In this paper we discuss properties of the KdV equation under periodic boundary conditions, especially those which are important to study perturbations of the equation. Next we review what is known now about long-time behaviour of solutions for perturbed KdV equations.

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We obtain two-sided sub-Gaussian estimates of heat kernels for strongly local Dirichlet forms on intervals, equipped with self-similar measures generated by iterated function systems (IFS's) that do not satisfy the open set condition (OSC) and have overlaps. We first give a framework for heat kernel estimates on intervals, and then consider examples of self-similar measures to illustrate this phenomenon. These examples include the infinite Bernoulli convolution associated with the golden ratio, and a family of convolutions of Cantor-type measures. We make use of Strichartz second-order identities defined by auxiliary IFS's to compute measures of cells on different levels. These auxiliary IFS's do satisfy the OSC and are used to define new metrics. The walk dimensions obtained under these new metrics are strictly greater than 2 and are closely related to the spectral dimension of fractal Laplacians.

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Laser scanning is an effective tool for acquiring geometric attributes of trees and vegetation, which lays a solid foundation for 3-dimensional tree modelling. Existing studies on tree modelling from laser scanning data are vast. However, some works cannot guarantee sufficient modelling accuracy, while some other works are mainly rule-based and therefore highly depend on user inputs. In this paper, we propose a novel method to accurately and automatically reconstruct detailed 3D tree models from laser scans. We first extract an initial tree skeleton from the input point cloud by establishing a minimum spanning tree using the Dijkstra shortest-path algorithm. Then, the initial tree skeleton is pruned by iteratively removing redundant components. After that, an optimization-based approach is performed to fit a sequence of cylinders to approximate the geometry of the tree branches. Experiments on various types of trees from different data sources demonstrate the effectiveness and robustness of our method. The overall fitting error (i.e., the distance between the input points and the output model) is less than 10 cm. The reconstructed tree models can be further applied in the precise estimation of tree attributes, urban landscape visualization, etc. The source code of this work is freely available at https://github.com/tudelft3d/adtree