Rainbow Thresholds

We extend a recent breakthrough result relating expectation thresholds and actual thresholds to include rainbow versions

Synthesis of novel double metal cyanide catalysts and polymerization of PO and CO2

Double metal cyanides (DMC) are a versatile group of complexes that find numerous applications in catalytic conversions, e.g. as catalysts for polycondensation of diols and diacids[1], for the ring-opening polymerization of epoxides[2] and their co- and terpolymerization with CO2[3] and cyclic anhydrides.[4] The DMC catalysts usually have a high selectivity; in case of propylene oxide ring opening polymerizations (and in contrast to e.g. alkali-based catalysts), products with low degrees of unsaturation and narrow molecular weight distributions are obtained. A major challenge in the application of DMC catalysts is that they generally feature an induction period of several minutes up to hours during which no substantial propagation is observed. The length of the induction period is affected for instance by the catalyst preparation itself but also by the presence of impurities.[6,7] Up to this date, no reliable model exists that allows the prediction of the length of this activation step. This does not only result in decreasing overall space-time yield but also is a serious safety issue as the spontaneous initiation at the end of the induction period causes an increase in temperature due to the exothermic polymerization reactions. Please click Additional Files below to see the full abstract

Characterizing catalyst performance of DMCs on PO homopolymerization

Double metal cyanide (DMC) complexes are known effective catalysts for the ring-opening polymerization of propylene oxide to generate polyether polyols (Scheme 1).1,2 The high activity of DMC catalysts relative to basic alkaline catalysts eliminates the need for expensive removal of residual catalyst from the product. Furthermore, the poly(propylene glycol) (PPG) products prepared by DMC catalysts have - contrary to products from alkaline catalysis - a low degree of unsaturation and narrow molecular weight distributions. Latter is advantageous with respect to the resulting low viscosities. A common challenge when applying DMC catalysts is the need for an activation procedure, leading to an induction period of unknown length (Figure 1).2,3 In a larger, usually semibatch process, PO monomer can only be added after the activation has been secured; the concentration of PO must not reach certain limits as its ring-opening is highly exothermal. Please click Additional Files below to see the full abstract

Locally finite graphs and their localization numbers

We study the Localization game on locally finite graphs trees, where each of the countably many vertices have finite degree. In contrast to the finite case, we construct a locally finite tree with localization number $n$ for any choice of positive integer $n$. Our examples have uncountably many ends, and we show that this is necessary by proving that locally finite trees with finitely or countably many ends have localization number at most 2. Finally, as is the case for finite graphs, we prove that any locally finite graph contains a subdivision where one cop can capture the robber

The kk-visibility Localization Game

We study a variant of the Localization game in which the cops have limited visibility, along with the corresponding optimization parameter, the $k$-visibility localization number $\zeta_k$, where $k$ is a non-negative integer. We give bounds on $k$-visibility localization numbers related to domination, maximum degree, and isoperimetric inequalities. For all $k$, we give a family of trees with unbounded $\zeta_k$ values. Extending results known for the localization number, we show that for $k\geq 2$, every tree contains a subdivision with $\zeta_k = 1$. For many $n$, we give the exact value of $\zeta_k$ for the $n \times n$ Cartesian grid graphs, with the remaining cases being one of two values as long as $n$ is sufficiently large. These examples also illustrate that $\zeta_i \neq \zeta_j$ for all distinct choices of $i$ and $j.

Subgraph Games in the Semi-Random Graph Process and Its Generalization to Hypergraphs

The semi-random graph process is a single-player game that begins with an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$ and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the problem of constructing a subgraph isomorphic to an arbitrary, fixed graph $G$. Let $\omega = \omega(n)$ be any function tending to infinity as $n \to \infty$. In (Omri Ben-Eliezer et al. "Semi-random graph process". In: Random Structures & Algorithms 56.3 (2020), pp. 648-675) it was proved that asymptotically almost surely one can construct $G$ in less than $n^{(d-1)/d} \omega$ rounds where $d \ge 2$ is the degeneracy of $G$. It was also proved that the result is sharp for $G = K_{d+1}$, that is, asymptotically almost surely it takes at least $n^{(d-1)/d} / \omega$ rounds to create $K_{d+1}$. Moreover, the authors conjectured that their general upper bound is sharp for all graphs $G$. We prove this conjecture here. We also consider a natural generalization of the process to $s$-uniform hypergraphs, the semi-random hypergraph process in which $r \ge 1$ vertices are presented at random, and the player then selects $s-r \ge 1$ vertices to form an edge of size~$s$. Our results for graphs easily generalize to hypergraphs when $r=1$; the threshold for constructing a fixed $s$-uniform hypergraph $G$ is, again, determined by the degeneracy of $G$. However, new challenges are mounting when $r \ge 2$; thresholds are not even known for complete hypergraphs. We provide bounds for this family and determine thresholds for some sparser hypergraphs