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1. The Flow Matrix Offers a Straightforward Alternative to the Problematic Markov Matrix

   https://doi.org/10.3390/land12071471  

   Strzempko, Jessica; Pontius, Robert Gilmore (July 2023, Land)

   The Flow matrix is a novel method to describe and extrapolate transitions among categories. The Flow matrix extrapolates a constant transition size per unit of time on a time continuum with a maximum of one incident per observation during the extrapolation. The Flow matrix extrapolates linearly until the persistence of a category shrinks to zero. The Flow matrix has concepts and mathematics that are more straightforward than the Markov matrix. However, many scientists apply the Markov matrix by default because popular software packages offer no alternative to the Markov matrix, despite the conceptual and mathematical challenges that the Markov matrix poses. The Markov matrix extrapolates a constant transition proportion per time interval during whole-number multiples of the duration of the calibration time interval. The Markov extrapolation allows at most one incident per observation during each time interval but allows repeated incidents per observation through sequential time intervals. Many Markov extrapolations approach a steady state asymptotically through time as each category size approaches a constant. We use case studies concerning land change to illustrate the characteristics of the Flow and Markov matrices. The Flow and Markov extrapolations both deviate from the reference data during a validation time interval, implying there is no reason to prefer one matrix to the other in terms of correspondence with the processes that we analyzed. The two matrices differ substantially in terms of their underlying concepts and mathematical behaviors. Scientists should consider the ease of use and interpretation for each matrix when extrapolating transitions among categories. 

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2. The matrix Bochner problem

   https://doi.org/10.1353/ajm.2022.0022  

   Casper, W. Riley; Yakimov, Milen (August 2022, American Journal of Mathematics)
3. The Microservice Dependency Matrix

   Amr S. Abdelfattah and Tomas Cerny (October 2023, Springer)
4. Variational Properties of Matrix Functions via the Generalized Matrix-Fractional Function

   https://doi.org/10.1137/18M1209660  

   Burke, James V.; Gao, Yuan; Hoheisel, Tim (January 2019, SIAM Journal on Optimization)
5. The no boundary density matrix

   https://doi.org/10.1007/JHEP02(2025)124  

   Ivo, Victor; Li, Yue-Zhou; Maldacena, Juan (February 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> We discuss a no-boundary proposal for a subregion of the universe. In the classical approximation, this density matrix involves finding a specific classical solution of the equations of motion with no boundary. Beyond the usual no boundary condition at early times, we also have another no boundary condition in the region we trace out. We can find the prescription by starting from the usual Hartle-Hawking proposal for the wavefunction on a full slice and tracing out the unobserved region in the classical approximation. We discuss some specific subregions and compute the corresponding solutions. These geometries lead to phenomenologically unacceptable probabilities, as expected. We also discuss how the usual Coleman de Luccia bubble solutions can be interpreted as a possible no boundary contribution to the density matrix of the universe. These geometries lead to local (but not global) maxima of the probability that are phenomenologically acceptable. 

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