
8. Dynamics in a map Let In+1 = f(xn), where f(x) = -(1+r)x - 72 -...
8. Problem 8. Let f and g map R into itself where /(x) and g(x) = Show that if f is conjugate to g via a homeomorphism h, then either h or h 1 is not differentiable.
Explanation and graphs are important here
8. Describe the dynamics of a linear map F: R-R, F(z)= Az, z E R2, whose matrix representation is 1/2 0 -2 0 a) A = 0 2 0 1 2 :) =ra (5)-
8. Describe the dynamics of a linear map F: R-R, F(z)= Az, z E R2, whose matrix representation is 1/2 0 -2 0 a) A = 0 2 0 1 2 :) =ra (5)-
1) Let f(x) = 1 sin x, x E R , and consider the discrete-time dynamics given by xn+1 = f(xn), n = 0.1, 2, . . . How many fixed points are there? Stable or unstable?
(9) Let E R" and let A E L(R"). Define a map f : R" -> R" by f (x) A,)v. Here (is the Euclidean inner product (a) Prove that f is a C1 map and find f'(x) (b) Prove that there exist two that f U V is a bijection on R" neighborhoods of the origin in R", U and V, such
(9) Let E R" and let A E L(R"). Define a map f : R" -> R"...
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
3. Let X1,..Xn be a sample with joint pdf (or pmf) f(x,0), 0 e 0 c R. Suppose that {f(x, 0) 0 e 0} has monotone likelihood ratio (MLR) in T(X,). Consider test function if T(xn)> c if T(xn) c if T(x)<c 0 E [0,1 and c 2 0 are constants. Prove that the power function of ¢(X,) is where non-decreasing in 0
3. Let X1,..Xn be a sample with joint pdf (or pmf) f(x,0), 0 e 0 c R....
2. Let X 1, , Xn be iid from the distribution modeled by 8-2 fx (1:0)-(9. θ):r"-"(1-2) dr where 0 < x < 1 and θ > 1 Find the MME (method of moments estimate/estimator) for 0
, xn be a sample with joint pdf (or pmf) f(Xn10), θ 3. Let Xi, Θ C R. Suppose that {f(x,10) : θ E Θ} has monotone likelihood ratio (MLR) in T(Xn). Consider test function if T(%) > c Xn if T(%) < c, where γ E [0, 1) and c 〉 0 are constants. Prove that the power function of φ(Xn) is non-decreasing in θ
, xn be a sample with joint pdf (or pmf) f(Xn10), θ 3. Let...
4) Let xn +1 =- + rzn for r > 0. (A) Find the fixed points (in terrns of r) and use the derivative to determine the values of r where they are stable. Use your calculator to verify your results. (B) Find the two-cycles of this map. (Hint: the equation has 2 solutions that are already known from part A.) Using the derivative, find the values of r where the two-cycle is stable. Use your calculator to verify your...
1 Let f: R R be a continuously differentiable map satisfying ilf(x)-FG) ll 리1x-vil, f Rn. Then fis onto 2. f(RT) is a closed subset of R'" 3, f(R") is an open subset of RT 4. f(0)0 or all x, y E 5) S= (xe(-1,4] Sin(x) > 0). Let of the following is true? I. inf (S).< 0 2. sup (S) does not exist Which . sup (S) π ,' inf (S) = π/2
1 Let f: R R be...