(1) A firm has the production function f(x_1, x_2) = ln(x_1) + ln(x_2). It operates in perfectly competitive markets, and the output price p and the input prices w_1 and w_2 are all strictly positive. (a) Solve for the input demand functions. (b) Solve for the supply function. (c) Formulate the profit function and apply Hotelling's lemma to check that the supply function in your previous answer was correct. (2) Solve for the conditional demand functions and the cost function of the firm in problem 1. (3) Calculate the conditional input demands and the cost function for the linear production function y = f(x) = the sum over i = 1, ..., m of alpha_i x_i, where alpha_i is a positive constant for all i = 1, 2, ..., m.
Consider the time preference model of O'Donoghue and Rabin (1999), in which an agent has to complete a task within T periods. (a) Assume that T = 4. Let time preferences be given by the (beta, delta)-model with delta = 1. The utility from completing the task is V, regardless of the completion date. While the utility occurs in the future, costs are immediate: C = 3 in t = 1, C = 5 in t = 2, C = 8 in t = 3, and C = 12 in t = 4. Time-inconsistent agents have beta = 1/3. Derive the perception-perfect strategies for the three types of agents (time-consistent, time-inconsistent sophisticates, and time-inconsistent naifs), and indicate for each type the period in which she effectively completes the task. Explain your answers. (b) Explain why sophisticates may behave in ways that seemingly contradict having present-biased preferences.
ABC is a university that has Learner Support Centers (LSCs) and Regional Centers (RCs) across the world. The university includes many departments, which offer various programs. Not all LSCs offer all programs. Counseling sessions are held on weekends, and attendance of 75% is required for practical sessions. Every entity, such as RCs, departments, programs, courses, and students, has a unique code or identifier. Semester-end exams are held at designated Examination Centers, and students are assigned to specific LSCs for counseling. Based on this scenario: (1) List the entities. (2) For each entity, list its attributes. (3) Define the relationships between the entities. (4) Draw the Entity-Relationship Diagram (ERD).
Answer the following speech and language processing questions. (a) Explain the Noisy Channel Model in the context of Automatic Speech Recognition (ASR), and why this model is important in the process of decoding speech signals. (b) What is the Viterbi Algorithm, and how does it function within Hidden Markov Models (HMMs) to assist in speech recognition? (c) Explain the concept of the spectral envelope in speech signal processing and how it is extracted from the speech spectrum. (d) Discuss the differences between Connectionist Temporal Classification (CTC) and RNN-Transducer models in neural network-based ASR systems. What are the advantages of using one model over the other?
Consider two relations A and B. A has 80,000 tuples and B has 100,000 tuples, and both relations store 100 tuples per page. Consider the SQL statement SELECT * FROM A INNER JOIN B ON A.a = B.a, an equijoin with the condition A.a = B.a. There are 102 buffer pages available for this operation. Both relations are stored as unsorted heap files, and neither has any index. Calculate the cost of each of the following join strategies, measured in the number of disk I/Os (pages). (a) Page-oriented nested loops join, with A as the outer relation. (b) Block-oriented nested loops join, with A as the outer relation. (c) Sort-merge join, assuming it can be done in 2 passes. (d) Hash join.
An energy company needs a control center to manage a distributed wind farm. Due to terrain and other considerations, the cost to reach a group of windmills is specific to each location. Locate the control center to minimize the cost, and return the minimum cost. The control center is located at integral coordinates (cx, cy). Wind farm i is located at integral coordinates (x[i], y[i]). The distance to farm i is the Manhattan distance, distance[i] = |x[i] - cx| + |y[i] - cy|. The cost to connect farm i is premium[i] * distance[i], and the total cost is the sum of the cost to connect each farm. Determine the minimum cost to connect all the farms to the control center, and design an efficient algorithm for it.
Consider the language B of strings of balanced parentheses: B contains the empty string, and B also contains any string of the form (x)y where x and y are arbitrary strings in B. You are given as input a string S = s_1 s_2 ... s_n consisting of "(", ")", and "*" characters. Give a dynamic programming algorithm to count the number of strings in B that can be created by replacing each "*" with "(" or ")". The running time of your algorithm should be O(n^2). (1) Define the entries of your table in words, for example "T(i, j) is ...". (2) State a recurrence for the entries of the table in terms of smaller subproblems, and briefly explain in words why it is correct. (3) Analyze the running time of your algorithm.
In this question, we study the limiting properties of the k-nearest neighbour algorithm. Suppose that we are given n data points X_1, ..., X_n, sampled uniformly and independently from the interval [0, 2], and one test point y = 1. Let Z_i = |X_i - y| denote the distance of y to the data point X_i, and let Z_(i) denote the distance of y to its i-th nearest neighbour, so that Z_(1) < ... < Z_(n). (a) Show that the random variable Z_i is uniformly distributed on the unit interval [0, 1]. (b) Show that E[Z_(1)] = 1/(n + 1); that is, the expected distance to the first nearest neighbour is 1/(n + 1). (Hint: it may help to first compute P{Z_(1) > t}. Note that Z_(1) = min over i of Z_i, so P{Z_(1) > t} = P{min over i of Z_i > t}.)
Consider the infinite square well potential V(x) = 0 for 0 < x < a and V(x) = infinity otherwise, where a > 0. The initial wave function is Psi(x, 0) = A sin^2(pi x / a). (a) Determine A. (b) Express Psi(x, t) as the infinite sum of the eigenfunctions psi_n(x) = sqrt(2/a) sin(n pi x / a), that is, Psi(x, t) = sum over n of c_n psi_n(x) e^(-i E_n t) (taking hbar = 1). You may find the following identity useful: the integral from 0 to 1 of sin^2(pi x) sin(n pi x) dx equals 4/((4n - n^3) pi) if n is odd, and 0 if n is even. (c) Evaluate the sum over all positive odd integers n of 1/(n^2 (n - 2)^2 (n + 2)^2).
In a modern physics course you learn that a hydrogen atom has allowed energies -13.6/n^2 eV, where n = 1, 2, 3, ..., which are the solutions of the time-independent Schrodinger equation. The wavefunction of the state with energy -13.6 eV is denoted psi_1(r), and that of the state with energy -13.6/4 eV is psi_2(r); both are normalized. (a) Suppose that at time t = 0 the system's wavefunction is Psi(r, 0) = psi_1(r). (i) What is Psi(r, t) at some later time t? (ii) Calculate the probability densities at t = 0 and at time t. Compare the two and comment on your results. (b) Repeat parts (a)(i) and (a)(ii) for Psi(r, 0) = psi_2(r). (c) Now suppose that at t = 0 the wavefunction is Psi(r, 0) = sqrt(1/5) psi_1(r) + sqrt(4/5) psi_2(r). (i) What is Psi(r, t) at some later time t? (ii) Calculate the probability densities at time 0 and at time t using your result from part (c)(i), and comment on your result.
