The aim of this paper is to introduce and study the notions of fuzzy upper $e$-limit set, fuzzy lower $e$-limit set and fuzzy $e$-continuously convergent functions. Properties and basic relationships among fuzzy upper $e$-limit set, fuzzy lower $e$- limit set and fuzzy $e$-continuity are investigated via fuzzy $e$-open sets.

In eigenvalue analysis, transformation from real systems to complex systems is very important. First, we clarify a necessary and sufficient condition that solutions of real nonlinear systems coincide with solutions of transformed complex nonlinear systems in the real subspace. Moreover, we propose a complex transformation such that a) real homogeneous systems of degree with respect to r are tra...

Let f : Gn,k−→Gm,l be any continuous map between two distinct complex (resp. quaternionic) Grassmann manifolds of the same dimension. We show that the degree of f is zero provided n,m are sufficiently large and l ≥ 2. If the degree of f is ±1, we show that (m, l) = (n, k) and f is a homotopy equivalence. Also, we prove that the image under f ∗ of elements of a set of algebra generators of H(Gm,...

0 1 classes are important to the logical analysis of many parts of mathematics. The 0 1 classes form a lattice. As with the lattice of computable enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality namely the notion of a thin class. We prove a number of results relating automorphisms, invariance and thin cla...

One of the recurrent themes in the area of the recursively enumerable (r.e.) degrees has been the study of the meet operator. While, trivially, the partial ordering of the r.e. degrees is an upper semi-lattice, i.e., the join ∗Lempp was partially supported by NSF grant DMS-0140120 and a Mercator Guest Professorship of the Deutsche Forschungsgemeinschaft. †Slaman was partially supported by the A...

It is shown that for any computably enumerable (c.e.) degree w, if w 6= 0, then there is a c.e. degree a such that (a ∨w)′ = a′′ = 0′′ (so a is low2 and a ∨w is high). It follows from this and previous work of P. Cholak, M. Groszek and T. Slaman that the low and low2 c.e. degrees are not elementarily equivalent as partial orderings.

Sacks [14] showed that every computably enumerable (c.e.) degree > 0 has a c.e. splitting. Hence, relativising, every c.e. degree has a Δ2 splitting above each proper predecessor (by ‘splitting’ we understand ‘nontrivial splitting’). Arslanov [1] showed that 0′ has a d.c.e. splitting above each c.e. a < 0′. On the other hand, Lachlan [9] proved the existence of a c.e. a > 0 which has no c.e. sp...

The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the Σk relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k ≥ 7, the Σk relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that Low1 is parameter definable, and we provide a new example of a ∅–...

We prove two antibasis theorems for Π1 classes. The first is a jump inversion theorem for Π1 classes with respect to the global structure of the Turing degrees. For any P ⊆ 2, define S(P ), the degree spectrum of P , to be the set of all Turing degrees a such that there exists A ∈ P of degree a. For any degree a ≥ 0, let Jump(a) = {b : b = a}. We prove that, for any a ≥ 0 and any Π1 class P , i...

We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ω-c.e. iff every set in it is weak truth-table reducible to a hypersimple, or ranked, set. We also show that ...