Algorithmic derivation of functional renormalization group equations and Dyson-Schwinger equations

We present the Mathematica application DoFun which allows to derive Dyson-Schwinger equations and renormalization group flow equations for n-point functions in a simple manner. DoFun offers several tools which considerably simplify the derivation of these equations from a given physical action. We discuss the application of DoFun by means of two different types of quantum field theories, namely a bosonic O(N) theory and the Gross-Neveu model.Comment: 40 pages, 12 figs.; corresponds to published versio

Community tracking in a cMOOC and nomadic learner behavior identification on a connectivist rhizomatic learning network

This article contributes to the literature on connectivism, connectivist MOOCs (cMOOCs) and rhizomatic learning by examining participant interactions, community formation and nomadic learner behavior in a particular cMOOC, #rhizo15, facilitated for 6 weeks by Dave Cormier. It further focuses on what we can learn by observing Twitter interactions particularly. As an explanatory mixed research design, Social Network Analysis and content analysis were employed for the purposes of the research. SNA is used at the macro, meso and micro levels, and content analysis of one week of the MOOC was conducted using the Community of Inquiry framework. The macro level analysis demonstrates that communities in a rhizomatic connectivist networks have chaotic relationships with other communities in different dimensions (clarified by use of hashtags of concurrent, past and future events). A key finding at the meso level was that as #rhizo15 progressed and number of active participants decreased, interaction increased in overall network. The micro level analysis further reveals that, though completely online, the nature of open online ecosystems are very convenient to facilitate the formation of community. The content analysis of week 3 tweets demonstrated that cognitive presence was the most frequently observed, while teaching presence (teaching behaviors of both facilitator and participants) was the lowest. This research recognizes the limitations of looking only at Twitter when #rhizo15 conversations occurred over multiple platforms frequented by overlapping but not identical groups of people. However, it provides a valuable partial perspective at the macro meso and micro levels that contribute to our understanding of community-building in cMOOCs

Testing an Optimised Expansion on Z_2 Lattice Models

We test an optimised hopping parameter expansion on various Z_2 lattice scalar field models: the Ising model, a spin-one model and lambda (phi)^4. We do this by studying the critical indices for a variety of optimisation criteria, in a range of dimensions and with various trial actions. We work up to seventh order, thus going well beyond previous studies. We demonstrate how to use numerical methods to generate the high order diagrams and their corresponding expressions. These are then used to calculate results numerically and, in the case of the Ising model, we obtain some analytic results. We highlight problems with several optimisation schemes and show for the best scheme that the critical exponents are consistent with mean field results to at least 8 significant figures. We conclude that in its present form, such optimised lattice expansions do not seem to be capturing the non-perturbative infra-red physics near the critical points of scalar models.Comment: 47 pages, some figures in colour but will display fine in B

Thin presentation of knots and lens spaces

This paper concerns thin presentations of knots K in closed 3-manifolds M^3 which produce S^3 by Dehn surgery, for some slope gamma. If M does not have a lens space as a connected summand, we first prove that all such thin presentations, with respect to any spine of M have only local maxima. If M is a lens space and K has an essential thin presentation with respect to a given standard spine (of lens space M) with only local maxima, then we show that K is a 0-bridge or 1-bridge braid in M; furthermore, we prove the minimal intersection between K and such spines to be at least three, and finally, if the core of the surgery K_gamma yields S^3 by r-Dehn surgery, then we prove the following inequality: |r| <= 2g, where g is the genus of K_gamma.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-23.abs.htm