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    Genealogical properties of spatial models in Population Genetics
    (Hannover : Technische Informationsbibliothek, 2023-09) Wirtz, Johannes
    At the interface between Phylo- and Population Genetics, and recently heavily inspired by Epidemonology, the discipline of Phylogeography comprises modelling techniques from classical theoretical biology and combines them with a spatial (2D or 3D) aspect, with the purpose of utilizing geographical information in the analysis to understand the evolutionary history of a biological system or aspects of virology such as directionality and seasonality in pandemic outbreaks [1, 2, 3, 4]. An prime example of this are datasets that take into account the sampling locations of its components (geo-referenced genomic data). In this project, we have focused on the model called "spatial Lambda-Fleming-Viot process" ( V [5, 6]) and analzed its statistical properties forward in time as well as in the ancestral (dual) process, with results that may be used for parameter inference. Of particlar interest was the spatial variance, denoted , a parameter controlling the speed at which genetic information is spread across space and therefore an analog of the reproduction number (R0) used in epidemonology e.g. to assess the infectiousness of differing viral strains. We explored the relation of this parameter to the time to coalescence between lineage pairs in this model and described methods of estimating it from sampled data under different circumstances. We have furthermore investigated similarities and differences between this model and classical models in Population Genetics, particularly Birth-Death processes, which are heavily used for all kinds of biological inference problems, but do not by themselves feature a spatial component. We compared the Vto a variant of the Birth-Death process where the location of a live individual changes over the course of its lifetime according to a Brownian motion. This process is not as easily viewed backward in time as the V, but the genalogical process is accessible by Markov-Chain Monte Carlosimulation, as the likelihoods of ancestral positions and branch lengths are easily calculated, making this model easily applicable to data. Our analysis highlights the analogy between the two processes forward in time as well as backward in time; on the other hand, we also observed a divergent behavior of the two models when no prior on the phylogenetic time scale was assumed. Lastly, this project has given rise to a study of combinatorial properties of tree shapes relevant to the V, the Birth-Death and other biological processes. In particular, we were able to identify the combinatorial class genealogical trees generated from these processes belong to and verify a conjecture regarding their enumeration. Preliminary versions of software tools for the aforementioned inference have also been provided.
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    Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs
    (Orlando, Fla. : Elsevier, 2023) Alphonse, Amal; Caetano, Diogo; Djurdjevac, Ana; Elliott, Charles M.
    We develop a functional framework suitable for the treatment of partial differential equations and variational problems on evolving families of Banach spaces. We propose a definition for the weak time derivative that does not rely on the availability of a Hilbertian structure and explore conditions under which spaces of weakly differentiable functions (with values in an evolving Banach space) relate to classical Sobolev–Bochner spaces. An Aubin–Lions compactness result is proved. We analyse concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we explicitly provide the definition of the time derivative and verify isomorphism properties with the aforementioned Sobolev–Bochner spaces. We conclude with the proof of well posedness for a class of nonlinear monotone problems on an abstract evolving space (generalising the evolutionary p-Laplace equation on a moving domain or surface) and identify some additional problems that can be formulated with the setting developed in this work.
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    Can One Series of Self-Organized Nanoripples Guide Another Series of Self-Organized Nanoripples during Ion Bombardment: From the Perspective of Power Spectral Density Entropy?
    (Basel : MDPI, 2023) Li, Hengbo; Li, Jinyu; Yang, Gaoyuan; Liu, Ying; Frost, Frank; Hong, Yilin
    Ion bombardment (IB) is a promising nanofabrication tool for self-organized nanostructures. When ions bombard a nominally flat solid surface, self-organized nanoripples can be induced on the irradiated target surface, which are called intrinsic nanoripples of the target material. The degree of ordering of nanoripples is an outstanding issue to be overcome, similar to other self-organization methods. In this study, the IB-induced nanoripples on bilayer systems with enhanced quality are revisited from the perspective of guided self-organization. First, power spectral density (PSD) entropy is introduced to evaluate the degree of ordering of the irradiated nanoripples, which is calculated based on the PSD curve of an atomic force microscopy image (i.e., the Fourier transform of the surface height. The PSD entropy can characterize the degree of ordering of nanoripples). The lower the PSD entropy of the nanoripples is, the higher the degree of ordering of the nanoripples. Second, to deepen the understanding of the enhanced quality of nanoripples on bilayer systems, the temporal evolution of the nanoripples on the photoresist (PR)/antireflection coating (ARC) and Au/ARC bilayer systems are compared with those of single PR and ARC layers. Finally, we demonstrate that a series of intrinsic IB-induced nanoripples on the top layer may act as a kind of self-organized template to guide the development of another series of latent IB-induced nanoripples on the underlying layer, aiming at improving the ripple ordering. The template with a self-organized nanostructure may alleviate the critical requirement for periodic templates with a small period of ~100 nm. The work may also provide inspiration for guided self-organization in other fields.
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    Optimality Conditions and Moreau-Yosida Regularization for Almost Sure State Constraints
    (Paris : EDP Sciences, 2022) Geiersbach, Caroline; Hintermüller, Michael
    We analyze a potentially risk-averse convex stochastic optimization problem, where the control is deterministic and the state is a Banach-valued essentially bounded random variable. We obtain strong forms of necessary and sufficient optimality conditions for problems subject to equality and conical constraints. We propose a Moreau-Yosida regularization for the conical constraint and show consistency of the optimality conditions for the regularized problem as the regularization parameter is taken to infinity.
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    On the Regularity of Weak Solutions to Time-Periodic Navier–Stokes Equations in Exterior Domains
    (Basel : MDPI, 2022) Eiter, Thomas
    Consider the time-periodic viscous incompressible fluid flow past a body with non-zero velocity at infinity. This article gives sufficient conditions such that weak solutions to this problem are smooth. Since time-periodic solutions do not have finite kinetic energy in general, the well-known regularity results for weak solutions to the corresponding initial-value problem cannot be transferred directly. The established regularity criterion demands a certain integrability of the purely periodic part of the velocity field or its gradient, but it does not concern the time mean of these quantities.