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Hjorth and Nies proposed notions of randomness corresponding to the higher recursion setting. In particular, they defined the notion of Π 1 1 -Martin-Löf randomness. In this article we present examples of Π 1 1 -Martin-Löf random reals which are obtained as measures of Π 1 1 open sets. [ABSTRACT FROM AUTHOR]

Jockusch and Lewis (J Symb Logic 78:977-988, ) proved that every DNC function computes a bi-immune set. They asked whether every DNC function computes an effectively bi-immune set. We construct a DNC function that computes no effectively bi-immune set, thereby answering their question in the negative. [ABSTRACT FROM AUTHOR]

Abstract: Franek et al. recently described four constructions for large sets of -intersecting Steiner triple systems of order (STS ) [F. Franek, M.J. Grannell, T.S. Griggs, A. Rosa, On the large sets of -intersecting Steiner triple systems of order , Des. Codes Cryptogr. 26 (2002) 243–256]. In this study we focus on large sets of -intersecting STS . Some recursive constructions are presented that involve three-wise balanced designs. By applying these constructions we obtain some new infinite classes of such large sets, and the large sets of -intersecting KTS are used to produce some new large sets of Kirkman triple systems. [Copyright &y& Elsevier]

Abstract: This article introduces the notions of prolongations and prolongational limit sets of control systems. It is shown how the prolongational limit sets play in studying recursive and dispersive concepts of control systems. [Copyright &y& Elsevier]

Abstract: Let denote a graph formed from unlabelled graphs and a labelled graph replacing every vertex of by the graph and joining the vertices of with all the vertices of those of whenever . For unlabelled graphs , let stand for the class of all graphs taken over all possible orderings of . A prime inductive class of graphs, , is said to be a set of all graphs, which can be produced by recursive applying of where is a graph from a fixed set of prime graphs and are either graphs from the set of prime graphs or graphs obtained in the previous steps. Similar inductive definitions for cographs, -trees, series–parallel graphs, Halin graphs, bipartite cubic graphs or forbidden structures of some graph classes were considered in the literature (Batagelj (1994) Drgas-Burchardt et al. (2010) and Hajós (1961) ). This paper initiates a study of prime inductive classes of graphs giving a result, which characterizes, in their language, the substitution closed induced hereditary graph classes. Moreover, for an arbitrary induced hereditary graph class it presents a method for the construction of maximal induced hereditary graph classes contained in and substitution closed. The main contribution of this paper is to give a minimal forbidden graph characterization of induced hereditary prime inductive classes of graphs. As a consequence, the minimal forbidden graph characterization for some special induced hereditary prime inductive graph classes is given There is also offered an algebraic view on the class of all prime inductive classes of graphs of the type . [Copyright &y& Elsevier]

Abstract: Given a set of vertices of a connected graph , the metric representation of a vertex of with respect to is the vector , where , denotes the distance between and . is a resolving set for if for every pair of distinct vertices of , . The metric dimension of , , is the minimum cardinality of any resolving set for . Let and be two graphs of order and , respectively. The corona product is defined as the graph obtained from and by taking one copy of and copies of and joining by an edge each vertex from the th-copy of with the th-vertex of . For any integer , we define the graph recursively from as . We give several results on the metric dimension of . For instance, we show that given two connected graphs and of order and , respectively, if the diameter of is at most two, then . Moreover, if and the diameter of is greater than five or is a cycle graph, then . [Copyright &y& Elsevier]

Abstract: It has been previously observed that for many -learnable computable families of computably enumerable (c.e. for short) sets all their computable numberings are evidently -equivalent, i.e. are equivalent with respect to reductions computable in the halting problem. We show that this holds for all -learnable computable families of c.e. sets, and prove that, in general, the converse is not true. In fact there is a computable family of c.e. sets such that all computable numberings of are computably equivalent and is not -learnable. Moreover, we construct a computable family of c.e. sets which is not -learnable though all of its computable numberings are -equivalent. We also give a natural example of a computable -learnable family of c.e. sets which possesses non--equivalent computable numberings. So, for the computable families of c.e. sets, the properties of -learnability and -equivalence of all computable numberings are independent. [Copyright &y& Elsevier]

Abstract: This work initiates a systematic investigation into the matrix forms of the Pascal triangle as mathematical objects in their own right. The present paper is especially devoted to the so-called G-matrices, i.e. the set of the twelve triangular matrix forms that can be derived from the Pascal triangle expanded to the level . For , the G-matrix set reduces to a set of four distinct matrices. The twelve G-matrices are defined and the classic Pascal recursion is reformulated for each of the twelve G-matrices. Three sets of matrix transformations are then introduced to highlight different relations between the twelve G-matrices and for generating them from appropriately chosen subsets. [Copyright &y& Elsevier]

The article presents a research study which defines the notion called k-immunity which can be found between immunity and hyperimmunity in strength. It examines the findings that ϕ' does not k-tt reduce to a k-immune set which justified the concepts of k-immunity. It also applies the result to MIN, the set of minimal programs, to show that ϕ' cannot btt-reduce to MIN.

We prove that there are two incomplete d.r.e.\ degrees (the Turing degrees of differences of two recursively enumerable sets) such that every non-zero recursively enumerable degree cups at least one of them to ${\bf 0}2$, the greatest recursively enumerable (Turing) degree. [ABSTRACT FROM AUTHOR]

The article describes a theorem for certain classes of recursive infinitary formulas. It considers the following generalization of the notion of a structure recursive relative to a set X. It demonstrates that a certain, fairly obvious, description of classes C', of recursive infinitary formulas has a property.

The article discusses the basic recursive notions for fuzzy sets. It tackles on the general theory of recursive equivalence types of fuzzy sets and fuzzily recursively infinite sets. It defines the union, intersection and complement using S and T as fuzzy sets with characteristic functions. It tackles on the stronger version of reducibility and equivalence as well as fuzzy isolation.