The problem we study in this paper originated from a simple high school math competition (AMC 12) question on a geometric probability model: “A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog’s final position is no more than 1 meter from its starting position?” [1] From here, we changed the number of jumps the frog made, the length of each jump and the dimensions of the space. Using recursion, we obtained the recursive formula of the probability of a frog landing x meters within its original spot after jumping m 1-meter jumps in an N -dimension space by integrating the probability distribution of the former step. We also suggested a concept “the intensity of the probability field” to describe the relative probability of each spot and gave an expression of the intensity of each spot of a 2-dimensional space when the frog makes m jumps.

Since the traffic congestion become more and more serious in modern society due to the sharp increasing of private cars, how to improve the transportation efficiency and utilize the current road network more effectively has become a crucial issue. In this paper, a new dynamic route guidance algorithm was proposed in order to provide travelers humanized “optimal route” and to alleviate the loss caused by traffic jams. The study built a graph theory model for Beijing’s ring road transportation system, and proposed a evaluation function &σ=V_f/[k×(t+m^(ρ-ρ0))]& to describe the real time complex traffic flow, and realized the route searching by timed recomputation of classic Dijkstra algorithm. Meanwhile, due to the investigation of special features of ring roads, the study improved the priority of ring road nodes during the searching process. Comparing with Dijkstra algorithm, the time-complexity of this new algorithm decreases to 1/(16k^2) (k is the number of ring road in the road network), and extra mileage is less than 5%, which is more effective applying in large scale ring-road networks. The algorithm was realized by C++ language and connected to Google Earth’s map database with easy operation interface. (The operation of the algorithm program was manifested in the appended video)

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.

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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.