Satellite communication promises potential methods for providing global communication. In particular, by the development of a Low Earth Orbital (LEO) satellite constellation, both global coverage and broadband communication will be accessible. Problems arise in situations where various traffic types in broadband communication require different levels of quality of service (QoS). Traffic control is required to make sure that each traffic demand may receive the expected QoS. Another problem is that the dynamic topology of a LEO satellite network requires a traffic allocation control, which is able to allocate traffic demand into the Inter Satellite Links (ISLs) between LEO satellites.In this thesis, traffic allocation strategy in a dynamic LEO satellite communication network is studied and analyzed. The delivery of Quality of Service (QoS) is an important objective. Traffic allocation control is performed in the LEO satellite constellation to provide a near optimal utilization of these ISLs. An alternative solution is proposed in this research, in which a combination of two algorithms will be used to allocate traffic in this dynamic satellite network. The first algorithm allocates traffic during small time intervals, based on an assumption that the topology is unchanged during these intervals. The second algorithm allocates traffic after topology updating has been accomplished. Traffic allocation respects some constraints including QoS (due to multiservice requirements), capacity constraints, traffic distribution, and availability constraints. Both theoretical and empirical studies have been undertaken to examine the performance of the proposed algorithm, denoted GALPEDA (Genetic Algorithm Linear Programming and Extended Dijkstra Algorithm). The proposed algorithm provides privileges to a class of high priority traffic, including benefits for traffic allocation of multiclass traffic in LEO satellite communication. It provides a novel traffic allocation mechanism to cope with the dynamic topology of a LEO satellite; moreover this algorithm distributes multiservice traffic evenly over the network. Simulations results are provided.

The high pace at which many of the world's energy markets have gradually been opened to competition have generated a significant amount of new financial activity. Both academicians and practitioners alike recently started to develop the tools of energy derivatives pricing/hedging as a quantitative topic of its own. The energy contract structures as well as their underlying asset properties set the energy risk management industry apart from its more standard equity and fixed income counterparts. This thesis naturaly contributes to these broad market developments in participating to the advances of the mathematical tools aiming at a better theory of energy contingent claim pricing/hedging. We propose many realistic two-factor and three-factor models for spot and forward price processes that generalize some well known and standard modeling assumptions. We develop the associated pricing methodologies and propose stable calibration algorithms that motivate the application of the relevant modeling schemes.

Let $M$ be a smooth compact manifold equipped with a co-oriented subbundle $E\subset TM$. We suppose that a compact Lie group $G$ acts on $M$ preserving $E$, such that the $G$-orbits are transverse to $E$. If the fibres of $E$ are equipped with a complex structure then it is possible to construct a $G$-invariant Dirac operator $\dirac$ in terms of the resulting almost CR structure. We show that there is a canonical equivariant differential form with generalized coefficients $\mathcal{J}(E,X)$ defined on $M$ that depends only on the $G$-action and the co-oriented subbundle $E$. Moreover, the group action is such that $\dirac$ is a $G$-transversally elliptic operator in the sense of Atiyah \cite{AT}. Its index is thus defined as a generalized function on $G$. Beginning with the equivariant index formula of Paradan and Vergne \cite{PV3}, we obtain an index formula for $\dirac$ computed as an integral over $M$ that is free of choices and growth conditions. This formula necessarily involves equivariant differential forms with generalized coefficients and we show that the only such form required is the canonical form $\mathcal{J}(E,X)$. In certain cases the index of $\dirac$ can be interpreted in terms of a CR analogue of the space of holomorphic sections, allowing us to view our index formula as a character formula for the $G$-equivariant quantization of the almost CR manifold $(M,E)$. In particular, we obtain the ``almost CR'' quantization of a contact manifold, in a manner directly analogous to the almost complex quantization of a symplectic manifold.

Around 1990, Arthur proved a local (ordinary) trace formula for real or p-adic connected reductive groups. The local trace formula is a powerful tool in the local harmonic analysis of reductive groups. One of the aims of this thesis is to establish a local twisted trace formula for certain non-connected reductive groups, which is a twisted version of Arthur’s local trace formula. As an application of the local twisted trace formula, we will prove some twisted orthogonality relations, which are generalizations of Arthur’s results about orthogonality relations for tempered elliptic characters. To establish these relations, we will also give a classification of twisted elliptic representations.

Ten years ago, J. Villadsen constructed the first examples of simple C*-algebras with stable rank other than one or infinity. Villadsen's examples all had a unique tracial state. It is natural to ask whether examples can be found of simple C*-algebras with higher stable rank and more than one tracial state; by building on Villadsen's construction, we describe such examples that admit arbitrary tracial state spaces.

This thesis is devoted to the study of foliations that come from dynamical systems. In the first part we study foliations of Stein manifolds locally given by vector fields. The leaves of such foliations are Riemann surfaces. We prove that for a generic foliation all leaves except for not more than a countable number are homeomorphic to disks, the rest are homeomorphic to cylinders. We also prove that a generic foliation is complex Kupka-Smale. In the second part of the thesis we study complex H\'non maps. The sets of points $U^+$ and $U^-$ that have unbounded forward and backwards orbits correspondingly, is naturally endowed with holomorphic foliations $^+$ and $^-$. We describe the critical locus -- the set of tangencies between these foliations -- for H\'{e}non maps that are small perturbations of quadratic polynomials with disconnected Julia set.

Near the beginning of the last century, R. A. Fisher and Sewall Wright devised an elegant, mathematically tractable model of gene reproduction and replacement that laid the foundation for contemporary population genetics. The Wright-Fisher model and its extensions have given biologists powerful tools of statistical inference that enabled the quantification of genetic drift and selection. Given the utility of these tools, we often forget that their model - for reasons of mathematical tractability - makes assumptions that are violated in many real-world populations. In particular, the classical models assume fixed population sizes, held constant by (unspecified) sampling mechanisms. Here, we consider an alternative framework that merges Moran’s continuous time Markov chain model of allele frequencies in haploid populations of fixed size with the density dependent models of ecological competition of Lotka, Volterra, Gause, and Kolmogorov. This allows for haploid populations of stochastically varying – but bounded – size. Populations are kept finite by resource limitation. We show the existence of limits that naturally generalize the weak and strong selection regimes of classical population genetics, which allow the calculation of fixation times and probabilities, as well as the long-term stationary allele frequency distribution.

In the early 70's Parshin introduced his notion of the multidimensional residues of meromorphic top-forms on algebraic varieties. Parshin's theory is a generalization of the classical one-dimensional residue theory. The main difference between the Parshin's definition and the one-dimensional case is that in higher dimensions one computes the residue not at a point but at a complete flag of irreducible subvarieties. Parshin, Beilinson, and Lomadze also proved the Reciprocity Law for residues: if one fixes all elements of the flag, except for one, and consider all possible choices of the missing element, then only finitely many of these choices give non-zero residues, and the sum of these residues is zero. Parshin's constructions are completely algebraic. In fact, they work in very general settings, not only over complex numbers. However, in the complex case one would expect a more geometric variant of the theory. In my thesis I study Parshin residues from the geometric point of view. In particular, the residue is expressed in terms of the integral over a smooth cycle. Parshin-Lomadze Reciprocity Law for residues in the complex case is proved via a homological relation on these cycles. The thesis consists of two parts. In the first part the theory of Leray coboundary operators for stratified spaces is developed. These operators are used to construct the cycle and prove the homological relation. In the second part resolution of singularities techniques are applied to study the local geometry near a complete flag of subvarieties.

In type II noncommutative geometry, the geometry on a C∗-algebra A is given by an unbounded Breuer–Fredholm module (ρ,N,D) over A. Here ρ:A→N is a ∗-homomorphism from A to the semi-finite von Neumann algebra N⊂B(H), and D is an unbounded Breuer–Fredholm operator affiliated with N that satisfies certain axioms. Each Breuer–Fredholm module assigns an index to a given element in the K-theory of A. The Breuer–Fredholm index provides a real-valued pairing between the K-homology and the K-theory of A. When N=B(H), a construction of Jaffe-Lesniewski-Osterwalder associates to the module (ρ,N,D) a cocycle in the entire cyclic cohomology group of A for D is theta-summable. The JLO character and the K-theory character intertwine the K-theoretical pairing with the pairing of entire cyclic theory. If (ρ,N,F) is a finitely summable bounded Breuer–Fredholm module, Benameur-Fack defined a cocycle generalizing the Connes's cocycle for bounded Fredholm modules. On the other hand, given a finitely-summable unbounded Breuer–Fredholm module, there is a canonically associated bounded Breuer–Fredholm module. The first main result of this thesis extends the JLO theory to Breuer–Fredholm modules (possibly N does not equal B(H)) in the graded case, and proves that the JLO cocycle and Connes cocycle define the same class in the entire cyclic cohomology of A. This extends a result of Connes-Moscovici for Fredholm modules. An example of an unbounded Breuer–Fredholm module is given by the noncommutative space of G-connections due to Aastrup-Grimstrup-Nest. In their original work, the authors limit their construction to the case that the group G=U(1) or G=SU(2). Another main result of the thesis extends AGN’s construction to any connected compact Lie group G; and generalizes by considering connections defined on sequences of graphs, using limits of spectral triples. Our construction makes it possible to equip the module (ρ,N,D) with a Z_2-grading. The last part of this thesis studies the JLO character of the Breuer–Fredholm module of AGN. The definition of this Breuer–Fredholm module depends on a divergent sequence. A concrete condition on possible perturbations of the sequence ensuring that the resulting JLO class remains invariant is established. The condition implies a certain functoriality of AGN’s construction.

The high pace at which many of the world's energy markets have gradually been opened to competition have generated a significant amount of new financial activity. Both academicians and practitioners alike recently started to develop the tools of energy derivatives pricing/hedging as a quantitative topic of its own. The energy contract structures as well as their underlying asset properties set the energy risk management industry apart from its more standard equity and fixed income counterparts. This thesis naturaly contributes to these broad market developments in participating to the advances of the mathematical tools aiming at a better theory of energy contingent claim pricing/hedging. We propose many realistic two-factor and three-factor models for spot and forward price processes that generalize some well known and standard modeling assumptions. We develop the associated pricing methodologies and propose stable calibration algorithms that motivate the application of the relevant modeling schemes.