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

In the MacOS system, the keychain password has to match the user password in order to use keychain appropriately. In my case, they have to match in order to allow Apple Watch to unlock the Mac.

The official way to change the keychain password can be found HERE. However, in my case, the "Change password for keychain login" option is in gray and can not be selected. One solution is the following:

Open terminal and type in

`security set-keychain-password`

Then, type in old password and then new password. The password is then changed. This does not require admin access.

- MacOS
- Keychain password

- 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