## Institut für Mathematik

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Purpose Systematic comparison of analysis methods of clinical microdialysis data for impact on target-site drug exposure and response. Methods 39 individuals received a 500 mg levofloxacin short-term infusion followed by 24-h dense sampling in plasma and microdialysate collection in interstitial space fluid (ISF). ISF concentrations were leveraged using non-compartmental (NCA) and compartmental analysis (CA) via (ii) relative recovery correction at midpoint of the collection interval (midpoint-NCA, midpoint-CA) and (ii) dialysate-based integrals of time (integral-CA). Exposure and adequacy of community-acquired pneumonia (CAP) therapy via pharmacokinetic/pharmacodynamic target-attainment (PTA) analysis were compared between approaches. Results Individual AUC(ISF) estimates strongly varied for midpoint-NCA and midpoint-CA (>= 52.3%CV) versus integral-CA (<= 32.9%CV) owing to separation of variability in PK parameters (midpoint-CA = 46.5%-143%CVPK, integral-CA = 26.4%-72.6%CVPK) from recovery-related variability only in integral-CA (41.0%-50.3%CVrecovery). This also led to increased variability of AUC(plasma) for midpoint-CA (56.0%CV) versus midpoint-NCA and integral-CA (<= 33.0%CV), and inaccuracy of predictive model performance of midpoint-CA in plasma (visual predictive check). PTA analysis translated into 33% of evaluated patient cases being at risk of incorrectly rejecting recommended dosing regimens at CAP-related epidemiological cut-off values. Conclusions Integral-CA proved most appropriate to characterise clinical pharmacokinetics- and microdialysis-related variability. Employing this knowledge will improve the understanding of drug target-site PK for therapeutic decision-making.

To study the forced variability of atmospheric circulation regimes, the use of model ensembles is often necessary for identifying statistically significant signals as the observed data constitute a small sample and are thus strongly affected by the noise associated with sampling uncertainty. However, the regime representation is itself affected by noise within the atmosphere, which can make it difficult to detect robust signals. To this end we employ a regularised k-means clustering algorithm to better identify the signal in a model ensemble. The approach allows for the identification of six regimes for the wintertime Euro-Atlantic sector and leads to more pronounced regime dynamics, compared to results without regularisation, both overall and on sub-seasonal and interannual time-scales. We find that sub-seasonal variability in the regime occurrence rates is mainly explained by changes in the seasonal cycle of the mean climatology. On interannual time-scales relations between the occurrence rates of the regimes and the El Nino Southern Oscillation (ENSO) are identified. The use of six regimes captures a more detailed response of the circulation to ENSO compared to the common use of four regimes. Predictable signals in occurrence rate on interannual time-scales are found for the two zonal flow regimes, namely a regime consisting of a negative geopotential height anomaly over the Norwegian Sea and Scandinavia, and the positive phase of the NAO. The signal strength for these regimes is comparable between observations and model, in contrast to that of the NAO-index where the signal strength in the observations is underestimated by a factor of 2 in the model. Our regime analysis suggests that this signal-to-noise problem for the NAO-index is primarily related to those atmospheric flow patterns associated with the negative NAO-index as we find poor predictability for the corresponding NAO- regime.

As a concrete setting where stochastic partial differential equations (SPDEs) are able to model real phenomena, we propose a stochastic Meinhardt model for cell repolarization and study how parameter estimation techniques developed for simple linear SPDE models apply in this situation. We establish the existence of mild SPDE solutions, and we investigate the impact of the driving noise process on pattern formation in the solution. We then pursue estimation of the diffusion term and show asymptotic normality for our estimator as the space resolution becomes finer. The finite sample performance is investigated for synthetic and real data.

In this paper we introduce new birth-and-death processes with partial catastrophe and study some of their properties.
In particular, we obtain some estimates for the mean catastrophe time, and the first and second moments of the distribution of the process at a fixed time t.
This is completed by some asymptotic results.

Natural gas can be temporarily stored in a variety of underground facilities, such as depleted gas and oil fields, natural aquifers and caverns in salt rocks. Being extensively monitored during operations, these systems provide a favourable opportunity to investigate how pressure varies in time and space and possibly induces/triggers earthquakes on nearby faults. Elaborate and detailed numerical modelling techniques are often applied to study gas reservoirs. Here we show the possibilities and discuss the limitations of a flexible and easily formulated tool that can be straightforwardly applied to simulate temporal pore-pressure variations and study the relation with recorded microseismic events. We use the software POEL (POroELastic diffusion and deformation) which computes the poroelastic response to fluid injection/extraction in a horizontally layered poroelastic structure. We further develop its application to address the presence of vertical impermeable faults bounding the reservoir and of multiple injection/extraction sources. Exploiting available information on the reservoir geometry and physical parameters, and records of injection/extraction rates for a gas reservoir in southern Europe, we perform an extensive parametric study considering different model configurations. Comparing modelled spatiotemporal pore-pressure variations with in situ measurements, we show that the inclusion of vertical impermeable faults provides an improvement in reproducing the observations and results in pore-pressure accumulation near the faults and in a variation of the temporal pore-pressure diffusion pattern. To study the relation between gas storage activity and recorded local microseismicity, we applied different seismicity models based on the estimated porepressure distribution. This analysis helps to understand the spatial distribution of seismicity and its temporal modulation. The results show that the observed microseismicity could be partly linked to the storage activity, but the contribution of tectonic background seismicity cannot be excluded.

We introduce the concept of TRAP (Traces and Permutations), which can roughly be viewed as a wheeled PROP (Products and Permutations) without unit. TRAPs are equipped with a horizontal concatenation and partial trace maps.
Continuous morphisms on an infinite-dimensional topological space and smooth kernels (respectively, smoothing operators) on a closed manifold form a TRAP but not a wheeled PROP.
We build the free objects in the category of TRAPs as TRAPs of graphs and show that a TRAP can be completed to a unitary TRAP (or wheeled PROP).
We further show that it can be equipped with a vertical concatenation, which on the TRAP of linear homomorphisms of a vector space, amounts to the usual composition. The vertical concatenation in the TRAP of smooth kernels gives rise to generalised convolutions.
Graphs whose vertices are decorated by smooth kernels (respectively, smoothing operators) on a closed manifold form a TRAP. From their universal properties we build smooth amplitudes associated with the graph.

We prove that optimal lower eigenvalue estimates of Zhong-Yang type as well as a Cheng-type upper bound for the first eigenvalue hold on closed manifolds assuming only a Kato condition on the negative part of the Ricci curvature.
This generalizes all earlier results on Lp-curvature assumptions.
Moreover, we introduce the Kato condition on compact manifolds with boundary with respect to the Neumann Laplacian, leading to Harnack estimates for the Neumann heat kernel and lower bounds for all Neumann eigenvalues, which provides a first insight in handling variable Ricci curvature assumptions in this case.

Each completely regular semigroup is a semilattice of completely simple semigroups. The more specific concept of a strong semilattice provides the concrete product between two arbitrary elements.
We characterize strong semilattices of rectangular groups by so-called disjunctions of identities. Disjunctions of identities generalize the classical concept of an identity and of a variety, respectively.
The rectangular groups will be on the one hand left zero semigroups and right zero semigroups and on the other hand groups of exponent p is an element of P, where P is any set of pairwise coprime natural numbers.

Reentrant tensegrity
(2021)

We present a three-periodic, chiral, tensegrity structure and demonstrate that it is auxetic. Our tensegrity structure is constructed using the chiral symmetry Pi(+) cylinder packing, transforming cylinders to elastic elements and cylinder contacts to incompressible rods. The resulting structure displays local reentrant geometry at its vertices and is shown to be auxetic when modeled as an equilibrium configuration of spatial constraints subject to a quasi-static deformation. When the structure is subsequently modeled as a lattice material with elastic elements, the auxetic behavior is again confirmed through finite element modeling. The cubic symmetry of the original structure means that the auxetic behavior is observed in both perpendicular directions and is close to isotropic in magnitude. This structure could be the simplest three-dimensional analog to the two-dimensional reentrant honeycomb. This, alongside the chirality of the structure, makes it an interesting design target for multifunctional materials.

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.