2. Prove the following: Lemma 1. Consider a function f, defined for all positive integers. Suppose...
Problem: "A function is defined by f(1) = 1 and, for all x ≥ 1, Prove that the range of f is . Provide a clear proof, explaining and justifying all steps taken." HINN $(2x) = f(x) f (2x + 1) = f(1) + f(x+1) We were unable to transcribe this image
I need help to prove iii.
Lemma 8.1 Let n be a positive integer, and let s and t be integers. Then the following hold. (i) We have s et mod n if and only if n dividest - s. (ii) We have pris + t) = Hn (s) +Mn(t) mod n. (iii) We have Hr(st) = Hn (3) Men(t) mod n. Proof. The definition of fr gives us integers u, v such that S = nu + An (8)...
5.7. Let n an E C be a multiplicative function defined by a 1 and ifn-2, pθ and 0 < θ < 1 . Prove that, as x → oo, we have where lEpl ano(a) for some constant r.
5.7. Let n an E C be a multiplicative function defined by a 1 and ifn-2, pθ and 0
6.59. Let f be a continuous function on [a, b]. Suppose that there exists a positive constant K such that If(x) <K for all x in [a, b]. Prove that f(x) = 0 for all x in [a, b]. *ſ isoidi,
For an integer n > 0, consider the positive integer F. = 22 +1. (a) Use induction to prove that F. ends in digit 7 whenever n 2 is an integer (b) Use induction to prove that F= 2 + IT- Fholds for all neN. (c) Use (b) to prove that ged(F, F.) = 1 holds for all distinct nonnegative integers m, na (d) Use (e) to give a quick proof that there must be infinitely many primes! That is...
Prove: If n and a are positive integers and n=(a^2+ 1)/2, then n is the sum of the squares of two consecutive integers (that is, n=k^2+ (k+1)^2 for some integer k).
1 versus H:λ 2. Find a 6. Consider Neyman-Pearson Lemma. Consider testing Ho:λ suitable number k so that this lemma can be applied. Do you see any change in k if we replace 1 and 2 above by 4 and 51 our X is still Poisson from number 5; choose any meaningful alpha for number 6 and do the problem.) (Question 5. For the random variable X following a Poisson distribution with mean 2 Consider testing Ho: λ 1 versus...
Let n be a positive integer, and let s and t be integers. Then
the following hold.
I need the prove for (iii)
Lemma 8.1 Let n be a positive integer, and let s and t be integers. Then the following hold. (i) We have s et mod n if and only if n dividest - s. (ii) We have pris + t) = Hn (s) +Mn(t) mod n. (iii) We have Hr(st) = Hn (3) Men(t) mod n. Proof....
Lemma. If two vector spaces have the same dimension then they are isomorphic Proof. To show that any two spaces of dimension n are isomorphic, we can simply show that any one is isomorphic to R. Then we will have shown that they are isomorphic to each other, by the transitivity of isomorphism (which was established in the first Theorem of this section) Theorem 1 Isomorphism is an equivalence relation among ctor spaces Let v be n--dimensional. Fix a basis...
Applied Mathematics Laplace Transforms
1. Consider a smooth function f(t) defined on 0 t<o, with Laplace transform F(s) (a) Prove the First Shift Theorem, which states that Lfeatf(t)) = F(s-a), where a is a constant. Use the First Shift Theorem to find the inverse trans- form of s2 -6s 12 6 marks (b) Prove the Second Shift Theorem, which states that L{f(t-a)H(t-a))-e-as F(s), where H is the Heaviside step function and a is a positive constant. Use the First and...