First Order Linear Systems

A fundamental occupation of a mathematician is to describe a physical situation by a set of equations in order to solve real life problems. Most natural events can be expressed as differential and difference equations. In this respect, Ordinary Differential Equations (ODE) are one of the most useful parts of mathematics for theory and applications. The objective of this project is to study systems of linear differential and difference equations. First, we compare two solution forms for the first order matrix differential equation Y\u27=AY+YB. The first form, due to Neudecker, utilizes the Kronecker products of matrices to convert an n x n matrix differential equation into an n x n vector ODE. The second form, due to Murty, finds the solution in terms of the fundamental matrix solutions of two n x u vector ODE. This allows dealing with relatively much larger matrices than in Neudecker\u27s solution. Second, we present some basic results on the relation between the kth order difference equation and the companion matrix equation; these results are not available in the literature. Then, we present a set of necessary and sulfirient conditions for the complete controllability and observability of the general first order difference system. Examples are provided to illustrate many of the theoretical results

Cheminformatics and Machine Learning Approaches to Assess Aquatic Toxicity Profiles of Fullerene Derivatives

Fullerene derivatives (FDs) are widely used in nanomaterials production, the pharmaceutical industry and biomedicine. In the present study, we focused on the potential toxic effects of FDs on the aquatic environment. First, we analyzed the binding affinity of 169 FDs to 10 human proteins (1D6U, 1E3K, 1GOS, 1GS4, 1H82, 1OG5, 1UOM, 2F9Q, 2J0D, 3ERT) obtained from the Protein Data Bank (PDB) and showing high similarity to proteins from aquatic species. Then, the binding activity of 169 FDs to the enzyme acetylcholinesterase (AChE)—as a known target of toxins in fathead minnows and Daphnia magna, causing the inhibition of AChE—was analyzed. Finally, the structural aquatic toxicity alerts obtained from ToxAlert were used to confirm the possible mechanism of action. Machine learning and cheminformatics tools were used to analyze the data. Counter-propagation artificial neural network (CPANN) models were used to determine key binding properties of FDs to proteins associated with aquatic toxicity. Predicting the binding affinity of unknown FDs using quantitative structure–activity relationship (QSAR) models eliminates the need for complex and time-consuming calculations. The results of the study show which structural features of FDs have the greatest impact on aquatic organisms and help prioritize FDs and make manufacturing decisions