We find the sum.
) Compute п 1 E k=1 kvk +1+(k+1) VK Hint: rationalize the denominator, i.e. multiply by...
2/10 To rationalize the denominator of , 1. you should multiply the expression by which fraction? /10 10 11 11 2-10 2-10 3-11 3-11
Which choice is equivalent to the fraction below when x2 1? Hint: Rationalize the denominator and simplify. le 2x -1
Let exp(-т*) + vk Yk where dent M and V N(0, o2 are mutually indepen R, k = 1, (a) Construct the likelihood T(y|x) and the negative log-likelihood. (b) Compute the maximum likelihood estimate îML (c) Bonus question: How does the estimate change if E(k) t0?
Let exp(-т*) + vk Yk where dent M and V N(0, o2 are mutually indepen R, k = 1, (a) Construct the likelihood T(y|x) and the negative log-likelihood. (b) Compute the maximum likelihood estimate...
4. Approximating Clique. The Maximum Clique problem is to compute a clique (i.e., a complete subgraph) of maximum size in a given undirected graph G. Let G = (V,E) be an undirected graph. For any integer k ≥ 1, define G(k) to be the undirected graph (V (k), E(k)), where V (k) is the set of all ordered k-tuples of vertices from V , and E(k) is defined so that (v1,v2,...,vk) is adjacent to (w1,w2,...,wk) if and only if, for...
2. Consider the torsional system shown in Figure 2. Assume damping is negligible, i.e., ok/J. From the free vibration response, it is observed that the natural frequency of the system is w Then increase the moment of inertia of the disk to JJ, and the free vibration response shows that the natural frequency is reduced to a,2. Calculate J and k Express Jand k in terms of ω 1 and ω". Hint: Apply a, = vk/J to both cases. 0,T...
A real symmetric matrix B e Rnxn (i.e. BT = B) is said to be positive definite if all of its eigenvalues 11, 12, ..., In are positive. (Recall that is an eigenvalue of B if and only if there exits a nonzero vector t such that Bt = it). Show that B-1 is also positive definite. That is, you need to show that all the eigenvalues of B-1 are also positive. (Hint: consider equation Bt; = liti for all...
7. Let E C R be nonempty, n E N, and K, L E Z such that K/n is an upper bound for E, but L/n is not an upper bound for E. (a) Show that there exists an for E, but (m - 1)/n is not an upper bound for E. (Hint: Prove by contradiction, and use induction. Drawing a picture might help) m < K such that m/n is an upper bound integer L (b) Show that m...
Compute the exponentiation x^e mod 29 of x = 5 with both variants of e from above* for n = 4. Use the square-and-multiply algorithm and show each step of your computation. *above, referring to the formulas e = 2^n + 1 and e = 2^n -1
Anharmonic oscillator. Hydrogen bromide, H8Br, vibrates approximately according to a Morse potential VM(r) = Dell-e-ck/2De)i/2(r-re) , with De= 4.810 eV, = 1.4144 A, and k= 408.4 N m-1. With a0-Vk/a, the energies of the stationary states in a Morse potential are En (n + 1/2)2. (A) On the same graph, plot the Morse potential and the harmonic potential as a function of bond length (from 0.7 to 2 %). Use the software of your choice to generate this plot. (B)...
2. Prove that the infinite series Ex=1(-1)k diverges. (Hint: Compute the first few terms of the sequence of partial sums and determine a formula for the nth partial sum, Sn. Using this, give a formal proof, starting with the definition for divergence of this series. (Additional reference: Workshop Week #7)