@article{21820, abstract = {{The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting.}}, author = {{Nüske, Feliks and Koltai, Péter and Boninsegna, Lorenzo and Clementi, Cecilia}}, issn = {{1099-4300}}, journal = {{Entropy}}, title = {{{Spectral Properties of Effective Dynamics from Conditional Expectations}}}, doi = {{10.3390/e23020134}}, year = {{2021}}, } @article{16867, abstract = {{In this article, we present an efficient descent method for locally Lipschitz continuous multiobjective optimization problems (MOPs). The method is realized by combining a theoretical result regarding the computation of descent directions for nonsmooth MOPs with a practical method to approximate the subdifferentials of the objective functions. We show convergence to points which satisfy a necessary condition for Pareto optimality. Using a set of test problems, we compare our method to the multiobjective proximal bundle method by M\"akel\"a. The results indicate that our method is competitive while being easier to implement. While the number of objective function evaluations is larger, the overall number of subgradient evaluations is lower. Finally, we show that our method can be combined with a subdivision algorithm to compute entire Pareto sets of nonsmooth MOPs.}}, author = {{Gebken, Bennet and Peitz, Sebastian}}, journal = {{Journal of Optimization Theory and Applications}}, pages = {{696--723}}, title = {{{An efficient descent method for locally Lipschitz multiobjective optimization problems}}}, doi = {{10.1007/s10957-020-01803-w}}, volume = {{188}}, year = {{2021}}, } @article{16295, abstract = {{It is a challenging task to identify the objectives on which a certain decision was based, in particular if several, potentially conflicting criteria are equally important and a continuous set of optimal compromise decisions exists. This task can be understood as the inverse problem of multiobjective optimization, where the goal is to find the objective function vector of a given Pareto set. To this end, we present a method to construct the objective function vector of an unconstrained multiobjective optimization problem (MOP) such that the Pareto critical set contains a given set of data points with prescribed KKT multipliers. If such an MOP can not be found, then the method instead produces an MOP whose Pareto critical set is at least close to the data points. The key idea is to consider the objective function vector in the multiobjective KKT conditions as variable and then search for the objectives that minimize the Euclidean norm of the resulting system of equations. By expressing the objectives in a finite-dimensional basis, we transform this problem into a homogeneous, linear system of equations that can be solved efficiently. Potential applications of this approach include the identification of objectives (both from clean and noisy data) and the construction of surrogate models for expensive MOPs.}}, author = {{Gebken, Bennet and Peitz, Sebastian}}, journal = {{Journal of Global Optimization}}, pages = {{3--29}}, publisher = {{Springer}}, title = {{{Inverse multiobjective optimization: Inferring decision criteria from data}}}, doi = {{10.1007/s10898-020-00983-z}}, volume = {{80}}, year = {{2021}}, } @phdthesis{32057, abstract = {{Ein zentraler Aspekt bei der Untersuchung dynamischer Systeme ist die Analyse ihrer invarianten Mengen wie des globalen Attraktors und (in)stabiler Mannigfaltigkeiten. Insbesondere wenn das zugrunde liegende System von einem Parameter abhängt, ist es entscheidend, sie im Bezug auf diesen Parameter effizient zu verfolgen. Für die Berechnung invarianter Mengen stützen wir uns für ihre Approximation auf numerische Algorithmen. Typischerweise können diese Methoden jedoch nur auf endlich-dimensionale dynamische Systeme angewendet werden. In dieser Arbeit präsentieren wir daher einen numerischen Rahmen für die globale dynamische Analyse unendlich-dimensionaler Systeme. Wir werden Einbettungstechniken verwenden, um das core dynamical system (CDS) zu definieren, welches ein dynamisch äquivalentes endlich-dimensionales System ist.Das CDS wird dann verwendet, um eingebettete invariante Mengen, also eins-zu-eins Bilder, mittels Mengen-orientierten numerischen Methoden zu approximieren. Bei der Konstruktion des CDS ist es entscheidend, eine geeignete Beobachtungsabbildung auszuwählen und die geeignete inverse Abbildung zu entwerfen. Dazu werden wir geeignete numerische Implementierungen des CDS für DDEs und PDEs vorstellen. Für eine nachfolgende geometrische Analyse der eingebetteten invarianten Menge betrachten wir eine Lerntechnik namens diffusion maps, die ihre intrinsische Geometrie enthüllt sowie ihre Dimension schätzt. Schließlich wenden wir unsere entwickelten numerischen Methoden an einigen bekannten unendlich-dimensionale dynamischen Systeme an, wie die Mackey-Glass-Gleichung, die Kuramoto-Sivashinsky-Gleichung und die Navier-Stokes-Gleichung.}}, author = {{Gerlach, Raphael}}, title = {{{The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems}}}, doi = {{10.17619/UNIPB/1-1278}}, year = {{2021}}, } @article{32016, author = {{Delarue, Benjamin and Ramacher, Pablo}}, journal = {{Journal of Symplectic Geometry}}, number = {{6}}, pages = {{1281 -- 1337}}, title = {{{Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions}}}, doi = {{10.4310/JSG.2021.v19.n6.a1}}, volume = {{19}}, year = {{2021}}, } @article{34042, author = {{Li, Jiaao and Ma, Yulai and Miao, Zhengke and Shi, Yongtang and Wang, Weifan and Zhang, Cun-Quan}}, issn = {{0095-8956}}, journal = {{Journal of Combinatorial Theory, Series B}}, keywords = {{Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Theoretical Computer Science}}, pages = {{61--80}}, publisher = {{Elsevier BV}}, title = {{{Nowhere-zero 3-flows in toroidal graphs}}}, doi = {{10.1016/j.jctb.2021.11.001}}, volume = {{153}}, year = {{2021}}, } @article{34786, abstract = {{A locally compact contraction group is a pair (G,α), where G is a locally compact group and α:G→G an automorphism such that αn(x)→e pointwise as n→∞. We show that every surjective, continuous, equivariant homomorphism between locally compact contraction groups admits an equivariant continuous global section. As a consequence, extensions of locally compact contraction groups with abelian kernel can be described by continuous equivariant cohomology. For each prime number p, we use 2-cocycles to construct uncountably many pairwise non-isomorphic totally disconnected, locally compact contraction groups (G,α) which are central extensions0→Fp((t))→G→Fp((t))→0 of the additive group of the field of formal Laurent series over Fp=Z/pZ by itself. By contrast, there are only countably many locally compact contraction groups (up to isomorphism) which are torsion groups and abelian, as follows from a classification of the abelian locally compact contraction groups.}}, author = {{Glöckner, Helge and Willis, George A.}}, issn = {{0021-8693}}, journal = {{Journal of Algebra}}, keywords = {{Contraction group, Torsion group, Extension, Cocycle, Section, Equivariant cohomology, Abelian group, Nilpotent group, Isomorphism types}}, pages = {{164--214}}, title = {{{Decompositions of locally compact contraction groups, series and extensions}}}, doi = {{https://doi.org/10.1016/j.jalgebra.2020.11.007}}, volume = {{570}}, year = {{2021}}, } @article{34795, author = {{Glöckner, Helge}}, issn = {{0025-584X}}, journal = {{Mathematische Nachrichten}}, number = {{1}}, pages = {{74–81}}, title = {{{Direct limits of regular Lie groups}}}, doi = {{10.1002/mana.201900073}}, volume = {{294}}, year = {{2021}}, } @unpublished{34806, abstract = {{Let $G$ be a Lie group over a totally disconnected local field and $\alpha$ be an analytic endomorphism of $G$. The contraction group of $\alpha$ ist the set of all $x\in G$ such that $\alpha^n(x)\to e$ as $n\to\infty$. Call sequence $(x_{-n})_{n\geq 0}$ in $G$ an $\alpha$-regressive trajectory for $x\in G$ if $\alpha(x_{-n})=x_{-n+1}$ for all $n\geq 1$ and $x_0=x$. The anti-contraction group of $\alpha$ is the set of all $x\in G$ admitting an $\alpha$-regressive trajectory $(x_{-n})_{n\geq 0}$ such that $x_{-n}\to e$ as $n\to\infty$. The Levi subgroup is the set of all $x\in G$ whose $\alpha$-orbit is relatively compact, and such that $x$ admits an $\alpha$-regressive trajectory $(x_{-n})_{n\geq 0}$ such that $\{x_{-n}\colon n\geq 0\}$ is relatively compact. The big cell associated to $\alpha$ is the set $\Omega$ of all all products $xyz$ with $x$ in the contraction group, $y$ in the Levi subgroup and $z$ in the anti-contraction group. Let $\pi$ be the mapping from the cartesian product of the contraction group, Levi subgroup and anti-contraction group to $\Omega$ which maps $(x,y,z)$ to $xyz$. We show: $\Omega$ is open in $G$ and $\pi$ is \'{e}tale for suitable immersed Lie subgroup structures on the three subgroups just mentioned. Moreover, we study group-theoretic properties of contraction groups and anti-contraction groups.}}, author = {{Glöckner, Helge}}, booktitle = {{arXiv:2101.02981}}, title = {{{Contraction groups and the big cell for endomorphisms of Lie groups over local fields}}}, year = {{2021}}, } @inproceedings{29421, author = {{Ober-Blöbaum, Sina and Vermeeren, M.}}, booktitle = {{7th IIFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC}}, pages = {{327--333}}, title = {{{Superconvergence of galerkin variational integrators}}}, volume = {{54(19)}}, year = {{2021}}, } @article{16294, abstract = {{Model predictive control is a prominent approach to construct a feedback control loop for dynamical systems. Due to real-time constraints, the major challenge in MPC is to solve model-based optimal control problems in a very short amount of time. For linear-quadratic problems, Bemporad et al. have proposed an explicit formulation where the underlying optimization problems are solved a priori in an offline phase. In this article, we present an extension of this concept in two significant ways. We consider nonlinear problems and - more importantly - problems with multiple conflicting objective functions. In the offline phase, we build a library of Pareto optimal solutions from which we then obtain a valid compromise solution in the online phase according to a decision maker's preference. Since the standard multi-parametric programming approach is no longer valid in this situation, we instead use interpolation between different entries of the library. To reduce the number of problems that have to be solved in the offline phase, we exploit symmetries in the dynamical system and the corresponding multiobjective optimal control problem. The results are verified using two different examples from autonomous driving.}}, author = {{Ober-Blöbaum, Sina and Peitz, Sebastian}}, journal = {{International Journal of Robust and Nonlinear Control}}, pages = {{380--403}}, title = {{{Explicit multiobjective model predictive control for nonlinear systems with symmetries}}}, doi = {{10.1002/rnc.5281}}, volume = {{31(2)}}, year = {{2021}}, } @article{29543, author = {{Djema, Walid and Giraldi, Laetitia and Maslovskaya, Sofya and Bernard, Olivier}}, issn = {{0005-1098}}, journal = {{Automatica}}, keywords = {{Electrical and Electronic Engineering, Control and Systems Engineering}}, publisher = {{Elsevier BV}}, title = {{{Turnpike features in optimal selection of species represented by quota models}}}, doi = {{10.1016/j.automatica.2021.109804}}, volume = {{132}}, year = {{2021}}, } @unpublished{31058, abstract = {{We consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic billiard system is hyperbolic on its trapped set and the latter is compact and non-grazing the techniques for open hyperbolic systems developed by Dyatlov and Guillarmou can be applied to a smooth model for the discontinuous flow defined by the non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent for the generator of the billiard flow. As an application we prove a meromorphic continuation of weighted zeta functions together with explicit residue formulae. In particular, our results apply to scattering by convex obstacles in the Euclidean plane.}}, author = {{Schütte, Philipp and Weich, Tobias and Delarue, Benjamin}}, title = {{{Resonances and weighted zeta functions for obstacle scattering via smooth models}}}, year = {{2021}}, } @misc{31385, author = {{Hoffmann, Max}}, booktitle = {{Mathematische Semesterberichte}}, pages = {{295–297}}, title = {{{Rezension: Hendrik Kasten und Denis Vogel: Grundlagen der ebenen Geometrie – Eine zugängliche aber exakte Einführung in die ebene Geometrie}}}, doi = {{10.1007/s00591-021-00299-3}}, volume = {{68}}, year = {{2021}}, } @inbook{31364, author = {{Hoffmann, Max}}, booktitle = {{ Lehrinnovationen in der Hochschulmathematik. praxisrelevant – didaktisch fundiert – forschungsbasiert}}, editor = {{Biehler, Rolf and Eichler, Andreas and Hochmuth, Reinhard and Rach, Stefanie and Schaper, Niclas}}, isbn = {{9783662628539}}, issn = {{2197-8751}}, pages = {{179–204}}, publisher = {{Springer Berlin Heidelberg}}, title = {{{Einsatz von Schnittstellenaufgaben in Mathematikveranstaltungen – Praxisbeispiele aus der Universität Paderborn}}}, doi = {{10.1007/978-3-662-62854-6_9}}, year = {{2021}}, } @article{31261, abstract = {{Abstract For a compact Riemannian locally symmetric space $\mathcal M$ of rank 1 and an associated vector bundle $\mathbf V_{\tau }$ over the unit cosphere bundle $S^{\ast }\mathcal M$, we give a precise description of those classical (Pollicott–Ruelle) resonant states on $\mathbf V_{\tau }$ that vanish under covariant derivatives in the Anosov-unstable directions of the chaotic geodesic flow on $S^{\ast }\mathcal M$. In particular, we show that they are isomorphically mapped by natural pushforwards into generalized common eigenspaces of the algebra of invariant differential operators $D(G,\sigma )$ on compatible associated vector bundles $\mathbf W_{\sigma }$ over $\mathcal M$. As a consequence of this description, we obtain an exact band structure of the Pollicott–Ruelle spectrum. Further, under some mild assumptions on the representations $\tau$ and $\sigma$ defining the bundles $\mathbf V_{\tau }$ and $\mathbf W_{\sigma }$, we obtain a very explicit description of the generalized common eigenspaces. This allows us to relate classical Pollicott–Ruelle resonances to quantum eigenvalues of a Laplacian in a suitable Hilbert space of sections of $\mathbf W_{\sigma }$. Our methods of proof are based on representation theory and Lie theory.}}, author = {{Küster, Benjamin and Weich, Tobias}}, issn = {{1073-7928}}, journal = {{International Mathematics Research Notices}}, keywords = {{General Mathematics}}, number = {{11}}, pages = {{8225--8296}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces}}}, doi = {{10.1093/imrn/rnz068}}, volume = {{2021}}, year = {{2021}}, } @article{31576, author = {{Häsel-Weide, Uta and Nührenbürger, Marcus}}, journal = {{Zeitschrift für Grundschulforschung (ZfG)}}, number = {{14}}, pages = {{49--65}}, publisher = {{Springer}}, title = {{{Inklusive Praktiken im Mathematikunterricht. Empirische Analysen von Unterrichtsdiskursen in Einführungsphasen.}}}, year = {{2021}}, } @article{31577, author = {{Häsel-Weide, Uta and Schöttler, Christian}}, issn = {{ 2701-9012}}, journal = {{Zeitschrift für Mathematikdidaktik in Forschung & Praxis (ZMFP)}}, number = {{2}}, title = {{{Das Dezimalsystem verstehen – Bedeutung, Erkenntnisse, Anregungen}}}, year = {{2021}}, } @article{32810, author = {{Li, Jiaao and Ma, Yulai and Shi, Yongtang and Wang, Weifan and Wu, Yezhou}}, issn = {{0195-6698}}, journal = {{European Journal of Combinatorics}}, keywords = {{Discrete Mathematics and Combinatorics}}, publisher = {{Elsevier BV}}, title = {{{On 3-flow-critical graphs}}}, doi = {{10.1016/j.ejc.2021.103451}}, volume = {{100}}, year = {{2021}}, } @article{33278, abstract = {{The kinetic Brownian motion on the sphere bundle of a Riemannian manifold M is a stochastic process that models a random perturbation of the geodesic flow. If M is an orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the L2-spectrum of the infinitesimal generator of a time-rescaled version of the process converges to the Laplace spectrum of the base manifold.}}, author = {{Kolb, Martin and Weich, Tobias and Wolf, Lasse}}, journal = {{Annales Henri Poincaré }}, number = {{4}}, pages = {{1283--1296}}, publisher = {{Springer Science + Business Media}}, title = {{{Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature}}}, volume = {{23}}, year = {{2021}}, }