In 1975, P. Erd\"{o}s proposed the problem of determining the
maximum number $f(n)$ of edges in a  graph with $n$ vertices in which
any two cycles are of different
 lengths. In this paper, it is proved that $$f(n)\geq n+\frac{107}{3}t+\frac{7}{3}$$
 for $t=1260r+169 \,\ (r\geq 1)$
 and $n \geq \frac{2119}{4}t^{2}+87978t+\frac{15957}{4}$. Consequently,
$\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 +
\frac{7654}{19071}},$   which is better than the previous bounds
$\sqrt 2$ [Y. Shi, Discrete Math. 71(1988), 57-71], $\sqrt {2.4}$
   [C. Lai,  Australas. J. Combin. 27(2003), 101-105].
   The conjecture $\lim_{n \rightarrow \infty} {f(n)-n\over \sqrt n}=\sqrt {2.4}$ is not true.

In 1975, P. Erd\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different  lengths. In this paper, it is proved that $$f(n)\geq n+32t-1$$ for $t=27720r+169 \,\ (r\geq 1)$  and $n\geq\frac{6911}{16}t^{2}+\frac{514441}{8}t-\frac{3309665}{16}$. Consequently, $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + {2562 \over 6911}}.$

设 Zn 是一个独立同分布环境下的上临界分支过程. 本文得到了两个关于 Zn 的非一致性 Berry-Esseen 估计. 该结果把 Grama et al. [Stochastic,Process.,Appl.,127(4),1255-1281,2017] 的 Berry-Esseen 估计推广到了非一致性的情形. 最后, 我们讨论了这些结果在区间估计方面的应用.

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 熵在理论物理学、化学信息学、水文学、环境气象学、生态学、农学、认知神经学、运动神经学、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、社会学、政治学，以及能源工程、土木工程、制造工程、可靠性工程、航空航天、通信工程、测绘工程和金融工程等领域的实际应用。

本文给出Erdos 1975年提出的确定n个顶点没有等长圈的图的最大可能边数f(n)的问题的下界。

This paper discusses the properties the spaces of fuzzy sets in a metric space equipped with the endograph metric and the sendograph metric, respectively. We fist give some relations among the endograph metric, the sendograph metric and the Γ-convergence, and then investigate the level characterizations of the endograph metric and the Γ-convergence. By using the above results, we give some relations among the endograph metric, the sendograph metric, the supremum metric and the dp* metric. On the basis of the above results, we present the characterizations of total boundedness, relative compactness and compactness in the space of compact positive α-cuts fuzzy sets equipped with the endograph metric, and in the space of compact support fuzzy sets equipped with the sendograph metric, respectively. Furthermore, we give completions of these metric spaces, respectively.

In this paper, we discuss the properties of  the spaces of fuzzy sets in a metric space with $L_p$-type $d_p$ metrics, $p\geq 1$. Firstly, we give the characterizations of compactness in fuzzy set space with $d_p$ metrics. Then we present the completions of fuzzy set spaces with $d_p$ metrics.
The $d_p$ metrics are well-defined if and only if a function induced by the Hausdorff metric is measurable. In this paper, we give some fundamental conclusions on the measurability of this function. This paper was submitted to Fuzzy Sets and Systems on 2022.05.06.

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

可分集 (PS) 与几乎可分集 (APS) 是组合设计理论中两类重要的组合构型, 与许多其它组合结构具有密切联系, 例如 Z-循环惠斯特竞赛图, 循环差阵, 不含邻点的循环平衡样本设计, 不交差族及光正交码等. 由于可分集与几乎可分集的要求比较严苛, 其存在性问题迄今远未解决. 本文针对 p≡7 (mod 8) 为素数的情形, 建立p² 阶可分集与 p 阶几乎可分集的新构造方法, 给出两类组合构型存在性的若干新结果. 特别地, 对 p≡7 (mod 8) 的素数 p, 本文确定 p<30000的绝大部分p² 阶PS的存在性, 给出特定条件下 p 阶APS的存在性和渐近存在性, 并得到 p<50000 除去16个可能例外的 p 阶APS的存在性.

In this paper, we propose and study a novel continuous-time model, based on the well-known constant elasticity of variance (CEV) model, to describe the asset price process. The basic idea is that the volatility elasticity of the CEV model can not be treated as a constant from the perspective of stochastic analysis. To address this issue, we deduce the price process of assets from the perspective of volatility elasticity,  propose the constant volatility elasticity (CVE) model, and further derive a more general variable volatility elasticity (VVE) model. Moreover, our model can describe the positive correlation between volatility and asset prices existing in the commodity markets, while CEV model can only describe the negative correlation. Through the empirical research on the financial market, many assets, especially commodities, often show this positive correlation phenomenon in some time periods, which shows that our model has strong practical application value. Finally, we provide the explicit pricing formula of European options based on our model. This formula has an elegant form convenient to calculate, which is similarly to the renowned Black-Scholes formula and of great significance to the research of derivatives market.