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.

The subject of counting positive lattice points in n-dimensional simplexes has interested mathematicians for decades due to its applications in singularity theory and number the- ory. Enumerating the lattice points in a right-angled simplex is equivalent to determining the geometric genus of a singularity of a weighted homogeneous complex polynomial. It is also a method to shed insight into large gaps in the sequence of prime numbers. Seeking to contribute to these applications, this research project proves the Yau Geometric Con- jecture in six dimensions, a sharp upper bound for the number of positive lattice points in a six-dimensional tetrahedron. The main method of proof is summing existing sharp upper bounds for the number of points in 5-dimensional simplexes over the cross sections of the six- dimensional simplex. This research 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 Geometric and Yau Number-Theoretic Conjectures in full generality.

Matrix factorization methods are widely used for extracting latent factors for low rank matrix completion and rating prediction problems arising in recommender systems of on-line retailers. Most of the exist- ing models are based on L2 delity (quadratic functions of factorization error). In this work, a coordinate descent (CD) method is developed for matrix factorization under L1 delity so that the related minimization is done one variable at a time and the factorization error is sparsely distributed. In low rank random matrix completion and rating predic- tion of MovieLens 100k datasets, the CDL1 method shows remarkable stability and accuracy under gross corruption of training (observation) data while the L2 delity based methods rapidly deteriorate. A closed form analytical solution is found for the one-dimensional L1-delity sub- problem, and is used as a building block of CDL1 algorithm whose con- vergence is analyzed. A connection with robust principal component analysis is drawn.

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