A New Discontinuous Galerkin Finite Element Method for Directly Solving the Hamilton-Jacobi Equations

In this paper, we improve upon the discontinuous Galerkin (DG) method for Hamilton-Jacobi (HJ) equation with convex Hamiltonians in (Y. Cheng and C.-W. Shu, J. Comput. Phys. 223:398-415,2007) and develop a new DG method for directly solving the general HJ equations. The new method avoids the reconstruction of the solution across elements by utilizing the Roe speed at the cell interface. Besides, we propose an entropy fix by adding penalty terms proportional to the jump of the normal derivative of the numerical solution. The particular form of the entropy fix was inspired by the Harten and Hyman's entropy fix (A. Harten and J. M. Hyman. J. Comput. Phys. 50(2):235-269, 1983) for Roe scheme for the conservation laws. The resulting scheme is compact, simple to implement even on unstructured meshes, and is demonstrated to work for nonconvex Hamiltonians. Benchmark numerical experiments in one dimension and two dimensions are provided to validate the performance of the method

The Female Perspective on Self-Worth

How do we determine the value of ourselves? It is intuitive that our perception of self-worth has an implicit effect on our development as individuals. This thesis explores the process through which women determine their self-worth and why it is problematic

The safety of antipsychotic use during pregnancy

Aim: To investigate the patterns of gestational antipsychotics use and whether exposure to antipsychotic medications in pregnancy is associated with gestational diabetes mellitus (GDM) in mothers and seizure, attention deficit/hyperactivity disorder (ADHD), autism spectrum disorder (ASD), preterm birth (PTB) and small for gestational age (SFGA) in subsequent children. Methods: Firstly, a methodological review was conducted to review the methodological characteristics of existing observational studies that investigate the association between prenatal central nervous system (CNS) drugs use and CNS disorders. Secondly, a systematic review and meta-analysis was conducted to evaluate the evidence-based association between gestational antipsychotic use and GDM. Thirdly, a cross-sectional study was conducted to investigate the patterns and trends of antipsychotics use during pregnancy in the United Kingdom (UK) and Hong Kong (HK). Lastly, seven cohort studies were conducted to investigate the association between antipsychotics use in pregnancy and the risk of above-mentioned outcomes, respectively. Results: 4.64% and 0.34% of pregnancies were prescribed at least one prescription of antipsychotic during pregnancy in the UK and HK, respectively. When women who continued using antipsychotics during pregnancy were compared with those who had stopped, there was no evidence to demonstrate an increased risk of GDM. No evidence supported prenatal exposure to antipsychotics can increase the risk of ADHD/ASD/SFGA. Children with prenatal antipsychotics exposure was associated with an increased risk of seizure (HR 1.49, 95% CI 1.11-1.99) and PTB (OR 1.40, 95%CI 1.13-1.75), comparing to those without. However, further sibling-matched analyses and negative control analyses indicated no evidence supported the above-mentioned associations. Conclusion: This PhD project did not suggest an increased risk of GDM in mothers or seizure/ADHD/ASD/PTB/SFGA in children regarding antipsychotics use during pregnancy. Women are not recommended to stop their regular antipsychotic prescription during pregnancy due to the risk of developing GDM or delivering an offspring with seizure/ADHD/ASD/PTB/SFGA

Application of Hyperbolic Paraboloid in Architectural Design

Hyperbolic paraboloid is a kind of ruled space surface with beautiful shape. It is often used in architectural design and can achieve a free and flexible appearance effect. Due to the complexity of curved surfaces, many architects do not know how to navigate them. The main purpose of this article is to explore how to use hyperbolic paraboloids in architectural design. Firstly, the formation principle of hyperbolic paraboloid is analyzed from a mathematical perspective. Then, through investigating examples, it expounds its application in architectural design. Hyperbolic paraboloids are mainly used in building roofs, especially in large span buildings. There are three uses of hyperbolic paraboloids in roofs, corresponding to three different architectural shapes.The first is to cut a hyperbolic paraboloid vertically with four planes, and the contour projection is a rectangle or parallelogram. The second is to cut the hyperbolic paraboloid vertically and horizontally with four planes, and the contour projection is a curved quadrilateral. The third is to cut hyperbolic paraboloid with elliptic surface, and the contour projection is an ellipse. Finally, the conclusion is drawn on how to flexibly use hyperbolic paraboloids in architectural design, and what are the advantages and disadvantages of hyperbolic paraboloids, which has important reference value for architects to carry out related designs