We develop a kernel density estimation method for estimating the density of points on a network and implement the method in the GIS environment. This method could be applied to, for instance, finding 'hot spots' of traffic accidents, street crimes or leakages in gas and oil pipe lines. We first show that the application of the ordinary two-dimensional kernel method to density estimation on a network produces biased estimates. Second, we formulate a 'natural' extension of the univariate kernel method to density estimation on a network, and prove that its estimator is biased; in particular, it overestimates the densities around nodes. Third, we formulate an unbiased discontinuous kernel function on a network, and fourth, an unbiased continuous kernel function on a network. Fifth, we develop computational methods for these kernels and derive their computational complexity. We also develop a plug-in tool for operating these methods in the GIS environment. Sixth, an application of the proposed methods to the density estimation of bag-snatches on streets is illustrated. Lastly, we summarize the major results and describe some suggestions for the practical use of the proposed methods.

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Abstract-We devised a method of fabricating easily transplantable scaffoldless 3D heart tissue, made with a novel cell-sheet (CS) technology from cultured cardiomyocytes using a fibrin polymer coated dish. In the present study, we tested in vivo electrical communication which is essential for improving heart function between the host heart and the grafted CS. The epicardial surface of the ventricle of an anesthetized open-chest nude rat was ablated by applying a heated metal. Bilayered CS was obtained from neonatal rat primary culture. CS was transplanted onto the injured myocardial surface (sMI) (sMIϩsheet group). The rats were allowed to recover for 1 to 4 weeks, to stabilize the grafts. Action potentials (APs) from the excised perfused heart were monitored by the fluorescence signal of di-4ANEPPS with a high speed charge-coupled device camera. The APs were observed under epicardial pacing of the host heart or the CS grafts. The pacing threshold of the current output was measured in the sMIϩsheet group and in the nongrafted sMI group at the center of the sMI and in the normal zone (Nz). Bidirectional AP propagation between the sMI and Nz was observed in the sMIϩsheet group (nϭ14), but was blocked at the marginal area of the sMI in the sMI group (nϭ9). The ratio of the pacing threshold (sMI/Nz) was significantly lower in the sMIϩsheet than in the sMI group (3.0Ϯ0.7, 19.0Ϯ6.1 respectively PϽ0.05). There were neither spontaneous nor pacing-induced arrhythmias in these two groups. Bidirectional smooth AP propagation between the host heart and the grafted CS was observed. This finding suggested functional integration of this CS graft with the host heart without serious arrhythmia.

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