Markov and signed Markov matrices for impact problems in layered elastic media.

What is it about?

Since their introduction in the early 1900s, Markov matrices have found wide use in various areas, particularly in probability-based models, where the matrix entries are interpreted as probability measures. As such, in a (right) Markov matrix, the non-negative entries along each row sum to one. This work identifies Markov and signed Markov matrices, in the context of impact problems in layered elastic media, where a homogeneous flyer hits a layered target attached to a half-space.

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Photo by Milad Fakurian on Unsplash

Photo by Milad Fakurian on Unsplash

Why is it important?

Our findings contribute to physical applications of Markov and signed Markov matrices, while discovering the wide occurrence of the latter. More specifically: - We establish necessary and sufficient conditions that detect and construct Markov matrices of any size, depending on the number of layers of the target and their material properties. - In most cases, however, while preserving the row sum of one, some of the matrix entries are negative, going beyond the traditional concept of probability measure. This leads to signed Markov matrices, not as widely found in the literature, but with interesting interpretations. - We provide physical insight in the creation and placement of the positive or negative entries of Markov and signed Markov matrices, in terms of the reflection and transmission coefficients. - We extend convergence properties of the matrix powers, from regular Markov to regular signed Markov matrix.

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