<p style='text-indent:20px;'>In this paper, we prove sharp gradient estimates for positive solutions to the weighted heat equation on smooth metric measure spaces with compact boundary. As an application, we prove Liouville theorems for ancient solutions satisfying the Dirichlet boundary condition and some sharp growth restriction near infinity. Our results can be regarded as a refinement of recent results due to Kunikawa and Sakurai.</p>

In this paper, we prove sharp gradient estimates for a positive solution to the heat equation u t = ∆u + au log u in complete noncompact Riemannian manifolds. As its application, we show that if u is a positive solution of the equation u t = ∆u and log u is of sublinear growth in both spatial and time directions then u must be constant. This gradient estimate is sharp since it is well-known that u(x, t) = e x+t satisfying u t = ∆u. We also emphasize that our results are better than those given by Jiang ([Jia16]), Souplet-Zhang ([SZ06]), Wu ([Wu15, Wu17]), and others.

Objective: The study aims to describe the employment status of graduate students majoring in Preventive Medicine from 2018 to 2020. Methods: A cross-sectional descriptive study on 218 students who graduated from the Doctor of Preventive Medicine at Hanoi Medical University in 2018, 2019 and 2020. Results: The percentage of graduated students having jobs in 2018,2019, and 2020 was 82.9%, 80%, and 75.4% respectively. After graduating, there were 91.8% of graduated students who found a job less than 6 months, 56.1% of those worked in the right field of training, and 32.7% of those worked in the training industry. The self-employed rate accounted for 71.3%. The employment rate in the public sector was 52%. The percentage of graduated students who received starting salary (less than 5 million VND/month) accounted for 69.3%; 50.5% of graduated students received an average of less than 5 million VND per month. Conclusions: In 3 years, the proportion of students who had jobs (78.4%); The majority of students had jobs less than 6 months after graduating (91.8%); were self-employed (71.3%), and did the right field of training (56.1%). Starting salary and average monthly income of over 5 million students increased over 3 years. Keywords: Employment, Doctor of Preventive Medicine, Hanoi Medical University.

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