Our investigation in the present paper is based on three important results. (1) In [12], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the qua
ntum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [4], Green found a homological formula for the representation category of the quiver and equipped Ringels Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simple representations realizes the nilpotent part of quantum group of any type. (3) In [9], Lusztig defined induction and restriction functors for the perverse sheaves on the variety of representations of the quiver which occur in the direct images of constant sheaves on flag varieties, and he found a formula between his induction and restriction functors which gives the comultiplication as algebra homomorphism for quantum group. In the present paper, we prove the formula holds for all semisimple complexes with Weil structure. This establishes the categorification of Greens formula.
Let $textbf{U}^+$ be the positive part of the quantum group $textbf{U}$ associated with a generalized Cartan matrix. In the case of finite type, Lusztig constructed the canonical basis $textbf{B}$ of $textbf{U}^+$ via two approaches. The first one is
an elementary algebraic construction via Ringel-Hall algebra realization of $textbf{U}^+$ and the second one is a geometric construction. The geometric construction of canonical basis can be generalized to the cases of all types. The generalization of the elementary algebraic construction to affine type is an important problem. We give several main results of algebraic constructions to the affine canonical basis in this ariticle. These results are given by Beck-Nakajima, Lin-Xiao-Zhang, Xiao-Xu-Zhao, respectively.
For quantum group of affine type, Lusztig gave an explicit construction of the affine canonical basis by simple perverse sheaves. In this paper, we construct a bar-invariant basis by using a PBW basis arising from representations of the corresponding
tame quiver. We prove that this bar-invariant basis coincides with Lusztigs canonical basis and obtain a concrete bijection between the elements in theses two bases. The index set of these bases is listed orderly by modules in preprojective, regular non-homogeneous, preinjective components and irreducible characters of symmetric groups. Our results are based on the work of Lin-Xiao-Zhang and closely related with the work of Beck-Nakajima. A crucial method in our construction is a generalization of that by Deng-Du-Xiao.
Deep learning has demonstrated radiograph screening performances that are comparable or superior to radiologists. However, recent studies show that deep models for thoracic disease classification usually show degraded performance when applied to exte
rnal data. Such phenomena can be categorized into shortcut learning, where the deep models learn unintended decision rules that can fit the identically distributed training and test set but fail to generalize to other distributions. A natural way to alleviate this defect is explicitly indicating the lesions and focusing the model on learning the intended features. In this paper, we conduct extensive retrospective experiments to compare a popular thoracic disease classification model, CheXNet, and a thoracic lesion detection model, CheXDet. We first showed that the two models achieved similar image-level classification performance on the internal test set with no significant differences under many scenarios. Meanwhile, we found incorporating external training data even led to performance degradation for CheXNet. Then, we compared the models internal performance on the lesion localization task and showed that CheXDet achieved significantly better performance than CheXNet even when given 80% less training data. By further visualizing the models decision-making regions, we revealed that CheXNet learned patterns other than the target lesions, demonstrating its shortcut learning defect. Moreover, CheXDet achieved significantly better external performance than CheXNet on both the image-level classification task and the lesion localization task. Our findings suggest improving annotation granularity for training deep learning systems as a promising way to elevate future deep learning-based diagnosis systems for clinical usage.
The GWAC-N is an observation network composed of multi-aperture and multi-field of view robotic optical telescopes. The main instruments are the GWAC-A. Besides, several robotic optical telescopes with narrower field of views provide fast follow-up m
ulti-band capabilities to the GWAC-N. The primary scientific goal of the GWAC-N is to search for the optical counterparts of GRB that will be detected by the SVOM. The GWAC-N performs many other observing tasks including the follow-ups of ToO and both the detection and the monitoring of variable/periodic objects as well as optical transients. To handle all of those scientific cases, we designed 10 observation modes and 175 observation strategies, especially, a joint observation strategy with multiple telescopes of the GWAC-N for the follow-up of GW events. To perform these observations, we thus develop an AOM system in charge of the object management, the dynamic scheduling of the observation plan and its automatic broadcasting to the network management and finally the image management. The AOM combines the individual telescopes into a network and smoothly organizes all the associated operations. The system completely meets the requirements of the GWAC-N on all its science objectives. With its good portability, the AOM is scientifically and technically qualified for other general purposed telescope networks. As the GWAC-N extends and evolves, the AOM will greatly enhance the discovery potential for the GWAC-N. In the first paper of a series of publications, we present the scientific goals of the GWAC-N as well as the hardware, the software and the strategy setup to achieve the scientific objectives. The structure, the technical design, the implementation and performances of the AOM will be also described in details. In the end, we summarize the current status of the GWAC-N and prospect for the development plan in the near future.
We propose a generative framework based on generative adversarial network (GAN) to enhance facial attractiveness while preserving facial identity and high-fidelity. Given a portrait image as input, having applied gradient descent to recover a latent
vector that this generative framework can use to synthesize an image resemble to the input image, beauty semantic editing manipulation on the corresponding recovered latent vector based on InterFaceGAN enables this framework to achieve facial image beautification. This paper compared our system with Beholder-GAN and our proposed result-enhanced version of Beholder-GAN. It turns out that our framework obtained state-of-art attractiveness enhancement results. The code is available at https://github.com/zoezhou1999/BeautifyBasedOnGAN.
This paper considers estimation and inference about tail features when the observations beyond some threshold are censored. We first show that ignoring such tail censoring could lead to substantial bias and size distortion, even if the censored proba
bility is tiny. Second, we propose a new maximum likelihood estimator (MLE) based on the Pareto tail approximation and derive its asymptotic properties. Third, we provide a small sample modification to the MLE by resorting to Extreme Value theory. The MLE with this modification delivers excellent small sample performance, as shown by Monte Carlo simulations. We illustrate its empirical relevance by estimating (i) the tail index and the extreme quantiles of the US individual earnings with the Current Population Survey dataset and (ii) the tail index of the distribution of macroeconomic disasters and the coefficient of risk aversion using the dataset collected by Barro and Urs{u}a (2008). Our new empirical findings are substantially different from the existing literature.
We prove the consistency of the $ell_1$ penalized negative binomial regression (NBR). A real data application about German health care demand shows that the $ell_1$ penalized NBR produces a more concise but more accurate model, comparing to the classical NBR.
This paper shows $$ sup_{fin H^s(mathbb{R}^n)}dim _Hleft{xinmathbb{R}^n: lim_{trightarrow0}e^{it(-Delta)^alpha}f(x) eq f(x)right}leq n+1-frac{2(n+1)s}{n} text{under} begin{cases} ngeq2; alpha>frac12; frac{n}{2(n+1)}<sleqfrac{n}{2} . end{cases} $$
In this paper, Theorems 1.1- 1.2 show that the Boussinesq operator $mathcal{B}_tf$ converges pointwise to its initial data $fin H^s(mathbb{R})$ as $tto 0$ provided $sgeqfrac{1}{4}$ -- more precisely -- on the one hand, by constructing a counterexampl
e in $mathbb{R}$ we discover that the optimal convergence index $s_{c,1}=frac14$; on the other hand, we find that the Hausdorff dimension of the disconvergence set for $mathcal{B}_tf$ is begin{align*} alpha_{1,mathcal{B}}(s)&=begin{cases} 1-2s& text{as} frac{1}{4}leq sleqfrac{1}{2}; 1 & text{as} 0<s<frac{1}{4}. end{cases} end{align*} Moreover, Theorem 1.3 presents a higher dimensional lift of Theorems 1.1- 1.2 under $f$ being radial.
