(7.2:al) Let 10 -9 10 10 Find Condı(A) 10-10 11 10)-find the bound . If A is changed to A' on the...
b) (10 pts) Let D(0, oo)) be the vector space of all bounded continuous functions from [0, oo) such that R If(x) dz 00. Give an example of a sequence {fn} of functions in D(0,00)) which (i) converges pointwise for E [0, oo) to the constant function f(z)0 (ii) does not converge to 0, neither with respect to the norm, nor the Hint: it may be helpful to contemplate the phrase "mass escaping to infinity". norm. b) (10 pts) Let...
1. a) Let X ∼ Exponential(λ). Using Markov’s inequality find an upper bound for P(X ≥ a), where a > 0. Compare the upper bound with the actual value of P(X ≥ a). b) Let X ∼ Exponential(λ). Using Chebyshev’s inequality find an upper bound for P(|X − EX| ≥ b), where b > 0.
Let A e Cpxp,A e C, and let 11-11 a multiplicative norm on Cpxp. Use Theorem 7.27 of your lecture notes to show that if Al > ll All , then 1. XIp A is invertible and Ip Theorem 7.27. If Il is a sub-multiplicative norm on C, then pl 1. Moreover, i X E CPXP and |X|l < 1, then 1. Ip X is invertible. 2. (1,-X)-1-).X' ; i.e., the sum converges j-0 SI Let A e Cpxp,A e...
Let N = a,, l0" + . . . + a2 I 02 + al 10 + ao, where 0 a、 9, be the decimal expan- sion of a positive integer N. (a) Prove that 7, 11, and 13 all divide N if and only if 7, 11, and 13 divide the integer (100a2 + 10a1 +ao) - (100as + 10a4 + a3) +(100as 10a7 + a6) -... M Let N = a,, l0" + . . . + a2...
Please do 10 & 11 Use Intermediate Value Theorem lial p(x) = x4 +7x = 9 has two real root. 8. df Open with Google Docs Then use your calculator to find the ro 9 Let f(z)2with € [0, 00). Find a positive mumber e and two sequences {xn} and {yn} such that lim-(nn) = 0 but |f(xn)- f(Yn)| 2 e. Then conclude that f(x) = x2 is not uniformly continuous on [0, ao) [0, oo). Show that f is...
Part D,E,F,G 10. Let p(x) +1. Let E be the splitting field for p(x) over Q. a. Find the resolvent cubic R(z). b. Prove that R(x) is irreducible over Q. c. Prove that (E:Q) 12 or 24. d. Prove: Gal(E/Q) A4 or S4 e. If p(x) (2+ az+ b)(a2 + cr + d), verify the calculations on page 100 which show that a2 is a root of the cubic polynomial r(x)3-4. 1. f. Prove: r(x) -4z 1 is irreducible in...
Problem 4. (i) Let R> 2/14Z and consider the polynomial ring R[d]. Let A(z) 4 + 2r3 + 3r2 + 4x + 5 and B(x) 37 be elements of R]. Find q(x) and r(x) in R] such that: A(x)-q(z)E(z) + r(z) and deg(r) < 2. (2pts) (ii) Let R- Z/11Z, write down the table of squares in R as follows. For every a E R (there are 11 such elements), find a2. Here you are required to express the final...
Problem 4. (i) Let R> 2/14Z and consider the polynomial ring R[d]. Let A(z) 4 + 2r3 + 3r2 + 4x + 5 and B(x) 37 be elements of R]. Find q(x) and r(x) in R] such that: A(x)-q(z)E(z) + r(z) and deg(r) < 2. (2pts) (ii) Let R- Z/11Z, write down the table of squares in R as follows. For every a E R (there are 11 such elements), find a2. Here you are required to express the final...
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f) = J0 [f(t)]2dt. Prove that Ψ : X → R is differentiable. and find Ψ(f). (7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X,...
4. (a) Let A [0, oo) and let f.g:AR be functions which are continuous at 0 and are such that f(0) 9(0)-1. Show that there exists some δ > 0 such that ifTE 0,d) then (b) Consider the function 0 l if z e R is rational, if zER is irrational f(z) Show that limfr) does not exists for any ceR. 4. (a) Let A [0, oo) and let f.g:AR be functions which are continuous at 0 and are such...