MathSciDoc: An Archive for Mathematician ∫

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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.

In this project we study the energy functional on the set of Lagrangian tori in CP2. The energy functional has been introduced in [2] as integral of the potential of 2D periodic Schrödinger operator associated to Lagrangian torus. It has been conjectured in [2] that the Clifford torus is the unique global minimum of energy functional (the statement is later referred to as the energy conjecture). Due to geometric interpretation of energy functional as linear combination of the volume andWillmore functionals, this conjecture can be seen as the CP2 analogue of the well-known Willmore conjecture for tori in R3, recently proved in [18]. The energy conjecture has been verified for two families of Hamiltonian-minimal Lagrangian tori in [2]. Results of [5] and [23] imply the conjecture for minimal Lagrangian tori of sufficiently high spectral genus and non-embedded minimal Lagrangian tori, respectively. In the present work we prove the energy conjecture for a family of Hamiltonian-minimal Lagrangian tori in CP2 constructed in [4]. In sharp distinction with cases considered in [2], the value of the energy functional for these tori can not be calculated exactly. The proof relies on analytic bounds for certain elliptic integrals arising from the induced metric of tori. Possible directions of further work are: 1. Consider local behaviour of the energy functional. Are the critical points of the energy functional governed by an integrable PDE, akin to Tzizeica equation describing minimal Lagrangian tori? The same questions for critical points under Hamiltonian variations. 2. Is there an analogue of the energy conjecture for other Kähler-Einstein surfaces? The case of K3 surface is of special interest as minimal Lagrangian tori in K3 can be related to elliptic fibrations (for instance [20]) making the conjecture amenable to algebrogeometric analysis. 3. Examples of monotone Lagrangian tori with trivial Floer cohomology were constructed in [21]. Do there exist critical points of the energy functional with trivial Floer cohomology?

Checkpoint/restart is an important mechanism for system fault tolerance. It saves the state of an executing program periodically and recovers it after a failure. As many applications involve file operations, supporting file rollbacks is essential for checkpoint/restart. Complete backup can restore files to the correct state, but its cost is too high. In this paper we propose a behavior based file checkpointing strategy (BBFC), which provides a correct recovery of file data and ensures consistency between file state and other states of a process when a rollback is done by restarting the program from the last checkpoint. BBFC classifies details of the file operation behaviors and provides a guidance on what to be saved during file checkpointing according to those behaviors. It dramatically minimizes the overhead of file checkpointing due to the reduction of file data which need to be saved. And it is transparent and easy to use.

The Hamiltonian cycle problem is an important problem in graph theory and computer science. It is a special case of the famous traveling salesman problem. A signicant amount of research has been done on the special cases of nding Hamiltonian cycles in subgraphs of triangular, square and hexagonal grids. However, there is little work on more complicated grids. In this paper, we investigate the hardness of Hamiltonian cycle problem in grid graphs of semiregular tessellations, which are tessellations formed by two or more kinds of regular polygons. There are only eight semiregular tessellations, and we prove that the Hamiltonian cycle problem in all of them are NP-complete by reducing from NP-complete problems, such as the Hamiltonian cycle problem in max degree 3 bipartite planar graphs. Knowing the NP-completeness of Hamiltonian cycle problem in semiregular grids indicates that there will not be any polynomial time algorithm that solves the Hamiltonian path problem in these tessellations if NP does not equal to P, which helps show the limits of ecient motion planning algorithms and provides new information about what makes problems computationally dicult to solve.

Knotted ribbons form an important topic in knot theory. They have applications in natural sciences, such as cyclic duplex DNA modeling. A flat knotted ribbon can be obtained by gently pulling a knotted ribbon tight so that it becomes flat and folded. An important problem in knot theory is to study the minimal ratio of length to width of a flat knotted ribbon. This minimal ratio is called the ribbonlength of the knot. It has been conjectured that the ribbonlength has an upper bound and a lower bound which are both linear in the crossing number of the knot. In the first part of the paper, we use grid diagrams to construct flat knotted ribbons and prove an explicit quadratic upper bound on the ribbonlength for all non-trivial knots. We then improve the quadratic upper bound to a linear upper bound for all non-trivial torus knots and twist knots. Our approach of using grid diagrams to study flat knotted ribbons is novel and can likely be used to obtain a linear upper bound for more general families of knots. In the second part of the paper, we obtain a sharper linear upper bound on the ribbonlength for nontrivial twist knots by constructing a flat knotted ribbon via folding the ribbon over itself multiple times to shorten the length.

3D shape classification plays a fundamental role in geometric big data analysis. This work proposes a novel method for shape classification based on optimal mass transportation theory. The Riemann surfaces are mapped onto canonical domains conformally based on uniformization theorem. The conformal factor function is treated as probability distribution on the canonical domain. For each pair of probability distributions, the optimal mass transportation map is computed by solving Monge-Amper´e equation. The transportation cost is theWasserstein distance between two distributions. By using this distance, geometric classification based on clustering can be performed. The method is applied to 3D human facial expression recognition, which demonstrates the efficiency and efficacy of the method.

▲With the rapid development of bicycle-sharing system since 2016, illegal parking has become one of the most significant social issues. While the existing management methods cannot meet the demand, object detection technology provides a possibility. The Video Surveillance System can be utilized as a rich source of information. However, there is technical challenge to handle complex scenarios. Faster R-CNN, chosen because of its excellent overall performance, cannot specify the proposals of shared bikes especially in distant view. Processing the consecutive images in videos is neither necessary nor efficient. The original results are limited by interference of moving bikes and blocking effects. ▲ In this paper, I introduce a Faster R-CNN over Attention (FoA) that combines the from-coarse-to-fine visual mechanism with deep neural networks. FoA realizes fine-grained detection of shared bikes in complex circumstances. It consists of an Attention Region Extraction (ARE) and an optimized Faster R-CNN. In ARE, I introduce the concept of Attention Region and define it as video background, then apply Gaussian Mixture Model. ARE processes the surveillance video into discrete regions, allowing concentration on certain areas and efficiency in detection. In Faster R-CNN, I put forward an anchor box optimization in regards of the remarkable region-based characteristic in R-CNN. The optimization applies k-means to refine the anchor boxes and generate more appropriate region proposals. ▲ I construct a categorized shared bike data set of 4,291 images and 12,697 labeled objects for FoA training and testing. FoA performs excellently with the detection and recognition ratios of 94.19% and 92.73% respectively. The optimization is proved to make a difference. ▲ The paper not only provides a new direction for solving illegal parking issue and managing bicycle-sharing system, but also generalizes FoA into a new detection framework of Attention Region Extraction plus Region-Based Convolutional Neural Network, which can be applied to certain object detection in various scenes. The mechanism that combines ARE and R-CNN organically allows the towards-real-time detection to possess both practicality and universality. FoA is one of its robust implements.

This work focuses on the problem of high quality style transfer for Chinese characters, converting Chinese printed with standard typograph into those in calligraphic style. Traditional approaches include employing derivative network structures of CNN, RNN, and GAN, yet generation results are not satisfactory enough. In this paper, we proposed an integrated system of pix2pix structure, auxiliary classifier, and WGAN. Trials were run to evaluate the performance of traditional CNN and GAN. Cross-comparative sets of experiments among GAN, AC-GAN, and our proposed AC-WGAN have been conducted to test the capabilities of our proposed model. Inference and interpolation tests are conducted as well. Our proposed model outperformed existing style transfer system in generation’s delicacy, similarity, and efficiency. Incorporated with WGAN, the model also demonstrated a strong ability in providing a truthful training indicator with its loss. Study also suggested that AC and pix2pix adaption hold huge significance in accelerating the training process. Research validates the viability of fulfilling Chinese character transformation with style transfer. With handwriting recognition network, fast, high-quality Chinese character style transfer forms the basis for instantaneous conversion between printed and written work. Optimization on algorithm may be another direction for further exploration.

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.