In this paper, we investigate the Dirichlet type 3 distribution. First, some main properties are elaborated and illustrated. Next, we set forward a representation which allows to compute many functionals in a closed form, making the Dirichlet type 3 distribution an exactly soluble model. Furthermore, we consider the Gibbs version of the Dirichlet type 3 distribution including selection. By using the representation mentioned above, we obtain the moment function of the geometrical average of the random variables according to the new distribution; special types of Bell polynomials are shown to be involved. Finally, we provide a concrete example to illustrate the performance of the Dirichlet type 3 distribution.

We study modules over quandles and classify irreducible quandle modules. The main result of this paper states that there is a correspondence between irreducible modules over a quandle $Q$ and irreducible modules over certain groups: more specifically, irreducible modules over the fundamental group of $Q$ and nontrivial irreducible modules over the associated group $\text{As}(Q)$. As an application, we classify irreducible modules over generalized dihedral quandles, the quandles obtained from generalized dihedral groups, and connected quandles in $S{L}_2({\mathbb{F}}_q)$ where ${\mathbb{F}}_q$ denotes the finite field of $q=p^f$ elements.

In the present paper, we study certain quotients of the étale fundamental group of a hyperbolic curve over a field. We prove that the action of the outer automorphism group of a certain quotient of the étale fundamental group of a hyperbolic curve over an algebraically closed field on its conjugacy classes of open subgroups is faithful. Also, we prove that, if $k$ is either a number field or a $p$-adic local field, then the outer Galois representation associated to a certain quotient of the geometric fundamental group of ${\mathbb{P}}_k^1\backslash \{0,1,\infty \}$ is injective.

For any finite type connected surface $S$, we give an infinite presentation of the fundamental group ${\pi }_1(S,*)$ of $S$ based at an interior point $*\in S$ whose generators are represented by simple loops. When $S$ is non-orientable, we also give an infinite presentation of the subgroup of ${\pi }_1(S,*)$ generated by elements which are represented by simple loops whose regular neighborhoods are annuli.

By making use of our previous result on a localization principle for biholomorphic mappings between equidimensional Fock-Bargmann-Hartogs domains in ${ℂ}^n\times {ℂ}^m$ with $m\ge 2$ and the same technique as in our previous study of the Fock-Bargmann-Hartogs domains in ${ℂ}^n\times ℂ$, in this paper we establish a characterization of biholomorphicity of holomorphic self-mappings of generalized Fock-Bargmann-Hartogs domains. As a special case of this, we obtain the main result of a recent paper by Guo, Feng and Bi.