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The integration of functional, integral and delay components of pantograph Volterra–Fredholm integro-differential equations provides a powerful framework for modeling complex systems with interdependent dynamics, particularly where nonlinearity influences and proportional feedback are essential. Numerical approximations to multi-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations present significant challenges due to the integration of nonlinearity, proportional delays and mixed integral terms, necessitating adaptive methods to achieve highly accurate approximations. In this work, we extend the Legendre spectral approximation to the one- and two-dimensional nonlinear pantograph Volterra–Fredholm integro-differential equations. In this method, the Legendre differentiation matrix and the pantograph operational matrix are used to manage the proportional delay terms inherent in these equations. To demonstrate the superiority of the proposed scheme, we present comparisons with other established spectral methods, highlighting the advantages of the Legendre spectral approach.

Stauffer⁷ has replaced the window automata rules — applied by de Boer, Segel and Perelson² for simulating the immunological shape space — by probabilistic rules, very similar to the ones of the ferromagnetic Ising Model. Based on this model, we extend the interaction range to more than nearest neighbors and find phase transitions not only in higher dimensions, but also in two dimensions, Biologically these phase transitions correspond to a transition from a reasonable to a non-reasonable behavior of the whole immune network.

At present, as a research hotspot for time series data (TSD), the deep clustering analysis of TSD has huge research value and practical significance. However, there still exist the following three problems: (1) For deep clustering based on joint optimization, the inevitably mutual interference existing between deep feature representation learning progress and clustering progress leads to difficult model training especially in the initial stage, the possible feature space distortion, inaccurate and weak feature representation; (2) Existing deep clustering methods are difficult to intuitively define the similarity of time series and rely heavily on complex feature extraction networks and clustering algorithms. (3) Multidimensional time series have the characteristics of high dimensions, complex relationships between dimensions, and variable data forms, thus generating a huge feature space. It is difficult for existing methods to select discriminative features, resulting in generally low accuracy of methods. Accordingly, to address the above three problems, we proposed a novel general two-stage multi-dimensional spatial features based multi-view deep clustering method 1DCAE-TSSAMC (One-dimensional deep convolutional auto-encoder based two-stage stepwise amplification multi-clustering). We conducted verification and analysis based on real-world important multi-scenario, and compared with many other benchmarks ranging from the most classic approaches such as K-means and Hierarchical to the state-of-the-art approaches based on deep learning such as Deep Temporal Clustering (DTC) and Temporal Clustering Network (TCN). Experimental results show that the new method outperforms the other benchmarks, and provides more accurate, richer, and more reliable analysis results, more importantly, with significant improvement in accuracy and spatial linear separability.

With the transformation of health concepts and medical models, traditional medicine is receiving more and more attention from people all over the world. Traditional medicine has become an important part of health promotion and global health management improvement. This paper briefly describes the connotation and development trend of traditional medicine in the new era from the perspective of philosophical theory. Through a thorough analysis of the rational thinking and realistic basis for the integration and development of traditional medicine in the new era from the perspectives of time, space, content and technology, we come to the conclusion that, under the modern medical model supported by Big Health and Big Data, the deepening and improvement of the consolidation of multi-dimensional integration and multi-collaborative system of traditional medicine will play a more fundamental role in mankind’s health and hygiene. This paper also presents the paths of integrative development for the traditional medicine under the new circumstances, aiming to provide theoretical ideas for further promoting the systematic development of traditional medicine and helping the revival of traditional Chinese medical civilization.

Integration of transcriptomic and proteomic data should reveal multi-layered regulatory processes governing cancer cell behaviors. Traditional correlation-based analyses have demonstrated limited ability to identify the post-transcriptional regulatory (PTR) processes that drive the non-linear relationship between transcript and protein abundances. In this work, we ideate an integrative approach to explore the variety of post-transcriptional mechanisms that dictate relationships between genes and corresponding proteins. The proposed workflow utilizes the intuitive technique of scatterplot diagnostics or scagnostics, to characterize and examine the diverse scatterplots built from transcript and protein abundances in a proteogenomic experiment. The workflow includes representing gene-protein relationships as scatterplots, clustering on geometric scagnostic features of these scatterplots, and finally identifying and grouping the potential gene-protein relationships according to their disposition to various PTR mechanisms. Our study verifies the efficacy of the implemented approach to excavate possible regulatory mechanisms by utilizing comprehensive tests on a synthetic dataset. We also propose a variety of 2D pattern-specific downstream analyses methodologies such as mixture modeling, and mapping miRNA post-transcriptional effects to explore each mechanism further. This work suggests that the proposed methodology has the potential for discovering and categorizing post-transcriptional regulatory mechanisms, manifesting in proteogenomic trends. These trends subsequently provide evidence for cancer specificity, miRNA targeting, and identification of regulation impacted by biological functionality and different types of degradation. (Supplementary Material - https://github.com/arunima2/PTRE_PSB_2020)