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 -