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/2 for n E N. Use the Monotone Convergent (2) Suppose that o E R and...
a. [8 marks] Recall that f e O(g) if and only if Ice R+, 3no E...
a. [8 marks] Recall that f e O(g) if and only if Ice R+, 3no E N, Vn E N, n > no + f(n) < cg(n). Prove that 5n1.5 + 7n +2 € O(n1.5 log2 n).
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)!...
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
i. (2nd Principle of Induction): Suppose that a1 = 2 and a2 = 4 and for...
i. (2nd Principle of Induction): Suppose that a1 = 2 and a2 = 4 and for n > 2, an = 5an-1 – 6an-2. Prove that for all n e N, an = 2". (This is easy. Show precisely where you need the 2nd Principle.)
5. Prove that U(2") (n > 3) is not cyclic.
5. Prove that U(2") (n > 3) is not cyclic.
Suppose that a sequence {Zn} satisfies Izn+1-Znl < 2-n for all n e N. Prove that...
Suppose that a sequence {Zn} satisfies Izn+1-Znl < 2-n for all n e N. Prove that {z.) is Cauchy. Is this result true under the condition Irn +1-Fml < rt Let xi = 1 and xn +1 = (Zn + 1)/3 for all n e N. Find the first five terms in this sequence. Use induction to show that rn > 1/2 for all n and find the limit N. Prove that this sequence is non-increasing, convergent,
5. Use Rice's Theorem to prove the undecidablity of the following language. P = {< M...
5. Use Rice's Theorem to prove the undecidablity of the following language. P = {< M > M is a TM and 1011 E L(M)}.
(Exponential martingales) Suppose O(t,w) = (01(t, w),...,On(t,w)) E R" with Ox(t,w) E VIO, T] for k...
(Exponential martingales) Suppose O(t,w) = (01(t, w),...,On(t,w)) E R" with Ox(t,w) E VIO, T] for k = 1,..., n, where T < 0o. Define 2. = exp{ jQ1, wydBlo) – 4 640,w.do}osist where B(s) ER" and 62 = 0 . 0 (dot product). a) Use Ito's formula to prove that d24 = 2:0(t,w)dB(t). b) Deduce that 24 is a martingale for t <T, provided that Z40x(t,w) € V[O,T] for 1 sk sn.
| Prove that for n e N, n > 0, we have 1 x 1!+ 2...
| Prove that for n e N, n > 0, we have 1 x 1!+ 2 x 2!+... tnx n! = (n + 1)! - 1.
1. Recall the following theorem. Theorem 1. Let a, b, m,n e N, m, n >...
1. Recall the following theorem. Theorem 1. Let a, b, m,n e N, m, n > 0 and ged(m,n) = 1. There erists a unique r e Zmn such that the following holds. x = a (mod m) x = b (mod n) please show that such solution is unique.
The Lucas numbers L(n) have almost the same definition as the Fibonacci numbers: If n =...
The Lucas numbers L(n) have almost the same definition as the Fibonacci numbers: If n = 1 if n- 2 L(n 1) L(n - 2) if n > 2. 12, as in Theorem 3.6. Prove that L(n)-α, β n for all n E N. Use strong induction Let α = 1 + v/5 and β-- Proof. First, note that and L(2) suppose as inductive hypothesis that L()-α4 β, for all i k, for some k > 2. Then l(k) =...
a. [8 marks] Recall that f e O(g) if and only if Ice R+, 3no E N, Vn E N, n > no + f(n) < cg(n). Prove that 5n1.5 + 7n +2 € O(n1.5 log2 n).
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
i. (2nd Principle of Induction): Suppose that a1 = 2 and a2 = 4 and for n > 2, an = 5an-1 – 6an-2. Prove that for all n e N, an = 2". (This is easy. Show precisely where you need the 2nd Principle.)
5. Prove that U(2") (n > 3) is not cyclic.
Suppose that a sequence {Zn} satisfies Izn+1-Znl < 2-n for all n e N. Prove that {z.) is Cauchy. Is this result true under the condition Irn +1-Fml < rt Let xi = 1 and xn +1 = (Zn + 1)/3 for all n e N. Find the first five terms in this sequence. Use induction to show that rn > 1/2 for all n and find the limit N. Prove that this sequence is non-increasing, convergent,
5. Use Rice's Theorem to prove the undecidablity of the following language. P = {< M > M is a TM and 1011 E L(M)}.
(Exponential martingales) Suppose O(t,w) = (01(t, w),...,On(t,w)) E R" with Ox(t,w) E VIO, T] for k = 1,..., n, where T < 0o. Define 2. = exp{ jQ1, wydBlo) – 4 640,w.do}osist where B(s) ER" and 62 = 0 . 0 (dot product). a) Use Ito's formula to prove that d24 = 2:0(t,w)dB(t). b) Deduce that 24 is a martingale for t <T, provided that Z40x(t,w) € V[O,T] for 1 sk sn.
| Prove that for n e N, n > 0, we have 1 x 1!+ 2 x 2!+... tnx n! = (n + 1)! - 1.
1. Recall the following theorem. Theorem 1. Let a, b, m,n e N, m, n > 0 and ged(m,n) = 1. There erists a unique r e Zmn such that the following holds. x = a (mod m) x = b (mod n) please show that such solution is unique.
The Lucas numbers L(n) have almost the same definition as the Fibonacci numbers: If n = 1 if n- 2 L(n 1) L(n - 2) if n > 2. 12, as in Theorem 3.6. Prove that L(n)-α, β n for all n E N. Use strong induction Let α = 1 + v/5 and β-- Proof. First, note that and L(2) suppose as inductive hypothesis that L()-α4 β, for all i k, for some k > 2. Then l(k) =...
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