Let z, y > 0. Prove that for any ε > 0 and p, q > 1 so that = 1 we have q + is 1s known as Young'...
Let P, Q ∈ Z[x]. Prove that P and Q are relatively prime in Q[x] if and only if the ideal (P, Q) of Z[x] generated by P and Q contains a non-zero integer (i.e. Z ∩ (P, Q) ̸= {0}). Here (P, Q) is the smallest ideal of Z[x] containing P and Q, (P, Q) := {αP + βQ|α, β ∈ Z[x]}. (iii) For which primes p and which integers n ≥ 1 is the polynomial xn − p...
real analysis proof Let 1 3 p S q and suppose that for 1 Kjs n we have j2 0. Prove that 1 1 np i-1 Let 1 3 p S q and suppose that for 1 Kjs n we have j2 0. Prove that 1 1 np i-1
Let 1 ≤ p ≤ q and suppose that for 1 ≤ j ≤ n we have xj ≥ 0. Prove that ( n ∑ j=1 (xj^q) )^ (1/q) ≤ ( n ∑ j=1 (xj^p) )^ (1/p) ≤ n^ (1/p - 1/q) ( n ∑ j=1 (xj^q) )^ (1/q)
Advanced Calculus (3) Let the function f(x) 0 if x Z, but for n e z we have f(n) . Prove that for any interval [a3] the function f is integrable and Ja far-б. Hint: let k be the number of integers in the interval. You can either induct on k or prove integrability directly from the definition or the box-sum criterion. (3) Let the function f(x) 0 if x Z, but for n e z we have f(n) ....
(P(x),Q(y), R(z)), where P depends only 2. Let S be any surface with boundary curve C, and let F(x,y, z) on r, where Q depends only on y, and where R depends only on z. Show that F.dr 0 C (P(x),Q(y), R(z)), where P depends only 2. Let S be any surface with boundary curve C, and let F(x,y, z) on r, where Q depends only on y, and where R depends only on z. Show that F.dr 0 C
Number Theory 13 and 14 please! 13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
6. (Bonus question: extra marks) Here we will show that ||p difficult property is the triangle inequality: ||la +b||p < || ||, + ||b||p . Here are the main steps: ( )P is a norm over J R" for p > 1.The only k-1 1 + q for a, b0 and 1 1. Prove Young's inequality: ab < b 19)/for any ak, bk E R and 2. Prove Holder's inequality: 1 lab (=1 |akP)"/? TI 1. You + k=1 can...
Prove that if X 20, Y 2 0 and 0 p1, then E(X +Y)] Show that for any real numbers x > 0 and y > 0, E(X)E(YP). HINT: Here is how you can show the above formula holds. Start off by letting 0y. Use the fact that the function g(z) - z is concave-down (i.e., "spills water") on (0, oo) and is thus bounded above by its tangent line at any particular point. Find the tanget line at the...
and Y ~ Geometric - 4 Let X ~ Geometric We assume that the random variables X and Y are statistically independent. Answer the following questions: a (3 marks) For all x E 10,1,2,...^, show that 2+1 P(X>x) P(x (3 = Similarly, for all y [0,1,2,...^, show that Show your working only for one of the two identities that are pre- sented above. Hint: You may use the following identity without proving it. For any non-negative integer (, we have:...
Abstract Algebra; Please write nice and clear. If we wanted to use the definition of isomorphism to prove that Z is not isomorphic to Q, we would have to show that there does not exist an isomorphism p : Z Q. In other words, we would have to show that every function that we could possibly define from Z to Qwould violate at least one of the conditions that define isomorphisms. To show this directly seems daunting, if not impossible....