In this paper, we study the notion of McCoy ring over the class of non-commutative rings of polynomial type known as skew Poincaré–Birkhoff–Witt extensions. As a consequence, we generalize several results about this notion considered in the literature for commutative rings and Ore extensions.

In this paper, we introduce a new lifetime distribution with increasing, decreasing and bathtub-shaped hazard rate function which is constructed by compounding of the Weibull and Chen distributions and is called Weibull–Chen (W–C) distribution. The new distribution is more flexible to model the bathtub-shaped hazard rate data, and its hazard rate function is simple. We study its statistical properties including quantiles, moments, order statistics and Renyi entropy. The estimation of parameters by maximum likelihood method is discussed, and a Monte Carlo simulation study is conducted to investigate the performance of the maximum likelihood estimators. Finally, it is shown that the proposed distribution fits two real lifetime data better than other distributions.

In the present paper, we study the structure of cyclic DNA codes of even length over the ring ${F}_2+u{F}_2+u^2{F}_2$ where $u^3=0$. We investigate two presentations of cyclic codes of even length over ${F}_2+u{F}_2+u^2{F}_2$ satisfying the reverse constraint and the reverse-complement constraint.

Regular vine copula provides rich models for dependence structure modeling. It combines vine structures and families of bivariate copulas to construct a number of multivariate distributions that can model a wide range dependence patterns with different tail dependence for different pairs. Two special cases of regular vine copulas, C-vine and D-vine copulas, have been extensively investigated in the literature. We propose the Python package, pyvine, for modeling, sampling and testing a more generalized regular vine copula (R-vine for short). R-vine modeling algorithm searches for the R-vine structure which maximizes the vine tree dependence in a sequential way. The maximum likelihood estimation algorithm takes the sequential estimations as initial values and uses L-BFGS-B algorithm for the likelihood value optimization. R-vine sampling algorithm traverses all edges of the vine structure from the last tree in a recursive way and generates the marginal samples on each edge according to some nested conditions. Goodness-of-fit testing algorithm first generates Rosenblatt’s transformed data ${\varvec{E}}$ and then tests the hypothesis $H_0^*: {\varvec{E}} \sim C_{\perp }$ by using Anderson–Darling statistic, where $C_{\perp }$ is the independence copula. Bootstrap method is used to compute an adjusted p-value of the empirical distribution of replications of Anderson–Darling statistic. The computing of related functions of copulas such as cumulative distribution functions, H-functions and inverse H-functions often meets with the problem of overflow. We solve this problem by reinvestigating the following six families of bivariate copulas: Normal, Student t, Clayton, Gumbel, Frank and Joe’s copulas. Approximations of the above related functions of copulas are given when the overflow occurs in the computation. All these are implemented in a subpackage bvcopula, in which subroutines are written in Fortran and wrapped into Python and, hence, good performance is guaranteed.

In this paper we mainly investigate the Coleman automorphisms and class-preserving automorphisms of finite AZ-groups and finite groups related to AZ-groups. For example, we first prove that $Out_c(G)$ of an AZ-group G must be a $2'$-group and therefore the normalizer property holds for G. Then we find some classes of finite groups such that the intersection of their outer class-preserving automorphism groups and outer Coleman automorphism groups is $2'$-groups, and therefore, the normalizer property holds for these kinds of finite groups. Finally, we show that the normalizer property holds for the wreath products of AZ-groups by rational permutation groups under some conditions.

In this article, we prove that there exists a unique local smooth solution for the Cauchy problem of the Navier–Stokes–Schrödinger system. Our methods rely upon approximating the system with a sequence of perturbed system and parallel transport and are closer to the one in Ding and Wang (Sci China 44(11):1446–1464, 2001) and McGahagan (Commun Partial Differ Equ 32(1–3):375–400, 2007).