Problem 7. Give an example of a function : (0,00) - R that is in L'((0,1))...
Problem Give an example of a periodic non-constant function (with a period 2π) such that it is not infinity or zero for all x (meaning for all x, f(x) is a finite positive number). If such a function does not exist, explain why you think so.
g) Consider the problem Ou(x, t) = Oxxu(x, t), u(x,0) = Q(x), 0,u(0,1) = 0,1(L,t) = 0, (x, t) (0, L) x (0,00), T ( [0, LG, te [0,00). with a given function 0. Show that the energy L 1 ENE() = 1 u? (x, t)da decays in time.
Question 2.1. . (i) Give an example of a function, f: R R, that is not bounded. (ii) Give an example of a function, f: (1.2) + R, that is not bounded. (iii) Give an example of a function, f: R → R. and a set. S. so that f attains its maximum on S. (iv) Give an example of a function, f: R R , and a set, S, so that f does not attain its maximum on S....
Consider creating an NFA for the following language L = { ww^(R) | w∈{0,1}* }. What problems do you encounter when attempting to create this automaton, why do you encounter them and what does that mean for the existence of this NFA? Please explain your answer in detail.
(6) Pretend you know that the natural logarithm function log : (0,00) + R exists and has the properties you remember, in particular log(24) = x log(2) for all x. Let p > 0. For what values of p does the series » - converge? log(n)P:n Use the Cauchy Condensation Test.
1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx) fn(x) 1+ nx (i) Guess the pointwise limit f of fn on (0,00) and justify your claim. [15 Marks] (ii) Show that fn + f uniformly on ſa, 00), Va > 0. [10 Marks) (iii) Show that fn does not converge uniformly to f on (0,00) [10 Marks] (Hint: Show that ||fr|| 21+(1/2) (b) Prove that a continuous function f defined on a closed...
7. (10%) Let f: [0,1] R be defined by _x xe[0,1]n f(x) 0 otherwise Is fe L[0,1]? If yes, find its Lebesgue integral. i) Is feR[O,1] ? If yes, find its Riemann integral. ii) ii) What is lim || |, ? 7. (10%) Let f: [0,1] R be defined by _x xe[0,1]n f(x) 0 otherwise Is fe L[0,1]? If yes, find its Lebesgue integral. i) Is feR[O,1] ? If yes, find its Riemann integral. ii) ii) What is lim ||...
1 Fix an integer N > 1, and consider the function f : [0,1] - R defined as follows: if 2 € (0,1) and there is an integer n with 1 <n<N such that nx € Z, choose n with this property as small as possible, and set f(x) := otherwise set f(x):= 0. Show that f is integrable, and compute Sf. (Hint: a problem from Homework Set 7 may be very useful for 0 this!)
/ 4Given p > 1, find a function f such that l e L'(0,1) for <p but fe L'(0,1]) for s p. Find another function g such that ge L'(0,1)) for r <p but g&L ([0,1]) for s > p.
L.11) Brand name distributions a) Give an example of a normally distributed random variable. b) Give an example of an exponentially distributed random variable. c) Give an example of a random variable with the Weibull distribution. d) Give an example of a random variable with the Pareto distribution. L.4) Sample means from BinomialDist[1, p] IfXl. X2. X3, Xn are independent random samples from a random variable with the BinomialDist[1, p] distribution, then what normal cumulative distribution function do you use...