分类: 数学 >> 数学(综合) 提交时间: 2021-12-24
摘要: 最近,一种全新的集合被提出,名为随机排列集(Random Permutation Set RPS)。随机排列集考虑了某一集合的所有可能的排列,并可以看作是证据理论的一种推广。不确定性是随机排列集的一个重要特征。一个简单的问题是如何测量随机排列集的不确定度。为了解决这一问题,本文提出了随机排列集的信息熵(Entropy of Random Permutation Set)。当忽略排列事件(Permutation Event)中元素的顺序时,随机排列集的信息熵退化为证据理论中的邓熵。当每个排列事件只包含一个元素时,随机排列集的信息熵退化为概率论中的香农熵。因此,随机排列集的信息熵可以看作是邓熵和香农熵的推广。 本文用数值算例说明了随机排列集的信息熵的有效性。
分类: 数学 >> 统计和概率 提交时间: 2021-12-16
摘要: The paper considers Wasserstein metric between the empirical probability measure of n discrete random variables and a continuous uniform one on the d-dimensional ball and give the asymptotic estimation of their expectation as $n \to \infty$. Further We considers the above problem on a mixed process, i.e., n discrete random variables are produced by the Poisson process.
分类: 数学 >> 应用数学 提交时间: 2021-12-15
摘要: We analyze properties of degree and clustering of a hyperbolic geometric model of complex networks in small parameter case $\tau<1, 2\sigma<1$. We find that the probability of k-degree goes to 0 and the global clustering coefficient goes to 0 in probability too as the number of nodes $N\to\infty$ for some specific growth $R(N)$ of the region radius. Here the scale-free degree is failed and the connection between neighbors are very weak. The transition of properties of the model with the parameter $\sigma$ changes seems to show that the mobility is important to keep society full and stable communication, otherwise a silence society. Some analysis technique and method are first applied for such model.
分类: 计算机科学 >> 计算机科学的集成理论 分类: 数学 >> 数学(综合) 提交时间: 2021-12-14
摘要: 最近,一种名为随机排列集(Random Permutation Set RPS)的集合被提出,它考虑了一个集合中所有元素的排列。 为了度量随机排列集的不确定度,随机排列集的熵被提出。 然而,随机排列集的最大熵原理却没有详细地论证。为了解决这一问题,本文提出了随机排列集的最大熵。随机排列集的最大熵的解析解及其对应的PMF条件得到了证明与讨论。我们用数值例子说明了随机排列集的最大熵的性质。实验结果表明,随机排列集的最大熵和最大邓熵、最大香农熵相互兼容。 当忽略排列事件(Permutation Event)中元素的顺序时,随机排列集的最大熵将退化为最大邓熵。当每个排列事件只包含一个元素时,随机排列集的最大熵将退化为最大香农熵。
分类: 数学 >> 控制和优化 提交时间: 2021-11-29
摘要: 在机器学习和数学优化研究领域, 深度学习优化问题易优性的数学解释极具挑战性. 损失函数存在高维、非凸、不光滑等特质性, 然而也能通过梯度下降法搜索到全局最优值. 损失函数地貌分析成为揭示深度学习优化问题易优性本质的重要研究方向. 为促进可解释、可信的深度学 习在更关键领域的应用, 本文回顾了损失函数地貌特征(局部极小点的数量和空间分布、最优点之间的连通性、临界点的最优性)、梯度下降法收敛性、以及损失函数地貌可视化等方面的研究进展和挑战.
分类: 数学 >> 应用数学 分类: 计算机科学 >> 计算机软件 分类: 信息科学与系统科学 >> 信息与系统科学其他学科 提交时间: 2021-10-11
摘要: 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.
分类: 数学 >> 应用数学 提交时间: 2021-09-22
摘要: 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.
分类: 数学 >> 数学(综合) 提交时间: 2021-08-10
摘要: In this paper, we give some properties related to platform points of a fuzzy set and their applications.
分类: 数学 >> 计算科学理论 提交时间: 2021-07-26
摘要: 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.
分类: 数学 >> 数学物理 提交时间: 2021-06-16
摘要: 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.
分类: 数学 >> 建模与仿真 提交时间: 2021-05-07
摘要: 近年来,灰色系统理论在各个领域中的应用愈发广泛.在灰色预测模型中,GM(1,1)模型最为核心与基础,但该模型对振荡序列难以进行模拟,且振荡序列往往无法通过准指数律检验,这削弱了GM(1,1)模型的应用范围.本文利用GM(1,1)模型对单调序列的模拟,对振荡序列的1-AGO序列建立 GM(1,1)模型.再通过引入修正项的IAGO操作,还原到对原振荡序列的模拟.建立了改进的GM(1,1)模型,弥补了传统GM(1,1)模型对振荡序列模拟的缺陷.
分类: 数学 >> 建模与仿真 提交时间: 2021-04-23
摘要: 灰色系统模型由于其具有简便、数学背景清晰等特点,在数学建模与近似计算等领域都有着十分广泛的应用。灰色系统模型的原理可以概括为利用时间响应函数来模拟数据序列的演化,而其中GM(1,1)模型作为最基础、最能体现灰色建模思想的一个模型,应用最广泛。传统的GM(1,1)模型的时间响应函数是以自然常数 的指数构造的函数,也正是由于这个特点,所以GM(1,1)模型具有较强的局限性,在实际应用中人们往往会将GM(1,1)模型的时间响应函数做一定的修正。其中,最重要的一种修正方法就是将原本的时间响应函数与其他能够描述数据序列性质的函数进行耦合,例如对于具有一定周期性或准周期性的数据序列,就可将原本的时间响应函数与三角函数耦合,构成GM(1,1)-三角函数耦合模型。本文便深入探讨了这种模型的可行性,提出了耦合模型时间响应函数中参数的估计方法,以及误差分析。
分类: 地球科学 >> 地理学 分类: 数学 >> 控制和优化 提交时间: 2021-02-24
摘要: 分区问题广泛应用于地理、经济、政治、商业、公共服务等领域。均等分区问题是其中一类问题,通常要求分区人口数量或任务量均等、几何形状紧凑和空间连续,应用于选区、销售区和巡逻区的划分。本文针对均等分区问题提出了一个混合整型线性规划模型,并设计了一个基于迭代局部搜索(ILS)的混合算法。该算法从三个方面扩展ILS:群解搜索、VND搜索及SPP模型改进。选择5个区域对模型和算法进行测试,结果表明:数学模型能够求解空间单元数量为324的案例;混合算法优化性能优异,鲁棒性强,计算效率较高。所提出的均等分区问题适用于政治分区等经典问题,在新冠疫情应急服务等领域也具有应用潜力。