Alternate Entropy Computations by Applying Recurrence Matrix Masking

In practicality, recurrence analyses of dynamical systems can only process short sections of signals that may be infinitely long. By necessity, the recurrence plot and its quantifications are constrained within a truncated triangle that clips the signals at its borders. Recurrence variables defined within these confining borders can be influenced more or less by truncation effects depending upon the system under evaluation. In this study, the question being asked is what if the boundary borders were tilted, what would be the effect on all recurrence variables? This question was prompted by the observation that line entropy values are maximized for highly periodic systems in which the infinitely long line elements are truncated to different unique lengths. However, by redefining the recurrence plot area to a 45-degree tilted box within the triangular area, the diagonal lines would consequently be truncated to identical lengths. Such masking would minimize the line entropy to 0.000 bits/bin. However, what new truncation influences would be imposed on the other recurrence variables? This question is examined by comparing recurrence variables computed with the triangular recurrence area versus boxed recurrence area. Examples include the logistic equation (mathematical series), the Dow Jones Industrial Average over a decade (real-word data), and a square wave pulse (toy series). Good agreement among the variables in terms of timing and amplitude was found for most, but not all variables. These important results are discussed.

Computational data analysis shows that key developments towards the periodic system occurred in the 1840s

The periodic system arose from knowledge about substances, which constitute the chemical space. Despite the importance of this interplay, little is known about how the expanding space affected the system. Here we show, by analysing the space between 1800 and 1869, how the periodic system evolved until its formulation. We found that after an unstable period culminating around 1826, the system began to converge to a backbone structure, unveiled in the 1860s, which was clearly evident in the 1840s. Hence, contrary to the belief that the ``ripe moment'' to formulate the system was in the 1860s, it was in the 1840s. The evolution of the system is marked by the rise of organic chemistry in the first quarter of the nineteenth-century, which prompted the recognition of relationships among main group elements and obscured some of transition metals, which explains why the formulators of the periodic system struggled accommodating them. We also introduced an algorithm to adjust the chemical space according to different sets of atomic weights, which allowed for estimating the resulting periodic systems of chemists using one or the other nineteenth-century atomic weights. These weights produce orderings of the elements very similar to that of 1869, while providing different similarity relationships among the elements, therefore producing different periodic systems. By analysing these systems, from Dalton up to Mendeleev, we found that Gmelin's atomic weights of 1843 produce systems remarkably similar to that of 1869, a similarity that was reinforced by the atomic weights on the years to come.