This is a supplementary notes for MATH211 ODE and Linear Algebra.

When I tried to manually adjust the axis limits of a graph, Matlab will automatically cut off the part of the graph that is beyond the axis limits. This didn't happen before (and before means 2014a or earlier versions). I took a little while to find the cure. Just try the following scripts. 

ax = gca;
ax.Clipping = 'off';

Here is the steps to change default mail app in OS X:

1. Open Mail app.
2. Select Preferences -> General. Under "Default email reader", select your preferred mail app.

I am running into a problem by following the steps: the default email reader changes back to the original mail app seconds after I made the change. Here is the solution that works for me. Open terminal. Copy and paste the following script.

/System/Library/Frameworks/CoreServices.framework/Versions/A/Frameworks/LaunchServices.framework/Versions/A/Support/lsregister -kill -r -all local,system,user

Then try step 1-2 again, and it won't change back again!

Alexander Kiselev, and Changhui Tan

Advances in Mathematics, Volume 325, pp. 34-55 (2018).

Abstract

In recent work of Luo and Hou, a new scenario for finite time blow up in solutions of 3D Euler equation has been proposed. The scenario involves a ring of hyperbolic points of the flow located at the boundary of a cylinder. In this paper, we propose a two dimensional model that we call “hyperbolic Boussinesq system”. This model is designed to provide insight into the hyperbolic point blow up scenario. The model features an incompressible velocity vector field, a simplified Biot–Savart law, and a simplified term modeling buoyancy. We prove that finite time blow up happens for a natural class of initial data.

	doi:10.1016/j.aim.2017.11.019
	Download the Published Version

Yan-jun Guo, Mao Pan, Fei Yan, Zhe Wang, Changhui Tan, and Tiao Lu

Journal of PLA University of Science and Technology (Natural Science Edition), Volume 26, No 1, pp. 185-206 (2016).

Abstract

To enhance the accuracy of three-dimensional geological model, emphasize the high local relevance characteristics of the complex geological bodies, and avoid complicated calculation and dependence on human experience in traditional interpolation methods, the natural neighbor interpolation (NNI) method was used for three-dimensional discrete data interpolation in the process of modeling. But the existing NNI method could not be applied to the boundary interpolation of finite fields, which was the most difficult problem of its application in three-dimensional geological modeling. Based on the geometry of Voronoi Cells and Delaunay Triangles, the shape function was constructed using non-Sibsonian (Laplace) interpolation method. The continuity of the boundary in NNI method was proven, the boundary interpolation was implemented and the computational complexity was reduced. The accuracy and validity of the method were proven by building the city geological model.

	doi:10.3969/j.issn.1009-3443.2009.06.026
	Download the Published Version

The paper is related to the undergraduate thesis: Numerical analysis and algorithm design in natural neighbor method.

Download the Undergraduate Thesis (In Chinese)

Eitan Tadmor, and Changhui Tan

"Nonlinear Partial Differential Equations", Proceedings of the 2010 Abel Symposium held in Oslo, Sep. 2010 (H. Holden & K. Karlsen eds.), Abel Symposia 7, Springer 2011, 255-269.

Abstract

We implement the hierarchical decomposition introduced in [Ta15], to construct uniformly bounded solutions of the problem \(\nabla\cdot U = F\), where the two-dimensional data is in the critical regularity space, \(F\in L^2_{\#}(\mathbb{T}^2)\). Criticality in this context, manifests itself by the lack of linear mapping, \(F\in L^2_{\#}(\mathbb{T}^2)\to U\in L^{\infty}(\mathbb{T}^2,\mathbb{R}^2)\) [BB03]. Thus, the intriguing aspect here is that although the problem is linear, the construction of its uniformly bounded solutions is not.

	doi:10.1007/978-3-642-25361-4_14
	Download the Published Version

Razvan C. Fetecau, Weiran Sun, and Changhui Tan

Physica D: Nonlinear Phenomena, Volume 325, pp. 146-163 (2016).

Abstract

We include alignment interactions in a well-studied first-order attractive–repulsive macroscopic model for aggregation. The distinctive feature of the extended model is that the equation that specifies the velocity in terms of the population density, becomes implicit, and can have non-unique solutions. We investigate the well-posedness of the model and show rigorously how it can be obtained as a macroscopic limit of a second-order kinetic equation. We work within the space of probability measures with compact support and use mass transportation ideas and the characteristic method as essential tools in the analysis. A discretization procedure that parallels the analysis is formulated and implemented numerically in one and two dimensions.

	doi:10.1016/j.physd.2016.03.011
	Download the Published Version

The purpose of this note is to clarify questions and statements in class, and a step-by-step way to find an LU decomposition by hand.

The purpose of this note is to clarify questions and statements in class, and to provide detailed procedures to find polynomial interpolations.

This note is taken in the PDE discussion group in 2012, on the topic of important spaces in fluid dynamics.

Lecture 2: Carleson measure