• Mathematics

    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 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…

  • Mathematics

    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…

  • Mathematics

    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,…