Impartial achievement and avoidance games for generating finite groups

We study two impartial games introduced by Anderson and Harary and further developed by Barnes. Both games are played by two players who alternately select previously unselected elements of a finite group. The first player who builds a generating set from the jointly selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. After the development of some general results, we determine the nim-numbers of these games for abelian and dihedral groups. We also present some conjectures based on computer calculations. Our main computational and theoretical tool is the structure diagram of a game, which is a type of identification digraph of the game digraph that is compatible with the nim-numbers of the positions. Structure diagrams also provide simple yet intuitive visualizations of these games that capture the complexity of the positions.Comment: 28 pages, 44 figures. Revised in response to comments from refere

Impartial avoidance games for generating finite groups

We study an impartial avoidance game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The first player who cannot select an element without making the set of jointly-selected elements into a generating set for the group loses the game. We develop criteria on the maximal subgroups that determine the nim-numbers of these games and use our criteria to study our game for several families of groups, including nilpotent, sporadic, and symmetric groups.Comment: 14 pages, 4 figures. Revised in response to comments from refere

Impartial avoidance and achievement games for generating symmetric and alternating groups

We study two impartial games introduced by Anderson and Harary. Both games are played by two players who alternately select previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. We determine the nim-numbers, and therefore the outcomes, of these games for symmetric and alternating groups.Comment: 12 pages. 2 tables/figures. This work was conducted during the third author's visit to DIMACS partially enabled through support from the National Science Foundation under grant number #CCF-1445755. Revised in response to comments from refere

The spectrum of nim-values for achievement games for generating finite groups

We study an impartial achievement game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The game ends when the jointly selected elements generate the group. The last player able to make a move is the winner of the game. We prove that the spectrum of nim-values of these games is $\{0,1,2,3,4\}$. This positively answers two conjectures from a previous paper by the last two authors.Comment: 11 pages, 5 figures, 2 table

Impartial achievement games for generating nilpotent groups

We study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for finite groups of the form $T \times H$, where $T$ is a $2$-group and $H$ is a group of odd order. This includes all nilpotent and hence abelian groups.Comment: 10 pages, 2 figure

The alignment of projects dealing with wetland restoration and alien control: A challenge for conservation management in South Africa

An inventory of wetland vegetation across the country generated a list of the most common invasive alien plants across South Africa. Many of the plants on that list do not correspond with the priorities in the programmes for alien control across the country, as they are not listed on a government produced list that guides the priorities for alien control. We explore the reasons for this situation. We argue that because wetlands are such important parts of the landscape, invasive aliens in wetlands are of special concern, and there should be more alignment between alien control programmes and wetland rehabilitation programmes. This alignment starts by considering the full number of species that form a threat to wetland habitats, but also considers which pesticides to use, erosion and recolonisation in wetlands, planting indigenous vegetation after aliens have been removed, and strategising by working from upstream to downstream. Existing alien control programmes for specific grasses (some relatively new to the country and in the phase of early detection) and floating aquatic plants may guide how to tackle the invasions of grasses and forbs that have been established in South African wetlands for an extended period of time. Significance:• Wetlands have a distinct set of alien invasive plants that affect their ecology and functioning and many of these plants are not listed as priorities in alien control programmes.• Many restoration projects have an element of removing invasive plants and revegetating. Wetland restoration and alien control need to be integrated to preserve water resources

Corrigendum: The alignment of projects dealing with wetland restoration and alien control: A challenge for conservation management in South Africa

Errors that appear in the Discussion of the Research Article by Sieben et al. are corrected here. Dr Graham Harding (Registered PCO, Invader Plant Specialists (Pty) Ltd) is acknowledged for drawing the authors’ attention to these errors

Application of Super-Resolution and Advanced Quantitative Microscopy to the Spatio-Temporal Analysis of Influenza Virus Replication

With an estimated three to five million human cases annually and the potential to infect domestic and wild animal populations, influenza viruses are one of the greatest health and economic burdens to our society, and pose an ongoing threat of large-scale pandemics. Despite our knowledge of many important aspects of influenza virus biology, there is still much to learn about how influenza viruses replicate in infected cells, for instance, how they use entry receptors or exploit host cell trafficking pathways. These gaps in our knowledge are due, in part, to the difficulty of directly observing viruses in living cells. In recent years, advances in light microscopy, including super-resolution microscopy and single-molecule imaging, have enabled many viral replication steps to be visualised dynamically in living cells. In particular, the ability to track single virions and their components, in real time, now allows specific pathways to be interrogated, providing new insights to various aspects of the virus-host cell interaction. In this review, we discuss how state-of-the-art imaging technologies, notably quantitative live-cell and super-resolution microscopy, are providing new nanoscale and molecular insights into influenza virus replication and revealing new opportunities for developing antiviral strategies

Categories of impartial rulegraphs and gamegraphs

The traditional mathematical model for an impartial combinatorial game is defined recursively as a set of the options of the game, where the options are games themselves. We propose a model called gamegraph, together with its generalization rulegraph, based on the natural description of a game as a digraph where the vertices are positions and the arrows represent possible moves. Such digraphs form a category where the morphisms are option preserving maps. We study several versions of this category. Our development includes congruence relations, quotients, and isomorphism theorems and is analogous to the corresponding notions in universal algebra. The quotient by the maximum congruence relation produces an object that is essentially equivalent to the traditional model. After the development of the general theory, we count the number of non-isomorphic gamegraphs and rulegraphs by formal birthday and the number of positions.Comment: 22 pages, 19 figure