Let f(n) = (n + a)b and g(n) = nb, for any real constants a and...
Let f(n) = (n + a)b and g(n) = nb, for any real constants a and b, where b > 0.
Using Θ, what is your conclusion for the relationship between f(n) and g(n)?
Homework Answers
f(n) = Θ (g(n)) means there are positive constants c1, c2, and k, such that 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) for all n ≥ n0. The values of c1, c2, and n0 must be fixed for the function f and must not depend on n. (n + a)b = Θ(nb) => c1(nb) <= (n + a)b <= c2(nb) => c1(nb) <= nb + ab <= c2(nb) Let's assume c1 = 0.5 and c2 = 2 => c1(nb) <= nb + ab <= c2(nb) => 0.5(nb) <= nb + ab <= 2(nb) => 0.5(nb) <= nb + ab and nb + ab <= 2(nb) => -ab <= 0.5 nb and ab <= nb => -2ab <= nb and ab <= nb => -2a <= n and a <= n so, (n + a)b = Θ(nb) given c1 = 1/2, c2 = 2 and n0 = a
Let f(n) = 5n^2. Prove that f(n) = O(n^3). Let f(n) = 7n^2. Prove that f(n)...
Let f(n) = 5n^2. Prove that f(n) = O(n^3). Let f(n) = 7n^2. Prove that f(n) = Ω(n). Let f(n) = 3n. Prove that f(n) =ꙍ (√n). Let f(n) = 3n+2. Prove that f(n) = Θ (n). Let k > 0 and c > 0 be any positive constants. Prove that (n + k)c = O(nc). Prove that lg(n!) = O(n lg n). Let g(n) = log10(n). Prove that g(n) = Θ(lg n). (hint: ???? ? = ???? ?)???? ?...
1. a) Let f(n) = 6n2 - 100n + 44 and g(n) = 0.5n3 . Prove...
1. a) Let f(n) = 6n2 - 100n + 44 and g(n) =
0.5n3 . Prove that f(n) = O(g(n)) using the definition
of Big-O notation. (You need to find constants c and n0).
b) Let f(n) = 3n2 + n and g(n) = 2n2 . Use
the definition of big-O notation to prove that
f(n) = O(g(n)) (you need to find constants c and n0) and
g(n) = O(f(n)) (you need to find constants c and n0).
Conclude that...
Let h(n) = n mod 3. Next, let f(n) = n and g(n) = nh(n). What...
Let h(n) = n mod 3. Next, let f(n) = n and g(n) = nh(n). What is the order relationship between f and g? Please explain with complete sentences!
1. Theorem 4.1 (Master Theorem). Let a 2 1 and b >1 be constants, let f(n)...
1. Theorem 4.1 (Master Theorem). Let a 2 1 and b >1 be constants, let f(n) be a function, and let T(n) be defined on the nonnegative integers by the recurrences T(n)- aT(n/b) + f(n) where we take n/b to be either 1loor(n/b) or ceil(n/b). Then T(n) has the following asymptotic bounds. 1. If f(n) O(n-ss(a)-) for some constant e > 0, then T(n) = e(n(a). 2. If f(n) e(n(a), then T(n)- e(nlot(a) Ig(n)). 3. If f(n)-(n(a)+) for some constant...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we ...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...
Let f (n) and g(n) be asymptotically nonnegative functions. Using the basic definition of _-notation, prove...
Let f (n) and g(n) be asymptotically nonnegative functions. Using the basic definition of _-notation, prove that max( f (n), g(n)) = Θ( f (n) + g(n)).
Let n E Z20. Let a, b є R with a < b. Let y-f(x) be a continuous real- valued function on a, b]. Let Ln and R be the left and right Riemann sums for f over a, b) with n subintervals, respectively....
Let n E Z20. Let a, b є R with a < b. Let y-f(x) be a continuous real- valued function on a, b]. Let Ln and R be the left and right Riemann sums for f over a, b) with n subintervals, respectively. Let Mn denote the Midpoint (Riemann) sum for fover la, b with n subintervals (a) Let P-o be a Riemann partition of a,b. Write down a formula for M. Make sure to clearly define any expressions...
A traveling wave is described by the differential equation, where a and b are real, positive...
A traveling wave is described by the differential equation,
where a and b are real, positive constants. Solve this equation
using the given trial solution, and describe the relationship
between k and w.
Suppose that a traveling wave is described by the differential equation where a and b are real, positive constants. Solve this equation using a trial solution f(x, t) Aei(kx-wt) = the relationship betweenk imaginary, or complex?
Letter f and g only. 44 Let X,..., X. be a random sample from (a) Find a sufficient statistic. (b) Find a maximum-likel...
Letter f and g only.
44 Let X,..., X. be a random sample from (a) Find a sufficient statistic. (b) Find a maximum-likelihood estimator of θ. (c) Find a method-of-moments estimator of θ. (d) Is there a complete sufficient statistic? If so, find it. (e) Find the UMVUE of 0 if one exists. (f) Find the Pitman estimator for the location parameter θ. (g) Using the prior density g(0)--e-n,๑)(8), find the posterior Bayes estimator Of θ.
44 Let X,..., X....
Question 8: For any integer n 20 and any real number x with 0
Question 8: For any integer n 20 and any real number x with 0<<1, define the function (Using the ratio test from calculus, it can be shown that this infinite series converges for any fixed integer n.) Determine a closed form expression for Fo(x). (You may use any result that was proven in class.) Let n 21 be an integer and let r be a real number with 0<< 1. Prove that 'n-1(2), n where 1 denotes the derivative of...
1. a) Let f(n) = 6n2 - 100n + 44 and g(n) =
0.5n3 . Prove that f(n) = O(g(n)) using the definition
of Big-O notation. (You need to find constants c and n0).
b) Let f(n) = 3n2 + n and g(n) = 2n2 . Use
the definition of big-O notation to prove that
f(n) = O(g(n)) (you need to find constants c and n0) and
g(n) = O(f(n)) (you need to find constants c and n0).
Conclude that...
1. Theorem 4.1 (Master Theorem). Let a 2 1 and b >1 be constants, let f(n) be a function, and let T(n) be defined on the nonnegative integers by the recurrences T(n)- aT(n/b) + f(n) where we take n/b to be either 1loor(n/b) or ceil(n/b). Then T(n) has the following asymptotic bounds. 1. If f(n) O(n-ss(a)-) for some constant e > 0, then T(n) = e(n(a). 2. If f(n) e(n(a), then T(n)- e(nlot(a) Ig(n)). 3. If f(n)-(n(a)+) for some constant...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...
Let n E Z20. Let a, b є R with a < b. Let y-f(x) be a continuous real- valued function on a, b]. Let Ln and R be the left and right Riemann sums for f over a, b) with n subintervals, respectively. Let Mn denote the Midpoint (Riemann) sum for fover la, b with n subintervals (a) Let P-o be a Riemann partition of a,b. Write down a formula for M. Make sure to clearly define any expressions...
A traveling wave is described by the differential equation,
where a and b are real, positive constants. Solve this equation
using the given trial solution, and describe the relationship
between k and w.
Suppose that a traveling wave is described by the differential equation where a and b are real, positive constants. Solve this equation using a trial solution f(x, t) Aei(kx-wt) = the relationship betweenk imaginary, or complex?
Letter f and g only.
44 Let X,..., X. be a random sample from (a) Find a sufficient statistic. (b) Find a maximum-likelihood estimator of θ. (c) Find a method-of-moments estimator of θ. (d) Is there a complete sufficient statistic? If so, find it. (e) Find the UMVUE of 0 if one exists. (f) Find the Pitman estimator for the location parameter θ. (g) Using the prior density g(0)--e-n,๑)(8), find the posterior Bayes estimator Of θ.
44 Let X,..., X....
Question 8: For any integer n 20 and any real number x with 0<<1, define the function (Using the ratio test from calculus, it can be shown that this infinite series converges for any fixed integer n.) Determine a closed form expression for Fo(x). (You may use any result that was proven in class.) Let n 21 be an integer and let r be a real number with 0<< 1. Prove that 'n-1(2), n where 1 denotes the derivative of...
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