Index set (recursion theory)

Abstract: A new class of distributional transformations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transformed distribution on the test function’s higher order derivatives. The class includes the size and zero bias transformations, and when specializing to weighting by polynomial functions, relates distributional families closed under independent addition, and in particular the infinitely divisible distributions, to the family of transformations induced by their associated orthogonal polynomial systems. For these families, generalizing a well known property of size biasing, sums of independent variables are transformed by replacing summands chosen according to a multivariate distribution on its index set by independent variables whose distributions are transformed by members of that same family. A variety of the transformations associated with the classical orthogonal polynomial systems have as fixed points the original distribution, or a member of the same family with different parameter.

Abstract: A new class of distributional transformations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transformed distribution on the test function's higher order derivatives. The class includes the size and zero bias transformations, and when specializing to weighting by polynomial functions, relates distributional families closed under independent addition, and in particular the infinitely divisible distributions, to the family of transformations induced by their associated orthogonal polynomial systems. For these families, generalizing a well known property of size biasing, sums of independent variables are transformed by replacing summands chosen according to a multivariate distribution on its index set by independent variables whose distributions are transformed by members of that same family. A variety of the transformations associated with the classical orthogonal polynomial systems have as fixed points the original distribution, or a member of the same family with different parameter.

Abstract: In a digital information processing system wherein a model of a finite state machine (FSM) receiving a plurality of FSM inputs and producing a plurality of FSM outputs is represented by a reduced-state trellis and wherein the FSM inputs are defined on a base closed set of symbols, a novel method is presented for updating soft decision information on the FSM inputs into higher confidence information whereby (1) the soft decision information is inputted in a first index set, (2) a forward recursion is processed on the input soft decision information based on the reduced-state trellis representation to produce forward state metrics, (3) a backward recursion is processed on the input soft decision information based on the reduced-state trellis representation to produce backward state metrics, wherein the backward recursion is independent of the forward recursion and (4) the forward state metrics and the backward state metrics are operated on to produce the higher confidence information.

Abstract: Publisher Summary This chapter discusses the method of alternating chains. A common construction occurring in contributions to the theory of models is the development of an alternating chain of structures, i.e. of a sequence of structures lying alternately in each of two given classes of structures. Although not always originally formulated in these terms, it has turned out that a substantial number of the applications of these chains can be formulated in terms of separability (or interpolability) tests. The word “hierarchy” is usually used informally in mathematical discussions, so it no doubt suggests more or less structure to different people. By a family of sets we mean a function to a class of sets. “Hierarchy” is most often used in the literature in reference to a special kind of family of sets —namely a family of sets (or classes) of sets. The inclusion relation on the classes of sets induces a partial order on the domain (or index set) of the hierarchy, and in most cases this is a partial well order if not indeed a well order.