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A large amount of abandoned Pb–Zn tailings piled in the open air is one of the major sources of heavy metal pollution in the surrounding environment because acid rains can create favorable conditions for the leaching of heavy metals (HMs). In this study, the leaching characteristics of HMs from Pb–Zn mine tailings were studied for 150 days by dynamic leaching experiments, and their remediation was performed using three montmorillonite species. The dynamic leaching experiments have shown that HMs (Zn, Mn, As, Pb) can be continuously released from tailings, and the maximum leaching amount of the selected HMs was arranged in the order Zn (14.36 mg/kg) > Mn (12.59 mg/kg) > As (4.05 mg/kg) > Pb (1.81 mg/kg) on the 150th day. The remediation results of HMs showed that natural Na- and Ca-montmorillonite materials exhibit a certain immobilization effect on HMs. The immobilization performance of Ca-montmorillonite could be improved significantly by a simple Na modification treatment, yielding an improvement of 27.27, 10.53, 18.19, and 30.91% for As, Mn, Pb, and Zn, respectively. The immobilization of HMs by montmorillonite was mainly achieved by cation exchange, complexation, and precipitation. The obtained results demonstrate the significant potential of artificial Na-modified montmorillonite for the remediation of HMs against acid rains because of its low cost and high availability. Highlights: • Dynamics column was used to study the release of heavy metals for 150 days. • Heavy metals can be continuously released from tailings. • Lower pH of simulated acid rain could promote release of heavy metals. • Sodium modification can optimize the persistence immobilization of montmorillonite. • Cation exchange, complexation, and precipitation are the immobilization mechanisms. [ABSTRACT FROM AUTHOR]

By use of H. C. Wang's bound on the radius of a ball embedded in the fundamental domain of a lattice of a semisimple Lie group, we construct an explicit lower bound for the volume of a quaternionic hyperbolic orbifold that depends only on dimension. [ABSTRACT FROM AUTHOR]

An important problem in quaternionic hyperbolic geometry is to classify ordered m-tuples of pairwise distinct points in the closure of quaternionic hyperbolic n-space, $$\overline{\mathbf{H}_{\mathbb H}^n}$$ , up to congruence in the holomorphic isometry group $$\mathrm{PSp}(n,1)$$ of $$\mathbf{H}_{\mathbb H}^n$$ . In this paper we concentrate on two cases: $$m=3$$ in $$\overline{\mathbf{H}_{\mathbb H}^n}$$ and $$m=4$$ on $$\partial \mathbf{H}_{\mathbb H}^n$$ for $$n\ge 2$$ . New geometric invariants and several distance formulas in quaternionic hyperbolic geometry are introduced and studied for this problem. The congruence classes are completely described by quaternionic Cartan's angular invariants and the distances between some geometric objects for the first case. The moduli space is constructed for the second case. [ABSTRACT FROM AUTHOR]

Jørgensen's inequality gives a necessary condition for a nonelementary two generator subgroup of $SL(2, {\mathbb C})$ to be discrete. By embedding $SL(2,{\mathbb C})$ into $\hat U(1,1; {\mathbb H})$, we obtain a new type of Jørgensen's inequality, which is in terms of the coefficients of involved isometries. We provide an example to show that this result gives an improvement over the classical Jørgensen's inequality. [ABSTRACT FROM AUTHOR]

Let $${{\bf H}^{\bf 2}_{\mathbb F}}$$ denote the two dimensional hyperbolic space over $${\mathbb F}$$ , where $${\mathbb F}$$ is either the complex numbers $${\mathbb C}$$ or the quaternions $${\mathbb H}$$ . It is of interest to characterize algebraically the dynamical types of isometries of $${{\bf H}^{\bf 2}_{\mathbb F}}$$ . For $${\mathbb F=\mathbb C}$$ , such a characterization is known from the work of Giraud-Goldman. In this paper, we offer an algebraic characterization of isometries of $${{\bf H}^{\bf 2}_{\mathbb H}}$$ . Our result restricts to the case $${\mathbb F=\mathbb C}$$ and provides another characterization of the isometries of $${{\bf H}^{\bf 2}_{\mathbb C}}$$ , which is different from the characterization due to Giraud-Goldman. Two elements in a group G are said to be in the same z-class if their centralizers are conjugate in G. The z-classes provide a finite partition of the isometry group. In this paper, we describe the centralizers of isometries of $${{\bf H}^{\bf 2}_{\mathbb F}}$$ and determine the z-classes. [ABSTRACT FROM AUTHOR]

In this paper, three new discreteness criteria for Möbius groups acting on are obtained; they are generalizations of known results using the information of two-generator subgroups. [ABSTRACT FROM AUTHOR]

This paper discusses the solvability of quaternion matrix equation A*X*B* ± B X A= D and obtains it’s general explicit solutions in terms of A,B,D and their Moore-Penrose inverses. [ABSTRACT FROM AUTHOR]