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Xingdu is an emerging mathematical game about searching for a corresponding polyline for a given polyline in a 2-dimensional or 3-dimensional mesh grid, meanwhile the searched polyline should have the same start and end points as those of the given polyline and the corresponding line segments on the searched and given polylines are perpendicular each other respectively. Studying from mathematical perspective the problem solving and designing of Xingdu is helpful of finding a general method for Xingdu problem solving, but still not attract much more attentions due to the relatively short history of Xingdu. Referring to the achievements in mathematical methods for Sudoku, we investigated preliminarily the possibility of using optimization theory to solve the problem of Xingdu. According to the definition of Xingdu, the points on the given polyline and the solution polyline have to be the grid nodes in the mesh grid so that their coordinates are integer. In addition, there should not be any points appearing repeatedly and any three sequential points on the solution polyline cannot be located on a same line. Due to these constraints, the optimization problem related to Xingdu belongs to integer programming with nonlinear constraints, which is very difficult and usually can only be solved by enumerating. For this reason, we solve the related problem first by neglecting the nonlinear constraints so that it can be converted as an integer programming with linear constraints only. Afterwards, we can check whether it fulfill the related nonlinear constraints if a feasible solution is obtained. Based on that, we proposed two integer programming models for Xingdu problem, which are the model using only line segments perpendicular property as constraints and the binary integer programming model using line segments perpendicular property together with no repeated points as constraints. The test results show that the proposed binary integer programming model has a very good performance in finding the solution of Xingdu problem. We also discussed the possibility of using the proposed binary integer programming model together with random simulation to design a Xingdu problem. Due to the challenge of discriminating the uniqueness of Xingdu solution, we finally presented an approach, inspired from the branch and bound method for integer programming, which solves and designs the Xingdu problem by constructing and pruning a search tree.

In view of such problems as lacking interaction between figures and text and difficulty in simultaneous positioning of explanation with figures in chinese geometry exercise solution, a Win32 C++ voice lecturing tool integrated with HTML5 dynamic geometry software is designed to realize free control of lecturing progress, random skipping through double-click, simultaneous interaction and highlighting of figure and text, and automatic synchronized reading. On the basis of JavaScript regular expression, the semantic comprehension of geometry exercise text is realized and the production rule set of automatic plotting is designed. During voice synthesising line by line, the mathematic language undergoes normalization processing and figure highlighting information extraction. The study can shed some light on the GeoGebra integrated development and semantic comprehension-based automatic plotting application, and it can be applied to innovative geometry teaching and knowledge base structuring fields.

Motivated by Gutkin curves arising in capillary floating and billiard ball problem, we introduce a new projection function for convex plane curves, and study the curves when the defined projection function is constant. These curves can be regarded as a generalization of curves with constant width. We give a necessary and sufficient condition for the existence of such curves, and provide an explicit expression for the radius of curvature. Properties of our curves and their relation to Gutkin curves are also discussed. Finally, we construct and exhibit some curves with constant projection function.

With the rapid development of infrastructure including power grids, managers in power industry these days are faced with an increasingly severe problem of theft of electricity. Theft of electricity has negative effects on many socioeconomic aspects, including impacting stable growth of economic for power enterprises and social development. The traditional anti-theft means require officers checking the integrity of kilowatt-hour meter and the correctness of wiring house by house, which requires enormous manpower and material resources. With the advancement of information collecting technology, power enterprises now possess relatively complete database of power consumption. As a result, performing data mining on existing database and identifying abnormal users has become a hot topic in the field of information technology. The purpose of this paper is to identify abnormal users with machine learning algorithms, providing power enterprises with means to detect theft of electricity at lower cost. The contributions of this paper are listed as follows: 1) Propose various features, providing means to describe users’ behaviors. First, monistic features (mean, difference, coefficient of variation, range, standard deviation), binary features (cosine similarity, Pearson product-moment correlation coefficient) and multivariate features are calculated based on user power consumption (monthly, seasonally, yearly, per holiday, per workday). Then principal component analysis is used to perform dimensionality reduction on the dataset. The dataset is finally scaled and oversampled to form the feature dataset. 2) Study various classification algorithms, optimize hyperparameters, providing models to detect theft of electricity. In this paper, the mechanism behind support vector machine, back-propagation neural network, random forest and XGBoost is studied and the hyperparameters are optimized. 3) Compare and evaluate different classifiers, and provide suggestions for real-world application. In this paper, different classifiers are evaluated with different metrics and the best classifier is recommended based on real-world dataset and different application scenarios.