This paper explores the Nash equilibria of a variant of the Colonel Blotto game, which we call the Asymmetric Colonel Blotto game. In the Colonel Blotto game, two players simultaneously distribute forces across n battlefields. Within each battlefield, the player that allocates the higher level of force wins. The payo↵ of the game is the proportion of wins on the individual battlefields. In the asymmetric version, the levels of force distributed to the battlefields must be nondecreasing. In this paper, we find a family of Nash equilibria for the case with three battlefields and equal levels of force and prove the uniqueness of the marginal distributions. We also find the unique equilibrium payo↵ for all possible levels of force in the case with two battlefields, and obtain partial results for the unique equilibrium payo↵ for asymmetric levels of force in the case with three battlefields.

A physical method is developed to analyze the Riemann zeta function and the L-function. First, a physical model of an elastic bar fixed at both ends and subjected to symmetric axial loads with respect to the mid-span is constructed. The axial force of the bar at the left end is calculated with two methods: the equilibrium of forces and the principle of minimum potential energy. With the help of the orthogonality of {sinix} (i 1, 2, 3, ) , the Parseval identity and the Bessel inequality of the physical model are obtained. Further, it is proven that, in all the possible displacements which satisfy the boundary conditions, the real one minimizes the total potential energy of the bar. With the equivalence of the two methods, an identity of a type of infinite series is derived. Based on the identity, L(1) , L(3) , a recurrence formula of L(2k 1) (k  3) ,  (2) ,  (4) , and a relationship between  (2k) (k  2) and L(2i 1) (i  1, 2, ..., k) are then deduced with power axial load functions. The upper and lower bounds for the Riemann zeta function  (2k -1) and the L-function L(2k) are given. The numerical results show that the upper and lower bounds are in excellent agreement with the accurate value. Finally, two improvement methods are proposed for estimating the circumference ratio. The numerical results show that the two methods can estimate the circumference ratio with high accuracy and that the L -function is more efficient than the Riemann zeta function in estimating the circumference ratio.

Nowadays, pictures have more and more applications in various fields. Because different situations may have different commands of the ratios of pictures, we present a program which can resize the pictures according to their contents. Compared with the traditional method which only cares about the geometric constraints, our program can reserve the important contents of the pictures, which means that the visual effect of the pictures will not be damaged after the resizing. In order to change the ratio of pictures while avoiding distorting their main contents, we use a method called seam carving. Based on the energy function used for finding useless pixels and dynamic programming used for finding some continuous seams which will be deleted later, our program can finally change the image to a new ratio perfectly. Due to some deficiencies of the basic program, we also make some improvements. The first one is the improvement of the algorithm: we add face detection to the basic program. Not only does this improvement can solve many problems of the basic program automatically, but also it is our own innovation of the study of content-aware resizing. The second one is the addition of customization: we allow users to select the specific contents that they want to reserve, which can meet the different needs of users. It’s also our own innovation which will be very useful in practical situations. Our program can have many practical applications. We have some ideas about how to put this algorithm into real use, which is one of our practical innovations of this topic. In terms of different ratios of photo printing, graphic designing and screen displaying, this program will have various uses, which will make a groundbreaking progress in practice.

Information Technology has enabled new levels of convenience to our daily life. While providing people with astonishingly good experiences, computer science has still a long way to meet the needs from the users for various scenarios. As a good example, path selection problem is an important and common problem which finds applications in location based services, internet routing and navigation systems. Choosing the most cost effective and time saving routines plays an essential role in both theory and engineering. Although a plethora of classical solutions have been made to the static shortest path problem, dynamic shortest path problem has not been thoroughly studied. To address this critical issue, this paper investigates the dynamic shortest path problem by optimizing existing work and conducting extensive experiments. The studies fall into three main variants of the problem: unweighted shortest paths, weighted shortest paths, and shortest paths on the map. We present a comprehensive comparison of our optimizations and existing solutions which demonstrates the effectiveness and efficiency of our algorithms.

Opinions and sentiments are essential to human activities and have a wide variety of applications. As many decision makers turn to social media due to large volume of opinion data available, efficient and accurate sentiment analysis is necessary to extract those data. Hence, text sentiment analysis has recently become a popular field and has attracted many researchers. However, extracting sentiments from audio speech remains a challenge. This project explored the possibility of applying supervised Machine Learning in recognizing sentiments in English utterances on a sentence level. In addition, the project also aimed to examine the effect of combining acoustic and linguistic features on classification accuracy. Six audio tracks were randomly selected to be training data from 40 YouTube videos (monologue) with strong presence of sentiments. Speakers expressed sentiments towards products, films, or political events. These sentiments were manually labelled as negative and positive based on independent judgement of 3 experimenters. A wide range of acoustic and linguistic features were then analyzed and extracted using sound editing and text mining tools respectively. A novel approach was proposed, which used a simplified sentiment score to integrate linguistic features and estimate sentiment valence. This approach improved negation analysis and hence increased overall accuracy. Results showed that when both linguistic and acoustic features were used, accuracy of sentiment recognition improved significantly, and that excellent prediction was achieved when the four classifiers were trained respectively, namely kNN, SVM, Neural Network, and Naïve Bayes. Possible sources of error and inherent challenges of audio sentiment analysis were discussed to provide potential directions for future research.