Because of its importance in number theory and singularity theory, the problem of nding a polynomial sharp upper estimate of the number of positive integral points in an n- dimensional (n  3) polyhedron has received attention by a lot of mathematicians. S. S.-T. Yau proposed the upper estimate, so-called the Yau Number Theoretic Conjecture. The previous results on the Yau Number Theoretic Conjecture in low dimension cases (n  6) have been proved by using the sharp GLY conjecture. Unfortunately, it is only valid in low dimension. The Yau Number Theoretic Conjecture for n = 7 has been shown with a completely new method in [19]. In this paper, the similar method has been applied to prove the Yau Number Theoretic Conjecture for n = 8, but with more meticulous analyses. The main method of proof is summing existing sharp upper bounds for the number of points in 7-dimensional simplexes over the cross sections of eight-dimensional simplex. This reasearch project paves the way for the proof of a fully general sharp upper bound for the number of lattice points in a simplex. It also moves the mathematical community one step closer towards proving the Yau Number Theoretic Conjecture in full generality. As an application, we give a sharper estimate of the Dickman-De Bruijn function (x; y) for 5  y < 23, compared with the result obtained by Ennola.

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.

Nowadays, China has become the country with the largest aging population. The increasing proportion of the elders inspires the development of related industries. Falling is extremely dangerous for the elders. Thus, it is of vital importance to detect it and send an SOS signal immediately when an elder falls. Existing methods in fall detection include vision based, wearable sensor-based, ambinent device-based techniques. However, there are drawbacks in these existing methods. The video detection cannot guarantee the privacy of the elderly. Monitoring the changing of the environment requires high cost and its accuracy may be degraded by similar phenomena like strenuous exercise or a heavy object dropping down. The wearable sensors need to be worn all the time which is uncomfortable. In this paper, widely deployed Wi-Fi system is used for detecting a fall. It has little influence on the daily life of the elderly and can make up the drawbacks of the methods above. Because human movements would affect wifi signal propagation, we could detect human behaviors, like falling, through wifi. We firstly extract Channel State Information(CSI) from the original wifi signals collected. After performing noise and dimensionality reduction via Principal Component Analysis (PCA), the characteristic data of the Wi-Fi signal is obtained. Then, the detection model is trained using Recurrent Neural Networks (RNN) based on the PyTorch deep learning. Finally, fall detection is achieved automatically based on the trained model. For further improvement of its function, a strategy based on Random State Reset (RSR) and RNN is brought up in this paper, called RSR-RNN. Related experiments have been conducted to verify the feasibility of the method developed in this paper. Six types of fall scenarios are designed according to the data collected under different situations. The results of the experiments show that the RNN model can detect a fall effectively at an accuracy of 84.67%. The average accuracy of detection is further increased to 85.83% using RSR-RNN, which indicates that this strategy is effective. Keywords:

No abstract uploaded!

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.