{pdf=http://arxiv.org/pdf/1905.05090v1.pdf|100%|600|google}

- This is a joint work with
*Yongki Lee*. - The preprint is available on arXiv:1905.05090.

{pdf=http://arxiv.org/pdf/1905.05090v1.pdf|100%|600|google}

- critical threshold
- traffic model
- global regularity
- finite time wave breakdown

- The preprint is available on arXiv:1901.02582.

{pdf=http://arxiv.org/pdf/1901.02582v1.pdf|100%|600|google}

- Euler alignment
- critical threshold
- Blow up
- Singular interaction

2013-2014 | Ann G. Wylie Dissertation Fellowship |

SIAM Student Travel Award | |

AMS Graduate Student Travel Grant | |

Kaplan Travel Grant | |

2011-2012 | Mark E. Lachtman Graduate Student Award |

Jacob K. Goldhaber Travel Award | |

2010-2011 | Kaplan Travel Grant |

2008-2010 | Graduate Fellowship in University of Maryland |

2007-2008 | First Honor Graduates in Beijing |

"Outstanding University Graduates" in Peking University | |

2006-2007 | Triple-Good Student in Peking University |

Chinese Economical Research Scholarship | |

2005-2006 | Triple-Good Student in Peking University |

Baogang Scholarship | |

2004-2005 | Triple-Good Student in Peking University |

Guanghua Scholarship |

2019.7.15-19 | International Congress on Industrial and Applied Mathematics, Valencia, Spain. |

Talk: Eulerian dynamics with alignment interactions. | |

2019.7.8-10 | 2019 TIANFU International Conference on Partial Differential Equations, Southwestern University of Finance and Economics, Chengdu, China. |

Talk: Eulerian dynamics with alignment interactions. | |

2019.6.20 | PDE Seminar, University of Electronic Science and Technology of China, Chengdu, China. |

Talk: Small scale singularity formations in Euler and related equations. | |

2019.6.12 | PDE Seminar, Chongqing University, Chongqing, China. |

Talk: Small scale singularity formations in Euler and related equations. | |

2019.5.24 | Colloquium, Renmin University, Beijing, China. |

Talk: Small scale singularity formations in Euler and related equations. | |

2019.5.23 | Computational and Applied Math Seminar, Peking University, Beijing, China. |

Talk: Asymptotic preserving schemes on kinetic models with singular limits. | |

2019.5.22 | Thematic Reports, Beijing Normal University, Beijing, China. |

Talk: Eulerian dynamics with nonlocal interactions. | |

2019.5.15 | Analysis Seminar, Tsinghua University, Beijing, China. |

Talk: Self-organized dynamics: aggregation and flocking. | |

2019.5.9-12 | Partial Differential Equations and Applications in Physics and Life Sciences, Chongqing, China. |

Talk: Eulerian dynamics with alignment interactions. | |

2019.3.17-20 | DASIV Center Spring School Series: Models and Data. |

Talk: Asymptotic preserving schemes on kinetic models with singular limits. | |

2019.3.12 | PDE Seminar, Georgia Institute of Technology. |

Talk: Eulerian dynamics with alignment interactions. | |

2019.3.5-6 | Mathematical Aspects of Collective Dynamics: Kinetic Description and Fractional Diffusion, University of Maryland, College Park. |

Talk: On Euler Alignment system with weakly singular interactions. | |

2019.3.1 | Graduate Seminar, Department of Civil Engineering, University of South Carolina. |

Talk: The mathematics in continuum mechanics. | |

2019.2.19 | Graduate Colloquium, University of South Carolina. |

Talk: Fluid dynamics: from stirring a cup of coffee to tracking a hurricane. |

2012.11.5-11.9 | Kinetic Description of Social Dynamics: From Consensus to Flocking, University of Maryland, College Park. |

2012.10.3-10.7 | Young Researchers Workshop: Kinetic Description of Model Scale phonomena, University of Wisconsin, Madison. |

Talk: Critical thresholds on compressible Euler dynamics with nonlocal viscosity. | |

2012.9-12 | RIT on Applied Partial Differential Equations, University of Maryland, College Park. |

Talk: Macroscopic flocking models. | |

2012.8.13-17 | Summer school: III Frontiers of Mathematics and Applications, UIMP, Santander, Spain. |

2012.7.4-7.6 | Summer School on Nonlocal Operators, University of Bielefeld, Germany. |

2012.6.25-6.29 | 14th International Conference on Hyperbolic Problems, Padova, Italy. |

2012.2.27-3.2 | Nonlocal PDEs, Variational Problems and their Applications, UCLA, Los Angeles. |

2011.11.29 | AMSC student seminar, University of Maryland, College Park. |

Talk: Spectral dynamics and critical thresholds for nonlinear conservation laws. | |

2011.11.4 | Mid Atlantic Numerical Analysis Day, Temple University, Philadelphia. |

Talk: Hierarchical construction of bounded solutions of divU=F in critical regularity spaces. | |

2011.8.15-19 | Summer school: Frontiers of Mathematics and Applications, UIMP, Santander, Spain. |

2011.8.3-12 | IMA Workshop: Mathematical Modeling in Industry XV, University of Minnesota, Minneapolis. |

Topic: Modeling aircraft hoses and flexible conduits | |

2011.5.23-27 | FRG2011, Kinetic Description of Multiscale Phenomena, University of Wisconsin, Madison. |

2011.5.2-6 | New Perspectives in Nonlinear PDE's, University of Michigan, Ann Arbor. |

2010.9-12 | RIT on Applied Partial Differential Equations, University of Maryland, College Park. |

Talk: Hierarchical construction of uniformly bounded solutions to divU=F. | |

2010.2-5 | RIT on Probability and Statistics for Computational Math, University of Maryland, College Park. |

2006.7.15-22 | 2nd Summer School in Symbolic Computation, Peking University, Beijing, China. |

- The preprint is available on arXiv:1708.09360.

{pdf=http://arxiv.org/pdf/1708.09360v2.pdf|100%|600|google}

- Euler alignment
- Porous medium
- quasigeostrophic equation
- Blow up

- This is a joint work with
*Alexander Kiselev*. - The paper has been published in
*SIAM Journal on Mathematical Analysis*doi:10.1137/17M1141515.

- The preprint is available on arXiv:1707.07296.

{pdf=http://arxiv.org/pdf/1707.07296v1.pdf|100%|600|google}

- Euler alignment
- EulerPoisson equation
- Singular interaction
- global regularity

- This is a joint work with
*Alina Chertock*and*Bokai Yan*. - The paper has been published in
*Kinetic and Related Models*doi:10.3934/krm.2018030.

- The preprint is available on arXiv:1706.09568.

{pdf=http://arxiv.org/pdf/1706.09568v1.pdf|100%|600|google}

- Asymptotic preserving scheme
- kinetic equation
- singular limit

- This is a joint work with
*Tam Do*,*Alexander Kiselev*and*Lenya Ryzhik*. - The paper has been published in
*Archive for Rational Mechanics and Analysis*doi:10.1007/s00205-017-1184-2.

- The preprint is available on arXiv:1701.05155.

{pdf=http://arxiv.org/pdf/1701.05155v1.pdf|100%|600|google}

- Euler alignment
- flocking
- Burgers
- global regularity
- modulus of continuity
- critical

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

In the lecture, we discover that a good Ansatz for $n$-th order linear homogeneous ODE with constant coefficient is exponential functions. Indeed, if the characteristic function has $n$ distinct real roots $r_1,\cdots,r_n$. Then, $\{e^{r_1t},\cdots,e^{r_nt}\}$ are all solutions of the equation. If they are linearly independent, then they form a basis of the solution space, and therefore the general solution of the equation reads

\[y(t)=C_1e^{r_1t}+\cdots C_ne^{r_nt}.\]

Let us try to check whether they are always linearly independent.

One way to prove linearly independency of a set of smooth functions is through Wronskian. In this case, we have

\[W[e^{r_1t},\cdots,e^{r_nt}](t)=\text{det}\begin{bmatrix}e^{r_1t}&e^{r_2t}&\cdots&e^{r_nt}\\r_1e^{r_1t}&r_2e^{r_2t}&\cdots&r_ne^{r_nt}\\ \vdots&\vdots&\ddots&\vdots\\ r_1^{n-1}e^{r_1t}&r_2^{n-1}e^{r_2t}&\cdots&r_n^{n-1}e^{r_nt}\end{bmatrix}\]

In particular, if we evaluate the Wronskian at $t=0$, what we get is

\[W[e^{r_1t},\cdots,e^{r_nt}](0)=\text{det}\begin{bmatrix}1&1&\cdots&1\\r_1&r_2&\cdots&r_n\\ \vdots&\vdots&\ddots&\vdots\\ r_1^{n-1}&r_2^{n-1}&\cdots&r_n^{n-1}\end{bmatrix}=\prod_{1\leq i<j\leq n}(r_j-r_i)\neq0.\]

Here, the matrix is realized as Vandermonde matrix [Wikipedia Page], and its determinant is always nonzero. So, we conclude that these functions are linearly independent and hence form a basis of the solution space.

The argument is also true for complex roots. For instance, one can prove $\{\cos(r_1t),\cdots,\cos(r_nt)\}$ are linearly independent using a similar argument.

- MATH211
- ODE
- 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';
```

- Matlab
- mesh
- surf
- plot
- cut
- axis
- missing