@article{16715,
author = {Bittracher, Andreas and Koltai, Péter and Klus, Stefan and Banisch, Ralf and Dellnitz, Michael and Schütte, Christof},
issn = {0938-8974},
journal = {Journal of Nonlinear Science},
pages = {471--512},
title = {{Transition Manifolds of Complex Metastable Systems}},
doi = {10.1007/s00332-017-9415-0},
volume = {28},
year = {2018},
}
@article{19935,
abstract = {A bifurcation is a qualitative change in a family of solutions to an equation produced by varying parameters. In contrast to the local bifurcations of dynamical systems that are often related to a change in the number or stability of equilibria, bifurcations of boundary value problems are global in nature and may not be related to any obvious change in dynamical behaviour. Catastrophe theory is a well-developed framework which studies the bifurcations of critical points of functions. In this paper we study the bifurcations of solutions of boundary-value problems for symplectic maps, using the language of (finite-dimensional) singularity theory. We associate certain such problems with a geometric picture involving the intersection of Lagrangian submanifolds, and hence with the critical points of a suitable generating function. Within this framework, we then study the effect of three special cases: (i) some common boundary conditions, such as Dirichlet boundary conditions for second-order systems, restrict the possible types of bifurcations (for example, in generic planar systems only the A-series beginning with folds and cusps can occur); (ii) integrable systems, such as planar Hamiltonian systems, can exhibit a novel periodic pitchfork bifurcation; and (iii) systems with Hamiltonian symmetries or reversing symmetries can exhibit restricted bifurcations associated with the symmetry. This approach offers an alternative to the analysis of critical points in function spaces, typically used in the study of bifurcation of variational problems, and opens the way to the detection of more exotic bifurcations than the simple folds and cusps that are often found in examples. },
author = {McLachlan, Robert I and Offen, Christian},
issn = {0951-7715},
journal = {Nonlinearity},
pages = {2895--2927},
title = {{Bifurcation of solutions to Hamiltonian boundary value problems}},
doi = {10.1088/1361-6544/aab630},
year = {2018},
}
@article{21942,
author = {Boninsegna, Lorenzo and Nüske, Feliks and Clementi, Cecilia},
issn = {0021-9606},
journal = {The Journal of Chemical Physics},
title = {{Sparse learning of stochastic dynamical equations}},
doi = {10.1063/1.5018409},
year = {2018},
}
@unpublished{16292,
abstract = {In a recent article, we presented a framework to control nonlinear partial
differential equations (PDEs) by means of Koopman operator based reduced models
and concepts from switched systems. The main idea was to transform a control
system into a set of autonomous systems for which the optimal switching
sequence has to be computed. These individual systems can be approximated very
efficiently by reduced order models obtained from data, and one can guarantee
equality of the full and the reduced objective function under certain
assumptions. In this article, we extend these results to continuous control
inputs using convex combinations of multiple Koopman operators corresponding to
constant controls, which results in a bilinear control system. Although
equality of the objectives can be carried over when the PDE depends linearly on
the control, we show that this approach is also valid in other scenarios using
several flow control examples of varying complexity.},
author = {Peitz, Sebastian},
booktitle = {arXiv:1801.06419},
title = {{Controlling nonlinear PDEs using low-dimensional bilinear approximations obtained from data}},
year = {2018},
}
@inproceedings{8575,
abstract = {The transition from high school to university mathematics has proven to be difficult for many students but especially for pre-service secondary teachers. To support these students at mastering this transition, various universities have introduced support measures of various kinds. The WiGeMath project developed a taxonomy that makes it possible to describe and compare these measures concerning their goals as well as their frame characteristics. We will exemplify the use of the taxonomy in the description of one specific innovative measure that was part of the WiGeMath evaluations. Moreover, we will present first results concerning the goal-fulfilment of this measure concerning affective characteristics of the student cohort and their predominant beliefs.},
author = {Kuklinski, Christiane and Leis, Elena and Liebendörfer, Michael and Hochmuth, Reinhard and Biehler, Rolf and Lankeit, Elisa and Neuhaus, Silke and Schaper, Niclas and Schürmann, Mirko},
booktitle = {Proceedings of the Second Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2018, 5-7 April 2018)},
editor = {Durand-Guerrier, V. and Hochmuth, R. and Goodchild, S. and Hogstad, N.M.},
keyword = {Beliefs., Motivational developments, Novel approaches to teaching, Teacher education, Transition to and across university mathematics},
pages = {527--536},
publisher = {INDRUM Network, University of Agder},
title = {{Evaluating Innovative Measures in University Mathematics – The Case of Affective Outcomes in a Lecture focused on Problem-Solving}},
year = {2018},
}
@article{10129,
abstract = {There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the 5-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for 3-edge-colorable cubic graphs, cubic graphs which are not 3-edge-colorable, often called snarks, play a key role in this context. Here, we survey parameters measuring how far apart a non 3-edge-colorable graph is from being 3-edge-colorable. We study their interrelation and prove some new results. Besides getting new insight into the structure of snarks, we show that such measures give partial results with respect to these important conjectures. The paper closes with a list of open problems and conjectures.},
author = {Fiol, M. A. and Mazzuoccolo, Guiseppe and Steffen, Eckhard},
journal = {The Electronic Journal of Combinatorics},
keyword = {Cubic graph, Tait coloring, Snark, Boole coloring, Berge's conjecture, Tutte's 5-flow conjecture},
number = {4},
title = {{Measures of Edge-Uncolorability of Cubic Graphs}},
volume = {25},
year = {2018},
}
@article{19943,
abstract = {In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we present a convenient, coordinate free framework to analyse separated Lagrangian boundary value problems which include classical Dirichlet, Neumann and Robin boundary value problems. The framework is then used to prove the existence of obstructions arising from conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems. Under non-degeneracy conditions, a group action by conformal symplectic symmetries has the effect that the flow map cannot degenerate in a direction which is tangential to the action. This imposes restrictions on which singularities can occur in boundary value problems. Our results generalise classical results about conjugate loci on Riemannian manifolds to a large class of Hamiltonian boundary value problems with, for example, scaling symmetries. },
author = {McLachlan, Robert I and Offen, Christian},
journal = {New Zealand Journal of Mathematics},
keyword = {Hamiltonian boundary value problems, singularities, conformal symplectic geometry, catastrophe theory, conjugate loci},
pages = {83--99},
title = {{Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci}},
volume = {48},
year = {2018},
}
@article{21943,
author = {Hruska, Eugen and Abella, Jayvee R. and Nüske, Feliks and Kavraki, Lydia E. and Clementi, Cecilia},
issn = {0021-9606},
journal = {The Journal of Chemical Physics},
title = {{Quantitative comparison of adaptive sampling methods for protein dynamics}},
doi = {10.1063/1.5053582},
year = {2018},
}
@article{10132,
author = {Jin, Ligang and Mazzuoccolo, Giuseppe and Steffen, Eckhard},
issn = {1234-3099},
journal = {Discussiones Mathematicae Graph Theory},
pages = {165--175},
title = {{Cores, joins and the Fano-flow conjectures}},
doi = {10.7151/dmgt.1999},
volume = {38},
year = {2018},
}
@unpublished{16293,
abstract = {Kernel transfer operators, which can be regarded as approximations of
transfer operators such as the Perron-Frobenius or Koopman operator in
reproducing kernel Hilbert spaces, are defined in terms of covariance and
cross-covariance operators and have been shown to be closely related to the
conditional mean embedding framework developed by the machine learning
community. The goal of this paper is to show how the dominant eigenfunctions of
these operators in combination with gradient-based optimization techniques can
be used to detect long-lived coherent patterns in high-dimensional time-series
data. The results will be illustrated using video data and a fluid flow
example.},
author = {Klus, Stefan and Peitz, Sebastian and Schuster, Ingmar},
booktitle = {arXiv:1805.10118},
title = {{Analyzing high-dimensional time-series data using kernel transfer operator eigenfunctions}},
year = {2018},
}