Spectral sensing involves a range of technologies for detecting, identifying chemicals and biological agents. An important application is in homeland security where a critical problem is identification of unknown explosives. Though the advances of modern spectroscopy technology have made it possible to classify pure chemicals by spectra, realistic data are often composed of mixtures of chemicals and environmental noise.

In most cases, one has to deal with a so called blind signal(source) separation (BSS) problem. Conventional approaches such as NMF and ICA are non-convex and too general to be robust and reliable in real-world applications. Based on a partial knowlege of the data (e.g. local spectral sparseness), we are able to reduce the problem to a series of convex sub-problems. Compressive sensing algorithms are also brought into play. The methods will be illustrated in processing of datasets from NMR, DOAS,and Raman spectroscopy.

We consider the heat equation in a multidimensional domain with nonlocal hysteresis feedback control in a boundary condition.
Thermostat is our prototype model.

By reducing the problem to a discontinuous infinite dynamical system, we construct all periodic solutions with exactly two switchings on the period and study their stability. In the problem under consideration, the hysteresis gap (the difference between the switching temperatures) is of especial importance.

If the hysteresis gap is large enough, then the constructed periodic solution is in fact unique and globally stable. For small values of hysteresis gap coexistence of several periodic solutions with different stability properties is proved to be possible.