Abstract. Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems. New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter proofs of known results.

The aim of this paper is to identify the main drivers of CO 2 emissions in Spain, using an enhanced Structural Decomposition Analysis (SDA) from an extended Input-Output Model, which would allow both the direct and indirect effects of possible drivers to be captured. Six factors are considered; for two of them at wo-level decomposition is conducted. The approach used is a multisectoral one that offers a fine analysis, which is interesting for policy discussion. Data came from the World Input-Output Database

In this paper, a generalization of the concept of electrical power for periodic current and voltage waveforms based on a new generalized complex geometric algebra (GCGA), is proposed. This powerful tool permits, in n-sinusoidal/ nonlinear situations, representing and calculating the voltage, current, and apparent power in a single-port electrical network in terms of multivectors. The new expressions result in a novel representation of the apparent power, similar to the Steinmetz's phasor model, based on complex numbers, but limited to the purely sinusoidal case. The multivectorial approach presented is based on the frequency domain decomposition of the apparent power into three components: the real part and the imaginary part of the complex-scalar associated to active and reactive power respectively, and distortion power, associated to the complex-bivector. A geometrical interpretation of the multivectorial components of apparent power is discussed. Numerical examples illustrate the clear advantages of the suggested approach.

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