• The nodes of the above graph are labeled with letters a,b,c,…, and each arrow between 2 nodes is assigned a weight. For instance, the arrow between node “a” and node “i” is assigned the weight 4. You need to shuffle the weights on the arrows so that, for any 3 nodes “label 1”, “label 2”, and “label 3”, if there is an arrow from “label 1” to “label 2” and an arrow from “label 2” to “label 3”, then the weight of the arrow from “label 1” to “label 2” is less than the weight of the arrow from “label 2” to “label 3”. For a graph without the weights,…

• 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 of a…

• 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 2-dimensional flat'' disk as the circle gets smaller. The surface of…

• With the coronavirus pandemic, you might find yourself separated from friends and family with no way to see them. Luckily, if you have access to a new smartphone, you have different options to get in touch with them. The best method is having a video call with them. There are many ways to have a video call on your iPhone 11 or Samsung Galaxy S20. iPhone 11 For the iPhone 11, there are a couple different methods you can use to make a video call. If the other person has an iPhone, you should use FaceTime. If they don’t have an iPhone, you can use WhatsApp to video call them.…

• 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…