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.

Proclus' Commentary on Plato's Parmenides

This is the first English translation of Proclus'commentary on Plato'sParmenides. Glenn Morrow's death occurred while he was less than halfway through the translation, which was completed by John Dillon. A major work of the great Neoplatonist philosopher, the commentary is an intellectual tour de force that greatly influenced later medieval and Renaissance thought. As the notes and introductory summaries explain, it comprises a full account of Proclus'own metaphysical system, disguised, as is so much Neoplatonic philosophy, in the form of a commentary

Procli Diadochi sphaera: astronomiam discere incipientibvs incipientibus vtilissima utilissima

Procli Diadochi sphaera: astronomiam discere incipientibvs incipientibus vtilissima utilissim

Tabulae institutionum astronomicarum : adiuncta est Sphaera Procli cum textu Graeco e regione in Latinam linguam converso

Georgii Henischi

Proclus: Commentary on Plato's Timaeus

Proclus' Commentary on the dialogue Timaeus by Plato (d.347 BC), written in the fifth century AD, is arguably the most important commentary on a text of Plato, offering unparalleled insights into eight centuries of Platonic interpretation. It has had an enormous influence on subsequent Plato scholarship. This edition nevertheless offers the first new translation of the work for nearly two centuries, building on significant recent advances in scholarship by Neoplatonic commentators. It will provide an invaluable record of early interpretations of Plato's dialogue, while also presenting Proclus' own views on the meaning and significance of Platonic philosophy. The book presents Proclus' unrepentant account of a multitude of divinities involved with the creation of mortal life, the supreme creator's delegation to them of the creation of human life, and the manner in which they took the immortal life principle from him and wove it together with our mortal parts to produce human beings.</jats:p

Procli sphaera

Thoma Linacro Britanno interpret

Proclus: Commentary on Plato’s Timaeus Volume 4: Book 3, Part 2: Proclus on the World Soul

In the present volume Proclus describes the 'creation' of the soul that animates the entire universe. This is not a literal creation, for Proclus argues that Plato means only to convey the eternal dependence of the World Soul upon higher causes. In his exegesis of Plato's text, Proclus addresses a range of issues in Pythagorean harmonic theory, as well as questions about the way in which the World Soul knows both forms and the visible reality that comprises its body. This part of Proclus' Commentary is particularly responsive to the interpretive tradition that precedes it. As a result, this volume is especially significant for the study of the Platonic tradition from the earliest commentators onwards.</jats:p