A plotless density estimator with a Norton-Rice distribution for ordered distances

A Norton-Rice distribution(NRD)is a versatile,flexible distribution for k ordered distances from a random location to the k nearest objects.In a context of plotless density estimation(PDE)with n randomly chosen sample locations,and distances measured to the k=6 nearest objects,the NRD provided a good fit to distance data from seven populations with a census of forest tree stem locations.More importantly,the three parameters of a NRD followed a simple trend with the order(1,…,6)of observed distances.The trend is quantified and exploited in a proposed new PDE through a joint maximum likelihood estimation of the NRD parameters expressed as a functions of distance order.In simulated probability sampling from the seven populations,the proposed PDE had the lowest overall bias with a good performance potential when compared to three alternative PDEs.However,absolute bias increased by 0.8 percentage points when sample size decreased from 20 to 10.In terms of root mean squared error(RMSE),the new proposed estimator was at par with an estimator published in Ecology when this study was wrapping up,but otherwise superior to the remaining two investigated PDEs.Coverage of nominal 95%confidence intervals averaged 0.94 for the new proposed estimators and 0.90,0.96,and 0.90 for the comparison PDEs.Despite tangible improvements in PDEs over the last decades,a globally least biased PDE remains elusive.

Distribution of Miss Distance for Pursuit-Evasion Problem

Miss distance is a critical parameter of assessing the performance for highly maneuvering targets interception(HMTI). In a realistic terminal guidance system, the control of pursuer depends on the estimate of unknown state, thus the miss distance becomes a random variable with a prior unknown distribution. Currently, such a distribution is mainly evaluated by the method of Monte Carlo simulation. In this paper, by integrating the estimation error model of zero-effort miss distance(ZEM) obtained by our previous work, an analytic method for solving the distribution of miss distance is proposed, in which the system is presumed to use a bang-bang control strategy. By comparing with the results of Monte Carlo simulations under four different types of disturbances(maneuvers), the correctness of the proposed method is validated. Results of this paper provide a powerful tool for the design, analysis and performance evaluation of guidance system.

On the ErdÖs Distance Conjecture in Geometry

ErdÖs asks if it is possible to have n points in general position in the plane (no three on a line or four on a circle) such that for every i (1≤i≤n-1 ) there is a distance determined by the points that occur exactly i times. So far some examples have been discovered for 2≤n≤8 [1] [2]. A solution for the 8 point is provided by I. Palasti [3]. Here two other possible solutions for the 8 point case as well as all possible answers to 4 - 7 point cases are provided and finally a brief discussion on the generalization of the problem to higher dimensions is given.

Exponential distance distribution of connected neurons in simulations of two-dimensional in vitro neural network development

The distribution of the geometric distances of connected neurons is a practical factor underlying neural networks in the brain. It can affect the brain＇s dynamic properties at the ground level. Karbowski derived a power-law decay distribution that has not yet been verified by experiment. In this work, we check its validity using simulations with a phenomenological model. Based on the in vitro two- dimensional development of neural networks in culture vessels by Ito, we match the synapse number saturation time to obtain suitable parameters for the development process, then determine the distri-bution of distances between connected neurons under such conditions. Our simulations obtain a clear exponential distribution instead of a power-law one, which indicates that Karbowski＇s conclusion is invalid, at least for the case of in vitro neural network development in two-dimensional culture vessels.

EXTENDED CLUSTERING COEFFICIENTS：GENERALIZATION OF CLUSTERING COEFFICIENTS IN SMALL-WORLD NETWORKS

The clustering coefficient C of a network, which is a measure of direct connectivity between neighbors of the various nodes, ranges from 0 （for no connectivity） to 1 （for full connectivity）. We define extended clustering coefficients C（h） of a small-world network based on nodes that are at distance h from a source node, thus generalizing distance-1 neighborhoods employed in computing the ordinary clustering coefficient C = C（1）. Based on known results about the distance distribution Pδ（h） in a network, that is, the probability that a randomly chosen pair of vertices have distance h, we derive and experimentally validate the law Pδ（h）C（h） ≤ c log N / N, where c is a small constant that seldom exceeds 1. This result is significant because it shows that the product Pδ（h）C（h） is upper-bounded by a value that is considerably smaller than the product of maximum values for Pδ（h） and C（h）. Extended clustering coefficients and laws that govern them offer new insights into the structure of small-world networks and open up avenues for further exploration of their properties.