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An Investigation into the Mathematical Nature of Electrophysiological Signals with Applications

In this work we have proposed a rigorous mathematical definition for the one dimensional time domain electrophysiological signals and established its relationship with two of the three Dirichlet's conditions. We have argues that any such signal can be represented as the trajectory of a particle moving in a force field with one degree of freedom. At point on the trajectory, that is, on the signal, the kinetic energy dissipated by the particle embeds semantic information into the trajectory or the signal in terms of giving its shape. We have shown that the rate of kinetic energy dissipation operator or the power operator P is of importance in shape analysis of the signal by considering its sign changes. Operating the P-operator on digital signals we have mathematically proved that its sign change can induce 13 different shapes to a three successive point configuration. In other words, semantic information at each point in a digital signal can be embedded by a syllable of 13 different letters. We have shown some preliminary applications.