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Strictly Monotone Weight Functions on Directed Graphs
OUTLINE Introduction General Definitions When can any weight function be turned into a strictly monotone weight function? Weighted Directed Graphs in Applications Autonomous Driving Quantum Walks Artificial Neural Networks SUMMARY: We define a directed graph as a space with any binary relation, and we define a strictly monotone weight function on any directed graph. We prove when no weight function on a directed graph can be turned into a strictly monotone weight function, and we give a characterization of any strictly monotone weight function. Finally, we mention the use of weighted directed graphs in autonomous driving research, quantum information, and deep learning. Introduction Here is an example…

Topological Manifolds
OUTLINE Introduction A generalization to infinite dimension Is there any relationship between an infinitedimensional manifold and a finitedimensional 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 finitedimensional manifold, and we show in what sense an infinitedimensional manifold is related to any finitedimensional 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 2dimensional ``flat'' disk as the circle gets smaller. The…

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Two Notions of an Infinite Chain in a Directed Graph
OUTLINE 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)…

The Tensor Product: from vector spaces to categories
OUTLINE Introduction General Definition The tensor product is not the same as the Cartesian product A jump to categories Applications TensorFlow Artificial Intelligence Quantum Optics SUMMARY: We show why the tensor product is not the same as the Cartesian product, and we extend that result to categories. We then mention the use of the tensor product in applications, such as artificial intelligence and quantum optics. Introduction For any vectors $\left\langle x_{1},x_{2}\right\rangle $ of $\mathbb{R}^{2}$ and $\left\langle y_{1},y_{2},y_{3}\right\rangle $ of $\mathbb{R}^{3}$, a product of these two vectors, which is denoted as $\left\langle x_{1},x_{2}\right\rangle \otimes\left\langle y_{1},y_{2},y_{3}\right\rangle $, is defined as the matrix \[ \left[\begin{array}{ccc} x_{1}y_{1} & x_{1}y_{2} & x_{1}y_{_{3}}\\ x_{2}y_{1} &…

A Generalization of Convex Sets and Convex Functions
OUTLINE Introduction The setting: a vector space over a field with a partial order A Convex set is the same as an orderconvex set A convex function defined in terms of a convex set A convex function in terms of orderconvexity Convex sets and convex functions in machine learning Convex Optimization Software Libraries SUMMARY: We define a convex set in a general framework of a vector space over a field with a partial order, and we show how the general notion is related to the usual notion of a convex set. Then we define a convex function in terms of that general notion of a convex set, and…