## Diploma Thesis

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- Fachbereich Mathematik (20) (remove)

Tropical geometry is a very new mathematical domain. The appearance of
tropical geometry was motivated by its deep relations to other mathematical
branches. These include algebraic geometry, symplectic geometry, complex
analysis, combinatorics and mathematical biology.
In this work we see some more relations between algebraic geometry and
tropical geometry. Our aim is to prove a one-to-one correspondence between
the divisor classes on the moduli space of n-pointed rational stable curves
and the divisors of the moduli space of n-pointed abstract tropical curves.
Thus we state some results of the algebraic case first. In algebraic geometry
these moduli spaces are well understood. In particular, the group of divisor
classes is calculated by S. Keel. We recall the needed results in chapter one.
For the proof of the correspondence we use some results of toric geometry.
Further we want to show an equality of the Chow groups of a special toric
variety and the algebraic moduli space. Thus we state some results of the
toric geometry as well.
This thesis tries to discover some connection between algebraic and tropical
geometry. Thus we also need the corresponding tropical objects to the
algebraic objects. Therefore we give some necessary definitions such as fan,
tropical fan, morphisms between tropical fans, divisors or the topical moduli
space of all n-marked tropical curves. Since we need it, we show that the
tropical moduli space can be embedded as a tropical fan.
After this preparatory work we prove that the group of divisor classes in
v
classical algebraic geometry has it equivalence in tropical geometry. For this
it is useful to give a map from the group of divisor classes of the algebraic
moduli space to the group of divisors of the tropical moduli space. Our aim is
to prove the bijectivity of this map in chapter three. On the way we discover
a deep connection between the algebraic moduli space and the toric variety
given by the tropical fan of the tropical moduli space.

Matrices with the consecutive ones property and interval graphs are important notations in the field of applied mathematics. We give a theoretical picture of them in first part. We present the earliest work in interval graphs and matrices with the consecutive ones property pointing out the close relation between them. We pay attention to Tucker's structure theorem on matrices with the consecutive ones property as an essential step that requires a deep considerations. Later on we concentrate on some recent work characterizing the matrices with the consecutive ones property and matrices related to them in the terms of interval digraphs as the latest and most interesting outlook on our topic. Within this framework we introduce a classiffcation of matrices with consecutive ones property and matrices related to them. We describe the applications of matrices with the consecutive ones property and interval graphs in different fields. We make sure to give a general view of application and their close relation to our studying phenomena. Sometimes we mention algorithms that work in certain fields. In the third part we give a polyhedral approach to matrices with the consecutive ones property. We present the weighted consecutive ones problem and its relation to Tucker's matrices. The constraints of the weighted consecutive ones problem are improved by introducing stronger inequalities, based on the latest theorems on polyhedral aspect of consecutive ones property. Finally we implement a separation algorithm of Oswald and Reinhelt on matrices with the consecutive ones property. We would like to mention that we give a complete proof to the theorems when we consider important within our framework. We prove theorems partially when it is worthwhile to have a closer look, and we omit the proof when there are is only an intersection with our studying phenomena.

The flow of a liquid into an empty channel is simulated. The simulation is based on a recently published model for general fluid/liquid/solid systems which eliminates the shear stress singularity at the moving contact line between the liquid/fluid interface and the solid. This model is carefully analyzed for low Reynolds and Capillary numbers, adapted to the channel inflow problem, and implemented. Very convincing numerical results are presented.

A hub location problem consists of locating p hubs in a network in order to collect and consolidate flow between node pairs. This thesis deals with the uncapacitated single allocation p-hub center problem (USApHCP) as a special type of hub location problem with min max objective function. Using the so-called radius formulation of the problem, the dimension of the polyhedron of USApHCP is derived. The formulation constraints are investigated to find out which of these define facets. Then, three new classes of facet-defining inequalities are derived. Finally, efficient procedures to separate facets in a branch-and-cut algorithm are proposed. The polyhedral analysis of USApHCP is based on a tight relation to the uncapacitated facility location problem (UFL). Hence, many results stated in this thesis also hold for UFL.

Satellite-to-satellite tracking (SST) and satellite gravity gradiometry (SGG), respectively, are two measurement principles in modern satellite geodesy which yield knowledge of the first and second order radial derivative of the earth's gravitational potential at satellite altitude, respectively. A numerical method to compute the gravitational potential on the earth's surface from those observations should be capable of processing huge amounts of observational data. Moreover, it should yield a reconstruction of the gravitational potential at different levels of detail, and it should be possible to reconstruct the gravitational potential from only locally given data. SST and SGG are modeled as ill-posed linear pseudodifferential operator equations with an injective but non-surjective compact operator, which operates between Sobolev spaces of harmonic functions and such ones consisting of their first and second order radial derivatives, respectively. An immediate discretization of the operator equation is obtained by replacing the signal on its right-hand-side either by an interpolating or a smoothing spline which approximates the observational data. Here the noise level and the spatial distribution of the data determine whether spline-interpolation or spline-smoothing is appropriate. The large full linear equation system with positive definite matrix which occurs in the spline-interplation and spline-smoothing problem, respectively, is efficiently solved with the help of the Schwarz alternating algorithm, a domain decomposition method which allows it to split the large linear equation system into several smaller ones which are then solved alernatingly in an iterative procedure. Strongly space-localizing regularization scaling functions and wavelets are used to obtain a multiscale reconstruction of the gravitational potential on the earth's surface. In a numerical experiment the advocated method is successfully applied to reconstruct the earth's gravitational potential from simulated 'exact' and 'error-affected' SGG data on a spherical orbit, using Tikhonov regularization. The applicability of the numerical method is, however, not restricted to data given on a closed orbit but it can also cope with realistic satellite data.

* naive examples which show drawbacks of discrete wavelet transform and windowed Fourier transform; * adaptive partition (with a 'best basis' approach) of speech-like signals by means of local trigonometric bases with orthonormal windows. * extraction of formant-like features from the cosine transform; * further proceedingings for classification of vowels or voiced speech are suggested at the end.

In an undirected graph G we associate costs and weights to each edge. The weight-constrained minimum spanning tree problem is to find a spanning tree of total edge weight at most a given value W and minimum total costs under this restriction. In this thesis a literature overview on this NP-hard problem, theoretical properties concerning the convex hull and the Lagrangian relaxation are given. We present also some in- and exclusion-test for this problem. We apply a ranking algorithm and the method of approximation through decomposition to our problem and design also a new branch and bound scheme. The numerical results show that this new solution approach performs better than the existing algorithms.