## 510 Mathematik

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Investigating plant/root-soil interactions at different scales is crucial to advance the understanding of soil structure formation in the rhizosphere. To better comprehend the underlying interwoven processes an explicit, fully dynamic spatial and image-based modeling at the pore scale is a promising tool especially taking into account experimental limitations. We develop a modeling tool to investigate how soil aggregation, root growth and root exudates mutually interact with each other at the micro-scale. This allows the simultaneous simulation of the dynamic rearrangement of soil particles, the input and turnover of particulate organic matter, root growth and decay as well as the deposition, redistribution and decomposition of mucilage in the rhizosphere. The interactions are realized within a cellular automaton framework. The most stable configuration is determined by the amount and attractiveness of surface contacts between the particles, where organo-mineral associations preferably lead to the formation of soil aggregates. Their break-up can be induced by root growth or the degradation of gluing agents previously created after the decomposition of particulate organic matter and mucilage. We illustrate the capability of our model by simulating a full life cycle of a fine root in a two-dimensional, horizontal cross section through the soil. We evaluate various scenarios to identify the role of different drivers such as soil texture and mucilage. We quantify the displacement intensity of individual particles and the variations in local porosity due to the change in available pore space as influenced by the root growth and observe compaction, gap formation and a biopore evolution. The simulation results support that the deposition of mucilage is an important driver for structure formation in the rhizosphere. Although mucilage is degraded within a few days after exudation, it leads to a persistent stabilization of the aggregated structures for both textures in the vicinity of the root within a time frame of 1000 days. Local porosity changes are quantified for exudation periods of 1, 10 and 100 days and are already pronounced for short-term exudation of mucilage. This stabilization is significantly different from the structures encountered when only POM could trigger the evolution of gluing spots, and is still present after complete degradation of the root.

Standard subspaces are a well studied object in algebraic quantum field theory (AQFT). Given a standard subspace V of a Hilbert space H, one is interested in unitary one-parameter groups on H with U(t)V⊆V for every t∈R+. If (V,U) is a non-degenerate standard pair on H, i.e. the self-adjoint infinitesimal generator of U is a positive operator with trivial kernel, two classical results are given by Borchers’ Theorem, relating non-degenerate standard pairs to positive energy representations of the aﬀine group Aﬀ(R) and the Longo–Witten Theorem, stating the the semigroup of unitary endomorphisms of V can be identified with the semigroup of symmetric operator-valued inner functions on the upper half plane.
In this thesis we prove results similar to the theorems of Borchers and of Longo–Witten for a more general framework of unitary one-parameter groups without the assumption that their infinitesimal generator is positive. We replace this assumption by the weaker assumption that the triple (H,V,U) is a so called real regular one-parameter group.
After providing some basic theory for regular one-parameter groups, we investigate regular one-parameter groups that are infinitesimally generated by operator-valued Pick functions. We provide criteria for a Pick function to generate a regular one-parameter group and in this case we provide a formula for the multiplicity space of the regular one-parameter group and show that the dimension of the multiplicity space is compatible with composition of Pick functions.
We also investigate reflection positive regular one-parameter groups and provide a normal form for them. We do this by linking them to reflection positive Hilbert spaces of the form (L²(R,K),H²(C+,K),θ(h)) with some complex Hilbert space K and some function h∈L∞(R,U(K)), where the involution θ(h) on L²(R,K) is given by (θ(h)f)(x)=h(x)·f(−x), x∈R. In the multiplicity free case K=R we give a full classification of all functions h∈L∞(R,T) for which the triple (L²(R,C),H²(C+),θ(h)) is a maximal reflection positive Hilbert space. Also we give an explicit description of the Osterwalder–Schrader transform of these reflection positive Hilbert spaces.
We apply these results to prove that every positive contractive Hankel operator on H²(C+) can be extended to an involution θ(h) on L²(R,C) for which the triple (L²(R,C),H²(C+),θ(h)) is a maximal reflection positive Hilbert space.
Finally, we apply our results about (reflection positive) regular one-parameter groups in the context of standard subspaces and use them to provide analogues of Borchers’ Theorem and the Longo–Witten Theorem to pairs (V,U) for which the triple (H,V,U) is a real regular one-parameter group or for which the quadruple (H,V,U,J(V)) is a real reflection positive regular one-parameter group respectively.

This thesis tries to enlarge the knowledge about non-collision singularities in n-body systems as, e.g. the case of celestial mechanics. Non-collision singularities are orbits, where the moment of inertia and the kinetic energy become unbounded while approaching a finite escape time. They are by now only well understood in the case of four particles.
One approach for understanding systems with more than four particles consists in considering a simplified billiard model in which the particles move on straight lines until some of them collide. At collision times, however, only the total momentum needs to be conserved, whereas the kinetic energy is allowed to change. Due to this effect, these models could be used to simulate corresponding non-collision orbits and we derive some lower bounds for the diameter/moment of inertia and the kinetic energy of the simplified nondeterministic model. As such results should hold in the real model as well, we believe these investigations could prove useful in the study of non-collision singularities.
The second part of this thesis deals with non-collision orbits where the particles can be decomposed in different classes of mainly interacting particles. If each of these classes consists of at most four particles, the known results from the literature could be applicable to these subsystems, if one can handle the small forces from the other classes. We will provide some assumptions under which this might be possible for a diverging system of four particles and extend the results from [Fleischer2018] to get quantitative control of the motion of the subsystem during one passage with these small forces. Finally, we will prove these assumptions in a special case of non-collision orbits consisting of two diverging four-particle subsystems. The main problem in composed systems lies in the fact that we lose many conserved quantities for the subsystems as the energy of a subsystem and the corresponding angular momentum. We derive some techniques to control the changes of these quantities, which might prove useful in similar situations as well.

Abstract
In this study, we formulated a mathematical model of hypoxia-inducible factor 1 (HIF-1) mediated regulation of cellular energy metabolism describing the reprogramming of cell metabolic processes from oxidative phosphorylation to glycolysis under reduced oxygen levels. The model considers the dynamics of fifteen biochemical species and the proton concentration with the underlying reaction processes localized in three intracellular compartments, i.e. the cytoplasm, mitochondrion and nucleus. More than sixty parameters of the model were calibrated using both the published data and the system steady-state based identification procedure. The model was validated by generating predictions which could be compared to empirical observations. The model behaviors representing the cell metabolism switching over in response to transitioning from a normoxic to hypoxic environment are consistent with the current views of the role of HIF-1 in hypoxia.

Abstract
The Schlögl system is governed by a nonlinear reaction-diffusion partial differential equation with a cubic nonlinearity. In this paper, feedback laws of Pyragas-type are presented that stabilize the system in a periodic state with a given period and given boundary traces. We consider the system both with boundary feedback laws of Pyragas type and distributed feedback laws of Pyragas and classical type. Stabilization to periodic orbits is important for medical applications that concern Parkinson’s disease. The exponential stability of the closed loop system with respect to the L2-norm is proved. Numerical examples are provided.

Joint Model of Probabilistic-Robust (Probust) Constraints Applied to Gas Network Optimization
(2021)

Abstract
Optimization problems under uncertain conditions abound in many real-life applications. While solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually applied in case solutions are sought that are feasible for all realizations of uncertainties within some predefined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we deal with a class of joint probabilistic/robust constraints. Focusing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for finding their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound with high probability.

Abstract
As a result of its liberalization, the European gas market is organized as an entry-exit system in order to decouple the trading and transport of natural gas. Roughly summarized, the gas market organization consists of four subsequent stages. First, the transmission system operator (TSO) is obliged to allocate so-called maximal technical capacities for the nodes of the network. Second, the TSO and the gas traders sign mid- to long-term capacity-right contracts, where the capacity is bounded above by the allocated technical capacities. These contracts are called bookings. Third, on a day-ahead basis, gas traders can nominate the amount of gas that they inject or withdraw from the network at entry and exit nodes, where the nominated amount is bounded above by the respective booking. Fourth and finally, the TSO has to operate the network such that the nominated amounts of gas can be transported. By signing the booking contract, the TSO guarantees that all possibly resulting nominations can indeed be transported. Consequently, maximal technical capacities have to satisfy that all nominations that comply with these technical capacities can be transported through the network. This leads to a highly challenging mathematical optimization problem. We consider the specific instantiations of this problem in which we assume capacitated linear as well as potential-based flow models. In this contribution, we formally introduce the problem of Computing Technical Capacities (CTC) and prove that it is NP-complete on trees and NP-hard in general. To this end, we first reduce the Subset Sum problem to CTC for the case of capacitated linear flows in trees. Afterward, we extend this result to CTC with potential-based flows and show that this problem is also NP-complete on trees by reducing it to the case of capacitated linear flow. Since the hardness results are obtained for the easiest case, i.e., on tree-shaped networks with capacitated linear as well as potential-based flows, this implies the hardness of CTC for more general graph classes.

Abstract
Modulated crystals and quasicrystals can simultaneously be described as modulated quasicrystals, a class of point sets introduced by de Bruijn in 1987. With appropriate modulation functions, modulated quasicrystals themselves constitute a substantial subclass of strongly almost periodic point measures. We re-analyze these structures using methods from modern mathematical diffraction theory, thereby providing a coherent view over that class. Similar to de Bruijn’s analysis, we find stability with respect to almost periodic modulations.

Abstract
This article addresses the numerical simulation and optimization of the optical properties of mono-layered nano-particulate films. The particular optical property of interest is the so-called haze factor, for which a model close to an experimental setup is derived. The numerical solution becomes very involved due to the resulting size of computational domain in comparison to the wave length, rendering the direct simulation method infeasible. As a remedy, a hybrid method is suggested, which in essence consistently combines analytical solutions for the far field with finite element-based solutions for the near field. Using a series of algebraic reformulations, a model with an off- and online component is developed, which results in the computational complexity being reduced by several orders of magnitude. Based on the suggested hybrid numerical scheme, which is not limited to the haze factor as objective function, structural optimization problems covering geometrical and topological parametrizations are formulated. These allow the influence of different particle arrangements to be studied. The article is complemented by several numerical experiments underlining the strength of the method.

Abstract
In this paper, we present DiscreteZOO, a project which illustrates some of the possibilities for computer-supported management of collections of finite combinatorial (discrete) objects, in particular graphs with a high degree of symmetry. DiscreteZOO encompasses a data repository, a website and a SageMath Package.

The objective of this work is an existence proof for variational solutions u to parabolic minimizing problems. Here, the functions being considered are defined on a metric measure space (X,d,μ). For such parabolic minimizers that coincide with Cauchy-Dirichlet data η on the parabolic boundary of a space-time-cylinder Ω×(0,T) with an open subset Ω⊂X and T>0, we prove existence in the parabolic Newtonian space Lp(0,T;N1,p(Ω)). In this paper we generalize results from Collins and Herán (Nonlinear Anal 176:56–83, 2018) where only time-independent Cauchy–Dirichlet data have been considered. We argue completely on a variational level.

Abstract
The Bayesian approach to inverse problems provides a rigorous framework for the incorporation and quantification of uncertainties in measurements, parameters and models. We are interested in designing numerical methods which are robust w.r.t. the size of the observational noise, i.e., methods which behave well in case of concentrated posterior measures. The concentration of the posterior is a highly desirable situation in practice, since it relates to informative or large data. However, it can pose a computational challenge for numerical methods based on the prior measure. We propose to employ the Laplace approximation of the posterior as the base measure for numerical integration in this context. The Laplace approximation is a Gaussian measure centered at the maximum a-posteriori estimate and with covariance matrix depending on the logposterior density. We discuss convergence results of the Laplace approximation in terms of the Hellinger distance and analyze the efficiency of Monte Carlo methods based on it. In particular, we show that Laplace-based importance sampling and Laplace-based quasi-Monte-Carlo methods are robust w.r.t. the concentration of the posterior for large classes of posterior distributions and integrands whereas prior-based importance sampling and plain quasi-Monte Carlo are not. Numerical experiments are presented to illustrate the theoretical findings.

Abstract
This paper presents a novel method for the solution of a particular class of structural optimzation problems: the continuous stochastic gradient method (CSG). In the simplest case, we assume that the objective function is given as an integral of a desired property over a continuous parameter set. The application of a quadrature rule for the approximation of the integral can give rise to artificial and undesired local minima. However, the CSG method does not rely on an approximation of the integral, instead utilizing gradient approximations from previous iterations in an optimal way. Although the CSG method does not require more than the solution of one state problem (of infinitely many) per optimization iteration, it is possible to prove in a mathematically rigorous way that the function value as well as the full gradient of the objective can be approximated with arbitrary precision in the course of the optimization process. Moreover, numerical experiments for a linear elastic problem with infinitely many load cases are described. For the chosen example, the CSG method proves to be clearly superior compared to the classic stochastic gradient (SG) and the stochastic average gradient (SAG) method.

Abstract
We extend and improve recent results given by Singh and Watson on using classical bounds on the union of sets in a chance-constrained optimization problem. Specifically, we revisit the so-called Dawson and Sankoff bound that provided one of the best approximations of a chance constraint in the previous analysis. Next, we show that our work is a generalization of the previous work, and in fact the inequality employed previously is a very relaxed approximation with assumptions that do not generally hold. Computational results demonstrate on average over a 43% improvement in the bounds. As a byproduct, we provide an exact reformulation of the floor function in optimization models.

Abstract
We develop an optimization model to provide a fair allocation of multiple resources to multiple users. All resources might not be suitable to all users. We develop a notion of fairness, and then provide a general class of functions achieving it. Next, we develop more restricted notions of fairness—special cases of which exist in literature. Finally, we distinguish between scarce and abundant resources, and show that if a resource is abundant, all users seeking it achieve the maximum possible coverage.

Abstract
Linear bilevel optimization problems are often tackled by replacing the linear lower-level problem with its Karush–Kuhn–Tucker conditions. The resulting single-level problem can be solved in a branch-and-bound fashion by branching on the complementarity constraints of the lower-level problem’s optimality conditions. While in mixed-integer single-level optimization branch-and-cut has proven to be a powerful extension of branch-and-bound, in linear bilevel optimization not too many bilevel-tailored valid inequalities exist. In this paper, we briefly review existing cuts for linear bilevel problems and introduce a new valid inequality that exploits the strong duality condition of the lower level. We further discuss strengthened variants of the inequality that can be derived from McCormick envelopes. In a computational study, we show that the new valid inequalities can help to close the optimality gap very effectively on a large test set of linear bilevel instances.

Abstract
We develop a complementarity-constrained nonlinear optimization model for the time-dependent control of district heating networks. The main physical aspects of water and heat flow in these networks are governed by nonlinear and hyperbolic 1d partial differential equations. In addition, a pooling-type mixing model is required at the nodes of the network to treat the mixing of different water temperatures. This mixing model can be recast using suitable complementarity constraints. The resulting problem is a mathematical program with complementarity constraints subject to nonlinear partial differential equations describing the physics. In order to obtain a tractable problem, we apply suitable discretizations in space and time, resulting in a finite-dimensional optimization problem with complementarity constraints for which we develop a suitable reformulation with improved constraint regularity. Moreover, we propose an instantaneous control approach for the discretized problem, discuss practically relevant penalty formulations, and present preprocessing techniques that are used to simplify the mixing model at the nodes of the network. Finally, we use all these techniques to solve realistic instances. Our numerical results show the applicability of our techniques in practice.

Abstract
In this paper, we consider a 5-dimensional Hindmarsh–Rose neuron model. This improved version of the original model shows rich dynamical behaviors, including a chaotic super-bursting regime. This regime promises a greater information encoding capacity than the standard bursting activity. Based on the Krasovskii–Lyapunov stability theory, the sufficient conditions (on the synaptic strengths and magnetic gain parameters) for stable chaotic synchronization of the model are obtained. Based on Helmholtz’s theorem, the Hamilton function of the corresponding error dynamical system is also obtained. It is shown that the time variation of this Hamilton function along trajectories can play the role of the time variation of the Lyapunov function—in determining the stability of the synchronization manifold. Numerical computations indicate that as the synaptic strengths and the magnetic gain parameters change, the time variation of the Hamilton function is always nonzero (i.e., a relatively large positive or negative value) only when the time variation of the Lyapunov function is positive, and zero (or vanishingly small) only when the time variation of the Lyapunov function is also zero. This, therefore, paves an alternative way to determine the stability of synchronization manifolds and can be particularly useful for systems whose Lyapunov function is difficult to construct, but whose Hamilton function corresponding to the dynamic error system is easier to calculate.

Modeling flexible generator operating regions via chance-constrained stochastic unit commitment
(2020)

Abstract
We introduce a novel chance-constrained stochastic unit commitment model to address uncertainty in renewables’ production in operations of power systems. For most thermal generators, underlying technical constraints that are universally treated as “hard” by deterministic unit commitment models are in fact based on engineering judgments, such that system operators can periodically request operation outside these limits in non-nominal situations, e.g., to ensure reliability. We incorporate this practical consideration into a chance-constrained stochastic unit commitment model, specifically by infrequently allowing minor deviations from the minimum and maximum thermal generator power output levels. We demonstrate that an extensive form of our model is computationally tractable for medium-sized power systems given modest numbers of scenarios for renewables’ production. We show that the model is able to potentially save significant annual production costs by allowing infrequent and controlled violation of the traditionally hard bounds imposed on thermal generator production limits. Finally, we conduct a sensitivity analysis of optimal solutions to our model under two restricted regimes and observe similar qualitative results.

We prove the existence of nonnegative variational solutions to the obstacle problem associated with the degenerate doubly nonlinear equation
∂tb(u)−div(Df(Du))=0,
where the nonlinearity b:R≥0→R≥0
is increasing, piecewise C1 and satisfies a polynomial growth condition. The prototype is b(u):=um with m∈(0,1). Further, f:Rn→R≥0 is convex and fulfills a standard p-growth condition. The proof relies on a nonlinear version of the method of minimizing movements.