Consider n nodes $\{X_i\}_{1 \le i \le n}$ independently and identically distributed (i.i.d.) across N cities located within the unit square S. Each city is modelled as an $r_n \times r_n$ square, and $\mathrm{{MSTC}}_n$ denotes the weighted length of the minimum spanning tree containing all the n nodes, where the edge length between nodes $X_i$ and $X_j$ is weighted by a factor that depends on the individual locations of $X_i$ and $X_j.$ We use approximation methods to obtain variance estimates for $\mathrm{{MSTC}}_n$ and prove that if the cities are well connected in a certain sense, then $\mathrm{{MSTC}}_n$ appropriately centred and scaled converges to zero in probability. Using the above proof techniques we also study $\mathrm{{MST}}_n,$ the length of the minimum weighted spanning tree for nodes distributed throughout the unit square S with location-dependent edge weights. In this case, the variance of $\mathrm{{MST}}_n$ grows at most as a power of the logarithm of n and we use a subsequence argument to get almost sure convergence of $\mathrm{{MST}}_n,$ appropriately centred and scaled.

Based on the improvement in establishing the relations of data, this study proposes a new fuzzy time series model. In this model, the suitable number of fuzzy sets and their specific elements are determined automatically. In addition, using the percentage variations of series between consecutive periods of time, we build the fuzzy function. Incorporating all these improvements, we have a new fuzzy time series model that is better than many existing ones through the well-known data sets. The calculation of the proposed model can be performed conveniently and efficiently by a MATLAB procedure . The proposed model is also used in forecasting for an urgent problem in Vietnam. This application also shows the advantages of the proposed model and illustrates its effectiveness in practical application.

The purpose of this paper is to define a new symbol class $\Lambda $ and discuss the theory of two different pseudo-differential operators (p.d.o.) involving Fourier–Jacobi transform associated with a single symbol in $\Lambda $. We also derive boundedness results for p.d.o.’s in Sobolev type space. A new pseudo-differential operator is developed using the product of symbols. Finally, norm inequality for commutators between two pseudo-differential operators is obtained.

One of the important issues in order to survey multivariate distribution or model dependency structure between interested variables is finding the proper copula function. Extensive studies have been done based on Akaike information criterion (AIC), copula information criterion (CIC), and pseudo-likelihood ratio and fitness test of the copula function. The previous methods of selecting copula functions when the sample size is too small are not satisfactory. Therefore, our method in this paper is based on tracking interval for the parametric copula function which is obtained using expected Kullback–Leibler risk between the two proposed non-nested parametric copula model. It can be find that optimal parametric copula between proposed copula functions in a good level of significance. Finally, efficiency and capability of our method using simulation and applied example have been shown.

In this paper, we first obtain the existence and uniqueness of solution u of elliptic equation associated with Brownian motion with singular drift. We then use the regularity of the weak solution u and the Zvonkin-type transformation to show that there is a unique weak solution to a stochastic differential equation when the drift is a measurable function.

Multiple testing has gained much attention in high-dimensional statistical theory and applications, and the problem of variable selection can be regarded as a generalization of the multiple testing. It is aiming to select the important variables among many variables. Performing variable selection in high-dimensional linear models with measurement errors is challenging. Both the influence of high-dimensional parameters and measurement errors need to be considered to avoid severely biases. We consider the problem of variable selection in error-in-variables and introduce the DCoCoLasso-FDP procedure, a new variable selection method. By constructing the consistent estimator of false discovery proportion (FDP) and false discovery rate (FDR), our method can prioritize the important variables and control FDP and FDR at a specifical level in error-in-variables models. An extensive simulation study is conducted to compare DCoCoLasso-FDP procedure with existing methods in various settings, and numerical results are provided to present the efficiency of our method.

Let $\sigma =\{\sigma _i |i\in I\}$ be some partition of all primes ${\mathbb {P}}$ and G a finite group. A subgroup H of G is said to be $\sigma $-subnormal in G if there exists a subgroup chain $H=H_0\le H_1\le \cdots \le H_n=G$ such that either $H_{i-1}$ is normal in $H_i$ or $H_i/(H_{i-1})_{H_i}$ is a finite $\sigma _j$-group for some $j \in I$ for $i = 1, \ldots , n$. We call a finite group G a $T_{\sigma }$-group if every $\sigma $-subnormal subgroup is normal in G. In this paper, we analyse the structure of the $T_{\sigma }$-groups and give some characterisations of the $T_{\sigma }$-groups.

In the past ten years, deep learning technology has achieved a great success in many fields, like computer vision and speech recognition. Recently, large-scale geometry data become more and more available, and the learned geometry priors have been successfully applied to 3D computer vision and computer graphics fields. Different from the regular representation of images, surface meshes have irregular structures with different vertex numbers and topologies. Therefore, the traditional convolution neural networks used for images cannot be directly used to handle surface meshes, and thus, many methods have been proposed to solve this problem. In this paper, we provide a comprehensive survey of existing geometric deep learning methods for mesh processing. We first introduce the relevant knowledge and theoretical background of geometric deep learning and some basic mesh data knowledge, including some commonly used mesh datasets. Then, we review various deep learning models for mesh data with two different types: graph-based methods and mesh structure-based methods. We also review the deep learning-based applications for mesh data. In the final, we give some potential research directions in this field.