Le sujet de cette thèse est l'étude des progressions arithmétiques dans les nombres entiers. Plus précisément, nous nous intéressons à borner inférieurement v(N), la taille du plus grand sous-ensemble des nombres entiers de 1 à N qui ne contient pas de progressions arithmétiques de 3 termes. Nous allons donc construire de grands sous-ensembles de nombres entiers qui ne contiennent pas de telles progressions, ce qui nous donne une borne inférieure sur v(N). Nous allons d'abord étudier les preuves de toutes les bornes inférieures obtenues jusqu'à présent, pour ensuite donner une autre preuve de la meilleure borne. Nous allons considérer les points à coordonnés entières dans un anneau à d dimensions, et compter le nombre de progressions arithmétiques qu'il contient. Pour obtenir des bornes sur ces quantités, nous allons étudier les méthodes pour compter le nombre de points de réseau dans des sphères à plusieurs dimensions, ce qui est le sujet de la dernière section. / The subject of this thesis is the study of arithmetic progressions in the integers. Precisely, we are interested in the size v(N) of the largest subset of the integers from 1 to N that contains no 3 term arithmetic progressions. Therefore, we will construct a large subset of integers with no such progressions, thus giving us a lower bound on v(N). We will begin by looking at the proofs of all the significant lower bounds obtained on v(N), then we will show another proof of the best lower bound known today. For the proof, we will consider points on a large d-dimensional annulus, and count the number of integer points inside that annulus and the number of arithmetic progressions it contains. To obtain bounds on those quantities, it will be interesting to look at the theory behind counting lattice points in high dimensional spheres, which is the subject of the last section.

A hierarchical control system is proposed for automated flexible manufacturing cells (FMC) that operate in a job shop flow setting. The control system is made up of a higher level scheduler/reactive scheduler, which optimizes the production flow within the cell, and a lower level supervisor that implements the decisions of the scheduler on the shop floor. Previous studies have regularly considered the production scheduling and the supervisory control as two separate problems. This has led to: i) deadlock-prone optimized schedules that cannot be implemented in an automated setting, ii) deadlock-free optimized schedules that lack the means to be transformed into shop floor supervisors, or iii) supervisors that can safely drive the system with no consideration for production performance. The proposed control system combines mathematical models and an insertion heuristic to solve the deadlock-free scheduling problem in job shops, a deadlock-free reactive scheduling heuristic that can revise the schedules upon the occurrence of a wide variety of disruptions, and a systematic procedure that can transform schedules into readily implementable Petri net (PN) supervisors. The integration of these modules into one control hierarchy guarantees a correct, optimized and agile behavior of the controlled system. The performances of the mathematical models, the scheduling and the reactive scheduling heuristics were evaluated by comparison to performances of previous approaches. Experimental results showed that the proposed modules performed consistently better than the other corresponding approaches. The supervisor realization procedure and the overall control architecture were validated by simulation and implementation in an experimental robotic FMC. The control system developed was capable of driving the experimental cell to satisfactorily complete the processing of different product mixes that featured complex processing routes through the cell.

While network data transmission was traditionally accomplished via routing, network coding (NC) broke this rule by allowing network nodes to perform linear combinations of the upcoming data packets. Network operations are performed in a specific Galois field of fixed size q. Decoding only involves a Gaussian elimination with the received network-coded packets. However, in practical wireless environments, NC might be susceptible to transmission errors caused by noise, fading, or interference. This drawback is quite problematic for real-time applications, such as multimediacontent delivery, where timing constraints may lead to the reception of an insufficient number of packets and consequently to difficulties in decoding the transmitted sources. At best, some packets can be recovered, while in the worst case, the receiver is unable to recover any of the transmitted packets.In this thesis, we propose joint source-network coding and decoding schemes in the purpose of providing an approximate reconstruction of the source in situations where perfect decoding is not possible. The main motivation comes from the fact that source redundancy can be exploited at the decoder in order to estimate the transmitted packets, even when some of them are missing. The redundancy can be either natural, i.e, already existing, or artificial, i.e, externally introduced.Regarding artificial redundancy, we choose multiple description coding (MDC) as a way of introducing structured correlation among uncorrelated packets. By combining MDC and NC, we aim to ensure a reconstruction quality that improves gradually with the number of received network-coded packets. We consider two different approaches for generating descriptions. The first technique consists in generating multiple descriptions via a real-valued frame expansion applied at the source before quantization. Data recovery is then achieved via the solution of a mixed integerlinear problem. The second technique uses a correlating transform in some Galois field in order to generate descriptions, and decoding involves a simple Gaussian elimination. Such schemes are particularly interesting for multimedia contents delivery, such as video streaming, where quality increases with the number of received descriptions.Another application of such schemes would be multicasting or broadcasting data towards mobile terminals experiencing different channel conditions. The channel is modeled as a binary symmetric channel (BSC) and we study the effect on the decoding quality for both proposed schemes. Performance comparison with a traditional NC scheme is also provided.Concerning natural redundancy, a typical scenario would be a wireless sensor network, where geographically distributed sources capture spatially correlated measures. We propose a scheme that aims at exploiting this spatial redundancy, and provide an estimation of the transmitted measurement samples via the solution of an integer quadratic problem. The obtained reconstruction quality is compared with the one provided by a classical NC scheme.

Media processing applications typically involve numerical blocks that exhibit regular floating-point computation patterns. For processors whose architecture supports only integer arithmetic, these patterns can be profitably turned into custom operators, coming in addition to the five basic ones (+, -, X, / and √), but achieving better performance by treating more operations. This thesis addresses the design of such custom operators as well as the techniques developed in the compiler to select them in application codes. We have designed optimized implementations for a set of custom operators which includes squaring, scaling, adding two nonnegative terms, fused multiply-add, fused square-add (x*x+z, with z>=0), two-dimensional dot products (DP2), sums of two squares, as well as simultaneous addition/subtraction and sine/cosine. With novel algorithms targeting high instruction-level parallelism and detailed here for squaring, scaling, DP2, and sin/cos, we achieve speedups of up to 4.2x for individual custom operators even when subnormal numbers are fully supported. Furthermore, we introduce the optimizations developed in the ST231 C/C++ compiler for selecting such operators. Most of the selections are achieved at high level, using syntactic criteria. However, for fused square-add, we also enhance the framework of integer range analysis to support floating-point variables in order to prove the required positivity condition z>= 0. Finally, we provide quantitative evidence of the benefits to support this selection of custom operations: on DSP kernels and benchmarks, our approach allows us to be up to 1.59x faster compared to the sole usage of basic ones.

We study bounds on the distances of roots of integer polynomials and applications of such results. The separation of complex roots for reducible monic integer polynomials of fourth degree is thoroughly explained. Lemmas on roots of polynomials in the p-adic setting are proved. Explicit families of polynomials of general degree as well as families in some classes of quadratic and cubic polynomials with very good separation of roots in the same setting are exhibited. The second part of the thesis is concerned with results on p-adic versions of Mahler's and Koksma's functions wn and w*n and the related classifications of transcendental numbers in Cp. The main result is a construction of numbers such that the two functions wn and w*n differ on them for every n and later on expanding the interval of possible values for wn-w*n. The inequalities linking values of Koksma's functions for algebraically dependent numbers are proved.

Die 2^(k – 1) Typen der ungeordneten Zahlpartitionen mit k Parts (k-Partitionen) werden hier mit Hilfe der geordneten Partitionen von k definiert. Für jeden Typ gibt es eine erzeugende Funktion der geschlossenen Form mit eindeutiger Nummerierung. Die bekannte erzeugende Funktion der k-Partitionen ist die Summe dieser 2^(k – 1) typspezifischen erzeugenden Funktionen. Die Expansion dieser typspezifischen erzeugenden Funktionen in (unendlich lange) Potenzreihen ist rekursiv möglich. Untersucht werden Zerlegungen von erzeugenden Funktionen der einfachen Typen in erzeugende Funktionen anderer Typen. Damit lassen sich Bijektionen zwischen den Partitionen verschiedener Typen aufspüren. Die typspezifischen Betrachtungen werden auf die geordneten Partitionen und auf ihre erzeugenden Funktionen ausgeweitet.

Jede ungeordnete Zahlpartition mit k Parts (k-Partiton) hat einen Typ, der mittels einer geordneten Partition von k definiert werden kann. Es können somit 2^(k - 1) Typen definiert werden. Pro Typ gibt es eine eindeutig nummerierbare erzeugende Funktion der geschlossenen Form. Mit Rekursionen können diese Funktionen in (unendlich lange) Potenzreihen expandiert werden. Mit diesen erzeugenden Funktionen lassen sich Bijektionen zwischen den Partitionsmengen verschiedener Typen aufspüren.

La programmation linéaire en nombres entiers est une approche robuste qui permet de résoudre rapidement de grandes instances de problèmes d'optimisation discrète. Toutefois, les problèmes gagnent constamment en complexité et imposent parfois de fortes limites sur le temps de calcul. Il devient alors nécessaire de développer des méthodes spécialisées afin de résoudre approximativement ces problèmes, tout en calculant des bornes sur leurs valeurs optimales afin de prouver la qualité des solutions obtenues. Nous proposons d'explorer une approche de reformulation en nombres entiers guidée par la relaxation lagrangienne. Après l'identification d'une forte relaxation lagrangienne, un processus systématique permet d'obtenir une seconde formulation en nombres entiers. Cette reformulation, plus compacte que celle de Dantzig et Wolfe, comporte exactement les mêmes solutions entières que la formulation initiale, mais en améliore la borne linéaire: elle devient égale à la borne lagrangienne. L'approche de reformulation permet d'unifier et de généraliser des formulations et des méthodes de borne connues. De plus, elle offre une manière simple d'obtenir des reformulations de moins grandes tailles en contrepartie de bornes plus faibles. Ces reformulations demeurent de grandes tailles. C'est pourquoi nous décrivons aussi des méthodes spécialisées pour en résoudre les relaxations linéaires. Finalement, nous appliquons l'approche de reformulation à deux problèmes de localisation. Cela nous mène à de nouvelles formulations pour ces problèmes; certaines sont de très grandes tailles, mais nos méthodes de résolution spécialisées les rendent pratiques. / Integer linear programming is a robust and efficient approach to solve large-scale instances of combinatorial problems. However, problems constantly gain in complexity and sometimes impose strong constraints on computation times. We must then develop specialised methods to compute heuristic primal solutions to the problem and derive lower bounds on the optimal value, and thus prove the quality of our primal solutions. We propose to guide a reformulation approach for mixed integer programs with Lagrangian relaxations. After the identification of a strong relaxation, a mechanical process leads to a second integer formulation. This reformulation is equivalent to the initial one, but its linear relaxation is equivalent to the strong Lagrangian dual. We will show that the reformulation approach unifies and generalises prior formulations and lower bounding approaches, and that it exposes a simple mechanism to reduce the size of reformulations in return for weaker bounds. Nevertheless, our reformulations are large. We address this issue by solving their linear relaxations with specialised methods. Finally, we apply the reformulation approach to two location problems. This yields novel formulations for both problems; some are very large but, thanks to the aforementioned specialised methods, still practical.

The accurate determination of gender of biological samples has valuable applications in medical and forensic investigations. Gender determination based on length variations in the X-Y homologous amelogenin gene, is part of most commercial multiplex DNA profiling kits. The first report of a failure of the amelogenin sex test was in 1998 when two normal males were typed as female. Subsequently, several amelogenin Y (AMELY) negative males have been reported. This study represents the first report of this phenomenon in the black South African population. This study determined the size of the Y-chromosome deletion that resulted in the failure of the amelogenin sex test in two black South African AMELY-negative males by typing specific DNA markers surrounding the amelogenin locus. Through deletion size and Y-chromosome microsatellite haplotypes, the relationship between the samples was investigated. The samples were sequenced at the amelogenin gene and typed for thirteen sites on the short arm of the Y-chromosome. In order to determine the Y-chromosome haplotypes, eleven Y-chromosome microsatellite markers were typed. These samples had the same size deletion of approximately 3 Mb. The Y-chromosome haplotypes indicated that these were probably independent events. The frequency of AMELY-negative males is rare in this population sample of 8,344 individuals, with a frequency of 0.065% in the black South African sample population. Notwithstanding, tests performed for detecting the presence of male DNA based on the presence of the amelogenin gene alone should be reconsidered, as this study confirms that these deletions do occur in the African population. The impact of the results generated in this study on the medical and forensic practise of DNA testing is significant. / Thesis (Ph.D. (Biochemistry))--North-West University, Potchefstroom Campus, 2011.

There are many practical decision problems that fall into the category of network flow problems: numerous examples of applications can be found in areas such as telecommunications, logistics, distributions, engineering, computer science and so on. One of the most popular and valuable tools to solve network flow problems of a topological nature is the use of linear programming models. An important extension of these models is that of integer programming models that deal with problems where some, or all, of the variables are required to assume integer variables. A significant application in this class of problems is the knapsack problem that arises in different contexts such as loading containers in aircraft or satisfying the demand for various lengths of cloth which must be cut from fixed length bolts of fabric. In this study, the feasibility of representing a network flow model in a tree network model and subsequently solving it using a tree knapsack approach is investigated. To compare and validate the proposed technique, a specific case study was chosen from the literature that can be used as a basis for the research project. The said study was an oil pipeline design problem, addressed by Brimberg et al. (2003). This focuses on the optimal design of an oil pipeline network for the South Gabon oil field in Africa. The objective was to reduce oil transportation costs to a major port. Following an overview of different network flow and knapsack models, an overview of the said matter is presented. A description of the proposed tree knapsack approach and the application of this approach to the given problem is given. Results have indicated that it is feasible to apply a tree knapsack approach to solve network flow problems. / Thesis (M.Sc. (Computer Science))--North-West University, Potchefstroom Campus, 2011.