510 Mathematik
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We study the spectral properties of curl, a linear differential operator of first order acting on differential forms of appropriate degree on an odd-dimensional closed oriented Riemannian manifold. In three dimensions, its eigenvalues are the electromagnetic oscillation frequencies in vacuum without external sources. In general, the spectrum consists of the eigenvalue 0 with infinite multiplicity and further real discrete eigenvalues of finite multiplicity. We compute the Weyl asymptotics and study the zeta-function. We give a sharp lower eigenvalue bound for positively curved manifolds and analyze the equality case. Finally, we compute the spectrum for flat tori, round spheres, and 3-dimensional spherical space forms. Published under license by AIP Publishing.
Background: Circulating infliximab (IFX) concentrations correlate with clinical outcomes, forming the basis of the IFX concentration monitoring in patients with Crohn's disease. This study aims to investigate and refine the exposure-response relationship by linking the disease activity markers "Crohn's disease activity index" (CDAI) and C-reactive protein (CRP) to IFX exposure. In addition, we aim to explore the correlations between different disease markers and exposure metrics.
Methods: Data from 47 Crohn's disease patients of a randomized controlled trial were analyzed post hoc. All patients had secondary treatment failure at inclusion and had received intensified IFX of 5 mg/kg every 4 weeks for up to 20 weeks. Graphical analyses were performed to explore exposure-response relationships. Metrics of exposure included area under the concentration-time curve (AUC) and trough concentrations (Cmin). Disease activity was measured by CDAI and CRP values, their change from baseline/last visit, and response/remission outcomes at week 12.
Results: Although trends toward lower Cmin and lower AUC in nonresponders were observed, neither CDAI nor CRP showed consistent trends of lower disease activity with higher IFX exposure across the 30 evaluated relationships. As can be expected, Cmin and AUC were strongly correlated with each other. Contrarily, the disease activity markers were only weakly correlated with each other.
Conclusions: No significant relationship between disease activity, as evaluated by CDAI or CRP, and IFX exposure was identified. AUC did not add benefit compared with Cmin. These findings support the continued use of Cmin and call for stringent objective disease activity (bio-)markers (eg, endoscopy) to form the basis of personalized IFX therapy for Crohn's disease patients with IFX treatment failure.
Entdeckendes Lernen
(2017)
Trotz der nachweislichen Popularität des Entdeckenden Lernens in der deutschsprachigen Mathematikdidaktik finden sich aktuell keine kritischen Beiträge, die dazu beitragen könnten, dieses grundlegende Unterrichtskonzept zu hinterfragen und auszuschärfen. In diesem Diskussionsbeitrag werden zunächst die Theorie und einige Umsetzungsbeispiele des Entdeckenden Lernens herausgearbeitet, um aufzuzeigen, dass das Entdeckende Lernen einem vagen Sammelbegriff gleicht, unter dem oft fragwürdige Unterrichtsumgebungen legitimiert werden. Anschließend werden an Hand erkenntnistheoretischer, lerntheoretischer, didaktischer und soziokultureller Betrachtungen Probleme des Entdeckenden Lernens im Mathematikunterricht und Möglichkeiten ihrer Überwindung thematisiert. Dabei zeigt sich, dass die Konzeption des Entdeckenden Lernens hinter dem aktuellen mathematikdidaktischen Erkenntnisstand zurückfällt und Lehrer sowie Schüler mit unmöglichen Forderungen konfrontiert, dass lerntheoretische Vorteile des Entdeckenden Lernens oft nicht nachweisbar sind, dass die Idee des Entdeckens auf einem problematischen platonistischen Verständnis von Erkenntnis beruht und dass Entdeckendes Lernen bildungsferne Schüler zu benachteiligen droht. Abschließend werden Forschungsdesiderata abgeleitet, deren Bearbeitung dazu beitragen könnte, die aufgezeigten Problemfelder zu überwinden.
The success of the ensemble Kalman filter has triggered a strong interest in expanding its scope beyond classical state estimation problems. In this paper, we focus on continuous-time data assimilation where the model and measurement errors are correlated and both states and parameters need to be identified. Such scenarios arise from noisy and partial observations of Lagrangian particles which move under a stochastic velocity field involving unknown parameters. We take an appropriate class of McKean-Vlasov equations as the starting point to derive ensemble Kalman-Bucy filter algorithms for combined state and parameter estimation. We demonstrate their performance through a series of increasingly complex multi-scale model systems.
A term, also called a tree, is said to be linear, if each variable occurs in the term only once. The linear terms and sets of linear terms, the so-called linear tree languages, play some role in automata theory and in the theory of formal languages in connection with recognizability. We define a partial superposition operation on sets of linear trees of a given type and study the properties of some many-sorted partial clones that have sets of linear trees as elements and partial superposition operations as fundamental operations. The endomorphisms of those algebras correspond to nondeterministic linear hypersubstitutions.
An efficient immunosurveillance of CD8(+) T cells in the periphery depends on positive/negative selection of thymocytes and thus on the dynamics of antigen degradation and epitope production by thymoproteasome and immunoproteasome in the thymus. Although studies in mouse systems have shown how thymoproteasome activity differs from that of immunoproteasome and strongly impacts the T cell repertoire, the proteolytic dynamics and the regulation of human thymoproteasome are unknown. By combining biochemical and computational modeling approaches, we show here that human 20S thymoproteasome and immunoproteasome differ not only in the proteolytic activity of the catalytic sites but also in the peptide transport. These differences impinge upon the quantity of peptide products rather than where the substrates are cleaved. The comparison of the two human 20S proteasome isoforms depicts different processing of antigens that are associated to tumors and autoimmune diseases.
We study corner-degenerate pseudo-differential operators of any singularity order and develop ellipticity based on the principal symbolic hierarchy, associated with the stratification of the underlying space. We construct parametrices within the calculus and discuss the aspect of additional trace and potential conditions along lower-dimensional strata.
For n∈N , let Xn={a1,a2,…,an} be an n-element set and let F=(Xn;<f) be a fence, also called a zigzag poset. As usual, we denote by In the symmetric inverse semigroup on Xn. We say that a transformation α∈In is fence-preserving if x<fy implies that xα<fyα, for all x,y in the domain of α. In this paper, we study the semigroup PFIn of all partial fence-preserving injections of Xn and its subsemigroup IFn={α∈PFIn:α−1∈PFIn}. Clearly, IFn is an inverse semigroup and contains all regular elements of PFIn. We characterize the Green’s relations for the semigroup IFn. Further, we prove that the semigroup IFn is generated by its elements with rank ≥n−2. Moreover, for n∈2N, we find the least generating set and calculate the rank of IFn.
Fractures serve as highly conductive preferential flow paths for fluids in rocks, which are difficult to exactly reconstruct in numerical models. Especially, in low-conductive rocks, fractures are often the only pathways for advection of solutes and heat. The presented study compares the results from hydraulic and tracer tomography applied to invert a theoretical discrete fracture network (DFN) that is based on data from synthetic cross-well testing. For hydraulic tomography, pressure pulses in various injection intervals are induced and the pressure responses in the monitoring intervals of a nearby observation well are recorded. For tracer tomography, a conservative tracer is injected in different well levels and the depth-dependent breakthrough of the tracer is monitored. A recently introduced transdimensional Bayesian inversion procedure is applied for both tomographical methods, which adjusts the fracture positions, orientations, and numbers based on given geometrical fracture statistics. The used Metropolis-Hastings-Green algorithm is refined by the simultaneous estimation of the measurement error’s variance, that is, the measurement noise. Based on the presented application to invert the two-dimensional cross-section between source and the receiver well, the hydraulic tomography reveals itself to be more suitable for reconstructing the original DFN. This is based on a probabilistic representation of the inverted results by means of fracture probabilities.