Random variables defined on the natural numbers may often be approximated by Poisson variables. Just as normal approximations may be improved by saddlepoint methods, Poisson approximations may be substantially improved by tilting, expansion, and other related methods. This work will develop and examine the use of these methods, as well as present examples where such methods may be needed. / Ph. D.

In this report we will plunge down in the fascinating world of the generating functions. Generating functions showcase the "power of power series", giving more depth to the word "power" in power series. We start off small to get a good understanding of the generating function and what it does. Also, off course, explaining why it works and why we can do some of the things we do with them. We will see alot of examples throughout the text that helps the reader to grasp the mathematical object that is the generating function. We will look at several kinds of generating functions, the main focus when we establish our understanding of these will be the "ordinary power series" generating function ("ops") that we discuss before moving on to the "exponential generating function" ("egf"). During our discussion on ops we will see a "first time in literature" derivation of the generating function for a recurrence relation regarding "branched coverings". After finishing the discussion regarding egf we move on the Hermite polynomials and show how we derive their generating function. Which is a generating function that generates functions. Lastly we will have a quick look at the "moment generating function".

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A combination of some well known nonparametric tests to detect trend in location is considered. Simulation results show that the power of this combination is remarkably increased. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

Consider a sequence (Formula presented.) of i.i.d. uniform random variables taking values in the alphabet set {1, 2,…, d}. A k-superpattern is a realization of (Formula presented.) that contains, as an embedded subsequence, each of the non-order-isomorphic subpatterns of length k. We focus on the (non-trivial) case of d = k = 3 and study the waiting time distribution of (Formula presented.). Our restricted set-up leads to proofs that are very combinatorial in nature, since we are essentially conducting a string analysis.

<p>Polya’s theorem can be used to enumerate objects under permutation groups. Using grouptheory, combinatorics and some examples, Polya’s theorem and Burnside’s lemma arederived. The examples used are a square, pentagon, hexagon and heptagon under theirrespective dihedral groups. Generalization using more permutations and applications tograph theory.Using Polya’s Enumeration theorem, Harary and Palmer [5] give a function whichgives the number of unlabeled graphs n vertices and m edges. We present their work andthe necessary background knowledge.</p>

The integrability of an m-component system of hydrodynamic type, Ut = v(u)ux, by the generalized hodograph method requires the diagonalizability of the m x m matrix v(u). The diagonalizability is known to be equivalent to the vanishing of the corresponding Haantjes tensor. This idea is applied to hydrodynamic chains - infinite-component systems of hydrodynamic type for which the 00 x 00 matrix v(u) is 'sufficiently sparse'. For such 'sparse' systems the Haantjes tensor is well-defined, and the calculation of its components involves only a finite number of summations. The calculation of the Haantjes tensor is done by using Mathematica to perform symbolic calculations. Certain conservative and Hamiltonian hydrodynamic chains are classified by setting Haantjes tensor equal to zero and solving the resulting system of equations. It is shown that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability of such sysyems. In the cases of the Hamiltonian hydrodynamic chains we were able to first construct one extra conservation law and later a generating function for conservation laws, thus establishing the integrability.

Dynamical maps for magnetic components are fundamental to studies of beam dynamics in accelerators. However, it is usually not possible to write down maps in closed form for anything other than simplified models of standard accelerator magnets. In the work presented here, the magnetic field is expressed in analytical form obtained from fitting Fourier series to a 3D numerical solution of Maxwell’s equations. Dynamical maps are computed for a particle moving through this field by applying a second order (with the paraxial approximation) explicit symplectic integrator. These techniques are used to study the beam dynamics in the first non-scaling FFAG ever built, EMMA, especially challenging regarding the validity of the paraxial approximation for the large excursion of particle trajectories. The EMMA lattice has four degrees of freedom (strength and transverse position of each of the two quadrupoles in each periodic cell). Dynamical maps, computed for a set of lattice configurations, may be efficiently used to predict the dynamics in any lattice configuration. We interpolate the coefficients of the generating function for the given configuration, ensuring the symplecticity of the solution. An optimisation routine uses this tool to look for a lattice defined by four constraints on the time of flight at different beam energies. This provides a way to determine the tuning of the lattice required to produce a desired variation of time of flight with energy, which is one of the key characteristics for beam acceleration in EMMA. These tools are then benchmarked against data from the recent EMMA commissioning.

This research investigates a batch arrival queueing system with a Bernoulli scheduled vacation and random system breakdowns. It is assumed that the repair process does not start immediately after the breakdown. Consequently there maybe a delay in starting repairs. After every service completion the server may go on an optional vacation. When the original vacation is completed the server has the option to go on an extended vacation. It is assumed that the system is equipped with a stand-by server to serve the customers during the vacation period of the main server as well as during the repair process. The service times, vacation times, repair times, delay times and extended vacation times are assumed to follow different general distributions while the breakdown times and the service times of the stand-by server follow an exponential distribution. By introducing a supplementary variable we are able to obtain steady state results in an explicit closed form in terms of the probability generating functions. Some important performance measures including; the average length of the queue, the average number of customers in the system, the mean response time, and the value of the traffic intensity are presented. The professional MathCad 2001 software has been used to illustrate the numerical results in this study.

Enumeration of pattern avoiding compositions of numbers Abstract The aim of this work was to find some new results for 3-regular compositions, i.e., for those compositions which avoid the set of patterns {121, 212, 11}. Those compositions can be regarded as a generalization of Carlitz composition. Based on the generating function of compositions avoiding the set of patterns {121, 11} and {212, 11} we derive an upper bound for the coefficients of the power series of the generating function of 3-regular compositions. Using the theory of finite automata we derive its lower bound. We develop this result further by defining 3-block compositions. For the generating function of 3-regular compositions we prove a recursive ralation. Besides that we also compute the generating function of compositions avoiding the set of patterns {312, 321} whose parts are in the set [d]. In the last section we prove that the generating function of Carlitz compositions is transcendental.