Consider the matrix K with rows (1, -1, -1), (1, 3, 1), and (-3, 1, -1). (a) Find the eigenvalues of K, denoted lambda_1, lambda_2, and lambda_3, ordered so that lambda_3 < lambda_2 < lambda_1. (b) Find the corresponding eigenvectors for these eigenvalues.
Suppose A, B are nonempty subsets of R that are bounded above. (a) What is the definition of the supremum sup A of A? (b) Show that sup(A union B) = max{sup A, sup B}. (c) Suppose the intersection of A and B is nonempty. Is it always true that sup(A intersect B) = min{sup A, sup B}? Either give a proof of the identity, or find a counterexample.
(a) Show that it is impossible to find continuous functions a_1, a_0: R -> R so that the differential equation x''(t) + a_1(t) x'(t) + a_0(t) x(t) = 0 has the solution x(t) = t^2. (Hint: check whether the existence and uniqueness theorem applies.) (b) Consider a function f: R x C -> C and the complex initial value problem dz/dt = f(t, z), z(0) = z_0, for a complex-valued function z: R -> C and a complex number z_0. By considering real and imaginary parts of this equation, formulate conditions on f for the problem to have a unique solution. Show that z(t) = e^(it) and z(t) = cos t + i sin t both satisfy dz/dt = iz, z(0) = 1, and deduce Euler's formula e^(it) = cos t + i sin t.
Two events are mutually exclusive when they cannot occur at the same time. Two events are independent when the occurrence of one event does not affect the occurrence of the other. Identify two examples of events in the contemporary art world. Describe a scenario in which the two events would be considered mutually exclusive. Describe a scenario in which they would be considered independent. What can you say about the main difference between mutually exclusive events and independent events? Cite any sources in APA style.
The following line of code in a C++ project causes a compiler warning: safe_strncpy(szBuffer, szBuffer+8, 26); where szBuffer is of type "char *" and safe_strncpy is defined by: #define safe_strncpy(a,b,n) do { strncpy((a),(b),(size_t)((n)-1)); (a)[(size_t)((n)-1)] = 0; } while (0). The compiler warning is about the call to strncpy, and reads: warning: 'char* strncpy(char*, const char*, size_t)' accessing 25 bytes at offsets 0 and 8 overlaps between 1 and 17 bytes at offset 8 [-Wrestrict]. Explain the warning and suggest how to resolve it.
Given a connected undirected graph G, a spanning tree is a connected subgraph that is a tree and includes all the vertices of G. Consider the following variant of the spanning tree problem: given a connected undirected graph G, decide whether there exists a spanning tree in which every node has degree at most five. Show that this problem is NP-hard. (Hint: try a reduction from the Hamiltonian cycle problem, but it is easier to go in two steps. First reduce Hamiltonian cycle to Hamiltonian path, then reduce Hamiltonian path to the spanning tree problem. For the second reduction, suppose V is the vertex set of the Hamiltonian path instance; for each vertex v in V, create three additional vertices and put edges between these new vertices and v.)
Consider an MDP with n states s1, s2, ..., sn and 2 actions a1, a2 at each state. At each state si with i < n, action a1 takes the agent to s(i+1), and for state sn, action a1 is a self-loop that keeps the agent in sn. At any state other than sn, action a2 takes the agent back to s1; at sn, a2 is again a self-loop. The reward R(si, aj) is 0 except for (sn, a1), which has a value of 1. The agent takes one action at each step. Under the uniform random policy, the agent at each state chooses one of the available actions uniformly at random. Show your work. (a) Compute the expected number of steps for the uniform random policy to go from state s1 to state sn. (b) In a discounted-reward setting with discount factor gamma, let Q(si, aj) be the action-value function of the uniform random policy. Prove that for all i < n, Q(si, a1) > Q(si, a2).
Given is a disjunctive form consisting of k monomials m1, ..., mk over n boolean variables x1, ..., xn. The task (Refutation) is to decide whether there is a truth assignment to the variables such that the truth value of the disjunctive form is false. You are given a decision algorithm A_d that solves this problem: for each instance, A_d answers YES if there is an assignment that makes the truth value of the disjunctive form false, and NO otherwise. Describe an algorithm A_o that solves the optimization problem, that is, one that finds a truth assignment making the disjunctive form false if one exists. The running time has to be polynomial in the input size, and A_o may make calls to A_d, where each call counts as one basic computational step.
A survey of a sample of 25 new cars being sold at a local auto dealer was conducted to see which of three popular options, air-conditioning (A), radio (R), and power windows (W), were already installed. The survey found: 15 had air-conditioning (A), 12 had radio (R), 11 had power windows (W), 5 had A and W, 9 had A and R, 4 had R and W, and 3 had all three options. Find the number of cars that had: (a) only W; (b) only A; (c) only R; (d) R and W but not A; (e) A and R but not W; (f) only one of the options; (g) at least one option; (h) none of the options.
Below are the winning times in the men's 10,000 m track event in the Olympic Games for the years 1984, 1988, 1992, 1996, and 2000. Times are in decimal form (for example, 28.4 = 28 minutes and 24 seconds): 27.8, 27.4, 27.8, 27.1, 27.3. Let the years be the x values and the times the y values. (a) Find the estimated regression line. (b) Predict the winning time in the Olympic year 2088.
