Problem 1. [20 points] The adjacency matrix A of a graph G with n vertices as defined in lecture is an n×n matrix in which Ai,j is 1 if there is an edge from i to j and 0 if there is not. In lecture we saw how the smallest k where Ai,j 6= 0 describes the length of the shortest path from i to j. Given a combinatorial interpretation of the following statements about the adjacency matrix in terms ...

The Student’s t Test. A very common problem in statistics is to estimate the population mean of a probability distribution given a sample, the subject of sections 7.1 and 7.2 of the statistics textbook. As explained there, this relates directly to the so-called t test. For large samples we may assume, due to the central limit theorem, that the sample mean is normally distributed; one often cons...

(b) If a process X(t) is strongly stationary, all of its moments have to be independent of t. The third moment of X(t), E(X(t)) = (EA)(cos(ωt) + sin(ωt)), is not a constant with respect to t unless EA = 0. Clearly we can find many examples of distributions for A so that the third moment is not zero while E(A) = 0. For example take A to be a discrete random variable with P (A = −1) = 1/3, P (A =...

Problem 3. The element 1 + 2 ∈ k[S3] is equal to (1 + ρ+ ρ2)/3. Thus M is the same as the image of (1 + ρ+ ρ2)/3, which I’ll denote by φ. Notice that if β ∈ L, then ρ(φβ) = (ρφ)β = φβ. This means that M is contained in the fixed field of ρ, i.e. M ⊆ L. Let’s show that in fact M = L. We need to prove the other inclusion, that L ⊆M . If β ∈ L, then ρ(β) = β, so φ(β) = (β + ρ(β) + ρ2(β))/3 = (β + ...

sueyoshi and sekitani in 2007 presented a paper that explores the measurement of returns to scale under a possible occurrence of multiple solution in a reference set and a supporting hyperplane. the occurrence ofmultiple solutions is classified into type i and type ii. type i is an occurrence of multiple solutions in a reference set. type ii is an occurrence of multiple solutions on a supportin...

a problem that sometimes occurs in multiobjective optimization is the existence of a large set of fairly effcient solutions. hence, the decision making based on selecting a unique preferred solution is diffcult. considering models with fair b-effciency relieves some of the burden from the decision maker by shrinking the solution set, since the set of fairly b-efficient solutions is contained with...

Problem 2. [15 points] In ”Die Hard: The Afterlife”, the ghosts of Bruce and Sam have been sent by the evil Simon on another mission to save midtown Manhattan. They have been told that there is a bomb on a street corner that lies in Midtown Manhattan, which Simon defines as extending from 41st Street to 59th Street and from 3rd Avenue to 9th Avenue. Additionally, the code that they need to defu...

The first case is trivial to prove (note that incrementing x̄ by 1 at each entry still maintains the vector in {0, 1, 2}k). The second case considers when the row i∗ + 1 does not have 2 blocks after the first n− 2: in this case the only way to tile the last block in the i∗-th row is to place a 2× 1 tile horizontally. Similarly, the third case assumes that also the (i∗+1)-th row has 2 ending bloc...

Problem 3-1. Prime Triangles. TA Notes: It is a little-known fact (to us, anyway) that the Java 1.4.1 class library contains a bug in part of its primality tester. Specifically, the Lucas-Lehmer test is implemented incorrectly, and sometimes reports that certain large numbers are composite when they are actually prime. This slowed down our own trianglefinding algorithm for large values of k, an...