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Let x0, x1, x2 be three vectors in Rn. Prove that these vectors are affine independent if and only if none of them is a convex combination of the other two.
Prove that vectors x0, x1,...,xk are affine independent in Rn if and only if every vector y ? conv{x0, x1,...,xk}
Prove that the boundary of a simplex S is the set of all the points y in S whose barycentric coordinate representation has at least one zero coordinate.
Prove that if H1 and H2 are two affine spaces of the same dimension k in Rn (that is, each one of them is spanned by a k-dimensional simplex).
For each of the following partitions of a two-dimensional simplex, determine whether or not it is a simplicial partition. Justify your answer.
Let S be a k-dimensional simplex in Rn, and T be a simplicial partition of S. Then S equals the union of all the k-dimensional simplices in T .
Is it possible for the winner of the election to be ranked least preferred by at least half of the residents? Justify your answer.
Let A be a set of alternatives, and let P N be a strict preference profile. Alternative a ? A is termed the Condorcet winner if for every alternative b = a.
Approval voting In this question, we consider the case in which the individuals are called upon to choose candidates for a task.
A matching is stable if there is no pair consisting of a man and a woman who have an objection to the matching.
For every pair of matchings there exist preference relations for which these are two stable matchings.
Dan is at the bottom of Donna’s preference list, and Donna is at the bottom of Dan’s preference list.
Prove that if Romeo and Juliet are matched to each other under both the men’s courtship and the women’s courtship algorithms.
Prove that if the result of the men’s courtship algorithm yields the same result as the women’s courtship algorithm.
Is it possible to find three pairs such that if the matching among them is changed, each man will be matched to a woman whom he prefers.
Is there necessarily a stable matching under which Julius is matched to Messalina? Either prove this statement or provide a counterexample.
Prove that if in stage t of the men’s courtship algorithm, a particular man is dismissed for the (n - 1)-th time, then the algorithm terminates at stage (t + 1)
The candidate who thus amasses the greatest number of points wins the election. If two or more candidates are tied for first place in the number of points.
Show that if P* is a strict preference relation then P := P* ? {(a, a): a ? A} is a preference relation.
Find the number of Condorcet points that each strict preference relation receives and determine the preference relation that the Condorcet Method chooses.
A committee comprised of 15 members is called upon to rank three colors: red, blue, and yellow, from most preferred to least preferred.
Let F be a social welfare function satisfying the unanimity property. For every a, b ? A, the coalition N is decisive for a over b and the empty coalition.
The accused is found guilty if at least five jurors, including Debbie and at leastone of her vice-foremen, declare him to be guilty.
Suppose that |A| = 3, and letF be a social welfare function satisfying the properties of unanimity and independence of irrelevant alternatives.
Suppose that |A| = 3 and that F satisfies the unanimity and independence of irrelevant alternatives properties.