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 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 between the vertices. That is an example of a directed graph. Definitions Directed Graph A directed graph is a set $V$ with a binary relation $\rightarrow$. By a binary relation we mean a subset of the Cartesian product $V\times…


The Tensor Product: from vector spaces to categories
Introduction General Definition The tensor product is not the same as the Cartesian product A jump to categories Applications TensorFlow Artificial Intelligence 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 \[ \left[\begin{array}{ccc} x_{1}y_{1} & x_{1}y_{2} & x_{1}y_{_{3}}\\ x_{2}y_{1} & x_{2}y_{2} & x_{2}y_{3} \end{array}\right]. \] The collection of all the products $\left\langle x_{1},x_{2}\right\rangle \otimes\left\langle y_{1},y_{2},y_{3}\right\rangle $, up to some equivalence, is called the tensor product of $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ and is denoted by $\mathbb{R}^{2}\otimes\mathbb{R}^{3}$. General Definition For any vector spaces $U,V,W$ over…

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 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 Introduction Convex sets are thought of as subsets of $\mathbb{R}^{n}$, with $n$ a nonnegative integer, and convex functions as realvalued functions on convex subsets of $\mathbb{R}^{n}$. For instance, see this paper, p. 11 and this one. We’ll show you that convex sets and convex functions can be seen in a more general framework. The setting: a vector space…

The Rank of a Matrix with some Association with Machine Learning and Quantum Computing
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 Lowrank 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…

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. 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, can be fun. Besides using softwares, you can show your students how to program a computer to graph trigonometric functions. Programming a computer to graph trigonometric functions Again, we think that it’s best to show your students how to graph trigonometric functions by hand before showing them how to use automated tools, and one reason is that…

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. After you’ve explained to your students that the sine and cosine are functions from the real numbers to the real numbers, you may need to mention that the tangent, cotangent, secant, and cosecant are also functions of a real variable but that are undefined for some real numbers. Explain to them the reason for the undefinedness: these last four functions are defined as ratios, where their denominators are variables, which cannot be zero, so all the angles for which these ratios have denominators equal…

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. After you’ve spent some time practicing with your students and give them some time to work on exercises on their own, you may need to move on to other topics. Now that you’re done working with evaluating trigonometric functions on an acute angle, it’s probably time to introduce evaluating trigonometric functions on an arbitrary angle. Much of the trigonometry you’ll cover may involve nonacute angles, so it’s wise to bring them in. Nonacute angles You may need to emphasize to your students the…

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. 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, you will need to make some assumptions about their knowledge of some facts. If you choose not to do that, you may run into complications to…

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. 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 that the teacher is distinct from the student. As far as we are concerned, the teacher can also be the student. If you are teaching yourself trigonometry, then it is not too farfetched to…

Why Math?
You may have had bad memories from mathematics at school, or maybe you liked it and stayed curious about it or even work in a related field. But in any case, you may ask yourselves: why research in mathematics is important? You may think that we already know all we need to know in mathematics as we are already able, for instance, to make the sophisticated computations required to successfully send a rocket into space to explore other planets. You might be surprised, but continuing research in mathematics can and will change our lives. And even save lots of human beings. How could mathematics save lives? A few years ago,…