Dijkstra and Bellman-ford are the two basic methods to solve Single-source shortest paths problem，and SPFA is the optimized version of Bellman-ford. However, since the algorithm time complexity is 0 (kn), SPFA’s time efficiency has some instabilities. This paper will compare the time efficiency between Dijkstra optimized with heap and three kinds of SPFA in different density graphs. Meanwhile, random data is created to test optimized Dijkstra and 3 types of SPFA. Finally, according to the calculation result of two methods’ time efficiency, some suggestion is given for their application under different situation.

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.

The log-optimal strategy in gambling and asset allocation problem attempts to maximize the expectation of the logarithmic rate of return but not the gross wealth itself. This strategy has been shown to have long termsuperiority overother strategies in theory. Another remarkable virtue is it allows updating the real-time market information quickly which could increase of the logarithmic rate of return. We will prove these properties ourselves in a simple way. We further consider how to do asset allocation by applying the log-optimal strategy in practice. The distribution function of the returns is usually unknown and need to be estimated from the history in a real world. The estimation error will affect the asset allocation directly, but such effect mayresult in a "butterfly effect" which could bring an investment disaster. Therefore, it is an important issue to answer whether there is logoptimal strategy resisted "butterfly effect". We develop a new log-optimal strategy based on the allocation utility which is a quadratic approximation to the expectation of the logarithmic rate of return. We show that the upper bound of the allocation utility deviation can be controlled by the l_1-norm of portfolio and l_∞-norm of the bias of variance matrix estimator. This turns out that our proposed strategy is robust. We apply our strategy for NYSE data. The results showed that our strategy has high performance in return.

In this report, we have mainly developed an abstract framework of k bit (t,n) secret sharing framework for cloud storage. Such framework is clean and it can be implemented. A successful implementation of the framework would provide users with protection when the system is under the attack on its confidentiality,integrity and reliability. Furthermore, such system has its own encryption by using permutations and tailor made error detection, location and data rescue. We make use of Lagrange polynomials and take the advantages of the algebraic property “t distinct points on the plane can uniquely determine a polynomial function of degree t-1” to design a k bit (t,n) -secret sharing distributed storage. We employ the set with unique factorization property (UFP) so that we simply need to calculate the y intercept of a Lagrange polynomial and then use a look up table to recover a secret.Moreover, the set which has minimum UFP would help us to design storage with smallest containers. In addition to the algebraic methods, we can utilize the geometric facts that three non collinear points determine a unique circle and four non coplanar points determine a unique sphere to construct k bit (3,n) and k bit (4,n) secret sharing storage respectively. To generalize to arbitrary case, it is straight forward if we have defined the Haar measure on the higher dimensional unit sphere. The last method is an application of Chinese Remainder Theorem (CRT) and we have designed k bit (t,n) secret sharing distributed storage and one of the designs can produce containers with the half size of the original secret. However, such k is no longer unrestricted and it has to be chosen from an interval. We have developed a C program for implementing both algebraic, geometric k bit (3,n) secret sharing distributed storages as well as the CRT method. The performances of both algebraic and geometric designs are satisfactory in term of processing time and compressed container size. The container size is even half of the size of the original secret in the CRT case and it is also very speedy.

The diffusive transport of passive tracers or particles can be enhanced by incompressible, turbulent flow fields. Analyzing the effective behavior is a challenging problem with theoretical and practical importance in many areas of science and engineering, ranging from the transport of mass, heat, and pollutants in geophysical flows to sea ice dynamics and turbulent combustion. The long time, large scale behavior of such systems is equivalent to an enhanced diffusion process with an effective diffusivity tensor D*. Two different formulations of integral representations for D* were developed for the case of time-independent fluid velocity fields, involving spectral measures of bounded self-adjoint operators acting on vector fields and scalar fields, respectively. Here, we extend both of these approaches to the case of space-time periodic velocity fields, allowing for chaotic dynamics, providing rigorous integral representations for D* involving spectral measures of unbounded self-adjoint operators.We prove the different formulations are equivalent. Their correspondence follows from a one-to-one isometry between the underlying Hilbert spaces. We also develop a Fourier method for computing D*, which captures the phenomenon of residual diffusion related to Lagrangian chaos of a model flow. This is reflected in the spectral measure by a concentration of mass near the spectral origin.