On a bosonic-parafermionic realization ofU_q (\widehat{sl(2)})

We realize the current algebra at an arbitrary level in terms of one deformed free bosonic field and a pair of deformed parafermionic fields. It is shown that the operator product expansions of these parafermionic fields involve an infinite number of simple poles and simple zeros, which then condensate to form a branch cut in the classical limitq1. Our realization coincides with those of Frenkel-Jing and Bernard when the levelk takes the values 1 and 2, respectively.     

A noncommutative geometric approach to left-right symmetric weak interactions

By combining the generalized exterior algebra of forms over a noncommutative algebra with the gauging of discrete directions and the associated Higgs fields, we consider the construction of the bosonic sector of left-right symmetric models of the form SU(2) _L SU(2) _R U(1). We see that within this formalism maximal use can be made of the gauge connection associated with the noncommutative graded algebra.     

Der Richtungsabstand

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 -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.

     

Perturbed unstable variational inequalities with unbounded operators on approximately given sets

The stability and convergence of the solutions of perturbed and regularized variational inequality to the solutions of the primary (unstable a priori) variational inequality with proper monotone operator are investigated. All the objects of inequality: the operatorA, the right-hand partf and the set of constrains are to be perturbed. At the same time no assumptions of boundedness and smoothness of the operatorA are used. The connection between the parameters of perturbations, which guarantees strong convergence of approximate solutions, is established. It is proved that the existence of the solution to the unperturbed variational inequality is necessary and sufficient condition for convergence of the regularized perturbed inequality solutions.This research was supported in part by the Ministry of Science Grant 3481-1-91 and by the Ministry of Absorption Center for Absorption in Science.     

A spectral Galerkin approximation of the Orr-Sommerfeld eigenvalue problem in a semi-infinite domain

Summary The convergence of a Galerkin approximation of the Orr-Sommerfeld eigenvalue problem, which is defined in a semi-infinite domain, is studied theoretically. In case the system of trial functions is based on a composite of Jacobi polynomials and an exponential transform of the semi-infinite domain, the error of the Galerkin approximation is estimated in terms of the transformation parametera and the numberN of trial functions. Finite or infinite-order convergence of the spectral Galerkin method is obtained depending on how the transformation parameter is chosen. If the transformation parameter is fixed, then convergence is of finite order only. However, ifa is varied proportional to 1/N with an exponent 0<<1, then the approximate eigenvalue converges faster than any finite power of 1/N asN. Some numerical examles are given.     

Alternating iteration and elliptic variational inequalities

Summary Free boundary value problems, too complicated for formulation as a variational inequality, are broken up into two problems on overlapping regions. On one region the problem is treated as an ordinary boundary value problem; on the second region, the free boundary part of the problem is reduced to a variational inequality. By solving the two problems successively it is shown that under certain conditions the successive solutions converge to a single function that gives a solution of the original problem. Application to a filtration problem is given.     

A vector-sum theorem in two-dimensional space

Given a finite setX of vectors from the unit ball of the max norm in the twodimensional space whose sum is zero, it is always possible to writeX = {x₁, , x_n} in such a way that the first coordinates of each partial sum lie in [–1, 1] and the second coordinates lie in [–C, C] whereC is a universal constant.     

γ-sets and other singular sets of real numbers

A family of $\text{J}$ of open subsets of the real line is called an ω-cover of a set X iff every finite subset of X is contained in an element of $\text{J}$. A set of reals X is a γ-set iff for every ω-cover $\text{J}$ of X there exists $\text{〈D}{}_{\mathrm{n}}\text{: n < ω〉?}\text{J}{}^{\mathrm{\omega }}$ such that

In this paper we show that assuming Martin's axiom there is a γ-set X of cardinality the continuum.     

On the support of midconvex operators

Summary LetX be a real vector space,D a convex subset ofX and (Y, K) an order complete ordered vector space. The following sandwich theorem holds: Iff: D Y is midconvex,g: D Y {– } is midconcave andg f onD, then there exists a Jensen mappingh: D Y {– } such thatg h f onD. Using this theorem we show that a mappingf: D Y is midconvex if and only if it has Jensen support at every point ofD. Moreover, ifX is a Baire topological vector space and (Y, K) is an ordered topological vector space satisfying some additional conditions, then a mappingf: D Y is continuous whenever it has continuous Jensen support at every point ofD. As an application of these results we obtain the equality of some set-classes connected with additive and midconvex operators.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth