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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Acta applicandae mathematicae 30 (1993), S. 67-93 
    ISSN: 1572-9036
    Keywords: 51N10 ; 51FXX ; 51G05 ; 46B40 ; 46G05 ; 47N30 ; Direction distance ; mixing distance ; measure cone ; mc-space ; organization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In the study of chemical structural phenomena, the idea of mixedness appears to provide most valuable information if this notion is understood as a quantity that counts for a natural distinction between more or less mixed situations. The search for such a concept was initiated by the need of a corresponding valuation of chemical molecules that differ in the type-composition of a system of varying molecular parts at given molecular skeleton sites. In other words, an order relation for the partitions of a finite set was sought that explains the extent of mixing in a canonical way. This and related questions led to the concepts of themixing character andmixing distance. Success in applying these concepts to further chemical and physical problems, to graph theory, to representation theory of the symmetric group, and to probability theory confirmed the hope that there is a common background in some basic mathematics that allows a systematic treatment. The expected concept summarizing the above-mentioned experience is called thedirection distance and the mathematics concerned is linear geometry with a normspecific metric or structural analysis of normed vector spaces, respectively. Direction distance is defined as a map that represents the total metric information on any pair of directions (= pair of half-lines with a common vertex or a corresponding figure in normed vector spaces). Generally, that metrical figure changes when the half-lines are interchanged. As a consequence thereof, Hilbert's congruence axioms do not permit a metric criterion for the congruence of angles except in particular cases. The metric figures of direction pairs, however, may be classified according to metric congruence, and the normspecific metric induces an order in the set of congruence classes. This order, as a rule, is partial; it proves to be total if and only if the vector spaces are (pre-) Hilbert spaces (Lemma 8). A thorough comparison of the direction distance with the conventional distance deepens the understanding of the novel concept and justifies the terminology. The results are summarized in a number of lemmata. Furthermore, so-calledd-complete systems of order-homomorphic functional (so-calledd-functionals) establish an alternative formulation of the direction distance order. If and only if the order is total,d-complete systems can be represented by singled-functionals. Consequently, the case that normed vector spaces are (pre-) Hilbert spaces is pinpointed by the fact that the negative scalar product is already ad-complete system. These particular circumstances allow a metric congruence relation for angles. Another family of normed vector spaces is traced out by the conditions under which the direction distance takes the part of the mixing distance. Roughly speaking, a subset of vectors may be viewed as representing mixtures if it has two properties. First, with any two vectors of this subset all positive linear combinations are vectors of it as well. Second, the length of these vectors is an additive property. Correspondingly, the definition of the mentioned family, the family of so-calledmc-spaces, is based on the concepts of ameasure cone (Def. 5 and Def. 5′) and an associated class ofmc- (= measure cone)norms being responsible for length additivity ofpositive vectors (= vectors of the measure cone) (Def. 6). Such norms provide congruence classes for positive vectors and positive direction pairs marked by the propertieslength andmixing distance, respectively. These congruence classes do not depend on the choice of the particularmc-norm within the class associated with a given measure cone, however, the mixing distance does. The consistency of the stipulated mathematical instrumentarium becomes apparent with Theorem 1 stating: The mixing distanceorder doesnot depend on the choice of a particular norm within the measure cone specific class; this order, together with the stipulated length of positive vectors, are properties necessary and sufficient for fixing the measure cone specific class ofmc-norms. Decreasing (or constant) mixing distance was found to describe a characteristic change in the relation between two probability distributions on a given set of classical events, a change in fact necessary and sufficient for the existence of alinear stochastic operator that maps a given pair of distributions into another given pair. This physically notable statement was originally proved for the space ofL 1-functions on a compact ℝ-interval, it was expected to keep its validity for probability distributions in the range of classical physics and, as a consequence of that, for measures of any type. Theorem 2 presents the said statement in terms ofmc-endomorphisms ofmc-spaces; after an extension of the original proof to a more general family ofL 1-spaces another method presented in a separate paper confirms Theorem 2 for bounded additive set functions and, accordingly, secures the expected range of validity. The discussion below is without reference to the validity range and primarily devoted to geometrical consequences without detailed speculations about physical applications. A few remarks on applications, however, illustrate the physical relevance of the mixing distance and its specialization, theq-character, in the particular context of Theorem 2. With reference to measure cones with such physical interpretations as statistical systems,mc-endomorphisms effect changes that can be described by linear stochastic operators and result physically either from an approach to some equilibrium state or from an adoption to a time-dependent influence on the system from outside. Theorem 2 provides a necessary and sufficient criterion for such changes. The discussion may concern phenomena of irreversible thermodynamics as well as evolving systems under the influence of a surrounding world summarized asorganization phenomena. Entropies and relative entropies of the Renyi-type ared-functionais which do not establishd-complete systems. The validity of Theorem 2 does not encompass the nonclassical case; the reason for it is of high physical interest. The full range of validity and its connection with symmetry arguments seems a promising mathematical problem in the sense of Klein'sErlanger Programm. From the point of mathematical history, the Hardy-Littlewood-Polya theorem should be quoted as a very special case of Theorem 2.
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