f: Sn->Z where Sn is the set of permutations. f(a.b)=f(a)f(b) for all a,b in Sn, then...
7. (10 points) Let Sym(Z) = \f : Z Z : f bijective) be the set of bijective functions from Z to Z. (Sym(Z),o) is a group, where o denotes the composition of functions. Let g: Z Z be the function 8(n) = {-1 nodd n+1 neven (a) Prove that g € Sym(Z). (b) Find the order of g. Heat: gog - composition of functions
2. (a) Prove the product rule for complex functions. More specifically, if f(z) and g(z) f(z)g(z) is also analytic, and that analytic prove are that h(z) h'(z)f(z)9() f(z)g'(z) (You may use results from the multivariable part of the course without proof.) = nz"- for n e N = {1,2,3,...}. Your textbook establishes that S z"= dz (b) Let Sn be the statement is true. With the help of (a), show that if Sn is true, then Sn+1 is true. Why...
11. (8 marks) Let F(x, y, z) = x'yz, where r, y,z E R and y, z 2 0. Execute the following steps to prove that F(z,y,2) < (y 11(a) Assume each of r, y, z is non-zero and so ryz= s, where s> 0. Prove that 3 F(e.y.) (y)( su, y su, z sw and refer back to Question (Hint: Set 10.) 11(b) Show that if r 0 or y0 or z 0, then F(z, y, z) ( 11(c)...
Let S be the set of all Cauchy sequences (sn) such that sn є Q for all n. Prove that the following is an equivalence relation on the set S: (%) ~ (h) if and only if (sn tn) converges to zero. Let R denote the set of equivalence classes of S under ~
By a, b, f, r, the following permutations of the set {1,2,3,4,5,6} are given. a) Determine the unknown permutations g and h if the equations for g ◦ h = r and h◦a = b apply. b) Find (f ◦g◦h)2 = (f ◦g◦h) ◦ (f ◦g◦h). 142635 b=(425163 r=(314562) 6 4 3 21' = 521), 142635 b=(425163 r=(314562) 6 4 3 21' = 521),
(14.3) Suppose that f()-OP0cman for z E C. Prove that, for all R. where ) n=0 (14.3) Suppose that f()-OP0cman for z E C. Prove that, for all R. where ) n=0
This is all about abstract algebra of permutation group. 3. Consider the following permutations in S 6 5 3 489721)' 18 73 2 6 4 59 (a) Express σ and τ as a product of disjoint cycles. (b) Compute the order of σ and of τ (explaining your calculation). (c) Compute Tơ and στ. (d) Compute sign(a) and sign(T) (explaining your calculation) e) Consider the set Prove that S is a subgroup of the alternating group Ag (f) Prove that...
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if r 0 0 if r <0 θ(z) = 2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if...
Q5. a) Let f(z) be an analytic function on a connected open set D. If there are two constants and C, EC, not all zero, such that cf(z)+ f(2)=0 for all z € D, then show that f(z) is [4] a constant on D. b) Evaluate the contour integral f(z)dz using the parametric representations for C, where f(2)= and the curve C is the right hand half circle 1z| = 2, from z=-2 to z=2i. [4] c) Evaluate the contour...