One family of 13315 stable periodic orbits of non-hierarchical unequal-mass triple systems

The three-body problem can be traced back to Newton in 1687,but it is still an open question today.Note that only a few periodic orbits of three-body systems were found in 300 years after Newton mentioned this famous problem.Although triple systems are common in astronomy,practically all observed periodic triple systems are hierarchical(similar to the Sun,Earth and Moon).It has traditionally been believed that non-hierarchical triple systems would be unstable and thus should disintegrate into a stable binary system and a single star,and consequently stable periodic orbits of non-hierarchical triple systems have been expected to be rather scarce.However,we report here one family of 135445 periodic orbits of non-hierarchical triple systems with unequal masses;13315 among them are stable.Compared with the narrow mass range(only 10-5)in which stable"Figure-eight"periodic orbits of three-body systems exist,our newly found stable periodic orbits have fairly large mass region.We find that many of these numerically found stable non-hierarchical periodic orbits have mass ratios close to those of hierarchical triple systems that have been measured with astronomical observations.This implies that these stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses quite possibly can be observed in practice.Our investigation also suggests that there should exist an infinite number of stable periodic orbits of non-hierarchical triple systems with distinctly unequal masses.Note that our approach has general meaning:in a similar way,every known family of periodic orbits of three-body systems with two or three equal masses can be used as a starting point to generate thousands of new periodic orbits of triple systems with distinctly unequal masses.

Mathematical Tools of Cluster Analysis

The paper deals with cluster analysis and comparison of clustering methods. Cluster analysis belongs to multivariate statistical methods. Cluster analysis is defined as general logical technique, procedure, which allows clustering variable objects into groups-clusters on the basis of similarity or dissimilarity. Cluster analysis involves computational procedures, of which purpose is to reduce a set of data on several relatively homogenous groups-clusters, while the condition of reduction is maximal and simultaneously minimal similarity of clusters. Similarity of objects is studied by the degree of similarity (correlation coefficient and association coefficient) or the degree of dissimilarity-degree of distance (distance coefficient). Methods of cluster analysis are on the basis of clustering classified as hierarchical or non-hierarchical methods.

Proposal for Generation of the Three-Way Perceptual Map Using Non-metric Multidimensional Scaling with Clusters

The generation of a perceptual map via three-way multidimensional scaling allows analysts to see the separation of objects in Euclidean space. The MDSvarext method incorporates the objects＇ confidence regions in this analysis, allowing for statistical inference in the difference between objects, but the confidence regions that are generated are very large because of the inherent variability among the evaluators. One solution to this problem is cluster generation prior to the application of the MDSvarext method in order to obtain homogeneous subgroups and to achieve greater control of the variance. This work is relevant to studies of perception which usually evaluate the difference between objects or stimuli in the point of view of different people that judge this difference using several dimensions. This study investigated the possibility of using a K-means algorithm to generate subgroups before the MDSvarext method was applied, evaluating the process with two quality indicators, one Ex-Ante and one Ex-Post. The experiments were conducted based on simulation of judgment matrix of different objects in multiple dimensions being evaluated by several judges. In this experiment, the matrix used was a 10 objects, in 10 features, judged by 10 people. The results are promising as possible interpretations of the perceptual map and the indicators generated.