du fait de l'explosion actuelle des télécommunications, les opérateurs sont victimes d'une crise de croissance les obligeant à installer toujours plus de relais, à découper les cellules (zone de couverture d'un relais) en micro-cellules dans les grandes villes, afin de faire face à la demande toujours grandissante de communications. Les concepteurs des nouveaux réseaux de transmission sont donc constamment à la recherche d'une utilisation plus efficace des ressources disponibles

Random walks describe diffusion processes, where movement at every time step is restricted only to neighbouring locations. Classical random walks are constructed using the non-relativistic Laplacian evolution operator and a coin toss instruction. In quantum theory, an alternative is to use the relativistic Dirac operator. That necessarily introduces an internal degree of freedom (chirality), which may be identified with the coin. The resultant walk spreads quadratically faster than the classical one, and can be applied to a variety of graph theoretical problems. We study in detail the problem of spatial search, i.e. finding a marked site on a hypercubic lattice in d-dimensions. For d=1, the scaling behaviour of classical and quantum spatial search is the same due to the restriction on movement. On the other hand, the restriction on movement hardly matters for d ≥ 3, and scaling behaviour close to Grover’s optimal algorithm(which has no restriction on movement) can be achieved. d=2 is the borderline critical dimension, where infrared divergence in propagation leads to logarithmic slow down that can be minimised using clever chirality flips. In support of these analytic expectations, we present numerical simulation results for d=2 to d=9, using a lattice implementation of the Dirac operator inspired by staggered fermions. We optimise the parameters of the algorithm, and the simulation results demonstrate that the number of binary oracle calls required for d= 2 and d ≥ 3 spatial search problems are O(√NlogN) and O(√N) respectively. Moreover, with increasing d, the results approach the optimal behaviour of Grover’s algorithm(corresponding to mean field theory or d → ∞ limit). In particular, the d = 3 scaling behaviour is only about 25% higher than the optimal value.

This PhD thesis consists of a summary and five papers which all deal with asymptotic problems on certain homogeneous spaces. In Paper I we prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. All our results are given with precise estimates on the rates of convergence to equidistribution. Papers II and III are concerned with statistical problems on the space of n-dimensional lattices of covolume one. In Paper II we study the distribution of lengths of non-zero lattice vectors in a random lattice of large dimension. We prove that these lengths, when properly normalized, determine a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity 1/2. In Paper III we complement this result by proving that the asymptotic distribution of the angles between the shortest non-zero vectors in a random lattice is that of a family of independent Gaussians. In Papers IV and V we investigate the value distribution of the Epstein zeta function along the real axis. In Paper IV we determine the asymptotic value distribution and moments of the Epstein zeta function to the right of the critical strip as the dimension of the underlying space of lattices tends to infinity. In Paper V we determine the asymptotic value distribution of the Epstein zeta function also in the critical strip. As a special case we deduce a result on the asymptotic value distribution of the height function for flat tori. Furthermore, applying our results we discuss a question posed by Sarnak and Strömbergsson as to whether there in large dimensions exist lattices for which the Epstein zeta function has no zeros on the positive real line.

The vibration of and wave propagation in periodic cellular structures are analyzed. Cellular structures exhibit a number of desirable multifunctional properties, which make them attractive in a variety of engineering applications. These include ultra-light structures, thermal and acoustic insulators, and impact amelioration systems, among others. Cellular structures with deterministic architecture can be considered as example of periodic structures. Periodic structures feature unique wave propagation characteristics, whereby elastic waves propagate only in specific frequency bands, known as "pass band", while they are attenuated in all other frequency bands, known as "stop bands". Such dynamic properties are here exploited to provide cellular structures with the capability of behaving as directional, pass-band mechanical filters, thus complementing their well documented multifunctional characteristics. This work presents a methodology for the analysis of the dynamic behavior of periodic cellular structures, which allows the evaluation of location and spectral width of propagation and attenuation regions. The filtering characteristics are tested and demonstrated for structures of various geometry and topology, including cylindrical grid-like structures, Kagom and eacute; and tetrhedral truss core lattices. Experimental investigations is done on a 2-D lattice manufactured out of aluminum. The complete wave field of the specimen at various frequencies is measured using a Scanning Laser Doppler Vibrometer (SLDV). Experimental results show good agreement with the methodology and computational tools developed in this work. The results demonstrate how wave propagation characteristics are defined by cell geometry and configuration. Numerical and experimental results show the potential of periodic cellular structures as mechanical filters and/or isolators of vibrations.

In this thesis, a fairly detailed survey on the q-classical orthogonal polynomials of the Hahn class is presented. Such polynomials appear to be the bounded solutions of the so called qhypergeometric difference equation having polynomial coefficients of degree at most two. The central idea behind our study is to discuss in a unified sense the orthogonality of all possible polynomial solutions of the q-hypergeometric difference equation by means of a qualitative analysis of the relevant q-Pearson equation. To be more specific, a geometrical approach has been used by taking into account every posssible rational form of the polynomial coefficients, together with various relative positions of their zeros, in the q-Pearson equation to describe a desired q-weight function on a suitable orthogonality interval. Therefore, our method differs from the standard ones which are based on the Favard theorem and the three-term recurrence relation.

The object of this research was to investigate dispersion in surface phononic crystals (PnCs) for application to a newly developed passive surface acoustic wave (SAW) ozone sensor. Frequency band gaps and slow sound already have been reported for PnC lattice structures. Such engineered structures are often advertised to reduce loss, increase sensitivity, and reduce device size. However, these advances have not yet been realized in the context of surface acoustic wave sensors. In early work, we computed SAW dispersion in patterned surface structures and we confirmed that our finite element computations of SAW dispersion in thin films and in one dimensional surface PnC structures agree with experimental results obtained by laser probe techniques. We analyzed the computations to guide device design in terms of sensitivity and joint spectral operating point. Next we conducted simulations and experiments to determine sensitivity and limit of detection for more conventional dispersive SAW devices and PnC sensors. Finally, we conducted extensive ozone detection trials on passive reflection mode SAW devices, using distinct components of the time dispersed response to compensate for the effect of temperature. The experimental work revealed that the devices may be used for dosimetry applications over periods of several days.

No description available.

In the past little work has been done to characterize the models of various mereotopological systems. This thesis focuses on Asher and Vieu's first-order mereotopology which evolved from Clarke's Calculus of Individuals. Its soundness and completeness proofs with respect to a topological translation of the axioms provide only sparse insights into structural properties of the mereotopological models. To overcome this problem, we characterize these models with respect to mathematical structures with well-defined properties - topological spaces, lattices, and graphs. We prove that the models of the subtheory RT− are isomorphic to p-ortholattices (pseudocomplemented, orthocomplemented). Combining the advantages of lattices and graphs, we show how Cartesian products of finite p-ortholattices with one multiplicand being not uniquely complemented (unicomplemented) gives finite models of the full mereotopology. Our analysis enables a comparison to other mereotopologies, in particular to the RCC, of which lattice-theoretic characterizations exist.

In the past little work has been done to characterize the models of various mereotopological systems. This thesis focuses on Asher and Vieu's first-order mereotopology which evolved from Clarke's Calculus of Individuals. Its soundness and completeness proofs with respect to a topological translation of the axioms provide only sparse insights into structural properties of the mereotopological models. To overcome this problem, we characterize these models with respect to mathematical structures with well-defined properties - topological spaces, lattices, and graphs. We prove that the models of the subtheory RT− are isomorphic to p-ortholattices (pseudocomplemented, orthocomplemented). Combining the advantages of lattices and graphs, we show how Cartesian products of finite p-ortholattices with one multiplicand being not uniquely complemented (unicomplemented) gives finite models of the full mereotopology. Our analysis enables a comparison to other mereotopologies, in particular to the RCC, of which lattice-theoretic characterizations exist.

Graphene, a single sheet of carbon atoms sp2-bonded in a honeycomb lattice, is a possible all-carbon successor to silicon electronics. Ballistic conduction at room temperature and a linear dispersion relation that causes carriers to behave as massless Dirac fermions are features that make graphene promising for high-speed, low-power devices. The critical advantage of epitaxial graphene (EG) grown on SiC is its compatibility with standard lithographic procedures. Surface X-ray diffraction (SXRD) and scanning tunneling microscopy (STM) results are presented on the domain structure, interface composition and stacking character of graphene grown on both polar faces of semi-insulating 4H-SiC. The data reveal intriguing differences between graphene grown on these two faces. Substrate roughening is more pronounced and graphene domain sizes are significantly smaller on the SiC (0001) Si-face. Specular X-ray reflectivity measurements show that both faces have a carbon rich, extended interface that is tightly bound to the first graphene layer, leading to a buffering effect that shields the first graphene layer from the bulk SiC, as predicted by ab initio calculations. In-plane X-ray crystal truncation rod analysis indicates that rotated graphene layers are interleaved in C-face graphene films and corresponding superstructures are observed in STM topographs. These rotational stacking faults in multilayer C-face graphene preserve the linear dispersion found in single layer graphene, making EG electronics possible even for a multilayer material.