5 research outputs found
M-level rook placements
No full textRook theory focuses on placements of non-attacking rooks on boards of various shapes. An important role is played by the rook numbers which count the number of non-attacking placements of a given number of rooks on a board. Ferrers boards,which are boards indexed by integer partitions, are of particular interest. Briggs and Remmel introduced a generalization of rook placements, called m-level rook placements, where a rook is able to attack a subset of the rows.This manuscript presents generalizations of many of the central results regarding rook placements to the case of m-level rook placements. Goldman, Joichi, and White defined the rook polynomial of a board to be the generating function for the rook numbers of that board in the falling factorial basis. By doing so, they were able to give an elegant factorization of the rook polynomial of a Ferrers board in terms of the various column heights. Briggs and Remmel were able to generalize this factorization to the m-level rook polynomial of a subset of Ferrers boards called singleton boards.We give two factorization theorems for the m-level rook polynomial of a Ferrers board. The first is a generalization of the factorization theorem of Briggs and Remmel, working from similar principles. The second relies on a generalization of transposition which we present, called the l-operator. We are also able to use the factorization to describe a unique representative in any m-level equivalence class of Ferrers boards and count the number of singleton boards in the class..When generalizing the factorization from singleton boards to all Ferrers boards, we preserve the definition of the m-level rook polynomial and alter the factorization to apply to all Ferrers boards. We also consider the dual of this problem, applying the factorization of Briggs and Remmel to all Ferrers boards, then trying to determine what is counted by the coefficients of the polynomial in the m-falling factorial basis. It turns out that the coefficients count weighted file placements on a Ferrers board. We also describe a unique representative in each weighted file placement equivalence class of Ferrers boards, as well as count of the number of Ferrers boards in a given weighted file placement equivalence class.Foata and Sch\ufc}tzenberger presented explicit bijections between rook placements on any two rook equivalent Ferrers boards as part of their construction of a unique representative in each equivalence class of Ferrers boards. A key tool in their construction was local transposition. We present analogous bijections between m-level rook placements on any two -level rook equivalent Ferrers boards using the local l-operator.The Garsia-Milne Involution Principle was first used in Garsia and Milne's bijective proof of the Rogers-Ramanujan identities. We use it to construct two types of explicit bijections. The first is an explicit bijection between m-level rook placements on any two m-level rook equivalent singleton boards. The second bijection is between the sets counted by the m-level analogue of hit numbers of any two m-level rook equivalent Ferrers boards, providing a bijective proof that -level equivalent Ferrers boards have the same hit numbers.(Ph. D.)--Michigan State University. Mathematics - Doctor of Philosophy, 2015Includes bibliographical reference
Bijections on m-level Rook Placements
No full textPartition the rows of a board into sets of m rows called levels. An m-level rook placement is a subset of squares of the board with no two in the same column or the same level. We construct explicit bijections to prove three theorems about such placements. We start with two bijections between Ferrers boards having the same number of m-level rook placements. The first generalizes a map by Foata and Schützenberger and our proof applies to any Ferrers board. The second generalizes work of Loehr and Remmel. This construction only works for a special class of Ferrers boards but also yields a formula for calculating the rook numbers of these boards in terms of elementary symmetric functions. Finally we generalize another result of Loehr and Remmel giving a bijection between boards with the same hit numbers. The second and third bijections involve the Involution Principle of Garsia and Milne. Résumé. Nous considérons les rangs d’un échiquier partagés en ensembles de m rangs appelés les niveaux. Un m-placement des tours est un sous-ensemble des carrés du plateau tel qu’il n’y a pas deux carrés dans la même colonne ou dans le même niveau. Nous construisons deux bijections explicites entre des plateaux de Ferrers ayant les mêmes nombres de m-placements. La première est une généralisation d’une fonction de Foata et Schützenberger et notre démonstration est pour n’importe quels plateaux de Ferrers. La deuxième généralise une bijection de Loehr et Remmel. Cette construction marche seulement pour des plateaux particuliers, mais ça donne une formule pour le nombre de m-placements en terme des fonctions symétriques élémentaires. Enfin, nous généralisons un autre résultat de Loehr et Remmel donnant une bijection entre deux plateaux ayant les mêmes nombres de coups. Les deux dernières bijections utilisent le Principe des Involutions de Garsia et Milne
Business Continuity Management and Strategic Planning: the Case of Jordan
No full textBusiness Continuity Management (BCM) is a process that focuses on counteracting organizational risk, disasters and crises. Placing Business Continuity Management in the context of Strategic Planning (SP) will help organizations to cope with a wide range of unexpected incidents before, during and after their occurrence. Subsequently, this will help to ensure the long-term survival of an organization.
The aim of this research is to develop an understanding of the significance of placing BCM in the context of SP. This requires studying BCM, its significance, role and practice; Strategic Planning, its significance, purpose and potential vulnerability; the rationale for placing BCM in the context of SP; the factors that are likely to influence placing BCM in the context of SP including driving factors and obstacles; and managers’ views of BCM and the placing of BCM in the context of SP.
This research was undertaken in the Jordanian context. Data was collected via interviewer-administered questionnaires which were conducted with general managers and other key managers from Jordanian organizations from the banking, insurance, industrial and services sectors. 110 questionnaires were collected. The questionnaires were followed by 10 semi-structured interviews in order to support the quantitative findings obtained by the questionnaires.
The research findings revealed that 80.9% of the surveyed organizations in Jordan used BCM. Those organizations that used BCM differed to some extent in their practice of BCM. 51.8% of the surveyed organizations had BCM placed in the context of SP. SP was important for achieving organizational purposes including those related to BCM. The approach to BCM, which is adopted in Jordanian organizations, helped to place BCM in the context of SP. There were a number of factors that discouraged some Jordanian organizations from placing BCM in the context of SP. However, there were also a number of factors that encouraged some other Jordanian organizations to place BCM in the context of SP. Managers had positive views regarding BCM. They either agreed or strongly agreed that BCM can be integrated with SP; BCM would help their organizations to cope with various types of disasters and crises if it is integrated with SP; BCM was an integral part of their organizations’ approach to risk; and BCM was not an extra burden to their businesses
Larval ecology of malaria vectors and the impact of larviciding on malaria transmission in The Gambia
No full textThe study reported in this thesis explored the ecology of aquatic stages of mosquitoes in the middle reaches of the Gambia River in order to assess the feasibility and impact of microbial larviciding on malaria transmission in large river ecosystems in sub- Saharan Africa. All accessible water bodies in four study zones covering 400 km(^2) were mapped and sampled for mosquitoes. Microbial larvicides were applied in the four zones in across-over design and the impact of larviciding on mosquito densities assessed. Anopheline and culicine mosquitoes were found in all sampled habitats, apart from those with moving water. Similarly, all habitats, except puddles and water channels, had similar larval and pupal densities. Anopheles gambiae sensu lato, the major malaria vector in Africa, exploited a wide range of habitats and despite a decrease in population density during the dry season, could be found in breeding sites throughout the year. Mosquitoes shared habitats with other invertebrates including their predators. A closer look at rice fields revealed that mosquitoes were abundant in rice fields closer to the landward edge of the floodplains where water is fresher and contains high quantifies of nutrients. Mosquitoes of The Gambia were highly susceptible to both Bacillus thuringiensis var. israelensis (Bti) and B. sphaericus microbials, however no residual activity against anopheline larvae was observed. The basic training of personnel in identification of habitats, calibration of application equipment and active larviciding proved to be successful. Routine larviciding was associated with > 91 % reducfion (p < 0.001) in anophelines late stage larval density and 72 % (p < 0.001) in culicines. Overall, larviciding was associated with a 28% (p = 0.005) reduction in the number of adult female Anopheles gambiae s.l. found indoors, although this rose to 42%, when the study zone with the greatest abundance of breeding sites was excluded from the analysis. No significant reduction in adult culicines was observed. Ground application of Bti in areas with extensive floodplains is unlikely to contribute to a substantial reduction in malaria transmission in The Gambia, therefore vector control in such areas should target adult mosquitoes
