One-point Set Research Articles

A class of ergodic transformations having simple spectrum

A class of ergodic, measure-preserving, invertible point transformations is defined, called class S. Any measurepreserving point transformation induces a unitary operator on the Hilbert space of 22-functions. A theorem is proved here which implies that the operator induced by any transformation in class S has simple spectrum. [It is then a known result that the transformations in class S have zero entropy.] Let (X, 9, ,u) be a measure space, isomorphic to the unit interval with Lebesgue measure. A measurable, measure-preserving, invertible point transformation of X is called an automorphism of (X, i, u). A class of automorphisms, called class S for brevity, is defined below (Definition (4)). The purpose of this paper is to prove the following theorem: (1) THEOREM. Let r be an automorphism in class S. Then there exist arbitrarily small sets whose characteristic functions each generate ?2(dM) under the action of the unitary operator UT, where Ur is defined by UfQ(rx) =f (x). In particular U7 has simple spectrum. (2) DEFINITION. Let H be a Hilbert space, a bounded normal operator on H. Let vE H. Let H(v) consist of the closure of the set of all elements of the form P(T, T*)v, where P(T, T*) denotes a polynomial in and T*. To say that a vector vEH generates H under the action of T means that H= H(v). (3) DEFINITION. Let t = {IAt 1 < i ? m } be a finite, ordered collection of mutually disjoint measurable sets. Then t is called a partition. The union of the members of t need not be the whole space. Let ik be a sequence of partitions with the property that for every measurable set E, there exists a sequence of sets Ek such that each Ek is a union of members of (k, and pA(E A Ek)-*O as k-oo. Then it will be said that ik--e. Here e denotes the partition of the whole space into one-point sets. (4) DEFINITION. Let r be an automorphism of (X, iY, IA), {={A i1Ii<m} a partition. If rAi=Ai+1 for 1<i_m-1, t will Received by the editors March 30, 1970. AMS 1969 subject classifications. Primary 2870; Secondary 4730.

Equilibrium systems of recurrent Markov processes

Let &, , n 3 0, be a Markov process with states in a measurable space (x, F), where F is a u-algebra containing all one-point sets, and let P”(x, B) be the nth step transition function of the process. In this paper we will always assume that P has at least one u-finite invariant measure h (henceforth taken as fixed), and that the process is recurrent in the sense that for any B of positive h measure,

Some results on crumpled cubes

Introduction. This paper deals with some problems on crumpled cubes. In [6] R. H. Bing raised the following questions: (1) Is the union of a crumpled cube and a 3-cell topologically S3? (2) Under what conditions will the union of two crumpled cubes sewn together along their boundaries be S3? (3) Under what conditions will the union of two copies of the same crumpled cube sewn together along their boundaries by the identity homeomorphism be S3? In [2] Bing proves that the union of two copies of the Solid Alexander Horned Sphere sewn together along their boundaries by the identity homeomorphism is S3. In [1] B. J. Ball proves that there exists a crumpled cube C such that two copies of C sewn together along their boundaries (not by the identity homeomorphism) is not topologically S3. In [7] Casler proves that the union of two copies of the Solid Alexander Horned Sphere sewn together along their boundaries is S3. In this paper it is proved in ?2 that question (1) has an affirmative answer. This result has recently been obtained independently by Norman Hosay [9]. Using this result, it is shown in ?3 that if C is a crumpled cube and B is a unit ball in E3, then there exists an upper semi-continuous decomposition G of B such that (1) B/G is homeomorphic to C, and (2) if g is a nondegenerate element of G, then g is point-like and the intersection of g with the boundary of B is a onepoint set. Corollary 2 of ?3 shows that if C and D are crumpled cubes, S is a tame 2-sphere in S3, and C and D are sewn together along their boundaries, then there exists an upper semi-continuous decomposition G of S3 such that the intersection of each nondegenerate element of G with S is a one-point set and S3/ G is homeomorphic to the union of C and D. In ?4 it is shown that if two copies of a particular bad crumpled cube are sewn together by the identity homeomorphism along their boundaries the result iS S3, and it is shown that if two copies of a particular nice crumpled cube are sewn together by the identity homeomorphism along their boundaries the result is not S3.

Some Properties of Redundant Constraints and Extraneous Variables in Direct and Dual Linear Programming Problems

Model equivalences may sometimes be used to replace “realistic” but unwieldy initial formulations with simpler counterparts. This can involve sophisticated uses of prototypes, quasi models, etc., or it may involve only simpler ideas of redundancy elimination, removal of extraneous variables, etc. In either case questions can arise concerning the properties of these models when further analyses are to be conducted via parameterizations, duality, etc. These topics are examined in the general context of direct and dual linear programming problems with special reference to boundedness properties of the associated solution sets. It is shown that a bounded solution set in one problem implies an unbounded solution set in the dual problem, unless both are one-point sets. The ideas of projection equivalence are then developed to suggest a possible route for utilizing these one-point solution properties for analyzing or solving linear programming problems. These possibilities might prove useful when, for example, it is desired to simplify an initial formulation while achieving a solution that has additional properties—e.g., boundedness—that are also considered desirable.