Problem 1. Suppose you have seven dice—each a different color of the rainbow; otherwise the dice are standard, with faces numbered 1 to 6. A roll is a sequence specifying a value for each die in rainbow (ROYGBIV) order. For example, one roll is .3; 1; 6; 1; 4; 5; 2/ indicating that the red die showed a 3, the orange die showed 1, the yellow 6,. . . . For the problems below, describe a bijection...

2) We prove that Half-Clique is NP-hard by reducing 3-SAT to it. The reduction is very similar to the one from 3-SAT to Clique described in CLR. We start from an instance φ = C1 ∧ C2 ∧ . . . ∧ Ck of 3-SAT. Without loss of generality we can suppose that for r = 1, . . . , k Cr contains three different literals lr 1, l r 2, l r 3. Now we have to construct a graph that has a clique of size |V |/2 ...

Of this algebra really only the first line should be unfamiliar, as the second line simply says that Pμ and Kμ are vectors and the last line is just the Lorentz algebra SO(p, q). Now to put this into a more intuitive form we imagine that we are in a bigger space with two extra coordinates which we call −1 and 0 (the remaining coordinates are then 1...p+ q). Now consider the algebra defined by t...

Problem 1a. Ω = {T,H} b. Let set A = {1, 2, 3, ..., N} be the set of N balls numbered 1 to N. We have two choices for Ω. In the first choice, an outcome specifies which ball was picked first and which was picked second. With this choice, Ω = Ω1 := {(x, y) | x, y ∈ A, x 6= y}. In the second choice, an outcome only specifies the two balls that were picked, but not their order. With this choice, Ω...

Problem 1. We’re going to characterize a large category of games as a recursive data type and then prove, by structural induction, a fundamental theorem about game strategies. The games we’ll consider are known as deterministic games of perfect information. Checkers, chess, and GO, for example, all fit this description. It’s useful to regard each game situation as a game in its own right. For e...

For some problems with many solutions, like planning and phylogeny reconstruction, one way to compute more desirable solutions is to assign weights to solutions, and then pick the ones whose weights are over (resp. below) a threshold. This paper studies computing weighted solutions to such problems in Answer Set Programming. We investigate two sorts of methods for computing weighted solutions: ...

In Year 1, there are only ten factories, which can employ a maximum of 100 workers. Hence most farms will still have all five siblings working there. The total farm income will be $100, so the average product of labor is $20 = 100 5 . The marginal product is 0, since when there are at least three siblings working on the farm, any additional worker does not increase farm income. Consequently two...

And hence d(x, 0) ≥ lim inf d(0, xn)− lim sup d(xn, x) = 1 Thus x ∈ S. Hence S is closed. Clearly S is bounded since it is contained in B2(0). Define en = (0, 0, . . . , 0, 1, 0, . . . ) be the sequence with all 0’s except a 1 in the nth place. Notice that en ∈ S (for `, `∞, c0). However, notice also that since, in every case, d(en, em) ≥ 1 whenever n 6= m, then the sequence en has no convergen...