A (k,l)-colouring of a graph is a partition of its vertex set into k independent sets and l cliques. The bichromatic number chi^b of a graph is the minimum r$ such that the graph is (k,l)-colourable for all k+l=r. The bichromatic number is related to the cochromatic number, which can also be defined in terms of (k,l)-colourings. The bichromatic number is a fairly recent graph parameter that arises in the study of extremal graphs related to a classical result of Erd\H{o}s, Stone and Simonovits, and in the study of the edit distance of graphs from hereditary graph classes. While the cochromatic number has been well studied in the literature, there are only few known structural results for the bichromatic number. A main focus of this thesis is to establish a foundation of knowledge about the bichromatic number. The secondary focus is on $(k,l)$-colourings of certain interesting graph classes. Two known bounds for the bichromatic number are $\chi^b \leq \chi + \theta - 1$, where $\chi$ is the chromatic number and $\theta$ the clique covering number of the graph, and $\chi^b \geq \sqrt{n}$, where $n$ the number of vertices of the graph. We give a complete characterization of all graphs for which equality holds in the first bound, and show that the second bound is best possible by constructing graphs for square numbers $n$ such that equality holds in the bound. We investigate graphs for which the bichromatic number equals the cochromatic number and prove a Brooks-type theorem for the bichromatic number. Regarding $(k,l)$-colourings, we find a new algorithm for calculating the $(k,l)$-colourability of cographs and show that cographs have a particularly nice representation with regard to $(k,l)$-colourings. For proper circular arc graphs, we provide a method for $(k,l)$-colouring if $l \geq 1$, and establish an algebraic characterization for all maximally $(k,0)$-colourable proper circular arc graphs. Finally, we investigate the bichromatic number and cochromatic with respect to lexicographic products and show several nice bounds. / Graduate

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There are only a few detailed measurements of the cross-shore variation in the net sediment transport and beach evolution for single or multiple swash events, and no data showing the influence of long waves and wave groups on swash zone morphology. Novel laboratory experiments and numerical modeling have been performed to study the influence of long waves and bichromatic wave groups on sediment transport and beach morphodynamics in the swash zone. Due to complex processes, difficulties in measuring, and very significant difficulties in isolating the morphodynamic processes induced by long waves and wave groups on natural beaches, a laboratory study was designed to measure in very high detail the bathymetric evolution of model sand beaches under monochromatic waves, long wave and short wave composites (free long waves), and bichromatic wave groups (forced long waves). Net sediment transport, Q(x), and beach morphology changes under the monochromatic waves were analyzed and compared to conditions with and without the free long waves, and then compared with the bichromatic wave groups. A range of wave conditions, e.g., high energy, moderate energy, and low energy waves, were used to obtain beach evolution ranging from accretionary to erosive, and including intermediate beach states. Hydrodynamics parameters, e.g., instantaneous water depths, wave amplitudes, run-up and rundown, were also measured to study and test a sediment transport model for the swash zone, based on modifying the energetic-bedload based sediment transport equations with suspended sediment. The experimental data clearly demonstrate that for the monochromatic wave conditions, beach evolution develops erosion for high steepness waves and accretion for lower steepness waves. The model beach profile evolutions are similar to natural beaches, and form and develop bars and berms over time. Adding a free long wave to the short wave in the composite wave results in changes to the overall trend of erosion/accretion of the beach profile, but the net transport pattern does not change significantly. The short wave strongly dominates beach behavior and the net transport rate, instead of the free long wave in the composite wave. The free long wave, however, carries more water and sediment onshore, leading to an increase in shoreline motion and wave run-up further landward. The long wave influences the structure and position of the swash bar/berm, which generally tends to move onshore and forms a larger swash bar/berm for higher long wave amplitudes. The free long wave also increases overall onshore sediment transport, and reduces offshore transport for erosive conditions. The long wave tends to protect the beach face and enhances onshore transport for accretive conditions, especially in the swash zone. In contrast, for bichromatic wave groups having the same mean energy flux as their corresponding monochromatic wave, the influence on sediment transports is generally offshore in both the surf and swash zone instead of onshore. The swash berm is, however, formed further landward compared with the berm of the corresponding monochromatic wave. The sediment transport patterns (erosion or accretion) generated by the bichromatic wave group or corresponding monochromatic wave are similar, but differ in magnitude. The numerical model, starting in the inner surf zone to reduce the effect of poor breaker description in the non-linear shallow water equations, can produce a good match between observed data and the modeled hydrodynamics parameters in the SZ. The sediment transport model shows the important role of suspended sediment in the swash zone. In contrast with the observed data, energetic-based bed-load models predict offshore sediment transport for most wave conditions because of negative skewness. The modified sediment transport model, with added suspended sediment terms and optimized coefficients, produces a good match between model results and observed data for each wave condition, especially for low frequency monochromatic waves. The optimized coefficient set corresponding to particular monochromatic wave conditions can be used to predict the net sediment transport quite well for some composite wave conditions. Overall, the same optimized coefficient sets can be applied to predict the correct overall trend of net transport for most composite wave conditions. However, the predicted net transport for the bichromatic wave groups does not match well with the overall net transport patterns. There is no set of single transport coefficients that can be used to predict sediment transport for all wave conditions. This suggests that the present sediment transport models cannot predict evolution correctly, even for conditions which represent only perturbation from those for which they were calibrated.

There are only a few detailed measurements of the cross-shore variation in the net sediment transport and beach evolution for single or multiple swash events, and no data showing the influence of long waves and wave groups on swash zone morphology. Novel laboratory experiments and numerical modeling have been performed to study the influence of long waves and bichromatic wave groups on sediment transport and beach morphodynamics in the swash zone. Due to complex processes, difficulties in measuring, and very significant difficulties in isolating the morphodynamic processes induced by long waves and wave groups on natural beaches, a laboratory study was designed to measure in very high detail the bathymetric evolution of model sand beaches under monochromatic waves, long wave and short wave composites (free long waves), and bichromatic wave groups (forced long waves). Net sediment transport, Q(x), and beach morphology changes under the monochromatic waves were analyzed and compared to conditions with and without the free long waves, and then compared with the bichromatic wave groups. A range of wave conditions, e.g., high energy, moderate energy, and low energy waves, were used to obtain beach evolution ranging from accretionary to erosive, and including intermediate beach states. Hydrodynamics parameters, e.g., instantaneous water depths, wave amplitudes, run-up and rundown, were also measured to study and test a sediment transport model for the swash zone, based on modifying the energetic-bedload based sediment transport equations with suspended sediment. The experimental data clearly demonstrate that for the monochromatic wave conditions, beach evolution develops erosion for high steepness waves and accretion for lower steepness waves. The model beach profile evolutions are similar to natural beaches, and form and develop bars and berms over time. Adding a free long wave to the short wave in the composite wave results in changes to the overall trend of erosion/accretion of the beach profile, but the net transport pattern does not change significantly. The short wave strongly dominates beach behavior and the net transport rate, instead of the free long wave in the composite wave. The free long wave, however, carries more water and sediment onshore, leading to an increase in shoreline motion and wave run-up further landward. The long wave influences the structure and position of the swash bar/berm, which generally tends to move onshore and forms a larger swash bar/berm for higher long wave amplitudes. The free long wave also increases overall onshore sediment transport, and reduces offshore transport for erosive conditions. The long wave tends to protect the beach face and enhances onshore transport for accretive conditions, especially in the swash zone. In contrast, for bichromatic wave groups having the same mean energy flux as their corresponding monochromatic wave, the influence on sediment transports is generally offshore in both the surf and swash zone instead of onshore. The swash berm is, however, formed further landward compared with the berm of the corresponding monochromatic wave. The sediment transport patterns (erosion or accretion) generated by the bichromatic wave group or corresponding monochromatic wave are similar, but differ in magnitude. The numerical model, starting in the inner surf zone to reduce the effect of poor breaker description in the non-linear shallow water equations, can produce a good match between observed data and the modeled hydrodynamics parameters in the SZ. The sediment transport model shows the important role of suspended sediment in the swash zone. In contrast with the observed data, energetic-based bed-load models predict offshore sediment transport for most wave conditions because of negative skewness. The modified sediment transport model, with added suspended sediment terms and optimized coefficients, produces a good match between model results and observed data for each wave condition, especially for low frequency monochromatic waves. The optimized coefficient set corresponding to particular monochromatic wave conditions can be used to predict the net sediment transport quite well for some composite wave conditions. Overall, the same optimized coefficient sets can be applied to predict the correct overall trend of net transport for most composite wave conditions. However, the predicted net transport for the bichromatic wave groups does not match well with the overall net transport patterns. There is no set of single transport coefficients that can be used to predict sediment transport for all wave conditions. This suggests that the present sediment transport models cannot predict evolution correctly, even for conditions which represent only perturbation from those for which they were calibrated.

Consider a set B of blue points and a set R of red points in the plane such that R ∪ B is in general position. A graph drawn in the plane whose edges are straight-line segments is called a geometric graph. We investigate the problem of drawing non-crossing properly colored geometric graphs on the point set R ∪ B. We show that if ||B| − |R|| ≤ 1 and a subset of R forms the vertices of a convex polygon separating the points of B, lying inside the polygon, from the rest of the points of R, lying outside the polygon, then there exists a non-crossing properly colored geometric path on R∪B covering all points of R ∪ B. If R∪B lies on a circle, the size of the longest non-crossing geometric path is related to the size of the largest separated matching; a separated matching is a non-crossing properly colored geometric matching where all edges can be crossed by a line. A discrepancy of R ∪ B is the maximal difference between cardinalities of color classes of intervals on the circle. When the discrepancy of R ∪ B is at most 2, we show that there is a separated matching covering asymptotically 4 5 of points of R ∪ B. During this proof we use a connection between separated matchings and the longest common subsequences between two binary sequences where the symbols correspond to the colors of the points.

Les méthodes optiques passives de mesures de température comme la thermographie ou la pyrométrie optique sont intéressantes car elles permettent une mesure non intrusive de l’objet cible à condition de connaître le facteur d’émission. La connaissance de ce facteur est critique pour déterminer la température de surface par le rayonnement thermique émis dans un domaine spectral. Le pyromètre bichromatique impulsionnel permet de surmonter la connaissance de ce paramètre à condition que le choix des valeurs de longueurs d’onde soit fait avec précautions. Lorsque l’objet à mesurer est situé dans un environnement industriel, de telles méthodes optiques sont fortement perturbées par la présence d’un milieu optiquement absorbant. C’est également le cas pour des objets situés dans des environnements très chauds qui émettent d’intenses radiations interférentes. Dans cette thèse, nous présentons une méthode radiométrique active bichromatique pour mesurer la température d’une surface en milieu industriel. Cette méthode est basée sur une excitation locale par une source laser modulée dans l’infrarouge. La détection de la température qui est corrélé avec l’excitation permet d’extraire par détection synchrone le signal modulé noyé dans un bruit jusqu’à un million de fois supérieur. Travailler à courtes longueurs d’onde (domaine visible et proche infrarouge) offre une grande dynamique et minimise l’erreur induite par les variations d’émissivité avec la longueur d’onde. Ce système collecte le rayonnement émis par l’objet à distance, de quelques mètres à plusieurs dizaines de mètres selon la configuration du système optique. Le principe de la méthode de mesure, le système optique et l’appareillage sont présentés dans ce rapport ainsi que les études théoriques et expérimentales sur la sensibilité, sa calibration et les résultats obtenus sur différents sites industriels. / Optical passive methods of temperature measurements such as thermography or optical pyrometry are very interesting because they allow a non-intrusive measurement with the target object provided to know the emission factor. The knowledge of this factor is critical for determining the actual temperature of a surface from the thermal radiation emitted in a wavelength band. The bichromatic pulsed pyrometer allows to overcome the knowledge of this parameter provided that precautions are taken in the choice of the values of wavelengths. When the object to be measured is placed in industrial environments, such passive optical methods are greatly disturbed by the presence of an optically absorbing medium. It is also distorted for objects located in very hot environments emitting intense interfering radiation. In this thesis, we present an active bichromatic radiometric method for measuring the temperature of a surface in harsh environments. The method is based on a localized excitation by a modulated laser source in the infrared range. Detecting the temperature range which is correlated with the excitation allows a synchronous detection to extract the signal embedded in a noise up to 106 times superior. Working at short wavelengths (visible range and near infrared range) offers a large dynamic range and minimizes the error due to variations in emissivity with the wavelength. This system collects the radiation emitted by the object at a distance from a few meters up to dozens of meters depending on the configuration of the optical system. The principle of the measurement method, the optical wavelength separation system and the telemetry apparatus are presented in this report as well as the theoretical and experimental study of the sensitivity of the device, its calibration and the results obtained in different industrial sites.