Perpetually Dominating Large Grids

In the m-\emph{Eternal Domination} game, a team of guard tokens initially occupies a dominating set on a graph $G$. An attacker then picks a vertex without a guard on it and attacks it. The guards defend against the attack: one of them has to move to the attacked vertex, while each remaining one can choose to move to one of his neighboring vertices. The new guards' placement must again be dominating. This attack-defend procedure continues eternally. The guards win if they can eternally maintain a dominating set against any sequence of attacks, otherwise, the attacker wins. The m-\emph{eternal domination number} for a graph $G$ is the minimum amount of guards such that they win against any attacker strategy in $G$ (all guards move model). We study rectangular grids and provide the first known general upper bound on the m-eternal domination number for these graphs. Our novel strategy implements a square rotation principle and eternally dominates $m \times n$ grids by using approximately $\frac{mn}{5}$ guards, which is asymptotically optimal even for ordinary domination.Comment: latest full draft versio

Perpetually Dominating Large Grids

In the Eternal Domination game, a team of guard tokens initially occupies a dominating set on a graph G. A rioter then picks a node without a guard on it and attacks it. The guards defend against the attack: one of them has to move to the attacked node, while each remaining one can choose to move to one of his neighboring nodes. The new guards' placement must again be dominating. This attack-defend procedure continues perpetually. The guards win if they can eternally maintain a dominating set against any sequence of attacks, otherwise the rioter wins. We study rectangular grids and provide the first known general upper bound for these graphs. Our novel strategy implements a square rotation principle and eternally dominates m x n grids by using approximately (mn)/5 guards, which is asymptotically optimal even for ordinary domination

Visual Culture Archaeology: A Criti/Politi/cal Methodology of Image and Identity

This study argues the efficacy of the phenomenological cultural work of a visual culture archaeology that liberates a political and critical identity, resistant to domination, authoring social change and its own agency through multiple and incommensurable positions. Built upon Foucauldian premises, visual culture archaeology is developed as a methodology for discursive un-naming and re-naming, and emerges from the inherence and attenuation of inscripted meanings in the reinterpretation of identity during a postmodern confluence of ideas and images. The hybridized representation of the African American in Western visual culture has been unique in the effort by some to define us over significant periods as less than human, less than American, or less than statistically significant in the purpose to maintain an unequal relation of economic and political power. This paper continues the author’s effort to establish the efficacy of a poststructural and poetic aesthetic in qualitative research writing

Eternal Domination in Grids

In the eternal domination game played on graphs, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices on their turn. The goal is to determine the eternal domination number $\gamma^{\infty}_{all}$ of a graph which is the minimum number of guards required to defend against an infinite sequence of attacks.This paper continues the study of the eternal domination game on strong grids $P_n\boxtimes P_m$. Cartesian grids $P_n \square P_m$ have been vastly studied with tight bounds existing for small grids such as $k\times n$ grids for $k\in \{2,3,4,5\}$. It was recently proven that $\gamma^{\infty}_{all}(P_n \square P_m)=\gamma(P_n \square P_m)+O(n+m)$ where $\gamma(P_n \square P_m)$ is the domination number of $P_n \square P_m$ which lower bounds the eternal domination number [Lamprou et al., CIAC 2017]. We prove that, for all $n,m\in \mathbb{N^*}$ such that $m\geq n$, $\lfloor \frac{n}{3} \rfloor \lfloor \frac{m}{3} \rfloor+\Omega(n+m)=\gamma_{all}^{\infty} (P_{n}\boxtimes P_{m})=\lceil \frac{n}{3} \rceil \lceil \frac{m}{3} \rceil + O(m\sqrt{n})$ (note that $\lceil \frac{n}{3} \rceil \lceil \frac{m}{3} \rceil$ is the domination number of $P_n\boxtimes P_m$). Our technique may be applied to other ``grid-like" graphs

Dialectical Reasoning and Developing Responsive Models Toward Political Ecology

In this thesis, I seek out the modes of thought that we have developed for making sense of the world and elucidate how the logic of domination and reduction of reason to a calculative tool has led to the climate crisis. Throughout my research, I look for models to overcome mechanized thought and find two useful remedies that will require time and effort to implement: critical self-reflection and storytelling skills. Self-reflection involves dialectically thinking or considering alternative approaches to how we understand the world rather than accepting the standard norms for thinking and using them without question. Storytelling involves the skills of communicating and also of listening to other persons, which include human persons and nonhuman persons like plants and animals. All of this is aimed at implementing a novel form of political ecology – a politics built around ecological intelligences and the democratic deliberations of all persons from all understandings of the world

Mobility Problems in Distributed Search and Combinatorial Games

This thesis examines a collection of topics under the general notion of mobility of agents. We examine problems where a set of entities, perceived as robots or tokens, navigate in some given (discrete or continuous) environment to accomplish a goal. The problems we consider fall under two main research fields. First, Distributed Search where the agents cooperate to explore their environment or search for a specific target location within it. Second, Combinatorial Games, in the spirit of Pursuit-Evasion, where the agents are now divided into two groups with complementary objectives competing against each other. More specifically, we consider three distinct problems: disk evacuation, exploration of dynamic graphs and eternal domination. In Disk Evacuation, two robots with different speeds aim to discover an unknown exit lying on the boundary of a unit disk. For a wide range of speeds, we provide matching upper and lower bounds. In Dynamic Graph Exploration, we analyze the exploration time for a randomly-walking agent wishing to visit all the vertices of a stochastically-evolving graph. In Eternal Domination, we consider rectangular grid graphs and upper bound the amount of guard agents needed to perpetually defend the vertices against an attacker

Securing Populations: Foucault and the Cartography of Natural Bodies

The concept of biopolitics tends towards universal applicability and thus analytical impotency. By examining Foucault’s lecture seminars that address this concept directly and indirectly, this project aims to delimit its coordinates for future use. To do so, I begin by looking at the way biopolitical discourses on the population constituted liberal governmentality in the eighteenth century. This analysis will be supplemented by a cartography of the surfaces on which biopolitics emerges before and within liberalism, affecting its formation. I will therefore map out the formation of two objects that characterize modern biopower: the ‘natural’ body of the individual and the ‘natural’ body of the population. This will open up the problem of normalization and the constitution of a regime of truth run through with relations of power. Counter-posed to the disciplines, the dispositif of security will be shown to produce normalizing effects at the level of probable events, congealing them into realities of their own. The naturalness of the population will be normalized as such a reality constituting biopolitics as a politics of the chance event of life

After the Golden Age: Romantic Pianism and Modern Performance by Kenneth Hamilton

Winter reviews and critiques Hamilton\u27s work