Fuzzy relations and fuzzy functions

In this paper, we apply the standard element free mathematical definitions of equivalence relation and function as relations to the case of fuzzy relations. One of the surprising results of this approach is that fuzzy functions take their values from the complemented elements of a Brouwerian lattice. Thus, in the common case of the values being taken from the unit interval, the only fuzzy functions of this type are the ordinary functions. This might suggest that fuzzy set theory is more than just a generalization of ordinary set theory.     

Continuous Hermitian-indefinite functions

Some classes of Hermitian-indefinite functions and their integral representations are considered, as well as spaces with an indefinite metric that are formed with the aid of Hermitian-indefinite functions.Translated from Matematicheskie Zametki, Vol. 13, No. 2, pp. 303–310, February, 1973.The author expresses his sincere gratitude to I. S. Iokhvidov for posing the problem and for his interest.     

Continuous planar functions

This paper deals with continuous planar functions and their associated topological affine and projective planes. These associated (affine and projective) planes are the so-called shift planes and in addition to these, in the case of planar partition functions, the underlying (affine and projective) translation planes. We introduce a method that allows us to combine two continuous planar functions ? → ? into a continuous planar function ?² → ?². We prove various extension and embedding results for the associated affine and projective planes and their collineation groups. Furthermore, we construct topological ovals and various kinds of polarities in the associated topological projective planes.     

Linear relations betweenE-functions and Bessel functions

In this paper, some new linear relations for MacRobert'sE-Functions are established. They are formulae (1), (10), (15), (20), (22) and (28) below. For the definitions and properties of these functions, the reader is referred to MacRobert, “Functions of a Complex Variable” (3rd ed., London 1946), p. 348. This work will be denoted by the letters C.V. Also some expansions of Bessel Functions are deduced.     

On relations between usual continuous functions and generalized continuous functions

Summary I give here a theorem of realization of homomorphism in Cech homology theory, which improves some previous results of mine.     

Rational functions with linear relations

We find all polynomials over a field such that and are linear and . We also solve the same problem for rational functions , in case the field is algebraically closed.

     

Continuous itinerary functions and dendrite maps

We examine the “kneading sequence” theory of maps on dendrites, concentrating on those maps having one “turning point” and the “unique itinerary property” (i.e., distinct points have distinct itineraries). This theory has major overlaps with the theories of polynomial Julia Sets and Hubbard Trees, but also has significant differences from those two theories (for which the unique itinerary property does not always hold). We show that the unique itinerary property is a powerful property, and allows a simple classification of such dendrite maps with respect to their kneading sequences (up to conjugacy if there is no nontrivial invariant subdendrite).One of the major tools introduced here is the continuous itinerary function. If one takes the set of all sequences of the symbols used to define the itineraries with respect to a partition of a space, there is a natural topology which forces the itinerary function (from the original space into the space of sequences of symbols) to be continuous, although this often leads to a non-Hausdorff itinerary topology. Despite this apparent drawback, we show that this itinerary topology is a useful tool for analyzing the dynamics of continuous maps on metric spaces.Characterizations in terms of kneading sequences are given for topological properties of these maps, including various transitivity properties, (in)decomposability of inverse limits, and the existence of certain “piecewise linearizations” of such maps which are a natural generalization of “tent” maps on the interval. The itinerary topology provides a natural topology for the parameter space of all kneading sequences, a natural subspace of which will be shown to have a one-point compactification that is a dendrite, with an interesting connection to the Mandelbrot Set.     

Integral relations for heun-class special functions

New integral relations are obtained for eigenfunctions produced by Heun-class equations. These relations demonstrate the duality property of the eigenfunctions with different behaviors at singularities, the eigenfunctions being defined at different intervals. The obtained relations form two hierarchies, such that in each of them, the equations are produced one from another by a confluence of singular points.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 3, pp. 388–396, June, 1996.     

Recurrence relations for discrete hypergeometric functions

We present a general procedure for finding linear recurrence relations for the solutions of the second order difference equation of hypergeometric type. Applications to wave functions of certain discrete system are also given.