Relating Operator Spaces via Adjunctions

This chapter uses categorical techniques to describe relations between various sets of operators on a Hilbert space, such as self-adjoint, positive, density, effect and projection operators. These relations, including various Hilbert-Schmidt isomorphisms of the form tr(A-), are expressed in terms of dual adjunctions, and maps between them. Of particular interest is the connection with quantum structures, via a dual adjunction between convex sets and effect modules. The approach systematically uses categories of modules, via their description as Eilenberg-Moore algebras of a monad

Improved asymptotic upper bounds for the minimum number of pairwise distinct longest cycles in regular graphs

We study how few pairwise distinct longest cycles a regular graph can have under additional constraints. For each integer $r \geq 5$, we give exponential improvements for the best asymptotic upper bounds for this invariant under the additional constraint that the graphs are $r$-regular hamiltonian graphs. Earlier work showed that a conjecture by Haythorpe on a lower bound for this invariant is false because of an incorrect constant factor, whereas our results imply that the conjecture is even asymptotically incorrect. Motivated by a question of Zamfirescu and work of Chia and Thomassen, we also study this invariant for non-hamiltonian 2-connected $r$-regular graphs and show that in this case the invariant can be bounded from above by a constant for all large enough graphs, even for graphs with arbitrarily large girth.Comment: Submitted for publicatio

Counterexamples to conjectures on the occupancy fraction of graphs

The occupancy fraction of a graph is a (normalized) measure on the size of independent sets under the hard-core model, depending on a variable (fugacity) $\lambda.$ We present a criterion for finding the graph with minimum occupancy fraction among graphs with a fixed order, and disprove five conjectures on the extremes of the occupancy fraction and (normalized) independence polynomial for certain graph classes of regular graphs with a given girth.Comment: 8 pages, 4 figure

Prepolymerization for the gas phase polymerization of propylene

Since the development of Ziegler-Natta catalysts in the 1950s, continuous improvements were performed with regard to higher catalyst activity and higher stereo-selectivity. However, especially at the polymerization start, when the pure catalyst is injected under high rate conditions, the high activity of the catalyst may cause particle overheating and/or lead to an uncontrolled catalyst fragmentation resulting in lower catalyst activity and bad particle morphology. One possibility to solve the problem is to apply a prepolymerization step before the main polymerization. In the prepolymerization, the reaction starts at low rate conditions (mild reaction temperatures, low monomer concentration) in order to realize a controlled catalyst fragmentation and to improve heat removal conditions for the main polymerization by creating a higher heat transfer area. The current contribution deals with the prepolymerization of propylene in lab-scale under industrially relevant conditions. The main gas phase polymerization is conducted in a 5 l horizontal stirred tank reactor. Prepolymerization is performed in-situ, meaning in the same reactor by injecting the catalyst at a low initial temperature, or externally in a 250 ml calorimeter. The latter case offers the benefit of controlled prepolymerization conditions with access to prepolymerization kinetics and defined starting conditions for the main polymerization. Experiments were performed for varying reaction conditions. The two prepolymerization methods are compared and the effect of prepolymerization on the activity of the main polymerization is discussed

Metal sources for the Katanga Copperbelt deposits (DRC) insights from Sr and Nd isotope ratios

The ore deposits of the Central African Copperbelt formed during a multiphase mineralisation process. The basement underlying the Neoproterozoic Katanga Supergroup that hosts the ore, demonstrates the largest potential as metal source. Various ore deposits that formed during different mineralisation phases are taken as case studies, i.e. Kamoto, Luiswishi, Kambove West, Dikulushi and Kipushi (Democratic Republic of Congo, DRC). The Sr and Nd isotopic compositions of gangue carbonates associated with these deposits is determined and compared with those of rocks from several basement units, bordering or underlying the Copperbelt, to infer the metal sources. The mineralising fluid of diagenetic stratiform Cu-Co mineralisation interacted with felsic basement rocks underlying the region. The Co from these deposits is most likely derived from mafic rocks, but this is not observed in the isotopic signatures. Syn-orogenic, stratabound Cu-Co mineralisation resulted mainly from remobilisation of diagenetic sulphides. A limited, renewed contribution of metals from felsic basement rocks might be indicated by the isotope ratios in the western part of the Copperbelt, where the metamorphic grade is the lowest. The mineralising fluid of syn- and post-orogenic, vein-type mineralisations interacted with local mafic rocks, and with felsic basement or siliciclastic host rocks

A heuristic algorithm using tree decompositions for the maximum happy vertices problem

We propose a new methodology to develop heuristic algorithms using tree decompositions. Traditionally, such algorithms construct an optimal solution of the given problem instance through a dynamic programming approach. We modify this procedure by introducing a parameter $W$ that dictates the number of dynamic programming states to consider. We drop the exactness guarantee in favour of a shorter running time. However, if $W$ is large enough such that all valid states are considered, our heuristic algorithm proves optimality of the constructed solution. In particular, we implement a heuristic algorithm for the Maximum Happy Vertices problem using this approach. Our algorithm more efficiently constructs optimal solutions compared to the exact algorithm for graphs of bounded treewidth. Furthermore, our algorithm constructs higher quality solutions than state-of-the-art heuristic algorithms Greedy-MHV and Growth-MHV for instances of which at least 40\% of the vertices are initially coloured, at the cost of a larger running time.Comment: 31 pages, to appear in Journal of Heuristic