Introduction: “Convenient Notation” In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred  to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta.  The Kronecker delta is simply the indexed components of the identity operator in matrix algebra: [tex]\delta^j_k =\left\{\begin{array}{lcl}1&\text{ if }...

Abstract This article will build on the hints about treating the complex numbers as a branched surface, briefly described and pictured in section 4.2 of https://www.physicsforums.com/insights/views-on-complex-numbers/#The-Radish. Using a particular set of conventions, all the problems described in https://www.physicsforums.com/insights/things-can-go-wrong-complex-numbers/  can be removed, and the rules described there as applying only to reals generalized to complex numbers. A...

Abstract Fermat’s Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book...

The Concept A vector space is an additively written abelian group together with a field that operates on it. Intuitive picture vs abstract definition Vector spaces are often described as a set of arrows, i.e. a line segment with a direction that can be added, stretched, or compressed. That’s where the term linear to describe...

Abstract Topology as a branch of mathematics is a bracket that encompasses many different parts of mathematics. It is sometimes even difficult to see what all these branches have to do with each other or why they are all called topology. This article aims to shed light on this question and briefly summarize the content...

Stochastic calculus has come a long way since Robert Brown described the motion of pollen through a microscope in 1827. It’s now a key player in data science, quant finance, and mathematical biology. This article is drawn from notes I wrote for an undergraduate statistical physics course a few months ago. There won’t be any...

Abstract Why do we need yet another article about complex numbers? This is a valid question and I have asked it myself. I could mention that I wanted to gather the many different views that can be found elsewhere – Euler’s and Gauß’s perspectives, i.e. various historical views in the light of the traditionally parallel...

Preamble The classical mathematician practically by instinct views the continuous process as the “real” process, and the discrete process as an approximation to it. The mathematics of finance and certain topics in the modern theory of stochastic processes suggest that, in some cases at least, the opposite is true. Continuous processes are, generally speaking, the...

A division by zero is primarily an algebraic question. The reasoning therefore follows the indirect pattern of most algebraic proofs: What if it was allowed? Then we would get a contradiction, and a contradiction is the greatest enemy of mathematical rigor. Many students tried to find a way to divide by zero once in their...

Introduction Series play a decisive role in many branches of mathematics. They accompanied mathematical developments from Zeno of Elea (##5##-th century BC) and Archimedes of Syracuse (##3##-th century BC), to the fundamental building blocks of calculus from the ##17##-th century on, up to modern Lie theory which is crucial for our understanding of quantum theory....