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Institute
- Mathematik (373) (remove)
We present an overview of the mathematics underlying the quantum Zeno effect. Classical, functional analytic results are put into perspective and compared with more recent ones. This yields some new insights into mathematical preconditions entailing the Zeno paradox, in particular a simplified proof of Misra's and Sudarshan's theorem. We empahsise the complex-analytic structures associated to the issue of existence of the Zeno dynamics. On grounds of the assembled material, we reason about possible future mathematical developments pertaining to the Zeno paradox and its counterpart, the anti-Zeno paradox, both of which seem to be close to complete characterisations. PACS-Klassifikation: 03.65.Xp, 03.65Db, 05.30.-d, 02.30.T . See the corresponding presentation: Schmidt, Andreas U.: "Zeno Dynamics of von Neumann Algebras" and "Zeno Dynamics in Quantum Statistical Mechanics"
Rezension zu: George G. Szpiro : Mathematik für Sonntagmorgen : 50 Geschichten aus Mathematik und Wissenschaft, NZZ Verlag, Zürich 2006, ISBN 978-3-03823-353-4 ; 240 Seiten, 26 Euro/38 CHF George G. Szpiro : Mathematik für Sonntagnachmittag : Weitere 50 Geschichten aus Mathematik und Wissenschaft, NZZ Verlag, Zürich 2006, ISBN 978-3-03823-225-4 ; 236 Seiten, 26 Euro/38 CHF
Using limit linear series on chains of curves, we show that closures of certain Brill-Noether loci contain a product of pointed Brill-Noether loci of small codimension. As a result, we obtain new non-containments of Brill-Noether loci, in particular that dimensionally expected non-containments hold for expected maximal Brill-Noether loci. Using these degenerations, we also give a new proof that Brill-Noether loci with expected codimension −ρ≤⌈g/2⌉ have a component of the expected dimension. Additionally, we obtain new non-containments of Brill-Noether loci by considering the locus of the source curves of unramified double covers.
Sensitivity of output of a linear operator to its input can be quantified in various ways. In Control Theory, the input is usually interpreted as disturbance and the output is to be minimized in some sense. In stochastic worst-case design settings, the disturbance is considered random with imprecisely known probability distribution. The prior set of probability measures can be chosen so as to quantify how far the disturbance deviates from the white-noise hypothesis of Linear Quadratic Gaussian control. Such deviation can be measured by the minimal Kullback-Leibler informational divergence from the Gaussian distributions with zero mean and scalar covariance matrices. The resulting anisotropy functional is defined for finite power random vectors. Originally, anisotropy was introduced for directionally generic random vectors as the relative entropy of the normalized vector with respect to the uniform distribution on the unit sphere. The associated a-anisotropic norm of a matrix is then its maximum root mean square or average energy gain with respect to finite power or directionally generic inputs whose anisotropy is bounded above by a >= 0. We give a systematic comparison of the anisotropy functionals and the associated norms. These are considered for unboundedly growing fragments of homogeneous Gaussian random fields on multidimensional integer lattice to yield mean anisotropy. Correspondingly, the anisotropic norms of finite matrices are extended to bounded linear translation invariant operators over such fields.
We show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure p and a volume form σ on an oriented surface M equip the total space of a certain disk bundle Z→M with a pair (Jp,Jp,σ) of almost complex structures. A conformal structure on M corresponds to a section of Z→M and p is metrisable by the metric g if and only if [g]:M→Z is a pseudo-holomorphic curve with respect to Jp and Jp,dAg.
Mixed volumes, mixed Ehrhart theory and applications to tropical geometry and linkage configurations
(2009)
The aim of this thesis is the discussion of mixed volumes, their interplay with algebraic geometry, discrete geometry and tropical geometry and their use in applications such as linkage configuration problems. Namely we present new technical tools for mixed volume computation, a novel approach to Ehrhart theory that links mixed volumes with counting integer points in Minkowski sums, new expressions in terms of mixed volumes of combinatorial quantities in tropical geometry and furthermore we employ mixed volume techniques to obtain bounds in certain graph embedding problems.
Die anaerobe Fermentation beschreibt den Abbau organischen Materials unter Ausschluss von Sauerstoff und setzt sich aus vier Prozessphasen (Hydrolyse, Acidogenese, Acetogenese und Methanogenese) zusammen. Im Rahmen dieser Arbeit konnte die Aufteilung dieser vier Prozessphasen auf die beiden Stufen eines zweistufigen zweiphasigen Biogas-Reaktors genau bestimmt werden. Die Aufteilung ist von entscheidender Bedeutung für zukünftige Arbeiten, da dadurch genau festgelegt werden kann, welche Stoffe bei den Messungen und bei der Modellierung berücksichtigt werden müssen.
Im Jahre 2002 wurde von der IWA Taskgroup das ADM1-Modell, welches alle vier Prozessphasen der anaeroben Fermentation berücksichtigt, veröffentlicht. In der vorliegenden Arbeit wird ein räumlich aufgelöstes Modell für die anaerobe Fermentation erarbeitet, in dem das ADM1-Modell mit einem Strömungsmodell gekoppelt wird. Anschließend wird ein reduziertes Simulationsmodell für acetoklastische Methanogenese in einem zweistufigen zweiphasigen Biogasreaktor erstellt. Anhand von Messdaten wird gezeigt, dass der Abbau von Essigsäure zu Methan innerhalb des Reaktors durch das Simulationsmodell gut wiedergegeben werden kann.
Anschließend wird das validierte Modell verwendet um Regeln für eine optimale Steuerung des Reaktors herzuleiten und weiterhin wird mit Hilfe der lokalen Methanproduktion die Effektivität des Reaktors bestimmt. Die erlangten Informationen können verwendet werden, um den Biogas-Reaktor zu optimieren.
We deal with the shape reconstruction of inclusions in elastic bodies. For solving this inverse problem in practice, data fitting functionals are used. Those work better than the rigorous monotonicity methods from Eberle and Harrach (Inverse Probl 37(4):045006, 2021), but have no rigorously proven convergence theory. Therefore we show how the monotonicity methods can be converted into a regularization method for a data-fitting functional without losing the convergence properties of the monotonicity methods. This is a great advantage and a significant improvement over standard regularization techniques. In more detail, we introduce constraints on the minimization problem of the residual based on the monotonicity methods and prove the existence and uniqueness of a minimizer as well as the convergence of the method for noisy data. In addition, we compare numerical reconstructions of inclusions based on the monotonicity-based regularization with a standard approach (one-step linearization with Tikhonov-like regularization), which also shows the robustness of our method regarding noise in practice.
In 1957, Craig Mooney published a set of human face stimuli to study perceptual closure: the formation of a coherent percept on the basis of minimal visual information. Images of this type, now known as “Mooney faces”, are widely used in cognitive psychology and neuroscience because they offer a means of inducing variable perception with constant visuo-spatial characteristics (they are often not perceived as faces if viewed upside down). Mooney’s original set of 40 stimuli has been employed in several studies. However, it is often necessary to use a much larger stimulus set. We created a new set of over 500 Mooney faces and tested them on a cohort of human observers. We present the results of our tests here, and make the stimuli freely available via the internet. Our test results can be used to select subsets of the stimuli that are most suited for a given experimental purpose.