We propose methods of both slow and rapid post-processing of signals for erasure of artifacts that arise in the process of thresholding and quantization. We use wavelets as tools to define constraints and variational functionals as measures of complexity of signals. The methods come from analyses of different possibilities of blending variational calculus and wavelet multiresolution in ways that appear to be natural.

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The Mesoscopic Mechanics (MeM), as introduced in [5], is relevant to the electron gas confined to two spatial dimensions. It predicts a special way of collective response of correlated electrons to the external magnetic field. The dynamic variable of this theory is a finite-dimensional operator, which is required to satisfy the mesoscopic Schrödinger equation, cf. (2) below.In this article, we describe general solutions of the mesoscopic Schrödinger equation. Our approach is specific to the problem at hand. It relies on the unique structure of the equation and makes no reference to any other techniques, with the exception of the geometry of unitary groups. In conclusion, a surprising fact comes to light. Namely, the mesoscopic dynamics "filters" through the (microscopic) Schrödinger dynamics as the latter turns out to be a clearly separable part, in fact an autonomous factor, of the evolution. This is a desirable result also from the physical standpoint.

This article is concerned with the existence, status and description of the so-called emergent phenomena believed to occur in certain principally planar electronic systems. In fact, two distinctly different if inseparable tasks are accomplished. First, a rigorous mathematical model is proposed of emergent character, which is conceptually bonded with Quantum Mechanics while apparently non-derivable from the many-body Schrödinger equation. I call the resulting conceptual framework the Mesoscopic Mechanics (MeM). Its formulation is spaceindependent and comprises a nonlinear and holistic extension of the free electron model. Secondly, the question of relevancy of the proposed "emergent mechanics" to the actually observed phenomena is discussed. In particular, I postulate a probabilistic interpretation, and indicate how the theory could be applied and verified by experiment.The Mesoscopic Mechanics proposed here has been deduced from the Nonlinear Maxwell Theory (NMT)-a classical in character nonlinear field theory. This latter theory has already been shown to provide a consistent phenomenological model of such phenomena as superconductivity, charge stripes, magnetic vortex lattice, and magnetic oscillations. The NMT, which arose from geometric considerations, has long been awaiting an explanation as to its ties with the fundamental principles. I believe the MeM provides at least a partial explanation to this effect.

Hysteretic distortion is the common feature of signals generated by magnetic quantum-and nano-systems. In light of recent advances in nano-electronics, particularly the assent of memristance, it is expected that hysteretic features will become the trademark of next-generation electronics. Consequently, signal processing tasks, such as compression or denoising, will benefit from the use of bases which adapt to the hysteretic distortion. In this paper I describe a basis that consists of periodic functions with an inherent hysteretic feature. These functions originate from a certain nonlinear eigenvalue problem which, it is demonstrated here, relates to oscillators with memristance.