Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs

We study Markov chains for $\alpha$-orientations of plane graphs, these are orientations where the outdegree of each vertex is prescribed by the value of a given function $\alpha$. The set of $\alpha$-orientations of a plane graph has a natural distributive lattice structure. The moves of the up-down Markov chain on this distributive lattice corresponds to reversals of directed facial cycles in the $\alpha$-orientation. We have a positive and several negative results regarding the mixing time of such Markov chains. A 2-orientation of a plane quadrangulation is an orientation where every inner vertex has outdegree 2. We show that there is a class of plane quadrangulations such that the up-down Markov chain on the 2-orientations of these quadrangulations is slowly mixing. On the other hand the chain is rapidly mixing on 2-orientations of quadrangulations with maximum degree at most 4. Regarding examples for slow mixing we also revisit the case of 3-orientations of triangulations which has been studied before by Miracle et al.. Our examples for slow mixing are simpler and have a smaller maximum degree, Finally we present the first example of a function $\alpha$ and a class of plane triangulations of constant maximum degree such that the up-down Markov chain on the $\alpha$-orientations of these graphs is slowly mixing

Biochar as Additive in Biogas-Production from Bio-Waste

Previous publications about biochar in anaerobic digestion show encouraging results with regard to increased biogas yields. This work investigates such effects in a solid-state fermentation of bio-waste. Unlike in previous trials, the influence of biochar is tested with a setup that simulates an industrial-scale biogas plant. Both the biogas and the methane yield increased around 5% with a biochar addition of 5%-based on organic dry matter biochar to bio-waste. An addition of 10% increased the yield by around 3%. While scaling effects prohibit a simple transfer of the results to industrial-scale plants, and although the certainty of the results is reduced by the heterogeneity of the bio-waste, further research in this direction seems promising

BDNF Methylation and Maternal Brain Activity in a Violence-Related Sample

It is known that increased circulating glucocorticoids in the wake of excessive, chronic, repetitive stress increases anxiety and impairs Brain-Derived Neurotrophic Factor (BDNF) signaling. Recent studies of BDNF gene methylation in relation to maternal care have linked high BDNF methylation levels in the blood of adults to lower quality of received maternal care measured via self-report. Yet the specific mechanisms by which these phenomena occur remain to be established. The present study examines the link between methylation of the BDNF gene promoter region and patterns of neural activity that are associated with maternal response to stressful versus non-stressful child stimuli within a sample that includes mothers with interpersonal violence-related PTSD (IPV-PTSD). 46 mothers underwent fMRI. The contrast of neural activity when watching children-including their own-was then correlated to BDNF methylation. Consistent with the existing literature, the present study found that maternal BDNF methylation was associated with higher levels of maternal anxiety and greater childhood exposure to domestic violence. fMRI results showed a positive correlation of BDNF methylation with maternal brain activity in the anterior cingulate (ACC), and ventromedial prefrontal cortex (vmPFC), regions generally credited with a regulatory function toward brain areas that are generating emotions. Furthermore we found a negative correlation of BDNF methylation with the activity of the right hippocampus. Since our stimuli focus on stressful parenting conditions, these data suggest that the correlation between vmPFC/ACC activity and BDNF methylation may be linked to mothers who are at a disadvantage with respect to emotion regulation when facing stressful parenting situations. Overall, this study provides evidence that epigenetic signatures of stress-related genes can be linked to functional brain regions regulating parenting stress, thus advancing our understanding of mothers at risk for stress-related psychopathology

Über die Mischzeit der Face-Flip- und up/down Markov Kette einiger Familien von Graphen

This dissertation studies two natural Markov chains, each living on a family of finite distributive lattices and some related results on the enumeration of lattice paths. In the first part of this body of work, we introduce a coupling method – the block coupling technique – and apply it to the k-heights of some families of plane graphs. This gives some insight into the behaviour of the up/down Markov chain on these distributive lattices. Espe- cially it yields some upper bounds to the mixing time of the face flip Markov chain for the sets of 2-heights on some families of graphs, which is bounded by a polynomial in the number of vertices of the graph (and therefor polynomial logarithmic in the size of the state space / set of 2-heights). The second part is accepted for publication by the Journal of Integer Sequences. It focuses on the enumeration of bounded lattice paths and uses transfer matrices and methods from linear algebra to do so. Using these techniques we were able to enumerate a rather general class of lattice paths, which reduce to a lot of different well known integer sequences as special cases, and therefore group them together. Also this method looks promising to enumerate other families of sequences, as it links various fields (Graph Theory, Linear Algebra, and Lattice Path Enumeration) in an easily applicable way. Finally, the third part, considers the distributive lattices of α-orientations on planar graphs and the face flip Markov chain on these sets. It uses tower moves, another coupling method, to show that the the up/down Markov chain is rapid mixing on the set of 2-orientations of planar quadrangulations of maximum degree 4. In addition, this part contains results using the congestion of some carefully chosen families of graphs to prove, that for graphs containing vertices with high degree the Markov chain is not rapid mixing, i.e. that the mixing time can not be bound from above by some polynomial in the logarithm of the size of the state space.Diese Dissertation untersucht zwei natürliche Markov Ketten, die jeweils auf einer Familie endlicher distributiver Verbände leben. Weiterhin enthält sie einige Verwandte Resultate über das Zählen von Gitter-Pfaden. Im ersten Teil der Arbeit stellen wir die Block Coupling Methode vor und wenden sie auf k-heights einiger Familien planarer Graphen an. Dieses gibt etwas Einblick in das Verhalten der up/down-Markov Kette auf diesen distributiven Verbänden. Insbesondere liefern die Untersuchungen einige obere Schranken für die Mischzeit der FaceFlip Markov Kette auf der Menge der 2-heights einiger Graphen-Familien. Wir zeigen dass diese beschränkt ist durch ein Polynom in den Anzahl der Knoten der Graphen (was implizit bedeutet, dass sie polynomiel logarithmisch ist in der Anzahl der 2-heights). Der zweite Teil der Arbeit beschäftigt sich mit dem Zählen beschränkter Gitter-Pfade. Hier werden Transfer Matrizen und Methoden der Linearen Algebra angewendet um diese Pfade zu zählen. Auf diese Art war es möglich, viele bereits bekannte Integer-Sequenzes über die untersuchten beschränkten Gitter-Pfade zu beschreiben und sie daher in einen Zusammenhang zu stellen. Insbesondere scheint diese Methode eine Möglichkeit darzustellen, weitere Familien von Folgen zu untersuchen und sie verbindet einige verschiedene Gebiete der Mathematik. Die Ergebnisse die in diesem Kapitel vorgestellt werden wurden auch vom Journal of Integer Sequences zur Publikation akzeptiert. Der dritte Teil schliesslich beschäftigt sich mit den distributiven Verbänden von Alpha-Orientierungen planarer Graphen und der Face Flip Markov Kette auf diesen. Es werden Tower Movers, eine weitere coupling Methode, genutzt um zu zeigen, dass die up/down Markov Kette für 2-Orientierungen planarer Quadrangulierungen mit maximal Grad 4 schnell mischend ist. Zusätzlich enthält dieser Teil Resultate, die mithilfe sorgfältig konstruierter Graphenfamilien zeigen, dass für Graphen mit hohem Grad nicht erwartet werden kann, dass die untersuchte Markov Kette schnell mischt

Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs

We study Markov chains for $\alpha$-orientations of plane graphs, these are orientations where the outdegree of each vertex is prescribed by the value of a given function $\alpha$. The set of $\alpha$-orientations of a plane graph has a natural distributive lattice structure. The moves of the up-down Markov chain on this distributive lattice corresponds to reversals of directed facial cycles in the $\alpha$-orientation. We have a positive and several negative results regarding the mixing time of such Markov chains. A 2-orientation of a plane quadrangulation is an orientation where every inner vertex has outdegree 2. We show that there is a class of plane quadrangulations such that the up-down Markov chain on the 2-orientations of these quadrangulations is slowly mixing. On the other hand the chain is rapidly mixing on 2-orientations of quadrangulations with maximum degree at most 4. Regarding examples for slow mixing we also revisit the case of 3-orientations of triangulations which has been studied before by Miracle et al.. Our examples for slow mixing are simpler and have a smaller maximum degree, Finally we present the first example of a function $\alpha$ and a class of plane triangulations of constant maximum degree such that the up-down Markov chain on the $\alpha$-orientations of these graphs is slowly mixing