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Symmetric polynomials are polynomials that are invariant under the action of the symmetric group, and they play an integral role in mathematics. The space of quasiin- variant polynomials, polynomials that are invariant under the action of the symmetric group to a certain order, were introduced by Feigin and Veselov. These spaces are modules over the ring of symmetric polynomials, and their Hilbert series in elds of characteristic 0 were also computed by Feigin and Veselov. In this paper, we study the Hilbert series of these spaces in elds of positive char- acteristic. Braverman, Etingof, and Finkelberg recently introduced spaces of twisted quasiinvariant polynomials, a generalization of quasiinvariant polynomials in which the space is twisted by a monomial. We extend some of their results to spaces twisted by a product of smooth functions and compute the Hilbert series of the space in certain cases.

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.

This paper researches on the judgment theorem and proof of the equivalency condition of a class of symmetric inequalities. By controlling two elementary symmetric polynomials and using the monotonicity of functions and Jensen inequality, it finds the necessary and sufficient condition of the equivalency a class of three-variable and n-variables symmetric inequalities. And we illustrate the application of this method in proof of these inequalities. Then we obtain several judgment theorems on symmetric and cyclic inequalities.

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The traditional kaleidoscope uses the prism as a basic unit. Through the reflections of the mirrors on the inner surfaces of a prism, we can see many symmetric axles and patterns. However, traditional kaleidoscope only presents two-dimensional images. The three-dimensional kaleidoscope we designed uses the pyramid as a basic unit. We can see three-dimensional images via mirror faces on the inner surfaces of the pyramid. Our examination shows that not all pyramids are able to show three-dimensional images but only pyramids with specific angles do. The kaleidoscope we designed differs from the traditional kaleidoscope in the following ways: ①Our three-dimensional kaleidoscope not only shows three-dimensional space, but also shows it through two-dimensional planes. ②Our three-dimensional kaleidoscope is based on three kinds of pyramid: three-sided pyramid (tetrahedron), four-sided pyramid (square pyramid) and five-sided pyramid (pentagonal pyramid). Each kind of pyramid has its own charm in showing its reflected images. ③Each kind of pyramid is separated from its entire geometry. However, it can present the whole view of the geometry after mirrored. Our exploration shows that two dimensional and three dimensional worlds can change into one another under specific conditions. Perhaps before long we can discover more dimensions and explain the relationships between each kind of dimension. This treatise mainly involves the principles of Euclidean Geometry and Projective Geometry.

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Motivated by an extra credit problem from our Linear Algebra class, we study the invertibility probability of binary matrices (the number of invertible binary matrices divided by the total number of binary matrices). Binary matrices are of interest in combinatorics, information theory, cryptology, and graph theory. It is known that the invertibility probability of n × n binary matrices goes to 1 as n→∞. We conjecture that this probability monotonically increases as the size of the binary matrix increases, and we investigate this by exploring how n×n binary matrices of rank n and rank (n−1) can be enlarged to (n + 1) × (n + 1) invertible binary matrices. Calculating this explicitly for the identity matrix, we obtain a probable bound that would show that, in a sense, our conjecture is asymptotically true. With the use of a computer, we also computed how many (n + 1) × (n + 1) invertible binary matrices can be enlarged from n×n matrices of rank n and rank (n−1) for small n. In addition, we study the invertibility probability of matrices with entries in Z_q.

Tangram is one of the oldest puzzle games in China. It can make up volatile and wonderful patterns. To master the game, it would be ideal if the players can go through several complexity levels from the easiest to the most complicated. However, tangram today does not have the difficulty grade. Only a similar puzzle game—Japanese jigsaw puzzle has the difficulty grade which is not very reasonable. And the grading process is not provided. Therefore, we try to provide the theoretical basis of dividing the difficulty grades by means of probability of the patterns. We hope we can make some complements to this ancient game. Among the complicated influential factors, we sum up three key elements: the number of answers，the length of side, the basic figures. Based on these points, we have devised three plans, the procedural thinking of computer, the length-of-side principle, division principle, to calculate the probability. Then, we made comprehensive analysis on the probability. Finally, we got the difficulty grade of some classical tangram patterns. Moreover, we summarize some skills of piecing together the patterns.

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