Inst. Mathematik

398 Items

Recent Submissions
Diffusivity estimation for activator–inhibitor models: Theory and application to intracellular dynamics of the actin cytoskeleton

Pasemann, Gregor ; Flemming, Sven ; Alonso, Sergio ; Beta, Carsten ; Stannat, Wilhelm (2021-05-03)

A theory for diffusivity estimation for spatially extended activator–inhibitor dynamics modeling the evolution of intracellular signaling networks is developed in the mathematical framework of stochastic reaction–diffusion systems. In order to account for model uncertainties, we extend the results for parameter estimation for semilinear stochastic partial differential equations, as developed in...

Time-periodic weak solutions to incompressible generalized Newtonian fluids

Abbatiello, Anna (2021-05-20)

In this study we are interested in the Navier–Stokes-like system for generalized viscous fluids whose viscosity has a power-structure with exponent q. We develop an existence theory of time-periodic three-dimensional flows.

The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functions

Dio, Philipp J. di ; Kummer, Mario (2021-03-24)

In this paper we improve the bounds for the Carathéodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, Rn, and [0,1]n. We also treat moment problems with small gaps. We find that for every ε>0 and d∈N there is a n∈N such that we can construct a moment functional L:R[x1,⋯,xn]≤d→R w...

Miquel dynamics, Clifford lattices and the Dimer model

Affolter, Niklas C. (2021-05-02)

Miquel dynamics was introduced by Ramassamy as a discrete time evolution of square grid circle patterns on the torus. In each time step every second circle in the pattern is replaced with a new one by employing Miquel’s six circle theorem. Inspired by this dynamics we consider the local Miquel move, which changes the combinatorics and geometry of a circle pattern. We prove that the circle cente...

Thermal cutting of steel plates

Arenas Jaén, Manuel Jesús (2021)

Steel is commonly delivered as coils, sheets or plates at the final stages of production. The plates have to be cut to the right dimensions and this is usually done by means of thermal cutting. Flame cutting is a common and versatile method to produce steel plates but it can cause undesired side effects. Till date, the flame cutting process has not been thoroughly studied using mathematical mo...

Solving parametric PDEs with neural networks: unfavorable structure vs. expressive power

Raslan, Mones Konstantin (2021)

This cumulative dissertation extends the theory of neural networks (NNs). In the first part of this thesis, [PRV20] in Appendix A, we provide a general analysis of the hypothesis class of NNs from a structural point of view. Here, we examine the algebraic and topological properties of the set of NNs with fixed architecture. We establish that this set is never convex, hardly ever closed in class...

Tropical bisectors and diameters of simplicial complexes

Criado Gallart, Francisco (2021)

Tropical geometry is a discrete analogue of algebraic geometry where addition is replaced by taking the minimum (or maximum) and multiplication is replaced by addition. It is an active research eld where tropical analogues of geometric concepts, like linear spaces, varieties, polytopes or volume, are in different stages of development. The interest of these analogues is twofold: their combinat...

On a discretization of confocal quadrics. I. An integrable systems approach

Bobenko, Alexander I. ; Schief, Wolfgang K. ; Suris, Yuri B. ; Techter, Jan (2016-08-04)

Confocal quadrics lie at the heart of the system of confocal coordinates (also called elliptic coordinates, after Jacobi). We suggest a discretization which respects two crucial properties of confocal coordinates: separability and all two-dimensional coordinate subnets being isothermic surfaces (that is, allowing a conformal parametrization along curvature lines, or, equivalently, supporting or...

Discrete confocal quadrics and checkerboard incircular nets

Techter, Jan (2021)

Confocal quadrics constitute a special example of orthogonal coordinate systems. In this cumulative thesis we propose two approaches to the discretization of confocal coordinates, and study the closely related checkerboard incircular nets. First, we propose a discretization based on factorizable solutions to an integrable discretization of the Euler-Poisson-Darboux equation. The constructed so...

Duopoly price competition with limited capacity

Bërdëllima, Arian (2021-02-11)

We study a variation of the duopoly model by Kreps and Scheinkman (1983). Firms limited by their capacity of production engage in a two stage game. In the first stage they commit to levels of production not exceeding their capacities which are then made common knowledge. In the second stage after production has taken place firms simultane- ously compete in prices. Solution of this sequential ga...

Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs

Petrera, Matteo ; Vermeeren, Mats (2020-11-09)

We investigate the relation between pluri-Lagrangian hierarchies of 2-dimensional partial differential equations and their variational symmetries. The aim is to generalize to the case of partial differential equations the recent findings in Petrera and Suris (Nonlinear Math. Phys. 24(suppl. 1):121–145, 2017) for ordinary differential equations. We consider hierarchies of 2-dimensional Lagrangia...

Vanishing viscosity limit for the compressible Navier–Stokes system via measure-valued solutions

Basarić, Danica (2020-10-21)

We identify a class of measure-valued solutions of the barotropic Euler system on a general (unbounded) spatial domain as a vanishing viscosity limit for the compressible Navier–Stokes system. Then we establish the weak (measure-valued)–strong uniqueness principle, and, as a corollary, we obtain strong convergence to the Euler system on the lifespan of the strong solution.

On orthogonal symmetric chain decompositions

Däubel, Karl ; Jäger, Sven ; Mütze, Torsten ; Scheucher, Manfred (2019-09-27)

The n-cube is the poset obtained by ordering all subsets of {1,…,n} by inclusion, and it can be partitioned into (n⌊n/2⌋) chains, which is the minimum possible number. Two such decompositions of the n-cube are called orthogonal if any two chains of the decompositions share at most a single element. Shearer and Kleitman conjectured in 1979 that the n-cube has ⌊n/2⌋+1 pairwise orthogonal decompos...

Triangulations, discriminants, and Teichmüller theory

Löwe, Robert (2021)

A famous construction of Gel’fand, Kapranov and Zelevinsky associates to each finite point configuration A its secondary fan. This polyhedral fan stratifies the space of height functions by means of the induced regular subdivisions of A. A key result says that the secondary fan arises as the normal fan of a convex polytope, the secondary polytope of A. We apply the theory of regular triangulati...

Some aspects of integrability of birational maps

Zander, René (2021)

In dieser Arbeit behandeln wir verschiedene Aspekte zur Integrabilität von birationalen Abbildungen, vorrangig im Kontext von Abbildungen, welche als Kahan-Diskretisierung von Systemen von quadratischen gewöhnlichen Differentialgleichungen gegeben sind. Integrabilität von birationalen Abbildungen kann durch das Vorhandensein von geometrischen Eigenschaften, wie zum Beispiel eine ausreichende An...

Flip distances between graph orientations

Aichholzer, Oswin ; Cardinal, Jean ; Huynh, Tony ; Knauer, Kolja ; Mütze, Torsten ; Steiner, Raphael ; Vogtenhuber, Birgit (2020-07-27)

Flip graphs are a ubiquitous class of graphs, which encode relations on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon. For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is the mini...

Tensor-free proximal methods for lifted bilinear/quadratic inverse problems with applications to phase retrieval

Beinert, Robert ; Bredies, Kristian (2021-01-04)

We propose and study a class of novel algorithms that aim at solving bilinear and quadratic inverse problems. Using a convex relaxation based on tensorial lifting, and applying first-order proximal algorithms, these problems could be solved numerically by singular value thresholding methods. However, a direct realization of these algorithms for, e.g., image recovery problems is often impractica...

On the expansion of solutions of Laplace-like equations into traces of separable higher dimensional functions

Yserentant, Harry (2020-07-31)

This paper deals with the equation - Δ u + μ u = f on high-dimensional spaces R^m where μ is a positive constant. If the right-hand side f is a rapidly converging series of separable functions, the solution u can be represented in the same way. These constructions are based on approximations of the function 1/ r by sums of exponential functions. The aim of this paper is to prove results of simi...

Controlling transient gas flow in real-world pipeline intersection areas

Hennings, Felix ; Anderson, Lovis ; Hoppmann-Baum, Kai ; Turner, Mark ; Koch, Thorsten (2020-10-03)

Compressor stations are the heart of every high-pressure gas transport network. Located at intersection areas of the network, they are contained in huge complex plants, where they are in combination with valves and regulators responsible for routing and pushing the gas through the network. Due to their complexity and lack of data compressor stations are usually dealt with in the scientific lite...

(p, q)-equations with singular and concave convex nonlinearities

Papageorgiou, Nikolaos S. ; Winkert, Patrick (2020-09-29)

We consider a nonlinear Dirichlet problem driven by the ( p ,  q )-Laplacian with 1 < q < p . The reaction is parametric and exhibits the competing effects of a singular term and of concave and convex nonlinearities. We are looking for positive solutions and prove a bifurcation-type theorem describing in a precise way the set of positive solutions as the parameter varies. Moreover, we show the ...