On rough convergence of complex uncertain sequences

Complex uncertain variables are measurable functions from an uncertainty space to the set of complex numbers and are used to model complex uncertain quantities. The main purpose of this paper is to introduce rough convergence of complex uncertain sequences and study some convergence concepts namely rough convergence in measure, rough convergence in mean, rough convergence in distribution of complex uncertain sequences. Lastly some relationship between them have been investigated.

A finer singular limit of a single-well Modica–Mortola functional and its applications to the Kobayashi–Warren–Carter energy

An explicit representation of the Gamma limit of a single-well Modica–Mortola functional is given for one-dimensional space under the graph convergence which is finer than conventional L 1 L^{1} -convergence or convergence in measure. As an application, an explicit representation of a singular limit of the Kobayashi–Warren–Carter energy, which is popular in materials science, is given. Some compactness under the graph convergence is also established. Such formulas as well as compactness are useful to characterize the limit of minimizers of the Kobayashi–Warren–Carter energy. To characterize the Gamma limit under the graph convergence, a new idea which is especially useful for one-dimensional problems is introduced. It is a change of parameter of the variable by arc-length parameter of its graph, which is called unfolding by the arc-length parameter in this paper.

Lacunary Statistical Convergence in Measure for Double Sequences of Fuzzy Valued Functions

Based on the concept of lacunary statistical convergence of sequences of fuzzy numbers, the lacunary statistical convergence, uniformly lacunary statistical convergence, and equi-lacunary statistical convergence of double sequences of fuzzy-valued functions are defined and investigated in this paper. The relationship among lacunary statistical convergence, uniformly lacunary statistical convergence, equi-lacunary statistical convergence of double sequences of fuzzy-valued functions, and their representations of sequences of α -level cuts are discussed. In addition, we obtain the lacunary statistical form of Egorov’s theorem for double sequences of fuzzy-valued measurable functions in a finite measurable space. Finally, the lacunary statistical convergence in measure for double sequences of fuzzy-valued measurable functions is examined, and it is proved that the inner and outer lacunary statistical convergence in measure are equivalent in a finite measure set for a double sequence of fuzzy-valued measurable functions.

Lacunary ideal convergence in measure for sequences of fuzzy valued functions

We investigate the concepts of pointwise and uniform I θ -convergence and type of convergence lying between mentioned convergence methods, that is, equi-ideally lacunary convergence of sequences of fuzzy valued functions and acquire several results. We give the lacunary ideal form of Egorov’s theorem for sequences of fuzzy valued measurable functions defined on a finite measure space ( X , M , μ ) . We also introduce the concept of I θ -convergence in measure for sequences of fuzzy valued functions and proved some significant results.

Convergence of Complex Uncertain Double Sequences

Complex uncertain variables are measurable functions from an uncertainty space to the set of complex numbers and are used to model complex uncertain quantities. This paper introduces the convergence concepts of convergence almost surely (a.s.), convergence in measure, convergence in mean, convergence in distribution and convergence uniformly almost surely complex uncertain double sequences. In addition, relationships among the introduced classes of sequences have been introduced.

Convergence in Measure and in Category

W. Orlicz in 1951 has observed that if {fn(·, y)}n∈N converges in measure to f(·, y) for each y ∈ [0, 1], then {fn}n∈N converges in measure to f on [0, 1] × [0, 1]. The situation is different for the convergence in category even if we assume the convergence in category of sequences {fn(·, y)}n∈N for each y ∈ [0, 1] and {fn(x, ·)}n∈N for each x ∈ [0, 1].