On the optimality of the uniform random strategy

The concept of biased Maker-Breaker games, introduced by Chv\'atal and Erd{\H o}s, is a central topic in the field of positional games, with deep connections to the theory of random structures. For any given hypergraph ${\cal H}$ the main questions is to determine the smallest bias $q({\cal H})$ that allows Breaker to force that Maker ends up with an independent set of ${\cal H}$. Here we prove matching general winning criteria for Maker and Breaker when the game hypergraph satisfies a couple of natural `container-type' regularity conditions about the degree of subsets of its vertices. This will enable us to derive a hypergraph generalization of the $H$-building games, studied for graphs by Bednarska and {\L}uczak. Furthermore, we investigate the biased version of generalizations of the van der Waerden games introduced by Beck. We refer to these generalizations as Rado games and determine their threshold bias up to constant factors by applying our general criteria. We find it quite remarkable that a purely game theoretic deterministic approach provides the right order of magnitude for such a wide variety of hypergraphs, when the generalizations to hypergraphs in the analogous setup of sparse random discrete structures are usually quite challenging.Comment: 26 page

A Note on the Minimum Number of Edges in Hypergraphs with Property O

An oriented $k$-uniform hypergraph is said to have Property O if for every linear order of the vertex set, there is some edge oriented consistently with the linear order. Recently Duffus, Kay and R\"{o}dl investigated the minimum number $f(k)$ of edges in a $k$-uniform hypergaph with Property O. They proved that $k! \leq f(k) \leq (k^2 \ln k) k!$, where the upper bound holds for $k$ sufficiently large. In this short note we improve their upper bound by a factor of $k \ln k$, showing that $f(k) \le \left(\lfloor \frac{k}{2} \rfloor +1 \right) k! - \lfloor \frac{k}{2} \rfloor (k-1)!$ for every $k\geq 3$. We also show that their lower bound is not tight. Furthermore, Duffus, Kay and R\"{o}dl also studied the minimum number $n(k)$ of vertices in a $k$-uniform hypergaph with Property O. For $k=3$ they showed $n(3) \in \{6,7,8,9\}$, and asked for the precise value of $n(3)$. Here we show $n(3)=6$.Comment: 6 pages, 1 figur

Diversities, affinities and diasporas: a southern lens and methodology for understanding multilingualisms

We frame multilingualisms through a growing interest in a linguistics and sociology of the ‘south’ and acknowledge earlier contributions of linguists in Africa, the Américas and Asia who have engaged with human mobility, linguistic contact and consequential ecologies that alter over time and space. Recently, conversations of multilingualism have drifted in two directions. Southern conversations have become intertwined with ‘decolonial theory’, and with ‘southern’ theory, thinking and epistemologies. In these, ‘southern’ is regarded as a metaphor for marginality, coloniality and entanglements of the geopolitical north and south. Northern debates that receive traction appear to focus on recent ‘re-awakenings’ in Europe and North America that mis-remember southern experiences of linguistic diversity. We provide a contextual backdrop for articles in this issue that illustrate intelligences of multilingualisms and the linguistic citizenship of southern people. In these, southern multilingualisms are revealed as phenomena, rather than as a phenomenon defined usually in English. The intention is to suggest a third direction of mutual advantage in rethinking the social imaginary in relation to communality, entanglements and interconnectivities of both South and North

An acetylated form of histone H2A.Z regulates chromosome architecture in Schizosaccharomyces pombe

Histone variant H2A.Z has a conserved role in genome stability, although it remains unclear how this is mediated. Here we demonstrate that the fission yeast Swr1 ATPase inserts H2A.Z (Pht1) into chromatin and Kat5 acetyltransferase (Mst1) acetylates it. Deletion or an unacetylatable mutation of Pht1 leads to genome instability, primarily caused by chromosome entanglement and breakage at anaphase. This leads to the loss of telomere-proximal markers, though telomere protection and repeat length are unaffected by the absence of Pht1. Strikingly, the chromosome entanglement in pht1Delta anaphase cells can be rescued by forcing chromosome condensation before anaphase onset. We show that the condensin complex, required for the maintenance of anaphase chromosome condensation, prematurely dissociates from chromatin in the absence of Pht1. This and other findings suggest an important role for H2A.Z in the architecture of anaphase chromosomes

Probleme in Positionsspielen und der extremalen Kombinatorik

This thesis consists of five Chapters. The main purpose of the first chapter is to serve as an introduction. We will define all necessary concepts and discuss the problems that are studied in this thesis as well as stating the main results of it. This thesis deals with two different directions of extremal graph theory. In the first part we consider two kinds of positional games, the so-called strong games and the Maker-Breaker games. In the second chapter we consider the following strong Ramsey game: Two players take turns in claiming a previously unclaimed hyperedge of the complete k-uniform hypergraph on n vertices until all edges have been claimed. The first player to build a copy of a predetermined k-uniform hypegraph is declared the winner of the game. If none of the players win, then the game ends in a draw. The well-known strategy stealing argument shows that the second player cannot expect to ever win this game. Moreover, for sufficiently large n, it follows from Ramsey’s Theorem for hypergraphs that the game cannot end in a draw and is thus a first player win. Now suppose the game is played on the infinite k-uniform complete hypergraph. Strategy stealing and Ramsey’s Theorem still hold and so we might ask the following question: is this game still a first player win or a draw. In this chapter we construct a 5-uniform hypergraph for which the corresponding game is a draw. In the third chapter we consider biased (1 : q) Maker-Breaker games: Two players called Maker and Breaker alternate in occupying previously unoccupied vertices of a given hyper- graph H. Maker occupies 1 vertex per round and Breaker occupies q vertices. Maker wins if he fully occupies a hyperedge of H and Breaker wins otherwise. One of the central questions in this area is to find (or at least approximate) the maximal value of q that allows Maker to win the game. In this chapter we prove two new general winning criteria -one for Maker and one for Breaker- and apply them to two types of games. In the first type, the target is a fixed uniform hypergraph and in the second it is a solution to an arbitrary but fixed linear system of inhomogeneous equations. The second part of this thesis deals with two types of questions from extremal combi- natorics. The first type is of the following form: suppose we are given a certain property of hypergraphs, what is the minimum possible number of edges a k-uniform hypergraph can have such that it does have the property of interest. The second type goes in the different direction. If a certain family of hypergraphs has the minimum number of edges satisfying a certain property, i.e. if the family is extremal in that sense, then how can we characterise it and what operations can we perform while maintaining extremality? In the fourth chapter, we investigate the minimum number f(k) of edges a k-uniform hypergaph having Property O can have. Duffus, Kay and Rödl showed that k! ≤ f(k) ≤ (k2lnk)k!, where the upper bound holds for sufficiently large k. We improve the upper bound by a factor of klnk showing f(k) ≤ ⌊k⌋+1 k!−⌊k⌋(k−1)! for every k ≥ 3. We also answer a question regarding the minimum number n(k) of vertices a k-uniform hypergaph having Property O can have. In the fifth chapter, we consider shattering extremal set systems. A set system F ⊆ 2[n] is said to shatter a given set S⊆[n] if 2^S ={F ∩ S: F ∈ F}. TheSauer-ShelahLemma states that in general, a set system F shatters at least |F| sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly |F| sets. The so- called elimination conjecture, independently formulated by Mészáros and Rónyai as well as by Kuzmin and Warmuth, states that if a family is shattering- extremal then one can delete a set from it and the resulting family is still shattering-extremal. We prove this conjecture for a class of set systems defined from Sperner systems and for Sperner systems of size at most 4. Furthermore we continue the investigation of the connection between shattering extremal set systems and Gröbner bases.Diese Dissertation besteht aus 5 Kapiteln. Das erste Kapitel dient als Einleitung und stellt die in der Dissertation behandelten Themen und Ergebnisse vor. Das zweite Kapitel befasst sich mit sogenannten ”strong Ramsey games”, bei denen zwei Spieler ab- wechselnd Kanten des vollst ̈andigen (Hyper-)Graphen beanspruchen. Gewinner des Spiels ist der erste Spieler, welcher einen zuvor fest gewählten (Hyper-)Graphen vollst ̈andig für sich beanspruchen kann. Hat der vollst ̈andige Graph hinreichend viele Knoten, so kann gezeigt werden, dass für den beginnen- den Spieler stets eine Strategie existiert, die es ihm ermöglicht, das Spiel zu gewinnen. Entgegen der allgemeinen Vermutung, dass dies auch auf unendliche vollständige Graphen zutrifft, konstruieren wir einen 5-uniformen Hypergraphen, für den der zweite Spieler ein Unentschieden erzwingen kann. Das dritte Kapitel befasst sich mit sogenannten ‘biased (1 : q) Maker-Breaker’ Spielen. Zwei Spieler, genannt Maker und Breaker, beanspruchen abwechselnd Knoten eines gegebenen Hypergraphen. Maker beansprucht einen Knoten pro Runde und Breaker q Knoten. Maker gewinnt, falls er eine Hyperkante vollständig für sich beanspruchen kann, andernfalls gewinnt Breaker. Eine der zentralen Fragen auf diesem Gebiet ist, die sogenannte threshold bias zu finden. Wir beweisen allgemeine Gewinnkriterien, eins für Maker und eins für Breaker, und wenden diese auf zwei Klassen von Spielen an. Zum Einen verallgemeinern wir ein bekanntes Resultat von Bednarska und Luczak auf Hypergraphen. Zum Anderen bestimmen wir, bis auf eine Konstante, die threshold bias, wenn das Ziel des Spiels eine Lösung zu einem beliebigen, aber festen linearen System von inhomogenen Gleichungen ist. Das vierte Kapitel beschäftigt sich mit der sogenannten Ordnungseigenschaft von geordneten Hyper- graphen, welche kürzlich von Duffus, Kay und Rödl eingeführt worden ist. Wir verbessern die von jenen bewiesene obere Schranke um einen Faktor k ln k. Das letzte Kapitel befasst sich mit dem Begriff des Zerschmetterns (,shatter’). Ein Mengensystem heißt s-extremal, wenn es die Ungleichung von Sauer und Shelah mit Gleichheit erfüllt. Eine Vermutung von Mészáros und Rónyai besagt, dass man zu einem s-extremalen Mengensystem stets eine Menge hinzufügen kann, sodass das resultierende Mengensystem wiederum s-extremal ist. Wir beweisen diese Vermutung für bestimmte Mengensysteme, welche durch sogenannte Sperner Familien definiert werden und für den Fall, dass die Sperner Familie aus höchstens vier Mengen besteht. Zusätzlich beweisen wir ein neues Resultat, welches den Zusammenhang mit Gröbnerbasen weiter beleuchtet

Shattering-extremal set systems from Sperner families

We say that a set system F subset of 2([n]) shatters a given set S subset of [n] if 2(S) = {F boolean AND S: F is an element of F}. The Sauer-Shelah lemma states that in general, a set system F shatters at least vertical bar F vertical bar sets. We concentrate on the case of equality and call a set system shattering-extremal if it shatters exactly vertical bar F vertical bar sets. Here we discuss an approach to study these systems using Sperner families and prove some preliminary results based on an earlier algebraic approach. (C) 2019 Elsevier B.V. All rights reserved

Road to type approval

The Road to Type Approval with its new UNECE WP.29 cybersecurity requirements still needs to be mastered. Drawing on experience in implementing cybersecurity management systems for OEMs and suppliers worldwide, we identify best practices. The topics range from the relation with information security, key standards like the ISO/SAE 21434, to implications for secure future E/E architectures. We present actionable insights that may guide CSMS implementations and the journey to Type Approval

Allosteric signaling in C-linker and cyclic nucleotide-binding domain of HCN2 channels

Opening of hyperpolarization-activated cyclic nucleotide-modulated (HCN) channels is controlled by membrane hyperpolarization and binding of cyclic nucleotides to the tetrameric cyclic nucleotide-binding domain (CNBD), attached to the C-linker disk (CL). Confocal patch-clamp fluorometry revealed pronounced cooperativity of ligand binding among protomers. However, by which pathways allosteric signal transmission occurs remained elusive. Here, we investigate how changes in the structural dynamics of the CL-CNBD of mouse HCN2 upon cAMP binding relate to inter- and intrasubunit signal transmission. Applying a rigidity theory-based approach, we identify two intersubunit and one intrasubunit pathways that differ in allosteric coupling strength between cAMP binding sites or towards the CL. These predictions agree with results from electrophysiological and patch-clamp fluorometry experiments. Our results map out distinct routes within the CL-CNBD that modulate different cAMP binding responses in HCN2 channels. They signify that functionally relevant submodules may exist within and across structurally discernable subunits in HCN channels