## P vs. NP and what is a Turing Machine (TM)?

P or NP This article deals with the complexity of calculations and in particular the meaning of ##P\stackrel{?}{\neq}NP## Before we explain what P and NP actually are, we have to solve a far bigger problem: What is a calculation? And how do we measure its complexity? Many people might answer, that a calculation is an…

## The History and Importance of the Riemann Hypothesis

Riemann Hypothesis History RH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line. ERH: All zeros of L-functions to complex Dirichlet characters of finite cyclic groups within the critical strip lie on the critical line. Related Article: The Extended Riemann Hypothesis and Ramanujan’s Sum: Shortest Possible Explanation The history of the Riemann…

## The Extended Riemann Hypothesis and Ramanujan’s Sum

Riemann Hypothesis and Ramanujan’s Sum Explanation RH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line. ERH: All zeros of L-functions to complex Dirichlet characters of finite cyclic groups within the critical strip lie on the critical line. Related Article: The History and Importance of the Riemann Hypothesis The goal of this…

## The Amazing Relationship Between Integration And Euler’s Number

We use integration to measure lengths, areas, or volumes. This is a geometrical interpretation, but we want to examine an analytical interpretation that leads us to Integration reverses differentiation. Hence let us start with differentiation. Weierstraß Definition of Derivatives ##f## is differentiable at ##x## if there is a linear map ##D_{x}f##, such that \begin{equation*} \underbrace{D_{x}(f)}_{\text{Derivative}}\cdot…

## 10 Math Things We All Learnt Wrong At School

The title is admittedly clickbait. Or a joke. Or a provocation. It depends on with whom you speak, or who reads it with which expectation. Well, I cannot influence any of that. I can only tell how I mean it, namely as an entertaining collection of simple truths which later on turn out to be…

## How to Write a Math Proof and Their Structure

Proofs in mathematics are what mathematics is all about. They are subject to entire books, created entire theories like Fermat’s last theorem, are hard to understand like currently Mochizuki’s proof of the ABC conjecture, or need computer assistance like the 4-color-theorem. They are sometimes even missing, although everybody believes in the statement like the Riemann…

## Lie Algebras: A Walkthrough The Representations

Part III: Representations   10. Sums and Products. Frobenius began in ##1896## to generalize Weber’s group characters and soon investigated homomorphisms from finite groups into general linear groups ##GL(V)##, supported by earlier considerations from Dedekind. Representation theory was born, and it developed fast in the following decades. The basic object of interest, however, has…

## Learn Lie Algebras: A Walkthrough The Structures

Part II: Structures 5. Decompositions. Lie algebra theory is to a large extend the classification of the semisimple Lie algebras which are direct sums of the simple algebras listed in the previous paragraph, i.e. to show that those are all simple Lie algebras there are. Their counterpart are solvable Lie algebras, e.g. the Heisenberg algebra ##\mathfrak{H}=\langle…

## Learn Lie Algebras: A Walkthrough The Basics

Part I: Basics 1. Introduction. This article is meant to provide a quick reference guide to Lie algebras: the terminology, important theorems, and a brief overview of the subject. Physicists usually call the elements of Lie algebras generators, as for them they are merely differentials of trajectories, tangent vector fields generated by some operators. Thus…

## Match the Scientist with the Story Quiz

Among the most famous people are often geniuses. It’s hard to tell whether this is the reason for the many anecdotes which are told about them, or whether this is just incidentally true. Doubts are allowed, since most scientists are quite ordinary people. But some of them are cranky and all kind of stories circulate…

## Learn the Basics of Hilbert Spaces and Their Relatives: Operators

Operators. The Maze Of Definitions. We will use the conventions of part I (Basics), which are ##\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}##, ##z \mapsto \overline{z}## for the complex conjugate, ##\tau## for transposing matrices or vectors, which we interpret as written in a column if given a basis, and ##\dagger## for the combination of conjugation and transposition, the adjoint…

## Learn the Basics of Hilbert Spaces and Their Relatives

Basics Language first: There is no such thing as the Hilbert space. Hilbert spaces can look rather different, and which one is used in certain cases is by no means self-evident. To refer to Hilbert spaces by a definite article is like saying the moon when talking about Jupiter, or the car on an automotive…

## A Journey to The Manifold SU(2): Representations

Part 1   Representations Image source:    6. Some useful bases of ##\mathfrak{su}(2,\mathbb{C})## Notations can differ from author to author: the numbering of the Pauli matrices ##(\text{I 4}), (\ref{Pauli-I})##, the linear combinations of them in the definition to basis vectors ##\mathfrak{B}## of ##\mathfrak{su}(2,\mathbb{C}) \; (\text{I 5}), (\ref{Pauli-II}), (\ref{Pauli-III})##, the embedding of the orthogonal groups ##(\text{I…

## A Journey to The Manifold SU(2): Differentiation, Spheres, and Fiber Bundles

Part 2   Differentiation, Spheres, and Fiber Bundles Image source:  The special unitary groups play a significant role in the standard model in physics. Why? An elaborate answer would likely involve a lot of technical terms as Lie groups, Riemannian manifolds or Hilbert spaces, wave functions, generators, Casimir elements, or irreps. This already reveals…

## 10 Math Tips to Save Time and Avoid Mistakes

Exam situations are always situations of stress. It comes with our endeavor to be as good as possible together with our fear of failure. Some students handle these situations better than others. But there are some tricks I encountered over the years tutoring young students. I’m sure everybody has developed their own ways to get…

## How to Tell Operations, Operators, Functionals, and Representations Apart

All these concepts belong to the toolbox of physicists. I read them quite often on our forum and their usage is sometimes a bit confusing. Physicists learn how to apply them, but occasionally I get the impression, that the concepts behind are forgotten. So what are they? Especially when it comes to the adjoint…

## What Is a Tensor? The mathematical point of view.

Let me start with a counter-question. What is a number? Before you laugh, there is more to this question than one might think. A number can be something we use to count or more advanced an element of a field like the real numbers. Students might answer that a number is a scalar. This is…

## The Pantheon of Derivatives – Part V

Important Theorems – biased, of course Implicit Function Theorem  Jacobi Matrix (Chain Rule). Let ## (x_0,y_0 ) ## be a point in $$U_1 \times U_2 = \{x \in \mathbb{R}^k\,\vert \,||x-x_0||< \varepsilon_1 \} \times \{y \in \mathbb{R}^m\,\vert \,||y-y_0||< \varepsilon_2\}$$ and ## f: U_1 \times U_2 \rightarrow \mathbb{R}^m ## a function with  ##f(x_0,y_0)=0## which is totally…

## The Pantheon of Derivatives – Part IV

Lie Derivatives A Lie derivative is in general the differentiation of a tensor field along a vector field. This allows several applications since a tensor field includes a variety of instances, e.g. vectors, functions, or differential forms. In the case of vector fields, we additionally get a Lie algebra structure. This is, although formulated…

## The Pantheon of Derivatives – Part III

Some Topology Whereas the terminology of vector fields, trajectories, and flows almost by itself suggests its origins and physical relevance, the general treatment of vector fields, however, requires some abstractions. The following might appear to be purely mathematical constructions, and I will restrict myself to a minimum, but they actually occur in modern physics:…

## The Pantheon of Derivatives – Part II

Generalizations Beyond ##\mathbb{R}## and ##\mathbb{C}## As mentioned in the section on complex functions (The Pantheon of Derivatives – Part I), the main parts of defining a differentiation process are a norm and a direction. So to extend the differentiation concepts on normed vector spaces seems to be the obvious thing to do. Fréchet Derivative…

## The Pantheon of Derivatives – 5 Part Series

Differentiation in a Nutshell I want to gather the various concepts in one place, to reveal the similarities between them, as they are often hidden by the serial nature of a curriculum. There are many terms and special cases, which deal with the process of differentiation. The basic idea, however, is the same in…

## Linear Representations and Why Precision is Important in Math

First of all: What is a representation? It is the description of a mathematical object like a Lie group or a Lie algebra by its actions on another space 1). We further want this action to preserve the given structure because its structure is exactly what we’re interested in. And this other space here should…