AbstractThe early 20th century witnessed an explosion of activity, much of it centered at Harvard, on rigorizing the property of ergodicity first proposed by Boltzmann in his 1898 Ergodic Hypothesis for ideal gases. Earlier, in the 1880’s, Henri Poincaré and Felix Klein had also initiated a study of discrete groups of hyperbolic isometries. The geodesics in hyperbolic manifolds were discovered...
AbstractThe gut microbiota is vital for digestive health and immune function. However, current research tends to focus on individual microbial strains, missing the broader understanding of the entire ecosystem. Constructing microbial networks is key to understanding how these communities form, but existing methods have limitations, prompting the need for new approaches. By merging concepts from...