0 and 0, and let a E Z. Prove that [a],m C [a]n if and only if n | Let m,EN with m TT 0 and 0, and let a E Z. Prove that [a],m C [a]n if and only if n | Let m,EN with m TT
1. Prove that there are no Let m, n E Z with m, n > 3 and gcd(m, n) of mn. primitive roots 1. Prove that there are no Let m, n E Z with m, n > 3 and gcd(m, n) of mn. primitive roots
Question 7 (Chapters 6-7) 2+2+2+3+2+4+4-19 mark Let 0メs c Rn and fix r' E S. For a R" consider the following optimization problem: (Pa) min ar res and define the set K(S,) (aER z" is a solution of (Pa)) (e) If z' e int(S), prove that K(S, (0) (1) If possible, find a set S CR" and s* E S such that K(S,) (g) Let SB, 0.1] (rR l2l3 1) (the closed (, unit ball) and consider (1,0)7. Prove that...
5. Let Zli_ {a + bi l a,b E Z. i2--1} be the Gaussian integers. Define a function for all a bi E Zi]. We call N the norm (a) Prove that N is multiplicative. This is, prove that for all a bi, c+di E Z[i] (b) Prove that if a + r є z[i] is a unit of Zli], then Ma + bi)-1. (c) Find all of the units in Zli 5. Let Zli_ {a + bi l a,b...
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n E Z. Find [n]. equivalence relation in [N, the equivalence class of 3. We defined a relation on sets A B. Prove that this relation is an (In this view, countable sets the natural numbers under this equivalence relation). exactly those that are are 2. Consider the relation E on Z defined by...
Question 8 (Chapters 6-7) 12+2+2+3+2+4+4-19 marks] Let 0メS C Rn and fix E S. For a E R consider the following optimization problem: (Pa) min a r, and define the set K(S,x*) := {a E Rn : x. is a solution of (PJ) (a) Prove that K(S,'). Hint: Check 0 (b) Prove that K(S, r*) is a cone. (c) Prove that K(S,) is convex d) Let S C S2 and fix eS. Prove that K(S2, ) cK(S, (e) Ifx. E...
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn by, for all a є z /([a]mn) = ([a]rn , [a]n). (a) Prove that f is well-defined. (b) Let m- 4 and n - 7. Find a Z such that f ([al28) (34,(517). (c) Prove that f is a bijection.2 (HINT: To prove that f is onto, given (bm, [cm) E Zm x Zn, consider z - cmr + bns, where 1 mr +ns.)
2. Let a,b,c E Z. Prove the following. If aſb then g.c.d(b, c) = 1 implies g.c.d(a, c) = 1.
Let p E [0,1] with pメ, and let (Xn)n=o b l e the Markov chain on with initia [0,1] given by distribution δο and transition matrix 11: Z Z ify=x-1 p 0 otherwise. Use the strong law of large numbers to show that each state is transient. Hint: consider another Markov chain with additional structure but with the same distribution and transition matrix Let p E [0,1] with pメ, and let (Xn)n=o b l e the Markov chain on with...
Let F, C R be defined by F.-{x | x 20 and 2-1/n-x2〈 2+1/n). Show that n-&メ2. Use this to show the existence of V2. 18. Let F, C R be defined by F.-{x | x 20 and 2-1/n-x2〈 2+1/n). Show that n-&メ2. Use this to show the existence of V2. 18.