The asymptotic plurality rule, apl, is a consensus function which maps each profile P of length n... more The asymptotic plurality rule, apl, is a consensus function which maps each profile P of length n (i.e., each sequence of n bases appearing at an aligned position of n molecules) to a set apl (P) of consensus results (i.e., ambiguity codes) that is a descriptive summary of P. Our main result is to characterize each consensus result X = apl (P) in terms of the frequencies with which the bases in P occur. We then use these chamcterizations to investigate features (e.g., strong consistency, length independence) of apl that researchers may find useful for the interpretation of apl's consensus results.
The committee election problem is to choose from a finite set S of candidates a nonempty subset T... more The committee election problem is to choose from a finite set S of candidates a nonempty subset T of committee members as the consequence of an election in which each voter expresses a preference for a candidate in S. Solutions of this problem can be modelled by functions which map each partition of 1 (i.e., normalized vote tallies of candidates who have been ordered canonically by tally) into a nonempty subset of positive integers (i.e., sizes of committees). To solve this problem, we recently described a parameterized voting scheme, the ratio-of-sums or rasp consensus rule, in which p controls the degree to which votes must be concentrated in elected committees. It is desirable to identify the attainable results of such rules so as to understand their properties and to facilitate their comparison. For all p, we characterize the attainable rasp results in the general csse where the partition's parts are real, and in the special case where p as well as its parts are rational.
Working simultaneously in two teams [1,2], we have independently discovered essentially the same ... more Working simultaneously in two teams [1,2], we have independently discovered essentially the same concept and many common results. As expected, each team used its own notation and terminology but the results are easily transformed between the two systems. We plan to publish our full papers separately, but present the results here.
Let G be a graph and U, L' two vertices of G. Then the interval from K to 2' consists of all thos... more Let G be a graph and U, L' two vertices of G. Then the interval from K to 2' consists of all those vertices that lie on some shortest u -1; path. Let S be a set of vertices in a connected graph G. Then the Steiner distance d,(S) of S in G is the smallest number of edges in a connected subgraph of G that contains S. Such a subgraph is necessarily a tree called a Steiner tree for S. The Steiner interval I,(S) of S consists of all those vertices that lie on some Steiner tree for S. Let S be an n-set of vertices of G and suppose that k < n. Then the k-intersection interval of S. denoted by I,(S) is the intersection of all Steiner intervals of all k-subsets of S. It is shown that if s = ;a,, u2, . ..) u,j \ is a set of n 3 2 vertices of a graph G and if the 2-intersection interval of S is nonempty and x~l,(S), then d(S) = C:'= 1 d (u,,x). It is observed that the only graphs for which the ?-intersection intervals of all n-sets, n > 4, are nonempty are stars. Moreover. for every II > 4, those graphs with the property that the 3-intersection interval of every n-set is nonempty are completely characterized.
The asymptotic plurality rule, apl, is a consensus function which maps each profile P of length n... more The asymptotic plurality rule, apl, is a consensus function which maps each profile P of length n (i.e., each sequence of n bases appearing at an aligned position of n molecules) to a set apl (P) of consensus results (i.e., ambiguity codes) that is a descriptive summary of P. Our main result is to characterize each consensus result X = apl (P) in terms of the frequencies with which the bases in P occur. We then use these chamcterizations to investigate features (e.g., strong consistency, length independence) of apl that researchers may find useful for the interpretation of apl's consensus results.
The committee election problem is to choose from a finite set S of candidates a nonempty subset T... more The committee election problem is to choose from a finite set S of candidates a nonempty subset T of committee members as the consequence of an election in which each voter expresses a preference for a candidate in S. Solutions of this problem can be modelled by functions which map each partition of 1 (i.e., normalized vote tallies of candidates who have been ordered canonically by tally) into a nonempty subset of positive integers (i.e., sizes of committees). To solve this problem, we recently described a parameterized voting scheme, the ratio-of-sums or rasp consensus rule, in which p controls the degree to which votes must be concentrated in elected committees. It is desirable to identify the attainable results of such rules so as to understand their properties and to facilitate their comparison. For all p, we characterize the attainable rasp results in the general csse where the partition's parts are real, and in the special case where p as well as its parts are rational.
Working simultaneously in two teams [1,2], we have independently discovered essentially the same ... more Working simultaneously in two teams [1,2], we have independently discovered essentially the same concept and many common results. As expected, each team used its own notation and terminology but the results are easily transformed between the two systems. We plan to publish our full papers separately, but present the results here.
Let G be a graph and U, L' two vertices of G. Then the interval from K to 2' consists of all thos... more Let G be a graph and U, L' two vertices of G. Then the interval from K to 2' consists of all those vertices that lie on some shortest u -1; path. Let S be a set of vertices in a connected graph G. Then the Steiner distance d,(S) of S in G is the smallest number of edges in a connected subgraph of G that contains S. Such a subgraph is necessarily a tree called a Steiner tree for S. The Steiner interval I,(S) of S consists of all those vertices that lie on some Steiner tree for S. Let S be an n-set of vertices of G and suppose that k < n. Then the k-intersection interval of S. denoted by I,(S) is the intersection of all Steiner intervals of all k-subsets of S. It is shown that if s = ;a,, u2, . ..) u,j \ is a set of n 3 2 vertices of a graph G and if the 2-intersection interval of S is nonempty and x~l,(S), then d(S) = C:'= 1 d (u,,x). It is observed that the only graphs for which the ?-intersection intervals of all n-sets, n > 4, are nonempty are stars. Moreover. for every II > 4, those graphs with the property that the 3-intersection interval of every n-set is nonempty are completely characterized.
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Papers by Ewa Kubicka