Homework Answers
4.a) True
b. ) False
c) True
d) False
e) True
f) True
6) T(N) =O(f(N) +g(N)) which implies T(N)<=c*f(N) +c*g(N) for some positive constant c>0.......(i)
As g(N)=o(f(N)) which implies g(N) < k*f(N) for any positive integer k >0...........(ii)
So T(N) <= c* f(N) + c*k*f(N) [from i and ii ]
Hence T(N)<= r* f(N) where r =c+c*k
Hence T(N)=O(f(N)). [proved]
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4. True or false? (+1 for each correct answer, -1 for each incorrect answer, 0 for...
4. Let fín) and g(n) be asymptotically positive functions. Prove each of the following statements...
4. Let fín) and g(n) be asymptotically positive functions. Prove each of the following statements A. fin)-O(g(n)) if and only if fin) *gn)g(n)) B. fn) - Og(n if and only if fin)2- O(g(n)?)
4. Let fín) and g(n) be asymptotically positive functions. Prove each of the following statements A. fin)-O(g(n)) if and only if fin) *gn)g(n)) B. fn) - Og(n if and only if fin)2- O(g(n)?)
6) This question is part of Exercise 3-4 on page 62 of CLRS, but the letters...
6) This question is part of Exercise 3-4 on page 62 of CLRS, but the letters aren't all the same. Answer TRUE or FALSE for each of the following statements. TRUE means that the statement is TRUE for any asymptotically positive functions f(n) and g(n). Otherwise, answer FALSE. You don't have to Prove or Disprove these statements... but you should learn how to do that. f(n)- O( g(n)) implies g(n) O( f(n)) b) This is part a in Exercise 3-4....
1. Answer each of the following statements as true, false, or unknown. a. The set of...
1. Answer each of the following statements as true, false, or unknown. a. The set of nonnegative even integers is well ordered. b. The sequence of Mersenne numbers forms a geometric progression. c. The sequence {na +1} contains infinitely many primes. d. The sequence {n" +1}.contains infinitely many composites. D) - logo) e. The Prime Number Theorem implies that lim ++00 f. There exist infinitely many pairs of primes that differ by less than 300. g. The number V110520191105201911052019 is...
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is...
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is differentiable on (0,1) and f(x) < 1 for all x € (0,1) and n e N. (a) If (fn (0))nen converges to a number A, prove that lim sup|fn(x) = 1+|A| for all x € [0, 1]. n-too : (b) Suppose that (fr) converges uniformly on [0, 1] to a function F : [0, 1] + R. Is F necessarily differentiable on (0,1)? If...
Please answer it step by step and Question 2. uniformly converge is defined by *f=0* clear...
Please answer it step by step and Question 2. uniformly
converge is defined by *f=0* clear handwritten,
please, also, beware that for the x you have 2 conditions , such as
x>n and 0<=x<=n
1- For all n > 1 define fn: [0, 1] → R as follows: (i if n!x is an integer 10 otherwise Prove that fn + f pointwise where f:[0,1] → R is defined by ſo if x is irrational f(x) = 3 11 if x...
Problem I (10 points) Determine whether the following statements are True or False. Please circle your...
Problem I (10 points) Determine whether the following statements are True or False. Please circle your answer. You don't need to justify. (1) (T or F) Poisson processes are the only type of stochastic processes which have independent increment property. (2) (T or F) Let X; ~ Exp(1), 1 <i<n, be iid random variables. Then X1 +...+ Xn ~ Exp(nl). (3) (T or F) Any Brownian motion satisfies the Markov property. (4) (Tor F) Let S = X1 + X2...
Part I. (30 pts) (10 pts) Let fin) and g(n) be asymptotically positive functions. Prove or...
Part I. (30 pts) (10 pts) Let fin) and g(n) be asymptotically positive functions. Prove or disprove each of the following statements T a、 f(n) + g(n)=0(max(f(n), g(n))) 1. b. f(n) = 0(g(n)) implies g(n) = Ω(f(n)) T rc. f(n)- o F d. f(n) o(f(n)) 0(f (n)) f(n)=6((f(n))2)
1. Answer True or False, and give a brief justification for each answer: a) If lim...
1. Answer True or False, and give a brief justification for each answer: a) If lim 2 = 5 then the series i converges to 5. b) If = 5 then lime = 5. c) If S. and lim.- S.-5, then 10 -5. d) The series 5-5+5-5+... is divergent. e) If = 0 = 5 and the = 5, then 20 - 5 f) The Divergence Test can be used to prove a series is convergent.
10. (18 points total: 3 points for each correct answer, 0 points for incorrect answers or...
10. (18 points total: 3 points for each correct answer, 0 points for incorrect answers or no answer Answer "True" or "False” for each of the following: (i) If 8,9:R + R and are both continuous at a number c, then the composition function fog is continuous at c. (ii) If functions hi, h2: R + R and are both uniformly continuous on a non-empty set of real numbers E, then the product h h2 is uniformly continuous on E....
QUESTION G ONWARDS 13.2 State whether each of the following is true or false. If false,...
QUESTION G ONWARDS
13.2 State whether each of the following is true or false. If false, explain why: a) In general, a node's size should be defined explicitly. b) A node's position should be defined relative to its parent node and the other nodes in its parent. c) Most JavaFX layout panes use fixed positioning d) RadioButtons function as mutually exclusive options. e) Layout panes are container nodes that arrange their child nodes in a scene graph relative to one...
4. Let fín) and g(n) be asymptotically positive functions. Prove each of the following statements A. fin)-O(g(n)) if and only if fin) *gn)g(n)) B. fn) - Og(n if and only if fin)2- O(g(n)?)
4. Let fín) and g(n) be asymptotically positive functions. Prove each of the following statements A. fin)-O(g(n)) if and only if fin) *gn)g(n)) B. fn) - Og(n if and only if fin)2- O(g(n)?)
6) This question is part of Exercise 3-4 on page 62 of CLRS, but the letters aren't all the same. Answer TRUE or FALSE for each of the following statements. TRUE means that the statement is TRUE for any asymptotically positive functions f(n) and g(n). Otherwise, answer FALSE. You don't have to Prove or Disprove these statements... but you should learn how to do that. f(n)- O( g(n)) implies g(n) O( f(n)) b) This is part a in Exercise 3-4....
1. Answer each of the following statements as true, false, or unknown. a. The set of nonnegative even integers is well ordered. b. The sequence of Mersenne numbers forms a geometric progression. c. The sequence {na +1} contains infinitely many primes. d. The sequence {n" +1}.contains infinitely many composites. D) - logo) e. The Prime Number Theorem implies that lim ++00 f. There exist infinitely many pairs of primes that differ by less than 300. g. The number V110520191105201911052019 is...
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is differentiable on (0,1) and f(x) < 1 for all x € (0,1) and n e N. (a) If (fn (0))nen converges to a number A, prove that lim sup|fn(x) = 1+|A| for all x € [0, 1]. n-too : (b) Suppose that (fr) converges uniformly on [0, 1] to a function F : [0, 1] + R. Is F necessarily differentiable on (0,1)? If...
Please answer it step by step and Question 2. uniformly
converge is defined by *f=0* clear handwritten,
please, also, beware that for the x you have 2 conditions , such as
x>n and 0<=x<=n
1- For all n > 1 define fn: [0, 1] → R as follows: (i if n!x is an integer 10 otherwise Prove that fn + f pointwise where f:[0,1] → R is defined by ſo if x is irrational f(x) = 3 11 if x...
Problem I (10 points) Determine whether the following statements are True or False. Please circle your answer. You don't need to justify. (1) (T or F) Poisson processes are the only type of stochastic processes which have independent increment property. (2) (T or F) Let X; ~ Exp(1), 1 <i<n, be iid random variables. Then X1 +...+ Xn ~ Exp(nl). (3) (T or F) Any Brownian motion satisfies the Markov property. (4) (Tor F) Let S = X1 + X2...
Part I. (30 pts) (10 pts) Let fin) and g(n) be asymptotically positive functions. Prove or disprove each of the following statements T a、 f(n) + g(n)=0(max(f(n), g(n))) 1. b. f(n) = 0(g(n)) implies g(n) = Ω(f(n)) T rc. f(n)- o F d. f(n) o(f(n)) 0(f (n)) f(n)=6((f(n))2)
1. Answer True or False, and give a brief justification for each answer: a) If lim 2 = 5 then the series i converges to 5. b) If = 5 then lime = 5. c) If S. and lim.- S.-5, then 10 -5. d) The series 5-5+5-5+... is divergent. e) If = 0 = 5 and the = 5, then 20 - 5 f) The Divergence Test can be used to prove a series is convergent.
10. (18 points total: 3 points for each correct answer, 0 points for incorrect answers or no answer Answer "True" or "False” for each of the following: (i) If 8,9:R + R and are both continuous at a number c, then the composition function fog is continuous at c. (ii) If functions hi, h2: R + R and are both uniformly continuous on a non-empty set of real numbers E, then the product h h2 is uniformly continuous on E....
QUESTION G ONWARDS
13.2 State whether each of the following is true or false. If false, explain why: a) In general, a node's size should be defined explicitly. b) A node's position should be defined relative to its parent node and the other nodes in its parent. c) Most JavaFX layout panes use fixed positioning d) RadioButtons function as mutually exclusive options. e) Layout panes are container nodes that arrange their child nodes in a scene graph relative to one...
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