In this article we introduce the notion of multi-Koszul algebra for the case of a nonnegatively graded connected algebra with a finite number of generators of degree 1 and with a finite number of relations, as a generalization of the notion of (generalized) Koszul algebras defined by R. Berger for homogeneous algebras, which were in turn an extension of Koszul algebras introduced by S. Priddy. Our definition is in some sense as closest as possible to the one given in the homogeneous case. Indeed, we give an equivalent description of the new definition in terms of the Tor (or Ext) groups, similar to the existing one for homogeneous algebras, and also a complete characterization of the multi-Koszul property, which derives from the study of some associated homogeneous algebras, providing a very strong link between the new definition and the generalized Koszul property for the associated homogeneous algebras mentioned before. We further obtain an explicit description of the Yoneda algebra of a multi-Koszul algebra. As a consequence, we get that the Yoneda algebra of a multi-Koszul algebra is generated in degrees 1 and 2, so a K2 algebra in the sense of T. Cassidy and B. Shelton. We also exhibit several examples and we provide a minimal graded projective resolution of the algebra A considered as an A-bimodule, which may be used to compute the Hochschild (co)homology groups. Finally, we find necessary and sufficient conditions on some (fixed) sequences of vector subspaces of the tensor powers of the base space V to obtain in this case the multi-Koszul property in the case we have relations in only two degrees.Mathematics subject classification 2010: 16E05, 16E30, 16E40, 16S37, 16W50.

Remotely sensed data are essential for understanding environmental dynamics, for their forecasting, and for early detection of disasters. Microwave remote sensing sensors complement the information provided by observations in the optical spectrum, with the advantage of being less sensitive to adverse atmospherical conditions and of carrying their own source of illumination. On the one hand, new generations and constellations of Synthetic Aperture Radar (SAR) sensors provide images with high spatial and temporal resolution and excellent coverage. On the other hand, SAR images suffer from speckle noise and need specific models and information extraction techniques. In this sense, the G0 family of distributions is a suitable model for SAR intensity data because it describes well areas with different degrees of texture. Information theory has gained a place in signal and image processing for parameter estimation and feature extraction. Entropy stands out as one of the most expressive features in this realm. We evaluate the performance of several parametric and non-parametric Shannon entropy estimators as input for supervised and unsupervised classification algorithms. We also propose a methodology for fine-tuning non-parametric entropy estimators. Finally, we apply these techniques to actual data.

The ultimate purpose of the statistical analysis of ordinal patterns is to characterize the distribution of the features they induce. In particular, knowing the joint distribution of the pair entropy-statistical complexity for a large class of time series models would allow statistical tests that are unavailable to date. Working in this direction, we characterize the asymptotic distribution of the empirical Shannon’s entropy for any model under which the true normalized entropy is neither zero nor one. We obtain the asymptotic distribution from the central limit theorem (assuming large time series), the multivariate delta method, and a third-order correction of its mean value. We discuss the applicability of other results (exact, first-, and second-order corrections) regarding their accuracy and numerical stability. Within a general framework for building test statistics about Shannon’s entropy, we present a bilateral test that verifies if there is enough evidence to reject the hypothesis that two signals produce ordinal patterns with the same Shannon’s entropy. We applied this bilateral test to the daily maximum temperature time series from three cities (Dublin, Edinburgh, and Miami) and obtained sensible results.

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