The aim of this paper is to investigate different radicals (Wedderburn radical, lower nil radical, Levitzky radical, upper nil radical, the set of all nilpotent elements, the sum of all nil left ideals) of the noncommutative rings known as skew Poincaré–Birkhoff–Witt extensions. We characterize minimal prime ideals of these rings and prove that the Köthe’s conjecture holds for these extensions. Finally, we establish the transfer of several ring-theoretical properties (reduced, symmetric, reversible, 2-primal) from the coefficients ring of a skew PBW extension to the extension itself.

An M-estimation of the parameters in an undamped exponential signal model was proposed in Wu and Tam (IEEE Trans Signal Process 49(2):373–380, 2001), and the estimation was shown to be consistent under mild assumptions. In this paper, the limiting distributions of the M-estimators are investigated. It is shown that they are asymptotically normally distributed under similar conditions as assumed in Wu and Tam (IEEE Trans Signal Process 49(2):373–380, 2001). In addition, a recursive algorithm for computing the M-estimators of frequencies is proposed, and the strong consistency of these estimators is established. Monte Carlo simulation studies using Huber’s $\rho $ function are also provided.

In this paper, we consider Marshall and Olkin’s family of distributions. The parent (baseline) distribution is taken to be a scaled family of distributions. Two models: (i) modified proportional hazard rate scale and (ii) modified proportional reversed hazard rate scale, are considered. Some stochastic comparison results in terms of the usual stochastic, hazard rate and reversed hazard rate orders are studied to compare order statistics formed from two sets of independent observations following these models. Most of the sufficient conditions are obtained depending on various majorization-type partial orderings. Further, the setup with multiple-outlier model is taken. Various stochastic orders between the smallest and largest order statistics are developed. Several numerical examples are provided to illustrate the effectiveness of the established theoretical results.

In this work we generalise various recent results on the evolution and monotonicity of the eigenvalues of certain family of geometric operators under some geometric flows. In an attempt to understand the arising similarities we formulate two conjectures on the monotonicity of the eigenvalues of Schrödinger operators under geometric flows. We also pose three questions which we consider to be of a general interest.

We introduce the notion of symmetric covariation, which is a new measure of dependence between two components of a symmetric $\alpha $-stable random vector, where the stability parameter $\alpha $ measures the heavy-tailedness of its distribution. Unlike covariation that exists only when $\alpha \in (1,2]$, symmetric covariation is well defined for all $\alpha \in (0,2]$. We show that symmetric covariation can be defined using the proposed generalized fractional derivative, which has broader usages than those involved in this work. Several properties of symmetric covariation have been derived. These are either similar to or more general than those of the covariance functions in the Gaussian case. The main contribution of this framework is the representation of the characteristic function of bivariate symmetric $\alpha $-stable distribution via convergent series based on a sequence of symmetric covariations. This series representation extends the one of bivariate Gaussian.

A finite non-abelian group G is called metahamiltonian if every subgroup of G is either abelian or normal in G. If G is non-nilpotent, then the structure of G has been determined. If G is nilpotent, then the structure of G is determined by the structure of its Sylow subgroups. However, the classification of finite metahamiltonian p-groups is an unsolved problem. In this paper, finite metahamiltonian p-groups are completely classified up to isomorphism.