Subjects: Mathematics >> Mathematics (General) submitted time 2021-12-24
Abstract: Recently, a new kind of set, named as Random Permutation Set (RPS), is proposed. RPS takes the permutation of a certain set into consideration, which can be regarded as a generalization of evidence theory. Uncertainty is an important feature of RPS. A straightforward question is how to measure the uncertainty of RPS. To address this problem, the entropy of RPS is presented in this paper. When the order of elements in permutation event is ignored, the proposed RPS entropy degenerates into Deng entropy in evidence theory. When each permutation event is limited to containing just one element, the proposed RPS entropy degenerates into Shannon entropy in probability theory. Hence the RPS entropy can be regarded as the generalization of Deng entropy and Shannon entropy. Numerical examples are illustrated the efficiency of the proposed RPS entropy.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Statistics and Probability submitted time 2021-12-16
Abstract: 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Applied Mathematics submitted time 2021-12-15
Abstract: 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.
Peer Review Status:Awaiting Review
Subjects: Computer Science >> Integration Theory of Computer Science Subjects: Mathematics >> Mathematics (General) submitted time 2021-12-14
Abstract: Recently, a new type of set, named as random permutation set (RPS), is proposed by considering all the permutations of elements in a certain set. For measuring the uncertainty of RPS, the entropy of RPS is presented. However, the maximum entropy principle of RPS entropy has not been discussed. To address this issue, in this paper, the maximum entropy of RPS is presented. The analytical solution for maximum entropy of RPS and its corresponding PMF condition are respectively proofed and discussed. Numerical examples are used to illustrate the maximum entropy RPS. The results show that the maximum entropy RPS is compatible with the maximum Deng entropy and the maximum Shannon entropy. When the order of the element in the permutation event is ignored, the maximum entropy of RPS will degenerate into the maximum Deng entropy. When each permutation event is limited to containing just one element, the maximum entropy of RPS will degenerate into the maximum Shannon entropy.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. submitted time 2021-11-29
Abstract: In the filed of machine learning and mathematical optimization, it is a challenge to mathematically explain optimality of loss function for deep learning. Loss function is high-dimensional, non-convex, and non-smooth. It was, however, observed that gradient descent could reach zero training loss of this highly non-convex function. Loss landscape analysis is critical to reveal reasons why deep networks are easily optimizable. We reviewed the advance on loss landscape analysis, such as landscape features (number and spatial distribution of local minima, connectivity between global optima, and global optimality of critical points), convergence of gradient descent, and visualization of loss landscape. This survey aimed to promote interpretable and reliable deep learning in critical applications. "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Applied Mathematics Subjects: Computer Science >> Computer Software Subjects: Information Science and Systems Science >> Other Disciplines of Information Science and Systems Science submitted time 2021-10-11
Abstract: "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Applied Mathematics submitted time 2021-09-22
Abstract: 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. " " "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics (General) submitted time 2021-08-10
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Peer Review Status:Awaiting Review
Subjects: Mathematics >> Theoretical Computer Science submitted time 2021-07-26
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Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematical Physics submitted time 2021-06-16
Abstract: 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. "
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Modeling and Simulation submitted time 2021-05-07
Abstract: In recent years, Grey system theory has been widely used in various fields. Among the Grey Prediction Models, the GM (1,1) model is the core and foundation. However, due to the exponential time response sequence, GM (1,1) model is difficult to simulate the oscillation sequence, and the oscillation sequence can not pass the GM (1,1) pre modeling test. These factors weaken the application of GM (1,1) model. The paper establishes GM (1,1) model for the 1-AGO sequence of oscillation sequence, by using the advantage of monotone sequence simulation of GM (1,1) model. Then, the IAGO operation with correction term is introduced to restore the simulation of oscillation sequence. Finally, an improved GM (1,1) model is established to make up for the defect of the traditional GM (1,1) model in the simulation of oscillation sequence.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Modeling and Simulation submitted time 2021-04-23
Abstract: Grey system model is widely used in mathematical modeling and approximate calculation because of its simplicity and clear mathematical background. The principle of grey system model can be summarized as the use of time response function to simulate the evolution of data series. Among them, GM (1,1) model is the most basic and can reflect the idea of grey modeling most widely. The time response function of the traditional GM (1,1) model is constructed by the exponent of natural constant . Also, for this characteristic, the GM (1,1) model has strong limitations. In practical application, people often modify the time response function of GM (1,1) model. One of the most important correction methods is to couple the original time response function with other functions that can describe the properties of data series. For example, for data series with certain periodicity or quasi periodicity, the original time response function can be coupled with trigonometric function to form GM (1,1) - trigonometric function coupling model. In this paper, the feasibility of this model is deeply discussed, and the estimation method of the parameters in the time response function of the coupled model and the error analysis are proposed.
Peer Review Status:Awaiting Review
Subjects: Geosciences >> Geography Subjects: Mathematics >> Control and Optimization. submitted time 2021-02-24
Abstract: Districting problems have been widely applied in geography, economics, environmental science, politics, business, public service and many other areas. The equal districting problem (EDP) arises in applications such as political redistricting, police patrol area delineation, sales territory design and some service area design. The important criteria for these problems are district equality, contiguity and compactness. A mixed integer linear programming (MILP) model and a hybrid algorithm are proposed for the EDP. The hybrid algorithm is designed by extending iterative local search (ILS) algorithm with three schemes: population-based ILS, variable neighborhood descent (VND) local search, and set partitioning. The performance of the algorithm was tested on five areas. Experimentation showed that the instances could be solved effectively and efficiently. The potential applications of the EDP in emergency services are also discussed.
Peer Review Status:Awaiting Review