Mathematics

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…

Introduction The concept of a group is as simple as it gets: a set with a binary operation like addition and a couple of natural laws like the requirement…

Abstract It took until the last century for physicists and mathematicians in the Netherlands to question the Euclidean concept of dimension as length,…

A division by zero is primarily an algebraic question. The reasoning therefore follows the indirect pattern of most algebraic proofs:What if it was…

Abstract "Mathematics is the native language of nature." is a phrase that is often used when it comes to explaining why mathematics is all around in natural…

Abstract Richard Pierce describes the intention of his book [2] about associative algebras as his attempt to prove that there is algebra after Galois…

Abstract My school teacher used to say "Everybody can differentiate, but it takes an artist to integrate." The mathematical reason behind this phrase…

This is not meant as a full introduction to recursion relations but it should suffice for just about any level of the student.Most of us remember recursion…

I asked myself why different scientists understand the same thing seemingly differently, especially the concept of a metric tensor. If we ask a topologist,…

Riemann Hypothesis and Ramanujan's Sum ExplanationRH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line. ERH: All…

In calculus classes when you are asked to evaluate a trig function at a specific angle, it's 99.9% of the time at one of the so-called special angles we…

So, why only odd powers? Mostly because the even powers were solved by Leonard Euler in the 18th century. Since the “mathematical toolbox” at that…

Confessions of a moderate Bayesian, part 4 Bayesian statistics by and for non-statisticiansRead part 1: How to Get Started with Bayesian StatisticsRead…

Confessions of a moderate Bayesian, part 2 Read Part 1: Confessions of a moderate Bayesian, part 1Bayesian statistics by and for non-statisticianshttps://www.cafepress.com/physicsforums.13280237 Background One…

Introduction In this brief Insight article the analytic continuations of the Lerch Transcendent and Riemann Zeta Functions are achieved via the Euler's…

What is SOHCAHTOA SOHCAHTOA is a mnemonic acronym used in trigonometry to remember the relationships between the sides and angles of right triangles.…

A fibre bundle is a mathematical structure consisting of a total space that looks locally like a simple product of two spaces but may have a different…

A parabola is a U-shaped curve where every point is equidistant from a fixed point called the focus and a fixed line called the directrix. Its standard…

A Lie algebra is the vector space of infinitesimal generators attached to a Lie group at its identity element, equipped with a bilinear antisymmetric commutator…

  Part III: Representations  Sums and Products. Frobenius began in ##1896## to generalize Weber's group characters and soon investigated…

The fundamental theorem of calculus states that integration and differentiation are reverse operations of one another. This article gives an intuitive,…

  Operators. The Maze Of Definitions. We will use the conventions of part I (Basics), which are ##\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}##,…

Intransitive dice are sets of dice that don't follow the usual rules for "is better/larger than". If A<B and B<C, then A<C. If Bob runs faster…

Part 2  Differentiation, Spheres, and Fiber Bundles Image source: [24]The special unitary groups play a significant role in the standard…

Introduction: In this Insight article,  Parseval's theorem will be applied to a sinusoidal signal that lasts a finite period of time.  It will be shown…

  Lie Derivatives A Lie derivative is in general the differentiation of a tensor field along a vector field. This allows several applications…

  Differentiation in a Nutshell I want to gather the various concepts in one place, to reveal the similarities between them, as they are often…

Introduction It is a remarkable fact that consideration of very elementary concepts in geometry often leads quickly into deep and unexpected mathematical…

Differentiation is usually taught quite well. Perhaps that's because it is the first introduction to calculus, which is considered a big step in a student's…

The paradox where integration seems to prove 1 = 0 arises from treating an indefinite integral as a single function instead of a set of antiderivatives.…

Ramsey's theorem guarantees that in any sufficiently large gathering, a specific pattern of mutual relationships must appear, no matter how the relationships…

Direct answer: After finishing analysis on real numbers and n-dimensional real space, the recommended path for self-study is Carothers' Real Analysis for…

A monographic substitution cipher works by replacing individual characters of plaintext with corresponding characters of ciphertext. It is perhaps the…