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.

Human intestinal alkaline phosphatase--release to the blood is linked to lipid absorption, but removal from the blood is not linked to lipoprotein clearance.

To evaluate a possible quantitative relationship between the rise in intestinal alkaline phosphatase (IAP) activity and triglyceride-rich lipoproteins in the blood after an oral fat intake, a specific and sensitive immunocatalytic assay was used. First, day to day variation of the basal IAP activity in the blood of eight volunteers was evaluated. One group of subjects with high basal IAP activity and great variations from one day to the other was distinguished from a group with low basal IAP activity and small day to day variations. The subjects with high basal IAP activities were all secretors of red blood cell antigens, while those with low basal IAP activities were non-secretors. The IAP activity in the blood rose after fat rich meals but not after fat free meals. To further investigate this, IAP activities were measured in the blood of 28 men before 3, 6, 9 and 12 h after a standardized oral fat load of a mixed meal type. The activity rose in all individuals and showed a similar time course as the rise in plasma TG concentration. The elevations of the IAP activity were, however, not quantitatively correlated with the elevations in TG concentration, nor were there any relations in time to peak levels. Subjects with high basal IAP increased their IAP activity more than those with low basal IAP, but within neither group there was any correlation between the basal IAP activity and the rise in IAP activity. IAP did not float with lipoproteins on ultracentrifugation of plasma, nor did IAP bind to lipid droplets from a fat emulsion added to plasma.(ABSTRACT TRUNCATED AT 250 WORDS