# Mathematics Articles

Mathematics as the study of “relationships” rather than “patterns” but they are obviously closely related(!). There is a field of mathematics called “category theory” that is just about as abstract as you can get (the textbook, in the preface, said category theory is often called “abstract nonsense” with no sense of that being derogatory at all).

A category has “objects” and “relations”. The collection of all sets is a category with sets as objects and functions between them as “relations”. The collection of topological spaces is a category with the topological spaces being the objects and continuous functions from one topological space to another being the relations.

### Posts ### The Analytic Continuation of the Lerch and the Zeta Functions

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Introduction In this brief Insight article the analytic continuations of the Lerch Transcendent and Riemann Zeta Functions are achieved via the Euler's… ### A Path to Fractional Integral Representations of Some Special Functions

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0. Introduction As for the reference material I have used the text Special Functions by Askey, Andrews, and Roy which covers much of the theorems here… /
Preface My first experience with derivatives was seeing how they are obtained from the usual definition $$f'(x)=\underset{\text{\Delta x}\to 0}{\text{Lim}}\frac{f… ### How to Find Potential Functions? A 10 Minute Introduction / Definition/Summary Given a vector field ##\vec F(x,y,z)## that has a potential function, how do you find it? Equations$$\nabla \phi(x,y,z) = \vec F(x,y,z)… ### What is a Linear Equation? A 5 Minute Introduction

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Definition/Summary A first-order polynomial equation in one variable, its general form is $Mx+B=0$ where x is the variable. The quantities… ### What are Significant Figures? A 5 Minute Introduction

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Definition/Summary Significant figures (commonly called "sig figs") are the number of figures (digits) included when rounding-off a number.For example,… ### How Most Proofs Are Structured and How to Write Them

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... or the answer to: "I have no idea where to start!" Proofs in mathematics are what mathematics is all about. They are subject to entire books, created… ### What is a Fibre Bundle? A 5 Minute Introduction

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Definition/Summary Intuitively speaking, a fibre bundle is space E which 'locally looks like' a product space B×F, but globally may have a different… ### What is a Real Number? A 5 Minute Introduction

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Definition/Summary The real numbers are the most commonly encountered number system, familiar to the layman via the number line, and as the number system… ### What is a Parabola? A 5 Minute Introduction

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Definition/Summary A parabola has many definitions, a classical one being, "A Parabola is the locus of all points equidistant from a given point (called… ### What Is a Limit of a Function? A 5 Minute Introduction

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Definition/Summary Limits are a mathematical tool that is used to define the 'limiting value' of a function i.e. the value a function seems to approach… ### What is a Tangent Line? A 5 Minute Introduction

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Definition/Summary The tangent to a curve in a plane at a particular point has the same Gradient as the curve has at that point.More generally, the… ### Lie Algebras: A Walkthrough The Representations

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Part III: Representations  10. Sums and Products. Frobenius began in ##1896## to generalize Weber's group characters and soon investigated… ### Learn Lie Algebras: A Walkthrough The Structures

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Part II: Structures5. Decompositions.Lie algebra theory is to a large extend the classification of the semisimple Lie algebras… ### Learn Lie Algebras: A Walkthrough The Basics

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Part I: Basics 1. Introduction. This article is meant to provide a quick reference guide to Lie algebras: the terminology, important theorems,… ### How to Self Study Abstract Algebra

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There are three big parts of mathematics: geometry, analysis, and algebra. In this insight, I will try to give a roadmap towards learning basic abstract… ### An Interesting Ramsey Theory Riddle

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Ramsey theory has its origins in a very nice riddle Consider a party of 6 people. Any two of these 6 will either be meeting each other for the first time… ### How to Self Study Intermediate Analysis

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If you wish to follow this guide, then you should know how to do analysis on ##\mathbb{R}## and ##\mathbb{R}^n##. See my previous insight if you wish to… ### Intro to the Millennium Prize Problems

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IntroductionIn this Insight, I will go over the background information for the Millennium Prize problems and briefly describe three of them. A future… ### An Intro on Real Numbers and Real Analysis

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It is important to realize that in standard mathematics, we attempt to characterize everything in terms of sets. This means that notions such as natural… ### Learn About Matrix Representations of Linear Transformations

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Let X and Y be finite-dimensional vector spaces. Let ##T:X\to Y## be a linear transformation. Let ##A=(e_1,\dots,e_n)## and ##B=(f_1,\dots,f_m)## be ordered… ### Why Do People Say That 1 And .999 Are Equal?

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Why do people say 1 and 0.999... are equal? Aren't they two different numbers?No, they really are the same number, though this is often very counterintuitive… ### Is There a Rigorous Proof Of 1 = 0.999…?

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Yes.First, we have not addressed what 0.999... actually means. So it's best first to describe what on earth the notation $$b_0.b_1b_2b_3...$$… ### The History and Concept of the Number 0

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The goal of this FAQ is to clear up the concept of 0 and specifically the operations that are allowed with 0.The best way to start this FAQ is to look… ### How I Overcame Learning Challenges That I Faced Studying Science

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Introduction For the past few days, during my summer break, I have been intensively self-studying mathematics (namely number theory) for several hours…