Chromatic number of Euclidean plane

If the chromatic number of Euclidean plane is larger than four, but it is known that the chromatic number of planar graphs is equal to four, then how does one explain it? In my opinion, they are contradictory to each other. This idea leads to confirm the chromatic number of the plane about its exact value

M-estimation in high-dimensional linear model

We mainly study the M-estimation method for the high-dimensional linear regression model, and discuss the properties of M-estimator when the penalty term is the local linear approximation. In fact, M-estimation method is a framework, which covers the methods of the least absolute deviation, the quantile regression, least squares regression and Huber regression. We show that the proposed estimator possesses the good properties by applying certain assumptions. In the part of numerical simulation, we select the appropriate algorithm to show the good robustness of this methodComment: 16 pages,3 table

Can Up FCNC solve the ΔACP\Delta A_{CP} puzzle?

We investigate the attempt using flavor violation gauge interaction in the up sector to explain the LHCb recently observed large $\Delta A_{CP}$ ($A_{CP}(D^{0}\to K^{+}K^{-})-A_{CP}(D^{0}\to \pi^{+}\pi^{-})$). We study an Abelian model that only right-handed up quarks is charged under it and the 1-3 coupling is maximized. The simultaneous 1-3 2-3 mixing is realized by a quark mixing of 1-2 generation. Given the easy identification of top quark, the model can be directly tested by $\Delta F=1$ and $\Delta F=2$ processes at the hadron colliders as associated top production $g c \to t Z^{\prime}$ or same-sign top scattering $u u\to t t$. The direct search bounds are still consistent with the assumption that $ut$ and $ct$ couplings are equal but the same-sign top scattering bound is expected to be reached very soon. However, since there is no CKM-like suppression, the corresponding parameter space for generating $\Delta A_{CP}$ is completely excluded by the $D^{0}-\bar{D}^{0}$ mixing. We conclude that the up FCNC type models cannot explain the $\Delta A_{CP}$ while to be consistent with the $D^{0}-\bar{D}^{0}$ mixing constraint at the same time. On the other hand, a model as SM with fourth family extension has better chance to explain the large $\Delta A_{CP}$ consistently.Comment: 10 pages, 2 figure