BackgroundType 2 diabetes mellitus (T2DM) is a major public health burden. Self-management of diabetes including maintaining a healthy lifestyle is essential for glycemic control and to prevent diabetes complications. Mobile-based health data can play an important role in the forecasting of blood glucose levels for lifestyle management and control of T2DM.ObjectiveThe objective of this work was to dynamically forecast daily glucose levels in patients with T2DM based on their daily mobile health lifestyle data including diet, physical activity, weight, and glucose level from the day before.MethodsWe used data from 10 T2DM patients who were overweight or obese in a behavioral lifestyle intervention using mobile tools for daily monitoring of diet, physical activity, weight, and blood glucose over 6 months. We developed a deep learning model based on long short-term memory–based recurrent neural networks to forecast the next-day glucose levels in individual patients. The neural network used several layers of computational nodes to model how mobile health data (food intake including consumed calories, fat, and carbohydrates; exercise; and weight) were progressing from one day to another from noisy data.ResultsThe model was validated based on a data set of 10 patients who had been monitored daily for over 6 months. The proposed deep learning model demonstrated considerable accuracy in predicting the next day glucose level based on Clark Error Grid and ±10% range of the actual values.ConclusionsUsing machine learning methodologies may leverage mobile health lifestyle data to develop effective individualized prediction plans for T2DM management. However, predicting future glucose levels is challenging as glucose level is determined by multiple factors. Future study with more rigorous study design is warranted to better predict future glucose levels for T2DM management.

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For a commutative ring R with identity, the ideal-based zero-divisor graph, denoted by ?I (R), is the graph whose vertices are {x ? R\I|xy ? I for some y ? R\I}, and two distinct vertices x and y are adjacent if and only if xy?I. In this paper, we investigate an annihilator ideal-based zero-divisor graph, denoted by ?Ann(M)(R), by replacing the ideal I with the annihilator ideal Ann(M) for an R-module M. We also study the relationship between the diameter of ?Ann(M) (R) and the minimal prime ideals of Ann(M). In addition, we determine when ?Ann(M)(R) is complete. In particular, we prove that for a reduced R-module M, ?Ann(M) (R) is a complete graph if and only if R ? Z2?Z2 and M ? M1?M2 for M1 and M2 nonzero Z2-modules.

Let R be a commutative ring with identity and M be a unitary R-module. A torsion graph of M, denoted by Γ(M), is a graph whose vertices are the non-zero torsion elements of M, and two distinct vertices x and y are adjacent if and only if [x : M][y : M]M = 0. In this paper, we investigate the relationship between the diameters of Γ(M) and Γ(R), and give some properties of minimal prime submodules of a multiplication R-module M over a von Neumann regular ring. In particular, we show that for a multiplication R-module M over a Bézout ring R the diameter of Γ(M) and Γ(R) is equal, where M T(M). Also, we prove that, for a faithful multiplication R-module M with |M| 4, Γ(M) is a complete graph if and only if Γ(R) is a complete graph.

Multicomplex Taylor series expansion (MCTSE) is a numerical method for calculating higher-order partial derivatives of a multivariable realvalued and complex-valued analytic function based on Taylor series expansion without subtraction cancelation errors. The implementation has been facilitated using Cauchy-Riemann matrix representation of multicomplex variables. In this paper, we show steps for finding these matrices and, in addition, that the number of appearances of the k th derivatives follows the Pascal's triangle. Also, the situations where the MCTSE is not applicable is determined. Finally, we investigate the application of the method for complex-valued functions.