Let$X$be a rearrangement-invariant Banach function space on$R$^{$n$}and let$V$^{1}$X$be the Sobolev space of functions whose gradient belongs to$X$. We give necessary and sufficient conditions on$X$under which$V$^{1}$X$is continuously embedded into BMO or into$L$_{∞}. In particular, we show that$L$_{$n, ∞$}is the largest rearrangement-invariant space$X$such that$V$^{1}$X$is continuously embedded into BMO and, similarly,$L$_{$n$, 1}is the largest rearrangement-invariant space$X$such that$V$^{1}$X$is continuously embedded into$L$_{∞}. We further show that$V$^{1}$X$is a subset of VMO if and only if every function from$X$has an absolutely continuous norm in$L$_{$n, ∞$}. A compact inclusion of$V$^{1}$X$into$C$^{0}is characterized as well.

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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.