Applications of deferred Cesàro statistical convergence of sequences of fuzzy numbers of order (ξ, ω)

The purpose of this article is to study deferred Cesrào statistical convergence of order (ξ, ω) associated with a modulus function involving the concept of difference sequences of fuzzy numbers. The study reveals that the statistical convergence of these newly formed sequence spaces behave well for ξ ≤ ω and convergence is not possible for ξ > ω. We also define p-deferred Cesàro summability and establish several interesting results. In addition, we provide some examples which explain the validity of the theoretical results and the effectiveness of constructed sequence spaces. Finally, with the help of MATLAB software, we examine that if the sequence of fuzzy numbers is bounded and deferred Cesàro statistical convergent of order (ξ, ω) in (Δ, F, f), then it need not be strongly p-deferred Cesàro summable of order (ξ, ω) in general for 0 < ξ ≤ ω ≤ 1.

A Novel Analog Circuit Soft Fault Diagnosis Method Based on Convolutional Neural Network and Backward Difference

This paper develops a novel soft fault diagnosis approach for analog circuits. The proposed method employs the backward difference strategy to process the data, and a novel variant of convolutional neural network, i.e., convolutional neural network with global average pooling (CNN-GAP) is taken for feature extraction and fault classification. Specifically, the measured raw domain response signals are firstly processed by the backward difference strategy and the first-order and the second-order backward difference sequences are generated, which contain the signal variation and the rate of variation characteristics. Then, based on the one-dimensional convolutional neural network, the CNN-GAP is developed by introducing the global average pooling technical. Since global average pooling calculates each input vector’s mean value, the designed CNN-GAP could deal with different lengths of input signals and be applied to diagnose different circuits. Additionally, the first-order and the second-order backward difference sequences along with the raw domain response signals are directly fed into the CNN-GAP, in which the convolutional layers automatically extract and fuse multi-scale features. Finally, fault classification is performed by the fully connected layer of the CNN-GAP. The effectiveness of our proposal is verified by two benchmark circuits under symmetric and asymmetric fault conditions. Experimental results prove that the proposed method outperforms the existing methods in terms of diagnosis accuracy and reliability.

I−LACUNARY STATISTICAL CONVERGENCE OF ORDER β OF DIFFERENCE SEQUENCES OF FRACTIONAL ORDER

In this paper, we introduce the concepts of ideal ∆α−lacunary statis- tical convergence of order β with the fractional order of α and ideal ∆α−lacunary strongly convergence of order β with the fractional order of α ( where 0 < β ≤ 1and α be a fractional order) and give some relations about these concepts.

On the convergence difference sequences and the related operator norms

In this note, we discuss the definitions of the difference sequences defined earlier by Kızmaz (1981), Et and Çolak (1995), Malkowsky et al. (2007), Başar (2012), Baliarsingh (2013, 2015) and many others. Several authors have defined the difference sequence spaces and studied their various properties. It is quite natural to analyze the convergence of the corresponding sequences. As a part of this work, a convergence analysis of difference sequence of fractional order defined earlier is presented. It is demonstrated that the convergence of the fractional difference sequence is dynamic in nature and some of the results involved are also inconsistent. We provide certain stronger conditions on the primary sequence and the results due to earlier authors are substantially modified. Some illustrative examples are provided for each point of the modifications. Results on certain operator norms related to the difference operator of fractional order are also determined.

Rough variation on lacunary quasi Cauchy triple difference sequences

In the present paper, we extend the notion of rough ideal lacunary statistical quasi Cauchy triple difference sequence of order α using the concept of ideals, which automatically extends the earlier notions of rough convergence and rough statistical convergence. We prove several results associated with this notion.