Spin polynomial invariants, as denned by Pidstrigatch and Tyurin, are used to define differentiable invariants for Dolgachev surfaces. In order to define and compute these invariants, some machinery is developed to study the wall structure for Spin polynomials, to deal with the fact that the moduli spaces have the wrong dimension and to handle nonreduced structures. The invariants are then computed.

Speech signals are often produced or received in the presence of noise, which is known to degrade the performance of a speech recognition system. In this paper, a perception-and PDE-based nonlinear transformation was developed to process spoken words in noisy environment. Our goal is to distinguish essential speech features and suppress noise so that the processed words are better recognized by a computer software. The nonlinear transformation was made on the spectrogram (short-term Fourier spectra) of speech signals, which reveals the signal energy distribution in time and frequency. The transformation reduces noise through time adaptation (reducing temporally slowly varying portions of spectra) and enhances spectral peaks (formants) by evolving a focusing quadratic fourth-order PDE. Short-term spectra of speech signals were initially divided into three (low, mid and high) frequency bands based

We carry out the programme of R. Liouville [19] to construct an explicit local obstruction to the existence of a Levi–Civita connection within a given projective structure [..] on a surface. The obstruction is of order 5 in the components of a connection in a projective class. It can be expressed as a point invariant for a second order ODE whose integral curves are the geodesics of [..] or as a weighted scalar projective invariant of the projective class. If the obstruction vanishes we find the sufficient conditions for the existence of a metric in the real analytic case. In the generic case they are expressed by the vanishing of two invariants of order 6 in the connection. In degenerate cases the sufficient obstruction is of order at most 8.

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