• Basic,  Mathematics,  Premium

    Topological Manifolds

      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…

  • Basic,  Mathematics,  Premium

    Two Notions of an Infinite Chain in a Directed Graph

    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…

  • Basic,  Mathematics,  Premium

    A Generalization of Convex Sets and Convex Functions

    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…

  • Basic,  Mathematics,  Premium

    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.     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 explain to them. But you will also need to remind them of the importance of prerequisite knowledge. To effectively communicate the technicalities to your students,…

  • Basic,  Mathematics,  Premium

    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…