Easy quantum groups : linear independencies, models and partition quantum spaces

This work presents results in the context of (unitary) easy quantum groups. These are compact matrix quantum groups featuring a rich combinatorial structure given by partitions (of sets). This thesis reports on three topics within this area. Topic 1: Linear independence of the intertwiner maps Tp in the free case: Given a suitable collection of partitions p, there exists by definition a connection to easy quantum groups via intertwiner maps Tp. A sufficient condition for this correspondence to be one-to-one are particular linear independences on the level of maps Tp. In the case of non-crossing partitions, a proof of this linear independence can be traced down to a matrix determinant formula, developed by W. Tutte. We present a revised and adapted version of Tutte's work and the link to the problem above, trusting that this self-contained workout will assist others in the field of easy quantum groups. In particular, we fixed some errors in the original work and adapted in this sense notations, definitions, statements and proofs. Topic 2: A chain of models for C(S+ N): For any given natural number N 2 N 4 we present a chain of models (Bn;Mn)n2N for the C -algebra C(S+ N) which allows an inverse limit (B1;M1). For small n the elements in the chain have a quite concrete and comprehensive structure. In the inverse limit we obtain a compact matrix quantum group G = (B1;M1) that fulfils SN ( G S+ N. For N 2 f4; 5g it holds G = S+ N. Topic 3: Partition quantum spaces: Analogous to the construction of an easy quantum group G from a given set of partitions, we propose a definition for partition quantum spaces X, which are tuples of quantum vectors inspired by the first d columns of matrices in G. However, we define them as universal C -algebras, independently of those quantum groups. The central result is the reconstruction of G from X as its quantum symmetry group QSymG(X), at least if the number d is sufficiently large. In the case of non-crossing partitions, the minimal value for d to permit this reconstruction is proved to be one or two.Diese Arbeit widmet sich Forschungsergebnissen im Bereich der (unitären) easy Quantengruppen. Dies sind kompakte Matrix-Quantengruppen mit stark kombinatorisch geprägter Struktur, welche durch Partitionen auf Mengen gegeben ist. Die vorliegende Arbeit beschäftigt sich mit drei Themen innerhalb dieses Bereichs. Thema 1: Lineare Unabhängigkeit von Intertwiner-Abbildungen Tp im freien Fall: Per definitionem existiert ein Zusammenhang zwischen geeigneten Familien von Partitionen p und easy Quantengruppen, der durch Intertwiner-Abbildungen Tp hergestellt wird. Eine hinreichende Bedingung für die Eineindeutigkeit dieses Zusammenhangs sind gewisse lineare Unabhängigkeiten auf Ebene der Abbildungen Tp. Im Falle nichtkreuzender Partitionen können diese linearen Unabhängigkeiten mittels einer Matrixdeterminanten-Formel, wie sie von W. Tutte entwickelt wurde, bewiesen werden. Wir präsentieren eine überarbeitete, an obige Fragestellung angepasste Version der Arbeit Tuttes und ebenso die Argumentationsschritte zwischen ursprünglichem Problem und erwähnter Determinantenformel. Insbesondere korrigiert die vorliegende Arbeit einige Fehler in der ursprünglichen Quelle und ändert in Folge dessen weitere Schreibweisen, Definitionen, Behauptungen und Beweise ab. Thema 2: Eine Folge von Modellen für C(S+ N): Für eine gegebene natürliche Zahl N 2 N 4 konstruieren wir eine Folge von Modellen (Bn;Mn) der C -Algebra C(S+ N), die die Konstruktion eines inversen Limes (B1;M1) erlaubt. Für kleine n haben die Folgenglieder sehr konkrete und anschauliche Strukturen. Der inverse Limes dieser Folge liefert eine kompakte Matrixquantengruppe G = (B1;M1), welche SN ( G S+ N erfüllt. Im Falle N 2 f4; 5g gilt G = S+ N. Thema 3: Partition quantum spaces: Analog zur Konstruktion einer easy Quantengruppe G auf Grundlage einer gegebenen Menge an Partitionen, stellen wir eine Definition für partition quantum spaces, Partitionen-Quantenräume, vor. Deren Elemente sind Tupel von Quantenvektoren, angelehnt an die ersten d Spalten der Quantenmatrizen in G. Wir definieren jedoch diese Quantenräume als universelle C -Algebren, zunächst ohne direkten Bezug zu diesen Quantenmatrizen. Das Hauptresultat ist die Rekonstruktion der easy Quantengruppe G aus dem Quantenraum X als dessen Quantensymmetriegruppe QSymG(X), zumindest für hinreichend große Spaltenzahl d. Im Falle nicht-kreuzender Partitionen wird gezeigt, dass der kleinstm ögliche Wert für d, der diese Rekonstruktion erlaubt, entweder eins oder zwei ist.European Research Council, NCDF

Do Social Bots Dream of Electric Sheep? A Categorisation of Social Media Bot Accounts

So-called 'social bots' have garnered a lot of attention lately. Previous research showed that they attempted to influence political events such as the Brexit referendum and the US presidential elections. It remains, however, somewhat unclear what exactly can be understood by the term 'social bot'. This paper addresses the need to better understand the intentions of bots on social media and to develop a shared understanding of how 'social' bots differ from other types of bots. We thus describe a systematic review of publications that researched bot accounts on social media. Based on the results of this literature review, we propose a scheme for categorising bot accounts on social media sites. Our scheme groups bot accounts by two dimensions - Imitation of human behaviour and Intent.Comment: Accepted for publication in the Proceedings of the Australasian Conference on Information Systems, 201

Parameterized Complexity Results for General Factors in Bipartite Graphs with an Application to Constraint Programming

The NP-hard general factor problem asks, given a graph and for each vertex a list of integers, whether the graph has a spanning subgraph where each vertex has a degree that belongs to its assigned list. The problem remains NP-hard even if the given graph is bipartite with partition U+V, and each vertex in U is assigned the list {1}; this subproblem appears in the context of constraint programming as the consistency problem for the extended global cardinality constraint. We show that this subproblem is fixed-parameter tractable when parameterized by the size of the second partite set V. More generally, we show that the general factor problem for bipartite graphs, parameterized by |V|, is fixed-parameter tractable as long as all vertices in U are assigned lists of length 1, but becomes W[1]-hard if vertices in U are assigned lists of length at most 2. We establish fixed-parameter tractability by reducing the problem instance to a bounded number of acyclic instances, each of which can be solved in polynomial time by dynamic programming.Comment: Full version of a paper that appeared in preliminary form in the proceedings of IPEC'1

Neural Networks for Structural Optimisation of Mechanical Metamaterials

Mechanical metamaterials are man‐made designer materials with unusual properties, which are derived from the micro‐structure rather than the base material. Thus, metamaterials are suitable for tailoring and structural optimisation to enhance certain properties. A widely known example for this class of materials are auxetics with a negative Poisson's ratio. In this work an auxetic unit cell is modified with an additional half strut.During the deformation this half strut will get into contact with the unit cell and provide additional stability. This leads to a higher plateau stress and consequently to a higher energy absorption capacity. To achieve the maximum energy absorption capacity, a structural optimisation is carried out. But an optimisation exclusively based on finite element simulations is computationally costly and takes a lot of time. Therefore, in this contribution neural networks are used as a tool to speed up the optimisation. Neural networks are one of many machine learning methods and are able to approximate any arbitrary function on a highly abstract level. So the stress‐strain behaviour and its dependency from the geometry parameters of a type of microstructure can be learned by the neural network with only a few finite element simulations of varying geometry parameters. The modified auxetic structure is optimised with respect to the mass specific energy absorption capacity. As a result a qualitative trend for the optimal geometry parameters is obtained. However, the Poisson's ratio for this optimisation is close to zero