The simplest frequency formulation for conservative oscillators was proposed in 2019 (Results Phys 2019;15:102546). However, it becomes invalid for non-conservative oscillators. This work suggests the simplest amplitude-period formulation for non-conservative oscillators. The existence of a periodic solution of a second-order ordinary differential equation is given, and the periodic orbits are easily obtained. To the best of the authors’ knowledge, such a powerful result is not available in the literature, providing a tool to determining periodic orbits/limit cycles in the most general scenario.

In complex incomplete pattern classification, the classification results produced by a single classifier and used for decision-making may be quite unreliable and uncertain due to the random distribution of missing data. This paper proposes a new evidence integration credal classification algorithm (EICA) for multiple classifiers working on different attributes, aiming to reduce the negative impact on incomplete pattern classification by modeling the missing values locally. In EICA, the dataset is first grouped into several subsets, and missing values in each subset are estimated by similar subpatterns with different weights. The similarity is measured by discounting the overall similarity of subpatterns and the local similarity of attributes on the basis of fully exploiting the distribution characteristics of the attributes. The greater the variation in distribution across classes, the greater the weight.The classification results of the edited subpatterns with different discounting factors obtained by the optimization function can often provide (more or less) useful information for the classification of the query pattern. Thus, these discounted pieces of evidence (outputs) represented by basic belief assignments (BBAs) are globally fused to classify the query pattern on the basis of evidence theory. The validity has been demonstrated with various real datasets.

This paper considers a heat conduction problem of a common continuum-type stochastic mathematical model in an engineering field. The approximate solution is calculated with the Markov Chain Monte Carlo algorithm for the heat conduction problem. Three examples are given to illustrate the solution process of the method.

Transfer learning (TL) has grown popular in recent years. It is effective to improve the classification accuracy in the target domain by using the training knowledge in the related domain (called source domain). However, the classification of missing data (or incomplete data) is a challenging task for TL because different strategies of imputation may have strong impacts on learning models. To address this problem, we propose credal transfer learning (CTL) with multi-estimation for missing data based on belief function theory by introducing uncertainty and imprecision in data imputation procedure. CTL mainly consists of three steps: Firstly, the query patterns are reasonably mapped into multiple versions in source domain to characterize the uncertainty caused by missing values. Afterwards, the multiple mapping patterns are classified in the source domain to obtain the corresponding outputs with different discounting factors. Finally, the discounted outputs, represented by the basic belief assignments (BBAs), are submitted to a new belief-based fusion system to get the final classification result for the query patterns. Three comparative experiments are given to illustrate the interests and potentials of CTL method.