Subfactors of index less than 5, part 1: the principal graph odometer

In this series of papers we show that there are exactly ten subfactors, other than $A_\infty$ subfactors, of index between 4 and 5. Previously this classification was known up to index $3+\sqrt{3}$. In the first paper we give an analogue of Haagerup's initial classification of subfactors of index less than $3+\sqrt{3}$, showing that any subfactor of index less than 5 must appear in one of a large list of families. These families will be considered separately in the three subsequent papers in this series.Comment: 36 pages (updated to reflect that the classification is now complete

The blob complex

Given an n-manifold M and an n-category C, we define a chain complex (the "blob complex") B_*(M;C). The blob complex can be thought of as a derived category analogue of the Hilbert space of a TQFT, and as a generalization of Hochschild homology to n-categories and n-manifolds. It enjoys a number of nice formal properties, including a higher dimensional generalization of Deligne's conjecture about the action of the little disks operad on Hochschild cochains. Along the way, we give a definition of a weak n-category with strong duality which is particularly well suited for work with TQFTs.Comment: 106 pages. Version 3 contains many improvements following suggestions from the referee and others, and some additional materia

Fixing the functoriality of Khovanov homology

We describe a modification of Khovanov homology (math.QA/9908171), in the spirit of Bar-Natan (math.GT/0410495), which makes the theory properly functorial with respect to link cobordisms. This requires introducing `disorientations' in the category of smoothings and abstract cobordisms between them used in Bar-Natan's definition. Disorientations have `seams' separating oppositely oriented regions, coming with a preferred normal direction. The seams satisfy certain relations (just as the underlying cobordisms satisfy relations such as the neck cutting relation). We construct explicit chain maps for the various Reidemeister moves, then prove that the compositions of chain maps associated to each side of each of Carter and Saito's movie moves (MR1238875, MR1445361) always agree. These calculations are greatly simplified by following arguments due to Bar-Natan and Khovanov, which ensure that the two compositions must agree, up to a sign. We set up this argument in our context by proving a result about duality in Khovanov homology, generalising previous results about mirror images of knots to a `local' result about tangles. Along the way, we reproduce Jacobsson's sign table (math.GT/0206303) for the original `unoriented theory', with a few disagreements.Comment: 91 pages. Added David Clark as co-author. Further detail on variations of third Reidemeister moves, to allow treatment of previously missing cases of movie move six. See changelog section for more detai

Developing Mental Toughness in Collegiate Football: A Phenomenological Study

The purpose of this study was to enhance the phenomenon of mental toughness, by addressing the literature gap of how mental toughness can be developed in college football. This qualitative study used a tri-level snowball method, first interviewing head coach participants recommended from a collegiate football expert. Then, interviewing assistant coaches who were recommended by each head coach. Lastly, interviewing a focus group of current collegiate football athletes, who were recommended by both head coaches and assistant coaches. This study addressed four research questions surrounding the development of mental toughness in college football athletes, the differences between coaches’ and athletes’ perceptions of mental toughness, the incorporation of the four Cs (Clough et al., 2002) of mental toughness, comparing football coaches’ definitions of mental toughness, and applying mental toughness found to other sport programs. The following was found: Coaches felt mental toughness was necessary for both life and football and can be developed through techniques both on and off the field involving football; Athletes and coaches agree on multiple aspects of mental toughness but have differing opinions of how it is developed; The four Cs played an important role in the development of mental toughness; That definitions are subjective dependent upon characteristics that each coach deems important, and that mental toughness can be applied to other sports using some of the same techniques that football coaches would use

Les droits culturels dans la grammaire du développement

Les droits culturels permettent de comprendre, de façon théorique et pratique, la place fondamentale des facteurs culturels pour un développement libre et équilibré. Un tel développement ne peut que se fonder sur ce qui devrait être considéré comme ses premières ressources: les ressources culturelles, humaines et non humaines. Les droits culturels garantissent l'accès des personnes, des groupes et des communautés à la diversité des savoirs qui sont nécessaires pour utiliser, développer, créer les capacités de développement personnel et social. Pas de développement social sans une culture des liens ; pas de développement économique sans une culture adaptée des choix et des échanges ; pas de développement écologique sans un accroissement des savoirs sur les écosystèmes. Les facteurs culturels ne viennent pas après comme un quatrième pilier, ils sont les facteurs et les leviers fondamentaux, transversaux et permanents de tout développement fondé sur les capacités des personnes et de leurs écosystèmes culturels. Ils sont comme une grammaire de tout développement démocratique, dans la mesure où leur réalisation permet de garantir que le choix des objectifs et des moyens est issu d'un débat largement participatif et instruit, de façon à repérer et favoriser les meilleures connexions entre les ressources, et donc aussi la meilleure économie. Des indicateurs de connexions adéquates apparaissent alors comme des outils très pertinents d'évaluation.Los derechos culturales permiten entender, de manera tanto teórica como práctica, el lugar fundamental de los factores culturales para un desarrollo libre y equilibrado. Dicho desarrollo sólo puede basarse en lo que se deberían considerar sus primeros recursos: los recursos culturales humanos y no humanos. Los derechos culturales garantizan el acceso de las personas, de los grupos y de las comunidades a la diversidad de conocimientos necesarios para utilizar, desarrollar y crear las capacidades de desarrollo personal y social. No hay desarrollo social sin una cultura de conexiones; no hay desarrollo económico sin una cultura adaptada a los cambios y a los intercambios; no hay desarrollo ecológico sin una expansión del saber sobre los ecosistemas. Los factores culturales no son un cuarto pilar, son factores y engranajes fundamentales, transversales y permanentes de todo desarrollo cimentado sobre las capacidades de las personas y de sus ecosistemas culturales. Son como una gramática de todo desarrollo democrático en la medida en que su realización permite garantizar que la elección de objetivos y medios ha surgido de un debate ampliamente participativo e instruido para percibir y favorecer las mejores conexiones entre los recursos y, por tanto, también la mejor economía. Los indicadores de conexiones adecuadas, en consecuencia, resultan ser herramientas muy pertinentes de evaluación.Els drets culturals permeten entendre, de manera tant teòrica com pràctica, el lloc fonamental dels factors culturals per a un desenvolupament lliure i equilibrat. El dit desenvolupament només pot basar-se en el que s'haurien de considerar els seus primers recursos: els recursos culturals humans i no humans. Els drets culturals garanteixen l'accés de les persones, dels grups i de les comunitats a la diversitat de coneixements necessaris per a utilitzar, desenvolupar i crear les capacitats de desenvolupament personal i social. No hi ha desenvolupament social sense una cultura de connexions; no hi ha desenvolupament econòmic sense una cultura adaptada als canvis i als intercanvis; no hi ha desenvolupament ecològic sense una expansió del saber sobre els ecosistemes. Els factors culturals no són un quart pilar, són factors i engranatges fonamentals, transversals i permanents de tot desenvolupament fonamentat sobre les capacitats de les persones i dels seus ecosistemes culturals. Són com una gramàtica de tot desenvolupament democràtic en la mesura que la seua realització permet garantir que l'elecció d'objectius i mitjans ha sorgit d'un debat àmpliament participatiu i instruït per a percebre i afavorir les millors connexions entre els recursos i, per tant, també la millor economia. Els indicadors de connexions adequades, en conseqüència, resulten ser eines molt pertinents d'avaluació.Cultural rights enable us to understand, both theoretically and practically, the fundamental role of cultural factors for free and balanced development. Such development can only be based on what should be regarded as primary resources: human and non-human cultural resources. Cultural rights guarantee individuals, groups and communities access to the diversity of knowledge necessary to use, develop and create the capacities for personal and social development. There can be no social development without a culture of connections; no economic development without a culture adapted to changes and exchanges; no ecological development without the spread of knowledge about our ecosystems. Cultural factors are not a fourth pillar; they are fundamental, transversal and permanent factors and mechanisms for all development grounded on the capacities of people and their cultural ecosystems. They are like the grammar for all democratic development in the way their implementation guarantees that the choice of objectives and means has come out of a broad participative and informed debate to perceive and foster the optimum connections among resources and, therefore, also the optimum economy. Indicators of suitable connections are, in consequence, highly pertinent evaluation tools