TY - JOUR A1 - Gerlach, Moritz A1 - Glück, Jochen A1 - Kunze, Markus T1 - Stability of transition semigroups and applications to parabolic equations JF - Transactions of the American Mathematical Society N2 - This paper deals with the long-term behavior of positive operator semigroups on spaces of bounded functions and of signed measures, which have applications to parabolic equations with unbounded coefficients and to stochas-tic analysis. The main results are a Tauberian type theorem characterizing the convergence to equilibrium of strongly Feller semigroups and a generalization of a classical convergence theorem of Doob. None of these results requires any kind of time regularity of the semigroup. KW - Transition probabilities KW - strong Feller property KW - asymptotic KW - behavior KW - invariant measure KW - parabolic equations Y1 - 2023 U6 - https://doi.org/10.1090/tran/8620 SN - 0002-9947 SN - 1088-6850 VL - 376 IS - 1 SP - 153 EP - 180 PB - American Mathematical Soc. CY - Providence ER - TY - JOUR A1 - Dimitrova, Ilinka A1 - Koppitz, Jörg T1 - On relative ranks of the semigroup of orientation-preserving transformations on infinite chain with restricted range JF - Communications in algebra N2 - Let X be an infinite linearly ordered set and let Y be a nonempty subset of X. We calculate the relative rank of the semigroup OP(X,Y) of all orientation-preserving transformations on X with restricted range Y modulo the semigroup O(X,Y) of all order-preserving transformations on X with restricted range Y. For Y = X, we characterize the relative generating sets of minimal size. KW - Order-preserving transformations KW - orientation-preserving KW - transformations KW - relative rank KW - restricted range KW - transformation KW - semigroups on infinite chain Y1 - 2022 U6 - https://doi.org/10.1080/00927872.2021.2000998 SN - 0092-7872 SN - 1532-4125 VL - 50 IS - 5 SP - 2157 EP - 2168 PB - Taylor & Francis Group CY - Philadelphia ER - TY - JOUR A1 - Dimitrova, Ilinka A1 - Koppitz, Jörg T1 - On relative ranks of the semigroup of orientation-preserving transformations on infinite chains JF - Asian-European journal of mathematics N2 - In this paper, we determine the relative rank of the semigroup OP(X) of all orientation-preserving transformations on infinite chains modulo the semigroup O(X) of all order-preserving transformations. KW - Transformation semigroups on infinite chains KW - order-preserving KW - transformations KW - orientation-preserving transformations KW - relative rank Y1 - 2020 U6 - https://doi.org/10.1142/S1793557121501461 SN - 1793-5571 SN - 1793-7183 VL - 14 IS - 08 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Kretzschmar, Mirjam E. A1 - Ashby, Ben A1 - Fearon, Elizabeth A1 - Overton, Christopher E. A1 - Panovska-Griffiths, Jasmina A1 - Pellis, Lorenzo A1 - Quaife, Matthew A1 - Rozhnova, Ganna A1 - Scarabel, Francesca A1 - Stage, Helena B. A1 - Swallow, Ben A1 - Thompson, Robin N. A1 - Tildesley, Michael J. A1 - Villela, Daniel Campos T1 - Challenges for modelling interventions for future pandemics JF - Epidemics N2 - Mathematical modelling and statistical inference provide a framework to evaluate different non-pharmaceutical and pharmaceutical interventions for the control of epidemics that has been widely used during the COVID-19 pandemic. In this paper, lessons learned from this and previous epidemics are used to highlight the challenges for future pandemic control. We consider the availability and use of data, as well as the need for correct parameterisation and calibration for different model frameworks. We discuss challenges that arise in describing and distinguishing between different interventions, within different modelling structures, and allowing both within and between host dynamics. We also highlight challenges in modelling the health economic and political aspects of interventions. Given the diversity of these challenges, a broad variety of interdisciplinary expertise is needed to address them, combining mathematical knowledge with biological and social insights, and including health economics and communication skills. Addressing these challenges for the future requires strong cross disciplinary collaboration together with close communication between scientists and policy makers. KW - Mathematical models KW - Pandemics KW - Pharmaceutical interventions KW - Non-pharmaceutical interventions KW - Policy support Y1 - 2022 U6 - https://doi.org/10.1016/j.epidem.2022.100546 SN - 1755-4365 SN - 1878-0067 VL - 38 PB - Elsevier CY - Amsterdam ER - TY - THES A1 - Sareeto, Apatsara T1 - Algebraic properties of a subsemigroup of the symmetric inverse semigroup Y1 - 2024 ER - TY - JOUR A1 - Gerlach, Moritz Reinhardt A1 - Glück, Jochen T1 - On a convergence theorem for semigroups of positive integral operators JF - Comptes Rendus Mathematique N2 - We give a new and very short proof of a theorem of Greiner asserting that a positive and contractive -semigroup on an -space is strongly convergent in case it has a strictly positive fixed point and contains an integral operator. Our proof is a streamlined version of a much more general approach to the asymptotic theory of positive semigroups developed recently by the authors. Under the assumptions of Greiner's theorem, this approach becomes particularly elegant and simple. We also give an outlook on several generalisations of this result. Y1 - 2017 U6 - https://doi.org/10.1016/j.crma.2017.07.017 SN - 1631-073X SN - 1778-3569 VL - 355 SP - 973 EP - 976 PB - Elsevier CY - Paris ER - TY - JOUR A1 - Gerlach, Moritz Reinhardt T1 - Convergence of dynamics and the Perron-Frobenius operator JF - Israel Journal of Mathematics N2 - We complete the picture how the asymptotic behavior of a dynamical system is reflected by properties of the associated Perron-Frobenius operator. Our main result states that strong convergence of the powers of the Perron-Frobenius operator is equivalent to setwise convergence of the underlying dynamic in the measure algebra. This situation is furthermore characterized by uniform mixing-like properties of the system. Y1 - 2018 U6 - https://doi.org/10.1007/s11856-018-1671-7 SN - 0021-2172 SN - 1565-8511 VL - 225 IS - 1 SP - 451 EP - 463 PB - Hebrew univ magnes press CY - Jerusalem ER - TY - JOUR A1 - Gerlach, Moritz Reinhardt A1 - Glück, Jochen T1 - Convergence of positive operator semigroups JF - Transactions of the American Mathematical Society N2 - We present new conditions for semigroups of positive operators to converge strongly as time tends to infinity. Our proofs are based on a novel approach combining the well-known splitting theorem by Jacobs, de Leeuw, and Glicksberg with a purely algebraic result about positive group representations. Thus, we obtain convergence theorems not only for one-parameter semigroups but also for a much larger class of semigroup representations. Our results allow for a unified treatment of various theorems from the literature that, under technical assumptions, a bounded positive C-0-semigroup containing or dominating a kernel operator converges strongly as t ->infinity. We gain new insights into the structure theoretical background of those theorems and generalize them in several respects; especially we drop any kind of continuity or regularity assumption with respect to the time parameter. KW - Positive semigroups KW - semigroup representations KW - asymptotic behavior KW - kernel operator Y1 - 2019 U6 - https://doi.org/10.1090/tran/7836 SN - 0002-9947 SN - 1088-6850 VL - 372 IS - 9 SP - 6603 EP - 6627 PB - American Mathematical Soc. CY - Providence ER - TY - JOUR A1 - Edeko, Nikolai A1 - Gerlach, Moritz Reinhardt A1 - Kühner, Viktoria T1 - Measure-preserving semiflows and one-parameter Koopman semigroups JF - Semigroup forum N2 - For a finite measure space X, we characterize strongly continuous Markov lattice semigroups on Lp(X) by showing that their generator A acts as a derivation on the dense subspace D(A)L(X). We then use this to characterize Koopman semigroups on Lp(X) if X is a standard probability space. In addition, we show that every measurable and measure-preserving flow on a standard probability space is isomorphic to a continuous flow on a compact Borel probability space. KW - Measure-preserving semiflow KW - Koopman semigroup KW - Derivation KW - Topological model Y1 - 2019 U6 - https://doi.org/10.1007/s00233-018-9960-3 SN - 0037-1912 SN - 1432-2137 VL - 98 IS - 1 SP - 48 EP - 63 PB - Springer CY - New York ER - TY - JOUR A1 - Gerlach, Moritz Reinhardt A1 - Glück, Jochen T1 - Lower bounds and the asymptotic behaviour of positive operator semigroups JF - Ergodic theory and dynamical systems N2 - If (T-t) is a semigroup of Markov operators on an L-1-space that admits a nontrivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as t -> infinity. In this article we generalize and improve this result in several respects. First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result, we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalize a theorem of Ding on semigroups of Frobenius-Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results. Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups. Y1 - 2017 U6 - https://doi.org/10.1017/etds.2017.9 SN - 0143-3857 SN - 1469-4417 VL - 38 SP - 3012 EP - 3041 PB - Cambridge Univ. Press CY - New York ER - TY - JOUR A1 - Gerlach, Moritz Reinhardt A1 - Glück, Jochen T1 - Mean ergodicity vs weak almost periodicity JF - Studia mathematica N2 - We provide explicit examples of positive and power-bounded operators on c(0) and l(infinity) which are mean ergodic but not weakly almost periodic. As a consequence we prove that a countably order complete Banach lattice on which every positive and power-bounded mean ergodic operator is weakly almost periodic is necessarily a KB-space. This answers several open questions from the literature. Finally, we prove that if T is a positive mean ergodic operator with zero fixed space on an arbitrary Banach lattice, then so is every power of T . KW - positive operators KW - weakly almost periodic KW - order continuous norm KW - KB-space KW - mean ergodic Y1 - 2019 U6 - https://doi.org/10.4064/sm170918-20-3 SN - 0039-3223 SN - 1730-6337 VL - 248 IS - 1 SP - 45 EP - 56 PB - Polska Akademia Nauk, Instytut Matematyczny CY - Warszawa ER - TY - CHAP A1 - Hiort, Pauline A1 - Hugo, Julian A1 - Zeinert, Justus A1 - Müller, Nataniel A1 - Kashyap, Spoorthi A1 - Rajapakse, Jagath C. A1 - Azuaje, Francisco A1 - Renard, Bernhard Y. A1 - Baum, Katharina T1 - DrDimont: explainable drug response prediction from differential analysis of multi-omics networks T2 - Bioinformatics N2 - Motivation: While it has been well established that drugs affect and help patients differently, personalized drug response predictions remain challenging. Solutions based on single omics measurements have been proposed, and networks provide means to incorporate molecular interactions into reasoning. However, how to integrate the wealth of information contained in multiple omics layers still poses a complex problem. Results: We present DrDimont, Drug response prediction from Differential analysis of multi-omics networks. It allows for comparative conclusions between two conditions and translates them into differential drug response predictions. DrDimont focuses on molecular interactions. It establishes condition-specific networks from correlation within an omics layer that are then reduced and combined into heterogeneous, multi-omics molecular networks. A novel semi-local, path-based integration step ensures integrative conclusions. Differential predictions are derived from comparing the condition-specific integrated networks. DrDimont's predictions are explainable, i.e. molecular differences that are the source of high differential drug scores can be retrieved. We predict differential drug response in breast cancer using transcriptomics, proteomics, phosphosite and metabolomics measurements and contrast estrogen receptor positive and receptor negative patients. DrDimont performs better than drug prediction based on differential protein expression or PageRank when evaluating it on ground truth data from cancer cell lines. We find proteomic and phosphosite layers to carry most information for distinguishing drug response. Y1 - 2022 U6 - https://doi.org/10.1093/bioinformatics/btac477 SN - 1367-4803 SN - 1367-4811 VL - 38 SP - ii113 EP - ii119 PB - Oxford Univ. Press CY - Oxford ER - TY - JOUR A1 - Julien, Bärenzung A1 - Matthias, Holschneider A1 - Saynisch-Wagner, Jan A1 - Thomas, Maik T1 - Kalmag: a high spatio-temporal model of the geomagnetic field JF - Earth, planets and space N2 - We present the extension of the Kalmag model, proposed as a candidate for IGRF-13, to the twentieth century. The dataset serving its derivation has been complemented by new measurements coming from satellites, ground-based observatories and land, marine and airborne surveys. As its predecessor, this version is derived from a combination of a Kalman filter and a smoothing algorithm, providing mean models and associated uncertainties. These quantities permit a precise estimation of locations where mean solutions can be considered as reliable or not. The temporal resolution of the core field and the secular variation was set to 0.1 year over the 122 years the model is spanning. Nevertheless, it can be shown through ensembles a posteriori sampled, that this resolution can be effectively achieved only by a limited amount of spatial scales and during certain time periods. Unsurprisingly, highest accuracy in both space and time of the core field and the secular variation is achieved during the CHAMP and Swarm era. In this version of Kalmag, a particular effort was made for resolving the small-scale lithospheric field. Under specific statistical assumptions, the latter was modeled up to spherical harmonic degree and order 1000, and signal from both satellite and survey measurements contributed to its development. External and induced fields were jointly estimated with the rest of the model. We show that their large scales could be accurately extracted from direct measurements whenever the latter exhibit a sufficiently high temporal coverage. Temporally resolving these fields down to 3 hours during the CHAMP and Swarm missions, gave us access to the link between induced and magnetospheric fields. In particular, the period dependence of the driving signal on the induced one could be directly observed. The model is available through various physical and statistical quantities on a dedicated website at https://ionocovar.agnld.uni-potsdam.de/Kalmag/. KW - geomagnetic field KW - lithospheric field KW - secular variation KW - magnetospheric field KW - induced field KW - assimilation KW - Kalman filter KW - machine learning Y1 - 2022 U6 - https://doi.org/10.1186/s40623-022-01692-5 SN - 1880-5981 VL - 74 IS - 1 PB - Springer CY - New York ER - TY - JOUR A1 - Tian, Peibo A1 - Liang, Yingjie T1 - Material coordinate driven variable-order fractal derivative model of water anomalous adsorption in swelling soil JF - Chaos, solitons & fractals N2 - The diffusion process of water in swelling (expansive) soil often deviates from normal Fick diffusion and belongs to anomalous diffusion. The process of water adsorption by swelling soil often changes with time, in which the microstructure evolves with time and the absorption rate changes along a fractal dimension gradient function. Thus, based on the material coordinate theory, this paper proposes a variable order derivative fractal model to describe the cumulative adsorption of water in the expansive soil, and the variable order is time dependent linearly. The cumulative adsorption is a power law function of the anomalous sorptivity, and patterns of the variable order. The variable-order fractal derivative model is tested to describe the cumulative adsorption in chernozemic surface soil, Wunnamurra clay and sandy loam. The results show that the fractal derivative model with linearly time dependent variable-order has much better accuracy than the fractal derivative model with a constant derivative order and the integer order model in the application cases. The derivative order can be used to distinguish the evolution of the anomalous adsorption process. The variable-order fractal derivative model can serve as an alternative approach to describe water anomalous adsorption in swelling soil. KW - water adsorption KW - variable order KW - fractal derivative KW - swelling soil KW - material coordinate Y1 - 2022 U6 - https://doi.org/10.1016/j.chaos.2022.112754 SN - 0960-0779 SN - 1873-2887 VL - 164 PB - Elsevier CY - Oxford ER - TY - THES A1 - Reimann, Hans T1 - Towards robust inference for Bayesian filtering of linear Gaussian dynamical systems subject to additive change T1 - Hin zu einer robusten Inferenz für Bayes'sche Filterung linearer gaußscher dynamischer Systeme bei additiver Veränderung N2 - State space models enjoy wide popularity in mathematical and statistical modelling across disciplines and research fields. Frequent solutions to problems of estimation and forecasting of a latent signal such as the celebrated Kalman filter hereby rely on a set of strong assumptions such as linearity of system dynamics and Gaussianity of noise terms. We investigate fallacy in mis-specification of the noise terms, that is signal noise and observation noise, regarding heavy tailedness in that the true dynamic frequently produces observation outliers or abrupt jumps of the signal state due to realizations of these heavy tails not considered by the model. We propose a formalisation of observation noise mis-specification in terms of Huber’s ε-contamination as well as a computationally cheap solution via generalised Bayesian posteriors with a diffusion Stein divergence loss resulting in the diffusion score matching Kalman filter - a modified algorithm akin in complexity to the regular Kalman filter. For this new filter interpretations of novel terms, stability and an ensemble variant are discussed. Regarding signal noise mis-specification, we propose a formalisation in the frame work of change point detection and join ideas from the popular CUSUM algo- rithm with ideas from Bayesian online change point detection to combine frequent reliability constraints and online inference resulting in a Gaussian mixture model variant of multiple Kalman filters. We hereby exploit open-end sequential probability ratio tests on the evidence of Kalman filters on observation sub-sequences for aggregated inference under notions of plausibility. Both proposed methods are combined to investigate the double mis-specification problem and discussed regarding their capabilities in reliable and well-tuned uncertainty quantification. Each section provides an introduction to required terminology and tools as well as simulation experiments on the popular target tracking task and the non-linear, chaotic Lorenz-63 system to showcase practical performance of theoretical considerations. N2 - Modelle im Zustandsraum finden breite Anwendung in der mathematischen und statistischen Modellierung verschiedener Disziplinen und Forschungsgebiete. Häufige Lösung von Problemen der Schätzung und Vorhersage von latenten Signalen wie der populäre Kalman Filter benötigen hierbei eine Reihe von starken annahmen wie Linearität der Dynamiken des Systems und Normalität der Fehlerterme. Wir untersuchen Hürden in der Modellierung durch Mis-Spezifizierung der Fehlerterme, hier des Signalfehlers und des Beobachtungsfehlers, bezüglich schweren Rändern. Die wahre Dynamik produziert häufig Beobachtungsausreißer oder plötzliche Sprünge des Signals als Realisationen dieser schweren Ränder, diese werden aber nicht durch das Modell berücksichtigt. Wir schlagen eine Formalisierung der Mis-Spezifizierung des Beobachtungsfehlers im Sinne von Hubers Epsilon-Kontaminierung sowie eine rechnerisch kosteneffiziente Lösung vor. Diese Lösung durch generalisierte Bayes'sche a-posteriori Verteilungen mit einer Diffusion Stein Divergenz resultiert in dem Diffusion-Score-Matching Kalman Filter - ein modifizierter Algorithmus ähnlich dem regulären Kalman Filter in Komplexität. Für diesen neuen Filter diskutieren wir Interpretationen neuer Terme, der Langzeitstabilität und eine Ensemble Variante. Bezüglich der Mis-Spezifizierung des Signalfehlers schlagen wir eine Formalisierung durch Wechselpunktdetektierung vor und verknüpfen Ideen des populären CUSUM Algorithmus mit Ideen der Bayes'schen Online Wechselpunktdetektierung um Eigenschaften in Zuverlässigkeit und Online-Inferenz zu erhalten. Das Ergebnis ist eine Gaussian-Mixture-Model Variante aus mehreren Kalman Filtern. Hierbei nutzen wir explizit die Open-End sequentielle Probability-Ratio-Tests zwischen den verschiedenen Kalman Filtern, um Inferenzen unter Plausibiltätsargumenten zu aggregieren. Beide vorgeschlagenen Ansätze werden gemeinsam untersucht für den Fall der zweifachen Mis-Spezifizierung und die Ergebnisse werden diskutiert bezüglich ihrer Zuverlässigkeit und Unsicherheitsquantifizierung. Jedes Kapitel beinhaltet eine Einführung in die notwendige Terminologie und mathematischen Werkzeuge, sowie Simulationsexperimente für das populäre Beispiel des Target Trackings und das nicht-lineare, chaotische Lorenz-63 System, um die praktische Performance die Theorie zu veranschaulichen. KW - statistical model Mis-specification KW - robust filtering KW - state space change point detection KW - Bayesian filtering KW - Bayesian modelling KW - Bayes'sche Modellierung KW - robustes Filtern KW - Zustandsraum Wechselpunktdetektierung KW - Mis-Spezifizierung statistischer Modelle Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-649469 ER - TY - JOUR A1 - Hanisch, Florian A1 - Strohmaier, Alexander A1 - Waters, Alden T1 - A relative trace formula for obstacle scattering JF - Duke mathematical journal N2 - We consider the case of scattering by several obstacles in Rd for d ≥ 2. In this setting, the absolutely continuous part of the Laplace operator Δ with Dirichlet boundary conditions and the free Laplace operator Δ0 are unitarily equivalent. For suitable functions that decay sufficiently fast, we have that the difference g(Δ) - g(Δ0) is a trace-class operator and its trace is described by the Krein spectral shift function. In this article, we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles, we consider the Laplace operators Δ1 and Δ2 obtained by imposing Dirichlet boundary conditions only on one of the objects. Our main result in this case states that then g(Δ) - g(Δ1) - g(Δ2) C g(Δ0) is a trace-class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman–Krein formula. In case g(x) D x 2 , 1 the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators. Such integrals have been derived in the physics literature using nonrigorous path integral derivations and our formula provides both a rigorous justification as well as a generalization. KW - Birman–Krein formula KW - Casimir energy KW - obstacle scattering KW - trace formula Y1 - 2022 U6 - https://doi.org/10.1215/00127094-2022-0053 SN - 0012-7094 SN - 1547-7398 VL - 171 IS - 11 SP - 2233 EP - 2274 PB - Duke Univ. Press CY - Durham, NC ER - TY - JOUR A1 - Maoutsa, Dimitra Despoina A1 - Opper, Manfred T1 - Deterministic particle flows for constraining stochastic nonlinear systems JF - Physical Review Research / American Physical Society N2 - Devising optimal interventions for constraining stochastic systems is a challenging endeavor that has to confront the interplay between randomness and dynamical nonlinearity. Existing intervention methods that employ stochastic path sampling scale poorly with increasing system dimension and are slow to converge. Here we propose a generally applicable and practically feasible methodology that computes the optimal interventions in a noniterative scheme. We formulate the optimal dynamical adjustments in terms of deterministically sampled probability flows approximated by an interacting particle system. Applied to several biologically inspired models, we demonstrate that our method provides the necessary optimal controls in settings with terminal, transient, or generalized collective state constraints and arbitrary system dynamics. Y1 - 2022 U6 - https://doi.org/10.1103/PhysRevResearch.4.043035 SN - 2643-1564 VL - 4 IS - 4 PB - American Physical Society CY - College Park ER -