Abstract:The notions of the neutrosophic triplet and neutrosophic duplet were introduced by Florentin Smarandache. From the existing research results, the neutrosophic triplets and neutrosophic duplets are completely different from the classical algebra structures. In this paper, we further study neutrosophic duplet sets, neutrosophic duplet semi-groups, and cancellable neutrosophic triplet groups. First, some new properties of neutrosophic duplet semi-groups are funded, and the following important result is proven: there is no finite neutrosophic duplet semi-group. Second, the new concepts of weak neutrosophic duplet, weak neutrosophic duplet set, and weak neutrosophic duplet semi-group are introduced, some examples are given by using the mathematical software MATLAB (MathWorks, Inc., Natick, MA, USA), and the characterizations of cancellable weak neutrosophic duplet semi-groups are established. Third, the cancellable neutrosophic triplet groups are investigated, and the following important result is proven: the concept of cancellable neutrosophic triplet group and group coincide. Finally, the neutrosophic triplets and weak neutrosophic duplets in BCI-algebras are discussed.

Cage stability directly affects the dynamic performance of rolling bearing, which, in turn, affects the operating state of rotating equipment. The random collision between the rolling elements and the cage pocket is the main reason for cage instability. In this paper, from the perspective of the relative sliding velocity between the rolling elements and the bearing raceway, the interactions of the rolling elements and the cage pockets were analyzed, and the four zones with different collision features were defined. On this basis, and on the basis of the bearing dynamics model, the interaction of two adjacent rolling elements and the cage pockets in the a’–b’ area is discussed, and the peak impact force of the adjacent two balls and the cage pockets was investigated in terms of the rotation speed, radial load, acceleration/deceleration, and materials. When the ball runs close to the loaded zone, the probability of multiball random collision increases, which leads to an increase in the cage instability. At the entrance of the loaded zone, the peak impact force has the greatest impact on the cage stability during the acceleration process. Compared to the radial load applied to the bearing, the peak impact force is more sensitive to the bearing speed changes. The multiball collision analysis method provides a new idea for the research of cage stability.

Pooling layers appear widely in deep networks for its aggregating information in a local region and fast downsampling. Due to the reason that the closer to the output layer, the more the network learns is the high-level semantic information related to classification, the global average pooling would inhibit the contribution of local high magnitudes features in the global region. Besides, the gradient of the distinctive features is considerably attenuated due to the large region size of global average pooling. In this paper, we propose a global learnable pooling operation to enhance the distinctive high-level features in the global region, which is codenamed as GLPool. Because it is located before the classification layer, our GLPool is more sensitive to network performance. Besides, GLPool is not a hand-crafted pooling operation, which has the characteristic of adapting to any size of the input. With few parameters is added, GLPool is also a plug-and-play layer. The visualization via class activation map (CAM) on GoogLeNet and ShuffleNet-v2 also shows that GLPool can learn more concentrated and high-level distinctive features than global average pooling. The experiments on several classical deep models demonstrate the significant performance improvements on ImageNet32 and CIFAR100 datasets, which is exceeding obvious for lightweight networks.

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