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EUCLIDEAN DISTANCE GEOMETRY AND APPLICATIONS
"... Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We surv ..."
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Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of its most important applications, including molecular conformation, localization of sensor networks and statics. Key words. Matrix completion, barandjoint framework, graph rigidity, inverse problem, protein conformation, sensor network.
Evenness Preserving Operations on Musical Rhythms
"... Abstract In this paper we define four operations on musical rhythms that preserve a property called maximal evenness. The operations we define are shadow, complementation, concatenation, and alternation. The proofs of the theorems are omitted from this abstract. ..."
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Abstract In this paper we define four operations on musical rhythms that preserve a property called maximal evenness. The operations we define are shadow, complementation, concatenation, and alternation. The proofs of the theorems are omitted from this abstract.
CRESCENT CONFIGURATIONS
"... Abstract. In 1989, Erdős conjectured that for a sufficiently large n it is impossible to place n points in general position in a plane such that for every 1 ≤ i ≤ n − 1 there is a distance that occurs exactly i times. For small n this is possible and in his paper he provided constructions for n ≤ 8 ..."
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Abstract. In 1989, Erdős conjectured that for a sufficiently large n it is impossible to place n points in general position in a plane such that for every 1 ≤ i ≤ n − 1 there is a distance that occurs exactly i times. For small n this is possible and in his paper he provided constructions for n ≤ 8. The one for n = 5 was due to Pomerance while Palásti came up with the constructions for n = 7, 8. Constructions for n = 9 and above remain undiscovered, and little headway has been made toward a proof that for sufficiently large n no configuration exists. In this paper we consider a natural generalization to higher dimensions and provide a construction which shows that for any given n there exists a sufficiently large dimension d such that there is a configuration in ddimensional space meeting Erdős ’ criteria. 1.