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Self-similar pattern in coupled parabolic systems as non-equilibrium steady states
Get PDFWe consider reaction-diffusion systems and other related dissipative systems on unbounded domains which would have a Liapunov function (and gradient structure) when posed on a finite domain. In this situation, the system may reach local equilibrium on a rather fast time scale but the infinite amount of mass or energy leads to persistent mass or energy flow for all times. In suitably rescaled variables the system converges to a steady state that corresponds to asymptotically self-similar behavior in the original system
On forward and inverse uncertainty quantification for a model for a magneto mechanical device involving a hysteresis operator
Get PDFModeling real world objects and processes one may has to deal with hysteresis effects but also with uncertainties. Following D. Davino, P. Krejčí, and C. Visone: Fully coupled modeling of magneto-mechanical hysteresis through `thermodynamic' compatibility. Smart Mater. Struct., 22(9), (2013) 0950099, a model for a magnetostrictive material involving a generalized Prandtl- Ishlinskiĭ-operator is considered here. Using results of measurements, some parameters in the model are determined and inverse Uncertainty Quantification (UQ) is used to determine random densities to describe the remaining parameters and their uncertainties. Afterwards, the results are used do perform forward UQ and to compare the results with measured data. This extends some of the results from O. Klein, D. Davino, and C. Visone. On forward and inverse uncertainty quantification for models involving hysteresis operators. Math. Model. Nat. Phenom. 15 (2020) 53
Time-reversible dynamics in a system of two coupled active rotators
Get PDFWe study two coupled active rotators with Kuramoto-type coupling and focus our attention to specific transitional regimes where the coupling is neither attractive nor repulsive. We show that certain such situations at the edge of synchronization can be characterized by the existence of a time-reversal symmetry of the system. We identify two different cases with such a time-reversal symmetry. The first case is characterized by a non-reciprocal attractive/repulsive coupling. The second case is a reciprocal coupling exactly at the edge between attraction and repulsion. We give a detailed description of possible different types of dynamics and bifurcations for both cases. In particular, we show how the time-reversible coupling can induce both oscillation death and oscillation birth to the active rotators. Moreover, we analyse the coexistence of conservative and dissipative regions in phase space, which is a typical feature of systems with a time-reversal symmetry. We show also, how perturbations breaking the time-reversal symmetry and destroying the conservative regions can lead to complicated types of dissipative dynamics such as the emergence of long-period cycles showing a bursting-like behavior
Reduced Basis Approach for Convection-Diffusion Equations with Non-linear Boundary Reaction Conditions
No full textThis paper aims at an efficient strategy to solve drift-diffusion problems with non-linear boundary conditions as they appear, e.g., in heterogeneous catalysis. Since the non-linearity only involves the degrees of freedom along (a part of) the boundary, a reduced basis ansatz is suggested that computes discrete Green’s-like functions for the present drift-diffusion operator such that the global non-linear problem reduces to a smaller non-linear problem for a boundary method. The computed basis functions are completely independent of the non-linearities. Thus, they can be reused for problems with the same differential operator and geometry. Corresponding scenarios might be inverse problems in heterogeneous catalysis but also modeling the effect of different catalysts in the same reaction chamber. The strategy is explained for a mass-conservative finite volume method and demonstrated on a simple numerical example for catalytic CO oxidation
Energie effizienter speichern
No full textWie Batterien grundsätzlich funktionieren, ist schon lange bekannt. Sie exakt zu beschreiben, gestaltet sich allerdings als Mammutaufgabe. Mathematische Modellierung hilft dabei, die Geräte zu verstehen und dadurch zu verbessern
Absence of Percolation in Graphs based on Stationary Point Processes with Degrees Bounded by Two
No full textFoundations of Modern Statistics
No full textThis book contains contributions from the participants of the international conference “Foundations of Modern Statistics” which took place at Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, during November 6–8, 2019, and at Higher School of Economics (HSE University), Moscow, during November 30, 2019. The events were organized in honor of Professor Vladimir Spokoiny on the occasion of his 60th birthday. Vladimir Spokoiny has pioneered the field of adaptive statistical inference and contributed to a variety of its applications. His more than 30 years of research in the field of mathematical statistics had a great influence on the development of the mathematical theory of statistics to its present state. It has inspired many young researchers to start their research in this exciting field of mathematics. The papers contained in this book reflect the broad field of interests of Vladimir Spokoiny: optimal rates and non-asymptotic bounds in nonparametrics, Bayes approaches from a frequentist point of view, optimization, signal processing, and statistical theory motivated by models in applied fields. Materials prepared by famous scientists contain original scientific results, which makes the publication valuable for researchers working in these fields. The book concludes by a conversation of Vladimir Spokoiny with Markus Reiβ and Enno Mammen. This interview gives some background on the life of Vladimir Spokoiny and his many scientific interests and motivations
Chaos spectrum -- semiconductor laser with delayed optical feedback
Get PDFMaximizing the rf bandwidth associated with the chaotic output from tailored operation of nonlinear semiconductor laser systems is an ongoing research effort. The early pioneering research was done in semiconductor laser with delayed optical feedback systems, which continue to be researched. We report numerical simulations of this system, using a travelling wave model. The results provide new insights into the impact of key device parameters affecting the chaos bandwidth and spectrum. Linewidth enhancement factor and the nonlinear gain saturation parameter are found to be the most important parameters when seeking to optimize the chaotic output. We reassess a standard definition being used to report chaos bandwidth. We propose that more spectral information should be reported if numerical and experimental research results are to be of most value into the future. A database from previous experimental study is also analyzed to connect with the predictions of the numerical simulations. This elucidates the links between the chaos bandwidth achieved in real systems and the semiconductor laser parameters. The results inform recommendations for semiconductor laser parameters that will better support broadband chaos generation in whatever semiconductor-gain-medium-based nonlinear system approach is being used. They elucidate the physics of both the envelope and the fine structure of the rf spectrum of coherence collapse
Generalized Iterated-Sums Signatures
No full textWe explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F. Király and H. Oberhauser. In particular, we show how to recover the character property of the associated linear map over the tensor algebra by considering a deformed quasi-shuffle product of words on the latter. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties
Trajectorial Dissipation of -Entropies for Interacting Particle Systems
No full textA classical approach for the analysis of the longtime behavior of Markov processes is to consider suitable Lyapunov functionals like the variance or more generally Φ -entropies. Via purely analytic arguments it can be shown that these functionals are indeed non-increasing in time under quite general assumptions on the process. We complement these classical results via a more probabilistic approach and show that dissipation is already present on the level of individual trajectories for spatially-extended systems of infinitely many interacting particles with arbitrary underlying geometry and compact local spin spaces. This extends previous results from the setting of finite-state Markov chains or diffusions in Rn to an infinite-dimensional setting with weak assumptions on the dynamics