1) Let f(x) = 1 sin x, x E R , and consider the discrete-time dynamics...
8. Dynamics in a map Let In+1 = f(xn), where f(x) = -(1+r)x - 72 - 2x3. d) What is the long-term behavior of orbits that start near x* = 0, both for r <0 and r > 0.
Problem 4. (Discrete time dynamical system ). Consider the following discrete time dynamical system: Assume xo is given and 0.5 0.5 0.2 0.8 (a) Find eigenvalues of matrix A (b) For each eigenvalue find one eigenvector. (c) Let P be the matrix that has the eigenvectors as its columns. Find P-1 (d) Find P- AP (e) Use the answer from part (d) to find A" and xn-A"xo. (Your answers wl be in terms of n (f) Find xn and limn→ooXn...
(1 point) Consider the discrete-time dynamical system x+1 = 2x,(1 - x). If x = 1, Xr+1 = 0.5 Is x = an equilibrium for this system (yes/no)? yes What is the updating function f(x)? Compute the derivative: f'(x) = Evaluate the derivative at the equilibrium: Is the equilibruim stable, unstable or neither? stable
(1) Let f : [n] [n] be a permutation. A fixed point of f is an element x e [n] such that f(x) - x. Now consider random permutations of [n] and let X be the random variable which represents the number of fixed points of a given permutation. (a) What is the probability that X 0? (b) What is the probability that X-n -2? (c) What is the probability that X-n-1? (d) What is the expectation of X? (Hint:...
Let a be a positive real number. Consider a discrete-time echo system (called system 1) given by the difference equation y[n] = v[n] + av[n – 4). Here v is the input signal and y is the output signal. A. (1 mark) Determine the systems' transfer function H1 (2). B. (1 mark) What are the pole(s) of this system? Plot the pole(s) in the complex plane. C. (2 marks) Is this system stable? Explain your answer. D. (2 marks) Determine...
Let f [n]n] be a permutation. A fixed point of f is an element x e [n] such that f(x)-x. Now consider random permutations of [n] and let X be the random variable which represents the number of fixed points of a given permutation. (a) What is the probability that X 0? (b) What is the probability that X 2? (c) What is the probability that X--1? (d) What is the expectation of X? (Hint: As usual, express X as...
(0, 1) given by f (x) - sin (). Is f Let f b e the function t on the domain uniformly continuous? Explain. (You may take it as given that sin is a continuous function) Suppose that f [0, oo) -R is a continuous function, and suppose also that lim, ->oo f (x)- 0. Prove that f is uniformly continuous Just to be clear: to say that lim,->o f (x) - 0 means that
Power function sample with joint pdf (or pmf) f (x |0), 0 e 0 c R. Suppose Let X1,..., X,n be a that {f(xn0) : 0 E 0} has monotone likelihood ratio (MLR) in T(X). Consider test function if T(xn)> c 1 if T(Xn) (Xn) C if T(xn)c 0 where y E [0, 1] and c > 0 are constants. Prove that the power function of ø(X,,) is non-decreasing in 0 sample with joint pdf (or pmf) f (x |0),...
() At)x()B(f)u() Consider the following time-varying system y(t) C(f)x(t) where x) R", u(t)E R R 1 1) Derive the state transition matrix D(t,r) when A(f) = 0 0 sint 2) Assume that x(to) = x0 is given and u(f) is known in the interval [to, 4] Based on these assumptions, derive the complete solution by using the state transition matrix D(f, r). Also show that the solution is unique in the interval [to, 4]. 3) Let x(1) 0 and u(f)...
(a) Consider a discrete-time signal v[n] satisfying vn0 except if n is a multiple of some fixed integer N. i.e oln] -0, otherwise where m is an integer. Denote its discrete-time Fourier transform by V(eJ"). Define y[nl-v[Nn] Express Y(e) as a function of V(e). Hint : If confused, start with N-2 (b) Consider the discrete-time signal r[n] with discrete-time Fourier transform X(e). Now, let z[n] be formed by inserting two zeroes between any two samples of x[n]. Give a formula...