Numerical and combinatorial applications of generalized Appell polynomials

This thesis studies properties and applications of different generalized Appell polynomials in the framework of Clifford analysis. As an example of 3D-quasi-conformal mappings realized by generalized Appell polynomials, an analogue of the complex Joukowski transformation of order two is introduced. The consideration of a Pascal n-simplex with hypercomplex entries allows stressing the combinatorial relevance of hypercomplex Appell polynomials. The concept of totally regular variables and its relation to generalized Appell polynomials leads to the construction of new bases for the space of homogeneous holomorphic polynomials whose elements are all isomorphic to the integer powers of the complex variable. For this reason, such polynomials are called pseudo-complex powers (PCP). Different variants of them are subject of a detailed investigation. Special attention is paid to the numerical aspects of PCP. An efficient algorithm based on complex arithmetic is proposed for their implementation. In this context a brief survey on numerical methods for inverting Vandermonde matrices is presented and a modified algorithm is proposed which illustrates advantages of a special type of PCP. Finally, combinatorial applications of generalized Appell polynomials are emphasized. The explicit expression of the coefficients of a particular type of Appell polynomials and their relation to a Pascal simplex with hypercomplex entries are derived. The comparison of two types of 3D Appell polynomials leads to the detection of new trigonometric summation formulas and combinatorial identities of Riordan-Sofo type characterized by their expression in terms of central binomial coefficients.

Advanced energy efficient coordinated multi-point transmission for OFDMA

The ever-growing energy consumption in mobile networks stimulated by the expected growth in data tra ffic has provided the impetus for mobile operators to refocus network design, planning and deployment towards reducing the cost per bit, whilst at the same time providing a signifi cant step towards reducing their operational expenditure. As a step towards incorporating cost-eff ective mobile system, 3GPP LTE-Advanced has adopted the coordinated multi-point (CoMP) transmission technique due to its ability to mitigate and manage inter-cell interference (ICI). Using CoMP the cell average and cell edge throughput are boosted. However, there is room for reducing energy consumption further by exploiting the inherent exibility of dynamic resource allocation protocols. To this end packet scheduler plays the central role in determining the overall performance of the 3GPP longterm evolution (LTE) based on packet-switching operation and provide a potential research playground for optimizing energy consumption in future networks. In this thesis we investigate the baseline performance for down link CoMP using traditional scheduling approaches, and subsequently go beyond and propose novel energy e fficient scheduling (EES) strategies that can achieve power-e fficient transmission to the UEs whilst enabling both system energy effi ciency gain and fairness improvement. However, ICI can still be prominent when multiple nodes use common resources with di fferent power levels inside the cell, as in the so called heterogeneous networks (Het- Net) environment. HetNets are comprised of two or more tiers of cells. The rst, or higher tier, is a traditional deployment of cell sites, often referred to in this context as macrocells. The lower tiers are termed small cells, and can appear as microcell, picocells or femtocells. The HetNet has attracted signiffi cant interest by key manufacturers as one of the enablers for high speed data at low cost. Research until now has revealed several key hurdles that must be overcome before HetNets can achieve their full potential: bottlenecks in the backhaul must be alleviated, as well as their seamless interworking with CoMP. In this thesis we explore exactly the latter hurdle, and present innovative ideas on advancing CoMP to work in synergy with HetNet deployment, complemented by a novel resource allocation policy for HetNet tighter interference management. As system level simulator has been used to analyze the proposed algorithm/protocols, and results have concluded that up to 20% energy gain can be observed.

Efficient domination through eigenvalues

The paper begins with a new characterization of (k,τ)(k,τ)-regular sets. Then, using this result as well as the theory of star complements, we derive a simplex-like algorithm for determining whether or not a graph contains a (0,τ)(0,τ)-regular set. When τ=1τ=1, this algorithm can be applied to solve the efficient dominating set problem which is known to be NP-complete. If −1−1 is not an eigenvalue of the adjacency matrix of the graph, this particular algorithm runs in polynomial time. However, although it does not work in polynomial time in general, we report on its successful application to a vast set of randomly generated graphs.