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- Draw the graph of the following map: xn+1= {3xn if x<1/3 {3(xn- 1/3)/2 if x>1/3 Show also that the periodic orbits are unstable and dense in the unit interval.How do you interprete the main theorem of Galois Thoery in terms of subgroup and subfield diagrams?Prove that the mapping from R under addition to SL(2,R) that takes x to [ cos x sin x -sin x cos x] is a group homomorphism. Find the kernel.
- (Show your work) compute the girth of all generalized Coxeter graphs with parameter Pn,u,v where n is less or equal to121. Let O1 and O2 be topologies on X. (1) Show that the identity map idX : (X, O1) → (X, O2) is continuous if O2 ⊂ O1. (2) Show that the identity map idX : (X, O1) → (X, O2) is not continuous if O1⊂≠O2.Let G = <x, y | x = (xy)3, y = (xy)4>. To what familiar group is Gisomorphic?
- Why is inversion [that is, Φ(x, y) = (-x, -y)] not listed as one ofthe four kinds of isometries in R2?Given the tent map T(x) = {2x for x<= 1/22-2x for x> 1/2}Prove that the set of all periodic points of T is dense in [0, 1] and determine the number of points with least periods and their distinct orbits.Consider the group G = ℚ* × ℤ with operation * on G that can be expressed as: (w, x) * (y, z) = (wy + 1, xz - 1), for all (w, x), (y, z) ∈ ℚ* × ℤ. Is the group <G, *> abelian?