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How the New iPhone SE (2nd Generation) Compares with the iPhone 11 and iPhone 11 Pro
This article has affiliate links. For related products, scroll to the end. ↓↓↓ The new iPhone SE (2nd generation) is Apple’s sleek new economical alternative to their iPhone 11 line. Apple first introduced the SE in 2016 and now in 2020 we finally have a second generation SE. How does the iPhone SE stack up against the iPhone 11 and iPhone 11 Pro? The new SE is designed for users who are not content creators and prefer a phone that gives them that solid Apple experience at a solid price point. The SE is for users who want to post nice videos and photos to social media…

Strictly Monotone Weight Functions on Directed Graphs
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…

Topological Manifolds
For related products, scroll to the end. ↓↓↓ Introduction A generalization to infinite dimension Is there any relationship between an infinitedimensional manifold and a finitedimensional 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 finitedimensional manifold, and we show in what sense an infinitedimensional manifold is related to any finitedimensional 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…

How to Videoconference on your iPhone 11 and Android 11 Phone
This article has affiliate links. For related products, scroll to the end. ↓↓↓ 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…

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…

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 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} &…

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 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…

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. 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, 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…