80 research outputs found
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 ,
these pathologies occur in a full neighborhood of the low-temperature part of the first-order
phase-transition surface. For block-averaging transformations applied to the
Ising model in dimension , the pathologies occur at low temperatures
for arbitrary magnetic-field strength. Pathologies may also occur in the
critical region for Ising models in dimension . 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.
Existence theorems concerning simple integrals of the calculus of variations for discontinuous solutions
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46210/1/205_2004_Article_BF00276912.pd
Existence theorems for multiple integrals of the calculus of variations for discontinuous solutions
The authors prove existence theorems for the minimum of multiple integrals of the calculus of variations with constraints on the derivatives in classes of BV possibly discontinuous solutions. To this effect the integrals are written in the form proposed by Serrin. Usual convexity conditions are requested, but no growth condition. Preliminary closure and semicontinuity theorems are proved which are analogous to those previously proved by Cesari in Sobolev classes. Compactness in L 1 of classes of BV functions with equibounded total variations is derived from Cafiero-Fleming theorems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47939/1/10231_2005_Article_BF01766143.pd
Satistical problems of point processes
Wykłady niniejsze wygłoszone były we wrześniu 1976 r. w Międzynarodowym Centrum Matematycznym im. Stefana Banacha w Warszawie, w ramach Semestru ze Statystyki Matematycznej.This is a set of lectures delivered at the International Mathematical Centre, Warsaw, in 1976. The text contains three sections. Section 1 is devoted to basic notions. In the second section inference from a family of Poisson processes is considered. The last section deals with the problems of filtering of Cox processes (doubly stochastic processes)
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