Michael's theory of continuous selections. Development and applications

Abstract

A study has been made of the residual resistivity and magnetoresistivity at 4.2 K, the Hall coefficient between 4.2 and 7 7 K and the deviation from Matthiessen's rule at 7 7 K in AI-210 PPM Ti and AI-522 PPM Ti in both the slow-cooled condition and when quenched from the melt. It is shown that in the slow-cooled state, clustering/precipitation of solute atoms occurs creating electron scattering entities which. while appearing to behave like point defects. nevertheless possess a scattering anisotropy entirely different from conventional point defects. As a result these clusters produce considerably enhanced deviations from Matthiessen's rule, Hall and magnetoresistance coefficients. Values of the anisotropy in the mean free paths are calculated using a three-group model and a simple clustering model is invoked which demonstrates why the scattering becomes more 'interstitial' ill the clustered alloys. This leads to a scattering anisotropy for which the mean free path of electrons in the third zone becomes greater than that for holes in the corners of the second zone.

Continuous selections of set-valued mappings and approximation in asymmetric and semilinear spaces

“…There have been several attempts to obtain selection theorems for spaces more general than metrizable spaces and to find a satisfactory combination of metrical and convex type conditions. We note in this regard the studies of Michael [19], Beer [20], and the survey paper by Repovš and Semenov [17]. § 3.…”

“…Such sets, which are more general than convex sets, will be considered as values of set-valued functions. Among numerous extensions of Michael's theorem, we mention the survey [17] and the paper [18]. As a corollary, we will obtain a fixed-point theorem for lower semicontinuous mappings Ψ : K → 2 K with B-infinitely connected values defined on a B-infinitely connected compact set (Corollary 1).…”

Multivalued maps, selections and dynamical systems

“…For further references dealing with hyperspaces, multivalued maps and selections we recommend the books [12] and [17] and the papers [15,16,18]. A connection between multivalued maps, selections, coselections and shape theory has been studied in [20].…”

Continuous Selections of Lower semicontinuous Set-valued Mappings

“…Assume that Y is countable and regular, X is first countable, and ϕ : Y → (P(X) \ {∅}) is l.s.c.. Then ϕ has a continuous selection. Some applications and developments of Michael's theory of continuous selection are discussed in [6]. One of the problems in selection theory is what conditions should be imposed on X and Y such that any l.s.c.…”