Game Brush Number

We study a two-person game based on the well-studied brushing process on graphs. Players Min and Max alternately place brushes on the vertices of a graph. When a vertex accumulates at least as many brushes as its degree, it sends one brush to each neighbor and is removed from the graph; this may in turn induce the removal of other vertices. The game ends once all vertices have been removed. Min seeks to minimize the number of brushes played during the game, while Max seeks to maximize it. When both players play optimally, the length of the game is the game brush number of the graph $G$, denoted $b_g(G)$. By considering strategies for both players and modelling the evolution of the game with differential equations, we provide an asymptotic value for the game brush number of the complete graph; namely, we show that $b_g(K_n) = (1+o(1))n^2/e$. Using a fractional version of the game, we couple the game brush numbers of complete graphs and the binomial random graph $\mathcal{G}(n,p)$. It is shown that for $pn \gg \ln n$ asymptotically almost surely $b_g(\mathcal{G}(n,p)) = (1 + o(1))p b_g(K_n) = (1 + o(1))pn^2/e$. Finally, we study the relationship between the game brush number and the (original) brush number.Comment: 20 pages, 3 figure

Plants traditionally used to make brooms in several European countries

<p>Abstract</p> <p>Background</p> <p>The research was carried out within the course of two years (2005–2006) in four countries from southern, southeast and eastern parts of Europe: Bulgaria, Italy, Macedonia and Romania. The data are collected mainly from Bulgaria and Italy and are compared with those from Macedonia and Romania.</p> <p>Methods</p> <p>The information was gathered largely from literature as well as field collected data and interviewed informants. A brief questionnaire, referring to the vernacular name, plant description, providing specimens from the plants and brooms, details on their use has been prepared and applied.</p> <p>Results</p> <p>The total number of species as brooms in the study areas is about 108. The list includes two fungi taxa which caused the so-called "Witches' brooms". A high species diversity of 106 taxa of vascular plants, belonging to 37 families and 74 genera, is established in the research area. The investigation includes data about scientific name, family, vernacular name, life form, status (wild or cultivated), used parts and place of use. The relations between the plant characteristics and broom specific shape and working qualities, details of the traditionally broom planting and making, the broom as a part of folklore, traditions and religious rituals are discussed.</p> <p>Conclusion</p> <p>Collected data show how ecological, geographical features and different cultures are related with the variety of plants traditionally used as brooms as well as details for their uses. The data about the variety of plants traditionally used to make brooms and the ways in which they are used according to the specific characteristics of the areas are important for ethnobotanical knowledge.</p

Parallel cleaning of a network with brushes

AbstractWe consider the process of cleaning a network where at each time step, all vertices that have at least as many brushes as incident, contaminated edges, send brushes down these edges and remove them from the network. An added condition is that, because of the contamination model used, the final configuration must be the initial configuration of another cleaning of the network. We find the minimum number of brushes required for trees, cycles, complete bipartite networks; and for all networks when all edges must be cleaned on each step. Finally, we give bounds on the number of brushes required for complete networks

Sharon M. Weinstein v. Sinclair Oil Corporation, a Wyoming corporation, dba, Little America : Brief of Appellee

APPELLEE\u27S BRIEF On Appeal from the Summary Judgment of the Third Judicial District Court Salt Lake County, State of Utah Honorable Richard H. Moffat, Presidin

Public Health Rep

19313832PMCnul

Health and Safety Laboratory Reports

From Introduction: "The purpose of this report is to summarize the information gathered during the design and operation of these facilities so that it may be used in providing for the safe handling of beryllium materials.

Appraisal of OSHA's analytic approach

This report reviews the roles that analyses of control technology and regulatory impacts play in OSHA’s standard setting process, and evaluates the various methods and resources on which the agency draws in conducting these efforts. In addition, based on findings from close examinations of a number of OSHA’s past rule-makings, the report provides a critical appraisal of how well these analyses seem to be helping the agency achieve its basic occupational safety and health mission