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 the mixing character and mixing 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 the direction 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-called d-complete systems of order-homomorphic functionals (so-called d-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 single d-functionals. Consequently, the case that normed vector spaces are (pre-) Hilbert spaces is pinpointed by the fact that the negative scalar product is already a d-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-called mc-spaces, is based on the concepts of a measure cone (Def. 5 and Def. 5') and an associated class of mc- (= measure cone) norms being responsible for length additivity of positive vectors (= vectors of the measure cone) (Def. 6). Such norms provide congruence classes for positive vectors and positive direction pairs marked by the properties length and mixing distance, respectively. These congruence classes do not depend on the choice of the particular mc-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 distance order does not 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 of mc-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 a linear stochastic operator that maps a given pair of distributions into another given pair. This physically notable statement was originally proved for the space of L1-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 of mc-endomorphisms of mc-spaces; after an extension of the original proof to a more general family of L1-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, the q-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 as organization phenomena. Entropies and relative entropies of the Renyi-type are d-functionals which do not establish d-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's Erlanger Programm. From the point of mathematical history, the Hardy-Littlewood-Polya theorem should be quoted as a very special case of Theorem 2.

Let |·| be a norm on Rn, and s a compact convex semigroup of linear |·|-contractions. Given two k-tuples of n-vectors, (x(1),…,x(k)) and (y(1),…,y(k)), we seek conditions for the existence of a contraction S ϵ s that simultaneously takes x(i) to y(i), that is, Sx(i) = y(i), for all i = 1,…,k!. Straightforward application of the separation theorem for convex sets provides a general but abstract result, in the form of a system of inequalities. Specializing |·| to the 1-norm and the ∞-norm, respectively, and s to comprise either all contractions or those contractions that preserve a particular linear form, it is possible to evaluate the characteristic functionals arising from the separation theoem. Therby the abstract results can be rduced to a tractable form, which turns out to be of the Hahn-Banach type.

Given real m × n-matrixces X and Y, when does there exist a stochastic matrix S such that SX = Y? The existence of S is characterized by the validity of an inequality which must hold for all positively homogeneous, convex, piecewise linear functions (so-called ph cpl functions) or, equivalently, for all extremals in the cone of all such functions. Every ph cpl function induces canonically a fan, that is a stratification of Rn by closed convex cones. We develop a cohomology theory for fans which characterizes these stratifications and also those which belong to extremals in the cone of all ph cpl functions. A centersymmetric version of the cohomology theory answers the same questions for the cone of piecewise linear seminorms. For n = 2, our theory implies a geometric interpretation of earlier work by Hardy, Littlewood, and Pólya, as well as by Ruch, Schranner, and Seligman, based on the fact that in case n = 2 a ph cpl function is extremal only if it is the supremum of at most three linear functions. In case n ≥ 3, the situation changes drastically: For every k, the supremum of k linear functions is almost always an extremal ph cpl function. As another application of our theory we discuss a counterexample to a conjecture by Kakutani, constructed by A. Horn, and show how our theory could have been used to build large classes of such counterexamples.

We introduce ’’bilateral classes’’, a new concept for classifying the elements of groups. Bilateral classes are orbits in a group G under the action of any given subgroup of the direct product G×G. The classification concept presented here encompasses conjugacy classes, cosets, double cosets, and Ree’s σ-classes as particular cases. It has an interpretation as a classification scheme for bijections between G-sets under the aspect of symmetry equivalence due to symmetries in both the domain and the range. While double cosets and conjugacy classes correspond to the case of no or complete correlation between operations of the two symmetry groups, our concept also covers the general case of partial correlation. The scope of generalization corresponds to applications in physics. Expressions for the number of bilateral classes are given.

Information extent and information distance are introduced to describe the quantitative aspect of statistical information and gain of information without using prejudices that apply only under special conditions. It is shown that these concepts are closely related to mixing character and mixing distance as defined and discussed in previous papers (Refs. 1, 2, and 5). Several examples are treated to show the use of these concepts and their superiority in some respects over statistical entropy concepts of the conventional type. In a second part graphical illustrations are given to develop an intuitive understanding of partially ordered properties.

Mixing character denotes a property concerning the composition of sets. It was first used in order to present criteria for ligand systems in molecules that assure the existence of chiral isomers. Since then the concept of mixing character has been found to have further fields of applications in physics and pure mathematics. Various theorems have been given in the meantime that are reviewed here mostly in the form of equivalent definitions. The mixing distance is another, still more general concept suggested by a study of the time dependent behaviour of nonisolated thermodynamical systems. It describes a dissimilarity of probability distributions, the decrease of which characterizes irreversible phenomena. In the present paper a variety of equivalent definitions and theorems concerning the mixing distance are presented that are of use for further applications. Moreover, a theorem is deduced which is a generalization of a fundamental theorem by Hardy, Littlewood, and Polya on inequalities. Applications of the results given in the present paper will be explored in a subsequent publication that is mostly concerned with the concept of statistical information.

AbstractProblems in permutational isomerism are discussed in terms of double cosets of permutation groups. A short account of chirality vs. achirality of permutational isomers with pairwise mirror image chiral ligands is presented without mathematical detail.

Zeitschrift für Elektrochemie, Berichte der Bunsengesellschaft für physikalische ChemieVolume 66, Issue 2 p. 192-193 Buchbesprechung Handbuch der Physik. Band III/1: Prinzipien der klassischen Mechanik und Feldtheorie. Herausgegeben von S. Flügge. Springer-Verlag, Berlin-Göttingen-Heidelberg 1960. 902 Seiten, 106 Figuren. Preis: DM 158,– E. Ruch, E. RuchSearch for more papers by this author E. Ruch, E. RuchSearch for more papers by this author First published: März 1962 https://doi.org/10.1002/bbpc.19620660222AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinkedInRedditWechat No abstract is available for this article. Volume66, Issue2März 1962Pages 192-193 RelatedInformation