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## 1. chinaXiv:202107.00011 [pdf]

Subjects: Mathematics >> Mathematics （General）

 This paper discusses the properties the spaces of fuzzy sets in a metric s- pace equipped with the endograph metric and the sendograph metric, re- spectively. We fist discuss the level characterizations of the Γ-convergence and the endograph metric, and point out the elementary relationships among Γ-convergence, endograph metric and the sendograph metric. On the basis of these results, we present the characterizations of total boundedness, rel- ative compactness and compactness in the space of compact positive α-cuts fuzzy sets equipped with the endograph metric, and in the space of com- pact support fuzzy sets equipped with the sendograph metric, respectively. Furthermore, we give completions of these two kinds of spaces, respectively.

## 2. chinaXiv:202105.00070 [pdf]

Subjects: Statistics >> Mathematical Statistics
Subjects: Computer Science >> Computer Application Technology
Subjects: Information Science and Systems Science >> Basic Disciplines of Information Science and Systems Science

 统计独立性是统计学和机器学习领域的基础性概念，如何表示和度量统计独立性是该领域的基本问题。Copula理论提供了统计相关性表示的理论工具，而Copula熵理论则给出了度量统计独立性的概念工具。本文综述了Copula熵的理论和应用，概述了其基本概念定义、定理和性质，以及估计方法。介绍了Copula熵研究的最新进展，包括其在统计学四个基本问题（结构学习、关联发现、变量选择和时序因果发现等）上的理论应用。讨论了四个理论应用之间的关系，以及其对应的深层次的相关性和因果性概念之间的联系，并将Copula熵的（条件）独立性度量框架与基于核函数和距离的相关性度量框架进行了对比。简述了Copula熵在水文学、环境气象学、生态学、化学信息学、认知神经学、计算神经学、系统生物学、生物信息学、临床诊断学、老年医学、公共卫生学、经济政策学、政治学，以及能源工程、制造工程、可靠性工程、航空工程、通信工程、测绘工程和金融工程等领域的实际应用。

## 3. chinaXiv:202110.00083 [pdf]

Subjects: Mathematics >> Mathematics （General）

 In this paper, we give characterizations of compactness in fuzzy set space with Lpmetric. Based on these results, we present completions of fuzzy set spaces with Lpmetric. The fuzzy sets discussed in this paper are fuzzy sets in general metric space. We also point out that the results in this paper generalize the results given in our previous paper.

## 4. chinaXiv:202110.00056 [pdf]

Subjects: Mathematics >> Applied Mathematics
Subjects: Information Science and Systems Science >> Other Disciplines of Information Science and Systems Science

 The shortest path problem (SPP) is a classic problem and appears in a wide range of applications. Although a variety of algorithms already exist, new advances are still being made, mainly tuned for particular scenarios to have better performances. As a result, they become more and more technically complex and sophisticated. Here we developed a novel nature-inspired algorithm to compute all possible shortest paths between two nodes in a graph: Resonance Algorithm (RA), which is surprisingly simple and intuitive. Besides its simplicity, RA turns out to be much more time-efficient for large-scale graphs than the extended Dijkstra's algorithm (such that it gives all possible shortest paths). Moreover, RA can handle any undirected, directed, or mixed graphs, irrespective of loops, unweighted or positively-weighted edges, and can be implemented in a fully decentralized manner. These good properties ensure RA a wide range of applications.

## 5. chinaXiv:202109.00058 [pdf]

Subjects: Mathematics >> Applied Mathematics

 In the early days of the epidemic of coronavirus disease 2019 (COVID-19), due to insufficient knowledge of the pandemic, inadequate nucleic acid tests, lack of timely data reporting, etc., the origin time of the onset of COVID-19 is difficult to determine. Therefore, source tracing is crucial for infectious disease prevention and control. The purpose of this paper is to infer the origin time of pandemic of COVID-19 based on a data and model hybrid driven method. We model the testing positive rate to fit its actual trend, and use the least squares estimation to obtain the optimal model parameters. Further, the kernel density estimation is applied to infer the origin time of pandemic given the specific confidence probability. By selecting 12 representative regions in the United States for analysis, the dates of the first infected case with 50% confidence probability are mostly between August and October 2019, which are earlier than the officially announced date of the first confirmed case in the United States on January 20, 2020. The experimental results indicate that the COVID-19 pandemic in the United States starts to spread around September 2019 with a high confidence probability. In addition, the existing confirmed cases are also used in Wuhan City and Zhejiang Province in China to infer the origin time of COVID-19 and provide the confidence probability. The results show that the spread of COVID-19 pandemic in China is likely to begin in late December 2019.

## 6. chinaXiv:202108.00116 [pdf]

Subjects: Mathematics >> Mathematics （General）

 In this paper, we discuss the properties of functions generated using Hausdorff metric.

## 7. chinaXiv:202108.00117 [pdf]

Subjects: Psychology >> Cognitive Psychology
Subjects: Biology >> Neurobiology

 近年来，越来越多研究者使用神经网络研究认知和计算。由于研究背景和目的的差异，研究者使用神经网络的方式不尽相同。本文总结了几种“使用神经网络研究认知”的常见思路，并且列举了具体的实例，帮助大家对这些思路进行深入理解。同时，我们也探讨了每种研究思路需要研究者具备哪些能力。最后，对目前“神经网络研究认知”领域中所面临的问题以及如何建立通用人工智能，进行了简单探讨。

## 8. chinaXiv:202108.00057 [pdf]

Subjects: Mathematics >> Mathematics （General）

 In this paper, we give some properties related to platform points of a fuzzy set and their applications.

## 9. chinaXiv:202107.00070 [pdf]

Subjects: Mathematics >> Theoretical Computer Science

 In this paper, we theoretically propose a new hashing scheme to establish the sparse Fourier transform in high-dimensional space. The estimation of the algorithm complexity shows that this sparse Fourier transform can overcome the curse of dimensionality. To the best of our knowledge, this is the first polynomial-time algorithm to recover the high-dimensional continuous frequencies.

 In this paper, we mainly discuss the copositivity of 4th order symmetric tensor defined by scalar dark matter stable under a $\mathbb{Z}_{3}$ discrete group, and obtain an analytically necessary and sufficient condition of the copositivity of such a class of tensors. Furthermore, this analytic expression may be used to verify the vacuum stability for $\mathbb{Z}_{3}$ scalar dark matter.