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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 and irreducible modules over certain groups: more specifically, irreducible modules over the fundamental group of and nontrivial irreducible modules over the associated group . As an application, we classify irreducible modules over generalized dihedral quandles, the quandles obtained from generalized dihedral groups, and connected quandles in where denotes the ﬁnite ﬁeld of 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 is either a number ﬁeld or a -adic local ﬁeld, then the outer Galois representation associated to a certain quotient of the geometric fundamental group of is injective.
For any ﬁnite type connected surface , we give an inﬁnite presentation of the fundamental group of based at an interior point whose generators are represented by simple loops. When is non-orientable, we also give an inﬁnite presentation of the subgroup of 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 with and the same technique as in our previous study of the Fock-Bargmann-Hartogs domains in , 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.