Browsing by Author "Boadi, Richard Kena"

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    New Cone Metrics on the Sphere
    (2011-06-13) Boadi, Richard Kena
    We give an explicit construction of lattices in P U (1, 2). A family of these lattices was originally constructed by Livn´e [15]. Parker [19] constructed these lattices of Livn´e as the modular group of certain Euclidean cone metrics on the sphere. In this work we give a construction of these lattices which includes that of Parker’s as the modular group of certain Euclidean cone metrics on the sphere. Our cone metrics on the sphere had five cone points with cone angles (π − θ + 2φ, π + θ, π + θ, π + θ, 2π − 2θ − 2φ) Where θ > 0, φ > 0 and θ + φ < π. These corresponds to a group of five tuples lattices generated by Thurston [27] in his paper Shapes of Polyhedra and Triangulations of the Sphere . Hence our choice of θ and φ in order to obtain discreteness are as follows: θ 2π/3 2π/3 2π/3 2π/4 2π/4 (2π/5) 2π/5 2π/6 φ π/4 π/5 π/6 π/3 π/4 (2π/5) π/3 π/3 Certain automorphisms which we considered on our cone metrics yielded unitary matrices R1, R2 and I1. Using these matrices, we obtained our fundamental polyhe- dron D by constructing our vertices, edges and faces to define the polyhedron. Our vertices were obtained by the degeneration of certain cone metrics. The polyhedron D is contained in bisectors whose intersection give us the edges of the polyheron. The faces are also contained in the bisectors. Then finally we proved using Poincar´e’s polyhedron theorem that the group Γ generated by the side pairings of D is a dis- crete subgroup of P U (1, 2) with fundamental domain D and presentation: J 3 = Rp = Rp = (P −1J )k = I , \ 1 2 Γ = J, P, R1, R2 : R2 = P R1P −1 = J R1J −1, P = R1R2
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    New cone metrics on the sphere
    (2011-10-05) Boadi, Richard Kena
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    On the dynamics of the Tent function-Phase diagrams
    (Modem Science Publishers, 2016-05) Kwabi P. A.; Obeng Denteh, William; Boadi, Richard Kena; Ayekple, Yao Elikem
    This paper focuses on the study of a one dimensional topological dynamical system, the tent function. We give a good background to the theory of dynamical systems while establishing the important asymptotic properties of topological dynamical systems that distinguishes it from other fields by way of an example - the tent function. A precise definition of the tent function is given and iterates are clearly shown using the phase diagrams. By this same method, chaos in the tent map is shown as iterates become higher. We also show that the tent map has infinitely many chaotic orbits and also express some important features of chaos such as topological transitivity, boundedness and sensitivity to change in initial conditions from a topological viewpoint.