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In the literature, there are at least two equivalent two-factor Gaussian models for the instantaneous short rate. These are the original two-factor Hull White model (see [3]) and the G2++ one by Brigo and Mercurio (see [1]). Both these models first specify a time homogeneous two-factor short rate dynamics and then by adding a deterministic shift function '(·) fit exactly the initial term structure of interest rates. However, the obtained results are rather clumsy and not intuitive which means that a special care has to be taken for their correct numerical implementation.
Selfish Bin Coloring
(2009)
We introduce a new game, the so-called bin coloring game, in which selfish players control colored items and each player aims at packing its item into a bin with as few different colors as possible. We establish the existence of Nash and strong as well as weakly and strictly Pareto optimal equilibria in these games in the cases of capacitated and uncapacitated bins. For both kinds of games we determine the prices of anarchy and stability concerning those four equilibrium concepts. Furthermore, we show that extreme Nash equilibria, those with minimal or maximal number of colors in a bin, can be found in time polynomial in the number of items for the uncapcitated case.
This study deals with the optimal control problems of the glass tube drawing processes where the aim is to control the cross-sectional area (circular) of the tube by using the adjoint variable approach. The process of tube drawing is modeled by four coupled nonlinear partial differential equations. These equations are derived by the axisymmetric Stokes equations and the energy equation by using the approach based on asymptotic expansions with inverse aspect ratio as small parameter. Existence and uniqueness of the solutions of stationary isothermal model is also proved. By defining the cost functional, we formulated the optimal control problem. Then Lagrange functional associated with minimization problem is introduced and the first and the second order optimality conditions are derived. We also proved the existence and uniqueness of the solutions of the stationary isothermal model. We implemented the optimization algorithms based on the steepest descent, nonlinear conjugate gradient, BFGS, and Newton approaches. In the Newton method, CG iterations are introduced to solve the Newton equation. Numerical results are obtained for two different cases. In the first case, the cross-sectional area for the entire time domain is controlled and in the second case, the area at the final time is controlled. We also compared the performance of the optimization algorithms in terms of the solution iterations, functional evaluations and the computation time.
Product development with end-user integration is not an end in itself but a logical necessity due to divergent types of knowledge of the user and the developer of a product. While the user is an expert in regard to the product’s usage the developer is an expert in the product’s construction and functioning. For the development of high-end products both types of expertises were a prerequisite at all times. The efficient and throughout integration of the user’s perspective into existing product development approaches is the core of user-centred product development. Activities that are the basic ingredient of just any user-centred development approach can be roughly categorized into analysis, design and evaluation activities. Research and practice prove the early integration of real end-users within those activities to add significant and sustainable value to product innovation. The instrumental, methodological and procedural impact of globalization tendencies, on modern user-centred product development in particular, is the primary research focus of the field of cross-cultural user-centred product development. This research aims at the further advancement of the methodological foundations of cross-cultural user centred product development approaches based on a stabile and profound theoretical basis. Primary research objects are established user-analysis methodologies, which are mainly based on Western concepts and theories, and their applicability in disparate cultural contexts of the Far East (China and Korea in particular). For facilitating the adaptation of abstract method characteristics to the situational context of method application as foundation of cross-cultural methodological advancement, a model of method localization was developed. In alignment with internationalization and localization activities within product development processes, a framework for localizing user-centred methodologies was developed. Equivalent to internationalization activities of real product development, the abstraction of method traits from specific methodologies is a necessity in a first step. Methodological adaptation with the primary objective of optimizing situational application of a methodology is to be done in a second step – the step of method-localization. This model of method localization and its underlying theories and principles were tested within an extensive empirical study in Germany, China and Korea. Within this study the applicability of six distinct user-centred product development methodologies, each with its very own profile of abstract method traits, was tested with 248 participants in total. Results clearly back the basic hypothesis of method-localization, i.e. that the application of a user-centred methodology rises and falls with the alignment of its characteristic traits with the cross-cultural application context. Beyond, applicability-influencing factors identified within this study could be proven to be valid indicators of adaptation-necessities and –potentials of user-centred product development methodologies.
This dissertation deals with two main subjects. Both are strongly related to boundary problems for the Poisson equation and the Laplace equation, respectively. The oblique boundary problem of potential theory as well as the limit formulae and jump relations of potential theory are investigated. We divide this abstract into two parts and start with the oblique boundary problem. Here we prove existence and uniqueness results for solutions to the outer oblique boundary problem for the Poisson equation under very weak assumptions on boundary, coefficients and inhomogeneities. Main tools are the Kelvin transformation and the solution operator for the regular inner problem, provided in my diploma thesis. Moreover we prove regularization results for the weak solutions of both, the inner and the outer problem. We investigate the non-admissible direction for the oblique vector field, state results with stochastic inhomogeneities and provide a Ritz-Galerkin approximation. Finally we show that the results are applicable to problems from Geomathematics. Now we come to the limit formulae. There we combine the modern theory of Sobolev spaces with the classical theory of limit formulae and jump relations of potential theory. The convergence in Lebesgue spaces for integrable functions is already treated in literature. The achievement of this dissertation is this convergence for the weak derivatives of higher orders. Also the layer functions are elements of Sobolev spaces and the surface is a two dimensional suitable smooth submanifold in the three dimensional space. We are considering the potential of the single layer, the potential of the double layer and their first order normal derivatives. Main tool in the proof in Sobolev norm is the uniform convergence of the tangential derivatives, which is proved with help of some results taken from literature. Additionally, we need a result about the limit formulae in the Lebesgue spaces, which is also taken from literature, and a reduction result for normal derivatives of harmonic functions. Moreover we prove the convergence in the Hölder spaces. Finally we give an application of the limit formulae and jump relations. We generalize a known density of several function systems from Geomathematics in the Lebesgue spaces of square integrable measureable functions, to density in Sobolev spaces, based on the results proved before. Therefore we have prove the limit formula of the single layer potential in dual spaces of Soboelv spaces, where also the layer function is an element of such a distribution space.
In its rather short history robotic research has come a long way in the half century since it started to exist as a noticeable scientic eld. Due to its roots in engineering, computer science, mathematics, and several other 'classical' scientic branches,a grand diversity of methodologies and approaches existed from the very beginning. Hence, the researchers in this eld are in particular used to adopting ideas that originate in other elds. As a fairly logical consequence of this, scientists tended to biology during the 1970s in order to nd approaches that are ideally adapted to the conditions of our natural environment. Doing so allows for introducing principles to robotics that have already shown their great potential by prevailing in a tough evolutionary selection process for millions of years. The variety of these approaches spans from efficient locomotion, to sensor processing methodologies and all the way to control architectures. Thus, the full spectrum of challenges for autonomous interaction with the surroundings while pursuing a task can be covered by such means. A feature that has proven to be amongst the most challenging to recreate is the human ability of biped locomotion. This is mainly caused by the fact that walking,running and so on are highly complex processes involving the need for energy efficient actuation, sophisticated control architectures and algorithms, and an elaborate mechanical design while at the same time posting restrictions concerning stability and weight. However, it is of special interest since our environment is favoring this specic kind of locomotion and thus promises to open up an enormous potential if mastered. More than the mere scientic interest, it is the fascination of understanding and recreating parts of oneself that drives the ongoing eorts in this area of research. The fact that this is not at all an easy task to tackle is not only caused by the highly dynamical processes but also has its roots in the challenging design process. That is because it cannot be limited to just one aspect like e.g. the control architecture, actuation, sensors, or mechanical design alone. Each aspect has to be incorporated into a sound general concept in order to allow for a successful outcome in the end. Since control is in this context inseparably coupled with the mechanics of the system, both has to be dealt with here.
In this article we combine the modern theory of Sobolev spaces with the classical theory of limit formulae and jump relations of potential theory. Also other authors proved the convergence in Lebesgue spaces for integrable functions. The achievement of this paper is the L2 convergence for the weak derivatives of higher orders. Also the layer functions F are elements of Sobolev spaces and a two dimensional suitable smooth submanifold in R3, called regular Cm-surface. We are considering the potential of the single layer, the potential of the double layer as well as their first order normal derivatives. Main tool is the convergence in Cm-norm which is proved with help of some results taken from [14]. Additionally, we need a result about the limit formulae in L2-norm, which can be found in [16], and a reduction result which we took from [19]. Moreover we prove the convergence in the Hölder spaces Cm,alpha. Finally, we give an application of the limit formulae and jump relations to Geomathematics. We generalize a density results, see e.g. [11], from L2 to Hm,2. For it we prove the limit formula for U1 in (Hm,2)' also.
Using a stereographical projection to the plane we construct an O(N log(N)) algorithm to approximate scattered data in N points by orthogonal, compactly supported wavelets on the surface of a 2-sphere or a local subset of it. In fact, the sphere is not treated all at once, but is split into subdomains whose results are combined afterwards. After choosing the center of the area of interest the scattered data points are mapped from the sphere to the tangential plane through that point. By combining a k-nearest neighbor search algorithm and the two dimensional fast wavelet transform a fast approximation of the data is computed and mapped back to the sphere. The algorithm is tested with nearly 1 million data points and yields an approximation with 0.35% relative errors in roughly 2 minutes on a standard computer using our MATLAB implementation. The method is very flexible and allows the application of the full range of two dimensional wavelets.
This thesis is devoted to the study of tropical curves with emphasis on their enumerative geometry. Major results include a conceptual proof of the fact that the number of rational tropical plane curves interpolating an appropriate number of general points is independent of the choice of points, the computation of intersection products of Psi-classes on the moduli space of rational tropical curves, a computation of the number of tropical elliptic plane curves of given degree and fixed tropical j-invariant as well as a tropical analogue of the Riemann-Roch theorem for algebraic curves. The result are obtained in joint work with Hannah Markwig and/or Andreas Gathmann.