## 510 Mathematik

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We compute exact values respectively bounds of dissimilarity/distinguishability measures–in the sense of the Kullback-Leibler information distance (relative entropy) and some transforms of more general power divergences and Renyi divergences–between two competing discrete-time Galton-Watson branching processes with immigration GWI for which the offspring as well as the immigration (importation) is arbitrarily Poisson-distributed; especially, we allow for arbitrary type of extinction-concerning criticality and thus for non-stationarity. We apply this to optimal decision making in the context of the spread of potentially pandemic infectious diseases (such as e.g., the current COVID-19 pandemic), e.g., covering different levels of dangerousness and different kinds of intervention/mitigation strategies. Asymptotic distinguishability behaviour and diffusion limits are investigated, too.

This paper develops the generalized causality algorithm and applies it to a multitude of data from the fields of economics and finance. Specifically, our parameter-free algorithm efficiently determines the optimal non-linear mapping and identifies varying lead–lag effects between two given time series. This procedure allows an elastic adjustment of the time axis to find similar but phase-shifted sequences—structural breaks in their relationship are also captured. A large-scale simulation study validates the outperformance in the vast majority of parameter constellations in terms of efficiency, robustness, and feasibility. Finally, the presented methodology is applied to real data from the areas of macroeconomics, finance, and metal. Highest similarity show the pairs of gross domestic product and consumer price index (macroeconomics), S&P 500 index and Deutscher Aktienindex (finance), as well as gold and silver (metal). In addition, the algorithm takes full use of its flexibility and identifies both various structural breaks and regime patterns over time, which are (partly) well documented in the literature.

Boosting algorithms were originally developed for machine learning but were later adapted to estimate statistical models—offering various practical advantages such as automated variable selection and implicit regularization of effect estimates. The interpretation of the resulting models, however, remains the same as if they had been fitted by classical methods. Boosting, hence, allows to use an advanced machine learning scheme to estimate various types of statistical models. This tutorial aims to highlight how boosting can be used for semi-parametric modelling, what practical implications follow from the design of the algorithm and what kind of drawbacks data analysts have to expect. We illustrate the application of boosting in the analysis of a stunting score from children in India and a high-dimensional dataset of tumour DNA to develop a biomarker for the occurrence of metastases in breast cancer patients.

The aim of this paper is to investigate the use of a Landweber-type method involving the Shannon entropy for the regularization of linear ill-posed problems. We derive a closed form solution for the iterates and analyze their convergence behaviour both in a case of reconstructing general nonnegative unknowns as well as for the sake of recovering probability distributions. Moreover, we discuss several variants of the algorithm and relations to other methods in the literature. The effectiveness of the approach is studied numerically in several examples.

Soil functions are closely related to the structure of soil microaggregates. Yet, the mechanisms controlling the establishment of soil structure are diverse and partly unknown. Hence, the understanding of soil processes and functions requires the connection of the concepts on the formation and consolidation of soil structural elements across scales that are hard to observe experimentally. At the bottom level, the dynamics of microaggregate development and restructuring build the basis for transport phenomena at the continuum scale. By modeling the interactions of specific minerals and/or organic matter, we aim to identify the mechanisms that control the evolution of structure and establishment of stationary aggregate properties. We present a mechanistic framework based on a cellular automaton model to simulate the interplay between the prototypic building units of soil microaggregates quartz, goethite, and illite subject to attractive and repulsive electrostatic interaction forces. The resulting structures are quantified by morphological measures. We investigated shielding effects due to charge neutralization and the aggregate growth rate in response to the net system charge. We found that the fraction as well as the size of the interacting oppositely charged constituents control the size, shape, and amount of occurring aggregates. Furthermore, the concentration in terms of the liquid solid ratio has been shown to increase the aggregation rate. We further adopt the model for an assessment of the temporal evolution of aggregate formation due to successive formation of particle dimers at early stages in comparison to higher order aggregates at later stages. With that we show the effect of composition, charge, size ratio, time, and concentration on microaggregate formation by the application of a mechanistic model which also provides predictions for soil aggregation behavior in case an observation is inhibited by experimental limitations.

The hepatitis C virus (HCV) RNA replication cycle is a dynamic intracellular process occurring in three-dimensional space (3D), which is difficult both to capture experimentally and to visualize conceptually. HCV-generated replication factories are housed within virus-induced intracellular structures termed membranous webs (MW), which are derived from the Endoplasmatic Reticulum (ER). Recently, we published 3D spatiotemporal resolved diffusion–reaction models of the HCV RNA replication cycle by means of surface partial differential equation (sPDE) descriptions. We distinguished between the basic components of the HCV RNA replication cycle, namely HCV RNA, non-structural viral proteins (NSPs), and a host factor. In particular, we evaluated the sPDE models upon realistic reconstructed intracellular compartments (ER/MW). In this paper, we propose a significant extension of the model based upon two additional parameters: different aggregate states of HCV RNA and NSPs, and population dynamics inspired diffusion and reaction coefficients instead of multilinear ones. The combination of both aspects enables realistic modeling of viral replication at all scales. Specifically, we describe a replication complex state consisting of HCV RNA together with a defined amount of NSPs. As a result of the combination of spatial resolution and different aggregate states, the new model mimics a cis requirement for HCV RNA replication. We used heuristic parameters for our simulations, which were run only on a subsection of the ER. Nevertheless, this was sufficient to allow the fitting of core aspects of virus reproduction, at least qualitatively. Our findings should help stimulate new model approaches and experimental directions for virology.

In this article we study the connection of fractional Brownian motion, representation theory and reflection positivity in quantum physics. We introduce and study reflection positivity for affine isometric actions of a Lie group on a Hilbert space E and show in particular that fractional Brownian motion for Hurst index 0<H≤1/2 is reflection positive and leads via reflection positivity to an infinite dimensional Hilbert space if 0<H<1/2 . We also study projective invariance of fractional Brownian motion and relate this to the complementary series representations of GL2(R) . We relate this to a measure preserving action on a Gaussian L2 -Hilbert space L2(E) .

This thesis deals with mathematical modeling, analysis, and numerical realization of microaggregates in soils. These microaggregates have the size of a few hundred micrometers and can be understood as the fundamental building units of soil. Thus, understanding their dynamically evolving, three-dimensional structure is crucial for modeling and interpreting many soil parameters such as diffusivities and flow paths that come into play in CO2-sequestration or oil recovery scenarios.
Among others, the following aspects of the formation of microaggregates should be incorporated into a mathematical model and investigated in more detail: the spatial heterogeneity of the temporally evolving structure of microaggregates and the different processes that take place on different scales — temporal and spatial — within the so-called micro-scale itself. This work aims at formulating a process-based pore-scale model, where all chemical species are measured in concentrations. That is, we have a continuous model for reactive transport mainly in terms of partial differential equations (PDEs) with algebraic constraints. This continuous model is defined on a discrete and discretely moving domain whose geometry changes according to the rules of a cellular automaton method (CAM). These rules describe the restructuring of the porous matrix, growth and decay of biomass, and the resulting topological changes of a wetting fluid and a gas phase. The cellular automaton rules additionally imply stochastic aspects that are important on the pore-scale.
Moreover, effects and knowledge deduced from the model are transfered to scales which are more relevant for applications. The quality of these averaged models is of general interest, since simulations for the field-scale that resolve the pore-scale are not applicable for economical reasons. Thus, this book compares parameterizations of diffusivities with mathematically rigorous results and gives suggestions to improve the formulas that can be found in the literature.
The discrete movement of the microaggregates’ geometry at the micro-scale poses mathematical problems. The following question arises: Can the averaged quantities deduced from the pore-scale really be used for models on other scales or are the impacts of the artificial temporal jumps too detrimental for the solutions on other scales to be accurate? In the following, this problem is also dealt with, and the reliability of the obtained parameters is underlined.
Last but not least, it is imperative to apply a proper numerical method to implement the model in silico. The local discontinuous Galerkin (LDG) method seems to be suitable for this task, since it is locally mass-conservative and is stable for discontinuous data — that might, for example, originate from the discrete movement of the geometry or from the sharp boundaries between the different phases. Additionally, this method has no problems with complicated transfer conditions. These aspects are demonstrated in a mathematically rigorous way, and the method is improved upon by reducing the linear system of equations resulting from the discretization. This is a real enhancement, since it does not diminish the order of convergence but decreases the computational costs.

We study the existence and uniqueness of equilibria for perfectly competitive markets in capacitated transport networks. The model under consideration is rather general so that it captures basic aspects of related models in, e.g., gas or electricity networks. We formulate the market equilibrium model as a mixed complementarity problem and show the equivalence to a welfare maximization problem. Using the latter we prove uniqueness of the resulting equilibrium for piecewise linear and symmetric transport costs under additional mild assumptions. Moreover, we show the necessity of these assumptions by illustrating examples that possess multiple solutions if our assumptions are violated.

This work presents a novel combining method called ‘orthogonalizing ORing ◯∨’ which enables the building of the union of two conjunctions whereby the result consists of disjointed conjunctions. The advantage of this novel technique is that the results are already presented in an orthogonal form which has a significant advantage for further calculations as the Boolean Differential Calculus. By orthogonalizing ORing two calculation steps - building the disjunction and the subsequent orthogonalization of two conjunctions - are performed in one step. Postulates, axioms and rules for this linking technique are also defined which have to be considered getting correct results. Additionally, a novel equation, based on orthogonalizing ORing, is set up for orthogonalization of every Boolean function of disjunctive form. Thus, disjointed Sum of Products can be easily calculated in a mathematical way by this equation.

It is well-known that the full unitary group $\U(\SH)$ over a complex infinite-dimensional separable Hilbert space $\SH$ has Property FH. This means that the first-order cohomology spaces $H^1$ associated to its continuous unitary representations are trivial. This thesis sheds light on several important unitary subgroups of $\U(\SH)$, proving that they do not have Property FH. This is the direct limit group $\U(\infty)$ and its Banach-completions $\U_p(\SH)$, containing $\U(\infty)$ as a dense subgroup. The direct limit group is the countable union of the compact finite-dimensional unitary groups $\U(n)$ (for a positive integer $n$) viewed as increasing subgroups of $\U(\SH)$. A unitary operator $g$ belongs to $\U_p(\SH)$ if and only if the difference $g-\1$ is an operator of $p$th Schatten class (where $p$ is any positive real number greater or equal to $1$). The core of our thesis is the study of unitary highest weight representations of these infinite-dimensional unitary groups and the precise determination of those highest weight representations having a nontrivial first order cohomology space $H^1$. The criteria are simple and depend only on the highest weight of the representation. Thus, we are able to find, for each of the above unitary subgroups, infinitely many nonisomorphic irreducible unitary representations with infinite-dimensional first order 1-cohomology space $H^1$. This is a remarkable observation since it contrasts the well-known classical result that, for each finite-dimensional (connected) semisimple Lie group, there exist at most finitely many (nonisomorphic) continuous irreducible unitary representations with nontrivial 1-cohomology space $H^1$ and, in this case, the space $H^1$ is always finite-dimensional.

This thesis is concerned with the efficient and accurate simulation and optimization of linear Timoshenko beam networks subjected to external loads.
For this we develop a solution scheme, which is based on so-called analytic ansatz-functions known to provide analytic solutions
for the unknowns of a single beam subjected to given boundary data. In the present work we prove that this concept can be extended to a network of beams
which enables the analytic calculation of the unique deformation of the whole network.
Moreover, we show that this approach is equivalent to a finite element method where only one finite element with a particular shape function per
beam is required. With this we can provide the analytic solution on each node of the network, from which the state of the whole network can be uniquely determined - no further spatial discretization is needed.
Based on this analytic solution scheme for the simulation problem, we investigate a number of problems from structural optimization, more precisely
topology optimization, material and multi-material optimization, cross section optimization, geometry optimization and simultaneous versions of the before mentioned.
Hereby we provide the problem formulation, analytic formulae for first order derivatives, an implementation and numerical examples for each type of optimization model.
In the next part of this thesis we develop a novel approach for geometry, cross section and material optimization for a system of high-pressure steam pipes in a power plant.
This industrial application is efficiently modeled by including a special parametrization for the geometry of pipes.
Then we formulate
an optimization problem with an objective function considering monetary costs and a large number of constraints.
Moreover, we derive analytic expressions for the first order derivatives of the problem and provide an implementation. We demonstrate the
capabilities of our program with numerical tests for a real-world instance.

This paper presents a new approach for the determination of the wind speed distribution based on wind speed data. This approach is based on the fact that, in general, wind speed distributions restricted to seasons of year or months are different. Therefore, instead of one Weibull density function, a convex combination of Weibull density functions is considered for a calendar year. This model improves the maximum likelihood of the estimated wind speed distribution. Numerical results including a Kolmogorov–Smirnov test are given for a site at Jamaica. Numerical comparisons are carried out for different sites and various known methods for the estimation of the wind speed distribution.

In an example for the determination of publicly registered land values in urban building conflict situations an interval value duality and regression are applied. The result is incumbent upon a new interval value optimum – beginning with non-ordered point sets of publicly registered German land values for residential land. These two mathematical approaches with set-valued order relations describe the relation to the socio-economy in an urban building conflict situation.

This thesis deals with properties of singular orbits in multi-body-systems.
Here, systems are examined, in which the dynamics defining potential solely depends on the relative positions of the particles. Celestial mechanics, which is the motion of any number of particles due to Newton's law of universal gravitation, can be regarded as a standard example. With this in mind, the class of so-called long-ranged and moderated potentials are introduced, which among others also includes the Coulomb field of electrostatics. In these systems, the flow which is induced by the set of differential equations of motion, is generally not complete, and the non-global solutions are called singular. The goal is to show their improbability in due situations, that is to show that the set of initial conditions leading to non-global solution is a null-set in the sense of Lebesgue.
An important distinction of singular orbits are collision and non-collision orbits: the former are defined by the property that all particles have definitive limit points as time approaches the singularity. Previous results in the field of celestial mechanics show the improbability of collisions in general, and the improbability of non-collision singularities in the four-body-problem. Some generalizations to a wider class of potentials is possible with the techniques imposed there, however, certain restrictions like on the homogeneity of the force field cannot be dropped, and even within these restrictions optimal bounds cannot be shown.
Hence, the provision of a new technique seems to be necessary, in order to generalize the results to a wider class of systems, which is done within this work...

In this paper we present a chain of mathematical models that enables the numerical simulation of the airlay process and the investigation of the resulting nonwoven material by means of virtual tensile strength tests. The models range from a highly turbulent dilute fiber suspension flow to stochastic surrogates for fiber lay-down and web formation and further to Cosserat networks with effective material laws. Crucial is the consistent mathematical mapping between the parameters of the process and the material. We illustrate the applicability of the model chain for an industrial scenario, regarding data from computer tomography and experiments. By this proof of concept we show the feasibility of future simulation-based process design and material optimization which are long-term objectives in the technical textile industry.

Mathematical modeling of biochemical pathways is an important resource in Synthetic Biology, as the predictive power of simulating synthetic pathways represents an important step in the design of synthetic metabolons. In this paper, we are concerned with the mathematical modeling, simulation, and optimization of metabolic processes in biochemical microreactors able to carry out enzymatic reactions and to exchange metabolites with their surrounding medium. The results of the reported modeling approach are incorporated in the design of the first microreactor prototypes that are under construction. These microreactors consist of compartments separated by membranes carrying specific transporters for the input of substrates and export of products. Inside the compartments of the reactor multienzyme complexes assembled on nano-beads by peptide adapters are used to carry out metabolic reactions. The spatially resolved mathematical model describing the ongoing processes consists of a system of diffusion equations together with boundary and initial conditions. The boundary conditions model the exchange of metabolites with the neighboring compartments and the reactions at the surface of the nano-beads carrying the multienzyme complexes. Efficient and accurate approaches for numerical simulation of the mathematical model and for optimal design of the microreactor are developed. As a proof-of-concept scenario, a synthetic pathway for the conversion of sucrose to glucose-6-phosphate (G6P) was chosen. In this context, the mathematical model is employed to compute the spatio-temporal distributions of the metabolite concentrations, as well as application relevant quantities like the outflow rate of G6P. These computations are performed for different scenarios, where the number of beads as well as their loading capacity are varied. The computed metabolite distributions show spatial patterns, which differ for different experimental arrangements. Furthermore, the total output of G6P increases for scenarios where microcompartimentation of enzymes occurs. These results show that spatially resolved models are needed in the description of the conversion processes. Finally, the enzyme stoichiometry on the nano-beads is determined, which maximizes the production of glucose-6-phosphate.

We use the example of hot rolling to develop a comprehensive optimization approach that is based on a mathematical model of the underlying manufacturing process. More precisely, we study an optimal control problem that is designed to minimize cutting scrap while taking industrial specifications and technical limitations into account.
It is well known that the associated control-to-observation map is non-differentiable due to changes of state resulting from elasto-viscoplastic material behavior and frictional contact. However, we still want to apply gradient-based methods to solve the optimal control problem and therefore have to compute derivatives of cost functional and constraints. To resolve this issue, we first regularize all non-differentiabilities before computing sensitivity information via direct differentiation.
We moreover present solution techniques for the regularized problem and discuss numerical results for real-world examples.

Let K→X be a smooth Lie algebra bundle over a σ-compact manifold X whose typical fiber is the compact Lie algebra k. We give a complete description of the irreducible bounded (i.e., norm continuous) unitary representations of the Fréchet–Lie algebra Γ(K) of all smooth sections of K, and of the LF-Lie algebra Γc(K) of compactly supported smooth sections. For Γ(K), irreducible bounded unitary representations are finite tensor products of so-called evaluation representations, hence in particular finite dimensional. For Γc(K), bounded unitary irreducible (factor) representations are possibly infinite tensor products of evaluation representations, which reduces the classification problem to results of Glimm and Powers on irreducible (factor) representations of UHF C∗-algebras. The key part in our proof is the result that every irreducible bounded unitary representation of a Lie algebra of the form k⊗RAR, where AR is a unital real complete continuous inverse algebra, is a finite product of evaluation representations. On the group level, our results cover in particular the bounded unitary representations of the identity component Gau(P)0 of the group of smooth gauge transformations of a principal fiber bundle P→X with compact base and structure group, and the groups SUn(A)0 with A a complete involutive commutative continuous inverse algebra.