Algorithms, random tree models and combinatorial objects

The author thanks the editors of IMN for their invitation to give an overview about his recent research topics. The following paper exemplifies by concrete examples my main research directions, which concern the analysis of algorithms and data structures, random tree models and combinatorial objects. 1 1 Average-case analysis of Union-Find algorithms The concept of so-called "average case analysis of algorithms and data structures" has been introduced by Donald Knuth in his famous book series "The art of computer programming&quot

Computer-Aided Geometry Modeling

Techniques in computer-aided geometry modeling and their application are addressed. Mathematical modeling, solid geometry models, management of geometric data, development of geometry standards, and interactive and graphic procedures are discussed. The applications include aeronautical and aerospace structures design, fluid flow modeling, and gas turbine design

On the growth and structure of social systems following preferential attachment

L’inégalité est une caractéristique notoire des systèmes sociaux. Dans cette thèse, nous nous attarderons à la distribution et à la structure de la répartition de leurs ressources et activités. Dans ce contexte, leurs extrêmes iniquités tendent à suivre une propriété universelle, l’indépendance d’échelle, qui se manifeste par l’absence d’échelle caractéristique. En physique, les organisations indépendantes d’échelle sont bien connues en théorie des transitions de phase dans laquelle on les observe à des points critiques précis. Ceci suggère que des mécanismes bien définis sont potentiellement responsables de l’indépendance d’échelle des systèmes sociaux. Cette analogie est donc au coeur de cette thèse, dont le but est d’aborder ce problème de nature multidisciplinaire avec les outils de la physique statistique. En premier lieu, nous montrons qu’un système dont la distribution de ressource croît vers l’indépendance d’échelle se trouve assujetti à deux contraintes temporelles particulières. La première est l’attachement préférentiel, impliquant que les riches s’enrichissent. La seconde est une forme générale de comportement d’échelle à délai entre la croissance de la population et celle de la ressource. Ces contraintes dictent un comportement si précis qu’une description instantanée d’une distribution est suffisante pour reconstruire son évolution temporelle et prédire ses états futurs. Nous validons notre approche au moyen de diverses sphères d’activités humaines dont les activités des utilisateurs d’une page web, des relations sexuelles dans une agence d’escorte, ainsi que la productivité d’artistes et de scientifiques. En second lieu, nous élargissons notre théorie pour considérer la structure résultante de ces activités. Nous appliquons ainsi nos travaux à la théorie des réseaux complexes pour décrire la structure des connexions propre aux systèmes sociaux. Nous proposons qu’une importante classe de systèmes complexes peut être modélisée par une construction hiérarchique de niveaux d’organisation suivant notre théorie d’attachement préférentiel. Nous montrons comment les réseaux complexes peuvent être interprétés comme une projection de ce modèle de laquelle émerge naturellement non seulement leur indépendance d’échelle, mais aussi leur modularité, leur structure hiérarchique, leurs caractéristiques fractales et leur navigabilité. Nos résultats suggèrent que les réseaux sociaux peuvent être relativement simples, et que leur complexité apparente est largement une réflexion de la structure hiérarchique complexe de notre monde.Social systems are notoriously unfair. In this thesis, we focus on the distribution and structure of shared resources and activities. Through this lens, their extreme inequalities tend to roughly follow a universal pattern known as scale independence which manifests itself through the absence of a characteristic scale. In physical systems, scale-independent organizations are known to occur at critical points in phase transition theory. The position of this critical behaviour being very specific, it is reasonable to expect that the distribution of a social resource might also imply specific mechanisms. This analogy is the basis of this work, whose goal is to apply tools of statistical physics to varied social activities. As a first step, we show that a system whose resource distribution is growing towards scale independence is subject to two constraints. The first is the well-known preferential attachment principle, a mathematical principle roughly stating that the rich get richer. The second is a new general form of delayed temporal scaling between the population size and the amount of available resource. These constraints pave a precise evolution path, such that even an instantaneous snapshot of a distribution is enough to reconstruct its temporal evolution and predict its future states. We validate our approach on diverse spheres of human activities ranging from scientific and artistic productivity, to sexual relations and online traffic. We then broaden our framework to not only focus on resource distribution, but to also consider the resulting structure. We thus apply our framework to the theory of complex networks which describes the connectivity structure of social, technological or biological systems. In so doing, we propose that an important class of complex systems can be modelled as a construction of potentially infinitely many levels of organization all following the same universal growth principle known as preferential attachment. We show how real complex networks can be interpreted as a projection of our model, from which naturally emerge not only their scale independence, but also their clustering or modularity, their hierarchy, their fractality and their navigability. Our results suggest that social networks can be quite simple, and that the apparent complexity of their structure is largely a reflection of the complex hierarchical nature of our world

The mathematical modelling and numerical solution of options pricing problems

Accurate and efficient numerical solutions have been described for a selection of financial options pricing problems. The methods are based on finite difference discretisation coupled with optimal solvers of the resulting discrete systems. Regular Cartesian meshes have been combined with orthogonal co-ordinate transformations chosen for numerical accuracy rather than reduction of the differential operator to constant coefficient form. They allow detailed resolution in the regions of interest where accuracy is most desired, and grid coarsening where there is least interest. These transformations are shown to be effective in producing accurate solutions on modest computational grids. The spatial discretisation strategy is chosen to meet accuracy requirements as sell as to produce coefficient matrices with favourable sparsity and stability properties. In the case of single factor European options, a modified Crank-Nicolson, second order accurate finite difference scheme is presented, which uses adaptive upwind differences when the mesh Peclet conditions are violated. The resulting tridiagonal system of equations is solved using a direct solver. A careful study of grid refinement displays convergence towards the true solution and demonstrates a high level of accuracy can be obtained with this approach. Laplace inversion methods are also implemented as an alternative solution approach for the one-factor European option. Results are compared to those produced by the direct solver algorithm and are shown to be favourable. It is shown how Semi-Lagrange time-integration can solve the path-dependent Asian pricing problem, by integrating out the average price term and simplifying the finite difference equations into a parameterised Black-Scholes form. The implicit equations that result are unconditionally stable, second order accurate and can be solved using standard tridiagonal solvers. The Semi-Lagrange method is shown to be easily used in conjunction with co-ordinate transformations applied in both spatial directions. A variable time-stepping scheme is implemented in the algorithm. Early exercise is also easily incorporated, the resulting linear complementarity problem can be solved using a projection or penalty method (the penalty method is shown to be slightly more efficient). Second order accuracy has been confirmed for Asian options that must be held to maturity. A comparison with published results for continuous-average-rate put and call options, with and without early exercise, shows that the method achieves basis point accuracy and that Richardson extrapolation can also be applied

Program and Abstracts of the Annual Meeting of the Georgia Academy of Science, 2014

The annual meeting of the Georgia Academy of Science took place March 28-29, 2014, at Georgia Regents University, Augusta, Georgia. Presentations were provided by members of the Academy who represented the following sections: I. Biological Sciences II Chemistry III. Earth & Atmospheric Sciences IV. Physics, Mathematics, Computer Science, Engineering & Technology V. Biomedical Sciences VI. Philosophy & History of Science VII. Science Education VIII. Anthropology

Spin and magnetotransport properties of narrow gap semiconductors

Imperial Users onl