•   For related products, scroll to the end. ↓↓↓     Introduction A generalization to infinite dimension Is there any relationship between an infinite-dimensional manifold and a finite-dimensional manifold? Manifolds in applications Infection dynamics Topological data analysis Riemannian manifold optimization software library   SUMMARY: We define a manifold of an infinite dimension, which is an extension of a finite-dimensional manifold, and we show in what sense an infinite-dimensional manifold is related to any finite-dimensional manifold. We then mention the use of manifolds in applications. Introduction Given any point $x$ on the surface of a sphere and any circle drawn around $x$, the region inside the circle approaches the shape of a…
• Introduction Definitions Directed Graphs Infinite Chains (Def. 1 & Def. 2) When are Def. 1 and Def. 2 equivalent? A fix with the axiom of countable choice Directed Graphs in Applications Softwares Graph Neural Networks Quantum Information   SUMMARY: We introduce two notions of an infinite chain in a directed graph, and we show when these two notions are equivalent. We then mention the use of directed graphs in applications, such as artificial intelligence and quantum information. Introduction Consider this diagram $\begin{array}{ccccc} \bullet & \rightarrow & \bullet\\ \downarrow & & \downarrow\\ \bullet & \rightarrow & \bullet & \rightarrow & \bullet \end{array}$ which consists of vertices (the dots) and arrows…