation. The developed formal models allow the graphical and analytical description of the considered processes. Also, these models enable to evaluate some quantitative and qualitative properties of the considered processes. A platform, called TransportML, has been developed for collaboration and interoperability of different LBS. The obtained simulation results from the formal models are compared, on one hand, to those obtained from trials of the platform, and on the other hand, to those obtained from the real experimentations on the field.This work is a part of the FP7 European projects ASSET (2008-2011) and TeleFOT (2008-2012)
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Rimoldi, Anna. "On algebraic and statistical properties of AES-like ciphers". Doctoral thesis, Università degli studi di Trento, 2009. https://hdl.handle.net/11572/367817.
Texto completoResumen
The Advanced Encryption Standard (AES) is nowadays the most widespread block cipher in commercial applications. It represents the state-of-art in block cipher design and provides an unparalleled level of assurance against all known cryptanalytic techniques, except for its reduced versions.
Moreover, there is no known efficient way to distinguish it from a set of random permutations.
The AES (and other modern block ciphers) presents a highly algebraic structure, which led researchers to exploit it for novel algebraic attacks.
These tries have been unsuccessful, except for academic reduced versions.
Starting from an intuition by I. Toli, we have developed a mixed algebraic-statistical attack. Using the internal algebraic structure of any AES-like cipher, we build an algebraic setting where a related-key (statistical) distinguishing attack can be mounted. Our data reveals a significant deviation of the full AES-128 from a set of random permutations.
Although there are recent successful related-key attacks on the full AES-192 and the full AES-256 (with non-practical complexity), our attack would be the first-ever practical distinguishing attack on the full AES-128 (to the best of our knowledge).
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Ugolini, Simone. "Graded Lie algebras of maximal class in positive characteristic, generated by two elements of different weights". Doctoral thesis, Università degli studi di Trento, 2010. https://hdl.handle.net/11572/369223.
Texto completo
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Oriente, Francesco. "Classifying semisimple orbits of theta-groups". Doctoral thesis, Università degli studi di Trento, 2012. https://hdl.handle.net/11572/368303.
Texto completoResumen
I consider the problem of classifying the semisimple orbits of a theta-group. For this purpose, once a preliminary presentation of the theoretical subjects where my problem arises from, I first give an algorithm to compute a Cartan subspace; subsequently I describe how to compute the little Weyl group.
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Marcolla, Chiara. "On structure and decoding of Hermitian codes". Doctoral thesis, Università degli studi di Trento, 2013. https://hdl.handle.net/11572/368401.
Texto completoResumen
Given a linear code, it is important both to identify fast decoding algorithms and to estimate the rst terms of its weight distribution. Ecient decoding algorithms allow the exploitation of the code in practical situations, while the knowledge of the number of small-weight codewords allows to estimate its decoding performance. For ane-variety codes and its subclass formed by Hermitian codes, both problems are as yet unsolved. We investigate both and provide some solutions for special cases of interest.
The rst problem is faced with use of the theory of Gröbner bases for zero-dimensional ideals.
The second problem deals in particular with small-weight codewords of high-rate Hermitian codes. We determine them by studying some geometrical properties of the Hermitian curve, specically the intersection number of the curve with lines and parabolas.
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Pizzato, Marco. "Some Problems Concerning Polynomials over Finite Fields, or Algebraic Divertissements". Doctoral thesis, Università degli studi di Trento, 2013. https://hdl.handle.net/11572/367913.
Texto completoResumen
In this thesis we consider some problems concerning polynomials over finite fields.
The first topic is the action of some groups on irreducible polynomials. We describe orbits and stabilizers.
Next, we consider transformations of irreducible polynomials by quadratic and cubic maps and study the irreducibility of the polynomials obtained.
Finally, starting from PN functions and monomials, we generalize this concept, introducing k-PN monomials and classifying them for small values of k and for fields of order p, p^2 and p^4.
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Scarbolo, Claudio. "Graded Lie algebras of maximal class in characteristic p, generated by two elements of degree 1 and p". Doctoral thesis, Università degli studi di Trento, 2014. https://hdl.handle.net/11572/368969.
Texto completoResumen
Lie algebras of maximal class (or filiform Lie algebras) are the Lie-theoretic analogue of pro-p-groups of maximal class. In particular, they are 2-generated. If one further assumes that the algebras are graded over the positive integers, then over a field of characteristic p it has been shown that a classification is possible provided one generator has
degree 1 and the other has either degree 1 or 2. In this thesis I give a classification of graded Lie algebras of maximal class with generators of degree 1 and p, respectively.
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Piva, Matteo. "Algebraic methods for the distance of cyclic codes". Doctoral thesis, Università degli studi di Trento, 2014. https://hdl.handle.net/11572/368507.
Texto completoResumen
In this thesis we provide known and new results which explain the relationship between the actual minimum distance of cyclic codes, bounds that use only information on the defining sets of cyclic codes to lower bound the distance (root bounds) and bounds that also need the knowledge of the defining sets of all cyclic subcodes (border bounds).
We propose a new bound which is provably better of many known bounds and that can be computed in polynomial time with respect to the length of the code.
We sketch how to use the generalized Newton identities to give alternative proofs of known bounds.
Finally, we use Groebner bases to prove that the optimal root bound can be computed in finite time.
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Tinnirello, Claudia. "Cyclic Codes: Low-Weight Codewords and Locators". Doctoral thesis, Università degli studi di Trento, 2016. https://hdl.handle.net/11572/369123.
Texto completoResumen
Error correcting codes has become an integral part of the design of reliable data transmissions and storage systems. They are also playing an increasingly important role for other applications such as the analysis of pseudorandom sequences and the design of secure cryptosystems. Cyclic codes form a large class of widely used error correcting codes, including important codes such as the Bose-Chaudhuri-Hocquenghem (BCH) codes, quadratic residue (QR) codes and Golay codes. In this thesis I tackle two problems related to cyclic codes: finding low-weight codewords and decoding.
Computing efficiently low-weight codewords of a cyclic code is often a key ingredient of correlation attacks to LFSR-based stream ciphers. The best general purpose algorithm is based on the generalized birthday problem. In the first part of the thesis I show an alternative approach based on discrete logarithms which has - in some cases relevant for applications - much lower memory complexity requirements and a comparable time complexity. The second part of the thesis is devoted to some results concerning efficient bounded-distance decoding for cyclic codes.
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Meneghetti, Alessio. "Optimal Codes and Entropy Extractors". Doctoral thesis, Università degli studi di Trento, 2017. https://hdl.handle.net/11572/368164.
Texto completoResumen
In this work we deal with both Coding Theory and Entropy Extraction for Random Number Generators to be used for cryptographic purposes. We start from a thorough analysis of known bounds on code parameters and a study of the properties of Hadamard codes. We find of particular interest the Griesmer bound, which is a strong result known to be true only for linear codes. We try to extend it to all codes, and we can determine many parameters for which the Griesmer bound is true also for nonlinear codes. In case of systematic codes, a class of codes including linear codes, we can derive stronger results on the relationship between the Griesmer bound and optimal codes. We also construct a family of optimal binary systematic codes contradicting the Griesmer bound. Finally, we obtain new bounds on the size of optimal codes. Regarding the study of random number generation, we analyse linear extractors and their connection with linear codes. The main result on
this topic is a link between code parameters and the entropy rate obtained by a processed random number generator. More precisely, to any linear extractor we can associate the generator matrix of a linear code. Then, we link the total variation distance between the uniform distribution and the probability mass function of a random number generator with the weight distribution of the linear code associated to
the linear extractor. Finally, we present a collection of results derived while pursuing a way to classify optimal codes, such as a probabilistic algorithm to compute
the weight distribution of linear codes and a new bound on the size of codes.
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Civino, Roberto. "Differential attacks using alternative operations and block cipher design". Doctoral thesis, Università degli studi di Trento, 2018. https://hdl.handle.net/11572/368586.
Texto completoResumen
Block ciphers and their security are the main subjects of this work.
In the first part it is described the impact of differential cryptanalysis, a powerful statistical attack against block ciphers, when operations different from the one used to perform the key addition are considered on the message space. It is proven that when an alternative difference operation is carefully designed, a cipher that is proved secure against classical differential cryptanalysis can instead be attacked using this alternative difference.
In the second part it is presented a new design approach of round functions for block ciphers. The proposed round functions can give to the cipher a potentially better level of resistance against statistical attacks. It is also shown that the corresponding ciphers can be proven secure against a well-known algebraic attack, based on the action of the permutation group generated by the round functions of the cipher.
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Longo, Riccardo. "Formal Proofs of Security for Privacy-Preserving Blockchains and other Cryptographic Protocols". Doctoral thesis, Università degli studi di Trento, 2018. https://hdl.handle.net/11572/368971.
Texto completoResumen
Cryptography is used to protect data and communications.
The basic tools are cryptographic primitives, whose security and efficiency are widely studied.
But in real-life applications these primitives are not used individually, but combined inside complex protocols.
The aim of this thesis is to analyse various cryptographic protocols and assess their security in a formal way.
In chapter 1 the concept of formal proofs of security is introduced and the main categorisation of attack scenarios and types of adversary are presented, and the protocols analysed in the thesis are briefly introduced with some motivation.
In chapter 2 are presented the security assumptions used in the proofs of the following chapters, distinguishing between the hardness of algebraic problems and the strength of cryptographic primitives.
Once that the bases are given, the first protocols are analysed in chapter 3, where two Attribute Based Encryption schemes are proven secure.
First context and motivation are introduced, presenting settings of cloud encryption, alongside the tools used to build ABE schemes.
Then the first scheme, that introduces multiple authorities in order to improve privacy, is explained in detail and proven secure.
Finally the second scheme is presented as a variation of the first one, with the aim of improving the efficiency performing a round of collaboration between the authorities.
The next protocol analysed is a tokenization algorithm for the protection of credit cards.
In chapter 4 the advantages of tokenization and the regulations required by the banking industry are presented, and a practical algorithm is proposed, and proven secure and compliant with the standard.
In chapter 5 the focus is on the BIX Protocol, that builds a chain of certificates in order to decentralize the role of certificate authorities.
First the protocol and the structure of the certificates are introduced, then two attack scenarios are presented and the protocol is proven secure in these settings.
Finally a viable attack vector is analysed, and a mitigation approach is discussed.
In chapter 6 is presented an original approach on building a public ledger with end-to-end encryption and a one-time-access property, that make it suitable to store sensitive data.
Its security is studied in a variety of attack scenarios, giving proofs based on standard algebraic assumptions.
The last protocol presented in chapter 7 uses a proof-of-stake system to maintain the consistency of subchains built on top of the Bitcoin blockchain, using only standard Bitcoin transactions.
Particular emphasis is given to the analysis of the refund policies employed, proving that the naive approach is always ineffective whereas the chosen policy discourages attackers whose stake falls below a threshold, that may be adjusted varying the protocol parameters.
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Bonini, Matteo. "Intersections of Algebraic Curves and their link to the weight enumerators of Algebraic-Geometric Codes". Doctoral thesis, Università degli studi di Trento, 2019. https://hdl.handle.net/11572/368573.
Texto completoResumen
Channel coding is the branch of Information Theory which studies the noise that can occur in data transmitted through a channel. Algebraic Coding Theory is the part of Channel Coding which studies the possibility to detect and correct errors using algebraic and geometric techniques. Nowadays, the best performing linear codes are known to be mostly algebraic geometry codes, also named Goppa codes, which arise from an algebraic curve over a finite field, by the pioneering construction due to V. D. Goppa. The best choices for curves on which Goppa's construction and its variants may provide codes with good parameters are those with many rational points, especially maximal curves attaining the Hasse-Weil upper bound for the number of rational points compared with the genus of the curve. Unfortunately, maximal curves are difficult to find. The best known examples of maximal curves are the Hermitian curve, the Ree curve, the Suzuki curve, the GK curve and the GGS curve.
In the present thesis, we construct and investigate algebraic geometry codes (shortly AG codes), their parameters and automorphism groups.
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Rapa, Alessandro. "Simple objects in the heart of a t-structure". Doctoral thesis, Università degli studi di Trento, 2019. https://hdl.handle.net/11572/369193.
Texto completoResumen
Historically, the study of modules over finite dimensional algebras has started with the study of the ones with finite dimension. This is sufficient when dealing with a finite dimensional algebra of finite representation type, where there are only finitely many indecomposable modules of finite length. Indecomposable modules of infinite length occur when dealing with algebras of infinite representation type and the study of pure-injective modules over a finite dimensional algebra is crucial for the problem of describing infinite dimensional modules. In this talk, we consider a specific class of finite dimensional algebras of infinite representation type, called "tubular algebras". Pure-injective modules over tubular algebra have been partially classified by Angeleri Hügel and Kussin, in 2016, and we want to give a contribution to the classification of the ones of "irrational slope". In this talk, first, via a derived equivalence, we move to a more geometrical framework, ie. we work in the category of quasi-coherent sheaves over a tubular curve, and we approach our classification problem from the point of view of tilting/cotilting theory. More precisely, we consider specific torsion pairs cogenerated by infinite dimensional cotilting sheaves and we study the Happel-Reiten-Smalø heart of the corresponding t-structure in the derived category. These hearts are locally coherent Grothendieck categories and, in these categories, the pure-injective sheaves over the tubular curve become injective objects. In order to study injective objects in a Grothendieck category is fundamental the classification of the simple objects. In the seminar, we use some techniques coming from continued fractions and universal extensions to provide a method to construct an infinite dimensional sheaf of a prescribed irrational slope that becomes simple in the Grothendieck category given as the heart of a precise t-structure.