Kaiserslautern - Fachbereich Mathematik
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Understanding human crowd behaviour has been an intriguing topic of interdisciplinary research in recent decades. Modelling of crowd dynamics using differential equations is an indispensable approach to unraveling the various complex dynamics involved in such interacting particle systems. Numerical simulation of pedestrian crowd via these mathematical models allows us to study different realistic scenarios beyond the limitations of studies via controlled experiments.
In this thesis, the main objective is to understand and analyse the dynamics in a domain shared by both pedestrians and moving obstacles. We model pedestrian motion by combining the social force concept with the idea of optimal path computation. This leads to a system of ordinary differential equations governing the dynamics of individual pedestrians via the interaction forces (social forces) between them. Additionally, a non-local force term involving the optimal path and desired velocity governs the pedestrian trajectory. The optimal path computation involves solving a time-independent Eikonal equation, which is coupled to the system of ODEs. A hydrodynamic model is developed from this microscopic model via the mean-field limit.
To consider the interaction with moving obstacles in the domain, we model a set of kinematic equations for the obstacle motion. Two kinds of obstacles are considered - "passive", which move in their predefined trajectories and have only a one-way interaction with pedestrians, and "dynamic", which have a feedback interaction with pedestrians and have their trajectories changing dynamically. The coupled model of pedestrians and obstacles is used to discern pedestrian collision avoidance behaviour in different computational scenarios in a long rectangular domain. We observe that pedestrians avoid collisions through route choice strategies that involve changes in speed and path. We extend this model to consider the interaction between pedestrians and vehicular traffic. We appropriately model the interactions of vehicles, following lane traffic, based on the car-following approach. We observe how the deceleration and braking mechanism of vehicles is executed at pedestrian crossings depending on the right of way on the roads.
As a second objective, we study the disease contagion in moving crowds. We consider the influence of the crowd motion in a complex dynamical environment on the course of infection of pedestrians. A hydrodynamic model for multi-group pedestrian flow is derived from the kinetic equations based on a social force model. It is coupled along with an Eikonal equation to a non-local SEIS contagion model for disease spread. Here, apart from the description of local contacts, the influence of contact times has also been modelled. We observe that the nature of the flow and the geometry of the domain lead to changes in density which affect the contact time and, consequently, the rate of spread of infection.
Finally, the social force model is compared to a variable speed based rational behaviour pedestrian model. We derive a hierarchy of the heuristics-based model from microscopic to macroscopic scales and numerically investigate these models in different density scenarios. Various numerical test cases are considered, including uni- and bi-directional flows and scenarios with and without obstacles. We observe that in low-density scenarios, collision avoidance forces arising from the behavioural heuristics give valid results. Whereas in high-density scenarios, repulsive force terms are essential.
The numerical simulations of all the models are carried out using a mesh-free particle method based on least square approximations. The meshfree numerical framework provides an efficient and elegant way to handle complex geometric situations involving boundaries and stationary or moving obstacles.
A geoscientifically relevant wavelet approach is established for the classical (inner) displacement problem corresponding to a regular surface (such as sphere, ellipsoid, actual earth's surface). Basic tools are the limit and jump relations of (linear) elastostatics. Scaling functions and wavelets are formulated within the framework of the vectorial Cauchy-Navier equation. Based on appropriate numerical integration rules a pyramid scheme is developed providing fast wavelet transform (FWT). Finally multiscale deformation analysis is investigated numerically for the case of a spherical boundary.
The focus of this work has been to develop two families of wavelet solvers for the inner displacement boundary-value problem of elastostatics. Our methods are particularly suitable for the deformation analysis corresponding to geoscientifically relevant (regular) boundaries like sphere, ellipsoid or the actual Earth's surface. The first method, a spatial approach to wavelets on a regular (boundary) surface, is established for the classical (inner) displacement problem. Starting from the limit and jump relations of elastostatics we formulate scaling functions and wavelets within the framework of the Cauchy-Navier equation. Based on numerical integration rules a tree algorithm is constructed for fast wavelet computation. This method can be viewed as a first attempt to "short-wavelength modelling", i.e. high resolution of the fine structure of displacement fields. The second technique aims at a suitable wavelet approximation associated to Green's integral representation for the displacement boundary-value problem of elastostatics. The starting points are tensor product kernels defined on Cauchy-Navier vector fields. We come to scaling functions and a spectral approach to wavelets for the boundary-value problems of elastostatics associated to spherical boundaries. Again a tree algorithm which uses a numerical integration rule on bandlimited functions is established to reduce the computational effort. For numerical realization for both methods, multiscale deformation analysis is investigated for the geoscientifically relevant case of a spherical boundary using test examples. Finally, the applicability of our wavelet concepts is shown by considering the deformation analysis of a particular region of the Earth, viz. Nevada, using surface displacements provided by satellite observations. This represents the first step towards practical applications.
In this thesis, we have dealt with two modeling approaches of the credit risk, namely the structural (firm value) and the reduced form. In the former one, the firm value is modeled by a stochastic process and the first hitting time of this stochastic process to a given boundary defines the default time of the firm. In the existing literature, the stochastic process, triggering the firm value, has been generally chosen as a diffusion process. Therefore, on one hand it is possible to obtain closed form solutions for the pricing problems of credit derivatives and on the other hand the optimal capital structure of a firm can be analysed by obtaining closed form solutions of firm's corporate securities such as; equity value, debt value and total firm value, see Leland(1994). We have extended this approach by modeling the firm value as a jump-diffusion process. The choice of the jump-diffusion process was a crucial step to obtain closed form solutions for corporate securities. As a result, we have chosen a jump-diffusion process with double exponentially distributed jump heights, which enabled us to analyse the effects of jump on the optimal capital structure of a firm. In the second part of the thesis, by following the reduced form models, we have assumed that the default is triggered by the first jump of a Cox process. Further, by following Schönbucher(2005), we have modeled the forward default intensity of a firm as a geometric Brownian motion and derived pricing formulas for credit default swap options in a more general setup than the ones in Schönbucher(2005).
In 2006 Jeffrey Achter proved that the distribution of divisor class groups of degree 0 of function fields with a fixed genus and the distribution of eigenspaces in symplectic similitude groups are closely related to each other. Gunter Malle proposed that there should be a similar correspondence between the distribution of class groups of number fields and the distribution of eigenspaces in ceratin matrix groups. Motivated by these results and suggestions we study the distribution of eigenspaces corresponding to the eigenvalue one in some special subgroups of the general linear group over factor rings of rings of integers of number fields and derive some conjectural statements about the distribution of \(p\)-parts of class groups of number fields over a base field \(K_{0}\). Where our main interest lies in the case that \(K_{0}\) contains the \(p\)th roots of unity, because in this situation the \(p\)-parts of class groups seem to behave in an other way like predicted by the popular conjectures of Henri Cohen and Jacques Martinet. In 2010 based on computational data Malle has succeeded in formulating a conjecture in the spirit of Cohen and Martinet for this case. Here using our investigations about the distribution in matrixgroups we generalize the conjecture of Malle to a more abstract level and establish a theoretical backup for these statements.
In this paper we consider the stochastic primitive equation for geophysical flows subject to transport noise and turbulent pressure. Admitting very rough noise terms, the global existence and uniqueness of solutions to this stochastic partial differential equation are proven using stochastic maximal
-regularity, the theory of critical spaces for stochastic evolution equations, and global a priori bounds. Compared to other results in this direction, we do not need any smallness assumption on the transport noise which acts directly on the velocity field and we also allow rougher noise terms. The adaptation to Stratonovich type noise and, more generally, to variable viscosity and/or conductivity are discussed as well.
Abstract
The main theme of this thesis is about Graph Coloring Applications and Defining Sets in Graph Theory.
As in the case of block designs, finding defining sets seems to be difficult problem, and there is not a general conclusion. Hence we confine us here to some special types of graphs like bipartite graphs, complete graphs, etc.
In this work, four new concepts of defining sets are introduced:
• Defining sets for perfect (maximum) matchings
• Defining sets for independent sets
• Defining sets for edge colorings
• Defining set for maximal (maximum) clique
Furthermore, some algorithms to find and construct the defining sets are introduced. A review on some known kinds of defining sets in graph theory is also incorporated, in chapter 2 the basic definitions and some relevant notations used in this work are introduced.
chapter 3 discusses the maximum and perfect matchings and a new concept for a defining set for perfect matching.
Different kinds of graph colorings and their applications are the subject of chapter 4.
Chapter 5 deals with defining sets in graph coloring. New results are discussed along with already existing research results, an algorithm is introduced, which enables to determine a defining set of a graph coloring.
In chapter 6, cliques are discussed. An algorithm for the determination of cliques using their defining sets. Several examples are included.
We present a constructive theory for locally supported approximate identities on the unit ball in \(\mathbb{R}^3\). The uniform convergence of the convolutions of the derived kernels with an arbitrary continuous function \(f\) to \(f\), i.e. the defining property of an approximate identity, is proved. Moreover, an explicit representation for a class of such kernels is given. The original publication is available at www.springerlink.com
We consider a linearized kinetic BGK equation and the associated acoustic system on a network.
Coupling conditions for the macroscopic equations are derived from the kinetic conditions via an asymptotic analysis near the nodes of the network.
This analysis leads to the consideration of a fixpoint problem involving the solutions of kinetic half-space problems.
This work extends the procedure developed in [13], where coupling conditions for a simplified BGK model have been derived.
Numerical comparisons between different coupling conditions
confirm the accuracy of the proposed approximation.
Numerical Algorithms in Algebraic Geometry with Implementation in Computer Algebra System SINGULAR
(2011)
Polynomial systems arise in many applications: robotics, kinematics, chemical kinetics,
computer vision, truss design, geometric modeling, and many others. Many polynomial
systems have solutions sets, called algebraic varieties, having several irreducible
components. A fundamental problem of the numerical algebraic geometry is to decompose
such an algebraic variety into its irreducible components. The witness point sets are
the natural numerical data structure to encode irreducible algebraic varieties.
Sommese, Verschelde and Wampler represented the irreducible algebraic decomposition of
an affine algebraic variety \(X\) as a union of finite disjoint sets \(\cup_{i=0}^{d}W_i=\cup_{i=0}^{d}\left(\cup_{j=1}^{d_i}W_{ij}\right)\) called numerical irreducible decomposition. The \(W_i\) correspond to the pure i-dimensional components, and the \(W_{ij}\) represent the i-dimensional irreducible components. The numerical irreducible decomposition is implemented in BERTINI.
We modify this concept using partially Gröbner bases, triangular sets, local dimension, and
the so-called zero sum relation. We present in the second chapter the corresponding
algorithms and their implementations in SINGULAR. We give some examples and timings,
which show that the modified algorithms are more efficient if the number of variables is not
too large. For a large number of variables BERTINI is more efficient.
Leykin presented an algorithm to compute the embedded components of an algebraic variety
based on the concept of the deflation of an algebraic variety.
Depending on the modified algorithm mentioned above, we will present in the third chapter an
algorithm and its implementation in SINGULAR to compute the embedded components.
The irreducible decomposition of algebraic varieties allows us to formulate in the fourth
chapter some numerical algebraic algorithms.
In the last chapter we present two SINGULAR libraries. The first library is used to compute
the numerical irreducible decomposition and the embedded components of an algebraic variety.
The second library contains the procedures of the algorithms in the last Chapter to test
inclusion, equality of two algebraic varieties, to compute the degree of a pure i-dimensional
component, and the local dimension.