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•   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…
• 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} & x_{2}y_{2} &… • Introduction The setting: a vector space over a field with a partial order A Convex set is the same as an order-convex set A convex function defined in terms of a convex set A convex function in terms of order-convexity 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 we show… • Introduction General Definition Some results involving the rank of a matrix Theorem 5.1 (Ivanyos et al., 2018) Lemma 5.2 (Ivanyos et al., 2018) Rank of a matrix in machine learning Low-rank matrix approximation Apache Spark: a computer framework for matrix computation Rank of a matrix in quantum information Kraus Operators Introduction (back to outline) For a matrix \[A=\left[\begin{array}{ccc}0 & -1 & 1\\1 & 2 & -1\\1 & 1 & 3\end{array}\right],$ where each entry is a real number, the columns of $A$ form a linearly independent set of vectors of $\mathbb{R}^{3}$, so the subspace generated by the three columns is of dimension $3$, which is isomorphic to $\mathbb{R}^{3}$. The dimension…