Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, defined on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension $d \ge 3$, these pathologies occur in a full neighborhood $\{ \beta > \beta_0 ,\, |h| < \epsilon(\beta) \}$ of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension $d \ge 2$, the pathologies occur at low temperatures for arbitrary magnetic-field strength. Pathologies may also occur in the critical region for Ising models in dimension $d \ge 4$. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.

Model-Theoretic Optimization Approach to Triathlon Performance Under Comparative Static Conditions Results Based on the Olympic Games 2012

In Olympic-distance triathlon, time minimization is the goal in all three disciplines and the two transitions. Running is the key to winning, whereas swimming and cycling performance are less significantly associated with overall competition time. A comparative static simulation calculation based on the individual times of each discipline was done. Furthermore, the share of the discipline in the total time proved that increasing the scope of running training results in an additional performance development. Looking at the current development in triathlon and taking the Olympic Games in London 2012 as an initial basis for model-theoretic simulations of performance development, the first fact that attracts attention is that running becomes more and more the crucial variable in terms of winning a triathlon. Run times below 29:00 minutes in Olympic-distance triathlon will be decisive for winning. Currently, cycle training time is definitely overrepresented. The share of swimming is considered optimal