## 510 Mathematik

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Abstract
We provide an extension operator for weighted Sobolev spaces on polyhedral cones K involving a mixture of weights, which measure the distance to the vertex and the edges of the cone, respectively. Our results are based on Stein's extension operator for Sobolev spaces and generalize previous results of the first author.

In this paper, we derive a four-dimensional ordinary differential equation (ODE) model representing the main interactions between Sox9, Sox10, Olig2 and several miRNAs, which drive the process of (olygodendrocyte) differentiation. We utilize the Lyapunov–Andronov theory to analyze its dynamical properties. Our results indicated that the strength of external signaling (morphogenic gradients shh and bmp), and the transcription rate of mOlig2 explain the existence of stable and unstable (sustained oscillations) behavior in the system. Possible biological implications are discussed.

Definition
Within Lorenz models, the three major kinds of butterfly effects (BEs) are the sensitive dependence on initial conditions (SDIC), the ability of a tiny perturbation to create an organized circulation at large distances, and the hypothetical role of small-scale processes in contributing to finite predictability, referred to as the first, second, and third kinds of butterfly effects (BE1, BE2, and BE3), respectively. A well-accepted definition of the butterfly effect is the BE1 with SDIC, which was rediscovered by Lorenz in 1963. In fact, the use of the term “butterfly” appeared in a conference presentation by Lorenz in 1972, when Lorenz introduced the BE2 as the metaphorical butterfly effect. In 2014, the so-called “real butterfly effect”, which is based on the features of Lorenz’s study in 1969, was introduced as the BE3.

Micro-macro models pose a powerful tool to mathematically describe reactive flow and trans- port phenomena in porous media research. Honoring the inherent multiscale spatial structure of porous media such as natural rock, these approaches hold the potential to combine the well-known benefits of pure microscopic (pore-scale) and pure macroscopic (continuum-) models. As such, the micro-macro ansatz aims at achieving computational efficiency com- parable to macroscopic models while ensuring a detailed and comprehensive description of all relevant physical/chemical processes typical of pore-scale representations. This thesis contributes to the elimination of current restrictions to micro-macro approaches by considering a model incorporating the additional physics arising from the presence of two different mineral phases complemented by a powerful numerical scheme allowing for simulations of realistic complexity.
In a first step, a new sharp interface micro-macro model for reactive flow and transport in two-mineral evolving porous media is derived generalizing existing single-mineral mod- els. As such, our approach is capable of accurately capturing covering and encapsulation phenomena as typically occurring due to dissolution/precipitation reactions. Being obtained from an underlying pore-scale model by methods of formal periodic homogenization, the model consists of effective equations for flow and transport on the macro-domain supplemented by auxiliary cell-problems from which the required effective parameters are derived. Demanding conservation of mass, macroscopic concentration fields in turn dictate geometry evolution within the reference cells.
Secondly, analytical studies are performed proving the local-in-time existence of strong solutions to a simplified version of our micro-macro model. Smooth parameter dependence results obtained along the way indicate stability of the multiscale coupling and provide a theoretical motivation for the methods employed in our numerical approach.
Finally, an iterative solution scheme capable of efficiently handling the tight nonlinear coupling between both spatial scales is presented. Employing a specialized version of the Voronoi implicit interface method, a proper description and evolution of the resulting three-phase geometry is ensured. Besides the development of a suitable adaptivity scheme significantly reducing the number of evaluated auxiliary cell-problems, data-driven methods are employed to further decrease computational effort. More precisely, convolutional neural networks are trained in two and three spatial dimensions to predict the permeability directly from the representative cell geometry, evading costly solutions of pore-scale flow problems at acceptable loss of accuracy. The power of our approach is demonstrated by conducting numerical experiments of challenging self-enforcing processes such as wormholing phenomena. Moreover, the methods are verified by direct comparison to microscopic simulation results as well as to a related diffuse interface model being implemented in an independent numerical framework.

How Heterogeneous Pore Scale Distributions of Wettability Affect Infiltration into Porous Media
(2022)

Wettability is an important parameter that significantly determines hydrology in porous media, and it especially controls the flow of water across the rhizosphere—the soil-plant interface. However, the influence of spatially heterogeneous distributions on the soil particles surfaces is scarcely known. Therefore, this study investigates the influence of spatially heterogeneous wettability distributions on infiltration into porous media. For this purpose, we utilize a two-phase flow model based on Lattice-Boltzmann to numerically simulate the infiltration in porous media with a simplified geometry and for various selected heterogeneous wettability coatings. Additionally, we simulated the rewetting of the dry rhizosphere of a sandy soil where dry hydrophobic mucilage depositions on the particle surface are represented via a locally increased contact angle. In particular, we can show that hydraulic dynamics and water repellency are determined by the specific location of wettability patterns within the pore space. When present at certain locations, tiny hydrophobic depositions can cause water repellency in an otherwise well-wettable soil. In this case, averaged, effective contact angle parameterizations such as the Cassie equation are unsuitable. At critical conditions, when the rhizosphere limits root water uptake, consideration of the specific microscale locations of exudate depositions may improve models of root water uptake.

In this paper, we study exact boundary controllability for a linear wave equation with strong and weak interior degeneration of the coefficient in the principle part of the elliptic operator. The objective is to provide a well‐posedness analysis of the corresponding system and derive conditions for its controllability through boundary actions. Passing to a relaxed version of the original problem, we discuss existence and uniqueness of solutions, and using the HUM method we derive conditions on the rate of degeneracy for both exact boundary controllability and the lack thereof.

Abstract
We study uncertain linear complementarity problems (LCPs), that is, problems in which the LCP vector q or the LCP matrix M may contain uncertain parameters. To this end, we use the concept of Γ‐robust optimization applied to the gap function formulation of the LCP. Thus, this work builds upon Krebs and Schmidt (2020). There, we studied Γ‐robustified LCPs for ℓ1‐ and box‐uncertainty sets, whereas we now focus on ellipsoidal uncertainty sets. For uncertainty in q or M, we derive conditions for the tractability of the robust counterparts. For these counterparts, we also give conditions for the existence and uniqueness of their solutions. Finally, a case study for the uncertain traffic equilibrium problem is considered, which illustrates the effects of the values of Γ on the feasibility and quality of the respective robustified solutions.

Abstract
Noncommutative spacetimes are widely believed to model some properties of the quantum structure of spacetime at the Planck regime. In this contribution the construction of (anti-)de Sitter noncommutative spacetimes obtained through quantum groups is reviewed. In this approach the quantum deformation parameter z is related to a Planck scale, and the cosmological constant plays the role of a second deformation parameter of geometric nature, whose limit Λ → 0 provides the corresponding noncommutative Minkowski spacetimes.

The constant curvature spacetimes of 3d gravity and their associated symmetry algebras are shown to arise from the 6d Drinfel'd double that underlies the two-parametric 'hybrid' quantum deformation of the 𝔰𝔩(2, ℛ) algebra. Moreover, the quantum deformation supplies the additional structures (star structure and pairing) that enter in the Chern-Simons formulation of the theory, thus establishing a direct link between quantum 𝔰𝔩(2, ℛ) algebras and 3d gravity models. In this approach the flat spacetimes and Newtonian models arise as Lie algebra contractions that are governed by two dimensionful 𝔰𝔩(2, ℛ) deformation parameters, which are directly related to the cosmological constant and to the speed of light.

We derive a class of equations of state for a multi-phase thermodynamic system associated with a finite set of order parameters that satisfy an integrable system of hydrodynamic type. As particular examples, we discuss one-phase systems such as the van der Waals gas and the effective molecular field model. The case of N–phase systems is also discussed in detail in connection with entropies depending on the order parameter according to Tsallis' composition rule.

We review Poisson–Lie groups and their applications in gauge theory and integrable systems from a mathematical physics perspective. We also comment on recent results and developments and their applications. In particular, we discuss the role of quasitriangular Poisson–Lie groups and dynamical r-matrices in the description of moduli spaces of flat connections and the Chern–Simons gauge theory.

In this thesis a framework for modelling two coupled hypersurfaces evolving in a three-dimensional domain
that is filled with a fluid is developed in terms of partial differential equations that can be classified as free
boundary problems. In particular, the coupling of both hypersurfaces depends on particles of a species that
evolve on one of the surfaces. Since they influence the surface tension, they can be understood as surfactants.
We use two common approaches for modelling free boundary problems, namely sharp and diffuse interfaces
by using similar physical arguments for their derivation as well as a formal asymptotic analysis. From the
modelling framework, we specify particular models for the evolution of biomembranes, and, even more specific,
for studying the biological phenomenon of cell blebbing. Properties of these models are investigated by means
of rigorous mathematical analysis showing existence of solutions, existence and stability of stationary solutions
where possible, as well a singular limits. We also present numerical algorithms for simulative studies of cell
blebbing models and report and interpret results obtained from their computations. Thus we can validate that
our models qualitatively reproduce phenomena that have been reported in the biological literature.

The Weinstein equation
tΔu+k∂u∂t=0, with
k∈ℤ, considered in
ℝ3=(x,y,t), is a modification of the classical Laplace equation
Δu=0. Its solutions are called k‐modified harmonic functions. Whereas for positive integers k the Weinstein equation is relatively well understood, little is known if the parameter k is negative.
The main result of this article is the statement that in case the negative integers are even, i.e.,
k=−2ℓ,ℓ∈ℕ, we still have a Fischer‐type decomposition. For
k=0, the classical harmonic functions, this decomposition is well known. But also in case
k∈ℕ, a Fischer‐type decomposition holds true, a Fischer‐type decomposition holds true. Surprisingly in case
k=−3,k=−5, or
k=−7 and probably in all higher negative odd cases, the decomposition doesn't hold.
In case
k=−1, we give a complete description of the vector space
Hnk(ℝ3) of homogeneous k‐modified harmonic polynomials of degree n in
ℝ3. Such a result is also at hand in case
k∈ℕ. Finally, in case
k=0 of the classical harmonic functions, we give a description of the vector space
Hn(ℝ3)=Hn0(ℝ3).

In this article, a new numerical scheme for the solution of the multidimensional fragmentation problem is presented. It is the first that uses the conservative form of the multidimensional problem. The idea to apply the finite volume scheme for solving one-dimensional linear fragmentation problems is extended over a generalized multidimensional setup. The derivation is given in detail for two-dimensional and three-dimensional problems; an outline for the extension to higher dimensions is also presented. Additionally, the existing one-dimensional finite volume scheme for solving conservative one-dimensional multi-fragmentation equation is extended to solve multidimensional problems. The accuracy and efficiency of both proposed schemes is analyzed for several test problems.

Life Cycle Assessment (LCA) deals with factors that impact the environment, such
as the greenhouse gas potential. It considers the complete life cycle of a product or
service. In the first case, this means the analysis of raw material extraction, through
production and usage to end of life. The ISO norms 14040 and 14044 recommend a
precisely defined procedure: Goal and Scope Definition, Life Cycle Inventory Analysis,
Life Cycle Impact Assessment, and Interpretation.
So far, this instrument has mostly been applied only as a balancing method, in order to
determine the ecological footprint of an available product. In our further development of
this approach, we use the techniques of the mathematical optimization to influence the
design of the life cycle. This allows the improvement of the environmental properties for
both during development and existing products while at the same time including social
and financial considerations.
For Vitesco Technologies, this resulting optimized Integrated Life Cycle Sustainability
Assessment (ILCSA) is an important aspect of sustainability which goes beyond the
scope of LCA in the ordinary sense. We implemented a highly customizable mixed-integer
optimization model which provides the product manager guidance to choose,
for example, the locations for gaining the raw materials, production sites, and material
composition.
For its solution, we developed a software with a user-friendly interface and show the
benefit in a detailed case study at the example of a lithium-ion battery. It is a main part
of an electric vehicle and, therefore, it is of great significance in the current focus topic
of electromobility. Our method can be transferred to all other automotive components,
up to the optimization of the life cycle for the complete vehicle.

The orthogonalization of Boolean functions in disjunctive form, that means a Boolean function formed by sum of products, is a classical problem in the Boolean algebra. In this work, the novel methodology ORTH[ⴱ] of orthogonalization which is an universally valid formula based on the combination technique »orthogonalizing difference-building ⴱ« is presented. Therefore, the technique ⴱ is used to transform Sum of Products into disjoint Sum of Products. The scope of orthogonalization will be solved by a novel formula in a mathematically easier way. By a further procedure step of sorting product terms, a minimized disjoint Sum of Products can be reached. Compared to other methods or heuristics ORTH[ⴱ] provides a faster computation time.

We introduce the two‐stage stochastic minimum s − t cut problem. Based on a classical linear 0‐1 programming model for the deterministic minimum s − t cut problem, we provide a mathematical programming formulation for the proposed stochastic extension. We show that its constraint matrix loses the total unimodularity property, however, preserves it if the considered graph is a tree. This fact turns out to be not surprising as we prove that the considered problem is NP‐hard in general, but admits a linear time solution algorithm when the graph is a tree. We exploit the special structure of the problem and propose a tailored Benders decomposition algorithm. We evaluate the computational efficiency of this algorithm by solving the Benders dual subproblems as max‐flow problems. For many tested instances, we outperform a standard Benders decomposition by two orders of magnitude with the Benders decomposition exploiting the max‐flow structure of the subproblems.

This note is about the topology of the path space of linear Fredholm operators on a real Hilbert space. Fitzpatrick and Pejsachowicz introduced the parity of such a path, based on the Leray–Schauder degree of a path of parametrices. Here an alternative analytic approach is presented which reduces the parity to the Z2‐valued spectral flow of an associated path of chiral skew‐adjoints. Furthermore, the related notion of Z2‐index of a Fredholm pair of chiral complex structures is introduced and connected to the parity of a suitable path. Several non‐trivial examples are provided. One of them concerns topological insulators, another an application to the bifurcation of a non‐linear partial differential equation.

In this thesis we consider certain Poisson structures associated with ribbon graphs that arise in the context of moduli spaces of flat G-bundles and are also related to Kitaev's lattice models for quantum computers.
In the first part we define Poisson analogues of Kitaev models. The latter were defined ad hoc and not by quantizing a Poisson-geometrical theory. We replace the Hopf-algebraic data of a Kitaev model on a ribbon, or equivalently, embedded graph Γ with Poisson data from a Poisson-Lie group G. Each edge of Γ is assigned a copy of the Heisenberg double H(G). Each face (vertex) of Γ defines a Poisson action of G (of G*) on the product of these Heisenberg doubles. The actions for a vertex and incident face form a Poisson action of the double Poisson-Lie group D(G). We define counterparts of Kitaev's vertex and face operators and relate them via the Poisson bracket to the vector fields generating the actions of D(G).
The natural symplectic structure on the moduli space Hom(π_1(S),D)/D of flat D-bundles on an oriented surface S with boundary can be described by a Poisson structure introduced by Fock and Rosly. This Poisson structure is constructed from the Poisson-Lie group D and an embedded graph. We define a Poisson isomorphism between this Poisson structure for D = D(G) and a Poisson-geometrical Kitaev model for G on the same graph. This isomorphism decouples Fock and Rosly's Poisson structure and represents it as a product of Heisenberg doubles. We use this isomorphism to relate the Poisson-geometrical Kitaev model to the symplectic structure on the moduli space of flat D(G)-bundles on S.
In the second part of this thesis we prove that the mapping class group of the oriented surface associated with a ribbon graph acts by Poisson automorphisms on both Fock and Rosly's Poisson structure and on our analogues of Kitaev models. We work with two presentations of the mapping class group, a presentation in terms of Dehn twists introduced by Gervais and a presentation in terms of chord slides due to Bene. We associate Poisson automorphisms to the generators of these presentations. To obtain mapping class group actions, we show that these Poisson automorphisms satisfy the relations of Bene's presentation. In the case of Fock and Rosly's Poisson structure, we also show that Gervais' relations are fulfilled to obtain another mapping class group action.
In the third part we generalize Fock and Rosly's Poisson structure. The latter requires a Poisson-Lie group D equipped with a classical r-matrix. It is constructed by assigning a copy of D to every edge of a ribbon graph Γ and associating a Poisson action of D to every vertex. We replace the D-action at each vertex v by an action of a Poisson-Lie groupoid X_v equipped with a classical dynamical r-matrix. Instead of a Poisson-Lie group, every edge e is labelled with a Poisson manifold Y_e together with a Poisson action of the groupoids at its starting and target vertices. We show that the symplectic structure on the moduli space of flat D-bundles for the oriented surface associated with Γ can be obtained by forming the quotient with respect to the groupoid actions at the vertices. This allows for a description of the moduli space in cases when its symplectic structure is not associated with a classical r-matrix, but with a classical dynamical r-matrix.

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.