Generalizations of Weighted Trapezoidal Inequality for Monotonic Mappings and Its Applications

In this paper, we establish some generalizations of weighted trapezoid inequality for monotonic mappings, and give several applications for r − moment, the expectation of a continuous random variable and the Beta mapping

Feasibility of automated 3-dimensional magnetic resonance imaging pancreas segmentation.

PurposeWith the advent of MR guided radiotherapy, internal organ motion can be imaged simultaneously during treatment. In this study, we evaluate the feasibility of pancreas MRI segmentation using state-of-the-art segmentation methods.Methods and materialT2 weighted HASTE and T1 weighted VIBE images were acquired on 3 patients and 2 healthy volunteers for a total of 12 imaging volumes. A novel dictionary learning (DL) method was used to segment the pancreas and compared to t mean-shift merging (MSM), distance regularized level set (DRLS), graph cuts (GC) and the segmentation results were compared to manual contours using Dice's index (DI), Hausdorff distance and shift of the-center-of-the-organ (SHIFT).ResultsAll VIBE images were successfully segmented by at least one of the auto-segmentation method with DI >0.83 and SHIFT ≤2 mm using the best automated segmentation method. The automated segmentation error of HASTE images was significantly greater. DL is statistically superior to the other methods in Dice's overlapping index. For the Hausdorff distance and SHIFT measurement, DRLS and DL performed slightly superior to the GC method, and substantially superior to MSM. DL required least human supervision and was faster to compute.ConclusionOur study demonstrated potential feasibility of automated segmentation of the pancreas on MRI images with minimal human supervision at the beginning of imaging acquisition. The achieved accuracy is promising for organ localization

A simple proof of inequalities related to means

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Chronology of the Basalt Units Surrounding Chang’e-4 Landing Area

The Chang’e-4 (CE-4) lunar probe, the first soft landing spacecraft on the far side of the Moon, successfully landed in the Von Kármán crater on 3 January 2019. Geological studies of the landing area have been conducted and more intensive studies will be carried out with the in situ measured data. The chronological study of the maria basalt surrounding the CE-4 landing area is significant to the related studies. Currently, the crater size-frequency distribution (CSFD) technique is the most popular method to derive absolute model ages (AMAs) of geological units where no returned sample is available, and it has been widely used in dating maria basalt on the lunar surface. In this research, we first make a mosaic with multi-orbital Chang’e-2 (CE-2) images as a base map. Coupled with the elevation data and FeO content, nine representative areas of basalt units surrounding the CE-4 landing area are outlined and their AMAs are derived. The dating results of the nine basalt units indicate that the basalts erupted from 3.42 to 2.28 Ga ago in this area, a period much longer than derived by previous studies. The derived chronology of the above basalt units establishes a foundation for geological analysis of the returned CE-4 data