• directed graph math
    Mathematics

    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
    Mathematics

    Topological Manifolds

      OUTLINE 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…

  • Two Notions of an Infinite Chain in a Directed Graph
    Mathematics

    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
    Mathematics

    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} &…

  • math convex functions
    Mathematics

    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 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…

  • On Teaching Trigonometry: Graphing Trigonometric Functions (Part 2)
    Mathematics

    On Teaching Trigonometry: Graphing Trigonometric Functions (Part 2)

    Note: Thoughts expressed in this article are solely those of the author(s). Any advice given in this article may not work in all situations.   OUTLINE Programming a computer to graph trigonometric functions Selecting a programming language and libraries Python Python library: Matplotlib Python library: Numpy It's time for graphing! Questions about the computer-generated graphs   SUMMARY: We talk about how you can show your students to program a computer to graph trigonometric functions. We use the Python programming language as an example.    If you need to see Part 1 of this article, you can access it here. Showing your students how to graph trigonometric functions, using different techniques,…

  • On Teaching Trigonometry: Graphing Trigonometric Functions
    Mathematics

    On Teaching Trigonometry: Graphing Trigonometric Functions (Part 1)

    Note: Thoughts expressed in this article are solely those of the author(s). Any advice given in this article may not work in all situations. OUTLINE Domain and co-domain Motivating graphing of trigonometric functions Now it's time for graphing Sine and cosine It's time for practice Tangent and cotangent It's time for practice Secant and cosecant It's time for some games Use of technology to graph trigonometric functions Graphing calculators Geogebra Maple Wolfram Alpha And more...   SUMMARY:  We talk about how to introduce your students to graphing trigonometric functions.    After you’ve explained to your students that the sine and cosine are functions from the real numbers to the real…

  • On Teaching Trigonometry: Trigonometric Functions and the Unit Circle
    Mathematics

    On Teaching Trigonometry: Trigonometric Functions and the Unit Circle

    Note: Thoughts expressed in this article are solely those of the author(s). Any advice given in this article may not work in all situations.   OUTLINE Non-acute angles Generalizing the sine function Tangent, secant, and so on Now the unit circle The sine as a function on the real numbers SUMMARY: We talk about how to introduce your students to the general notion of a trigonometric function defined for any angle, acute or not, and we talk about how to use the unit circle to define trigonometric functions.   After you’ve spent some time practicing with your students and give them some time to work on exercises on their own,…

  • On Teaching Trigonometry: Angles and the Right Triangle
    Mathematics

    On Teaching Trigonometry: Angles and the Right Triangle

    Note: Thoughts expressed in this article are solely those of the author(s). Any advice given in this article may not work in all situations.   OUTLINE A good start with angles A note on practice Augmented Reality Motivating the right triangle Cosine, cotangent, and cosecant It's time for practice  On proof SUMMARY: We talk about how to start introducing your students to the trigonometry of the right triangle and to remind them of the importance of practice.   By now, many of your students may be all excited about trigonometry. You’ve managed to persuade them that it is worth to put the efforts to comprehend the technicalities that you will…

  • On Teaching Trigonometry: Introduction
    Mathematics

    On Teaching Trigonometry: Introduction

    Note: Thoughts expressed in this article are solely those of the author(s). Any advice given in this article may not work in all situations.     This is the first in a series of articles about ways that one could teach trigonometry: On Teaching Trigonometry: Angles and the Right Triangle On Teaching Trigonometry: Trigonometric Functions and the Unit Circle On Teaching Trigonometry: Graphing Trigonometric Functions (Part 1) On Teaching Trigonometry: Graphing Trigonometric Functions (Part 2)   We will not assume a specific setting in which the teaching will take place. So what we will say may be applicable to classroom settings or other settings. Also, we will not necessarily assume…