This paper is a survey of two kinds of compressed proof schemes, the emph{matrix method} and emph{proof nets}, as applied to a variety of logics ranging along the substructural hierarchy from classical all the way down to the nonassociative Lambek sy
stem. A novel treatment of proof nets for the latter is provided. Descriptions of proof nets and matrices are given in a uniform notation based on sequents, so that the properties of the schemes for the various logics can be easily compared.
Time-frequency representations such as the spectrogram are commonly used to analyze signals having a time-varying distribution of spectral energy, but the spectrogram is constrained by an unfortunate tradeoff between resolution in time and frequency.
A method of achieving high-resolution spectral representations has been independently introduced by several parties. The technique has been variously named reassignment and remapping, but while the implementations have differed in details, they are all based on the same theoretical and mathematical foundation. In this work, we present a brief history of work on the method we will call the method of time-frequency reassignment, and present a unified mathematical description of the technique and its derivation. We will focus on the development of time-frequency reassignment in the context of the spectrogram, and conclude with a discussion of some current applications of the reassigned spectrogram.
