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Blue horizontal-branch (BHB) stars are crucial for studying the structure of the Galactic halo. Accurate atmospheric parameters of BHB stars are essential for investigating the formation and evolution of the Galaxy. In this work, a data-driven technique named stellar label machine (SLAM) is used to estimate the atmospheric parameters of Large Sky Area Multi-Object Fiber Spectroscopic Telescope low-resolution spectra (LAMOST-LRS) for BHB stars with a set of A-type theoretical spectra as the training dataset. We add color indexes ($(BP-G), (G-RP), (BP-RP), (J-H)$) during the training process to constrain the stellar temperature further. Finally, we derive the atmospheric parameters ($T_\mathrm{eff}$, log\, $g$, [Fe/H]) for 5,355 BHB stars. Compared to existing literature results, our results are more robust, after taking the color index into account, the resulted precisoin of $T_\mathrm{eff}$, log\, $g$ is significantly improved, especially for the spectrum with low signal-to-noise ratio (S/N). Based on the duplicate observations with a S/N difference $< 20\%$, the random errors are around 30\,K, 0.1~dex, and 0.12~dex for $T_\mathrm{eff}$, log\,$g$, [Fe/H], respectively. The stellar labels provided by SLAM are also compared to those from the high-resolution spectra in literature. The standard deviation between the predicted star labels and the published values from the high-resolution spectra is adopted as \sout{to} the statistical uncertainty of our results. They are $\sigma$($T_\mathrm{eff}$) = 76\,K, $\sigma$(log\,$g$) = 0.04~dex, and $\sigma$([Fe/H]) = 0.09~dex, respectively.

Grafakos systematically proved that $A_\infty$ weights have different characterizations for cubes in Euclidean spaces in his classical text book. Very recently, Duoandikoetxea, Mart\'{\i}n-Reyes, Ombrosi and Kosz discussed several characterizations of the $A_{\infty}$ weights in the setting of general bases. By conditional expectations, we study $A_\infty$ weights in martingale spaces. Because conditional expectations are Radon-Nikod\'{y}m derivatives with respect to sub$\hbox{-}\sigma\hbox{-}$fields which have no geometric structures, we need new ingredients. Under a regularity assumption on weights, we obtain equivalent characterizations of the $A_{\infty}$ weights. Moreover, using weights modulo conditional expectations, we have one-way implications of different characterizations., Comment: This paper has now been accepted for publication in The Journal of Geometric Analysis. 22 pages; 22pages

We present analysis of the spatial density structure for the outer disk from 8$-$14 \,kpc with the LAMOST DR5 13534 OB-type stars and observe similar flaring on north and south sides of the disk implying that the flaring structure is symmetrical about the Galactic plane, for which the scale height at different Galactocentric distance is from 0.14 to 0.5 \,kpc. By using the average slope to characterize the flaring strength we find that the thickness of the OB stellar disk is similar but flaring is slightly stronger compared to the thin disk as traced by red giant branch stars, possibly implying that secular evolution is not the main contributor to the flaring but perturbation scenarios such as interactions with passing dwarf galaxies should be more possible. When comparing the scale height of OB stellar disk of the north and south sides with the gas disk, the former one is slightly thicker than the later one by $\approx$ 33 and 9 \,pc, meaning that one could tentatively use young OB-type stars to trace the gas properties. Meanwhile, we unravel that the radial scale length of the young OB stellar disk is 1.17 $\pm$ 0.05 \,kpc, which is shorter than that of the gas disk, confirming that the gas disk is more extended than stellar disk. What is more, by considering the mid-plane displacements ($Z_{0}$) in our density model we find that almost all of $Z_{0}$ are within 100 \,pc with the increasing trend as Galactocentric distance increases., Comment: 14 pages and 8 figures, Accepted by APJ for publication