# Math

## Math Summary

• Would be great to download all about math in a moment into our conciosness - don't know how, so split in essential parts, summarize and work out best presentation to understand and to memorize
• LR arithmetics

## Computer parts

Processor Memory I/O

## LaTeX

LaTeX Symbols artofproblemsolving.com - LaTeX Commands

The default math delimiters are $...$ and $$...$$ for displayed mathematics, and $$$...$$$ for in-line mathematics. Note in particular that the $...$ in-line delimiters are not used by default
\lt and \gt instead of < and > inside of TeX

TeX Demo 1:
When $a \ne 0$, there are two solutions to $ax^2 + bx + c = 0$ and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$

Demo 2:
$$(2\pi h)^{-d}\iint_{\{H(x,\xi) <\tau\}} dx d\xi$$

Demo 3 (display mode multi-size):
$$\sum \int \oint \prod \coprod \bigcap \bigcup \bigsqcup \bigvee \bigwedge \bigodot \bigotimes \bigoplus \biguplus$$

Demo 4 (inline mode):

$\sum \int \oint \prod \coprod \bigcap \bigcup \bigsqcup \bigvee \bigwedge \bigodot \bigotimes \bigoplus \biguplus$

Symbols - Mac keyboard German
√ = alt v --- ∫ = alt b --- ~ = alt n --- º = alt j --- ∆ = alt K --- – = alt - --- ∞ = alt , --- ≈ = alt x --- ª = alt h --- ƒ = alt f --- ∂ = alt d --- ± = alt + --- π = alt p --- ø = alt o --- Ω = alt z --- ∑ = alt w --- « = alt q --- ¿ = alt ? --- ≠ = alt = --- ¡ = alt 1 ---

### Greek Alphabet

Number Letter Name Spell ancient / modern

1 Α α alpha, άλφα /a/ /aː/
2 Β β beta, βήτα [b] / [v]
3 Γ γ gamma, γάμμα [ɡ] [ŋ] / [ɣ] ~ [ʝ]
4 Δ δ delta, δέλτα [d] / [ð]
5 Ε ε epsilon, έψιλον [e]

6 Ζ ζ zeta, ζήτα [zd] / [z]
7 Η η eta, ήτα [ɛː] / [i]
8 Θ θ theta, θήτα [tʰ] / [θ]
9 Ι ι iota, ιώτα [i] [iː] / [ʝ] [ɲ]
10 Κ κ kappa, κάππα [k] / ~ [c]

11 Λ λ lambda, λάμδα [l]
12 Μ μ mu, μυ [m]
13 Ν ν nu, νυ [n]
14 Ξ ξ xi, ξι [ks]
15 Ο ο omicron, όμικρον [o]

16 Π π pi, πι [p]
17 Ρ ρ rho, ρώ [r]
18 Σ σ/ς sigma, σίγμα [s] / ~ [z]
19 Τ τ tau, ταυ [t]
20 Υ υ upsilon, ύψιλον [y] [yː] / [i]

21 Φ φ phi, φι [pʰ] / [f]
22 Χ χ chi, χι [kʰ] / [x] ~ [ç]
23 Ψ ψ psi, ψι [ps]
24 Ω ω omega, ωμέγα [ɔː] / [o]

## Linear Algebra

Matrices
Matrix_(mathematics) WP - Matrices and arrays LaTeX wikibooks.org
matrix (plural matrices) rectangular array of numbers, symbols, expressions in rows and colums - dimensions of matrix below 2 x 3 (read 2 by 3) = 2 rows 3 colums: $\begin{bmatrix} 1 & 9 & -13 \\ 20 & 5 & -6 \end{bmatrix}$

## Abstract Algebra

Abstract algebra WP - Universal algebra WP - Bartel Leendert van der Waerden WP - Sophus Lie WP - Lie algebra WP - Lie Algebra (Lecture 1 of 10) YT
List of abstract algebra topics WP
What is the best introductory abstract algebra textbook? Why? quora.com
Abstract Algebra - 3rd Edition - by David S. Dummit and Richard M. Foote
Abstract Algebra: L1: a bit of history, definition of group, 8-29-2016 YT - Niels Henrik Abel 1802-1829 Norway WP - Évariste Galois1811-1832 France - solving polinomial by radicals - Galois theory and group theory, subfield Galois connections - WP - Group - Set WP - Georg Cantor 3.3.1845-1918 Germany WP - Complex number $\mathbb{C}$ --- a + bi or a + bj (a + ib or a + jb) WP - Fundamental theorem of algebra WP - Karl Weierstrass 31.10.1815-1897 Germany WP
Shortest abstract algebra book stackexchange.com
Elements of Abstract Algebra - 47.30 Edition by Allan Clark

Ring (mathematics) WP - Combinatorics WP - Game theory - John von Neumann (Hungarian Neumann, János Lajos) 28.12.1903-8.2.1957 - published over 150 papers in his life: about 60 in pure mathematics, 20 in physics, and 60 in applied mathematics
Niels Henrik Abel - Évariste Galois - Galois Theory WP - Field (mathematics) WP - Combinatorics WP

## Learning Modern Algebra

### From Early Attempts to Prove Fermat's Last Theorem

Learning Modern Algebra: From Early Attempts to Prove Fermat's Last Theorem by Al Cuoco, Joseph J. Rotman - 2013

Contents
Preface xiii
Notation xvii
1 Early Number Theory 1
2 Induction 45
3 Renaissance 81
4 Modular Arithmetic 131
5 Abstract Algebra 191
6 Arithmetic of Polynomials 233
7 Quotients, Fields, and Classical Problems 277
8 Cyclotomic Integers 329
9 Epilog 379
A Appendices 409
References 449
Index 451
About the Authors 459

### Preface

pXIII - First courses in abstract algebra usually cover number theory, groups, and commutative rings - first encounter with groups inadequate, we focus on number theory, polynomials, commutative rings - introduce groups in last chapter, earlier discussion of commutative rings allows explaining how groups are used to prove Abel's Theorem: there is no generalization of the quadratic, cubic, and quartic formulas giving the roots of the general quintic polynomial - proposal: undergraduate abstract algebra should be a sequence of two courses, with number theory and commutative rings in the first course, and groups and linear algebra (with scalars in arbitrary fields) in the second - natural direction for us is towards algebraic number theory, whereas the usual direction is towards Galois theory - 4000 years ago led to Pythagorean triples: positive integers a; b; c satisfying a2 + b2 = c2 - 2000 years ago all triplets found - Pierre de Fermat (between 31.10. and 6.12.1607 – 12.1.1665 France) searched for positive integer solutions to an + bn = cn for n > 2 --- called Fermat’s Last Theorem - end of 20th century, Andrew Wiles (11.4.1953 Cambridge, England) proved Fermat’s Last Theorem in 1995 (Guinness Book of World Records as the "most difficult mathematical problem" - Wiles recieved Norwegian Abel prize worth €600,000)
pIX - before solution challenge to mathematicians as climbing Mount Everest - number theory recorded in Euclid has similarities with behavior of polynomials, generalizations of prime numbers and unique factorization owe their initial study to attempts at proving Fermat’s Last Theorem - abstract algebra is not merely beautiful and interesting, but also a valuable, perhaps essential, topic for understanding high school mathematics
Some features of the book:
... connections ... sidenotes ... interspersed in the text are boxed “callouts” such as How to Think About It ... Historical Note ... - biographies based on www-history.mcs.st-andrews.ac.uk - Etymology traces out the origin of some mathematical terms - often helps to understand the ideas they name - example:

pXV - Etymology. The word mathematics comes from classical Greek; it means “knowledge,” “something learned.” But in ancient Rome through the thirteenth century, it meant “astronomy” and “astrology.” From the Middle Ages, it acquired its present meaning.
The word arithmetic comes from the Greek word meaning “the art of counting.”
The word geometry, in classical Greek, meant “science of measuring;” it arose from an earlier term meaning “land survey.”

acknowledgements ... - Keith Conrad's website
A Note to Students:
work through as many exercises as possible, especially those that appear difficult - quite often, you will learn something valuable even if you don’t solve it completely -
two special kinds of exercises:
• labeled Preview may seem to have little to do with the section at hand; they are designed to foreshadow upcoming topics, often with numerical experiments
• labeled Take it Further develop interesting ideas that are connected to the main themes of the text, but are somewhat off the beaten path - not essential for understanding what comes later in the text
- exercise marked with asterisk, such as 1.8*, means that it is either used in some proof or it is referred to elsewhere in the text. For ease of finding such exercises, all references to them have the form “Exercise 1.8 on page 6” giving both its number and the number of the page on which it occurs
A Note to Instructors:
We recommend giving reading assignments to preview upcoming material - this contributes to balancing experience and formality as described above, and it saves time - many important pages can be read and understood by students, and they should be discussed in class only if students ask questions about them - possible to use book as text for three hour one-semester course, but we strongly recommend that it be taught four hours per week

Notation
pXVII - from ∆ (a, b, c) - p4 - triangle with sides of lengths a; b; c --- to AT - p438 - transpose of matrix A

### 1 - Early Number Theory

p1 - Algebra, geometry, and number theory used for millennia - commerce, architecture, sailing, farming seasons
1.1 - Ancient Mathematics
quadratic formula important tool (1700 BCE) - x2 - x = 870 -
p3 - distances earth moon sun - example 1.1 China 2000 BCE: door (Diagonale = p) -> p2 = (p - 4)2 + (p - 2)2 ...
p4 - Definition: A triple ∆ (a, b, c) of positive integers with a2 + b2 = c2 is called a Pythagorean triple. --- 15 triplets, Babylon ... - Diophantus, ca. 250 CE, will find all Pythagorean triplets -
p5 -

### 3 - Renaissance

...
p92 - In this section and the next, we’ll develop complex numbers in a more careful and formal way, and we’ll see that complex numbers are as real as real numbers!

### 5 - Abstract Algebra

p191 -

Vocabulary
inadequate - unzulänglich, mangelhaft, unangemessen, unangepasst, unzureichend ... - pXIII
intersperse - vermischen, einstreuen, hier und da einfügen ... - pIX
merely - lediglich, bloß, nur - pIX

## Elements of Abstract Algebra

Foreword
pV - It seems to me far more interesting and profitable in an introductory study of modern algebra to carry a few topics to a significant depth ... firmly based on the historical development of the subject - book only three areas: group theory, Galois theory, and classical ideal theory - in each case there is more depth and detail - all three topics converge in the fundamental theorem of algebraic number theory for Galois extensions of the rational field, the final result of the book
Introduction
pIX - ... Algebra from Muhammad ibn Musa al-Khwarizmi about the year 825 A.D. - also in algorithm - next major advance 1545 publication of Artis Magnae sive de Regulis Algebraicis by Hieronymo Cardano (1501-1576), usually called Ars Magna, or "The Grand Art", gave complete solution of equations of the third and fourth degree - method we shall use is due to Hudde, about 1650
every complex number has precisely three cube roots - ! = 1 +Oi has the three cube roots ...

## A Book of Abstract Algebra

A Book of Abstract Algebra - 2nd Edition - by Charles C Pinter

Ch 1 - Why abstract algebra?
p12 - ...
p13 - Origins

## Contemporary Abstract Algebra

Contemporary Abstract Algebra - 8th Edition - by Joseph A. Gallian WP

## Abstract Algebra - Theory and Applications

Thomas W. Judson - Associate Professor, Department of Mathematics and Statistics - Stephen F. Austin State University - Piney Woods of East Texas - sagemath.org
Sage Edudays 3: Tom Judson -- Abstract Algebra

Preface
pVI - standard order: groups, rings, fields - no specic prerequisites - assume elementary knowledge of matrices and determinants - Sage advanced software system for math sagemath.org, ideal for assisting study of abstract algebra - use on own computer, local server, or on SageMathCloud

Vocabulary
assume - voraussetzen, annehmen, vermuten ... - pVII
prerequisites - Grundvoraussetzungen, Vorbedingungen, Erfordernisse, Bedingungen ... - p VII
sophistication - Erfahrenheit, Gewandtheit, Raffinesse, Kultiviertheit, Vollkommenheitsgrad, Sophistikation ... - p VII

## Algebra

Moderne Algebra by Bartel Leendert van der Waerden February 2.2.1903–12.1.1996 Netherlands WP

## Algebra

Michael Artin WP -

## Abstract Algebra Maths 113

Alexander Paulin - December 8, 2010 - Lectures 1-24 (most Lectures 4 pages, L8 + L10 + L19 3 pages, L2 + L12 + L1 + L14-L16 5 pages, total p99 end)

Lecture 1
p1 - book recommendations: Classic Algebra by P.M.Cohn - This is hard but is the gold standard in my opinion - Algebra by Michael Artin - fantastic and easier to digest
What is Algebra
p2 - Algebra is the abstract encapsulation of our intuition for how arithmetic behaves - summarize first 6 yrs of math education:
concept of unity: number 1 --- N := {1, 2, 3...}
...
- (mathematics, logic) The symbol used in predicate calculus, etc, to represent the universal quantifier, meaning "for all" - an upside-down capital letter A - introduced by Gerhard Gentzen who based it on the Latin letter A, by analogy with - synonym: universal quantifier: - LaTeX: \forall
- (mathematics, logic) The existential quantifier, meaning "there exists (at least one)" - synonym: existential quantifier: - LaTeX: \exists
- (set theory) n-ary union of sets - antonym: - LaTeX: \bigcap
- (set theory) n-ary intersection of sets - antonym: - LaTeX: \bigcup
- (mathematics) is an element in the set of… x ∈ ℕ denotes that x is within the set of natural numbers - from the Greek letter ϵ (lunate epsilon) - the coinage is commonly attributed to Giuseppe Peano, who used the letter epsilon for set membership in his two papers from 1889 - antonym: - LaTeX: \in
- (mathematics) is not an element in the set of… - antonym: - LaTeX: \ni

Vocabulary
permeate - durchdringen, durchsetzen, eindringen, einziehen L1p2

## How to Prove It

### A Structured Approach

Daniel J. Velleman - Department of Mathematics and Computer Science at Amherst College, three hours north of New York City - Ph.D., University of Wisconsin-Madison (1980) - M.A., University of Wisconsin-Madison (1977) - B.A., Dartmouth College (1976) - A.M. (honorary), Amherst College (1992) - Second Edition
Theorem WP - List of theorems WP - Fundamental theorem WP - Lemma (mathematics) WP - Formula WP - Expression (mathematics) WP - List of unsolved problems in mathematics WP

Proof writing artofproblemsolving.com - Proof Designer (Java)
Case Study: How I Got the Highest Grade in my Discrete Math Class

Content
Preface (pIX) - Introduction (p1) - 1 Sentential Logic (p8) - 2 Quantificational Logic (p55) - 3 Proofs (84) - 4 Relations (p163) - 5 Functions (226) - 6 Mathematical Induction (p260) - 7 Infinite Sets (p306) - Appendix 1: Solutions to Selected Exercises (p329) - Appendix 2: Proof Designer (p373) - Suggestions for Further Reading (p 375) - Summary of Proof Techniques (p376) - Index (p381-384 end of book)

Preface
pIX - What distinguishes correct from incorrect proof - students learn proofs in high school like in former computer science from "list of instructions" to present "structured programming" - we may say book teaches "structured proving" - structured progr. not only listing instructions, but combining basic structures (if-else, do-while) and also nesting them - example:
do    if [condition]       [List of instructions goes here.]    else       [Alternate list of instructions goes here.] while [condition] ... pXII end

Introduction
p1 - math as deductive reasoning - ...

1 - Sentential Logic
1.1. Deductive Reasoning and Logical Connectives
p8 - introduction showed proofs central role in math - deductive reasoning is foundation on which proofs are based - now begin LR how it works:
Exmpl. 1.1.1. 3 valid examples of ded. reasoning: 1. rain or snow? 2. work today? 3. today or tomorrow - in each case arrived conclusion from assumted true premises - conclusion only false if at least one premise false, or true and conclusion is forced - this is used standard for correctness of deductive reasoning, argument valid if premises -
p9 - invalid: butler or maid, maid or cook
p or q - not q - therfore p

## Precalculus

Precalculus WP - algebra and trigonometry
graphing calculator
Vollständige Induktion WP - Blaise Pascal
Unicode Character 'INTEGRAL' (U+222B) - (&int;) - ∫n - 1p + 2 - character 'AMPERSAND' & (&amp;) - character sigma Σ (&Sigma;) - Σn + 3
Albebra WP
College Algebra - Lecture 1 - Numbers - 1/39 with Professor Richard Delaware - YT
Trigonometry WP

### College Algebra - Richard Delmare

Lecture 1 - Numbers
Set of Objects, finite or infinite - $\mathbb{N Z Q R}$

$$n^2 + 5 = 30\text{ so we have }n=\pm5$$

$\mathbb{N Z Q R}$

The characteristic polynomial $f(\lambda)$ of the $3 \times 3$ matrix $\left( \begin{array}{ccc} a & b & c d & e & f g & h & i \end{array} \right)$ is given by the equation $f(\lambda) = \left| \begin{array}{ccc} \lambda - a & -b & -c -d & \lambda - e & -f -g & -h & \lambda - i \end{array} \right|.$

## Calculus

Calculus WP
Gottfried Wilhelm Leibniz * 1.7.1646 Leipzig; † 14. November 1716 in Hannover - WP
Isaac Newton * 4.1.1643 in Woolsthorpe-by-Colsterworth in Lincolnshire; † 31.3.1727 in Kensington - WP
The Calculus Controversy
What is Calculus? (Mathematics) - Socratica YT
Calculus -- The foundation of modern science YT
Calculus - The Fundamental Theorem, Part 1 YT
Calculus in 20 minutes - Reviewing Calculus YT
The Birth Of Calculus (1986) YT
Calculus I - Lecture 01 - A Review of Pre-Calculus - 1/31 with Professor Richard Delaware - YT
What are the best calculus books? quora.com
The most enlightening Calculus books math-blog.com
Thomas' Calculus: Multivariable - 13th Edition - George B. Thomas WP
Calculus - 4th edition - Michael Spivak WP
Calculus - Basic Concepts for High School - 1982 by L.V. Taraso
Best calculus textbook? physicsforums.com

## Calculus Made Easy

Silvanus P. Thompson 19.6.1851 York, England - 12.6.1916 WP - free online file -

## Calculus

Gilbert Strang WP - free online book Calculus - Gil Strang's Introduction to Calculus for Highlights for High School YT

## Calculus

To the Student
pXXIII - may have to R passages more than once - pencil paper calculator - R section before doing exercises - try to solve examples, too - part of aim of book is training logic -

## Calculus

1 - Basic Properties of Numbers
p3 - 12 properties in this chapter, first 9 fundamental operations + * --- (P1) if a, b, and c are any numbers, then a + (b + c) = (a + b) + c

## Mathematics

Pure mathematics WP
mathreference.com
Here’s How to Teach Yourself Physics and Math futurism.com - Terence Tao WP - Pulling back the curtain: Terence Tao on mathematics in the Internet age YT - Career advice
John C. Baez WP - John Baez on the number 5 YT - ♡♡♡ John Baez - The Mathematics of Planet Earth ♡♡♡ Stellenbosch 30.10.2012 YT - Network Theory - Seminar 1 YT - Advice for the Young Scientist - How to Learn Math and Physics
annals.math.princeton.edu - Transactions of the American Mathematical Society
Advice John Baez - books: set theory: Elements of Set Theory 1st Edition by Herbert B. Enderton
A Mathematical Introduction to Logic 3rd Edition by Herbert Enderton
abstract algebra: Symmetry (Princeton Science Library) Reprint Edition by Hermann Weyl - (Before diving into group theory, find out why it's fun.) - Galois Theory 4th Edition by Ian Nicholas Stewart (A fun-filled introduction to a wonderful application of group theory that's often explained very badly.) - Ian Stewart (mathematician) WP

Areas of mathematics WP - Fields of mathematics WP
Arithmetic - natural numbers, integers, fractions, decimals, + - *(·x --- · shift alt 9) :(/) --- also % xn √ log
• Number Theory
Algebra - abstraction x y etc.
• Geometry
• Algebraic Geometry
• Analysis
• Logic
• Combinatorics
• Calculus
• Discrete Mathematics
• Algorithms
• Theoretical Computer Science
• Probability
• Mathematical Physics
• ...

## The Princeton Companion to Mathematics

The Princeton Companion to Mathematics - first edition 2008 by Timothy Gowers 20.11.1963 (Editor), June Barrow-Green 3.6.1953 (Editor), Imre Leader 30.10.1963 (Editor) -
VIII.6 Advice to a Young Mathematician p1000: Béla Bollobás 3.8.1943 WP - Reversi/Othello WP - Michael Atiyah 22.4.1929 WP - Alain Connes 1.4.1947 WP - Dusa McDuff 18.10.1949 WP - Peter Sarnak 18.12.1953 WP - Timothy Gowers WS - Gower's Webblog - Mathematics related discussions - Timothy Gowers: The Importance of Mathematics 2000 YT - Proof - Trailer YT - Tim Gowers - Computational Complexity and Quantum Compuation Lectures 1-10 YT
Proof 2005 Movie PG-13 - Gwyneth Paltrow & Anthony Hopkins & Hope Davis - YT
What is the best mathematics reference book or site you have ever seen? quora.com
Which is the best mathematics textbook dealing with all major topics on mathematics? physicsforums.com
NIST Handbook of Mathematical Functions
The Princeton Companion to Applied Mathematics - 2015 by Nicholas J. Higham (Editor), Mark R. Dennis (Editor), Paul Glendinning (Editor), & 3 more
The Princeton Companion to Applied Mathematics press.princeton.edu - Sample p1-87 play.google.com - Making the Princeton Companion to Applied Mathematics - Nicholas J. Higham - October 2015
Frank Wilczek to edit The Princeton Companion to Physics - frankwilczek.com
mathworld.wolfram.com
What Is Mathematics? An Elementary Approach to Ideas and Methods - 2nd Edition by Richard Courant (Author), Herbert Robbins (Author), Ian Stewart (Editor)
List of unsolved problems in mathematics WP
List of unsolved problems in physics WP

Contents
Preface ix - Contributors pxvii - Part I Introduction p1 - Part II The Origins of Modern Mathematics p77 - Part III Mathematical Concepts p157 - Part IV Branches of Mathematics p315 - Part V Theorems and Problems p681 - PartVI Mathematicians p733 - Part VII The Influence of Mathematics p827 - Part VIII Final Perspectives - Index p1015-1034 end of book

### Part I - Introduction

I.1 What Is Mathematics About?
1 Algebra, Geometry, and Analysis
1.1 Algebra versus Geometry
p1 - high school pupils think of algebra: substitute letters for numbers, geometry as shapes (circle, triangle, cube, sphere etc.) and concepts (rotation, reflection, symmetry etc.) -
p2 contrasted with arithmetic, more direct study of numbers themselves - contrast not in advanced math - geometrical problems may be converted into algebra etc., Cartesian coordinates - example circle rotation, visualization or algebraic approach, matrices etc. - while one can distinct between algebra and geometry, but shouldn't imagine boundary between the two is sharply defined - branch called algebraic geometry (1V.4) - different mehtods of thinking: algebra more symbolic, geometry more pictorial -> influence which subjects mathematician chooses to pursue
1.2 Algebra versus Analysis
... not branches should be classified into algebra or analysis, but mathematical techniques - derivative of x3 is 3x2 - gradient at x is 3x2
p3 - a proof that x4 − x2 − 6x + 10 is positive for every real number x - ... proof ... - algebraic equivalent: (x2 − 1)2 + (x − 3)2 - ... - 1 analysis often involves limiting processes and algebra usually does not - 2 algebraists like to work with exact formulas and analysts use estimates - 3 algebraists like equalities and analysts like inequalities
2 The Main Branches of Mathematics
potential confusion: words “algebra,” “geometry,” and “analysis” refer both to specific branches of mathematics and to ways of thinking that cut across many different branches - keep in mind distinctions of previous section and be aware that they are in some ways more fundamental
2.1 Algebra
2.2 Number Theory
2.3 Geometry
2.4 Algebraic Geometry
2.5 Analysis
2.6 Logic
2.7 Combinatorics
2.8 Theoretical Computer Science
2.9 Probability
2.10 Mathematical Physics

I.2 The Language and Grammar of Mathematics
1 - Introduction
p8 -

I.3 Some Fundamental Mathematical Definitions
1 The Main Number Systems
2 Four Important Algebraic Structures
3 Creating New Structures Out of Old Ones
4 Functions between Algebraic Structures
5 Basic Concepts of Mathematical Analysis
6 What Is Geometry?

I.4 The General Goals of Mathematical Research
1 Solving Equations
2 Classifying
3 Generalizing
4 Discovering Patterns
5 Explaining Apparent Coincidences
6 Counting and Measuring
7 Determining Whether Different Mathematical Properties Are Compatible
8 Working with Arguments That Are Not Fully Rigorous
9 Finding Explicit Proofs and Algorithms
10 What Do You Find in a Mathematical Paper?

### Part II - The Origins of Modern Mathematics

II.1 From Numbers to Number Systems
II.2 Geometry
II.3 - The Development of Abstract Algebra
II.4 Algorithms
II.5 The Development of Rigor in Mathematical Analysis
II.6 The Development of the Idea of Proof
II.7 - The Crisis in the Foundations of Mathematics

### Part III - Mathematical Concepts

III.1 The Axiom of Choice p157
III.2 The Axiom of Determinacy
III.3 Bayesian Analysis
III.4 Braid Groups
III.5 Buildings
III.6 Calabi–Yau Manifolds
III.7 Cardinals
III.8 Categories
III.9 Compactness and Compactification
III.10 Computational Complexity Classes
III.11 Countable and Uncountable Sets
III.12 C ∗ -Algebras
III.13 Curvature
III.14 Designs
III.15 Determinants
III.16 Differential Forms and Integration
III.17 Dimension
III.18 Distributions
III.19 Duality
III.20 Dynamical Systems and Chaos
III.21 Elliptic Curves
III.22 The Euclidean Algorithm and Continued Fractions
III.23 The Euler and Navier–Stokes Equations
III.24 Expanders
III.25 The Exponential and Logarithmic Functions
III.26 The Fast Fourier Transform
III.27 The Fourier Transform
III.28 Fuchsian Groups
III.29 Function Spaces
III.30 Galois Groups
III.31 The Gamma Function
III.32 Generating Functions
III.33 Genus
III.34 Graphs
III.35 Hamiltonians
III.36 The Heat Equation
III.37 Hilbert Spaces
III.38 Homology and Cohomology
III.39 Homotopy Groups
III.40 The Ideal Class Group
III.41 Irrational and Transcendental Numbers
III.42 The Ising Model
III.43 Jordan Normal Form
III.44 Knot Polynomials
III.45 K-Theory
III.46 The Leech Lattice
III.47 L-Functions
III.48 Lie Theory
III.49 Linear and Nonlinear Waves and Solitons
III.50 Linear Operators and Their Properties
III.51 Local and Global in Number Theory
III.52 The Mandelbrot Set
III.53 Manifolds
III.54 Matroids
III.55 Measures
III.56 Metric Spaces
III.57 Models of Set Theory
III.58 Modular Arithmetic
III.59 Modular Forms
III.60 Moduli Spaces
III.61 The Monster Group
III.62 Normed Spaces and Banach Spaces
III.63 Number Fields
III.64 Optimization and Lagrange Multipliers
III.65 Orbifolds
III.66 Ordinals
III.67 The Peano Axioms
III.68 Permutation Groups
III.69 Phase Transitions
III.70 π
III.71 Probability Distributions
III.72 Projective Space
III.73 Quadratic Forms
III.74 Quantum Computation
III.75 Quantum Groups
III.76 Quaternions, Octonions, and Normed Division Algebras
III.77 Representations
III.78 Ricci Flow
III.79 Riemann Surfaces
III.80 The Riemann Zeta Function
III.81 Rings, Ideals, and Modules
III.82 Schemes
III.83 The Schrödinger Equation
III.84 The Simplex Algorithm
III.85 Special Functions
III.86 The Spectrum
III.87 Spherical Harmonics
III.88 Symplectic Manifolds
III.89 Tensor Products
III.90 Topological Spaces
III.91 Transforms
III.92 Trigonometric Functions
III.93 Universal Covers
III.94 Variational Methods
III.95 Varieties
III.96 Vector Bundles
III.97 Von Neumann Algebras
III.98 Wavelets
III.99 The Zermelo–Fraenkel Axioms

### Part IV - Branches of Mathematics

IV.1 Algebraic Numbers p315 - IV.40 General Relativity and Cosmology p579-590

### Part VIII - Final Perspectives p897-962

Vocabulary
crude - roh, unverarbeitet, unfein, barbarisch, geschmacklos, grob, grell, plump, simpel, undurchdacht, ungehobelt, unreif, ungekocht, krud, krude, ungehobelt ... - p3
pursue - verfolgen, betreiben, fortfahren, fortführen, weiterverfolgen, nachgehen ... - p2
substitute - austauschen, auswechseln, an die Stelle setzen, substituieren, vertreten, ersetzen (sth. for sth., etwas durch etw.), subst. for - jmdn. vertrenten, sth. - etw. wechseln ... - p1

## Concrete Mathematics

Dedicated to Leonhard Euler (1707 - 1783) - WP

### "...Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems."

Concrete Mathematics: A Foundation for Computer Science - 2nd Edition by Ronald L. Graham (Author), Donald E. Knuth (Author), Oren Patashnik (Author)
CM safaribooksonline.com
The Art of Computer Programming, Volumes 1-4A - Boxed Set 1st Edition by Donald E. Knuth
Concrete Mathematics: A Foundation for Computer Science - Community reviews - goodreads.com
What books do you recommend before 'Concrete Mathematics'? stackexchange.com
What are some opinions on Concrete Mathematics by Donald Knuth? quora.com - Stockholm syndrome
Donald Knuth - My advice to young people (93/97) YT -
Donald Knuth WP - TeX typesetting system WP - mathjax.org - MathJax (TeX for Web) - Getting Started - MathJax TeX and LaTeX Support - ctan.org
Surreal Numbers - 1st Edition by Donald E. Knuth - how two ex-students turned on to pure mathematics and found total happiness - a mathematical novelette - full text at archive.org - WP
John Horton Conway WP - On Numbers and Games

pV - based on course at Stanford University (California) since 1970 - CM born in dark and stormy decade - Knuth missed math tools, created this course - CM was antidote to Abstract Math (concrete classical results wiped out by New Math) -
pVI - CM blend of CONtinuous and disCRETE math - controlled manipulation of math formulas, using collection of techniques for solving problems - if material of book is learned, reader will solve everything only on paper with handwriting! - major topics treated in this book include sums, recurrences, elementary number theory, binomial coefficients, generating functions, discrete probability, and asymptotic methods - will become familiar with discrete math as student of calculus with continuous operations (like absolute-value function and infinite integration) - original course textbook was "Mathematical Preliminaries" from TAOCP - 110 pages quite terse, OP (author) drafted supplementary notes, this book is outgrowth -
pVII - book is kind of manifesto about our (authors) favorite way to do mathematics, turned out to be a tale of mathematical beauty and surprise - we think math is not cold and dry, but fun, no line between work and play needed - margins include direct quotations from famous mathematicians, actual words in which they announced some of their fundamental discoveries -
pVIII - more than 500 exercises in six catgories: warmups, basics, homework exercises, exam problems, bonus problems - research problems - answers in Appendix A - math typeface by Hermann Zapf, like handwriting, called AMS Euler - Concrete mathematics is Eulerian mathematics -
pIX - Thanks to: ... 16 years' lecture notes - notes to 2nd edition ... - $2.56 for mistakes pX - notation of symbols and pages where they are explained: lnx --- lgx --- log x --- ⌊x⌋ --- ⌈x⌉ --- x mod y --- {x} --- ∫f(x) δx --- ∫baf(x) δx --- xn --- xn --- n¡(! usidedown) --- Rz --- Iz --- Hn --- Hn(x) --- pXI - f(m)(z) --- [n(/)m] --- {n(/)m} --- <n(/)m> --- <<n(/)m>> --- (am, ...,a0)b --- K(a1,...,an) --- F(a,b(/)c|z) --- #A --- [zn] f(z) --- [α..β] --- [m=n] --- [m\n] --- [ m\\n ] --- [m⊥n] --- --- '...' for written, "..." for spoken text, string of 'string' is called "string" - expression 'a/bc' is same as 'a/(bc)' --- log x/log y = (log x)/(log y) --- 2n! = 2(n!) pXII-XIII - Contents - 1 Recurrent Problems (p1=20p) - 2 Sums (p21=45p) - 3 Integer Functions (p67=34p) - 4 Number Theory (p102=50p) - 5 Bionomial Coefficients (p153=103p) - 6 Special Numbers (p257=62p) - 7 General Functions (p320=60p) - 8 Discrete Probability (p381=57p) - 9 Asymtotics (p439=57p) - A Answers to Exercises (p497=106p) - B Bibliography (383 books) (p604=32p) - C Credits for Exercises (p632=4p) - Index (p637=19p) - List of Tables (p657=1p=end of book) ### 1 Recurrent Problems p1 - chapter 3 sample problems - all investigated repeatedly by mathematicians, solutions use recurrence, depending on solutions to smaller instances 1.1 The tower of Hanoi - invented by French math. Edouard Lucas 1883 (bk260) - tower of 8 disks, stacked in decreasing size on one of three pegs - transfer entire tower to one of the other pegs, moving only one disc at a time, never moving larger one onto smaller - Lucas furnishes to tower of Brahma, 64 disks of gold on 3 diamond needles, God asks at beginning of time priests to transfer according to same rules above, priests work day and night, when finish tower will crumble and world will end - p2 - best to do: how many moves necessary and sufficient to perform task - best way to tackle a question like this is to generalize it: Brahma tower 64 discs, Hanoi 8 - consider n discs - we'll see repeatedly in book: it's advantageous to "look at small cases" first - easy transfer tower of 1 ore 2 discs, then 3, and 0 - next step introduce appropriate notation: name and conquer - Tn is minimum number of moves to transfer n disks according to rules - then T1 = 1, T2 = 3, (T3 = 7 etc.) - T0 = 0 is smallest case - now think big: ### Vocabulary enfeeblement - Entkräftung, Schwächung - pV hotbed - Brutstätte, Frühbeet, Mistbeet ... - pV peg - Aufhänger, Stift, Stöpsel, Zapfen, Pflock, Wirbel (Musik), Dübel ... - p1 recurrent - periodisch, wiederkehrend, rekurrent ... - p1 rift - Riss, Kluft, Spalte, Graben - WP Carl Friedrich Gauss --- reef, ledge, shelf - Riff scrutiny - genaue Überprüfung/Untersuchung ... - pV spawn - vermehren, erzeugen, hervorbringen, laichen - pV tackle - bewältigen, anpacken, in Angriff nehmen, angehen, fertig werden ... - p2 trait - Eigenschaft, Zug, Charakterzug, Wesenszug, Gesichtszug, Charaktereigenschaft, Charakteristik, Merkmal - p1 ## Links Math Wiki Wikipedia WP - former Nupedia, Wikimedia Foundation WP - Wikia - Jimmy Wales WP, Co-Feounder of WP and Wikia - Wikipedia Co-Founder Jimmy Wales On Encryption And The Economy Of Content YT - Larry Sanger WP, Co-Founder of WP - larrysanger.org - watchknow.org • Session 1 • Tue 2017-1-31 Berlin 8:45-11:40, 15:10-18:15, 18:40-19:50, 20:50-22:25 start Concrete Mathematics WR pV-XIII ♡♡♡ + colors + formatting ♡♡♡ 23:35 start ch1 R p1-16 - WR p1-2 3:25 end of session ## Discrete Mathematics and Its Applications Discrete Mathematics and Its Applications 7th Edition - Kenneth Rosen - Solutions Guide How should I read Kenneth H. Rosen's Discrete Mathematics effectively? quora.com Discrete mathematics WP What is the best book for studying discrete mathematics? math.stackexchange.com Concrete Mathematics: A Foundation for Computer Science - 2nd Edition - Ronald L. Graham, Donald E. Knuth, Oren Patashnik Discrete and Combinatorial Mathematics: An Applied Introduction - scribd.com - Ralph Grimaldi WP Who has read Discrete Mathematics and Its Applications and did you feel it helped you learn discrete math well? reddit.com Discrete Mathematics with Applications - 4th Edition - Susanna S. Epp WP Chapter 1-13 - pages 1-903 - Appendixes - Books - Answers - Index 1 - The Foundations: Logic and Proofs 1.1 - Propositional Logic p1 - examples ... p3 - proposition p (or q, r, s ...) is T (true) or F (false) - ¬p is negation of p, read "not p" - Def 1: if p T, then ¬p F --- if p F, then ¬p T --- logical operators are called "connectives" p4 - Def 2: if p and q are T, the conjunction ∧ of p and q, p ∧ q, is also T, otherwise F - in conjunctions sometimes "but" is uses instead of "and": The sun is shining, but it is raining Def 3: disjunction ∨ (read or) is true if p or q are true, or both, but false if both are false --- p ∨ q p ∧ q has 4 possible results: TFFF - p ∨ q has 4 possible results: TTTF p5 - ## Algorithms and Data Structures Algorithms and Data Structures © N. Wirth 1985 (Oberon version: August 2004) Contents Preface - 7 1 Fundamental Data Structures - 11 2 Sorting - 45 3 Recursive Algorithms - 87 4 Dynamic Information Structures - 109 5 Key Transformations (Hashing) - 177 Appendices - 183 A The ASCII Character Set B The Syntax of Oberon Index Preface p7 - E.W. Dijkstra's "Notes on Structured Programming" - C.A.R. Hoare's "Axiomatic Basis of Computer Programming" and "Notes on Data Structuring" - the subjects of program composition and data structures are inseparably interwined - book in Pascal lang ... p9 - book "Systematic Programming" provides ideal background, based on Pascal notation - ... Preface To The 1985 Edition Pascal replaced by Modula-2 (Pascal is ancestor) - Modula-2 WP - ... - p10 - book was edited and laid out by the author with the aid of a Lilith computer and its document editor Lara Notation & denotes conjunction, pronounced and ~ denotes negation, pronounced not Boldface A and E used to denote the universal and existential quantifiers In the following formulas, the left part is the notation used and defined here in terms of the right part. Note that the left parts avoid the use of the symbol "...", which appeals to the readers intuition. Ai: m ≤ i < n : Pi ≡ Pm & Pm+1 & ... & Pn-1 The Pi are predicates, and the formula asserts that for all indices i ranging from a given value m to, but excluding a value n, Pi holds. Ei: m ≤ i < n : Pi ≡ Pm or Pm+1 or ... or Pn-1 The Pi are predicates, and the formula asserts that for some indices i ranging from a given value m to, but excluding a value n, Pi holds. MIN i: m ≤ i < n : xi = minimum(xm, ... , xn-1) MAX i: m ≤ i < n : xi = maximum(xm, ... , xn-1) ### 1. Fundamental Data Structures 1.1. Introduction Vocabulary abundantly - reichlich, mächtig, im Überfluss, in Hülle und Fülle ... - p7 amenable - zugänglich, empfänglich, biegsam, gefügig, zugängig, offen - p7 bearing - Bedeutung, Tragweite, Bezug, Zusammenhang ... - p7 crucial - entscheidend, kritisch, wichtig, ausschlaggebend, äußerst wichtig ... - p7 elaboration - Vervollkommnung, Ausarbeitung, sorgfältige Ausführung, ausführliche Darstellung ... - p9 hitherto - bisher, bisherig, wie bisher - p7 ### Program Development by Stepwise Refinement ### Links ## The Algorithm Design Manual Best algorithms book I ever read - Eric Wendelin WS - TW 1.2K - created stacktracejs.com - Who's Bigger? WP - whoisbigger.com - iTunes App The Algorithm Design Manual - 2nd edition 2008 by Steven S. Skiena WP - Jesus the Game Changer Steven Skiena Segment 1 YT - CSE373 2012 - Lecture 01 - Introduction to Algorithms Algorithms and Data Structures Course by Robert Sedgewick - YT What are the best books on algorithms and data structures? quora.com Preface V - Designing correct, efficient, and implementable algorithms for real-world problems: ## Structured Computer Organization ### Fifth edition - Andrew S. Tanenbaum Andrew S. Tanenbaum WP - Structured Computer Organization (6th Edition) - Amazon - MINIX WP - Andrew S. Tanenbaum: The Impact of MINIX YT Modern Operating Systems The Minix Book - (4th Edition) - Amazon - Computer Networks (5th Edition) - Amazon - Operating Systems - Design and Implementation (3rd Edition) Computer WP - Microprocessor WP - Microcontroller WP Pentium (WP) - x86-compatible microprocessors - Intel (WP) founded 1968 - x86 since 1993 - Santa Clara, California ARM architecture WP - Acorn/Advanced RISC Machine - start 1983 - in mobile devices UltraSPARC III - since 2001 - designed by Sun Microsystems and manufactured by Texas Instruments - SPARC64 V - in future will implement the ARMv8 architecture Intel MCS-51 - (commonly termed 8051) - since 1980 Instruction set WP - ISA (Architecture) Programming Languages -- Past Achievements and Future Challenges Niklaus Wirth u.a. - Euler Algol_W Pascal Modula Oberon Lilith_OS Oberon_OS Lola_digital_hardware ACM_Turing_Award PL/0_compiler_design 29:00 simpler but more powerful Systems Architecture, Design, Engineering, and Verification ## Links cs - computer science Harward CS50 David J. Malan ♡♡♡ c-howto.de - eliasfischer.de - lebeblog.de - YT - Wie finde ich das Ziel, das ich wirklich will? - TW - Antworten zu mir, meinem Leben und meiner Arbeit - FB - Wer bin ich wirklich? - Elias Fischer FB Why is my host name wrong at the Terminal prompt when connected to a public WiFi network? Result successful: Last login: Tue Jan 10 22:55:47 on ttys000 feroniba_macbook_pro:~ Feroniba$

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### Vocabulary

: - colon - Doppelpunkt
brevity - Kürze - C Preface
formalize - formalisieren, offiziell machen - C Preface
retain - behalten, sichern, beibehalten ... - C Preface
submit - einreichen, vorlegen, überreichen, unterbreiten ...
unambiguous - unzweideutig, eindeutig, unmissverständlich, widerspruchsfrei, eindeutig, unzweideutig - C Preface

## Sessions

• Session 1 • Sun 2016-6-26 0:45- Programming in C - start
• Session 2 • Thu 2017-1-12 23:15-5:35 LR The C Programming Language - alles bisherige rekapitulieren - cont. p13-16 - chapter 1.2 test ♡ - WR Summary 1-6, timer.c, 1.2.2_while_i_smaller_j.c u.a.       